JPRS ID: 9179 TRANSLATION MANUAL ON SHIP ACOUSTICS BY I.I. KLYUKIN AND I.I. BOGOLEPOV
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'   ON
BY
3 JULY 1980 1.1. KL YUKI AND I . I . BOGOLEfiOV : I OF G
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JPRS L/9179
3 July 1980
FOR OFFICL4L USE ONLY
Translation
 MANUAL ON SHIP ACOUSTiCS
By
I.I. Kiyukin and I.I. Bogolepov
F~I$ FOREIGN BROADCAST INFORMATION SERVICE
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JPRS L/9179
3 July 1980
MANUAL ON SfII P ACOUST I CS
Leningrad SPRAVOCHNIK PO SUDOVOY AKUSTIKE [Manual on Ship Acous
tics] in Russian 1978 signed to press 26 Oc.t 78 pp 291, 94103,
160480
[Excerpts from book edited by Doctor of Engi.neering Sciences,
Professor I.I. Klyukin and Candidate of Engineering Sciences
I.I. Bogolepov, Izdatel'stvo Sudostroyeniye, 6,600 copies]
CONTENTS
Foreword
Basic Definitions
Chapter 1. Acoustic Vibrations and iVaves (A. A. Kleshchev, I. I.
Klyukin, A. S. Nikiforov)
1.1. Basic Acoustic Field Equations
1.2. Plane, Spherical and Cylindrical Waves in Gases and Fluids
1.3. Wave Propagation in Elastic and Viscous Media
1.4. Elastic Waves in Rods, Plates and Cylindrical Shells
1.5. Niechanical Strength of Rods, Plates and Hull Structures.
Acoustic Irradiation of Plates and Shells
1.6. Electromechanical and Electroacoustic Analogies
1.7. Electrical, Mechanical and Acoustic nPoles
Bibliography
Chapter 2. Principles of hleasurement Acoustics (A. Ye. Kolesnikov)
2.1. Input and bietering Circuits. Calibration of Acoustic and
Vibration Meters
2.2. hieasurement of Noise and Vibration
2.3. Spectral and Correlation Analysis
2.4. Measurement of Vibration Power Radiated by D9achinery
2.5. Similitude and Dimensional TechniQues in Marine Acoustics
2.6. Determination of Reliability and Precision of Acoustic
bteasurements
2.7. Application of Electronic Digital Computers for Acoustic
bieasurements
Bibliography
 a  [I  USSR  G FOUO]
FGR OFFICIAL USE ONLY
~
1
3
5
5
7
14
22
34
42 
48
51
56
56
67
70
78
84
86
90
94
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Chapter 3. General Characteristics of Machinery as Vibration and
Noise Sources (V. I. Popkov, V. K. I1'kov) 97
3.1. Induced Mechanical_Yibrations 97
3.2. Mechanical Radiation of Vibration Energy 108
Bibliography 119
4.2. Standardization of Noise and Vibration on Ships 121
4.3. Monitoring of Ship Noise and Vibration 127
Bibliography 130
Chapter 6. Reduction of Noise of Steam Turbine and Gas Turbine
Power Plants (V. I. Zinchenko)
133
6.1.
Basic Sources of Noise of
STP and GTP
133
6.2.
Gas Turbine Engine Noise.
Turbocompressor Noise Reduction
135
6.3.
Elimination of Gas Dynamic Free Vibrations in Exhaust Pipes
141
6.4.
Gas Turbine Engine Intake
and Exhaust Mufflers and Their
Calculation
145
6.5.
Reducer Noise and Ways to
Reduce It
157
6.6.
Ship Pipe Valve Noise and
Ways to Reduce It
163
Bibliography
176
Chapter 7. Reduction of Noise of Ventilation and Air Conditioning
Systems (N. F. Yegorov, Yu. I. Petrov, G. A. Khoroshev) 179
7.1. Noise Sources in Ventilation and Air Conditioning System 179
7.2. Reduction of Turbulence Noise and Nonuniform Flow Noise
of Blowers 186
7.3. Acoustic Calculation of Ventilation and Air Conditioning
Systems , 202
7.4. Ways to Reduce Noise of Ventilation and Air Conditioning
Systems 214
7.5. Noise Mufflers of Ship Ventilation and Air Conditioning
Systems 217
Bibliography 222
Chapter 8. Noise Reduction of Ship Hydraulic Systems, Compressors
and Electrical Machinery (N. I. Duan, M. A. Fedorovich) 224
8.1. Causes of Noise in Ship Hydraulic Systems 224
 8.2. Hydrodynamic and Mechanical Noise Saurces in Pumps 231
8.3. Ways to Reduce Pump Noise 234
8.4. Noise and Vibration of Reciprocating Compressors 238
8.5. Basic Noise Sources of Electrical Machinery 241
8.6. Reduction of Noise of Electrical Machinery 251
Bibliography � 259
Chapter 9. Reduction of Noise of Operating Engines and Other
Systems (L. S. Boroditskiy) 261
9.1. Engine Noise in Ship Rooms 261
9.2. Methods of Reducing Screw and Other Engine Noise 269
9.3. Noise Interference on Bridge and Ways to Reduce It 272
Bibliography 286
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Chapter 10. Acoustic Insulation on Ships (I. I. Bogolepov,
E. I. Avferonok, K. I. Mal'tsev)
289
10.1.
Importance of Acoustic Insulation on Ships
289
10.2,
Basic Principles of Acoustic Insulation
289
10.3.
Acoustic Insulation of SingleWall Structures
294
10.4.
Acoustic Insulation of DoubleWall Structures
300
10.5.
Planning of Acoustic Insulation of Ship
306
10.6.
Soundproofing Shields and Booths
319
10.7.
Acoustic Insulation of Ship Machinery
325
10.8.
Materials for Marine Acoustic Insulation and Sound
proofing Structures
334
10.9.
P9easurement of Acoustic Insulation
336
10.10.
Personal Acoustic Insulation Gear
341
Bibliography
343
" Chapter 11. Soundproofing of Ships (K. A. Velizhanina, I. V.
Lebedeva) 346
11.1. Application of Soundproofing Structures 346
11.2. Wave Parameters. Porous Panel. Resonant SQUnd Absorbers 350
11.3. Laminated Soundproofing Structures. Space Absorbers.
 Membrane Acoustic Absorbers 355
11.4. Soundproofing Structures with Oblique and Diffusive
Incidence 361
11.5. Acoustic Absorption in High Acoustic Pressure Levels 363
11.6. Measurement of Acoustic Absorption 366
Bibliography 371
Chapter 12. Vibration Damping of Marine Machinery (I. I. Klyukin,
N. G. Belyakovskiy, B. P. Polonskiy, V. I. Popkov)
373
 12.1.
Purpose and Classification of VibrationDamping Shock
Absorbing Structures and Fasteners
373
12.2.
Calculation of Effectiveness of VibrationDamping Mounts
Using Electromechanical Analogies, Matrix Procedures and
Orthogonal Polynomials
395
12.3.
Energy Analysis of Effectiveness of ShockAbsorbing
Mounts
408
12.4.
Marine Shock Absorbers
415
 12.5.
Calculation of ShockAbsorbing Mounts of Marine Machinery
424
 12.6.
Vibration Damping of Nonbearing Connections of Machinery
428
12.7.
hleasurement of Vibration Damping of ShockAbsorbing
Mounts
434
,
Bibliography
436
Chapter 13.
Vibration Damping of Ship Hull Structures (L. S.
Nikiforov, V. T. Lyapunov)
438
13.1.
Basic Principles and Laws of Vibration Damping for
Bending iVaves
438
13.2.
Vibration Damping of VibrationImpeding Mases and
Stiffening Ribs
441
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13.3. Vibration Damping of Plate Joints 44~
13.4. Vibration Damping of Periodic Barriers 448
13.5. Influence of Features of Ship Structures on Vibration
Damping of Barrier 449
13.6. Vibration Damping as Function of Vibration Absorption 452
13.7. Efficiency Analysis of Ship VibrationDamping
Techniques 453
13.8. Personal Vibration Safety Gear 455
13.9. hieasurement of Vibration Damping 458
Bibliography 464
 Chapter 14.
Vibration Absorption on Ships (A. S. Nikiforov,
I. I. Klyukin)
465
14.1.
Absorption of Vibration Energy in Deformable Media
and Structures
465
14.2.
Hard VibrationAbsorbing Coatings
468
14.3.
Soft VibrationAbsorbing Coatings
472
14.4.
Reinforced VibrationAbsorbing Coatings
474
14.5.
VibrationAbsorbing Construction Materials
476
14.6.
Bulk Vibration Absorbers and Bituminous Damping Materials
478
14.7.
Local Vibration Absorbers
482
14.8.
Where to Install Vibration Absorbers in Ship Structures
491
14.9.
Analysis of Effectiveness of Vibration Absorption
Techniques in Ship Structures
493
14.10.
Measurement of Vibration Absorption
495
Bibliography
498
Chapter 15. Forecasting of Acoustic Situation and Combined
Application of Noise Control Techniques on Ships (V. M.
Spiridonov) 500
15.1. General Description of Problem 500
15.2. Forecasting of Noise Level on Ship in Design Stage 502
15.3. Basic Relations and General Calculation Procedure 504
15.4. Rec{uirements on Programming of Calculation of Anti
cipated Noise Level with Computer 513
15.5. Graphoanalytical Calculation of Anticipated Noise Level 518
15.6. Features of Combined Application of Noise Control
Technique, 530
Bibliography 539
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PUBLICATION DATA
English title : MANUAL ON SHIP ACOUSTICS
Russian title : SPRAVOCHNIK PO SUDOVOY AKUSTIKE
Author (s) .
Editor (s) : I. I. Klyukin, I. I. Bogolepov
Publishing House : Sudostroyeniye
Place of Publication ; Leningrad
Date of Publication : 1978
Signed to press ' : 26 Oct 78
Copies ; 6600
COPYRIGHT : Izdatel'stvo "Sudostroyeniye", 1978
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ANNOTATION
[Text] The results of scientific studies and developments on ship acoustics
are generalized in this manual. Sources of ship noises are examined. Basic
data, methods and information necessary for the design, manufacture and
testing of systems for noise and acoustic vibration abatement at the source,
on propagation paths and in ship rooms are presented. Problems of acoustic
insulation, sound absorption, vibration insulation and vibration damping and
of the combined application of noise abatement systems are examined in
detail.
Advanced Soviet experience in acoustical developments, undertaken for the
purpose of improving the inhabitability of ships and of work conditions on
them, and the results of foreign practice are reflected.
The manual is intended for scientific researchers at institutions, planning
 design offices and factories. It will be beneficial to students and post
graduate students.
1
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FOREWORD
hiodern machinery and mechanisms produce strong noises. The relentless
increase of their power is acccmpanied by a further increase of noise. At
the same time requirements on living conditions on ships are becoming
increasingly more rigid. Therefore noise abatement has taken on special
scientific and practical importance in recent years.
In view of the importance of marine acoustics problems Sudostroyeniye has
published in the last decade a number of books on that subject. Noteworthy,
in particular, are the following publications: "Rasprostraneniye i Poglo
shcheniye Zwkovoy Vibratsii na Sudakh" [Propagation and Absorption of
Acoustic Vibrations on Ships], 1968 (authors A. S. Nikiforov, S. V. Budrin);
"Akusticheskiye Izmereniya v Sudostroyenii" [Acoustic Measurements in Ship
building], 2nd edition, 1968 (authors I. I. Klyukin, A. Ye. Kolesnikov);
"Zvukoizolyatsiya na Sudakh" [Acoustic Insu.lation on Ships], 1970 (authors
I. I. Bogolepov, E. I. Avferonok); "Bor'ba s Shumom i Zvukovoy Vibratsiyey
na Sudakh" [Noise and Acoustic Vibration Abatement on Ships], 2nd edition,
1971 (author I. I. Klyukin); "Shum Sudovykh Sistem Ventilyatsii i Kon
ditsionirovaniya Vozdukha" [Noise of Ship Ventilation and Air Conditioning
Systems], 1974 (authors N. F. Yegorov, Yu. I. Petroy, G. A. Khoroshev);
"Vibroakusticheskaya Diagnostika i Snizheniye Vibroaktivnosti Sudovykh
Mekhanizmov" [Vibroacoustic Diagnostics and Reduction of Vibration Activity
of Ship Machinery], 1974 (author V. I. Popkov); "Snizheniye Strukturnogo
Shuma v Sudovykh Pomeshcheniyakh" [Reduction of Structural Noise in Ship
Rooms], 1974 (authors L. S. Boroditskiy, V. M. Spiridonov); "Vibroizo
lyatsiya v Sudovykh Konstruktsiyakh" [Vibration Insulation 'Ln Ship Con
structions], 1975 (authors V. T. Lyapunov, A. S. Nikiforov) and cextain
others.
The time has come to summarize the experience that has been compiled in the
field of marine acoustics, in a comprehensive manual, covering its various
aspects. This manual, which serves this purpose to a certain extent, in
addition to data from the abovecited and other books on acoustics, includes
information from scientific and trade periodicals that have appeared in
recent years in the USSR and abroad.
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The authors deemed it appropriate not to restrict the contents of the
manual to reference information; the book also includes materials of a methodological nature. Calculation data are given, as a rule, in the 
International Units System. In some cases the SGS [Centimetergramsecond] 
and b9KSS [D9eterkilogramsecondcandle] systems are used.
This book is the first manual on ship acoustics. Readers are invited to
send inquiries and comments to the following address: 191065, Leningrad,
ul. Gogolya, 8, izdatel'stvo "Sudostroyeniye."
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BASIC DEFINITIONS
A acoustic absorption
 a length
B bending rigidity of beam
b width
C longitudinal or linear rigidity
c speed of sound in medium
cb bending wave velocity
clo longitudinal wave velocity
csh shear wave velocity
D bending rigidity of plate; diameter
d thickness
E energy; Young's modulus
F force
 f frequency
fCr critical frequency of plate
G shear modulus; flow rate
g acceleration of gravity
h height; thickness
I acoustic intensity; moment of inertia; electric current
i imaginary unit; electric current
J moment of inertia
k wave number
L acoustic pressure level
LN acoustic poiver level
Z linear dimension
M momentum; mass
m mass per unit area or ].ength
N acoustic power
n rotation frequency
_ p acoustic pressure
Q generalized force; Qfactor; productivity
R sound insulation; radiation resistance
Ra active component of acoustic impedance
r radius, distance
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S area
T period; reverberation time; absolute temperature
t time; temperature; thickness
U voltage
V volume
v velocity
W specific acoustic impedance; moment of resistance
w acoustic energy density
Xa reactive part of acoustic impedance
x, x, z displacement, velocity and acceleration in x coordinate for
waves in solids
, y, y, y displacement, velocity and acceleration in y coo rdinat e for
wavzs in solids
Za acoustic impedance
Zm total mechanical impedance
a absorption coefficient; attenuation coefficient (constant)
reflection coefficient; wave number (phase constant)
S elastic deformation; thickness
AL acoustic pressure drop
n loss coefficient
6, 6 angle of incidence
_ X acoustic wavelength; Lame's first constant
u Lame's second constant
~ displacement, velocity and acceleration
p density of inedium
Q Poisson's ratio
ratio of vibration frequency of plate to critical frequency; velocity
potential; angle
w circular frequency
When symbols given in this list are used in the text of the book for ather
definitions additional explanation is given. Symbols not contained in the
list are also explained. 
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CHAPTER 1. ACC'JSTIC VIBRATIONS AND WAVES
1.1. Basic Acoustic Field Equations
Elastic and fluid media (including gases), examined in marine acoustics,
are considered to be continuous. This means physicall), that the length of
a wave propagating in such a medium vastly exceeds the size of the mole
cules, and the period of vibrations vastly exceeds their free flight time between collisions. A medium (elastic or fiuid) is perfect when internal
friction and thermal conductivity are ignored. Only a longitudinal
acoustic wave exists in a perfect fluid, and the particles of such a
medium in a plane acoustic wave are displaced in the direction of propaga
tion of the wave. The displacement of particles involves a change of
pressure (normal pressure) p and density p, which are transferred by the
wave. An acoustic wave has five parameters: pressure p, density p and
three components of velocity vector 1. The adiabatic equation of state of
a medium establishes the relationship between pressure p. and density p,
[3, 10] :
pE~(PE)'
Euler's nonlinear vectorial motion equation includes the following: velo
city of particles v, pressure p. and density p,:
;r
au ~ gradp..,
at  PE
The continuity equation derives from the requirement that a medium exhibit
continuity and is determined by the law of canservation of mass:
a
o(PEU)= a~
where 0 is Hamilton's vector operator.
Nlarine acoustics specialists are usually interested in sma11 amplitudes of
vibrations of particles. This enables them to linearize the equations =
derived above. 
5
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The linear equation of state acquires the form [15]
vn~.
P �P, (1.1.4)
 a;jo ) 5=col,st
where s is entropy; p, p are the deviations of the pressure and density in
an acoustic wave from equilibrium po, po, and here p� p0' p 1.05, where vcr y v12T'
In the range 0.95 < v< 1.05 the density of frequencies has to be computed
with expressions (1. 4.52) and (1.4.53).
1.5. Niechanical Strength of Rods, Plates and Hull Structures. Acoustic
Irradiation of Plates and Shellsl
_ Mechanical Strength of Rods, Plates and Hull Structures. The velocity of a
structure at an analyzed point when a unit of force is applied to the
structure is defined as mechanical strength. Mechanical strength is called
input if the point of application of a force coincides with the point where
velocity is measured, and it is called transiert otherwise. In the general
case, when a structure is excited at several points, its velocity at any
point is determined both by the input, and by the transient strengths of
the structure. The mechanical strength of a structure is represented
formally as a quadratic matrix, which connects the column matrix of the
forces that excite the structure2 and the row matrix of the same order,
which describes reciprocating and angular velocities at all points of
excitation [28]:
1This section was written by V. S. Konevalov and V. A. Svyatenko.
2Since three components of forces and three components of moments can act
upon each point of ex%:itation, matrix order is determined by the number of
excitation points, multiplied by six.
34
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[F] " fZ1 [61]. (1.5.1) _
In practice matrix [Z] is usually replaced with the inverse conductance
matrix [Y] _ [Z]1,
The frequency characteristics of inechaniGal strengths of structures are used
for determining their resonance properties and for detuning the resonance
frequencies from the fundamental frequencies of perturbing forces [28]. The
frequency characteristic of the mechanical strength of complex structures,
 with resonance valleys and antiresonance peaks, is estimated on the basis of
the characteristic mechanical strength [35]. The characteristic strength of
a structure corresponds to the mechanical strength of an analogous structure~
but with infinite dimensions or with a high internal loss coefficient, which
eliminates wave reflection from boundaries. The characteristic strengths of
� structures in relation to various perturbing forces, such as a rod [26], a
homogeneous plate [24, 25, 26], a homogeneous shell [18], orthotropic plate
[43], and a ribbed plate [48], are now known. These strengths are pre
 sented in Table 1.3.
Acoustic Irradiation of Plates and Shells. The radiation of sound by
ship structures is attributed to vibrations, which are usually defined as
bending vibrations of structures (plates or shells). The practical problem
is to determine acoustical presF~ure p in a medium, or to determine
acoustical power N radiated by a structure.
The radiating capacity of a structure is characterized by radiation
resistance R, which is related to the radiated acoustical power and mean
square vibration velocity of a plate, in terms of time and area, by
the equation [42, 49]
N=2(z=) ,
FOR OFFICIAL USE ONLY
or by the radiation loss coefficient
[H=b]
~x= wnSH'
(1.5.3)
where m is mass per unit area; Sb is the area of the surface of a plate or
shell.
The emissivity of structures is estimated variously, depending on critical
frequency fcr, on which the bending wavelength in a structure and
acoustical wavelength in the medium are
~
[kp=cr] fKp = ~n y B ' (1.5.4)
where m is the mass of a structure per unit area; B is cylindritical
rigidity; c is the speed of sound in the medium. 35
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p
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~ On frequencies above the critical frequency the bending wave velocity
exceeds the speed of sound in a medium, and on frequencies below critical
it is less, and the vibration waveforms are traditionally called acous
tically fast and acoustically slow modes, respectively.
Radiation as a Physical Process. When acoustically slow modes propagate
in an infinite plate radiation does not occur. The reason is that the
distance between the vibration node lines in a plate is shorter than one
half the wavelength in the medium. In this case the medium behaves as an
incompressible medium and the medium only flows along the surface of the
plate between adjacent sections, which move in antiphase (Figure 1.14).
\U/
I'
~
, c)
. ~
~ 1
~ i
Figure 1.14. Diagram of interaction of bendingvibrating
plate and medium: a infinite plate with free bending
wave; b infinite plate driven by point force; c
finite pl_ate in rigid screen. Solid arrows indicate flow
of inedium, dash arrows indicate acoustical radiation.
When acoustically fast modes are generated the distance between the
vibration node lines in a plate is greater than onehalf the wavelength in
the medium. In this case the medium exhibits properties of elasticity.
Compression of the medium occurs between adjacent inphase vibrating sec
tions of the plate, and consequently the entire surface effectively
radiates sound.
38
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The radiation resistance of an infinite plate per unit area, on which a
free bending wave propagates, is
 O f G jKp,
[kp=cr] R= p� f> FKP.
If 1  cpi .
where 0 = f/fcr'
If the vibration waveform of a plate is such that both projections of the
bending wavelength onto the coordinate axes, aligned with the edges of the
plate, are longer than the wavelength in the medium, the radiation of all
sections of the plate, vibrating in antiphase, is canceled out, with the
exception of quarterwave sections at the corners of the plate (piston
modes). If one of the projec*ions of the bending wavelength onto the
coordinate axes is longer, and the other shorter than the wavelength in the
medium, the radiation of the quarterwave strips along the edges of the
plate is not canceled out (band modes, Figure I.15).
Radiation of Infinite Plate Excited by Point Forces. When a plate is
excited by a point force the modulus of the amplitude of the acoustical
pressure in the far wave zone (kR0 >1) will be
kF cos 16
I PI = 2~[ro ~ c~m 2 i/2 ~
[ 1 (P ~ Cos2$ (1  ya sin4$)aI (1. 5 . 6)
where r0 is the distance between the point of application of the force and
the point of observation; 6 is the angle between the line to the point of
observation and the line perpendicular to the plate.
On frequencies below critical the cofactor (1 sin4 6) in the denomi
nator is approximately one.
On low frequencies in a medium with a high wave resistance radiation is
dipole in nature and the modulus of the acoustical pressure amplitude is
kF cos ~4
I P I  2rcro .
(1.5.7)
Radiation in a medium with low wave resistance (air) is nondirectional, and
the acoustic pressure is determined by the formula
kpcF
IPI � 2ncmro ' (1.5. 8)
_ The acoustic power radiated by a plate is:
on frequencies f < fcr
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a) ~ 71Hx~ a
~%~~~Jr'~ 1, 1. ~.1,~~~
~
i
~
v
~ i
~
I I I I ~ I I
, ~  1 . ~  ~
I  I  1  .i
J_ I I I. ~_LJ
b) ~N~>~ �
i
i ~
.
~
~Itrfr+~+~ t
~ I  ..L _ L J _ 4
'~._I_LJ._~
~r++++i++ H
5,'
Figure 1.15. Diagram of vibration waveforms of plate
(radiating portions crosshatched): a for piston
modes; b for strip modes. [H=b]
N  pck2F2 1  ()c arctg Om
4n (~rn)= C wn~ pc (1. 5.9)
' on frequencies f > fcr
pc
N _ Fa t~m{~I (P1
 16Vme 11 + PC ' (1.5.10)
wmv'1(P1
(?n frec{uencies close to critical the acoustic power can be determined by
numerical integration [6].
When a plate is driven by linear force F1 the acoustic pressure amplitude
modulus acquires the form
40
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k lV~ F,cos~4 .
Iv,l ~ ,
l 2nro L I + (pc )Y cos2 $ (1  y2 sin4 D)2
J
and the acoustic power radiated by the plate on frequencies f< fcr is
kpcF; ~
Nl  4(wrn)2 ~ 1_}_ / t,~m 12 (1. 5. 12)
` pc J
The subscript 1 in expressions (1.5.11) and (1.5.12) signifies that the
acoustic power and driving force are expressed in terms of unit length.
The acoustic power radiated by a plate driven by a point moment M on fre
quencies f < fcr is
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(1.5.11)
pckIM2 /
~ 
N 12rc (wr::)z ~l 2(~~ ri )2 2 W ri [ 1 ( W~nl2~ arctg ~ (1.5. 13)
~ /
Radiation of Finite Plate. Expressions for the radiation resistance of
_ single vibration waveforms of a plate are presented in the literature [48],
but they are rarely used in practice. If the linear dimensions of a
_ structure are shorter than the wavelength in the medium and the frequency
is higher than the first resonance frequency of the vibrations of a plate,
the radiation resistance of a hinged plate wil1 be
[kp=cr]
R apC3
=
n4fK, Pl~~ .
(1.5.14)
If the linear dimensions of a structure are longer than the wavelength in
the medium, then on frequencies f< fcr
a 2
9s
R = P 4i ((P) 1 /PCP Kp
fKp
(1.5.15)
where P is the perimeter of a plate,
_ f 4(12(p)
I n' {~cp V l  ~P
9i ((P)
0
f < 2 fhpt 2V~P (1  (p) In l~~P
�l  l~W
1 .
fKp~ 92 (~P) = 4nz (l  V')3/2
f >
2
(1.5.16)
For different boundary conditions the expression q2(~) wi11 acquire the
form
2V'cp}2(1cp)In i+~~ {(1cp)aresin ~
92 W 4a z (1  (03/2 41
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for a pinched plate;
21f p  2 (1  y) ln 1 + Y~ ( t  aresin 2
9~ (~P) t l~~p t (P
. 9,t=(  T).1/2
for a free plate;
 ?V(p  2I +(2)V~+ 2V~ ) In 1� l~~ I~'I f 2)Y 2yY) aresin 12~
[H=b] 92 (T) 4nz (1  (P)3/2
, . mo k"
Y = nt
for a plate, around whose contour is fastened a rib with linear mass m0;
kb is the wave number for the bending wave in a plate.
On frequencies f> fcr the boundary conditions have no effect on the
emissivity of plates, and the radiation resistance is
pcs
R V1 _ ~_1� � (1. 5. 20)
When the force that drives the structure is known the acoustic power
radiated by resonance vibration waveforms can be determined from the expres
sion
N = FZ 4x
16Vme 1 I qlf'
Mounting hardware exerts a negligible influence, in the first approxima
tion, on the radiating capacity of ship structures. This problem is solved
in various approximations in the literature [7, 32].
Radiation of Cylindrical Shell: The problem of the radiation of a cylin
drical shell is much more complicated than the problem of the radiation of
flat plates; a computer must be used to find the exact solution of this
 problem. However, in the case when the conditions f> f and f � f
a cr a
are satisfied, where f a = c/ffd, which is called the annular frec{uency, the
radiation resistance of a cylindrical shell is equal numerically to the
radiation resistance of an equivalent plate.of equal size. This subject is
examined in greater detail in the literature [45].
1.6. Electromechanical and Electroacoustic Analogies
_ Table of Analogies. By using differential equations of the identical arder
and form for vibrations in media with the same elastic constant (acoustic
processes), for mechanical (reciprocating and rotating) vibrations and
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electrical oscillations it is possible to establish analogies bath
between the parameters of all these processes, and between the parameters
and impedances of vibrating systems [12, 27, 31, 35, 39, 40, 47].
The convenient and elegant techniques that have been developed in electri
cal engineering for analyzing vibration processes and systems provide an
opportunity to conduct a similar analysis in mechanical and acoustic
 systems [47].
The most important analogies of the parameters of vibration processes and
the parameters of systems themselves and of their impedances are presented
in Table 1.4. The electrical equivalent diagrams for determining the
required parameters are drawn on the basis of the analogies presented in
the table.
Comparison of Equivalent Diagrams. A few rules and suggestions that are
helpful in the comparison of equivalent diagrams are presented below.
1. Elasticity (spring, pad) transmits all of a vibrating forGe, and there
fore an element or a combination of elements placed upon it should be
included in a diagram as a parallel system (then the voltages on them and
on the elasticity analog will be identical). Mass transmits all of a 
vibration velocity, and therefore an element or a set of elements that
comes after it should be seriesconnected in the diagram (then the currents
in them and in the mass analog will be identical).
2. The velocity of a spring is equal to the difference of the velocities of
its ends, and therefore the currents in the analogs of mass and of aspring
that comes after it can be equal only at zero velocity, i.e., when the far
end of the spring is rigidly attached. The force of mass is equal to the
difference of the forces acting upon its front and rear faces.
3. The part of a circuit with a mass or elasticity analog is shortened
when the mass or elasticity is equal to zero, and it is broken when the 
mass or elasticity is infinite.
4. A friction element in an equivalent diagram is seriesconnected to the
elasticity analog, since the total force transferred by an elastic pad is
equal to the sum of the forces transmitted by elasticity and friction.
Example 1. A mass is placed on an elastic pad (frictionless), which is
placed on another mass (Figure 1.16a). A vibrating force acts upon the
firs4: mas5.
The equivalent diagram is shown in Pigure 1.16b. The first mass is series
_ connected to a circuit consistir.g of two different elements, because the
force (electric voltage) on the circuit is equal to the difference of
forces original and lost on the mass. The last two elements are
43
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Table 1.4. Ntechanical, Electrical and Acoustic Analogies Sign
of Analogy)
Mechanical parameters (for
Electrical
Acoustic
reciprocating and rotation
vibrations)
arameters
P
parameters
Parameters of vibration processes
Vibration force F} t Voltage Acoustic pressure p
Vibration moment M (emf) U
Vibration velocity (for one
Volume vibration velocity
of coordinates) x }
f Current i
I
v, vibration velocity ~
Same for rotation vibra
,
tions $
Vibration displacement x
Volume vibration displace
Same for rotation vibra }
4 Charge q
ment E
, vibration dis
tions ~
v
placement ~
Parameters and impedances of systems
Mass m, M
Mass moment of inertia l}
Inductance L
Acoustic mass ma
Inertial impedance iwm,
iwM
Inductive
Inertial acoustic
Same for rotation vibra }
ance
f impw
impedance imaw
tions iwI
L
Flexibility (pliability)
C
m
Same for rotation vibra
f Ca.pacitance
Acoustic pliability C
a
tions Dt
c
e
Elasticity (rigidity)
C C= G:M  CM
1
Acoustic rigidity
Same for rotation vibra
F
C
C= 1/C
tions D = 1/D
e
a
m
Deformation resistance
C/jw
Capacitive
Acoustic reactance
Same for rotation vibra
f impedance
C
/jw
tion D/jw
1/iwC
a
e
44
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Table 1.4 (Continued)
Mechanical parameters (for Electrical Acoustic
reciprocating and rotation arameters
vibrations) P parameters
Friction resistance R Ohmic Active acoustic impedance
4 impedance R
a
R
e
Total mechanical strength f Total electri Total acoustic impedance
Zm ca1 impedance Ze Za
paral lelconnected, since elasticity transfers all the vibration force to
the second mass, i.e., the electrical voltages on these elements should be
identical.
a) F I/ b)
m,j y1 I,
C N
IZ cu iwm2 a
~
rn2 Fs ~ ~i
~SId
 Figure 1.16. Construction of ec{uivalent electrical cir
cuit for twomass mechanical system with intermediate
elasticity.
a) b) imm,
m, '
C ~ .�_u
icr�22
m7 UU~ ~
Figure 1.17. Same as Figure 1.16, but force applied to
bottom mass.
All the limit transitions confirm the validity of the diagram. When m2 =
_ = 0, for instance, elasticity exerts no "backwater effect"; nor does it
45
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exert any action in the circuit, since the right hand branch is shorted.
When c= 0 the left hand branch is shorted and only the first mass counter
acts the force.
Example 2. The force is transferred to the second mass (Figure 1.17a).
The two top elements now function in the antivibrator mode. In the
 equivalent diagram (Figure 1.17b) these elements are parallelconnected,
since the elasticity transfers all the force acting upon it to the top
mass.
Example 3. The equivalent electrical diagram is transformed inversely to a
mechanical system.
 In light of what we have said above both systems shown in Figure 1.18a, b
are electrically equivalent, since the currents in all the elements of the
circuit are identical. However, if the same arrangement of elements as in
the top circuit (Figure 1.18a) is used in the construction of the mechani
cal diagram, and a vibrating force is applied to the elastic element, the
identical vibration displacement of the masses and elasticity cannot be
achieved. By rearranging the elements as is shown in Figure 1.18b it is
possible to arrive at the desired mechanical system.
a) C m,
b) , F'U
/j N . m,
m,
_ m., C m
z
~ ^dt2 C
Figure 1.18. Equivalent elect.rical diagram with series
connected elements converted to corresponding mechanical
system.
Consider now a circuit in which all three elements are parallelconnected,
i.e., the voltages on them are identical. Here the identical force must be
applied to all three elements. It is easy to see by the lower left circuit
in Figure 1.19a that force must be applied to the spring. The desired
effect is achieved, in particular, by using a rigid equalarm lever with
hinges (Figure 1.19b).
The electromechanical analogy procedure can also be used for analyzing such
systems as, for example, elastic soundinsulating couplings on the shafts
of machinery. When torsional vibrations occur in the elastic elements of
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a)
U E7n, Drn
Ll C
C m, )7nz
b) Fu
Figure 1.19. Same as Figure 1.18 with parallelconnected
elements in electric circuit.
9z Iz
f', 1,
.r u
Figure 1,20. Analysis of equivalent electric circuits by
loop current method.
the coupling, depending on its design, torsional or shear vibrations occur,
which are described by secondorder differential equations.
Analysis of Equivalent Diagrams. It is helpful to use Kirchhoff's loop
current method for analyzing branched equivalent circuits. To do this the
currents in the loops are numbered and their possible voltages are indi
cated. For the loops with a voltage source the sum of the voltage drops on
 all the elements of each loop should be equal to the voltage applied, and
for the other laops it should be zero (Figure 1.20; for more details see
[38]). The resulting equation system gives all the currents, and on their
basis the voltage drops (equivalent to mechanical forces) in every part of
_ the circuit (see Chapter 12).
It has been noted [35] that the second system of analogies, in which the
current is equivalent to force, can be useful in complicated circuits, when
the first system of analogies leads to nonplanar electrical circuits.
Application of Graph Theory to Analysis of Complex Mechanical Systems.
Graph theory [S] can be used for analyzing complex branched mechanical
systems, for example threestage and group (unit) vibrationinsulating
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damping systems. Graph theory was first used for calculating electro
mechanical converters by G. hfasori [23] and V. A. Fedorovich.
1.7. Electrical, Mechanical and Acoustic nPo1es
Simple and Compound Quadrupoles. An extremely convenient formalized mathe
matical tool the theory of quadrupoles, which describes oscillation pro
cesses in individual electrical elements and circuits [1, 38], is often
used in electrical engineering. Quadrupole theory is valid for systems
that are described by secondorder differential equations. In considera
tion of electromechanical and electroacoustic analogies, quadrupole theory
can be applied to longitudinal waves in mechanical and acoustic systems,
in this case conventionally called mechanical (or acoustic) quadrupoles
[11, 12, 28, 50].
There are only six known kinds of equation systems of a quadrupole, of
which two are in the socalled A, B, C, D form, and four are solved in
terms of input and transient impedances or conductances of a system [8,
29]. The A, B, C, D equations are suitable for analyzing the passage of
vibrations through mechanical systems with distributed constants (elastic
pads, models of inechanisms and foundations), for which the acoustic
impedance of the material and geometric dimensions are known, and through
acoustic systems with distributed constants (Figure 1.21a).
a) I
I
Fp
ft
A,B~C,A Zm b)
Y~
~2
' ~ . Fi fp
Zf>> Zvz ~
F. 3~f Z?f ~ 772,~11~ l?
F 2
a ~
l
A,B,C,A I Iz9,
0 j.
L.1 yp .
9r
Figure 1.21. Homogeneous symmetric mechanical link with
distributed constants (for longitudinal waves) and equiva
lent electric quadrupole in A, B, C, A form (a); asymmetric
or symmetric link with given input and transient impedances
of equivalent quadrupole (no load resistance) (b).
For systems (for instance shock absorbers) , for which the mechanical
impedances or conductances are known as the result of testing or calcula
tion (Figure 1.21b), it is better to use tha quadrupole equations that con
tain these parameters.
48
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The matrix equation that connects the parameters af an input longitvdinal
wave with the analogous parameters at the output of a passive svn:T, ;;ric
(homogeneous) mechanical system with distributed constants (seE Figure
1.21a), is (for harmonic waves)
Cy~] [C B] [ye]
c1.7.1,
Equation (1.7.1) is equivalent to the system of two algebraic equations
Fi = AFZ + B y:1
yi = CFz f Ags�
(1.7.2)
The terms of the summary transition matrix (with the E symboi), in con
sideration of the continuity of vibration forces and vibration displace
ments, can be found by multiplying the component matrices (Figure 1.22):
Am BE n B` I
ME=f C~Dyl=~l[Aci
` a`J _ a) b)
A>>Bf~~>>A~ Ai,Bi,~i,Ai An,gn,~n,An t,8cD
Figure 1.22. Chainconnected mechanical homogeneous
(symmetric) quadrupoles (a) and equivalent asymmetric
quadrupole (b).
a) .
I
^i m ZIP
y+
I yr
. ~
jwm
F Fi
Z4v

o
y2 = y~
b~) Fr I Slf
T RI I C
T
Fz Zo #z
F2
F, 6 Z ~
R ~
Ye
Yi
Figure 1.23. Mechanical elements of mass (a) and of
e.lasticity with friction (b) and corresponding quadru
poles.
49
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Thus, a compound quadrupole, equivalent to several chainconnected
different homogeneous (symmetric) quadrupoles, is asymmetric, i.e., the
te?ms of the left diagonal are not equal to each other (Figure 1.22b).
The quadrupole equations in which the parameters of the transition matrices
are expressed as input and transient impedances (see Figure 1.21b) are
suitable for inhomogeneous links in a system. The matrix ec{uation
I_z21 z221 [Y21
(1.7.4) `
. is very often used. It is equivalent to a system of two algebraic equa
tions:
Fi = Zii9i + Zizys1
Fz = Zziyi + Z:29 2.
(1. 7.5)
In equations (1.7.4) and (1.7.5) Z11 = FilY10 is the input impedance at
input I with output II open; Z21 = F2/yio is the transient impedance from
input I to output II with output II open; Z12 = F2/yl0 is the transient
impedance from output II to input I with input I open; Z22 = F1/y20 is the
input impedance from output II with input I open; for a symmetric vibrating
system (which, incidentally, is quite rare in this class 3f equations)
Z22 = Z1l,
If the conditions of the problem stipulate that the input parameters of a
process not be expressed through the output parameters, but rather the
output through the input, then the equation for any chain structure must
include the inverse matrix of M~ [see (1.7.3)]:
f E 'L ~ 2~ I JaJ l `~~D'! B~~"= ~C~ ` A:;J ~ F~
9i
(for A,Dz  B,C, _f 0),
(1.7.6) 
_ If the system completes two independent vibrations simultaneously, each of
which is characterizPd by a secondorder differential equation (for
example, longitudinal and torsional vibrations or longitudinal vibrations
and shear vibrations), then the general matrix equation of the vibrations
of the system may be represented as
hll A� KP 0 O BE KP ~ti1z
[np=1o; FL 0 AZ np 33E np 0 FZ
kp=t] jl 0 CZ np D, lip 0 y2
~Q 1 _C' Hp O O Dy Kp (P2
50
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where the subscripts "lo" and "t" indicate the kind of vibration
(longitudinal or torsional); M1 and M2 are the torsional moments at the
input and output of the system; $1 and $2 are the corresponding vibration
velocities.
a) , b)
J
_ 1n~ f~ F 1 Y'zn f, f?
0
M2 IPa
72
Figure 1.24. Mechanical system with bending vibrations
(a) and corresponding eightterminal network (b).
Elements with Bunched Constants in FourTerminal and TwoTerminal Networks.
For relatively 1ow vibration frequencies, when the elastic wavelength of a
given form is substantially shorter than the dimensions of the element, the
dimensions are represented as a point mass, elasticity and friction. For
= point mass m(Figure 1.23a) the equation is
r F11 l iwm i(~ Fa 1 .
~ y~~ ~ o t J IYZJ ~ c~. ~.8~
where w is circular frequency.
For a combination of elements of elasticity C and friction R(elastic pad
with losses on low frequencies, Figure 1.23b) the equation will be
~ o
(1. 7. 9)
[Yi F1J c l 1 L y~].
_ tu~ +R
EightTerminal Networks. A straight rod (and plate), completing plane
bending vibrations (Figure 1.24a), is described by a fourthorder differen
tial equation and accordingly is represented by the equivalent eight
terminal network (Figure 1.24b), at the input and output of which figure,
in addition to vibrating forces of velocities for transverse deformations,
vibration moments and velocities of rotation vibrations. The parameters
of the transition matrices for it (quadratic fourthorder matrices) are
given in the literature [9, 42].
BIBLIOGRAPHY
1. Ango, A., "Matematika dlya Elektro i Radioinzhenerov" [Mathematics for
Electrical and Radio Engineers], Moscow, Nauka, 1964.
51
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2. Belousov, Yu. I., and A. V. RimskiyKorsakov, "Reciprocity Principle
in Acoustics and Its Application for Calculating Acoustic Fields of
Vibrating Objects. A Survey," AKUSTICHESKIY ZHURNAL [Acoustics
Journal], 1975, Vol 21, No 2, pp 161173.
3. Brekhovskikh, L. M., "Vo1ny v Sloistykh Sredakh" [Waves in Lamellar
Medi a], 2nd edition, Moscow, Nauka, 1973.
4. Viktorov, I. A., "Fizicheskiye Osnovy Primeneniya U1'trazvukovykh I/oln
Releya i Lemba v Tekhnike" [Physical Principles of Application of
Ultrasonic Rayleigh and Lamb Waves in Technology], Moscow, Nauka,
1966.
5. Gerlikh, A. Yu. and I. I. Klyukin, "Application of Graph Theory for
Analyzing Efficiency of Multistage and Unit Vibration Insulation
Systems of Mechanisms," TRUDY LKI [Proceedings of Leningrad Shipbuild
ing Institute], 1974, No 91, pp 1318.
6. Gutin, L. Ya., "Acoustic Radiation of Infinite Plate Driven by Normal
Point Force," AKUSTICHESKIY ZHURNAL, 1964, Vo1 10, No 4, pp 431434.
7. Yevseyev, V. N., "Acoustic Radiation of Infinite Plate with Periodic
Inhomogeneities," AKUSTICHESKIY ZHURNAL, 1973, Vol 19, No 3, pp 345
351.
8. Zelin ger, Dzh., "Osnovy Matrichnogo Analiza i Sinteza" [Fundamentals
of hiatrix Analysis and Synthesis], Moscow, Sovetskoye radio, 1970.
9. Ivovich, V. A., "Perekhodnyye Matritsy v Dinamike Uprugikh Sistem"
[Transition Matrices in Elastic Systems Dynamics], Moscow, Mashino
stroyeniye, 1969.
10. Isakovich, M. A., "Obshchaya Akustika" [General Acoustics], Moscow,
Nauka, 1973.
11. Klyukin, I. I., "On Theory of SoundInsulating Pads," ZHURNAL
TEKIINICHESKOY FIZIKI [Journal of Engineering Physics], 1950, Vol 20,
No 5, pp 579589.
12. Klyukin, I. I., "Bor'ba s Shumom i Zvukovoy Vibratsiyey na Sudakh"
[Control of Noise and Acoustic Vibrations on Ships], 2nd edition,
Leningrad, Sudostroyeniye, 1971.
13. Kol'skiy, G., "Volny Napryazheniy v Tverdykh Telakh" [Stress Waves in
Solids], Moscow, IL, 1956.
14. Lamb, G., "Gidrodinamikal' [Hydrodynamics], Moscow, Gostekhizdat, 1947.
52
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15. Landau, L. D. and Ye. M. Lifshits, "Mek}ianika Sploshnykh Sred"
[Mechanics of Continua], Moscow, GITTL, 1953.
16. Lyamshev, L. M., "On Reciprocity Principle in Acoustics," DOKLADY AN
SSSR [Reports of the USSR Academy of Sciences], 1959, Vol 125, No 6,
pp 12311234.
17. Lyapunov, V. T. and A. S. Nikiforov, "Vibroizolyatsiya v Sudovykh
Konstruktsiyakh" [Vibration Insulation in Ship Structures], Leningrad,
Sudostroyeniye, 1975.
18. Lyapunov, V. T. and T. D. Rozhinova, "Characteristic Impedance of
Cylindrical She11 in Relation to Point Force," AKUSTICHESKIY ZHURNAL,
1970, Vol 16, No 1, pp 156157.
19. Meyz, Dzh., "Teoriya i Zadachi Mekhaniki Sploshnykh Sred" [Theory and
Problems of Continuum Mechanics], bfoscow, Mir, 1974.
20. Morz, F., "Kolebaniya i Zvuk" [Oscillations and Sound], Moscow
Leningrad, GITTL, 1949.
21. Morz, F. and G. Feshbakh, "Metody Teoreticheskoy Fiziki" [Methods of
Theoretical Physics], Vol 1, Moscow, IL, 1958.
22. Morz, F. and G. Feshbakh, "Metody Teoreticheskoy Fiziki," Vol 2,
 Moscow, IL, 1960.
23. Meyson, S. and G. Tsimmerman, "Elektronnyye Tsepi, Signaly i Sistemy"
[Electronic Circuits, Signals and Systems], Moscow, IL, 1963.
24. Nikiforov, A. S., "Impedance of Infinite Plate in Relation to Torsion
Moment," AKUSTICHESKIY ZHURNAL, 1971, Vol 17, No 3, pp 484485.
25. Nikiforov, A. S., "Impedance of Infinite Plate in Relation to Force
_ Acting in Its Plane," AKUSTICHESKIY ZHURNAL, 1968, Vol 14, No 2,
PP 297298.
26. Nikiforov, A. S. and S. V. Budrin, "Rasprostraneniye i Pogloshcheniye
Zvukovykh Vibratsiy na Sudakh" [Propagation and Absorption of Acoustic
Vibrations on Ships], Leningrad, Sudostroyeniye, 1968.
_ 27. Olson, G., "Dinamicheskiye Analogii" [Dynamic Analogies], Moscow, IL,
1947.
28. Popkov, V. I., "Vibroakusticheskaya Diagnostika i Snizheniye Vibro
aktivnosti Sudovykh Mekhanizmov" [Vibroacoustic Diagnosis and Reduc
tion of Vibration Activity of Ship Mechanisms], Leningrad,
Sudostroyeniye, 1974.
53
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29. Reza, F. and S. Sili, "Sovremennyy Analiz Elektricheskikh Tsepey"
[Modern Analysis of Electric Circuits], Moscow, Energi.ya, 1964.
30. Rzhevkin, S. N., "Kurs Lektsiy po Teorii Zvuka" [Course of Lectures
on Acoustics Theory], Izdvo MGU, 1960.
31. RimskiyKorsakov, A. V., "Elektroakustika" [Electroacoustics], Moscow,
Svyaz', 1973.
32. Romanov, V. N., "On Acoustic Radiation of Infinite Plate with Rigidity
Ribs," AKUSTICHESKIY ZHURNAL, 1972, Vol 18, No 4, pp 602607.
33. Skuchik, Ye., "Osnovy Akustiki" [Fundamentals of Acoustics], Vols 1
and 2, bfoscow, IL, 19581959.
34. Skuchik, Ye., "Osnovy Akustiki," Vol 1, 2, Moscow, Mir, 1976.
_ 35. Skuchik, Ye., "Prostyye i Slozhnyye Kolebatel'nyye Sistemy" [Simple
and Compound Oscillating Systems], Moscow, Mir, 1971.
36. Stashkevich, A. P., "Akustika Morya" [D9arine Acoustics], Leningrad,
Sudostroyeniye, 1966.
37. Strett, Dzh. V. (Lord Rayleigh), "Teoriya Zvuka" [The Theory of
Sound], Vol 1, MoscowLeningrad, GITTL, 1955.
38. Tolstov, Yu. G., "Teoriya Lineynykh Elektricheskikh Tsepey" [Theory of
Linear Electrical Circuits], bioscow, Vysshaya shkola, 1978.
39. Furduyev, V. V., "Elektroakustika" [Electroacoustics], MoscowLenin
grad, OGIZ, 1948.
40. Kharkevich, A. A., "Izbrannyye Trudy" [Selected Works], Vol 1, Moscow,
Nauka, 1973.
41. Shenderov, Ye. L., "Volnovyye Zadachi Gidroakustiki" [tiVave Problems in
Hydroacoustics], Leningrad, Sudostroyeniye, 1972.
42. Cremer, L. and M. Heckl, "Korperschallphysikalische Grundlagen und
technische Anwendungen," Berlin, Springer Verlag, 1967.
_ 43. Heckl, M., "Untersuchungen an Ortotropen Platten," ACUSTICA, 1960,
Vol 10, No 2, pp 109115.
44. Heckl, bi., "Vibrations of PointDriven Cylindrical Shells," J. ACOUST.
SOC. AMERICA, 1962, Vol 34, No 10, pp 15531557.
45. Junger, M. and D. Feit, "Sound, Structures and Their Interaction,"
Cambridge, Mass., MIT Press, 1972.
' 54
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46. Kennard, E., "The New Approach to Shell Theory," J. OF APP. MECHANICS,
1953, Vol 20, No 1, pp 3340.
.
47. Kurtze, G., "Physik und Technik der Larmebekampfung," Karlsruhe, 1964.
48. Lamb, G., "Input Impedance of a Beam Coupled to a Plate," J. ACOUST.
SOC. AMERICA, 1961, Vol 33, No 5, pp 628632.
49. Maidanik, G., "Response of Ribbed Panels to Reverberant Acoustic
Fields," J. ACOUST. SOC. AMERICA, 1962, Vol 34, No 6, pp 809826.
50. Snowdon, J. C., "Mechanical FourPole Parameters and Their Applica
tion," J. SOUND VIB., 1971, Vol 15, No 4, pp 307324.
\
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CHAPTER 2. PRINCIPLES OF ~4EASUREMENT ACOUSTICS
2.1. Input and Metering Circuits. Calibration of Acoustic and Vibration
Meters
Metering circuits are used for measuring the characteristics of acoustic
and vibration fields. These circuits include the following basic units:
electroacoustic converter a sound or vibration receiver, amplifiers,
filters, voltage dividers, displays and recording instruments.
In laboratory research a metering circuit includes an input circuit, which
turns on a generator that produces electric oscillations of a given kind,
a power amplifier and an electroacoustic converter a source of sound or
 vibration. A block diagram of a typical acoustic channel for laboratory
studies is shown in Figure 2.1. A measurement space should satisfy a
number of requirements, determined by the purpose of investigations, fre
quency range and utilized equipment. Electronic units of audio channels
(amplifiers, generators, indicators and recording machines) are selected in
consideration of the frequency band and purpose of the investigation; they
satisfy typical requirements, imposed on radio electronic test instruments.
_ In most cases industrial instruments are used if their parameters meet
requirements.
An acoustic metering chsnnel must be calibrated (for instance, sensitivity
and concentration factor are determined). Calibration is done comparatively
rarely. Calibration is done several times for the purpose of checking the
performance of a channel during the measurement process. This procedure
determines how well the condition of a channel corresponds to its rated
parameters.
Through and twostep calibration procedures are used for metering channels.
Ia through calibration a known acoustic parameter, for instance p, is
supplied to the receiver part, the readings of dividers N1 and indicators
N2 of the channel are recorded and correction factor 0, which must be added
to their readings after measurement, is determined:
a =PNlN2, (2.2.1)
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where all the values are given in decibels (relative to an arbitrary
level).
Measured pressure pX is determined by the expression
pX = N ~ N., A, ( 2 .1. 2 )
where N~ and N2 are readings of the dividers and indicators of the channels
during measurement; pX is read in decibels relative to the same zero level.
Figure 2.1. Schematic diagram of typical acoustic channel
for laboratory research: 1 oscillator; 2 power ampli
fier; 3 electroacoustic transducer; 4 test room;
5 electroacoustic receiver; 6 amplifier; 7 voltage
' divider; 8 filter; 9 oscillograph; lU recorder.
In twostep calibration the sensitivity of the electroacoustic receiver is
determined separately and the el ectronic part of the metering channel is
calibrated. Acoustic pressure pX during measurements in this case is deter
mined by the formula
nX~ NifN,;,ti9TicN,Nzi (2.1.3)
where u is electrical voltage, V(in decibels during calibration of the
electronic part of the channel); M is receiver sensitivity, dB; Ni, N2,
N1, N2 are the same as in formulas (2.1.1) and (2.1.2).
The analogous expressions can be formulated for the calibration of vibra
tion metering channels.
During the assembly of audio metering channels from individual devices it 
is essential to consider the following requirements: the input impedance of _
= each successive instrument must be substantially greater than the output
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impedance of the one before it; the capacitance of connecting wires and
cables must be taken into consideration when connecting the instruments up.
_ Electrical calibration must be done for the entire channel to ensure that
all of the components have a linear response.
Methods used for calibrating electroacoustic converters are: calibration on
the basis of the reciprocity principle; electrostatic calibration; electro
 dynamic and piezocompensation calibration; vibrating liquid column calibra
tion. In addition, calibration by comparison with a reference converter is
used extensively.
Calibration Based on Reciprocity Principle. In accordance with the recipro
city principle the sensitivity of a linear twoway electroacoustic converter
in the receive and radiate modes is related as
M = TH,
where M is the sensitivity of the electroacoustic converter in the receive
mode (the ratio of the output voltage of the converter to the acoustic
pressure acting upon the instrument); T is the sensitivity of the electro
acoustic converter in the radiate mode (the ratio of the acoustic pressure
at a given distance to the electric current that drives the converter); H
is the reciprocity coefficient (parameter).
The reciprocity coefficient is determined by the conditions of radiation
and reception and by the nature of the generated acoustic field. When
spherical, cylindrical and plane waves are radiated and received the
reciprocity coefficients are, respectively, _
H_ 2r?. , N_ 2La., ' H, 2S (2 . 1. 4)
nC pC ~ PC e
where r is the distance to the receiver; a is the acoustic wavelength in
the medium; p, c are the density and speed of sound in the medium; L is
converter length; S is converter area.
Using three converters is the most common calibration technique. In addi
tion to a test converter it is necessary to have a twoway converter and an
auxiliary audio source. Calibration is done in three steps (Figure 2.2).
Step one: the auxiliary source radiates and voltage ul at the output of the
tested audio receiver is measured. Step two: a twoway convertex, operating
in the receive mode, is set up in place of the instrument to be tested. The
operating mode of the auxiliary audio source is not changed. Voltage u2 at
the output of the twoway converter is measured. Step three: the twoway
converter is set up in place of the source, is driven by current i and
generates acoustic pressure, which acts upon the tested receiver.
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. '
1) 3man 1~�T u
,
/
2
3man II ~r~~~ rf.~~
 2
1) 3man I!I Fu 3
.R .
Figure 2.2. Calibration of audio receivers with three
converters based on reciprocity principle: 1 auxiliary
source; 2 twoway converter; R small active
impedan ce; ulu4 measured voltages; X,(here and herein
after) tested receiver.
Key: 1. Stage
 Small resistor R is usually connected in series with the twoway converter.
Voltage u4 on that resistor (proportional to current i) and voltage u3 at
the output of the tested receiver are measured.
Sensitivity M, in V/Pa, is determined by the formula
_ M = 7 ~ RH
uzu;
If during calibration with voltage dividers the same signal appears on the
indicator, then sensitivity M, in dB, is determined from readings of klk4
on the scales of the dividers:
!Y1= 2 (ki�k:fkakaIRIH) (2.1.6)
(here a11 values are expressed in decibels relative to units of the Inter
national Units System, and scale values klk4 are expressed in terms of an
arbitrary unit).
This procedure can also be used for calibrating directional converters if
" they are aimed at the sound source in all calibration steps. The metering
channel must be kept linear in all calibration steps. Distance must corre
 spond to the inequality r> D2/;k, where D is the size of the converters.
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Test conditions should correspond to a boundless medium. Therefore
calibration is often done in the pulse mode; calibration can also be done
in a slightly deadened room, the dimensions of which do not exceed the
calibration distance by a factor of more than 20.
When complicated types of converters are used at short distances the
correctness of a selected distance must be determined on the basis of the
dependence of 1/p on r; segment Ar, cut on the r axis by the extension of
this function, is the correction factor.
To avoid repeated setups during the calibration process the converters are
arranged so that progression from one step to the next can be done with an
electric switch. A sample schematic diagram of a pulse metering channel
for calibrating converters set up "in line" is shown in Figure 2.3.
To switch from the first position to the second (i.e., to switch to the
se cond calibration step) an electric motor is turned on, which rotates the
radiator by 180�. If the radiator has a symmetric electricplane charac
te ristic or is nondirectional the radiator need not be rotated. The three
converter method is used on frequencies from hundreds of hertz to hundreds
of kilohertz.
Selfcalibration of reversible converters is based on the following
principle. The converter receives the signals it radiated after they
bounce off a reflector (reflecting surface), located at distance r/2. The
_ dimensions of the converters D, reflector Re and r are connected by the
re lations
z 2
[3=e] r ] 67~; 2D ~ ~ < 'R3 . (2. 1. 7)
The surface of water may be used as a reflector.
Calibration is done in the pulse mode. Measures must be taken to prevent
blocking of the amplifier when a driving pulse reaches it. During calib ra
tion voltage uX of the signal reflected from the reflector, and voltage uI
proportional to the driving current, are measured. Converter sensitivit y is
M= v ux !ZH . (2.1. 8)
ul
This procedure is used for high frequencies.
Re ciprocity calibration on low frequencies is done in small booths (the
calibration of microphones by this method is regulated by standards in many
countries). The booth must be substantially smaller than the acoustic wave
length in the medium, and the booth walls must be rigid. Then the recipro
city coefficient is H= kV/pc, where k is the wave number for the medium;
V is booth volume. A booth is a miniature setup, to two sides of which are
60
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connected radiators and microphones, which are replaced in accordance with
the abovedescribed procedure.
A hooth made of a piezoceramic ring, the walls of which function as an
auxiliary acoustic source (Figure 2.4), is used in order to convert from
multiple setups to electric switches. Large capacitor C0 is seriescon
nected to the reversible converter instead of a resistor. Sensitivity is
determined by the formula
_ M = Y ulu3 V
uxua CoW ' (2.1.9)
The frequency band for this method is about 1005,000 Hz.
Figure 2.5. Electrostatic c alibration of capacitive micro
phones: 1 AC voltage generator; 2 coupling capacitor; 
_ 3 DC source; 4 filter resistor; 5 extra electrode;
6 capacitive microphone; 7 amplifier; 8 filters;
J recorder.
Calibration of piezoconvertershydrophones by measurement of their electri
ca1 resistance in water Rr and in asr Ra is a modification of the recipro
city method. This procedure minimizes the measurement space and is
especially suitable for nondirectional receivers. On frequencies below the
_ mechanical resonance, during measurement on an AC bridge, sensitivity is
determined by the formula [17] 
4ncY (Rc  Ro) U'.
[c=r; e=a] M = [ ] , (2.1.10)
pWs
where y is the concentration coefficient of the hydrophone;to is angular
~ frequency.
Sensitivity, in V/Pa, for nondirectional hydrophones in water is
 M  0,7
f
YR` R�(2.1.11)
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An industrial instrument, or Qfactor meter, may be used for such measure
ments [16]. The precision of the procedure is 1 dB and the frequency range
is 430 kHz.
Electrostatic Calibration. This method is used for calibrating flat
membrane microphones in air (Figure 2.5). An electric potential in the
form of DC and AC voltages, with the DC voltage substantially higher than
the AC voltage, is applied to the gap between the sensitive element of the
microphone and the auxiliary electrode. Under the influence of the elec
tric field electrostatic forces act upon the membrane of a receiver, and
the pressure p of these forces, in Pa, is
8,851tou�1012
p = d= , (2.1.12)
where u0 is DC volti.be, in V; u is the amplitude of AC voltage, in V; d is
the gap.
The electrode is perforated to eliminate the influence of the elasticity of
water in the gap; in this case a correction factor is added to the value of '
d according to formula (2.1.12), _
The frequency range is 510,000 Hz. Precision is 1 dB.
Electrodynamic Compensation Calibration [3]. The pressure, developed in a
small chamber in which is placed an instrument that senses acoustic pres
sure in water, or a hydrophone, is compensated by the electrodynamic force
 that drives a membrane, mounted in one of the walls of the chamber. At the
time of compensation, which is achieved by selecting the proper amplitude
and phase of the driving current, the membrane does not vibrate, and this
fact is shown by an optical indicator. The constant of the chamber is
determined by calibrating the chamber to a known static pressure pst. The
sensitivity of the tested hydrophone is determined by the formula
[ct=5t] ,1q = uxlo _
'XnCT ' (2.1.13)
where 1 0 is the static pressure compensation current; Ix is the acoustic `
pressure compensation current.
The frequency range is 0.11,000 Hz. When laser systems are used high pre
cision can be achieved in the monitoring of the compensation time. We note
_ that it is not necessary to perform compensation during measurements; it is
sufficient to use compensation for deteimining the constant of the chamber
and to drive the electrodynamic converter with a given current during cali
bration. The precision of the method is t1/2 dB.
Piezocompensation Calibration. A small chamber is made of two elastically
coupled piezoelectric cylinders, the ends of which are capped (Figure 2.6).
An auxiliary radiator is inserted in one of the end caps. The hydrophone _
to be tested is placed in the chamber.
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~
1r 7 ,
'
!i
4.1
4 f
Figure 2.6. Piezocompensation calibration of hydrophones:
 1 radiator; 2 null member; 3 electric oscillator;
4 amplifier; 5 cam drive; 6 volt meter; 7
reservoir; 8 null indicator; 9 tested hydrophone.
_ According to the literature [9] the calibration procedure consists in the
following. Some pressure on a given frequency is produced with radiator 1
in the chamber. Voltage uX at the input of the test receiver is measured,
and then the pressure is compensated by changing the amplitude and phase of
the drive voltage of one of the cylinders (null member 2). The compensation
time is determined by the weakest signal picked up by the second cylindrical
converter (null indicator 3).
The sensitivity of the procedure is determined by the formula
hf = uK Mo '
f!" (2.1.14)
where M0 is the constant of the setup, determined by compensating pressure
 po in ths chamber, developed by a variable column of liquid on low fre
quencies [3].
The frequency band for which the described method is suitable is 1 to 3,000
Hz. In this band M0 does not depend on frequency.
The calibration procedure can be simplified substantially by virtue of the
fact that po is directly proportional to the drive voltage of the null
member. Therefore calibration in the working frequency band can be done by
maintaining voltage uk on the null member constant and by measuring the
voltage uX on the hydrophone output. Sensitivity is also determined by
, formula (2.1.14), and constant MO, as before, should be determined by com
pensating the pressure developed by a variable column of liquid. Precision
is �0.3 dB.
A piezocompensation chamber, consisting of two piezocylinders, can also be
used for calibrating hydrophones by the reciprocity principle (three
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converters in a small chamber). The null member here is an extra source,
and the null indicator is a reversible converter. The following voltages
are measured: ul, u3 of the tested receiver, u2 of the reversible
converter and u4, which is proportional to the drive current of the null
indicator. Sensitivity is determined by the formula
M _ r a1u,R11 \Il"
(2.1.15)
` u2ua ~
Here H= wCO, where C0 is the total flexibility of the measurement space,
characteristic of a given setup. Its value is determined by calibrating
the instrument with a colwnn of liquid or with a waterair resonator [15].
Calibration with Variable Column of Liquid over Hydrophone. This method is
used on frequencies not higher than 23 Hz. The hydrostatic pressure can
be changed by vertically shifting the hydrophone in a vessel of water or by
changing the water level over the stationary hydrophone. The dash line in
Figure 2.6 explains how the second method is done. The cavity of the test
chamber is connected to an open vessel, which completes vertical motions
through a cam drive. The sensitivity of the hydrophone being calibrated is
determined by the formula
A1 = uX
pgll ' ( 2 .1.16 )
where p is the density of water; g is the acceleration of gravity; h is the
_ amplitude of the vertical displacement of the water level. The frequency range is fractions of a hertz to 1 Hz. The error is about
1 dB.
Relative Calibration of Acoustic Pressure Receivers. This method is based
on comparison of the readings of a reference and test receivers when acted
upon by a given acoustic field. There are the comparison method, whereby
the source radiation acts simultaneously upon the reference and tested
receivers, and the substitution method, whereby the receivers are placed
one after another at the same point of the field. The second method is more
accurate if the source is kept in the same excitation mode and the same
electronic meter is used. The use of a rotating system that places the
receivers in the same place in the field (Figure 2.7) speeds up the com
 parison procedure. In relative calibration of ar.oustic radiators equal
cond'Ltions of electrical excitation are created and the pressures developed
by the radiator are compared with an auxiliary test instrument.
Calibration of Vibration Receivers. The basic method is calibration on
vibration stands, in which vibrations of a given frequency are generated on
a massive (in comparison with the receiver to be calibrated) vibration
bench. The parameter of vibration (displacement, vibration velocity,
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vibration acceleratic~:) is usually determined by the contactless method,
i.e., optically or capa,.itively, with the aid of eddy currents, the
Mossbauer effect, etc. Relative calibration by comparison of the signals
from a reference and test receivers, reacting to the vibrations from the
same vibration source, is used extensively.
Stationary systems for calibrating acceleration receivers on frequencies
of 0.1 Hz to 20,000 Hz are being manufactured. The most precise methods of
determining vibration levels are procedures that use interference laser
setups, by strobe interference, by counting the number of times the inter
ference bands disappear, etc. They provide a displacement measurement
error of about 0.1 micron. VVhen working on vibration stands it is very
important to keep the vibrations linear and to prevent spurious vibrati.ons
of the vibration bench.
\
2
4
n
Ds
5 , n
7
lil R81
Figure 2.7. Relative calibration of audio receivers: 1
electric oscillator; 2 pulse generator; 3 radiator;
4 receivers; 5 switch; 6 amplifier; 7 filters;
8 recorder.
Automated portable vibration stands provide constant vibration accelerations
of the vibration bench in the working frequency range. This is accomplished
by using electromechanical feedback with the aid of a reference vibration
receiver, set up symmetrically with respect to the one being tested on the
opposite side of the vibration bench [7].
When calibrating by comparing the readings of the reference and tested
vibration receivers it is important to maintain the excitation mode
constant and to keep external conditions (temperature, pressure and nature
of fastening) consistent with the mode of calibration of the reference
receiver. The best results are obtained when the receivers are fastened
alternately to the sarne point on a structure. The use of one meter,
switched to the reference and tested receivers, improves the accuracy of
the comparison if the outputs of the receivers are correctly matched to the
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amplifier input. The effect of transverse vibrations is checked by
fastening the receiver at a right angle to the direction of the basic
vibrations.
Reciprocity calibration of vibration receivers [14] is comparatively rarely
utilized in connection with the high level of electrical induction from the
driving generator. However, calibration on frequencies higher than 3035
kHz can be done only by this method.
2.2. Measurement of Noise and Vibration
Quantitative measurement of noises and vibrations requires the selection of
the proper audio receivers (microphones, hydrophones, vibration receivers),
meters, measurement facilities (when possible), proper setup and determina
tion of the number of ineasurement points and the corresponding methodo
logical evaluation of the results obtained.
bteters that receive acoustic pressure in air microphones should meet
rigid (in comparison with their popular household counterparts) requirements
on performance stability in time and under various external conditions, and
on the dynamic and working frequency ranges. Electrostatic microphones,
with high and uniform sensitivity in a wide frequency range, are used for
the measurements. The sensitivity of the microphones, their directivity
and intrinsic noise level should be taken into consideration during measure
ments.
A noise meter, an instrument with an amplifier with a given frequency
characteristic and an indicator, in addition to a microphone, is used for
measuring noise in air. Modern noise meters are capable of using different
_ frequency characteristics (A, B, C), regulated by IEC [International
Electrotechnical Commission], of octave analysis and of transmitting infor
mation to recording and analyzing systems. The requirements on the charac
teristics of noise meters are set forth in international recommendations of
 IEC.
Piezoceramic watertight spherical and cylindrical receivers, and disk
receivers on high ultrasonic frequencies, are used as measuring hydrophones
during measurements in water. Systems that measure vibration displacement,
vibration velocity and vibration acceleration are used for analyzing the
parameters of a vibration process.
Displacement is measured with a microscope and capacitive receivers, and
vibration velocity is measured with inductive receivers velocity meters
[11]. The piezoelectric vibration acceleration receivers (accelerometers),
which utilize lead zirconatetitanate piezoceramic (TsTS19, TsTS23) and
other compounds, are the most popular [8]. The working band of these
receivers is 1 IIz to tens of kilohertz. Accelerometers with integrated
circuits can measure vibration velocity and displacement. High stability
and seiisitivity, compactness, light weight, wide working temperature range,
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the ability to function in liquids and gases are typical features of
piezoceramic vibration receivers.
The characteristics of modern microphones and vibration receivers are pre
sented in Table 2.1 [13].
Table 2.1. Characteristics of Modern Mic.rophones and Vibration
Receivers
9)
12)
Key: 1. Type and brand of instrument
2. Working frequency range, Hz
3. Sensitivity
4. Dynamic range, dB
S. Microphone capsule dimen
sions, mm
6. Temperature reaction, dB/1�
7. Weight, g
8. Manufacturer
Acoustic parameters are usually measured on logarithmic scale in deci
bels, because o� the difficulty of using linear scales in a wide range of
change of noise: from the audibility threshold of 2�105 to the discomfort
threshold of 106 Pa, i.e., a factor of 1011.
The decibel scale for sound intensity I and pressure p is determined by the
expression
IA6=db ] NArl 10 ig o~ 20 19 n,
68
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2) ~
3)
4)K
o
~ 7X
61a
7)
8) ~
1
1 J
r
"
.
K
Titn ii ,tapKa
C.
�
111,1160(1:1
OL
!V
rIA
C
rH
~
~
ah
M
z '
~Y
"0.
~
n
Cu
G
'n
"p
02
V
Q
K S
GI 2
0. S
C. H
~G
U
n~it~tpaq)oI 11,1:
9131
20I8 000
30 ntl3;Ila
ISIAS
25,2
0,2

Jjaun~
4134
1040 OOU
13 MIl/Ila
28160
12,6
0,1

J~OIIHH1~
4136
5100000
2,8 m D!fla
60174
6.3
0,1

T1atilt,t
4138
5140000
0,1 ec8llla
65181
3,17
0,1

Jlnirna
A1K 6
2040 000
10 tt13!Ila
40I54
15
0,1

CCCP 1;
lIAiK 6
202U U00

5015�1
17
0.1

CCCfl
Bt:rponpHemunKtc
4332
06000
50 M[3/g


0,1
30
J.1a H11A
4335
4
08 000
19 mii;9


0.1
13
'
Aamin
336
023 000
�ti DtB /g


0~1
2�
'AaHNH
4339
010000
10 m n/,q�




n;,H1,n
Au
:io 000
ao Mn"g




cccN 1.
13) ' I10,31511pMcSI crpuro i1octosiHiioA.
I)
)
9. Microphones
10. Denmark
11. USSR
12. Vibration receivers
13. *Set strictly c~)nstant
[MB=mV; f1a=Pa; HMH=IMK; A=D]
(2.2.1)
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where N is the noise level in terms of intensity and pressure; Io, p0 are
the zero (or threshold) levels of the examined values.
The international threshold intensity in air is I= 1012 W�m2. The
 acoustic pressure threshold is p0 = 2�105 Pa; threshold p0 is used in
water. The acoustic pressure level is routinely expressed in decibels and
 it is not stipulated, but understood that the zero level is po = 2�10~ Pa.
Vibrations are also measured in decibels relative to the acceleration
threshold (acceleration of gravity) g= 9.80665 m/s2. The sensitivity of
accelerometers is expressed as M= u/g and is measured in decibels in terms
of 1 mV/g, The zero vibration velocity threshold is sometimes 5�108
m/s, and the vibration acceleration threshold is 3�102 m/s .
 Arbitrary zero levels, usually equal to a unit uf ineasurement, are often
used for calculations. For example, 1 m/s2, Eo = 1 m/s, p0 = 1 Pa.
There are many advantages in representing results in decibels, but non
standard zero levels that are used must be indicated.
Measurement of noises produced by different sources requires that several
conditions be satisfied in regard to the measurement space, number and
location of ineasurement points. The most precise measurements can be made
in specially equipped sound chambers, the interior of which is lined with
soundabsorbing coverings (usually wedges), and measures are taken during
the construction of the building to prevent the infiltration of exterior
noises. The quality of sound chambers is usually expressed as the acoustic
ratio at the test point the ratio of the total energy of reflected sig
nals to the energy of the incident signal. For satisfactory measurements
this ratio should be smaller than 0.1 (then the error introduced by reflec
tions will be less than S%). Sound chambers are also evaluated in terms of
the nature of change of a signal as the distance between the receiver and
the source increases: deviation from the ratio 1/r should not exceed 0.5 dB.
In cases when the requirements of boundless space cannot be satisfied the
acoustic energy density is measured in a room, in which a noise source is
installed. Reverberation chambers with surfaces that reflect the incident
sound serve well for this purpose.
The procedure �or measuring noise generated by electrical machinery is
regulated by GOST 1127066. There are also several international recommen
dations of IECISO on various aspects of atmospheric noise measurement:
R1996, R1680, R495, R392; R357, R354, etc. [37] (these recommenda
tions are called "ISO Standards" since 1972).
To obtain reliable results the noises of machinery on ships should be
 measured at several points around a machine at a distance of about 1 m from
 the liiall, and the results are averaged [24].
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During acoustic vibration measurements the quality of the results depends
on how the vibration receivers are fastened and on how much influence it
exerts on the source of vibrations. The best results are obtained when
the vibration receivers are firmly fastened to the vibrating surface. The
vibration receiver may be assumed to exert no influence on the vibrations
of a surface if its mass is negligible in comparison with the linear mass
of the vibrating surface per unit length. Therefore low mass receivers,
from 20 to 3 g, are used [7].
2.3. Spectral and Correlation Analysis
Virtually all modern acoustic analyzers are electronic instruments, i.e.,
they work with electrical signals that are proportional to the acoustical
values that are measured. {Vhen all the elements of the metering channel
(electroacoustic converters, amplifiers, indicators) are linear acoustic
signals may be replaced with electric signals, and acoustic values can be
assigned to the results.
The determination of the amplitudes and frequencies (or frequency bands) of
 vibrations that are contained in a measured signal is ealled spectral
analysis. Analysis consists in the experimental determination with fre
quency selective elements filters for individual components of a compound
signal under analysis. The types of analysis are sequential, when the
spectrum of a signal is found by sequentially changing the frequency
proper.*.ies of the filters while "watching" the entire analyzed band (Figure
2.8a), and simultaneous analysis, when a signal is fed to a bank of
parallelconnected filters, so that the analyzed frequency components
appear at their outputs all at the same time (Figure 2.8b). The performance
of each filter in simultaneous analysis is similar to the performance of the
filter in the corresponding position c:uring sequential analysis.
K a) �
NT ir'
4701   ~  p
/ 1 ~
/
� b)
~0~
0 0
Figure 2.8. Position of filter passbands in sequential and
simultaneous spectral analysis.
The only practical difference between sequential and simultaneous analysis
_ is the fact that the number of filters in the second case is limited and the
 seams between the filters are analyzed with less precision, and therefore
the results of simultaneous analysis are not quite as good as in the case
of sequential analysis.
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The main element of spectral analysis is the filter, which is characterized
by the passband, transadmittance in the pass and opaque bands, and trans
conductance. Passband Af of a filter is determined by the difference of
the top and bottom frequencies, on which the transadmittance of the filter
(the ratio of the output voltage to the input voltage) decreases by 3 dB.
Transadmittance may vary within 3 d6 in the passband.
The transconductance of a filter is usually evaluated by attenuation out _
side of the passband for a given frequency error. In onethird octave
analysis with more than 20 dB attenuatian on the middle frequency of the
adjacent filter, the error related to the reception of signals from the
nopass band does not exceed 2 dB. The requirements on the attenuation
outside of the passband are considerably more rigid (more than 60 dB for a
threeoctave frequency difference) in the case of analysis of a wide pass
band.
The analyzing capabilities of a channel as a whole are characterized by _
resolution, dynamic range and analysis time.
Resolution expresses the capability of the analyzer to distinguish two
sidebyside frec{uency components of an analyzed signal and is expressed
as interval OF between the frequencies of two sini.isoidal signals with dif
ferent amplitudes, separated with a valley in the frequency characteristic,
equal to 500 of the maximum (Figure 2.9). The smaller AF the higher the
resolution.
0,
Figure 2.9. Determination of resolution of analyzer. _
Resolution depends on the properties of the filter and on analysis condi
tions. In simultaneous analysis it is related to the passband and
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transconductance of the filters, and in sequential analysis it is also
related to the speed of analysis. For an LCR circuit resolution is 4Af,
and for a bandpass filter it is approximately 3Af.
Analysis time is determined by the transient time of each filte r, which is turned on during analysis for only a certain length of time. Transient
time Att is related to passband Af by the ratio AttAf = A, where the 
coefficient A is determined by measurement conditions and by the character
istic of the filter. For an LCR circuit A= 1 for the time it takes the
signal to build up to 0.95 of steady state.
 e
Filters with a bellshaped frequency characteristic, for example weak
coupled LCR circuits [2], have the shortest transient time.
Signal analysis is done with the passband held constant in the entire fre
quency range (Af = const), or with the relative passband Af/f0 = const kept
constant, where f0 i_s the middle frequency of the filter.
In simultaneous analysis Af/f0 = const lowpass filters have the longest
transient time.
In sequential analysis the speed of analysis must be selected correctly.
The dynamic frequency characteristic of a filter with a continuously
changing middle frequency differs from static because the peak of the curve
falls and shifts toward a change of frequency, and the passband gets wider
(the characteristic becomes asymmetric). This is due to the inf luence of
inertial elements of the filter and of the interaction of its induced and
natural vibrations. To obtain undistorted results the rate of change of
frequency v to which the filter is tuned in sequential analysis must be
determined by the expression
v < No (A/)=,
(2. 3.1)
where u0 is a constant, which depends on the requirements impose d on the
analysis and on filter characteristics [27].
VVhen filter characteristics are not know an analysis is general in nature;
to determine v, in Hz/s, one may use the approximate expression
v = (0,25 1,0) (Af)=� (2. 3. 2)
The total time of sequential analysis is determined by the quotient of the
bandwidth of the analyzer, divided by the analysis speed. The analysis time
is many times longer than of simultaneous analysis.
'I'he dynamic range of an analyzer the ratio of the peaks of th e individual
components that can be distinguished by the instrument in its wo rking
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freQuency band, to the valleys characterizes the uncorrectable error of
analysis (for a given instrument), caused by deviation of the frequency
characteristics of perfect filters from real ones in the nopass band. For
successful analysis the dynamic range of the signal should be narrower than
the dynamic ran ge of the analyzer.
Spectral analysi s of random processes discloses the characteristics of
stable averages, and therefore the values obtained must be averaged
appropriately in time, which is called integration. Signal integration
time At is determined an the basis of the acceptable relative mean square
error 8 of the results of analysis of a random process:
b = (Al1AJAf)2,
(2. 3. 3)
where A1 is a co efficient that is determined by the type of filter, detec
tion and integration methods; A1 = 0.040.1.
Calculation of the time of analysis of random processes shows that the time
spent on such analysis is considerably longer than the time spent on the
analysis of dete rministic signals. When commercial instruments with a com
paratively wide p assband (octave, thirdoctave) are used for spectral
analysis of random processes larger errors are acceptable, if features of
the averaging of a random signal are ignored.
{Vhen selecting the instruments and conditions for conducting an analysis it
is important to consider the nature of the spectrum of the signal to be
analyzed, the purpose of the studies and the acceptable time of analysis.
For finding noise sources the analysis should have a constant passband,
which, however, should not be too narrow, since industrial noise sources
inevitably fluctuate during operation. Analysis with a constant relative
passband is bette r for noise abatement or sound insulation, since rather
wide frequency ranges are usually soundinsulated.
One way to reduce the analysis time is to transpose the signals during
recording on magnetic tape. Then the signal is recorded at a slower tape
transport speed and is played back at a higher speed; the band of the
reproduced signal is wider and the analysis time is shorter. Lowfrequency
signals are also easier to transpose because the spectrum of the signal,
transferred to a higher frequency range, can be analyzed with commercial
instruments, designed for the medium and highfrequency ranges [20]. Note
worthy among the spectrum analyzers for transposition analysis are the 7001,
7003 and 7004 magnetic recording meters, manufactured by the (Bryul' and
yer) firm.
The basic techni cal specifications of industrial analyzers and spectro
meters are liste d in Table 2.2.
Correlation analysis determines the strength of the relationship between
random phenomena on the basis of probability analysis of processes that
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are taking place. A mathematical measure of the rela*.ion of processes is
the correlation function, which expresses the probability relation of
samples of processes, separated by an interval of time.
Correlation analysis consists in the experimental determination of the
 dependence of correlation functions on the time delay of one of the
parameters of an analyzed process.
The crosscorrelation function of two processes x(t) and y(t) is written as
T
~Fxu (i) = liin t~.r (t) y(t dt = x(t)1(t ti)+ 2. 3. 4
r>;~ : ~ )
u
where T is the time delay of one function in relation to another; the
superscore over the product indicates averaging in time.
The autocorrelation function, which determines the probability relation of
one process at different moments of time, is written as
r
(PXx(i) lim J.r(:)x(t �i)dt. (2.3.5)
o
The practical importance of T is determined by the tolerable error and the
type of instrument used.
Correlation functions are connected by Fourier transforms to the enPrgy
spectrum of the processes:
co co
�Xy (T ) _ n r (~j (4)) COS lUi C1W fl r {(il) = 2 J ~Xy (T ) COS filt CIT, (2 , 3 . 6 )
J
0 0
where G(w) is the relative energy spectrum of the analyzed processes.
Hence it follows that correlation analysis methods cannot give any more
information than spectral analysis. But correlation methods in some cases
make it possible to simplify, speed up or facilitate an analysis.
Normal values crosscorrelation coefficient RXY(T) and autocorrelation
R(T) are used for eliminating the influence of the numerical character
istics of processes on the analysis:
x (t) y (I � (T) ,  y (1) X (t � r)
Ray  (T)  _ ~ R (T) (2. 3. 7)
X= (i) yZr) x2
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 If processes are independent their normal crosscorrelation function is
zero. However, even mutually related processes, under certain conditions,
_ have correlation coefficients that are equal to zero.
The crosscorrelation and autocorrelation coefficients cliange from +1 to
1. Equality to one means that there is a linear rela tionship between
processes: x(t) = ny(t), where n is a numerical coeffi cient. The auto
~ correlation function for T= 0 is equal to the mean square of the function,
i.e., it expresses the poiver of an analyzed process. The autocorrelation
coefficient for T= 0 is equal Lo unity.
The correlation functions of different signals used fo r analysis are
characterized by the coherence (or correlation) interval, i.e., the time
delay between phenomena, after which they may be assumed to be independent.
The time delay, after which the envelope of thf; autocorrelation function
cannot exceed a prescribed value (usually 0.1) after a further increase of
the time delay, is used in practice as the correlation interval.
The autocorrelation function of periodic processes is also periodic, and
the correlation interval is infinite.
The correlation interval is inversely proportional to the signal bandwidth.
Signals with the shortest correlation interval (for example, random signals
with a bellshaped spectrum) are of special importance in correlation
analysis.
The typical correlation analyzer [22] contains two amplifier and filter
 c}iannels, a variable time delay unit in one of the channels, a multiplier
and an integrating unit. Because it is so difficult t o perform multiplica
tion in modern correlation analyzers a system is used for estimating the
probability that analyzed signals, after being amplitudelimited, will have
the same sign. The delay of these instruments can be assigned by digital
componerits shift registers, controlled by a clock pulse generator, which
_ makes it much easier to set the delay [23]. Correlation analyzers, com
 bined with relative spectrum ar.alyzers, are also import ant [1].
Let us examine the application of correlation analysis byivay of example of
the determination of the acoustic ratio in a room, i.e. , the ratio of the
elergy of all reflected signals to the energy of the in cident signal at the
point of reception. A schematic diagram of a setup fo r measuring is shown
in Figure 2.10. When a noise signal is radiated one of the channels of the
correlator picks up the signal radiated in the room, received with a micro
phone and amplified in the electronic channel, and the other channel,
eqtiipped with a variable time delay, receives the signal directly from the
\ iioise generator. The signal received by the microphone is produced not
only by tfie direct sound from the radiator, but also by reverberations from
ttic walls, floor and ceiling of the room. The path of the sound from the
radiator to the receiver is shorter than the path of any reflected signal.
Tllerefore the parameters of the analyzer and the characteristics of the
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signal are selected such that coherence interval T0 of the
here expresses the length of a coherent parcel in space Z
shorter than the time difference of the propagation of the
direct p ath from the radiator to the receiver Z1, and via
from the reflector Z 2 :
io <  tr
.
c
signal (which
= 2T0 c) will be
sound via the
the shortest path
F .
I ~
in
~ `t t 7 8 = 9
Figure 2.10. Schematic diagram of setup for measuring 
acoustic ratio in room by correlation method: 1 noise
_ band generator; 2 room to be tested; 3 radiator;
4 receiver; 5 amplifier; 6 filters; 7 delay;
8 multiplier; 9 integrator; 10 indicator.
The acoustic ratio R is given by the formula
1
R R= (Tl)  1' (2. 3.8)
 where T1 = Z1/c is the delay corresponding to the direct path.
Correlation analyzers are used extensively for detecting periodic signals
in random interference.
The correlation procedure measures sound insulation, sound absorption and
 reflections, and because of its high sensitivity it is especially suitable
for rooms with poor sound insular.ion and with a high noise level.
Correlation analysis decermines the contribution of each operating machine
_ to the total acoustic or vibration field without having to turn the
machinery off.
Correlation analysis is good for determining the acoustic propagation
velocity in a material on the basis of data on the correlation function as
77
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a function of the delay. Longitudinal waves, bending waves and Rayleigh
and Lamb waves, propagating at different velocities, can be identified.
To do this the reference wavelength should be selected in accordance with
the coherence interval of the utilized signals and the expected difference
in propagation velocities.
2.4. bieasurement of Vibration Power Radiated by hlachineryl
The vibration processes of most mechanisms are stationary random processes.
Vibration power N, radiated by such mechanisms into bearing and nonbearing
connections is the scalar product of force pressure P(s, t), averaged in
time, from the machine and the vibration velocity of the plane of contact
q(s, t) :
r,
N= lim 1 j ) p(s,t)9(S, t).
r, S (2.4.1)
A machine and its shock absorber and foundation may be viewed as a single
system with a certain number of flat contact surfaces. In this case N is
the sum of vibration powers Ni, radiated through indivi.dual plane contact
surfaces with bearing and nonbearing connections by individual components
of the generalized forces coming from the machine. Spectral density
Ni(w) of the vibration energy flux is
[ao=ef] N; (w) = Qn9~, (co)q~'94) (w) cos u, (2.4. 2)
where Qi ef(w) and Qi ef(w) are the effective spectral components of the
force from the machine and the vibration velocity of the nth part in the
ith direction; a is the phase shift between force and velocity.
In the frequency band Aw
N'~1 (Aw) = Qn34, (AW) (Aco) ReRn, (2.4.3)
 where ReRn is the real part of the correlation coefficient between Qi and
qi in frec{uency band Aw. _
In the first approximation
[0=af [9i :,(b (Aw) ]2 Re T,'~`�~ (Aw), (2 . 4. 4)
1This section was written by V. I. Popkov.
78
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where ReZii af(Aw) is the real part of the impedance of the shock absorber
foundation system in the frequency band.
The vibration power of a machine can be determined experimentally by two
methods: direct as the scalar product, averaged in time, of force Qi and
corresponding velocity qi, and indirect on the basis of the vibration
velocities of the machine and mechanical impedances of bearing and non
bearing couplings.
Vibration power, radiated by f:,rces normal to bearing surfaces, is measured
directly.
A schematic diagram of an instrument that measures vibration power directly
is shovn in Figure 2.11. The Soviet IKM69 meter is built on this design.
The meter performs synchronous and cophasal analysis of signals, propor
tional to force and velocity, multiplies the signals in the passband of the
analyzer and averages the product in time. The zero levels for determining
(in d6) force, velocity and power are 2�109 N; 7�106 cm�s and 1016 W.
_ Figure 2.11. Schematic diagram of vibration power meter:
a diagram; b position o� dynamic pickup under machine;
1 dynamic pickup; 2 acceleration pickup; 3 ampli
fiers; 4 integrator; 5 dualchannel analyzer; 6
multiplier; 7 recorders. 
The dynamic pickup is installed in the bolt connection and serves as an
element, through which force is transmitted from the machine to bearing and
nonbearing couplings. This sensor can be installed between the foot and
foundation (Figure 2.11b), or between the foot and nut (Figure 2.11a). In
79
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the former case 70 to 900 of the force acting on the foundation is picked
up, and in the latter case approximately onetenth [26].
Vibration powers radiated as a result of excitation of any vibration
components linear and rotation, can be determined indirectly. The
impedances of the bearing and nonbearing couplings in relation to forces
and moments are measured by methods that will be explained below.
Signals from the sensors are fed to preamplifiers and then to a common
meter amplifier. The phase characteristics of the sensors should differ by
180�, and the gains of the preamplifiers kai and ka2 are set so that the
equality
tyik y i = Z'y2k yi
is satisfied.
Sensitivity to torsional vibrations is
 ~c~ =kyx.4~r.
(2,4.5)
hteasurement of Nfechanical Impedance. The exact mechanical impedance in
relation to force ZF is determined as the combined ratio of force F, applied
to a linear system, to the velocity component of the point of application of
that force qF:
ZF= F
9F
(2.4.6)
A diagram of a system that measures mechanical impedance is shown in Figure
2.12. An electrical signal from audio frequency generator 1 passes through
compression unit 2 and power amplifier 3 to vibrator 8. The frequency of
the signal is gradually changed, and the voltage is controlled by the com
pression unit so that the velocity level of the generated vibrations is
maintained constant in the entire frequency band. In this case the applied
force is proportional to the modulus of the impedance of all the examined
structures, and the frequency characteristic of the effective force,
recorded on the tape of recording instrument 17, corresponds to the fre
quency characteristic of the modulus of impedance. The product of the
signals of force uF and velocity ua in consideration of the cosine of phase
sliift a between them is proportional to the real part ReZF of the impedance,
and the same product in consideration of the sine of the phase shift is the
imagi.nary part ImZF. A si.gnal of acceleration is converted by integrating
amplifier 15 to a signal that is proportional to the vibration velocity,
and uF uq and cos a are multiplied by electronic multiplier 16.
For measuring mechanical pliancy N1 (the inverse of inechanical impedance)
compressor unit 6 maintains the force constant, and the frequency _
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characteristic of the velocity of the generated vibrations is recorded on
tape as the modulus IMFI�
A block diagram of channels for measuring impedances and pliancies with an
instrument built by the Bryul'and h"ycr firm [7] is shown in Figure 2.13.
A combined sensor impedance head, which generates force and acceleration
signals, or instruments made up of dynamic and acceleration pickups, ar.s
used for measuring force and velocity. Rigid requirements are imposed on
the design of combined pickups and instruments and on how they are attached
to test products, because of the wide ranges of frequencies and mechanical
impedances.
The b151C factors that influence t}le precision with which Zf and Mf are
measured are:
the mass of ttle transition part (adapter) between the dynamic pickup and
analyzed structure;
ttic rigidity of the adapter and of the fastener that holds the pickup to the
analyzed structures;
tlle strain sensitivity of the acceleration pickup.
The mass of the adapter restricts the lower limit of the dynamic impedance
measurement range, particularly on high frequencies, since the dynamic
pickup also measures the force spent to overcome the inertia of the adapter.
1'he rigidities of the adapter and contact and the strain sensitivity of the
acceleration pickup restrict the upper limit of the dynamic impedance
measurement range.
The search for designs of combined sensors and instruments is aimed at
correcting tlle abovementioned factors. Possible designs of force and
acceleration pickups and of combined pickups are illustrated in Figure 2.14
[26, 38]. The errors introduced by the adapter can be corrected by correct
ing the signals from the force and acceleration pickups with the aid of
electronic instruments [26]. ~
The mechanical impedance of structures in relation to moment Zm is deter
mincd by the ratio of combined bending moment M, applied to a harmonically
vibrating structure, to rotation velocity component Bending moment can
bc applied to an analyzed system with the aid of a cantilever T or Lbeam.
An instrument that de*_ermines impedance in relation to moment is s}iown in
Figure 2.14. 'I'ransverse force is applied to the free end of the beam. A
bending moment appears at the point where the beam is fastened. A segment
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a)
b)
NCMOvHJh' I7UMQHlIA1 J
5
~ 2J05 '
Figure 2.13. Schematic diagram of impedance meter channel
of BrytilK"yeri.nstrumcnts; a excited by pure tone;
1 vibrating product to be analyzed; 2 impedance head;
3 2623 preamplifier; 4, 7 meter amplifier; 5 level
recorder; G signal generator; 8 velocity signal; 9
acceleration signal; 10 preamplifier integrator; 11
vibrator; 12 force signal; b frequencyselective
meter: 1 vibrating product to be analyzed; 2 impedance
head; 3 preamplifier integrator; 4, 7 heterodyne
tracking filter; 5 level recorder; 6 signal generator;
8 preamplifier; 9 force signal; 10 vibration stand;
11 acceleration signal; 12 velocity signal; 13
power source.
Key: 1. Power source
of the beam that is short in comparison with the total length is assumed to
experience pure bending, and the longitudinal deformation of the boundary
layer is proportional to the moment acting upon its cross section. To
measure h1 it is sufficient to determine the deformation of the boundary
layer of the beam near its root section. Defoyii.ation pickups are cemented
to both ends of the beam in the plane of application of the moment and are
electrically counterwired to each other to eliminate the influence of
shear deformation and of the longitudinal wave on their readings.
83
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Figure 2.14. Determination of impedances in relation to
moment: 1 vibrators; 2 beam; 3 deformation pick
ups; 4 rotary vibration pickup; 5 product to be
analyzed; 6 preamplifiers; 7 meter amplifier.
2.5. Similitude and Dimensional Techniques in r'.arine Acoustics
Phenomena are called similar if the numerical values of parameters that
characterize the process in one system of objects can be found by multiply
ing the values of parameters corresponding to them in another system by
constant dimensionless factors. Similitude is used for establishing suit
able conditions for analyzing certain processes, a fullscale evaluation of
which is difficult, expensive or unreliable. A substitution of this kind
is called modeling.
Geometric, physical and mathematical modeling are employed. Geometric
modeling is used for demonstrating or showing the principle of operation,
general character of a process or kind of acoustic field. Physical modeling
is intended for determining the numerical values of parameters that describe
the behavior of a process in full scale, by measuring the corresponding
parameters in a model. Mathematical models are based on a mathematical
description of phenomena and on the numerical solution of equations that
describe their real behavior.
, Before phenomena and processes can be similar in physical modeling both the
basic equations that describe a process must be solved, and several boundary
conditions that make processes identical and unique in reality and in the
model, must be satisfied. The ratios of the corresponding values that
characterize similar processes at the corresponding points and moments of
time are called similitude constants. In physical modeling these abstract
numbers are called modeling scales. The dimensionless sets of parameters
that have the identical values in similar phenomena are called similitude
criteria. Such criteria for vibrations of an isotropic solid, for example,
are
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t~ idem; d_� idctn; G= idcm, (2.5. 1)
_ nE_i
wllere c, Z, t, p, p, d are the dimensions of velocity, length, time, pres
sure, density and volume of expansion, and Ef, G are Young's modulus and
shear modulus.
If the equations that describe investigated phenomena are not known, but
the physical values that describe them are known, the similitude criteria
can be found on the basis of dimensional analysis.
Some of the modeling scales can be selected arbitrarily for experimental
convenience, and the others are determined through the similitude criteria
and requirements of uniqueness.
The physical modeling of processes with geometric similitude intact (i.e.,
the dimensions of the model can be reduced or increased in comparison with
the ori ginal) is of greatest importance in acoustic research. The scales
for ce rtain parameters (for example, length and acceleration) are assigned
for convenience; all other parameters are determined in accordance with the
similit ude criteria and on the basis of dimensional analysis. Dissipative
losses due to internal friction are analyzed on the basis of the combined
representation of the Young's and shear moduli; the similitude conditions
are un changed if the Young's moduli do not depend on the amplitude and fre
quency of viUrations.
Dissipative losses in gases due to heat transfer, viscosity and molecular 
absorption are not simulated.
The modeling scale for linear dimensions ML should be equal to time scale
61T and inversely proportional to frequency scale htf, i.e., ML = MT = 1/Mf,
for physical modeling using the same materials in real life and in the
model (i.e., the constants p, Ef) G are maintained identical). The propaga
_ tion velocity of longitudinal and transverse waves remains unchanged in the
model. Bending wave propagation velocity also remains constant, in spite
of a change of size, since a decrease uf thickness is compensated by an
increase of frequency. In this kind of modeling an arbitrary ratio of the
scale of &formations to t}ie scale of linear dimensions may be used. For
xample, the velocity scale in reality and in a model with the same
~ a.,cele ration should be ML, and the displacement modeling scale is M~.
The choice of numerical values of the main modeling scales depends on the
specifi c purpases of the analyses. The linear scales for geometrically
similar reality and model are determined by the feasibility of imparting to
the model all thc features of the original and by convenience of ineasure
ment on the model. The linear scal.e usually varies from 1:5 to 1:200. Too
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large a modeling scale will not produce the desired scientific and
economic effect the models turn out too expensive and unwieldy, even if
they are quite exact. Too small a scale does tiot correctly reproduce
reality, and measurements on a small model produce gross errors.
The biggest problems in the modeling of acoustic processes in shipbuilding
are related to dissipative losses, since modeling is valid only if the
losses are identical on the frequencies of the model and the original.
The correct setup of ineasurements under conditions of smaller linear
dimensions and higher frequency requires the selection of instruments and
techniques that correspond to the model frequency band: the audio and
vibration receivers must be substantially smaller than the shortest
acoustic wavelengths. They must not exert any influence on the manner in
which an acoustic process takes place through their dissipative and sound
damping properties.
Electriual modeling of many acoustic processes, on the basis of the
generality of the differential equations that describe acoustic propaga
tion through structures and the flow of current in complicated electrical
circuits, is also used successfully [13].
2.6. Determination of Reliability and Precision of Acoustic Measurementsl
Before obtaining auantitative data for any acoustic measurements it is
important to make sure of the reliability of the expected result and to
estimate the anticipated precision. Otherwise gross errors may occur, and
a collection of random variables that bear no relationship to the process
being analyzed, measured or tested, may be accepted as a valid result.
A reliability check gives assurance that the metering channel receives
precisely the acoustic parameters being analyzed, and not different types
of interference (electrical induction, noises from a neighboring facility,
etc.). Noteworthy among the various check methods is the procedure whereby
the useful signal source is turned off, it:s power is attenuated with
insulating screens and the distance between the radiator and receiver is
changed. {Vhen the measurements are reliable the output signal of the
receiving channel changes exactly in accordance with a change of the
acoustic value.
A reliability evaluatiin must also include the characteristics of inter
ference, i.e., readings of the metering channel in the absence of the
signal to be aniilyzed.
'fhc reliability of acoustic measurements also depends on the measurement
procedure, wliich must correspond to the nature of the parameter to be
lI. I. Bogolepov contributed to thc writing of this section.
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measured. For instance, measurements of a fluctuating variable require
some averaging in time; measurements of noise from a moving source with a
stationary receiver require the selection of the correct measurement time
and, if need be, consideration of the Doppler effect, etc.
Visual monitoring of a measured value by oscillograph is also very
important, because it often helps to evaluate the nature of interference,
which may be of both acoustic and electrical origin.
hleasurement precision is determined in all cases by specific conditions.
Dleasurement precision is characterized by error AR, which expresses the
difference between the true value RD and the measured value R of the
parameter being measured:
eR=RoR.
(2.6.1)
The acoustic measurement errors of the typical acoustic channel can be
represented as follows.
The general expressions for a measured acoustic pressure or vibration
acceleration are of the form
P = Mk (I) S Miki Si ( 2 . 6 . 2 )
where u, ul are readings on the scales of the voltage dividers and indi
cator; D1, bli are the sensitivities of the receivers; k(f), kl(f) are the
summary transadmittance of the metering channel; s(f), sl(f) are functions
that characterize the influence of ineasurement conditions on thE results
(for instance, the influence of surrounding surfaces, how the vibration
receivers are fastened, etc.).
The measurement error of u, ui on the scales of the instruments is deter
mined by the class of the instruments and by how well they correspon4 to
 the naturc of the measured signals (particularly during the recording of
brief and fluctuating signals). The divider calibration error can be sub
stantial. Therefore it is better to avoid switching the dividers during
the measurement process. IIowever, the error can be reduced to an accept
able level, if need be, by means of electrical calibration of the channel.
The error introduced by the instability of the transadmittance of the
channel for k(f) and kl(f) depends on the quality of the power sources and
on the charactEr of heat transfer between the instruments and the environ
ment. 'fhe measurement instruments must be turned on in advance for warmup
for at least 30 min before the beginning of ineasurements to assure the
proper thermal balance.
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The calibration error (in terms of h1 and M1) is usually specified in the
technical documentation of the receiver along with calibration condi
tions temperature, pressure, orientation, etc. If calibration condi
tions do not correspond to measurement conditions appreciable errors can
occur, since the sensitivity of many receivers depends on temperature,
orientation, etc.
Errors attributed to the influence of ineasurement conditions, expressed in
terms of s(t) and sl(t), are often the most serious, since they include the
influence of diffraction and wave guide phenomena, reflections from the
enclosure of the measurement space, the fastening of the receiver, etc.
The correct choice of distance r between the sound source and the measure
ment point is very important for high measurement precision; it is desir
able that the measurements be done in the zone where the electric plane
characteristic of the source is formed; r> 2C2/a, where D is the size of
the radiator.
Acoustic measurement errors consist of the sum of systematic errors and
random errors of the measured value.
Systematic errors are usually known before measurements begin. They are
determined by standard procedures and are taken into account by incorporat
ing the appropriate correction factors in the measurement results. It is
desirable to use an acoustic reference to eliminate systematic errors [4].
To do this it is helpful to reduce the measurements to relative measure
ments, taken with one channel, when working out the measurement procedure.
Errors caused by random factors cannot be ascertained before measurements,
even in individual measurements. The' v can be taken into consideration
only by statistical processing of repeated measurements, and the larger
the series of ineasurements the more complete the information about random
errors.
Random error AR with a normal distribution and probability N(t) are
connected by the relation
!3 (I A!2 I < Ivo) ~ 24) (t),
t uS
where a0 is the mean square deviation; m(i) V2n f e 7 ~f=,
0
 function.
If 11 is measured n times and the result is averaged
1 :
R n ~ Rr,
r ,
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(2.6.3)
is a Laplace
(2.6.4)
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then
a (I e2 n < t LT" 2m (r).
~ n (2.6.5)
To calculate the random error by formulas (2.6.3) and (2.6.5) it is
necessary to know 60 the mean square deviation of the measured value.
The mean square deviation of random variable S, based on data of n samples,
is
S (R~  R)= � (2. 6. 6
n  1 )
If the samples are taken from a normal group, then nS2/6o has a X2distri
bution with k= n 1 degrees of freedom. Here Q0 is the desired mean
square deviation.
The probability of the X2distribution is calculated by the formula
B (X2>Yq)=~V (2.6.7)
where
M
k/21 X12
X e dx'
~
2k/2
J :2;
YZ .
r
/ cv
I' + ~_1 = (e 'dl,
~l/ J
0
Let us estimate 60 with the aid of the distribution probability
 ~ S~ n ) _ ~N (~g)� (2. 6. 8)
We use with probability ~(Xg), close to unity, the largest of the possible
values of 60 as the desired mean square deviation (top estimate):
Sxan . (2. 6. 9)
If there are k> 30 degrees of freedom formula (2.6.8) may be simplified!
B r Qo S tl2n 0,5 E ID(t) . (2 . 6.10)
tzktt/
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i   ON 0.
BY
3 JULT 1980 I . I . KLYUKI AND I . I . BOGOLEfiOV 2 OF 6
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Consequently we obtain with a given probability,' close to unity, the top
estimate of the mean square deviation :
S ~2n
(To 
r2n3t (2.6. 11)
_ Formulas (2.6.9) and (2.6.11) are the calculation relations for determining
G 0'
If the assumed probability is close to unity we obtain, by substitutiiig o'Q
into formulas (2.6.3) and (2.6.5), the following calculation relations:
e (IARI < di = Qi3~p (w) 9 3q, w cos aQ Q, (3. 2. 3)
0
where Qi ef an3 ai ef are the effective (mean square) values of force and
velocity; aQa is the angle of phase shift between active forces and vibra
tion velocity.
If the combined amplitudes of force and velocity are defined as Qi(w) and
qin(w), respectively, then the expression for the radiated power may be
written as
N r(0)) = 2 Re (Qn (~)4~*
(3.2.4)
where the asterisk denotes complex conjugation.
Formulas (3.2.3) and (3.2.4) are suitable for calculating the vibration
power radiated on discrete components of the spectrum of ship machinery.
However, vibrations of electrical and mechanical origin on frequencies
above 1,0003,000 Hz and hydrodynamic vibrations do not exhibit pronounced
discrete components and may be viewed as random stationary processes. In
this case the spectral density of the vibration power radiated on frequency
w is determined by the relative spectral density of active force and vibra
tion velocity:
Nn(co)=ReS(Q~, 9i)= .
T ' ~ . .
= Re lim lim i J Qi (t, w, Oc~)� qn (t, w, ew) dr. (3. 2.5)
eu~,ocz,~ OwT
0
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In practice force and velocity are usually measured in the frequency bands,
for instance, of 1/3 or 1/2octave, from which the frequ,ncy ch aracter...
istic of a process in the analyzed range is formed. Vibration p ower Nn(Aw),
i
radiated by a mechanism in frequency band Aw, is determined by the formula
ecu
0
[30=ef] N ' (Aw) = j y N' (w) do) = Q~',4' (AW) ' (Aw) � ReRQ Q(Ac)), (3. 2.6)
e~
2 .
where Re RQa(Aw) is the real part of the correlation coefficient between
force and velocity in frequency band Aw:
r,
~ Q~' (aw) � qn~ (Aw) dt
et I ~
RQ Q(Aq Qni t 1 ~
(A~) � qn, (Aw)
~
i
If the mechanism is i.n contact with bearing and nonbearing surfac es with
area SZaf, the radiated vibration power is
[as=af ] N(w) = Re J Pt (S, w) 9t .(S, (o)dS,
(3.2.7)
a ad)
where Pi(S, w) is the surface density of the force; qi(S, w) is t he vibra
tion velocity at point S on frequency w.
The values Ni(w) and Ni(Aw) will be positive if the energy flux goes from
the mechanism into bearing and nonbearing connections. On some frequencies
. vibration energy can enter a mechanism from nearby vibration sources.
There are also cases when vibration energy is radiated by a mechanism in
one zone of contact with shock absorbers and reenters the mechan ism
through other areas of contact. When the vibration energy flux goes into
the mechanism the values Ni(w) and Ni(Aw) are negative.
Thus, if the forces and vibration velocities are known in each area of con
 tact between a mechanism and bearing and nonbearing surfaces (or if the
surface distribution of the forces and velocities is known), and if the
phase and correlation relation between the forces and velocities are known,
it is then possible, on the basis of equations (3.2.3), (3.2.5), (3.2.6)
and (3.2.7), to determine the vibration power radiated by a mechanism. It
can also be found on the basis of information about mechanical impedances
or about the pliancies of bearing and nonbearing connections.
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In the case of unidirectional harmonic vibrations of a mechanism and
singlepoint contact with the shock absorberfoundation system, we obtain:
in terms of inechanical impedances
N~ (w) _[9t'cp ((,)12 ReZ'!i 14r ((0). (3. 2. 8)
i.e., the vibration power radiated by a mechanism is a function of the
square of the vibration velocity and the real part of the input mechanical
impedance of the shock absorberfoundation system;
in terms of inechanical pliancies
Nn (0)) _ [Q~3fp (w)12 ReM~i e~p (w), (3. 2.9)
i.e., the vibration power radiated by a mechanism may be expressed through
the square of the force and the real part of the input mechanical pliancy
of the shock absorberfoundation system.
Likewise, for the vibration power radiated by a mechanism in the frequency
band
N; (e(o)  [ 9; 3,~ (nw) 12 � ReZa'!la(~ (do)), (3. 2.10)
N"(Aw) [ Q~�'3~, (Aw) 12 � ReM j a~ (~1tu).
(3.2.11)
In the general case bearing and nonbearing connections of ship mechanisms
are multipoint vibrating systems, in which the vibration velocity of each
point is a function of all the forces exerted by the mechanism. In con
sideration of this dependence the expression for the radiated vibration
power may be written as follows:
in terms of inechanical impedances
m
1
N~ = 2~ I 9!` i~) I2�ReZtt a�, (W)
n=1
m m s . (3.2.12)
+ ~ E Y, RL, [Z~,ke(~ (w)�ql (w)�q~~ (~j~;'
n1 k=1 il
in terms of inechanical pliancies
m ~
N1 (w) = 2~ I Q! (w) 11 ReMit a~
n=l
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m ut 6 �
_ + . Re[Alii~. ~(G))�Q~'(w)�Qf~ (3.2.13)
n=1 k=i J=l
The second sums in the right hand sides of the equations are used for
evaluating the influence of the connectedness of vibrations at various
points and in different directions on the radiated vibration energy.
In the case when there is surface contact between a mechanism and bearing
and nonbearing connections the vibration energy radiated by the mechanism
can be determined only through mechanical pliancies:
s
A'' (W) J J m ii a(p (S, S'w) P! (S, w) P; (S', (o) dS'dS. (3. 2.14)
sa34, 0,14)
It follows from the above formulas that the vibration power radiated by a
mechanism may be determined experimentally in two different ways: directly,
as the scalar product, averaged in time, of force and t}ie corresponding
velocity, and indirectly on the basis of the vibration velocities of the
mechanism and mechanical impedances of bearing and nonbearing connections.
The direct method is used for measuring the vibration power radiated by
forces normal to bearing surfaces [6, 7].
The indirect method can be used, in principle, to determine all components
of vibration power.
Vibration power should be assumed to be the basic parameter for analyzing
the vibration activity of machinery. Vibration power contains information
that tells the vibration levels and forces that a mechanism exerts upon
bearing and nonbearing connections.
_ It is possible on the basis of radiated vibration power, first of all, to
compare different types of machinery and mechanisms, utilizing different
operating principles, and with different masses, sizes and installed in
different places as sources of vibrations. Such a comparison cannot be
 made on the basis of vibration levels. In fact, the forces generated in
_ different assemblies are spent to overcome mechanical impedances of
structures of a mechanism itself and of connected bearing and nonbearing
couplings. For large and heavy mechanisms and foundations even large
forces produce near the mechanism a vibration that does not exceed the
vibration of small mechanisms on pliant foundations [5].
Spectrograms of the vibration of an electric motor and ship reducer, and
also of a diesel engine and gas turbine, are presented in Figure 3.1. As
can be seen, the vibration of the reducer is about the same as the vibra
tion of the electric motor. At the same time it is obvious that the
reducer has substantially greater vibration activity. A comparison of the
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G,d6
900 �r 1
80 ?
60
G d610 /02 !O J~' ~q
'g0
60
>0
~
^
.
4
>0 ` 90' , P, Cq
Figure 3.1. Spectrograms of vibra
tion: 1 gas turbine; 2
 diesel engine; 3 electric motor;
4 MTRA [main turbine reducer
assembly]. [a6=0; F4=Hz]
N, Br
fp
19,11 ,
q0f
10 90Z f03 W
. P, r14
Figure 3.2. Spectrograms of radiated
vibration power: 1 electric motor;
2 MTRA. [Bt=W]
N; Br
f9a
!02
fo 
,
0,3
n .
A.t�
r ~
~ 2
~
>0 , 101 90' f04
r, r4
Figure 3.3.:Spectrogram8 of radiated
vibration power: 1 diesel; 2
gas turbine.
It is important, during the development of recommendations on the reduction
of vibration activity of inechanisms, to determine what component must be
reduced first. In connection with differences in the magnitude of linear
and rotary vibrations, and also in the impedances of bearing structures in
 relation to forces of different directions, this can be solved only after
the components of the vibration power radiated by a mechanism are separated.
Measurement results show that in application to ship machinery there is no
one kind of action on bearing surfaces that determines the radiation of
energy. The component vibrations have different significance in each
particular case, depending on the design features of the mechanism and on
the system of forces acting upon moving assemblies.
The frequency characteristics of the ratio of vibration power N3, radiated
by a vibration component normal to bearing surfaces, to the total radiated
113
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spectrograms of the radiated vibration
powers (Figure 3.2) confirms this: the
reducer radiates 23 times more vibra
tion power.
A comparison of the vibration powers
radiated by a diesel engine and gas
turbine (Figure 3.3) shows that the
gas turbine is preferable, in spite of
the fact that the diesel engine pro
duces lower levels of vibration.
It can be established, on the basis of
the spectrum of vibration power
radiated by a mechanism, what is the
main source a moving part or a
process.
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vibration power N,, are plotted in Figure 3.4 for several ship machines.
As can be seen, vibration power N3 has different values (in fractions of
the total radiated power). The vibration activity of a mechanism in rela
tion to bearing and nonbearing connections can be compared only in terms
of vibration power.
NjI Nij
4
BO
SO
3
.
V/
.
40
\
.
2
20
~
rV /V . /U4
s
Figure 3.4. Frequency characteristics of ratio N3/NE :
1 diesel generator; 2 generator; 3 steam tur
bine; 4 reducer. 
Information about radiated vibration powers can also be used for deter
mining mechanisms the basic sources of vibration activity of assembled
_ units. The influence of the causes of vibration not only on individual
frequencies, but also in the frequency band, is an important feature of
such diagnosis.
In some cases it is important to analyze the energy radiated by a shock
absorbing mechanism into foundation structures. Let us examine a harmonic
vibration process in the mechanismshock absorberfoundation system.
Each shock absorber can be represented as being a linear mechanical
 quadrupole with unidirectional vibrations. In the general case set n0 of 
shock absorbers, with all six components of vibration taken into account,
is a 24nopole. Vibration processes obviously should be described with
the aid of matrix coefficients.
The relationship between vibration velocities q in mechanismshock
absorber cross sections and qf in shock absorberfoundation cross sections,
and also between the forces of interaction of the shock absorber and
mechanism Q and foundation Qf, is given by the following matrix equations:
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Q = Zan:9 'I fi9 ~p;
Q4, _ za>sf~ q + Za(bq(b, (3. 2.15)
where quasidiagonal matrices Zam , Zam.f' Zam.f' Zaf describe the mechanical
impedance of the shock absorbers.
 By combining these equations with equation (3.1.11), which describes the
dynamic properties of a mechanism, and also considering that
za,a~,
(3.2.16)
where Zf is the matrix of inechanical impedances of the foundation, we
obtain the following matrix equation system that completely describes the
combined harmonic vibrations of the mechanism, shock absorber and founda
tion:
Qo = Zo9 + Q;
Q = 7am9 I laW4, 9b;
  ~ (3. 2.17)
Q~  Zach� 9i Za~ 9 cp;
Q~p =  Z~b9d~� ~
 The equations may also be written in dual form in terms of inechanical
pliancies:
_ q = moQo h1oQ I
q = r'1�IaNQ A1aa~Qcp;
q ,p lL
= ly�(b2 Ma4,Q(b; (3 . 2 . 18)
9 ,h =  M4,Q4,.
The equations of the propagation of vibrations through nonbearing connec
tions are of the analogous form if the impedances or pliancies of flexible
parts in nonbearing connections are defined, respectively, as impedances  Zam' Zamf' Zafm' Zaf or pliancies Mam' Mamf' Mafm' Maf'
From equations (3.2.17), after simple algebraic trar:sformations, we obtain
the equations of the balance of the vibration energy in the mechanism
shock absorberfoundation system [3]:
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RM = N~u~ 4" Natp;
N  ~ � ' ~ ' ' ,
 19] 9i Z12a~i4)'9(b14* ~
N~y= 2[Z~ 0. 60. 7) .
2
4
. i
I i
~ I
1
Figure 6.5. Sectional soundinsulating housing with
mufflers.
The reduction of noise due to the increase of axial clearances can be
evaluated through the formula
OLa = 2019 S/bao
(6.2.6)
where T a is the initial dimensionless axial clearance.
When T/ a= 2.0 it is possible to achieve a 46 dB noise reduction. The
axial clearance can reach the distance of two rotor blade chords.
It is recommended that the optimum ratio of rotor and stator blads numbtrs
be used for reducing noise. By optimizing this ratio it is possible to
reduce the noise level by 35 dB. By angling the blades at approximately
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10� compressor noise can be reduced by 34 dB. The blade pitch has
comparatively little effect on compressor noise; a change o� pitch by 5�
produces a 12 dB change of the acoustic level. Deviation of the mean
attack angle by 1� from optimum in the maximum thrust mode increases noise
 by 1 dB. Blade thickness has virtually no effect on compressor noise [13].
Gas turbine engine noise can be reduced by using a soundinsulating
 enclosure and the corresponding intake and exhaust mufflers.
Requirements on Gas Turbine Engine SoundInsulating Housing. The purpose
of a soundinsulating housing is to reduce audible noise emitted by the
gas turbine power plant housing. Onepiece enclosures may be used for
small GTP; sectional frame or frameless housings that provide access to the
GTE for service without dismantling the housing, are usually employed for
heavy marine gas turbine power plants.
It is extremely important to install intake and exhaust mufflers in the
immediate vicinity of the gas turbine engine, or even inside of the housing
itself. This precludes complicated and expensive measures for sound
' insulation of pipes.
A moduluar box type of housing, in which there are access doors for servic
ing the gas turbine power plant, is recommended for more ready access to
the engine. An example of this type of housing is illustrated in Figure
6.5. It consists of three removable sections. Exhaust muffler 5 and intake
muffler 2 are inserted in turbine section 4 and compressor section 1,
respectively. Power turbine section 4 and middle section 3 have access
doors. Profiled elastic gaskets, which should meet the requirements of
high oil, gasoline and heat resistance, should be used for sealing
hatches, covers and doors. This type of housing provides a noise reduction
on 500 Hz of 1525 dB. The noise reduction on low frequencies, for example
on the gas turbine engine rotation frequency (80150 Hz) is 58 dB.
6.3. Elimination of Gas Dynamic Free Vibrations in Exhaust Pipes
Gas Dynamic Free Vibrations as Physical Process. Under certain conditions
the gas flow in two heat exchangers, installed in the exhaust ducts of gas
turbine power plants and in the tail parts of steam boilers, become
unstable in relation to natural vibrations of gas exchange. This leads to
free vibrations, which are maintained by the kinetic energy of the gas flow.
This is often manifested externally as destructive vibration of the heat
exchanger and loud noise in the engine room and at the discharge of the
_ exhaust pi.pe (Figure 6.6).
Before the initial natural vibrations can occur zhere must be strong pres
sure pulsations in the flow around the exchanger bundles, and therefore the
�requency of flow separation processes should be close to the frequency of
fxcf: vibraticr.s in the flue. The frequency of free vibrations in the flue
is
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G,a6
,120
1
110
f00
90
80
ZO 30 100 100 300 1000 2000
Figure 6.6. Noise spectrum at smokestack discharge (r =
= 1 m,
~
O
�rl Q)
LIi
0
41
b
O
~t
Cd
Ri
~
ii
Cd
E
~O
U
U
H
~
II
~
N
iJ
~
0) 'C!
Q)
~
�
RS ~
w
U i4 ~q U
~ V.
+
~
~
o
o
o
~ ~ , .s4
cd
II
�
~
~ y
lH
r1
o~Na,
~
U +J v
o
Q) 4+ q \
Y'
U
~
H
.%4
~
O
N +J +J 0
c~d q cd M
+1 .a rt II
R' m
O
~
F+
7
� o a,~ L.
Q.~ Q) fA � ~
~ > H 4J LO 11
�H (D z o co
Lt.~ r1 {J 
IQ
~o
~
m
~
N
ti
~
~
~
2
~N,
~ ~j ^ 1~ ri N
N
VI
Cd
N' o � c~ U)
Z �
U
b4
fd �rl O
bA
ri
.a ~ C~
0
.54
d
~
O
bD I p I
N
;1
�rl ~
Cd
Li. cd tA ,.fl
II
r
u~ E
�a Cd
0 N (D
En
N N Cd
N fi
.
ii N
W r. 3a
�rI O E


t
~
,


\
~



~
\
~
\ .
~
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According to formula (6.6.1) the acoustic power N at subcri*_ical flow
rates 'Lncreases in proportion to the eightli power of the gas velocity at
the nozzle discharge. Valves have about the same ratio for subcritical
pressure drops pl and p2; in the case of supercritical drops this function
is proportional to vm  vm.
The calculated noise spectrum of a jet as a function of frequency ratio
f/f0 is sliown in Figure 6.17. Also given there is the typical noise
spectrum of a control valve, operating under a supercritical pressure drop:
pl = 21 kg/cm2 and p2 = 6.5 kg/cm2. The maximum frequency f0 of the spec
trum in terms of amplitude is given by the Strouhal number St
[3$=ef] st = io d3~ ,
(Jm
where def is the effective orifice; vm is the gas velocity in the critical
cross section.
For a jet discharged into open space with pressure ratio pl/p2 2 the
Strouhal number is St ^ 0.5. Strouhal number St decreases as the pressure
ratio pl/p2 increases according to the formula
[kp=cr] St 0,2 ( Pi  Pi
) .
P: PKp
Using ttie critical ratio pl/p2 = 1.89 and the critical flow coefficient Cf,
it is possible to find a more exact equation for determining the Strouhal
number for a valve [15]:
[ Kn=va ] StKn = 0,2 I PI/P:  ( 1  o147C~ )]112 (6 . 6 . 2 )
I.
Coefficient Cf represents the ratio of the valve transmission capacity Cv
under critical and under the examined conditions. By definition Cf =
AP cr~(p1 pV where Opcr = pl  p2 for a critical flow, pv is the
pressure in the critical cross section of the valve.
_ The coefficient Cf is connected to the different parameter of flow through
a valve by the expression [l]
G = 0,0207CDCI lf ep (pl P') P ,
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where G, in tons/hr, is the mass flow rate; p2, in kg/cm2, is the absolute
discharge pressure in critical flow; pl, in t/m3, is the density of gas
(steam) for working intake parameters; Op = pl  p2 is the pressure drop.
The pressure recovery coefficient Kp.r, numerically Kp r= Cf, is utilized
in Soviet practice. Coefficient Cf varies from design to design, depending
on the valve operating mode, within 0.21.0; in most cases Cf = 0.50.90.
The Strouhal number for valves according to calculated and experimental
data does not exceed 0.2; usually St = 0.100.18 [15].
The acoustic power Na of a valve is equal to the kinetic power Wk of the
jet, multiplied by the acoustic efficiency qva'
Na = WKYIKn (6. 6. 3)
where WK 8g pmt""+ds~' (6. 6.4)
pm is the density of the medium in the critical cross section. The value
vm in critical flows may be assumed equal to the speed of sound. The value
rlva depends on pressure drop pl/p2 and on the coefficient Cf.
The generation of acoustic energy in valves is a complicated physical pro
cess, particularly for critical and supercritical pressure drops. Since
the ratio pl/p2 in throttle valves usually is higher than critical it is
worthwhile to examine just that operating mode.
tiVhen a compressed medium is discharged with critical and supercritical
pressure ratio pI/p2 pulsating shock waves and flow turbulence, which cause
strong vibration of the discharge pipe, are the determining factors in the
generation of noise at the discharge from a valve. The change of rlva as a
function of pl/p2 and Cf is plotted in Figure 6.18. As Cf increases, i.e.,
as the pressure recovery coefficient after the valve decreases, noise
increases. This is physically explainable, since when Cf = 1 the kinetic
energy after the valve is not converted to potential energy, but is con
verted entirely to thermal arid acoustic energy, and when Cf 0 most of the
kinetic energy of the flow after the valve reverts to potential energy.
Valves that have small Cf and operate on a compressed medium make less
noise. Conversely, the flow of incompressible f.luids under the same condi
tions can cause cavitation phenomena, and valve noise increases.
166
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VKA
B
6
4
3
2
pJ
8
6
4
3
2
Oi
B
B
90
P10t
Figure 6.18. Function nva f(pl/p2) [15].
[Kn=va]
Acoustic efficiency nva at the critical pressure ratio pl/pz is not a
function of valve design; therefore all curves meet at the same point. At
approximately pl/p2 = 2.8 the curves bend, and after that acoustic power is
proportional to the flow velocity to approximately the sixth power.
The effective cross section def in a valve can be determined through the
coefficients Cv and C f :
d3,p = aVC�Cj , (6. 6. S)
where a is a constant.
The value Cv, like C.J. usually is indicated in the certification of a valve.
In Soviet practice the transmission capacity is sometimes denoted by the
symbol K. It represents the mass flow rate of a medium w3th a density of
1 t/m3 at a pressure drop of 1 kg/cm2 on a valve; numerically K= 0.86Cv.
The value Cv ran be calculated by empirical relations [1]: for pl/p2 > 2
the coefficient C is
v
for steam
C` _ 0,84Gk ~
n, (6.6.6)
167
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for gas
FOR OFFICIAL USE ONLY
G t
C�  0,0182 P11'Pi ,
(6.6. 7)
where G, in t/hr, is the mass flow rate; p, in t/m3, is the density of
steam (gas) for the working intake parameters; k is a correction factor for
heating of steam above the saturation point. For saturated steam k= 1,
and for superheated steam k 1.023 + 0.0007 At, where At, in �C, is the
temperature of the steam.
Bauman [15] proposed on the basis of equations (6.6.3) and (6.6.4) a
formula for calculating the acoustic level Lva, in dBA, in control valves
at a distance of 1 m for a blocked flow:
[Kn=va; Tp=p; R=d] L, 15
(where rl is blower efficiency),
reaches 80100 dB. The spectral
characteristics of the noise of some
blowers of the TsS series are listed
in Table 7.1.
The spectral makeup of the air ytoise
of centrifugal blowers usually
includes a continuous part and indi
vidual discrete components, which are
515 dB stronger than the continuous
spectrum. A typical air noise spec
trum of a centrifugal blower i.s shown
in Figure 7.2.
The sources of the continuous part of the air noise spectrum of a centri
fugal blower are eddy systems and individual eddies, formed in the ducts,
primarily in the flow around the blades of the squirrel cage. The initial
flow turbulence at the intake into the squirrel wheel exerts a substantial
influence on the intensity and size of eddies, and consequently on the
level and frequency of the noise. The source of the discrete components of
air noise in singlestage centrifugal blowers without guides and flow
straightening liardware is the interaction of the flow in tlie delivery cross
section of the squirrel cage, which has uneven velocity, with the tongue of
the discharge channel (snail). This noise component is called turbulent
flow noise.
L,db
80
70
60
506
Figure 7.2. Typical air noise spectrum of centrifugal
blower. [a6=dB; Fu,=Ela]
Axial fans are used considerably less frequently in ship ventilation and air
conditioning systems than centrifugal blowers. The spectral makeup of the
air noise of an axial fan, like that of a centrifugal blower, includes a
continuous part and individual discrete components (Figure 7.3).
180
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90
86
70
60
11, a6
3
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0 400 63
0 100 ;6
279
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7,7
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nn
arn
n
ou ,w zuv 373 Suu 800 Iz50 ZODO 3150 5000 rru
Figure 7.3. Typical air noise spectrum of axial fan at dif
ferent productivities: 1 Q=~ot; 2 Q>~ot; 3__
Q< Qpot'
The continuous part of the noise spectrum (eddy noise) is of the same
physical nature as of centrifugal blowers. In axial fans, which usually
have a different kind of flow in the profile grilles in comparison with
centrifugal blowers, and the flow past them is also virtually without
separation, the main sources of eddy noise are pressure pulsations (of
tltrust) on the blades, related to the sliding of eddies from the trailing
edge. The appearance of discrete components in the noise spectrum of axial
fans is attributed basically to the displacement of some volume of air by
the rotating blades (socalled rotation noise [1]) and to the interaction
of the air stream with the guide vanes of the stator.
Rotation noise usually occurs in axial fans wi'ch a few blades (z = 35)
[11]. The discrete noise components of fans with highdensity grilles,
b/t > 0.5, are on the same level as the continuous part of the spectrum.
The noise density and frequency depend on the interaction of the air flow
coming out of the working wheel of the rotor, with the guide vanes of the
stator, on the ratio of the number of working wheel blades and the blades
of the guide system, and also on the space between them.
'I'he acoustic characteristics of marine axial fans depend to a great extent
on their productivity: as productivity decreases (Q Qpot)
the noise on high frequencies (f > 1 kHz) increases by 57 dB and the dis
crete component due to flow inhomogeneity also increases,
182
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The main source of noise in an air conditioner is the builtin blower. Air
con ditioner blowers should be balanced very carefully in order to prevent
the appearance of high noise levels on low frequencies.
The refrigeration compressors of selfcontained room air conditioners are a
source of intense vibrations, and therefore considerable attention is
devoted to their vibration insulation.
'I'he humidifiers of air conditioners with steam humidifyinc, systems are a
strong source of air noise. For instance, the UVP72 humidifier, with a
ste am flow rate of 0.2 kg/s and pressure.of 4�105 Pa, has a total noise
level of 98 dB. htufflers greatly reduce the noise levels of a humidifier
in intermediate and highfrequency ranges (Figure 7.4). The basic charac
te ristics of a number of ship air conditioners are given in Table 7.2.
Marine ventilation and air conditioning systems are usually packed with
plumbing fittings and various heating equipment (throttle dampers, slide
gates and heaters). All these elements offer a certain amount of hydraulic
res istance to the flow, passage through which is accompanied by strong
turbulence, and consequently the emission of noise.
The acoustic power level of the noise generated by the passage of air
through the hardware depends primarily on the flow velocity in front of it.
Spe ctrograms of the discharge noise of the OVPCh brand refrigerator at
different air flow velocities are given in Figure 7.5. As can be seen in
the figure, an increase of velocity is accompanied by an increase of the
noi se level, particularly in the intermediate and high audio frequency
ran ges. The total noise level LN of the equipment is proportional to the
six th power of the air flow velocity [11]. Audio frequency components of
noi se, the acoustic power of which is proportional to the flow velocity to
the fourth power, occur in systems with short ducts. These audio frequency
components fall in the infrasound frequency range in long systems (air
duc ts longer than 6 m), and their fundamental frequency is inaudible.
'I'he tieat exchangers of ventilation systems represent a collection of
cylindrical rods, the flow separatiun around which produces noise. However,
bec ause the heat exchangers have many tubes the Strouhal frequency and its
}iarmonics in the noise spectrum of thtse machines are not manifested
exp licitly, particularly when the noise is analyzed in the 1, 1/2 and 1/3
octave bands.
"!^ise sources in ventilation and air conditioning systems, in addition to
hardware and lieating equipment, are shaped parts (corners, Tbranches, etc.)
of the air ducts, The data in Figure 7.6 show that the noise in the low
freq uency range, occurring in an intake Tbranch at a flow velocity of
30 m/s, is somewli3t louder than the noise of a blower with small parameters
Q and H.
183
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G, d6
70
1
90
70
2
60
50
G
nJ 100 160 250 400 610 1000 1600 250 40D0 6J00
au nJ .U47 315 S00 800 7750 1000 dtSO 5000 f / 14
 Figure 7.4. Spectrogram of airborne noise of UVP72
humidifier: 1 without muffler; 2 with muffler.
[aG=dB; Fu=Hz]
The noise generation process in air distribution equipment is of the same
character as in the duct work, except that the duct work noise occurs in
the main air channel, experiences reflection at the discharge from the
system and can be deadened with a muffler. The noise of air distribution
_ equipment is generated directly in the ventilated room as a result of the
flow of air around corners, through grates, grilles and other elements,
_ located in the plane of the discharge hole.
Various distribution and control systems are used extensively, in addition
to distribution hardware, in air conditioning systems. The noisiest of
them are decorative ejection air distributors [2], the high noise level of
which is attributed primarily to high velocities (up to 25 m/s) at the dis
charge from the air distribution apparatuses, necessary for forcing air
through the lieat exclianger. Even when soundabsorbing materials are used
~ in decorative ai r distributors their noi.se is comparatively high; the total
noise level is 65 dB,
The noise of air distribution equipment is the result of turbulence and
depends primarily on the air flow velocity. As in the case of duct work,
an increase of the flow velocity leads to a rapid increase of the noise
level in the medium and high audio frequency ranges. The design features
of air distributors also have an influence on the level of the noise which
tliey produce. For instance, the noise of a swivel air distributor with the
control in the "under plate" working position is 35 dB higher on inter
mediate and high frequencies than in the "onto plate" working position at
the same flow velocity.
184
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APPROVED FOR RELEASE: 2007/02/08: CIARDP8200850R0002001000099
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Air distribution systems of the "perforated deckhead" type are finding
increasing application in ship air conditioning systems. They sub
stantially reduce the amount of air being supplied to a conditioned room,
and consequently they reduce the flow velocity. This is why air distri
butors of this type produce relatively little noise [9]. Systems in which
air is supplied through a perforated grille at an intake air flow velocity
of about 10 m/s, and 3 m/s in the holes, are the quietest of all existing
systems.
G, d6
nn _
Figure 7.5. Spectrograms of air noise of OVPCh
air refrigerator at different intake flow velo
cities: 1 v= 20 m/s; 2 v= i5 m/s; 3
v = 10 in/s. [86=dB; fu=Hz]
The noise level in panel air distributcrs, in which air is supplied through
perforated side plates, is vastly hidher than in ordinary perforated panel
air distributors, particularly on i;ttermediate audio frequencies. Slotted
air distributors in the form of ar. ordinary slot, without windows and guide
nozzles, are comparatively quiet at air flow velocities of 10 m/s in the
channel in front of the slot, and not higher than 5 m/s in the slot. The
noise of all air distributors increases rapidly as the air velocity in the
intake channel increases.
7.2. Reduction of Turbulence Noise and Nonuniform Flow Noise of Blowers
Reduction of Turbulence Noise. The acoustic power of the air noise emitted
by blower vanes due to pressure pulsations on the vanes, is determined by
the expression [10]
N= 2Pzn~ f r! x 1,5 r r 2
 l~ ~ X
3hc3 1' ~ i s) rlr.
x
X ui (x) 12 (x) F'(x) r sin=~~ (x) 1dx (7. 2. 1)
~ Slfl`Prn ~
186
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7L'
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L,e6
5063
FOR OFFICIAL USE ONLY
2
1
ICJ WV JUU /UUU LUUU qUUU f,
Figure 7.6. Spectrogram of air noise of supply
Tbranch 1(branch angle 45�, branch channels
have equal areas) and of centrifugal blower 2
(Q = 0.222 m3/s, H= 2550 Pa).
where p is the density of air, in kg/m3; b, z are chordlength, in m, and
the number of fan blades; 1 1 is the size of an eddy along the blade (on
the x axis), in m; rl, rZ are the correlation radii of an eddy perpendic
ular to the blade surface and along the chord, in m; e, f are the power and
spectrum of boundary layer turbulence on the blade surface at the point of
flow separation; vl is the velocity of the air flowing onto a blade, in
m/s; S1' Sm are the entrance angle and mean geometric angle of flow in the
grille,
It follows from expression (7.2.1) that in order to reduce the acoustic
power it is necessary to reduce the flow velocity and the turbulence
characteristic and to optimize the blade array parameters for a given head
and capacity. The optimum blade array parameters, according to the
literature [10], have the following values:
Geometric entrance angle
Profile curvature
Array density
Angle of attack
Sbl = (2732)�
f= f/b = 0.070.08 for b/t < 1.0
f= 0.10.11 for b/t > 1.0
b/t = 1.5 for f> 0.10
b/t = 1.0 for f< 0.10
i= 3 5� for f< 0.07
i=03�forf>0.07
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BT
3 JUL Y 1980 1.1. KLYUKI N AND I . I . BOGOLEPO;r' 3 OF 6
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b
95
_
108 6 4 2 0 2 4.,6 10 a o 4 z u z 4 6 L
L �
41
X

70 G b 4 Z 0 Z 4 6 L
Figure
7.7. Aeroacoustic
characteristics of plane blade
arrays
(vl = 6
0 m/s). Arrays:
a TsZ8 T= 0.51, Xx T=
1.0);
b Ts
Z8 T= 1.5,
x_x T= 1.0); c Ts58 T
= 0.5,
x_x T=
1.0); d Ts58
T= 1.5, xx T= 2.0); e
 Ts78
T
= 0. 5, xx T= 1.0) ; f Ts78 T= 1.5, Xx
T=
= 2. 0);
g Tsll8 (xx
T= 1. 0) ; h Tsll8
5,
x_x 'r =
2. 0). [a6=dB]
[Figure
7.7 continued on
next page]
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c)
4ao
d.)
,ry
a6
1D B 6 4 p 0 ? 4 6 'a� 10 v o u 2 u z 4 a
g)  h) ,
L, 6
90
'
88
86
Cy
1,
1
0
.
Ar
,
Q8
~
Q6
4
4
4
O
4
~
'
y
a
)
S
�
'
C
Cx
108 S 2 0 2 4 6 i� :108 6 4 2 0 2 4 6 i.`
Figure 7.7 (continued).
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The relative profile thickness witiiin its range of change (0.060.12) has
practically no influence on the noise level of a blade array.
The following requirements must be met during the design of quiet fans:
1) use a higller than usual fan pressure coefficient; for instance, if the
recommended pressure coefficient of a stage for calculatinz the energy
characteristics of fans is H= 0.150.20 for c< 0.3 and H= 0.250.3 for
a 
ca < 0.5, then this coefficient for an axial fan with low noise levels
should be H > 0.30.35;
optimize the conditions of flow around the peripheral cross sections of
the fan blades, which are the strongest noise sources;
~
3) eliminate strong components from the fan noise spectru;i, produced by
flow inhomogeneity on the frequency f= knz/60 (k = 1, 2, 3, n is the
rotation frequency of the fan, rpm);
4) select blade array pa.rameters in accordance with the reccmmendations
presented above and in consideration of the aerodynamic and acoustic
characteristics of plane profile arrays, presented in Figi:re 7.7. Also
given in the figure are the characteristics of arrays wh:,se profile curva
ture varies from 3 to llo, the array density from T=0.5 to T= 2,0, and
the angle of attack from 9 to +5�.
The intensity of the air noise emitted by a fan or by a guide device can be
evaluated by the formula (for a point 1 m from the intake at an angle of
45� to its axis at vl = 60 m/s) :
lP. K=f; n=p; K=r] IP. K = 'K),
(7. 2. 2)
where rp, rr are the peripheral and root radii of the fan wheel; Z is blade
height; I p.c is the intensity of the noise emitted by a profile array,
equivalent to the periphe'ral cross section at vl = 60 m/s (determined by
Figure 7.7).
The air noise level of a fan or the guide apparatus of a fan at a relative 
air flow velocity onto the peripheral cross section of the blade of v1 =
var is determined by the expression
Lr~ = 10Ig 10=1 + 601g v1 sOvar . (7. 2. 3)
By calculating several versions of the blading of axial fans and by deter
mining through formula (7.2.3) their noise level it is possible to select
the quietest version of blade array. 
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An effective means of reducing turbulence noise of squirrel cage blowers,
in which flow separation usually occurs around the intake surfaces of the
blade, is to use turbulizer grilles, installed on the intake and discharge
edges of the working wheel and turning along with it. tiYhen turbulizers
are ins�alled on the intake edges of the vanes turbulization occurs in the
air flow next to the vane surface. lVhen this happens some of the energy
of the main flow of the betweenvane channel is transferred to the region
of the turbulent boundary layer. Consequently the boundary layer profile
changes and its pattern is filled out more completely, which leads to an
increase of the stability of the boundary layer and shift of the point of
separation doi,mstream. As a result of this fan turbulence noise is reduced.
Another positive effect of a turbulizer on the aeroacoustic characteristics
of blade arrays is the fact that the eddies formed during flow through the
turbulizer break large separation eddies down into smaller ones, the scale
of which is determined by the parameters of the turbulizer. Here the total
acoustic power, radiated by a blade array, is decreased.
The aeroacoustic efficiency of turbulizer grilles is determined by the
following factors:
conditions of flow around blade arrays, which depend, in turn, on the type
of profile, its curvature, angle of attack, density of the array, etc.;
parameters and dimensions of the turbulizers: wire diameter ps, effective
cross section coefficient S, and screen size of a turbulizer screen; 
_ location of turbulizer in relation to the blade chord and screen size (if a
. discontinuous turbulizer is used).
The conditions of flow around a profile in an array can be characterized by
one parameter the pattern of the distribution of velocity (pressure) on
a profile (maximum dimensionless profile velocity vm d= vm/vl and velocity
_ gradient near peak velocity). The acoustic efficiency of turbulizers in
various frequency ranges is plotted in Figure 7.8 as a function of vm.d'
Turbulizers produce the greatest acoustic effect when their optimum
parameters are determined by the expression
[c=s; m. 6=m. d] `o  tc 1 opt = 0,5  0,022 (ub. 6;"., (7. 2. 4)
i
where ts is the space separating the wires in the screen; S is the effective
cross section coefficient of the screen.
The optimum value of S for virtually all array parameters (vm.d = 411), for
arrays used in modern ship ventilators, falls within a rather narrow range
(Sopt = 0.50.6).
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When selecting turbulizer parameters it is important to consider tlie
nature of the noise spectrum of the fari. If the greatest reduction of
noise must be achieved in the low and intermediate frequency ranges (up to
23 kHz) it is advisable to use a screen turbulizer witll a relatively
large wire diameter (Ss = 11.5 mm). The noise levels in this case may
increase somewhat in the highfreauency range. To reduce the noise level
in the entire audio frequency band it is desirable to use a turbulizer
with a smaller wire diameter (ds = 0.40.8 mm), but the turbulizer will
have somewhat less acoustic efficiency. In consideration of the conditions
of air flow in centrifugal blowers it is desirable to install turbulizers
at points of the length of the vanes next to the front disk. Blower pres
sure and efficiency losses in this case are negligible (Figure 7.9). The
optimum screen turbulizer length for blowers of the TsS series is
(0.40.5)Z, where Z is the lengtll of the intake edge of the vane.
Turbulizers should also be installed on the delivery edges of the blades.
Although the acoustic efficiency of these turbulizers is low, they still
equalize the velocity field in betweenvane channels, thereby promoting a
certain amount of compensation of pressure and efficiency losses, wliich
occur when turbulizers are used.
9L,~7o
,i 4 5 6 7 6
vn.3
Semipermeable (perforated) blades
(Figure 7.10) may also be used, in addi
tion to turbulizers, for reducing
turbulence noise, generated by a fan
with separated flow past the blades. A
perforation forces air to flaw from the
delivery surface of a blade to the suc
tion surface. The point of flow separa
tion is shifted downstream, and con
seQuently the acoustic characteris:.cs
of the fan are improved.
Figure 7.8. Reduction of noise
To prevent a reduction of the noise
level from detracting from the aero
level by screen turbulizers as
dynamic characteristics of blade arrays
, function of maximum velocity on
the perforation parameters must be
profile: 1 Af = 201,350 Hz;
selected in accordance with the graphs
2 Af = 205,600 Hz; 3 Af =

presented in Figure 7.11. If the maxi
= 2015,000 Hz. [a6=dB; M.6=
=m.d]
mum noise reduction is required and some
sacrifice of the aerodynamic character
istics of the blowers can be tolerated,
the perforation coefficient k=
0.150.20 should be used. In cases when
the requirements on the energy c
haracteristics are rigid the perforation
coefficient must not exceed k=
0.030.12. Figure 7.12 shows, for
instance, that when k= 0.18 and
dhol 2'4 mm the noise reduction is 512
dB through the spectrum and the
loss of bluwer efficiency is negligible
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' (lo). When k= 0.1 and dhol  1.8 mm the noise reduction for f< 4,000 Hz
is 46 dB, but blower efficiency is 10% higher.
Figure 7.10. Diagram of perforated blade.
QS
0,4
Q~
0,~
0
dLx dG,dtp
dCy 6
5
dL
.
x
d
~ x
2
1
dCr .
O x
s
a
J�
.
0,2 '
J o
0,1
JOS
k
0,1 110,15
Figure 7.11. Effect of pacoustic characteristics
Lori.a Lperf.a' ACy
Lperf.a' A~  A~ perf.a
a
,rforation coefficient k on aero
of plane blade arrays. AL =
Lperf.a Lori.a' Acx Lori.a
 Uori,a'
For the air velocities that are customary in ship ventilators the hole
diameter in the blades should be in the range dhol = 12 mm, where the
smaller hole diameters correspond to arrays with better conditions of flow
past the blades.
The optimum angle of inclination of the holes lies in the range a= 4050�.
When the angles of inclination of the holes are larger (a > 60�) the aero 
_ dynamic parameters of blade 4rrays (Cy, Cx) deteriorate (Figure 7.13);
there is also a simultaneous reduction of acoustic efficiency. When
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a< 40 � the values AL, AC Y , ACX are virtually unchanged and perforated
blades with small a are much harder to manufacture.
The re lative area of the perforated surface of a blade (in relation to the
total blade area) S= Sperf/bZ, exerts a considerable effect on the aero
dynami c characteristics of perforated blades. Here b and Z are c}lord
length and blade length; SPerf (Xhol Xlhol Xsep)Zperf' where Zperf is
the len gth of a perforated blade; x has the corresponding i.ndex of the
chord c oordinate: xSep of the point of flow separation, Xlhol of the
first row of holes, Xhol of the last row of holes.
If a b 1 ade is perforated through the entire length the equality
S _ Xhol Xlhol Xsep
b
is valid.
G, 4
63 125 250 500 1C~9 206'0 4000 8000F/74
Figure 7.12. Spectrum of air noise of 40TsS17 fan:
original fan with unperforated blades (Q = 1.11 m3/s;
H= 1545 Pa; r1 = 0.688); A0 perforated blades (k = 0.1;
dhol 1'8 mm; H= 1505 Pa; 'n = 0.782); xx perforated
blades (k = 0.18; dhol  2.4 mm; H= 1500 Pa; n = 0.678).
[a6=dB; Fu=Hz]
The acoustic efficiency of perforated blades AL, plotted as a function of S
(Figure 7.14), shows that when S> 0.15 the value of AL remains virtua.lly
const an t at 56 dB of the total level. To take most complete advantage of
the effect of perforation of blades the first 23 rows of holes should be
in fron t of the points of flow separation, and the rest behind them.
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dG,d~Luc~ p nePaP
F
v
5 
4
3
~ dL
'  .
 dC _
C;,S
0,5
0,3 04
0,2 LI1 
0,1 ~ 
0
o d Cz
(?1
30 40 .S17 Fn 717 an on
Figure 7.13. Influence of inclination angle of hole in
blade on aeroacoustic characteristics of plane blade
arrays: [a6=dB; ucx=ori; p=a; nepo=perf]
dL,d6 
7
5
5
4
3
. 2
1
0

 r�'
�
o
Y
~
0,15 0,20 015 0,10 0,05  0 q05 Q10 0,15 020 Q25 0,3G s 
Figure 7.14. Effect of perforation area on acoustic
efficiency.
The parameters of the blade arrays themselves also influence acoustic
efficiency, in addition to perforation parai.ieters. As is shown in the
literature [10], perforated blades should be used when vm d> 2.5, i.e.,
when the flow past the blades is unsatisfactory. As profile thickness
increases the acoustic ef�iciency decreases. For instance, when the rela
tive profile thickness increases from 3 to 12% the reduction of noise AL
decreases from 6 to 3 dB at constant AC and AC .
y x
The acoustic efficiency of perforated blades also depends on the relative
blade extension T = Z/b. When Z< 1 and secondary eddies appear the
196
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coefficients Cy and CX decrease and AL decreases simultaneously. Per
forated blades should be used only when Z>:..25.
Reduction of Noise from Irregular Flow. Acoustic pressure p on frequencies
f= knz/60 (k = 1, 2, 3, the source of which is flow irregularity, is
determined by the expression
P=410'e rkT sin k 21 sin k 2= , (7.2.5)
i
where r is the distance to the point of ineasurement, in m;
r+er,  rtg e d,cose
Ti  2aR ;
IteB~l!/COS0 0.1, (7.2.6)
Tl  tz = aP ~
Ax3 is a parameter of the discharge flow velocity pattern, in m; t is the
~ space between the blades; Z is the height of the worlcing blades in the
delivery cross section; a is angular velocity; R is fan radius; 6 is the
ang?e of inclination of the body in the flow (the vanes of the flow
straightening device, snail tongues) relative to the delivery edges of the
blades; d is the thickness of the intake edge of the body in the flow.
The value of F, H is determined by the formula
 [Ct=St] F = 2~ (APcT.} P4JZ
(7.2.7)
where Op = p'  pI'is the maximum deficiency of static pressure in the
st st st
 aerodynamic medium;
Ov2 (c'i~)':
pst' v2 are the static pressure and flow velocity at the center of the
betweenblade channel; pst, v2 are the static pressure and flow velocity in
the center of the aerodynamic medium.
It follows from expressions (7.2.5)(7.2.7) that the following measures
should be taken to reduce irregular flow noise.
1. Reduce the dimensions Z, d of the intake parts of the flow str.aighten
ing devices. This technique, however, cannot always be used, since the
dimensions are usually determined by other considerations.
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2. Reduce Apst, Av, in particular, by increasing the space between the fan
and the discharge edges of the flow straightener the snail tongues. The
irregular flow noise level of axial fans can be reduced by AL = 1415 dB by
changing the relative axial space within the range ASb = 0,11.0.
The dependence between OSb and AL [10] is
AL
AS6 _ 10 20
(7.2.8)
V. I. Zinchenko's formula for centrifugal compressors can be used for
centrifugal blowers:
es
As6  ~z  o.� 0,22. (7. 2. s)
where D2 is the diameter of the working wheel.
However, an increase of the axial and radial spaces detracts somewhat Mom
the energy and weight and size parameters of blowers. For example, an
increase of the space from AS = 0.03D2 to AS = 0.12D2 leads to a decrease
of blower efficiency by 68o in addition to a decrease of the spectral
component of irregular flow noise.
3. Change parameters T1 and T2 so that the trigonometric functions entering
in formula (7,2.5) will be equal to zero. Since the parameters T1 and T2 ,
are determined by angle of inclination 6 of the body in the flow ( ' 7.2.6), ~
the trigonometric functions will be equal to zero for some value of 6 under 
otherwise identical conditions.
d.z dx
J "J
1~ ~
u
a ~
~5 x
~ x x. x
5 ~
'0005 QO7 Q015 0,02 a01,i 0103 Q035 404
dsa D
Figure 7.15. Parameter Ox3 as function of radial space.
By solving the equations kwTl/2 =7m; kwT2/2 =ffm, where m= 0, 1, 2,
for centrifugal blowers, we determine A corresponding to p= 0:
A arccos (t + Ax3) d !l/'(t Ax;,)a la  d' . ( 7. 2 .10 )
(t Ax3)a I 12 '
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It follows from formula (7.2.10) that the blade space, the length of the
body in the flow and the parameter Ax3 have the greatest effect on 8. The
thickness of the leading edge has little effect on the angle of inclina
tion, since d� Z in blowers.
To determine the parameter Ax3 we may use data obtained for ship centri
fugal blowers of the TsS series (Figure 7.15).
For axial blowers the angle of inclination of the blade of the flow
straightening device can be determined by the formula
[p.k=f; nep=per] A 180 Larctg '~Rep+ �x., udlCOSCA02.+.�1 ('7,2,11)
Zp. K I SIII C'~Op p where zf is the number of fan blades; tpeT is the blade perforation space; r
Ox2 =(0.10.2)tmid (tmid is the blade space in the middle cross section); 
_ a, al are values determined by the drawing of the fan blades and flow
guide (Figure 7.16); al has a plus sign when the vane of the flow guide is
inclined toward the concave surface, and a minus sign when it is inclined
toward the convex surface; 
< cAo, = so _ i so� ( 7. 2.12 )
Yp. K
The value of d in formulas (7.2.11) and (7.2.12) should be taken as the
maximum blade thickness.
The use of a flow guide and snail with incli.ned blades has practically no
effect on the energy characteristics of the machine, but it completely
eliminates from the air noise spectrum of axial (Figure 7.7) and
centrifugal (Figure 7.18) fans the component of noise due to flow
irregularity.
One way to reduce irregular flow noise of axial fans is to optimize the
 ratio of the number of fan blades zf to the flow guide device zf.
g.

According to data in the literature [3], the following ratios of 2f.g and
zf, depending on modulation index wTm, are the best:
(OT,n
I 2
I 3
I 4
I 5
I 67 I
R I
9 I
10
[c. a=f. g;
p.k=f]
(
zC.e  zp.x I
5 I
s I
7 I
s* I
io I
12 I
ts I
15
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The modulation index is determined by the expression
2:tnzp. K. r sin ~
~t,n 60 � C ~ ( 7 . 2 .13) 
where S is the angle between the radius vector, drawn from the center PK
to the measurement point, and the PK axis; r is the outside radius of the 
fan blades.
 � z
) 1 �
a
Figure 7.16. View of fan blades (a) and of guide vane
(b) of axial blower stage.
L.db d0
75
60
SD
5
Figure 7.17. Air noise spectrum of axial blower. Guide
vanes: serial; inclined. [86=0; Fu=Hz]
In blowe rs with rather large wTm it is difficult to satisfy the equality
zf.g  zf = 1015, since this involves a sacrifice of energy parameters.
For sing lemode blowers the number of fan blades and the number of guide
vanes may be determined by the expressions
JP' (wtm) _ 0;~
(7. 2.14)
!p;
where pi = zf g' zf' pl = zf� When condition (7.2.14) is satisfied the
value of zf.g  zf can be minimum.
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L,a6
80
70
60
/
SO
6J 100 150 150 400 SJD I"OO 16LI0 2500 4900 60 1000
80 125 100 J15 3D0 CCO 1150 1090 J150 5000 bCDD /q
Figure 7.18. Air noise spectrum of centrifugal blower.
straight tongue; oblique tongue.
7
(t3,MM
Figure 7.19. Graphs for determining reduction of noise
due to reflection from open end of air duct.
The amplitude of acoustic pressure on frequency f= knz/60 can also be
reduced by dephasing the pulses by using an uneven fan blade pitch [10].
For the first harmonic it is necessary that 1 0 (27r(x m/a0 ) = 0, where a0 _
= 27r/zp is the pitch for unevenly spaced blades; am is the amplitude of the
pitch nonuniformity. The first zero of the zeroorder Bessel functions
corresponds to an argument equal to 2.40; the second one to 5.52; the
third to 8.65, etc. [4]. The corresponding maximum values of the relative
amplitude of pitch nonuniformity are (am/a0)max  0.38, 0.88, 1.48.
The amount of reduction of noise ALn'n for a nonuniform pitch is plotted
in Figure 7.19 as a function of d
e .
In spite of the fact that the number of waves of pitch nonuniformity has
no effect on the amount of reduction of the noise level due to uneven flow,
much attention must be devoted to the choice of 1, since a determines the
balance of the fan. When a= 1 the blades are asymmetric, and the fan is
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unbalanced. When a> 1 the fans are balanced, which is preferable from
the standpoint of vibration and strength characteristics.
7.3. Acoustic Calculation of Ventilation and Air Conditioning Systems
Reduction of the aerodynamic noise of blowers at the source (see �7.2) in
most cases is not enough to assure acceptable noise conditions in rooms
serviced by ventilation and air conditioning systems (US and AC). Further
more, at the comparatively high wind velocities that occur in these systems,
parts of the duct work (branches and *_he elbows), hardware and air distri
bution sys tems are also sources of loud noise. In this connection the
designers o f VS and AC must solve the difficult problem of silencing these
systems b as i cally by selecting the best ventilation parameters, flow velo
cities, layouts and mufflers.
The acoustic calculation of VS and AC begins with the determination of the
sources of noise and quantitative analysis of their intensity. Then the
paths through which the noise propagates into rooms are found and acoustic
energy loss e s that can occur are determined. The influence of the acoustic
characteristics of these rooms on noise levels at given points is evaluated.
Then the ant icipated noise levels for a given room can be calculated and
compared w ith standards and the required amounts of noise reduction can be
determined.
As n4ise s ources US and AC systems are characterized by octave acoustic
power leve ls, which must be known in order to calculate the noise levels
that penet rate into a room through a branched ventilation system. The
acoustic power levels are determined on the basis of the acoustic pressure
levels, me as ured in accordance with the required conditions [7]. For rooms
in which the re are noise sources (ventilation units and air distributors),
calculation of the anticipated noise levels can be limited to the octave
acoustic p re ssure levels, measured on an acoustic test stand at a certain
di.stance (usually 0.5 or 1 m) from these sources.
The acoustic calculation of VS and AC includes determination of the anti 
cipatad nois a levels: in ventilator rooms, where ventilators and air con
ditioiiers are installed; in ventilated or air conditioned rooms; in rooms
that contain transit ducts.
Calculation of Noise Levels in Ventilator Rooms. Ventilators and central
air condit i oners should be installed in ships in groups in special rooms
ventilator rooms. This localizes the noise of the units and facilitates
attainment of acceptable noise levels in adj acent state rooms and service
rooms of the ship.
On the basis of design features that determine the propagation paths of
noise produced by ventilators and air conditioners it is possible to view
each of them as a set of three separate noise sources, which may be
assumed to b e independent, and to characterize them by their own noise
levels or acoustic power. Lu and LNu are, respectively, the acoustic
pressure level and acoustic power level, produced by a ventilator with 
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suction and delivery ducts leading into different rooms; LNsuc and LNV
are the acoustic power levels of aerodynamic noise, produced by a venti
lator or air conditioner in ducts from the directio.r. of the suction and
delivery ducts. Lsuc and Ldel charact2rize the noise levels near open
holes of the described ducts. Levels Lu and Lsuc are known from the
results of acoustic tests of ship ventilators, and L., Ldel and Lsuc are
~
 known from the results of acoustic tests of air conditioners. The rela
 tions that connect the acoustic power and noise levels of ventilation
 system (VS) and air conditioning (AC) equipment are presented below.
The values of Lu are the initial data for calculating the noise levels in
 ventilators, and LNsuc and LNdel are most often used for determining the
anticipated noise levels in rooms equipped with VS and AC, and on open
 decks of a ship where there are holes, through which air is drawn in and
discharged.
If the suction and delivery ducts of a ventilation unit are located outside
of the ventilation room noise level L. at a given point, in dB, the posi
tion of which is usually near the bulkhead that separates the ventilator
room from the adjacent room, can be determined by the relation
[f10M=Y'; ar=u] Lno� = Lar + ALno.%t� (7. 3. 1)
The correction factor ALr, in dB, characterizing the influence of a room on
the propagation of sound in it, is calculated by the formula
OL�O" 10 Ig 1 a 4 (1  a) ] const, ( 7 . 3 . 2 )
C 4Zr aS
where r is the distance between the unit and a given point in the venti
lator room, in m; S is the total area of the surfaces that enclose the
ventilator room, in m 2 ; a is the average sound absorption coefficient of
the surfaces.
_ Information about coefficients a for the interior surfaces of ship rooms
without special soundabsorbing wall coverings is presented in Chapter 11.
The value of const depends on the distance from the unit at which levels
Lu were determined under conditions of the shipbuilder's acoustic test 
stand. If this distance is 0.5 m, then const = 5 dB; for 1 m const = 11 dB.
If there are several units in the ventilator room it is necessary to
calculate Lrl, Lr2, Lr, which express the octave levels of the noise
generated at a given point by each of the individual units. The total
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octave levels Lr, in dB, of noise that occurs in the ventilator room
during the operation of all the units are determined by logarithmic addi
tion of the levels:
Lnom = 10 lg (100,1Lnoni 1+, IO0,iLnou 2+ 100,1LnoM n)(7. 3. 3)
In cases when a ventilation unit takes in air directly from the ventilator
room, the unit should be viewed as consisting of two independent noise
sources, characterized by levels Lu and Lsuc' These levels are usually
presented in the technical documentation on ventilators and air condi
tioners. i9hen there is one unit operating in a room with an open air
intake duct Lr, in dB, is determined by the function
[ec=suc] Lnoy = 101g 1100'1 (tOC;�tno+, i) + 100.1 (Lar f ALnob 1
where ALrl and ALr2 are determined by formula (7.3.2), respectively, for
the distances between a given point in a room and the plane of the air
intake opening of the suction duct of the unit, and the unit itself.
tiVhen there are several units in a ventilator room the total noise level is
given by formula (7.3.3).
The octave noise levels L at a point 1 m from the plane of zn open air
intake or delivery duct of the ventilator unit are connected to the acous
tic power levels LN, in dB, by the function [10]
LN L} A} 1 I, ( 7. 3. 4)
where A characterizes the attenuation of the acoustic power due to reflec
tion from the plane of the delivery opening when it propagates from the
duct into open space. It is determined by the graphs in Figure 7.19 on the
basis of given freQuency and equivalent diameter de = 1.12 d, where Sd is
the cross section area of the duct. For cylindrical ducts de is the same
as the diameter of the cross section.
The octave acoustic power levels L Nsuc, in dB, radiated by a centrifugal
blower into a duct from the suction side, can be calculated by the
empirical formula
[ B=b ] LIv ac = 10 Ig Qe f no 19 HB  40 Ig /I 148, ( 7. 3. 5)
where Qb is blower capacity, m3/s; Hb is total pressure, Pa; f is the
mean geometric frequencies of the octave bands, Hz; nb is a coefficient,
the values of which are presented below:
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Frequency, Hz 63 125 250 500 1,000 2,000 4,000 8,000
 nb 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0
{11ith these relations it is possible to calculate noise levels L of
suc
 centrifugal blowers. It is important to note that formula (7.3.5) cannot
be used for determining the noise levels of air conditioners, since they
have builtin soundabsorbing structures that substantially reduce the
aerodynamic noise levels of the blowers install.ed in them.
The noise levels in a room adjacent to the ventilator room L , in dB,
are calculated by the formula [S] a.r, _ [c. n=a. r; noM=r;
nep=bul ] Lc. n= Lnoa1  R 10 Ig S11ep  lO lg Ac. n. (7. 3. 6)
where R is the acoustic insulation of the bulkhead that separates the
ventilator room from the adjacent room, dB; Sbul is the area of the bulk
head, m2; Aa,r is the sound absorption in the adjacent room, m2.
Methods of determining the value R of ship bulkheads are described in
Chapter 10, and methods of determining sound absorption A are given in
Chapter 11. If the condition La.r Ltol must be satisfied, where Ltol are
the tolerable noise levels, then it is possible, by solving equation
(7.3.6) in terms of sound insulation R, to establish the values of R that
must be assigned to the bulkhead t}tat separates the ventilator room from
an adjacent state room. It is obvious here that the noise levels produced
by other sources must be at least 810 dB lower than the standard levels.
Calculation of Noise Levels in Rooms Equipped with VS and AC. The octave
noise levels in a ventilated (air conditioned) room Lr, in dB, are repre
sented in the general case by the logarithmic sum
[B=V; C=S; H=Cll LnoH = 101g (100  In+ 100,1LuoH. c,+ 100.11110m. K), (7. 3. 7)
where LrYv are the octave noise levels of an operating ventilation unit,
in dB; Sr.s are the octave noise levels related to the passage of air
through the ducts of the system, in dB; Lr.d are the octave noise levels
due to the passage of air through the air distribution devices, in dB.
In order to determine levels Lr.v it is necessary to know the acoustic
power levels radiated by the ventilator or air conditioner into the system,
and its losses as the noise propagates from the ventilator to an examined
room. Levels LNsuc and L Ndel are calculated by formula (7.3.4) on the
basis of known levels Lsuc and Ldel of air conditioners and ventilators.
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_ For ventilators Ldel usually are not measured, and therefore it is
necessary to use relations found by analyzing the characteristic dimensions
of the units, and on the basis of experimental data, in order to determine
acoustic power levels LNdel' For ship centrifugal blowers LNdel  LNsuc +
+ 4 dB in the entire octave frequency band, and for axial fans LNdel ~
~ LNsuc' Thus, if the suction noise levels are known it is possible to
determine the acoustic power radiated by a blower into the delivery duct.
Acoustic power levels LNsuc �f centrifugal blowers can also be calculated
by formula (7.3.5) on the basis of g.iven energy parameters Qb and Hb.
The great length of ventilation and air conditioning ducts, the branching
of the systems and the presence in them of many profiled elements, hard
ware, technological equipment and control systems, are responsible for
considerable losses of acoustic power in these systems. Data that
characterize the attenuation of noise in straight sections of ducts are
listed in Table 7.3 [10].
Table 7.3. Acoustic Power Losses in Straight Ducts, dB/m
1~ ~Aupn+a
~2) %l"s',+crp
nan nnlpi+ua
3)
9acrorw
oxtanuwx n
onoc, I'u
upoxoArioro
` ce~!emin
TpyGonpo
I ooAa, ntM
63
I 125
I 250
I 500
I_>1000 '
I~ar/ Soru ~ l..
[oTe=br; 0U78 = 10 lg f 101g ~
,
Mar=IDB] SoTn 4Sntar/ Sorn
where Sma and Sbr are the cross section areas of the main duct in front of
the branch and of the examined branch, respectively, in m2; ESbr is the
total cross section area of all the branches of an examined branch system,
in m2.
Reduction of the acoustic power level A c5, in dB, due to an abrupt ctiange
(decrease or increase) of the cross section area of a duct is calculated
by the formula
[cey=cs]
ece 60 acoustic noise and vibration have
levels that do not exceed L< 60 dB and L.. < 52 dB. Air noise and vibra
 x
tion with 6c > 60 are produced by turbulence, which precedes flow cavita
tion, and by vibrations of air bubbles, released in eddies and eddy zones.
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As 6c decreases from approximately 60 to the minimum value the noise and
vibration levels increase suddenly due to the onset and intensification of
cavitation.
L, a6 '
>00
90
dG
70
60
50
,q 6
100
90
80
70
60
SD
90
0 SO 100 150 ?OQ
. aK ~
l.jG, o�
Figure 8.1. Acoustic noise level of hydraulic system as
function of cavitation number. [36=dB; tt=c]
4~ ~ �
.
�
. . .
.
.
�
.
.
0
.
~
: 06
�
0 SD 100 950 200
6K ~
Figure 8.2. Vibration level of hydraulic system as func
tion of cavitation number.
Two curves that bound the maximum and minimum acoustic noise and vibration
levels of a system are given in Figures 8.1 and 8.2. The bottom curve
corresponds to a system with low local impedances = 0.250.4), and the
top to a system with high local impedances lt$5).
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The acoustic noise and vibration levels of ship equipment on the hydraulic
impedance coefficients for different values of csc are plotted in Figures
8.3 and 8.4. As can be seen, the noise and vibration levels for given
numbers 6c are higher in a system with high local impedances. And the
noise and vibration levels increase faster with local impedances as 6c
gets smaller. The reason is that cavitation develops more slowly in a
system with less cavitation impedance for the same values of a c.
G, d6 ,
90
166
76
60
500 > 2 3 4 5 B 7I
Figure 8.3. Acoustic noise level of system as function
of hydraulic impedances.
L~,d6
100
9a
84
7G
6G
50n > 2 j 4 5 6, 7~
Figure 8.4. Vibration level of system as function of
hydraulic impedances.
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a
G,~ =3
.
.
�
~
7
,
s
~
~
�
~
1C
.
Q=
'
�
.
7
�
17
16'
e
�
0
6K=?S
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Vibration spectrograms of individual kinds of cutoff and control devices
are shown in Figures 8.5 and 8.6, where it is obvious that highfrequency
components in the spectra play a dominant role.
GZ , d6
100
90
80
%C
60
50
40
:c
6G
40
d0
20
f, r4
Figure 8.5. Vibration spectrograms of cutoff valves:
1 check valve, Dy = 100 mm; 2 gate valve, DY =
= 100 mm; 3 sluice valve, Dy = 100 mm. [86=dB; FuA=Hz]
L~ , d6
e.~
r rq
Figure 8.6. Vibration spectrograms of control valves:
1 throttle valve, Dy = 60 mm; 2 throttle val.ve,
Dy = 40 mm; 3 reduction valve, DY = li0 mm.
Approximation of experimental data produced the following approximate rela
tions between the overall noise and vibration levels of marine hydraulic
equipment on the cavitation number:
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noise level Lmax' in dB
[k=c]
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Lmax = 35e0. 12aK 60;
vibration 1eve1 L x max, in dB
L x=max = 52e 0,09a X 52.
(8.1.2)
(8.1.3)
The strongest noise source of the three main types of valves check, dis
tribution and control, is control hardware, since the working process in a
control valve is based on the flow throttling principle.
The best ways to reduce the noise of hydraulic equipment is to limit the
pressure drops (per stage) to subcritical when cavitation begins to occur
and to perfect the profiles of the flow passages. Results that have been
achieved in this respect are extremely encouraging [1]. A few designs of
the passage parts of multistage throttle valves, which provide a sub
stantial reduction of noise and vibration in the entire spectrum in com
parison with singlestage valves, calculated for the same parameters, are
illustrated in Figure 8.7 as examples.
i
/
,
~
~
~
Figure 8.7. Design diagrams of flow passage parts of
quiet throttle valving.
228
.
O
i ~
/ . i
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Other strong noise and vib ration sources in hydraulic systems are certain
final control devices, used chiefly in hydraulic control systems. The
most vibrationactive elements of these systems are various kinds of
levers, slides and valves. Friction and impacts of contacting parts, in
particular impacts of valves on seats in the valve cases, impacts and
friction in couplings of the drives of slide valves, levers and in other
assemblies, are other sources of noise and vibration, in addition to
hydraulic factors (turbulence, pressure pulsations and cavitation).
Vibration and pressure pulsations, produced by pumps, fittings and 
hydraulic control mechanisms, arP transmitted by the lines that connect the
elements and also propagate through the fluid.
T'he propagation of perturbations in the flow, which takes place at the
speed of sound, does not depend on the velocity of the fluid in the line.
Pressure pulsations in hydraulic lines attenuate over very short distances
from the source. This applies principally to straight sections of lines.
Elbows, branches, flexible sleeves and various fittings in the path of the
flow promote reflection, dissipation and conversion of pulsation energy.
If these devices are not enough special antipulsation devices are connected
to the hydraulic line system.
The most common type of antipulsation device is the air box. The operating
principle of the air box is based on the accumulation of pressure pulses in
the flow and subsequent equalization of these pulses by periodic compres
sion and expansion of air, lockeci in a hox. In modern air box designs the
water and air cavities are separated by a rubber diaphragm, which prevents
air from dissolving in the water.
Various cavities in a system tanks, receivers, etc.,
to reduce the intensity of flow pulsations. Air traps,
highest parts of the lines, exert a doubleedged effect
the intensity of the flow pulsations they prevent in
flow and they damp pressure vibrations by virtue of the
air that accumulates in the air trap.
help considerably
installed in the
on the reduction of
terruption of the
elasticity of the
Antipulsation devices that reduce the activity of flow disturbances do not
prevent resonance vibrations of pipes and of columns of liquid standing in
them. To detune from resonance it is necessary that the ratio of the fre
quencies w of the disturbing forces to the natural vibration frequencies WD
of the pipes or of the fluid standing in them fall outside of the range
0.75 > w/w0 > 1.25. The e asiest way to accomplish resonance detuning is
to change the length of line sections and the rigidity of the bearing sur
faces.
Natural vibration frequencies fo, in Hz, of fluid in lines can be deter
mined by the following formulas:
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for a straight pipe with both ends open
[3HB=eq] fo = C (2n I);
213KB
for a straight pipe with one end open
c
f0 = QI3K6 (2n  1).
~
(8.1.4)
(8.1.5)
where c is the speed of sound in the liquid, in m/s; n is any positive
integer; Zea is the equivalent pipe length, in m, equa.l to Z+ AZ for a
pipe with one open end and Z+ 2AZ for a pipe with both ends open.
According to Rayleigh the correction factor is AZ =Trr/4, where r is pipe
radius, in m.
The natural vibration frequency f,,, in Hz, of a column of water in a pipe
~
with a box (collector) on the ;;nd can be found by the formula
f0  2~c ~Vi '
(8.1.6)
where s is the cross section of the pipe, m3 ; V is the volume of the box,
m 3 ; Z is pipe length, m.
iVays to determine the natural vibration frequencies of a liquid in compli
cated plumbing systems are described in the literature [9].
The natural transverse vibration frequency fp, in Hz, of individual sec
tions of pipes is expressed in the general case by the relation
az YTT
io = 2n12 in '
(8.1.7)
where a is acoefficient that characterizes how the ends of an examined
section of pipe are fastened; Z is the lengt:i of an examined section, in m;
E is the elasticity modulus, in Pa; J is the moment of inertia of the pipe
cross section, in m`'; m is the weight of 1 m of length, in kg/m.
If a pipe is continuous it usually may be viewed as an unbroken beam. For
multiple span beams with different distances between supports of identical
rigidity the coefficient a for determining funda;nental frequency is a=
= 3.14. For a pipe with intermediate supports, viewed as a beam with fixed
ends, this coefficient for fundamental vibrations, first and higher over
tones are, respectively, ao = 4.78; al = 7.85; an = 0.5[2(n + 1) + 1]Tr ~for
n > 1.
A pipe with one intermediate support in the middle, which permits rota.tion
in the cross section of the support, may be viewed as consisting of two
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beams, each of which has a fixed end and a hinged end. The coefficients
for the fundamental frequency, first and higher overtones of these beams
will be ao = 3.93; al = 7.07; an =4(n +41) + 1Tr for n> 1.
For a pipe with fixed ends and intermediate pivot supports, forming three,
four, five and six spans of identical length, the coefficients ao are,
respectively, 3.55, 3.39, 3.30, 3.26. As the number of intermediate
supp orts increases the dependence of vibrations on the manner in which the
ends of the pipe are fastened decreases. The waveforms of tlle vibrations,
inherent to a singlespan beam with pivot supports and with the character
istic dimension Z, equal to the span between supports, become the pre
dominant forms of vibration. The values of ao , al, an for such a beam are,
respectively, 7r, 2Tr, (n + 1)7r.
The methods whereby pipes are fastened to pumps and bearing structures
(parts of the frame, bulkheads and foundations) have a considerable effect
on the vibration and noise characteristics of pipes. To reduce the trans
mission of acoustic energy through pipes they are isolated from pumps with
soundinsulating elements (vibrationinsulating branches, siphons,
flexible sleeves).
The noise level of a ship hydraulic system depends to a considerable degree
on how well the operating mr,des of its elements are matched, since noise is
minimized when all the parts of a system operate in the optimum modes.
_ When designing ship hydraulic systems, therefore, it is important to make
sure first of all that the pumps used in them operate in the optimum
delivery mode. In particular, when several utilizers are connected in
parallel to the same pump with a constant rotation frequency and individual
utilizers are periodically turned off during the operation of the pump the
flow rate for which they are accountable should be compensated by means of
bypassing. Otherwise the turning on and off of recipients can lead to
devi ation of pump operation from the rated mode and, consequently, to an
increase of noise. Bypassing is especially important in systems with
impeller pumps, which run quietest at just one delivery rate, corresponding
to impactfree intake af the flow into the impeller blades.
To minimize the deviation of pump parameters from the optimum mode in a
system with a variable flow rate it is helpful to use variable speed pumps. _
8.2. Hyd.rodynamic and btechanical Noise Sources in Pumpsl
General Premises. The vibration and noise characteristics of pumps are
dete rmined by the pumps themselves, by the motors that drive them and by
tran smission mechanisms (reducers, hydraulic transformers, clutches, etc.).
1This section was written by 0. N. Sergeyeva.
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Accordingly the spectral composition of noise and vibrations of pumps is
attributed to hydrodynamic and mechanical sources. In many cases it
exhibits components of electromagnetic origin irom drive motors.
Impeller Pumps. The lowfrequency range of the noise and vibrations of
pumps of this type (centrifugal, axial, mixed f1ow, turbine) is determined
by two kinds of sources: a) mechanical from unbalanced forces of inertia
of the rotor, flaws in bearings and misalignment of rotors, which are mani
fested on the rotation frequency of a pump and on its harmonics; b) hydro
dynamic from the force interaction of the blades with a variable pres
 sure field, which can be seen in the spectra on the blade frequency and its
 higherorder components. The middle frequency range is filled basically
with harmonics of the blade frequency and with components produced by
turbulence in the flow passage part, and by vibrations of roller bearing
parts.
The highfrequency range of the noise and vibration spectra of impeller
pumps is determined principally by cavitation, and in precavitation modes
by turbulence in the flow part. Different forms of cavitation exert a
considerable influence on the intensity of the noise and vibration spectra
in a wide frequency range. The total noise level of an impeller pump is
plotted in Figure 8.8 as a function of the cavitation margin at a constant
impeller rotation speed and on the impeller rotation speed for a constant
_ cavitation margin. In the figure three critical modes are indicated,
corresponding to the following: Ocr development of cavitation in the
radial space; Icr developed profile cavitation, when the energy charac
te.ristics begin to deteriorate (the pressure drop is 20); IIcr pump
failure. The initial stages of cavitation are ascociated with the strongest
noise. The initial cavitation number in impellers in consideration of the
secondary influence of the basic parameters (tha scale effect) is expressed
by the relation
2gzA/i
[k=C] Q" u2+�`
where Ah is the absolute pressure in the flow; g is the acceleration of
gravity; u is the circumferential velocity of the rotor; a is a coefficient
that takes into account the influence of the scale effect (for impeller
pumps (x 1~t; 0.3).
A change of delivery exerts an extremely strong influence on the noise
level of impeller pumps. When the delivery deviates from the rated value
in either direction the noise and vibration levels increase due to angles
of attack different from 7ero at the impeller intake, which promotes
earlier onset of cavitation. The experimental dependence of the relative
initial cavitation number on the reduced delivery coefficient (Figure 8.9)
is an indirect c}iaracteristic of the change of the noise level of an axial
pump.
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a)
t,a6
1,h
fR D,4ho~ .
b) L,e6
dIL'M
fC" ,N/C
Figure 8.8. Total noise level of impeller pumps
as function of: a cavitation margin; b
 circumferential velocity of impeller.
[a6=dB; kp=cr; c=s]
G/an
7s 
z,o  _ I
~s
1,0
07 0,8 99 . 1,9 1,1 1,?
An examination of axial pumps that
have ogerated for a long time often
shows cavitation damage to the
peripheral parts of the impellers
and of the inner walls of the
impeller chamber. This is indica
tive of the occurrence of slotted
cavitation in the radial clearance.
Figure 8.9. Relative initial cavita The saturation of the spectrum with
tion number in axial pump as function strong components from turbuliza
of reduced deliver; caefficient. tion, pressure pulsations and cavi
tation creates conditions for the
excitation of resonances of the individual pump elements in a wide fre
quency range, and of free vibrations of the impeller blades.
The main cause of strong noise in centrifugal pumps is intensive turbuliza
tion in the suction branch and also high irregularity of velocities in the
discharge section of the suction branch. Consequently the flow strikes the
impeller with substantially differen.t pressures, velocities and directions.
Strong turbulence also occurs at the impeller discharge and at the entrance
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into the discharge branch. The formation of a hydrodynamic wake behind
the rotating impellers leads to flow irregularity at the entrance into the
flow guide or discharge branch,
Positive Displacement Pumps. The noise and vibrations of positive dis
_ placement (piston, gear) pumps are attributed to the nearly instantaneous
change from suction pressure to delivery pressure and to the pulsed nature
of the delivery of liquid into the discharge pipe. Components in the
spectra of noise and vibrations of piston pumps are produced by valve
impacts, pressure pulsations in the pumping chambers, pressure fluctuations
in ttie delivery pipe and cavitation. One cause of noise in this type of
pumps is the release of air in the stream of the pumped liquid. The
intensity of noise and vibration here increases as the discharge pressure
and rotation frequency increase.
A local increase of hydraulic pressure as the chamber between the gear
teeth passes the seal is also a source of strong noise in gear pumps.
Another cause of noise (in gear and screw pumps) is partial filling of the
gear tooth chambers with 1iQuid, which produces pulsation disturbances in
the working elements.
The causes of noise in individual kinds of pumps are related to their
design and technological features. Tliey include: alternating inertial
forces, caused by recipr.ocation of pistons in piston pumps and by the rota
tion of the teeth from the discharge to the suction cavity in gear pumps;
friction of working parts; elastic deformations and vibrations of mated
surfaces due to geometric errors in the machining of parts and assemblies.
_ Roller bearings are a common source of noise for all examined types of
pumps. The main cause of noise in roller bearings is wear of the inner and
outer races, and also flattening of the rollers. Friction bearings used in
pumps produce primarily lowfrec{uency components due to machining errors of
the journals and hydrodynamic processes in the space between the journal
and insert.
Jet Pumps. Noise in jet pumps is determined entirely by hydrodynamic pro
cesses. The main hydrodynamic processes are turbulization and turbulent
pressure pulsations. The noise and vibration spectra of jet pumps,
particularly in the highfrequency range, are uniform, and their levels
usually are low and meet existing standards.
 8.3. Ways to Reduce Pump Noise
Techniques for reducing pump noise and vibrations are classified in
accordance with the causes as design, technological and operational. The
most important opportunity for reducing pump noise during the design is to
reduce the speeds of working parts (rotation of impellers, gears, screws
and reciprocation pistons) and also the liquid flow velocity, since the
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vibration levels of pumps due to mechanical sources are proportional to the
square of velocity, and due to hydrodynamic sources to the fourthsixth
and higher power of velocity. The second opportunity is to make the most
complete utilization of design techniques to reduce the intensity of per
turbations or to completely eliminate them.
To improve the cavitation capacity of the impellers of centrifugal pumps,
especially of the first stage of multistage plamps, it is necessary to
reduce the diffusion of the annular suction channel, to use doublecurva
ture blades with channels of larger rotation radii and to install guides on
the seals to reduce turbulence in the flow as it enters the impeller from
the seals.
The radial space between the impellers and guide must be increased with
particular attention to reduction of the eccentricity of the radial
clearance around the periphery of the impeller in order to reduce the
intensity of discrete components on the blade frequency and its harmonics,
caused by irregular pressure at the impeller discharge.
To adjust the onset of cavitation in axial pumps the flow part must be
designed for a reduced powerspeed coefficient with the use of cavitation
resistant profiles. The calculated values of circulation along the radius
must be reduced sharply toward the periphery and gradually toward the radix.
The amount and eccentricity of the radial clearance between the ends of the
impeller blades and the chamber wall must also be minimized.
To reduce the amplitude of the pressure pulsations at the discharge of a
centrifugal pump it is helpful to install additional short blades in the
peripheral cross sections of the impeller blade channels. The suction flow
into the next stage can be improved to reduce flow irregularity, turbulence
and turbulent pressure pulsations by increasing the height of the guide
channel and by making in it a large bladeless axially symmetric annular
converging nozzle, by making an annular bladeless chamber around the
periphery of the guide, by reducing the diffusivity of return channels, and
in some cases by installing thin guide vanes in the return channels. How
ever, the best organized flow is achieved by using guides with bladeless
convergent return channels [6]. A schematic diagram of such a device is
shown in Figure 8.10.
To reduce the intensity of the discrete component from flow irregularity
after the impeller the discharge edges of the impeller blades and the
leading edges of the guide vanes should not be parallel, the numbers of
blades of the impeller and guide should not be multiples, and the number of
impeller blades should be made odd.
Adapters should be tapered and the larger radii rounded off to the extent
possible through the entire length of the flow part, including the suction _
and discharge branch pipes, to reduce turbulence noise.
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Figure 8.10. Guide of multistage
centrifugal pump with bladeless
annular convergent return channel:
1 case; 2 front wall; 3
rear wall; 4 seal; 5 blade
less annular convergent return
channel.
The following measures are used to
reduce the noise of piston pumps: the
impacts of the valves on the seats are
reduced by reducing the speed and
weight of the valves; pressure pulsa
tions in the delivery channel are
reduced by increasing the number of
cylinders and by installing air boxes
in the pump discharge; the rigidities
of the hydraulic unit and of parts that
connect the hydraulic unit to the
stand are increased.
The best way to reduce noise in positive
displacement rotor pumps is to perfect
production technology and especially to
increase the production precision of
the couplings of the moving parts (pis
tons, screws).
The third way to reduce pump noise is
to detune pump elements from their
natural vibration frequencies and the
entire unit as a whole from the fre
quencies of perturbing forces.
A fourth possibility of reducing noise is to assemble pump units
efficiently. 'I'he following measures must be taken into account here:
singleunit assembly of pump systems (the drive and the pump have a common
rotor and common chassis); the rotors are arranged vertically, are made
light as possible and rigidity is made symmetric with respect to the rota
tion axis.
There is one more possible way of reducing pump noise, and that is to use
vibration insulation and vibration damping techniques. The former include
internal shock absorption, and the latter consists in the manufacture of
case structures out of materials with a high decrement of vibrations and
 applying vibrationinsulating coatings on them.
An effective measure for pump units is to install under the bearings
elastic rings or special inserts, made of material with a high vibration
decrement (laminated and metal fiber materials, rubber and plastic). The
spectrogram in Figure 8.11 shows the effect of installing roller bearings
in a singlestage horizontal centrifugal pump with inserts made of pressed
copper fiber with a porosity of 70%, and the photograph in Figure 8.12
shows the position of the bearings in these inserts.
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cs,a6
79
~
60
sv I
~
40
~
, i' 2
70
10
5 6 7 8910 7 d 4 5 6 7 8910 1 3 4 5 6 789104
f,ru
Figure 8.11. Effect of installing roller bearings in
pressed copper fiber inserts on vibration level of centri
fugal pump: 1 bearings without inserts; 2 bearings
in inserts. [86=dB; fu=Hz]
Elasti.c pads, pneumatic tire and leaf spring carriages, installed between
the pump and pump stand, are used for absorbing acoustic energy on bearing
lugs and flanges. The transmission of acoustic energy in pipes is reduced
by using elastic inserts, elastic branch pipes, bellows, etc. (see
Chapter 12).
Figure 8.12. General view of roller bearings, inserted in
pressed copper fiber inserts: left 30% porosity;
right 70% porosity.
Pump noise is influenced to a great extent by productioa technology. Noise
can be reduced by improving the production nuality of pumps, precision af
the machining and assembly of parts and units, by using quiet bearings, by
balancing the rotors part by part and by balancing them in the assembled
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form. One of the most important requirements of technology is
standardization of the deviation of shapes and mating of joined surfaces
in accordance with the tolerances o� GOST 1035663, which meet the
requirements on pump noise.
Operational factors can predominate in pump noises under certain condi
tions. Therefore, to prevent excessive noise during operation it is
important to observe specific operating conditions and to eliminate
deformation and displacement of elements and assemblies under the influence
of temperatures and pressures.
8.4. Noise and Vibration of Reciprocating Compressorsl
The noise of reciprocating compressors is produced by the irregular speed
and turbulence of air in suction systems (suction noise), and by radiation
by the surfaces of parts and assemblies of the compressor, subjected to
variable forces.
Nonuniform suction velocity produces strong noise components with fre
quencies that are multiples of the shaft rotation frequency. Noise related
to turbulization is manifested in the medium and highfrequency ranges,
and it is strongest in the region corresponding to the eddy separation
frequency in the channels at the time of the maximum piston velocity.
Noise is especially strong when the frequency of turbulization and the
natural frequency of the resonator cavity (8.1.6), formed by the cavity of
the cylinder, coincide at specifically that time. Suction noise also
increases if the vibration frequency of the intake valves matches the fre
quency to which the cylinder cavity is tuned at the instant of seating.
Suction noise is readily reduced on frequencies above 200 Hz by using
mufflers in the form, for example, of an annular resonator with one or two
cells, tuned to the loudest noise. Lowfrequency noise, related to the
variable intake velocity, cannot be muffled this way because the cells of
an acoustic filter are ineffective within size limits that can be con
sidered acceptable. To reduce lowfrequency noise it is better to use a
muffler with friction elements in the form of slots and holes, representing
a parallel channel for the variable flow component.
One possible muffler design that can reduce intake noise to the level of
the noise radiated by the surface of a compressor is illustrated in Figure
8.13 [8]. The attenuation in the 100250 Hz band, inherent to this design,
is 2830 dB.
Intake noise can also be reduced without a muffler if the air is supplied
to the intake pipe through a flexible rubber or reinforced rubber sleeve.
The attenuation of noise for about a 20 caliber longsleeve is 1015 c'B in
a wide frequency range. Less attenuation (about 10 dB) can be achieved by
1This section was written by V. D. Kurnatov.
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 using a flexible metal sleeve. The amount of muffling is determined as the
difference of the noise levels without a muffler and with a muffler,
measured at a distance of 0.5 m from the intake pipe in the former case
and 0.5 m from the intake hole of the muffler or sleeve in the latter.
m 9Rn
The noise radiated by the surface of a compres
sor is produced by forces of inertia and pres
sure, by the periodically deformed cylinder
heads and walls, by pressure fluctuations in
nipes and assemblies of betweenstage connec
tions, acting upon the compressor block through
the crankconnecting rod assembly, and by
impacts in bearings, drive and auxiliary
mechanisms.
~
The design of a compressor and its arrangement
determine the magnitude of unbalanced forces,
their moments and the irregularity of the
tilting moment. Unbalanced forces and moments
are among the basic sources of lowfrequency
vibrations of ship compressors. The arrange
ment also influences the vibration and noise
levels in the medium and highfrequency ranges.
Figure 8.13. Intake noise In this case the influence of arrangement
muffler. depends on the distance between a source of
vibration and the bearing surfaces or radiating
surfaces, and on the degree of attenuation of vibrations along the propaga
tion path.
It may be assumed that lowlevel compressor noise and vibration can be
assured by selptting the proper arrangement. In view of rigid requirements
on vibration ar,d noise characteristics it is important that centrifugal
forces and forces of inertia of the first two orders from reciprocating
masses be balanced. These forces can be balanced by using a multibank,
including opposedpiston arrangement, or a special balancing mechanism. A
multibank arrangement provides equalization of the tilting moment and a
reduction of the vibration it produces. A moment with harmonics that do
 not exceed 1012a of the average may be assumed satisfactory.
_ The dynamic effect of the tilting moment can also be reduced by placing the
m4unts close to the principal inertia axis of the machine. This reduces
the rotation rigidity and natural vibration frequency fn. This measilre is
j ustifiLd in the range fn < 0.7f, where f is the frequency of the strongest
 harmonics of the moment. This rec{uirement corr�esponds to smaller coeffi
cients of dynamicity and transmission of forces in comparison with the
usual range n=(0.72.7)f fo.r rotation vibrations, so that both the
transmz.itted moment and the amplitude of the vibxations of a machine are
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reduced. In the limiting case of pivot support on the principal inertia
axis (fn 0), how much of the moment is transmitted is determined by the
amount of friction in the support.
One form of vibrations that are produced by st resses, characteristic of a
compressor, are bending vibrations of cantilever cylinders. The problem
of reducing them consists in detuning the natural frequency of vibrations
 of this form far enough away from the frequency range that includes strong
harmonics of normal force. The solution of the problem for a given rota
tion frequency consists in increasing the rigidity of the cantilever base.
For the purposes of estimation one may assume the rigidity to be inadequate
if the frequency of the indicated form of free vibrations in a crosshead
less machine is less than 25 times the shaft rotation frequency.
By examining the cylinder and head as a cantilever beam with mass m per
unit length and with mass M suspended on the end, the approximate free
vibration frequency can be determined by using relation (8.1.7). The
influence of the suspended mass is taken into account in this relation by
value a. In relation to point and distributed masses within M/mZ = 612
one may use a= 0.8 (here Z is the length of the beam).
_ Cylinder vibrations groduce noise, related to radiation not only by
cylinder surfaces, but to a lesser degree also by the surfaces of the crank
case and frame. Pipe vibrations are related to cylinder vibrations. They
can increase compressor noise if the pipes between fastening points are too
long. Additional fastening of pipes in a vibration loop eliminates the
source of noise in this case and improves the reliability of pipe connec
tions. The type of fastening that resists both transverse displacements of
the pipe and rotation vibrations at a fastening point is preferable here.
The vibrations of pipes and of parts connected to them can be caused by
pressure pulsations due to the periodic delivery of air. In this case
vibrations can be reduced by changing the rigidity and by detuning the
 natural frequencies of the pipe with extra fasteners. Another way to
reduce pipe vibrations is to lessen pressure pulsations. One way to do
this is to use a buffer box, installed as close as possible to the cylin
der. If the pipe connects two compressor stages the buffer box should be
installed on the delivery side, since a stronger compression wave is
generated than the rarefaction wave on the suction side.
If a buffer box is viewed as an acoustic filter of the expansion box type,
its size can be determined by assigning the amount of deadening AL by the
formula
OL lOlg 1E 4 m i 1 z sin 2:~ ~ l, (8.4.1)
/ J
_ where m= s2/sl is the ratio of the cross section areas of the chamber and
pipe; Ze is chamber length.
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A filter best attenuates waves for which the ratio Ze/X corresponds to
values 4 n, where n is a series of odd numbers, i.e., chamber length
corresponds to an odd number of quarter waves. Vibrations with an even
number of quarter waves through the length of the chamber are not
attenuated.
Pressure vibrations in a pipe can also be attenuated with a throttle disk
(diaphragm) or valve, installed at the point where the vibration velocity
of standing waves is maximum. The greatest effect is achieved when the
acoustic impedance of the disk is matche d with the wave impedance of the
pipe. The condition of matching is expressed by the relation [4]
9
LA=dl
d~ = j` ,
c
(s.a.a)
where c0 = cm(D/d)2 is the average velo city of the gas in the pipe, deter
mined by the average piston velocity c m ; dd, d, D are, respectively, the
diameters of the hole in the diaphragm, pipe and cylinder.
8.5. Basic Noise Sources of Electrical Machinery
Classification of Noise. Noise produced by electrical machinery (EM) is
subdivided into three categories: magnetic, mechanical and aerodynamic.
Magnetic noise is produced by electromagnetic forces, acting in the air gap
between the rotor and stator. The main sources of inechanical noise in EM
are .rotor disbalance, dual rigidity of the rotor, bearings and brushes.
Aerodynamic noise in EM occurs chiefly as a result of rotation of the rotor
and fan in air. This is the predominant source of noise in EM with a rotor
speed higher than 50 m/s.
A schematic diagram of a singlearmatixre DC motor with basic noise sources
is illustrated in Figure 8.14.
The noise of EM depends on its power, rotation frec{uency and design
execution. The noise level L, in dB, of selfventilated EM is proportional
to nominal power N and to the square of the rotation frequency:
L.::l01gN+201gn{K. (8.5.1)
The value of K depends on design and technological features of a machine,
and also on noise measuremPnt conditions (8.5.1).
The requirements on noise characteristics depend on the purpose of EM.
Accor.ding to GOST 1637270 di�ferent typ es of EM (with a nominal power more
than 0.25 k{9 and nominal rotation speed of up to 4,000 rpm) are divided
into five classes in terms of noise: 0, 1, 2, 3, 4. For ordinary class 1
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3
2
Figure 8.14. Basic noise sources of DC motor:
1 unbalanced rotor; 2 bearing noise; 3
brush noise; 4 magnetic noise; 5 fan noise.
AC and DC EM the average acoustic levels A, in dBA, at a distance of 1 m
from the machine should not exceed the values indicated in Table 8.1.
Different noise sources can predominate in the noise of a machine, depend
ing on the type, execution (roller bearings, friction bearings) and speed.
A classification of basic noise sources of EM is given in Table 8.2;
typical noise spectra are given with the fan not running.
Table 8.1. Noise Level of Electrical Machinery as Function of
Power and Rotation Frequency
~J HOMHH8116H8Si
MOI1AfIOCTb, K$T
OpHexTxpoeovFide 311aveHlfA }�ponHn wyHa. A6A,
~ 3JI2NTPH4fCKHX M8I11{IH RQH HOMN118:16110n 4.7CTOTC
Bp8l1ifH11A, 06/SI11,y
Ao d U00 I 1000~o I12~00 I23000 I 34000
0,251,50
64
68
70
I 71 _
75
1,504,00
67
72
74
76
80
4,0015,00
74
78
82
85
89
15,0045,00
80
85
87
89
93
45,00132,00
85
92
95
97
100
132,00400,00
90
96
98
100
104
400,001000,00
94
100
103
105
109
Key: 1. Rated power, kW
2. Approximate noise levels, dBA, of electrical
machinery at nominal rotation frequency, rpm
3. up to 1,000
Magnetic Noise. The magnetic noise of EM occurs under the influence of
alternating forces in the air gap (Figure 8.15). The forces in the air gap
act upon the rotor and upon the stator. The rotor, which isthe most
stable and invariant component of EM, resists deformation very well. Mag
netic noise in EM is produced chiefly by deformation of the stator. In
r
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n
a
o u
z.
a
~
I
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calculations the armature of a machine is usually represented as a
cylindrical case, acted upon by a system of radial and tangenti._ forces
with wave number r, periodically changing in time and symmetslc with
respect to the circumference.
blagnetic noise is a function of the magnetic flux density, the number and
form of poles, the number and form of Slots and the geometry of the air
space.
In the air space of an asynchronous motor there occurs, in addition to the
main magnetic field, which produces torque on the shaft, a large number of
higher field harmonics: higher harmonic winding fields, produced by non
sinusoidal distribution of the magnetic force in the air gap; toothlike
fields produced by the variable magnetic conductance in the air gap;
higher field harmonics due to various asy*nmetries in the magnetic circuit;
higher field harmonics due to asymmetric supply voltage; higher field
harmonics due to saturation of the magnetic conductor.
Figure 8.15. Effect of perturbing magnetic
forces F(radi al FX, tangential Fy, axial Fz)
on stator of electric motor.
In an asynchronous motor with a symmetric stator winding, with a whole
number of pole and phase slots, higher field harmonics occur with the
following numbers of pole pairs:
from the stator winding
v = P (2m19i (8 . 5 . 2)
from the rotor winding
p(2mr9j r I)  (8. 5. 3)
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in the case of a phase winding
lt =z9,,. i p  (8.5.4)
in the case of a shortcircuited winding ("squirrel cage");
from the sawtooth stator field
z
,~Z p t� P k 1. (8. 5. 5)
i
from the sawtooth rotor field
 I `Z  p~t � p k~ . (8.5.6)
In these formulas: ml, m2 are the numbers of winding phases; p is the
number of pole pairs of the main field; q l and q 2 =�1, �2, �3, Z1
and z2 are the numbers of stator and rotor slots; k= 1, 2, 3,
The circular frequencies are, respectively:
for the stator fields
[ct=st] (Ocr=wa=2nfo; (8.5.7)
for the rotor fields (in terms of the stator)
wu  a'o L p~1 (8.5.8)
where f0 is the power line frequency; s is the slip of the rotor relative
to the main rotating stator field.
The magnetic inductions of the higher field harmonics are determined by the
currents, winding parameters (number of turns, winding coefficients) and
magnetic circuit (the size of the air gap and Car.ter's coefficient). The
inductions of the serrated sawtoothlike harmonics are determined entirely
by the parameters of the serrated stator and rotor zones (the shape of the
_ slot and the size of the air gap). Favorable conditions are created for
the excitation or many natural vibration frequencies of the stator. Wave
number r is equal to the difference of the number of pole pairs uf the
_ interacting stator and rotor fields v and u:
r=Iv�l� (8.5.9)
The frequencies of the resulting forms of deformation of the stator ring
correspond to the orders of the active forces. The rigidity of the stator
ring depends on the geometric dimensions of the stator and on the order of
deformation (it is proportional to r4 for r> 2). Therefore lowerorder
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magnetic forces, for which the amplitudes of dynamic deformations are
maximum, play the biggest part in the generation of magnetic noise and
vibration. For small asynchronous motors these low orders may be assumed
 to be r< 4, and for medium and large motors they are larger values of r
(up to r< 12). Since the order of vibrations depends on the ratio of the
numbers of stator and rotor slots zl and z2 they must be selected such as
_ to obtain the highest possible value of r.
The radial magnetic forces that act upon a unit area of the stator boring
are determined by the expression
,
[MarH=magn] FMarx Bb (x, 1) 1
= [ 5ooU , # (8.5.10)
where BS(x, t) is the radial term of induction in the air gap.
The most important of the forces produced by the interaction of the stator
and rotor field harrnonics are those that are produced by the serrated
harmonics. These forces can develop considerable deformations of the
stator, particularly for loworder vibrations that occur in the case of
poor selection of the ratios of the numbers of stator and rotor slots.
Frequencies f, in Hz, of these forces are determined, respectively, by the
formulas
fi=jo[2I pk(1 s)l, (8.5.11)
.1
 fa=fo ZP (IS), (8.5.12)
where f0 is the line voltage frequency; z2 is the number of rotor slots; p
is the number of pole pairs of the main stator field; k=�1, �2, �3,
The process whereby magnetic noise is generated in synchronous motors is
analogous to the process that takes place in asynchronous motors. The
magnetic forces that subject the stator to periodic deformation occur due
to interference of higher harmonic fields in the air gap. The laws
responsible for winding and serrated harmonics of the stator in synchronous
motors are exactly the same as in asynchronous motors. The difference
between salientpole synchronous motors and asynchronous motors is
attributed to rotor design. In synchronous motors with massive rotors the
HF components of noise are weakened considerably by the larger air gap
between the stator and rotor and by the damping force of the rotor mass.
The rotor field harmonics are determined by the magnetic conductance of the
 air gap. If z2 = 2p in salientpole synchronous motors, we obtain the
following expression for the numbP, of pole pairs of the rotor field
harmonics:
~la = (1 p k) P (I � 2k),
(8.5.13)
where k= 1, 2, 3,
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By virtue of the symmetric design of the pole pieces of synchronous moto.rs
only equal phase harmonics of odd orders remain in the curve o� the rotor
field. These harmonics are determined by the following relation:
~a = P (293 (8. 5 .14)
where q3 = 1, 2, 3,
The circular frequency of these harmonics, expressed in terms of the
stator with synchronous rotation of the pole piece, is
Wo=2nfo(29a1)� (8.5.15)
As in asynchronous noise, magnetic noise in synchronous motors is produced
chiefly by magnetic force waves of lower orders r, occurring due to the
interaction of the stator and rotor field harmonics v and r1r. Magnetic
forces produced by the serrated stator field harmonics (particularly in
open slot motors) are chiefly responsible for noise in synchronous motors
in the idle mode. The resulting magnetic forces have the following orders:
i=2P(9ap (8.5.16)
where zl is the number of stator slots; 2p is the number of poles of the
main field; q3 = 1, 2, 3,
Frequencies fr, in Hz, of these forces, will be fr = 2fOq3, where q3 =
= 1, 2, 3,
Magnetic noise produced by the fundamental wave of magnetic field rotation
does not readily submit to suppression, particularly in twopole 50 Hz
asynchronous motors.
Magnetic noise also can occur, in principle, due to magnetostriction, i.e.,
due to a change of the shape and size of the ferromagnetic plates under the
influence of periodic magnetic fields. However, a change of size of the
active iron on acoustic frequencies in electric motors is usually insig
nificant, and magnetostriction noise does not reach the level of the noise
produced by the attraction of magnetic masses.
Periodic electromagnetic forces in DC motors are caused by a periodic
change of the magnetic conductance of the air gap under the poles as the
serrated armature rotates. The frequency of the magnetic noise fm, in Hz,
is fm 60 i, where z is the number of armature teeth; i= 1, 2, 3,
The only difference between magnetic noises of DC and AC motors is the fact
that in the former stator deformation occurs under the influence of point
forces, and in the latter it occurs under the influence of sinusoidally
distributed forces.
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Mechanical Noise. Mechanical noise is the result of many factors that
happen inside a motor. The most typical of them are unbalanced rotor
assemblies, roller bearings and the brushes on the commutator rings.
The level of the disbalance component of noise in electrical machinery
nearly entirely depends on the processwhereby the rotoris balanced. The type
of Ebi is important only from the standpoint of sensitivity to subsequent
changes in the distribution of mass. The front parts of the windings of
motors with a wound rotor is especially vulnerable to displacement under
the influence of centrifugal forces. Disbalance noise is manifested on
rotation frequency frot  n/60.
GOST 1232766 recommends residual disbalance for rigid rotors weighing
from 3 to 1,000 kg. The tolerable levels of vibration of rigid rotors
weighing more than 1,000 kg and of flexible rotors are usually established
by infactory standards.
In accordance with GOST 1692171 there are eight classes of motors weighing
from 0.25 to 2,000 kg: 0.28, 0.45, 0.7, 1.1, 1.8, 2.8, 4.5, 7.0. The index
corresponds to the maximum tolerable vibration velocity for a given class;
for example, 0.28 mm/s correspands to the vibration class of 0.28, 0.45
mm/s corresponds to class 0.45, etc.
Disbalance in electrical macllinery can change significantly, depending on
the thermal condition of the rotor (thermal disbalance). An investigation
of many different types of electrical machinery disclosed that when the
thermal asymmetry of the rotor is 0.5�C thermal disbalance can be 56
times greater than residual (mechanical) disbalance.
Basically the following factors are responsible for uncompensated residual
disbalance in the rotor: the absence of monolithic rotating windings
(particularly in DC and synchronous motors), which leads to a constant
change of residual disbalance, both in terms of magnitude and phase;
unavoidable thermal asymmetry due to different thicknesses of the slot
insulation; the existence of turn shorts in excitation windings; different
cooling conditions, leading to thermal deformation of the rotor, etc. The
most precise balancing obviously can be achieved in asynchronous motors
with shorted rotors, in which the first critical velocity is higher than
the working velocity. Highspeed synchronous turbogenerators with massive
rotors and tenons have higher residual disbalance per unit mass of the
rotor than asynchronous generators. hforeover, synchronous turbogenerators,
due to possible turn shorts in the excitation winding and failures of the
rotor cooling system during operation, are less stable in relation to
vibration (this is particularly characteristic of watercooled rotors).
The armatures of DC motors and the salientpole rotors of synchronous
motors have higher residual disbalance than the aboveexamined motors. The
strongest noise produced by disbalance occurs in motors with flexible
_ rotors, in which the working velocity is higher than the lst and 2nd
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critica 1 velocities of the rotor. The rotors of these motors are
particu 1 arly sensitive to thermal asymmetry and often require thermal
balance adjustment during operation in the rated mode.
Noise p roduced by 3ouble rigidity is not always heard, but it can be
signifi c ant, particularly in turbogenerators. Double rigidity occurs
because the large teetli in the slot zone of the turbogenerator barrel
imp art different rigidities to the rotor in the principal bending planes.
Two complete cycles of change of rigidity, and accordingly two complete
changes of bending, occur in each rotation of the rotor, causing noise with
a double rotation frequency. This noise does not occur in vertical motors.
Sometime s the rotor journals of electrical machinery are not manufactured
strictly symmetric; then the center of the rotor is displaced relative to
the rot ation axis and causes an inertial perturbing force, which acts upon
the be aring. Outofround journals exert the strongest influence on the
noise level in large highspeed EM with a journal diameter of 100120 mm.
Bearing noise depends on the type of electric motor, since the character
istics o f acoustic vibration and noise transmission vary from motor to
motor. Ball bearings with their numerous parts, which experience relative
displace ment, produce noise on many frequencies, which are proportional to
the rot ation frequency of the inside bearing race, i.e., fT = kTn/60, where
n is the shaft rotation frequency; kT is the proportionality coefficient
for the Tth perturbing harmonic, which depends on the microgeometry of the
bearing surfaces and on bearing design.
The freq uencies according to these formulas in application to electrical
machine ry can be calculated approximately. In real motors they can differ
somewhat due to the existence of numerous sources of excitation, the com
plexity of the elastic system of the motor and the effect of the phase
relations between the sources.
One sou rce of inechanical noise in electrical machinery are the brushes and
commutator. When the rotor rotates the commutator plates, striking the
brushes, impart vibrations to the parts of the brush clamps and to the
brushes themselves. The frequencies of the fundamental components of brush
noise are determined by the formul_a
f n
b = 60 zc"
where i= 1, 2, 3, are noise harmonics; z~ is the number of commutator
plates.
A frec{uency analysis shows that the brush noise range is 1,0008,000 Hz.
It is manifested particularly strongly in large slow (up to 350 rpm) DC
motors. In motors running at frequencies higher than 1,000 rpm this noise
is overl apped by magnetic and aerodynamic noises.
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Aerodynamic Noise. Aerodynamic, or fan noise in electric motors is
produced by the rotating rotor itself, by the fan blades and by air
passing through the ventilation channels and air cavities of the motor.
Ventilation noise has a wideband spzctrum. The fundamental frequency of
fan blade noise is ffan  nz/60, where z is the number of fan blades, and
this frequency is sometimes distinguishable in noise of random frequency.
The aerodynamic noise level is determined basically by the type of fan,
and not by the type of electric motor. For selfventilating motors, cooled
by a fan, pressed on the shaft of the motor, the aerodynamic noise level L,
in dB, at a distance of 0.5 m from the case is determined by the foxmula
L= 10 log N+ 20 log n+ 5; for sealed selfventilating motors L=
= 10 log N+ 20 log n; for sealed watercooled motors L= 10 log N+
1 20 log n 10, where N is motor power, in kW; n is the rotation fre
auency, in rpm. For independently ventilated (quiet running) motors, the
noise of which is determined by the noise of the external fan, L=
= 10 log N+ 80, where N is the power of the electric fan, in k{V.
8.6. Reduction of Noise of Electrical Machinery
Basic Principles. The noise of ele:ctrical machinery (EM) is reduced at the
present time in two basic ways. Vie first is to reduce perturbing forces
at the source, and the second is to reduce the noise on the propagation
paths by appropriately changing the parameters of the structural parts of
EM (rigidities, masses and damping elements).
Noise reduction technic{ues for Eb1 are classified in Table 8.3 [3]. The
design measures for reducing Eb1 noise are: rake the rotor slots; make the
_ roller bearings smaller; replace ball bearings with friction bearings;
_ reduce the mass ratio of rotating and nonmoving motor parts; change the
_ profile of the fan blades; reduce the rotor rotation frequency.
Perturbing inertial forces and noise produced by disbalance, outofround
journals, double rotor rigidity, misaligned shafts, etc., depend mostly on
manufacture precision and the quality of assembly.
= Noise of EM can be reduced by using vibration and soundinsulating
systems to reduce the transmission of perturbing forces to external sur
faces, reducing the noise radiated by the motor.
Reduction of Magnetic Noise. Magnetic noise is reduced primarily by
rrducing the periodic components of electromagnetic forces and by eliminat
ing resonance in the mechanical system consisting of the �rame and rotating
rotor, i.e., by mismatching the natural vibration frequency of the frame
and the frequency of perturbing �orces. This is especially important in
DC motors with the rotation frequency controlled in a wide range.
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Table 8.3. Classification of Noise Reduction Techniques for
Electrical Machinery
Cause of noise
Fundamental pertur
Noise reduction techniques
Effectiveness
bation fre uencies
of inethod, dB
Rotor dis
Static and dynamic
balance
balancing
1020
Elastic bearings
510
n/60
Elastic inertial bearings
1015
Vertical installation of
motor on horizontal shock
absorbers
1020
Double
Cut extra slots
1020
rigidity of
2n/60
Elastic damping bearings
1012
rotor
iElastic inertial bearings
1015
Outofround
~Careful maching of rotor
rotor
2n/60
journals
1020
journals
Elastic, elastic damping,
elastic inertial bearin s
1015
Friction
Careful machining of bear
bearings
n/60; tn/60
ing surface, selection of
ro er lubricant
1015
Roller bear
Use friction bearings
2025
ings
so ' so ' ~p ~
Spring bearing stays
1015
Elastic, elastic inertial
(J
bearings
1015
 p~ ~
o ' d~,
Use special quiet bearings
1020
Do  d�l n
2Uo 60 '
n
120 1 + w > ZW'
0
n ~ dw ~ ZeZ~u
60 Do ) 9
Brushes
Careful manufacture of
zc n
commutator and contact
60
ring relief
1015
Use appropriate soft
rades of brushes
810
Magnetic cir
cuit in
motors:
DC
Rake armature slots
1015
Eccentric air gap
612
zn
Step poles
1015
60
Magnetic wedges
1020
Smooth armature
1530
Elastic magnetic suspen
sion
1015
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Table 8.3 (continued)
Cause of noise
Fundamental pertur
Noise reduction techniques
Effectiveness
bation fre uencies
of inethod, dB
asynchro
.
Rake rotor or stator slots
1015
nous
fo� r 2+ Pk (I s)1 ;
Use proper stator and
~ ~
rotor slots
1015
fo zP (I S)
Elastic magnetic suspen
sion
1015
synchro
2f k
Rake stator slots
1015
nous
0
Elastic magnetic suspen
sion
1015
Fan
zfn
Use water cooling
1020
60
Install noise mufflers
1015
The basic measures for reducing magnetic vibration and noise in EM are: cut
raked rotor slots fo r all types of DC and AC electrical machinery; design
an eccentric air gap for DC motors; select the proper ratio of the number
of rotor and stator slots for asynchronous motors.
Raking the armature and rotor slots promotes a more uniform distribution of
magnetic flux in the air gap and reduces the intensity of the serrated
magnetic fields. The attainable reduction of magnetic noise in small and
mediumpower motors is
 = 201 sin 21
AL
gis� = 201g � as / ' (8.6.1)
p'2
where fsu is the skew factor; u is the order of the strongest field
harmonics; as is the rake angle of the slot; p is the number of pole pairs.
 In small and mediumpower motors it is customary to rake the slots by one
tooth divi sion (of the stator or rotor), which reduces the magnetic noise
and vibration by 1015 dB.
An eccentric air gap in DC motors reduces magnetic induction on the pole
edges, i.e., magnetic flux pulsation and perturbing forces. This reduces
magnetic noise by 25 dB. The use of a combination of raked slots and
eccentric air gap produces the greatest effect in terms of the reduction of
magnetic noise in DC motors.
The choice o� the proper ratio of the numbers of stator and rotor slots is
very important for reducing magnetic noise in asyrichronous electric motors.
There are many recommendations regarding the proper choice of slot ratio
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[2, 3, 8, 11]. However, there are no universal rules that are equally
suitable in terms oi the reduction of magnetic noise in small and large
motors. The reason is that the resonance properties of the iron packs
depend on the size of the motors. Therefore, when selecting the ratio
zl/z2 for the purpose of reducing magnetic noise it is necessary to
examine each individual small, medium and large motor separately.
Step poles, which reduce magnetic noise by 1015 dB, are used for DC motors.
Sometimes it is also recommended, for reducing magnetic vibration, to drive
magnetic wedges into the armature and rotor slots. This promotes a more
uniform distribution of induction in the air gap and thus reduces perturb
ing forces. Wedges reduce vibration in the serrated frequency range by up
to 18 dB without a sacrifice of the energy indices of the motor.
Reduction of hlechanical Noise. As was mentioned earlier, the disbalance
component very often determines the overall noise level in EM. The basic
method for reducing it is careful static and dynamic balancing. The rotor
must be balanced in its own bearings in order to reduce dynamic disbalance.
It is recommended that the rotor be balanced in assembled form, i.e., when
all parts are mounted on it, including the fan, rings or commutator, etc.;
otherwise there is no guarantee of the required precision, even if the parts
to be assembled are balanced separately. Extra slots (dummy slots) are cut
in the large teeth.
Elastic and elastic inertial EM mechanical and magnetic noise reduction
techniques are examined below.
Changing to friction bearings is a radical means of reducing bearing r:oise.
Friction bearings are already being used today in the construction of
numerous large machines by virtue of their great bearing capacity and
dependability. However, the use of friction bearings in small and medium
machines is difficult for design and operational considerations (in
particular, an efficient lubrication system is needed, etc.).
The problem of reducing bearing noise is solved in three independent steps:
development and use of roller bearings with improved noise characteristics;
vibration damping and absorption of vibrations transmitted to the frame of
a machine; creation of the most favorable operating conditions for the
bearings in a machine.
Experience in the electrical industry has shown that it is best from the
stan3point of noise reduction to use singlerace radial ball bearings in
 electrical machinery; other types of bearings, as a rule, produce a higher
noise level.
To reduce brush noise it is recommended that a highquality forged
commutator be used, carefully surface finished to minimize deviation from
cylindricity. The brush clamp must be sufficiently rigid, and the gaps
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d L,d6
50
40
JO
20
f0
f
2
D 50 f00 200 400 800 1600 3200 6400
f, r4
Figure 8.16. Noise mufflers of selfventilating electric
motor and their efficiency: 1 motor without muffler;
_ 2 motor with muffler. [86=0; fu=Hz]
ec,as
rn
VV
O
~
~
Y
30
,
2
20
10
0 50 100 200 400 800 1600 3200 6400
f, rq .
Figure 8.17. Noise muffler of electric fan and its effi
ciency: 1 fan without muffler; 2 fan with muffler.
between the brush and brush box must be minimized. Noise can be reduced by
810 dB by selecting soft grades of brushes, which wear in well.
Reduction of Aerodynamic Noise. In selfventilating large motors, in which
air is sucked in from the atmosphere and vented into the atmosphere, noise 
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can be reduced by 69 dB by replacing selfventilation with induced
ventilation using an external fan. The amount of noise reduction in
sealed selfventilating motors depends on the thickness and mass of the
walls of the air ducts. In large motors with thinwall brushes the brushes
must be covered on the inside with a muffler. In independently (induced)
ventilated motors noise is determined by the external fan. Therefore an
effort must be made to use quiet fans. In watercooled motors the power
bars that extend from the case and the slots in the case must be carefully
sealed. Water cooling provides 810 dB more reduction of aerodynamic noise
in comparison with air cooling.
6 5 View A
9
E
7
10
A
~
Figure 8.18. Fan muffler: 1 electric motor; 2 snail;
_ 3 fan blades; 4 muffler; 5 muffler case; 6
soundabsorbing material; 7 duraluminum muffler vanes;
 8 ball bearing; 9 cross piece; 10 metal screen.
One effective way to reduce aerodynamic noise in electrical machinery is to
install mufflers.(Figures 8.16 and 8.17). The efficiency of the mufflers
depends on the area and thickness of the soundabsorbing material. In many
mufflers the air flow is diverted by 90180� for the most effective deaden
ing of noise.
Illustrated in Figure 8.18 is a muffler, whic;h differs from familiar ones
in that a fan with vanes, which overlap each other and are covered on the
inside with a soundabsorbing material, is inserted in the air flow,
rotating on its axis under the influence of the air stream. The muffler is
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attached to an electric fan, which contains a motor and blower. {Vhen the
motor is turned on the fan is turned by the air flow without offering much
resistance to it; the fan serves at the same time as an acoustic screen.
A muffler of this design reduces the aerodynamic noise of an electric fan
by 810 d6.
Builtin Elastic Vibration Absorption. With this technique elastic and
elastic inertial elements, which substantially reduce mechanical and mag
neti c noise, are installed in a machine or outside of it. Local internal
vib ration and noiseinsulation are created as a result of the insertion of
additional elastic elements. These flexible couplers can be placed in the
rotor bearings or built into the bearing and magnetic assemblies.
Elastic magnetic suspension is used extensively in electrical machinery for
reducing magnetic noise [2]. Some versions of elastic suspensions are
shown in Figure 8.19. The efficiency of an elastic suspension increases as
its rigidity decreases. At the same time a reduction of the rigidity of a
suspension can lead to the appearance of intolerable eccentricity between
the rotor and stator, and for this reason deformation limiters must be
 incorporated in the designs.
In spite of the variety of designs, all suspensions share in common the
existence of a certain number of elastic elements (springs) with a
particuldr rigidity coPfficient K for a given type of machine. The sus _
pension designs have elastic elements with different cross sections
(usually made in the form of plates or webs that react to bending) and are
inse rted and fastened in a machine in different ways.
The main aspect of the determination of the parameters of an elastic sus
 pension is to find the natural frequency fn. To achieve the desired effect,
as is known, the natural frequency of a suspension must be different from
the frequency of the perturbing force by a factor of at least 34. The
minimum possible value of n can be determined by the formula f n = 5/a,
where S is the maximum sag of the elastic element of the suspension under
the mass of the stator, which does not exceed 100 of the air space.
To avoid resonance fn must differ considerably from the rotation frequency
of electrical machinery.
The total coefficient of radial rigidity is determined on the basis of the
known frequency fn: K= 47r2mfn, where m is the stator mass.
If an elastic element is broken down conditionally into N separate elements,
each of which represents a web, fastened on both sides, rigidity coeffi
cient K1 of such an element may be assumed to be K/2N. On the other hand,
as is known, the rigidity coefficient of such a web is determined by the
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formula K1 = 12EJ/13, where E is Young's modulus; J= bh3/12 is the moment
of inertia of the web; Z, b, h are the length, width and height of the web,
respectively.
If K1 is known it is possible to select the dimensions of the elastic ele
ment, which satisfy the requirements of structural strength, and also to
determine the effectiveness of the elastic susnension by the formula AL =
2
= 2rN log 1~y NK I, where M is the total weight of the machine; Y=
 = u/(1 + u)2, and u= m/M  m.
Experience shows that elastic suspensions can reduce magnetic vibrations
and noise by 1015 dB.
Une way to reduce acoustic vibration and noise on the rotation frequency,
aF is known, is to use elastic bearings. The resonance frequencies of a
system can be adjusted by changing the rigidity of the bearings, by
inserting elastic elements in them. Internal elastic inertial noise reduc
tion expands opportunities to reduce lowfrequency mechanical components of
electrical machinery noise. Inertial elements (antivibrators) are inserted
in elastic couplings. A builtin elastic inertial noiseinsulating unit
consists in the general case of a top elastic element, intermediate mass
and bottom elastic element. The elastic element of an antivibrator with
inertial mass is fastened to the intermediate mass.
BIBLIOGRAPHY
1. Blagov, E. Ye. and B. Ya. Ivnitskiy, "Drossel'noReguliruyushchaya
Armatura v Energetike" [ThrottleControl Hardware in Power Engineer
ing], Moscow, Energiya, 1974.
2. "Bor'ba s Shumom" [Noise Abatement], edited by Ye. Ya. Yudin, hioscow,
Stroyizdat, 1964.
3. "Vibratsiya Energeticheskikh hiashin. Spravochnoye Posobiye" [Vibration
of Electrical Machinery. A Textboak], edited by N. V. Grigor'yev,
Leningrad, Mashinostroyeniye, 1974.
4. "Kolebaniya i Vibratsii v Porshnevykh Kompressorakh" [Vibrations in
Reciprocating Compressors], Leningrad, Mashinostroyeniye, 1972.
Authors: Yu. A. Vidyakin, G. F. :ondrat'yeva, F. P. Petrova, et al.
5. Duan, N. I. and N. M. Yegorov, "Vibroacoustic Characteristics of Ship
Plumbing," SUDOSTROYENIYE [Shipbuilding], 1962, No 3, pp 1417.
6. Karelin, V. Ya., "Kavitatsionnyye Yavleniya v Tsentrobezhnykh i Osevykh
Nasosakh" [Cavitation Fhenomena in Centrifugal and Axial Pumps], Moscow,
Mashinostroyeniye, 1975.
259
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7. Kurnatov, D. V. and N. A. Stoyanova, "Noise Muffler for Variable
Velocity Flow," SUDOSTROYENIYE, 1972, No 12, pp 3335.
8. Lazaronts, D. F. and P. Bikir, "Shum Elektricheskikh b9ashin i
Transformatorov" [Noise of Electrical blachinery and Transformers],
bioscow, Energiya, 1973.
9. "Raschet i Proyektirovaniye Sistem Truboprovodov. Spravochnoye
Posobiye" [Calculation and Design of Plumbing Systems. A Textbook],
Dloscow, Gostoptekhizdat, 1961.
10. Timoshenko, S. P., "Kolebaniya v Inzhenernom Dele" [Vibrations in
Engineering], hioscow, Nauka, 1967.
11. Shubov, I. G., "Shum i Vibratsiya Elektricheskikh Mashin" [Noise and
Vibration of Electrical Machinery], Leningrad, Energiya, 1974.
A
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CHAPTER 9. REDUCTION OF NOISE OF OPERATING ENGINES AND OTHER SYSTEMS
9.1. Engine Noise in Ship Rooms
The engines of modern ship s are sources of loud noise in many aft
compartments. The following types of engines are considered to be sources
of loud noise: screws, ai r screws, fountain engines and rudder screws.
Each of the mentioned types of engines is characterized by its own noise
features.
Screws. The basic physical causes of noise during the operation flf a screw
are:
periodic forces that occur in water due to hydrodynamic loads on the blades
(the force component of the sound of rotation);
periodic displacement of water due to the corporeality of the screw blades
(the volume component of the sound of rotation);
generation of sound by turbulent flow past the screw blades (turbulent
_ noise) ;
acoustic pulses that accomp any the collapse of cavitation bubbles (cavita
_ tion noise). _
The stream that flows through a screw, for several reasons (the hull lines
of the stern are asymmetri c with respect to the axis of the screw, parts of
the hull protrude in front of the screw, etc.), is inhomogeneous in relation
both to the periphery, and to the radius of the screw disk. This leads to a
periodic change of the elements of the axial and tangential thrust and
pressure against the water. The frequency of change of these parameters is
nz (n is the rotation frec{uency of the screw, in revoluiio_is per second, and
z is the number of blades). Because the blades have finite cross section
dimensions their "force" and "volume" actions upon the water are impul
sive in nature and consequently can have harmonics with a frequency that is
a multiple of nz.
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The "force" and "volume" components of acoustic pressure can be calculated
at the present time if the circulation of the �lola velocity around the
blades is known as a function of radius and rotation angle (Figure 9.1).
In accordance with [8] the force
z component is
R,
~  ~ f P(r', t)D(x,Y)X ~ dr',
~ h. ~ ~ 2n Ri
~ B fo �
I
 ~ I y where x, r, y are the cylindrical
~ e p~~'r'r~ coordinates of a point in space in
the coordinate system that moves
~ Y__ along with the screw; t is time; p is
x the density of tlie liquid; R1 =[x2 +
~
_ ~Up \ + r2 + r'2 2rr' cos(y  tit)]112 is
the distance between the point in
Figure 9.1. Calculation of pulsating space and a point on the screw blade;
pressures. w is the angular velocity of the
screw; D(x, y) = wr' ~ ax is
a differential operator; vp is the axial component of the velocity of the
incident flow, averaged across the screw disk; T(r', t) is the circulation
of the flow velocity around an element of the screw blade, located at
radius r'.
The velocity circulation and the coefficients of its expansion into a
Fourier series with arguments kwt are assumed to be known.
The harmonic components of the force pressure of the screw, operating in a
uniform flow, are determined and tabulated in the literature [23]. The
components of pressure on a frequency that is a multiple of nz are
Pk = nR~ n~ Ak sIn ka dkcos kal , (9. 1.2)
~
w}iere P is screw thrust; ap = vp/nD is the relative approachability of the
screw; RD is screw radius; :y.i; the angle between a blade and a line to a
calculated point; K and Bk are the harmonic components of pressure; k=
= mz, where m= 1, 2, 3,
The coefficients for several first harmonics, camputed for several values of
x/R0 and r/R0, are listed in Tables 9.1 and 9.2 (x is tsle distance between
the center of the screw along its axis in the direction of the flow, and r
is the distance from the axis to a calculated point).
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Table 9.1. Expansion Coefficients Ak
x/Ro i //Ro I
As I
A. I
A, I
A, I
Ai
I Aie 
1,2
0,381
0,285
0,216
0,164
0,095
0,055
0
l,3
0,267
0,185
0,130
0,093
0,098
0,024
.
1,4
0,192
0,123
0,081
0,053
0,024
0,011
1,2
0,362
0,266
0,200
0,149
0,084
0,064
0,1
1,3
0,267
0,175
0, l22
0,086
0,043
0,031
1,4
0,186
0,118
0,077
0,050
0,022
0;015
1,2
0,313
0,220
0,158
0,114
0,060
0,031
0,2
1,3
0,230
0,152
0,102
0,069
0,033
0,015
,
1,4
0,171
0,105
0 ,066
0,042
0,018
0,008
Table 9.2. Expansion Coefficients Bk
x/Ro I
r/Ro I
Be I
B4 I
Bs I
Be
I Ba
I 8i�
1
0
JIroGoe
I
I 0
I 0
I 0 I
0 I
0 I
0
1,2
01085
O,OG5
0,049
0,037
0,021
0,011
0,1
1,3
0,045
0,033
0,023
O,Ol7
0,008
0,006
1,4
0,025.
0,017
0,011
0,008
0,004
0,002
1,2
0,128
0,094
0,069
0,051
0,027
0,019
0,2
1,3
0,073
0,051
0,035
0,025
0,012
0,008
1,4
0,043
0,028
0,018
0,012
0,005
0,003
Key: 1. Arbitrary
Example. A fivebladed tanker screw with n= 110 rpm produces 156 ton
thrust. Screw radius is 3.4 m. The pressure in the vicinity of the screw,
according to (9.1.2), should be n4< nR~ v ( n� A,) d {ak. For the funda
mental harmonic at a distance of 0.7 m from the screw disk and 4.5 m from
the screw axis A 5 = 0.102, B6 = 0.035, A10 = 0.015, B10 = 0.006.
At a sPeed of 18 knots the relative a roach is 18�0,515�60 _ 0,74.
PP P~ 110�6,8
The computations yield p5 = 1.82�103 Pa; plo = 2.93�102 Pa.
Due to the inhomogeneity of the flow these values increase by a factor of
34 [8]. In view of this, by converting to acoustic pressure levels we
3
obtain: on 9.2 Hz L= 20 log 1'822~i~_~ = 170 dB; on 18.4 Hz
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2
L= 20 log 2'23 4= 155 dB. As the example shows, the frequency of the
strongest components of rotation noise is in the infrasonic region (this is
typical of most screws). The amplitudes of HF (acoustic) harmonics
diminish rapidly as frequency increases. The fall is approximately 15 dB
per octave.
For real screws the amplitudes of the harmonics of the volume component of
rotation sound, the frequencies of which fall in the audio frequency range,
are vastly smaller than the amplitudes of the iiarmonics of the force
component. Therefore the former may be ignored during calculation of
acoustic pressures.
hleasurements of noise and acoustic vibration in the aft compartments show
that the noise spectrum of noncavitating screws is mixed, even in the low
frequency region; in addition to audio components there is noise with a
continuous spectrum. The flow velocity relative to the screw blades reaches
3050 m/s across much of the radius. The Reynolds numbe: for a flow around
blade elements becomes 107108. In this case the boundai�y layer of the
linuid, flowing past the blades, is an emitter of sound. The spectral
density of the acoustic pressure can be approximated thi~ough the
relation [8]
5s,s c wsi
P (f) = u.11 fe3's(fg*/Ii)o,szsj ' (9.1.3)
 where TW is the tangential force of friction on the blade surface; S* is
the thickness of the displaced turbulent boundary layer; uoo is flow velo
city.
The characteristics of flow past a plane surface may be used for estimating
the order of magnitude of pressure [8]:
Tw 0,1315nv 1/7ic3~7l~~?;
S* = 0,020G1�Rei 1/7, (9.1.4)
where ReZ is the Reynolds number, ReZ = vu (Z is profile length); u is flow
velocity; v is the kinematic viscosity, for water at 20�C v= 1.00106 m2/s.
Example. In the preceding example the circumferential velocity in the blade
cross section with R= 0.85R0 is u= 31 m/s. The blade width in this cross
section is Z= 1.0 m, and the Reynolds number is
30�1,0
3~10'.
Ret 1,0�106
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Substituting the necessary values into formulas (9.1.4), we obtain T=
w
= 1,000 Pa; 5* = 1.76�103 m. As u result of substitution of these values
into formula (9.1.3) and averaging in the octave bands we find the acoustic
 pressure levels in the 145155 dB range in the 1001,000 Hz frequency range.
Cavitating screws produce the strongest noise in rooms. The cause of the
 noise is the collapse of cavitation bubbles, which occur near the blade
surface at points where the pressure drops to some critical value and water
loses solidity. The popping of bubbles is accompanied by an acoustic pulse.
The intensity of cavitation noise depends on the number of bubbles and
their diameter, on the time intervals between the formation of bubbles and
. on the concentration of gas in water. All these factors are random, and
therefore cavitation noise should be viewed as a random pulse process [8].
1'he size of the cavitation volume is determined by the geometry of the
blades and by the velocity field in the screw disk. The inhomogeneity of
the velocity field in the screw disk causes a periodic change of the attack
, angle of any cross section of the blade (Figure 9.2), which leads to modu
lation of cavitation noise by frequency nz.
ne'3'je H�~
ob
0oabe
t
i~�~ ~Ui~ ~`al ~
1 vA ~
r�cu
. ~
. ~ ~
.
Figure 9.2. Diagram of velocities on blade.
Key: 1. Zero thrust line
Under otherwise identical conditions the inhomogeneity of the flow in the
screw disk amplifies cavitation and increases cavitation noisP.
Cavitation occurs not only on the screw blades, but also on certain protruding
parts of the hull (struts), located in front of the screw. As a result a
flow that represents a mixture of water and cavitation bubbles flows around
the blades at certain rotation angles. This phenomenon exacerbates even
more the calculation of cavitation noise.
If the acoustic pressure near a screw is known the acoustic vibration
levels of the part of the exterior hull above the screw can be estimated,
and then the noise levels in aft compartments can be calculated, using the
described part of the exterior llull as a reference bulkhead (see Chapter 15).
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The outer hull plating is located on the airwater interface. It can be
shown [16] that, excluding cases of very great thicknesses, the outer hull
plating may be viewed as a locally reacting surface, the resistance of
which is determined by impedance iwm, which does not depend on the angle of
, incidence (m is mass per unit area). Then the following relation will
exist between the acoustic pressure levels under the hull and the vibration
_ levels of the hull:
N, _ L 201g (1, 5� 108 i(om) 49, ( 9. 1. 5)
where Nv is the vibration velocity of the hull (relative to v0 = 5�108 m/s),
in dB; L is the acoustic pressure level under the outer hull, in dB; m is
mass per unit area of the outer hull plating, in kg/m2.
If the acoustic pressure levels are not known the vibration levels of the
outer hull Nv, in dB, located above the screw, operating in the strong
cavitation mode, can be estimated through the empirical formula
Nv = 54 Ig n Ro 0' 0" ( l a) E A,". ( 4. 1. 6)
where n is the rotation frequency, in revolutions per second; R0 is screw
radius, in m; A' is a correction factor that takes into account the fre
quency characteristic of the interaction of the cavitating screw and the
hull, in dB; A" is a cnrrection factor that takes into account the distance
between the screws and the hull and the angle of inclination of the screw
shaft, in dB; a is a coefficient that ta;ces into account the influence of
velocity with the screw and hull in different relative positions; 0"' is a
corrertion factor that takes into account the mass of the outer hull
plating, in dB.
Correction factor A' acquires the following values as a function of fre
Quen cy :
ru 63 125 250 500 1000 2000 4000 8000
~ A', Afi 54 50 47 40 32 28 15 10
Correction factor 0" and coefficient a are determined by the graphs in
Figures 9.3 and 9.4 as functions of the ratio r/R0 (r is the distance
between the center of the screw and the outer hull, and R0 is screw
radius), angle of inclination of the screw shaft and speed of the ship.
'I'1ie correction factor A"' is
3f
A 201g I, 1,6 � 10
1 14,4�10'4ut/ ' (9.1.7)
where m is the mass of the outer lining per unit area, in kg/m2.
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Table 9.3. Calculation of Outer Hull Vibration Levels
1 ~ 11
2) tfacTOrw oKTanttwr nonoc, I'u
�
apamcrp
~
63
I 125 I
250
I 500
I 1000
I 2000
1 4000
I 8000
/i  5919n Ro
48
48
48
98
48
48
48
48
nt
54
50
47
40
32
26
15
10
=e"(I 
>
i
~
i
i
t
>
>
16=A""
O
O
O
O
O
O
O
O
NU  11r 12 1 I3 Iq
103
99
96
89
81
75
64
59
Key: 1. Parameter 2. FreQuencies of octave bands, in Hz
Example. Calculate the vibration of a 10 mm thick steel hull plating,
located above a screw with a radius of 2,0 m(Figure 9.5). The rotation
frequency of the screw is n= 4 rev/s, the speed of the ship is 12 m/s, the
angle of inclination of the screw shaft is 3�, and the ratio r/R0 = 1.3
(Table 9.3).
The vibration levels in the last row of Table 9.3 for a bulkhead, located
above the screws, are used as the initial data for calculating the noise
levels in aft compartments. For example, determine the noise levels in room
III. The calculation can be done graphoanalytically (see Chapter 15), iVe
number the bulkheads; we call the bulkhead located immediately above the 
screw number 0. Using the formulas and nomograms in g15.5, we calculate the
vibration levels of the bulkheads that enclose room II, after which we
determine the noise levels, radiated by those bulkheads, and the total noise
levels in room II. The noise levels in room II are used as the initial data
_ for calculating the noise levels in examined room III by the procedure used
for distant compartments. The calculations yield the following levels of
the spectral components of noise in the octave frequency bands:
[ FL( =H Z; I'u 63 125 250 500 1000 2000 4000 8000
AE.,=dB ] L,Ab 100 95 88 80 76 71 60 50
 Air Screws. Air screws are one of the main sources of noise for ships that
utilize dynamic lift principles. Noise is produced by the periodic force
_ of the blades, acting upon a medium, and by the displacement of air masses
(the "force" and "volume" components of fan noise), and by the separation of
eddies from the trailing edges of the blades (eddy noise). A characteristic
feature of modern air screws is the fact that the frequencies of the first
harmonics of fan noise are in the audio frequency range, and not in the
infrasonic range, as in the case of screws.
To calculate the amplitude of acoustic pressure in the far field of an air
screw on frequency f= mn it is customary to use L. Ya. Gutin's formula
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n ;'Ob




~
rIRO~~t
~2




~.g
~1,6

j
0 1 2 3 4 5 6?1'
Figure 9.3. Correction factor A" as
function of angle of inclination of
screw shaft.
0,2
0 
_02J 6 9 17 15 18 u,M/c
Figure 9.4. Correction factor a as
function of ship speed.
3)nnaH r+umeu nonub
1/ 2 !If J
q 7 7. ? J
i 'i j j
1 1 1
1 1
Figure 9.5. Calculation of noise level.
Key: 1. Upper deck 3. Lower deck plan
2. Lower deck
mnz c 2nmt1z
[3=e ] P(f, r, O)  cor  P cos 9~ ~Li 2nnR3 Jn'z C r R3 sin 0~ , (9 .1. 8)
0
where m is the number of harmonic; n is the rotation frequency of the air
screw, in rev/s; z is the number af blades; r is the distance between the
center of the screw and the calculated point; c0 is the speed of sound in
air; P and M are, respectively, the thrust and torque of the screw; Re =
= 0.8R0 (Rp is screw radius); JmZ is a firstkind Bessel function; A is the
angle between a line to the calculated point and the screw axis (the axis
is aimed in the direction of travel of the screw).
Tfie results of calculation by formula (9.1.8) agree satisfactorily with the
experimental data for screws with peripheral velocities less than the speed
of sound. I; the case of high harmonics and near and supersonic circum
ferential velocities the calculated and measured acoustic, pressures differ
zbs
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substantially. The circumferential velocities of air screws use d on ships
sometimes reach (0.91.1)M (M is the btach number). In this case preference
should be given to initial data obtained from stand tests for ca lculating
noise.
Rudder Systems (RS). RS sometimes produce unacceptably highlevel noise in
nearby compartments. The total noise ievels at a distance of 1 m from
modern RS are 111110 dB, and vibration levels near RS reach 90 95 dB. RS
with an adjustable pitch screw are used most frequently in Soviet ship
building. The screw is usually driven by an electric motor with a constant
rotation frequency through an articulated reducer. The reducer and blade
pitch control mechanism are installed in a watertight gondola, which is
located in the RS shaft.
Narrowband analysis of RS noise and vibration and comparison of the
acoustic characteristics in different operating modes lead to the concl.usion
that noise and vibration on frequencies of up to 400500 Hz are produced in
approximately equal measure by the interaction of the gears of the articu
lated reducer and operation of the screw. On frequencies above 500 Hz
_ screw cavitation noise predominates. The results of acoustic te st stand
measurements at the shipyards may be used as the initial data for forecast
' ing the noise levels in ship compartments. A comparison of the results of
measurements on ships and on stands disclosed certain discrepancies in the
frequency characteristics (up to ? dB in individual bands) of noise and
vibrations and under natural and test stand conditions, but the integral
levels are virtually the same.
9.2. blethods of Reducing Screw and Other Engine Noise
There are two ways to reduce the noise produced by engines in ship compart
ments: reduce the acoustic pressure on the hull by cllanging engi ne operating
conditions; reduce hull vibration and the transmission of vibrati on to ship
compartments without changing engine operating mode.
'Che acoustic pressure under the hull is reduced by decreasing the inhomo
geneity of the flow, the maximum distance between the screw and hull and
screw shaft struts, and by reducing the downwash of the flow.
One way to reduce flow inhomogeneity is to secure the screw shaft without
struts, as recommended by the Leningrad Institute of Water Transp ort (LIVT)
for small ships. The reduction o� the vibration level of the outer hull
AN, in dB, for a cantilever screw (in comparison with one fastene d to a
strut) is characterized by the following figures:
[rq=11z; j, I'q 63 125 250 500 1000 2000 4000 8000
,r,D=dB] AN, nr 9 7 ? 6 5 4 3 3
Several techniques have been proposed for reducing acoustic vibrations of
the outer hull near the screws. Some of them utilize N. N. Babayev's
269
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concept [8], according to which the section of the bottom immediately
above the screw is replaced by a flexible element, which receives the
screw pressure and reduces the transmission of acoustic vibrat.ion to hu11
structures [11, 19].
The pressure may be absorbed by elastic skin, placed over the hull in a
recess (Figure 9.6), or by a membrane, covering a slot in the bottom
(Figure 9.7). Deep wells with a springloaded piston inside may ?lso be
used (Figure 9.8). High pressure is picked up by the piston, but vibration
is not transmitted to the hull due to the vibrationinsulating fastening
of the piston. The efficiency of these systems is 1012 dB in the entire
audio frequency band.
The mass of the part of the bottom directly over the screw can be increased
to reduce the vibration of the outer plating. To do this the bottom is
covered with concrete, bitumen or other massive material that is resistant
to vibration.
~
Figure 9.6. Vibrationdamping elas
tic skin: 1 outer skin; 2 Figure 9.7. Rubber membrane: 1
 elastic layer; 3 perforated skin. membrane; 2 keel.
\ I /
/
Figure 9.8. Spriiibloaded piston: 1 piston;
2 elastic element; 3. seal.
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The area over the screw to h, covered should be (1.21.4)D2, where D is
screw diameter. The average mass of the skin per unit area should be 45
times greater than the mass of the bottom in the vicinit y of the screw.
The thickness of the cover should decrease from the center to the periphery
by a ratio of 2.02.5. The use of massive coatings reduces noise in aft
compartments by 46 dB [4].
An air shroud may be used for reducing pressure pulsations picked up by the
outer skinl. Air is forced through several slots or holes in the outer
skin in front of the screw to create a shroud. It is entrained by the flow
toward the stern, forming a layer consisting of a mixture of air and water.
The acoustic impedance of this layer is much less than the impedance of
water. This produces a substantial soundinsulating effect of 810 dB on
low and intermediate frequencies of the audio frequency band.
Ordinary design measures may be used in addition to the abovedescribed
specific techniques for reducing screw noise: "room in room" sounding insu
lation, vibrationabsorbing coatings, vibration insulation of bulkhead
~ joints and other methods of reducing structural noise, like the ones
examined below.
According to the literature [1], the circumferential velocity of the blades
should be reduced and the number of blades increased in order to reduce the
noise of an air screw witil a given thrust. However, this recommendation
refers only to screws with subsonic velocities at the ends of the blades.
One important way to reduce noise is to use synchrophased screws. The
rotation freque~:cies of all screws are maintained equal to a high precision
with the aid of" a special system, and the blades of each screw are held in
a certain position in relation to the blades of the others. According to
the literature [1], the noise levels at individual points can be reduced by `
57 dB by means of synchrophasing. Lowfrequency noise can be reduced sub
stantially by placing the screws in an annular shroud. The effect, pro _
duced primarily by relieving the stress on the blades, is a reduction of
58 dB.
The noise of rudder systems can be reduced both by general design techniques
and by special measures. General d�sign techniques include increasing the
rigidity of the RS shaft and fastening the shaft to just the bottom bulk
head and sides without a rigid conneciion to the deck above the RS. T'he 
bottom in the vicinity of the foundation of RS should be poured with con 
crete, bitumen or other massive and vibrationresistant material for a
distance of five or six frame spaces, just as is recommended for reducing
screw noise.
1Satt att dampa buller fran fartygspropellrar (Method of Reducing Screw
Noise), Swedish Patent No. 322705, dated 13 April 1970, B63 No. 21/30.
 271 ~
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Another effective way to reduce noise is to line the inside of the rudder
system shaft with an elastic material [3]. hlaterials whose wave impedances
are much less than the wave impedance of water are used for this purpose:
sponge rubber and polyvinyl. chloride foam. To prevent separation of the
liner from the shaft the elastic material is covered witii thin sheet meta'.
The thickness af the elastic layer in constructions that are used in
practice is 1015 mm. The acouStic vibration levels of the rudder system
are rPduced by 1520 dB as a result of lining. The reduction of air noise
is less and is 810 dB.
9.3. Noise Interference ori Bridge and 111ays to Reduce Itl
Statistical Characteristics of Noise Interference and Acceptable Standards.
The types of noise interference on the bridge are noise levels in the wheel
house and naviga:or room and on the bri dge win gs. Noise interference on
the bridge should be viewed basically from the standpoint of navigation
safety, i.e., audibility of sound signals from oncoming ships, intelligi
_ bility of voice commands, etc. In accordance with the plan of CEMA
standards, the PS55 curve is used as the acceptable standard in rooms of
the bridga. However, this standard, as will be shown, does not assure
, audibility of acoustic signals from oncoming ships at certain distances.
The existing health standards call for the use of the PS40 limiting noise
spectrum for the bridge. The actual noise levels in the wheelhouse and
_ navigator room, and also on the bridge wings of seagoing transport ships,
based on TsNIIMF [Central Scientific Research Institute of the Maritime
Fleet] data, are given in Figure 9.9. Curves of the probability distribu
tion of acoustic pressure levels, on the b asis of which noise levels can be
 calculated, are given in Figure 9.10 for the same rooms. The data were
obtained as the result of ineasurements on 75 seagoing transport ships of
different types. The M curves in Figure 9.9 co rrespond to the universal
mean. As can be seen by the curves, the greatest excess above the PS45
curve occurs in the 5002,000 Hz octave bands.
The required reduction of noise in the wheelhouse and navigator room,
according to PS45, is 610 dB for 25 0 of the ships, 1317 dB for 50%, and
2025 dB for the remaining 25%.
Audibility of Acoustic Signals. The new International Rules on Collision
 Avoidance of Ships at Sea (MPPSS72), approved by the International Con 
vention (IMKO) and adopted on 1 January 1976, stipulate: the range of
_ audibility of acoustic signals, their frequency spectrum and acoustic pres 
sure levels, depending on the size of ships (Table 9.4). The Rules of the
USSR Register specify that acoustic signals should be audible at a distance
of 2 miles, irrespective of the size of ships.
 1Tiiis section was written by V. I. Zinchenko and L. A. Tatsi.
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03
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8000Tu 4000 2000 1000 500250125 61 df
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8000 ru 4000 2060 fOCO 500250 925 61 Jf
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841~1 ru 4000 200 f000 500 250 125 31
6J
6d
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Ju yu bU 60 70 80 .90 100 190 L;B6
Figure 9.10. Curves of probability distribution of
 acoustic pressure levels, P%, in bridge rooms: a in
wheelhouse; b in navigator room; c on bridge wing.
[36=dB; Fu,=Hz]
The audibility of a whistle depends on its acoustic pressure level,
attenuation in the atmosphexe and level of noise interference on the
receiving end. The receiving end is usually the wing of the bridge and
wheelhouse. The overall levels and frequencies of the fundamental tone of
 certain modErn signal systems, used on ships, are listed in Table 9.5.
According to MPPSS72 the acoustic level of a ship~s own signal at listen
ing stations should not exceed 110 dBA, and if possible not more than
100 dBA. In most cases a ship's whistle may be installed not farther than
2030 m from the bridge. At that distance the acoustic pressure level
will be 2530 dB lower than at a distance of 1 m. Therefore, in accordance
with MPPSS72 requirements, the acoustic signal level at 1 m should, if
possible, not exceed 130 dBA, and levels above 140 dBA are unacceptable.
On the basis of e xperimental investigations the optimum frequency band for
acoustic signals is 180700 Hz � 10%. T'herefore t3ie noise interference
level at listeni ng stations on board a ship in two octave frequency bands:
250 and 500 Hz, which cover the 177709 fIz range, is extremely important.
274
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Table 9.4. MPPSS 72 Requirements on Acoustic Signal Systems of
Seagoing Ships
Acoustic pressure
Length of
level in third
Recommended
Audibility rangi
ship, m
octavebands at
fundamental
in nautical
1 m from whistle,
frequency of
miles
in dB
whistle, Hz
200 and more
143
70200
2.0
75 and more,
but less than
200
138
130350
1.5
20 and more,
but less than
75
130
250700
1.0
Less than 20
120
0.5
The spectrum of a whistle is ruled (discrete) in character. The maximum
level of the frequency components depends on the design of the whistle and
on how it is tuned; it does not always appear on the fundamental frequency
(Figure 9.11).
G,aE
12!
19G
10l
9G
8G
70
6 7 8 9 f0 2 2 3 ~ 5 6 7 8 9 f03
.P, ru
Figure 9.11. Sarnple spectra of acoustic pressure levels
of ship foghorns. [a6=dB; c=s; fu=Hz]
Key: 1. (motor sliip "Sergey Eyzenshteyn")
Calculation of Audibilit y oi Acoustic Signals. According to the standards
of the acoustic signal committee of the International Association of
Beacon Services (IABS), the audibility range should be determined as a
function of the noise interference level inherent to 84o of the large
merchant ships [7].
275
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Table 9.5. Acoustic Characteristics of Certain Modern blarine
Acoustic Signal Systems [12]
M apha
oeLu~~n ypoec+b
2~ H8 pBCCTOAHfIN, M
3~~iactora
ocxoniioro
4)
C7 8H8�N3
o
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, = i
roUe, ru
ror
axTenb
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5) B03J(YllIH610 TNOHbI
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TB ] 00/ 1 SO
124 144
180
CCCP
T B 100/ 165
126 146
165
CCCP 6)
TB 15fl/90C
130 150
90
CCCP
IOOEA
15
EA
124 144
' 200
SInoNHA
0
 127 147
110
SlnoxNA
TA 100l200
124 144
200
IllaeueA
KT 150/165
KT 150/130
125 145
127 147
165
130
1IIneuxa�
IIleeuxA 8)
KT 230/75
131133 151153
75
W824NA
9) 3sexTpoTxcpoxa
T34
T3
2
'
125
145
110
CCCP 6)

126
196
90
CCCP
MH75
2
123
143
140
SInottHa 7)
T1H�2
0
128
148
113
SIrtoxxa
1tA 18/75
131135
151153
75
IlIeeueA
A1A 18/90
127131
147151
90
IlIeeuNA 8)
MA 18/130
123127
143147
130
WB24HA
10
)ilapoaM
e CSHCTK
N
230/75
131133
151153
75
IIIBeqNA
T300ME
129
f49
125
IlIaeui+A
T20
,
128
148
165
llleeuxA
Key: 1. Brand
6. USSR
2. Total level
at dis
7. Japan
tance, in m
8. Sweden
3. Fundamental
frequency,
9. Electric
fogho
rns
in Hz
10. Steam wh
istles
4. Manufacturer
S. Air foghorns
The results of ineasurements of the sound levels on the bridges of ships
with probabilities of 84% and 900, based on data of TsNIIMF [Central
Scientific Research Institute of the Maritime Fleet], IMKOSweden and the
IABS organization, are presented in Table 9.6. The data were obtained on
the basis of numerous measurements, and their results agree satisfactorily
in the 250 and 500 Hz octave ')ands. The interference levels in the wheel
house in the 250 and 500 Hz octave bands may be assumed, on the average, to
be 75 and 70 dB, respectively, and 83 and 76 dB on the bridge wings. In
thirdoctave bands, in consideration of the manner in which the spectra
fall off, the levels are 3 dB lower.
Curves that show the useful signal level that will be heard on the bridge
with a probability of SOo at the corresponding interference level in the
276
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4)
5)
6)
Table 9.6. Noise interference Levels on Bridges of Sea
going Ships, in dB [7]
MTO'IIINK :iII:WX
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Table 9.8. Necessary Signal Level, in dB, on Reception
End for Audibility with 90% Probability
Noise interfer
T e o
f fo horn
ence level
150EA
TA 100/130
TA 100/200
TV 150/90s
Actual on 840
of ships:
wheelhouse
85
85
74
68
bridge wings
95
95
86
76
Not more than
PS45
77
77
62
49
Not more than
PS55
81
81
71
58
The minimum necessary acoustic signal levels, audible with 90% probability
on a background of actual noise interference at listening stations, and on
a background of noise interference that does not exceed the PS45PS55
limiting spectra, are given in Table 9.8. A comp arison of the data in
Tables 9.8 and 9.7 shows that before acoustic signals can be audible in
accordance with the requirements of MPPSS and the Rules of the USSR
Register, the noise interference levels must not exceed the PS45 spectra.
Intelligibility of Speech. Interference noise levels on the bridge exert
a strong influence on the intelligibility of commands and instructiuns.
_ The mean interference noise levels in four octave frequency bands 500,
1,000, 2,000 and 4,000 Hz, are important for determining the intelligibility
of speech. 1'he maximum distances at which normal speech is assumed to be
satisfactorily intelligible (95% intelligibility, which corresponds to the
discrimina',ion of approximately SO% of the logatoms, or 900 of the words
[15, 18]), and the maximum distances at which rather loud speech can be
understood satisfactorily (shouting is excluded), are given in Table 9.9
as functions of the interference level.
On ships with the interference level indicated in Table 9.6, normal speech
is satisfactorily intelligible in wheelhouses at a distance of not farther
than 0.5 m, and on the bridge wings not farther than 0.15 m, and loud
voice not more than 0.85 m and 0.20 m, respectively. Before speech can be
satisfactorily intelligible when commands are conveyed by a louu voice from
a distance of 78 m, i.e., from a distance common for medium ships, between
the helmsman, standing at the wheel, and the navigator standing on the
bridge, the interference level must not exceed 4045 dB on frequencies of
500, 1,000, 2,000, 4,000 Hz, which corresponds approximately to PS45.
Ways to Reduce Noise Interference. The techniques for reducing engine and
_ boiler room ventilation and exhaust noise, the noise of air conditioning
systems and structural noise from screws and engines are examined in the
corresponding sections of this handbook. They consis� in the installation
of effective noise mufflers in the exhaust and intake of the ventilation
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1 Table 9.9. bfaximum Distances, m, at iVhich Speech is
Considered Satisfactorily Intelligible
_ Speech interference Kind of s eech
level*, dB normal voice loud voice
_ 35 7.5 11.5
40 4.5 8.4
45 2.3 4.6
50 1.3 2.6
55 0.75 ]..5
60 0.42 0.85
65 0.23 0.56
70 0.13 0.20
*Deternined on the basis of test results
system, vibration insulation of the exhaust stacks of diesel engines,
separation of the air intakes of the ventilation system and room exhausts
from the bridge by the maximum possible distance. To reduce structural
noise the wheelhouse and navigator room are installed on shock absorbers,
and other acoustic insulation techniques are used.
The strongest noise interference in the wheelhouse and navigator room on
frequencies above 250 Hz is produced by electronic and radio navigation
gear, in cluding the gyrocompass with repeaters or_ the bridge wings, course
recorder, automatic steeri.ng system, radar, radio direction finder, echo
sounder, log, windshield wipers, etc.
Consoles that include clusters of control instruments of a ship and its
power plant are installed in modern vessels. The results of ineasurement of
the noises of several navigation instr+.iments, most widely used on ships,
are presented in Figure 9.17; the avarage values of these instruments,
based on measurements on 1012 shirs, are presented. The noise level of
instruments of the same brand may vary in a wide range. The noise sources
of most navigation instrumeiits are contactless micromotors, principally
selsyn motors. The noise level of a contactless selsyn is not stable and
_ depends on the rotation angle of its rotor and varies strongly from
machine to machine. Particularly noisy selsyns account for about 200 of
the selsyns used in marine instruments. Cases when one noisy selsyn
exceeds the noise level in the wheelhouse or on the deck by 1015 dB are
not uncommon [2].
The choice of the right bearings for the friction moment is especially
important. The highest class of bearings, for example classes a and AB
with not more thsn 10micron radial play, and with subsequent spinning and
breakin of the bearings for 212 hours, should be used for selsyns.
Alternating electromagnetic forces are a direct cause of displacement of
the rotor in bearings. Therefore the micromotors must be carefully
adjusted in assembled form in the final analysis in ordcr to reduce the
noise in navigation instruments.
_ 'l84
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a)
C)
95
m

90
\
BG
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70
O
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so
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30
b)



,
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OO
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90
BO
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70
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o
o
50
1
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JO
J1 125 500 2600 8000 A,
93
90
80
70
60
50
40
JO
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d) 90
80
70
60
50
40
iD
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i J9 175 S00 2007 8000 rq
31 125 500 200 BOJO ry
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80
70
60
SO
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8~\

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_ Figure 9.17. Noise spectra of certain navigation instru
ments used on ships [12]: a indicator of NELS echo
sounder; b NELS recording echo sounder; c course
recorder (34A instrument); d log repeater (5D instru
ment); e type AR automatic steer.ing system.
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j  ON ' Ml
BY
3 JUL'r' 1980 1.1. KLYUKI N AND I . I . BOGOLEPOV 4 OF 6
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Vibrations are transmitted through the chassis and are amplified by the
case of an instrument. Therefore vibration insulation of the chassis from
the case is an effective way to reduce the noise of navigation instruments
at the source. Vibration insulation can be accomplished with shock
absorbers. Combining instruments ir.i consoles is quite effective on ships
if the instruments are acoustically insulated by the console cabinet, and
indicator instruments are tightly sealed by a soundinsulating transparent
 windoiv. The noise of navigation instruments can be reduced by 2530 dB
by using appropriately the measures explained above.
I'he windshield wiper is an instrument of the wheelhouse that is rarely
turned on. However, it is turnPd on, as a rule, during poor weather and
poor visibility, i.e., at times when special quiet is necessaxy for hearing
signals. Therefore its noise should not exceed the level of interference
on the bridge.
Noisy cooling fans are used in electronic and radio navigation gear. When
setting up these instruments mufflers should be installed on the suction
end discharge sides and quiet fans should be used.
Noise control at the source sliould include perfected electromechanical
systems and the development of fundamen�ally new devices on semiconductors
and integrated circuits.
BIBLIOGRAPHY
1. "Aviatsionnaya Akustika" [Aviation Acoustics], edited by A. G. Munin
and V. Ye. Kvitka, Moscow, btashinostroyeniye, 1973.
2. Grigor'yan, F. Ye., "Shum Sudovykh Navigatsionnykh Priborc;v" [Noise
of Nlarine Navigation Instruments], Leningrad, Transport, 1973.
(TRUDY TsNIIMF [Proceedings of the Central Scientific Research Insti
tute of the Niaritime Fleet], No. 171).
3. Daletskiy, K. P. and I. I. Klyukin, "Reduction of Screw Noise in Ship
Rooms," TRUDY LKI [Proceedings of Leningrad Shipbuilding Institute],
 1972, No 77, pp 1116.
4. Zefirov, L. B., "Experience in Noise Control on 'Volgodon' Type
 Ships," SUDOSTROYENIYE [Shipbuilding], 1967, No 2, pp 58.
5. Klyukin, I. I., "Bor'ba s Shumom i Zvukovoy Vibratsiyey na Sudakh"
[Noise and Acoustic Vibration Control on Ships], Leningrad,
Sudostroyeniye, 1971.
u
6. Kovrigin, S. D., "Bor'ba s Shumami v Grazhdanskikh Zdaniyakh (Udarnyye
i Strukturnyye Shumy)" [Noise Control in Civilian Buildings (Impact
and Structural Noises), Nloscow, Gosstroyizdat, 1969.
286
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7. Kovchegov, L. P. and N. A. Kubachev, "Vliyaniye Shumovykh Pomekh na
Dal'nost' Slyshimosti Zvukovykh Signalov" [Influence of Noise Intcr
ference on Audibility Range of Acoustic Signals], Moscow, Izdvo
Reklambyuro hiMIF, 1975 ("Sbornik Trudov 'Sudovozhdeniye"' [Symposium
 "Ship Navigation], No 16, pp 7885).
8. b4iniovich, I. Ya., A. D. Pernik and V. S. Petrovskiy, "Gidrodinami
cheskiye Istochniki Shuma" [Hydrodynamic Sources of Noise], Leningrad,
Sudostroyeniye, 1972.
9. Oguro Hideo, "Audibility of Ship Acoustic Signals," SEMPAKU, 1968,
Vol 12, No 41, pp 5864 (in Japanese).
10. Osipov, G. L., "Zashchita Zdaniy ot Shuma" [Protecting Buildings from
Noise], Moscow, Stroyizdat, 1972.
11. Panenko, S. rl., "Features of Noise Control on Dredges 'Chernoye More' 
and 'Baltiyskoye More'," SUDOSTROYENIYE, 1967, No 2, pp 811.
12. Prokhorov, Ye. S. and I. I. Epshteyn, "Zvukovaya Signalizatsiya Sudov
na Rekakh i Vodokhranilishchakh" [Acoustic Signaling of Ships on
Rivers and Reservoirs], Gor'kiy, GIVT, 1973.
13. "Sanitarnyye Normy i Pravila po Oranicheniyu Shuma na Territoriyakh i
v Pomeshcheniyakh Proizvodstvennykh Predpriyatiy" [Sanitation Rules on
Limitation of Noise on Land and in Rooms of Industrial Enterprises], .
No. 78569, Moscow, Minzdrav, 1970.
14. "Sredstva Zvukoizolyatsii i Zvukopogloshcheniya v Sudovykh
 Pomeshcheniyakh" [Methods of Acoustic Insulation and Acoustic Absorp
tion in Ship Compartments], LIVT, 1967.
15. French, N. R. and J. K. Steinberg, "Factors That Influence Speech
Intelligibility," JASA, 1949, iJo 19, pp 90119.
16. Shenderov, Ye. L., "Volnovyye Zadachi Gidroakustiki" [Wave Problems in
Hydroacoustics], Leningrad, Sudostroyeniye, 1972.
17. Yaskevich, A. P. and Yu. G. Zubarov, "Novyye Mezhdunarodnyye Pravila
_ Preduprezhdeniya Stolknoveniy Sudov v More (MPPSS)" [New International
Rules on Ship Collision Avoidance at Sea (MPPSS)], Moscow, Transport,
1975.
18. Beranek, L. L., "Noise and Vibration Control, New York, 1971.
19. Geicke, K., "Wirkung schallweicher Beschichtungen im Propellerbereich
auf die Einleitung von Schwingungen in der Ausenhaut," SCHIFF UND 
HAFEN, 1975, Vo1 27, No 3, pp 222223.
287
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20. Harris, A4., "Handbook of Noise Control," New YorkTorontoLondon,
1957. 1.
21. Hideo Ogura arid bfasayuki Tsuchiya, "Study of Acoustic Signal Propaga
tion over Ocean {Vater," Report of Ship Research Institute, 1973, 10,
 1, pp 117. 
22. Ort, A., "Korperschalldammung von elastisch gelagerten Propeller
Brunnen," SCHIFF UND 1IaFEN, 1975, Vol 27, No 3, pp 223224.
23. Pohl, K. H., "Das Instationare Druckfeld in der Umgebung eines Schiff
spropellers und die ihm benachbarten Platten erzeugten periodischen
Kraften," SCHIFFSTECHNIK, 1959, 32, pp 107116.
24. Winer, F. and D. Keat, "Experimental Study of Propagation of Sound
over Ground," JASA, 1959, Vol 31, No 6, p 724.
288
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A
CHAPTER 10. ACOUSTIC INSULATION ON SHIPS
10.1. Importance of Acoustic Insulation on Ships
Acoustic insulation is one of the most effective ways o.f reducing noise
produced by the propagation of acoustic vibrations from one room to another
(Figure 10.1). With acoustic insulatior it is possible to reduce the
transmission of noise through paths 1, 2 and 3 from room I to room II.
Bypass 4 reduces the effectiveness of acoustic insulation. SoundprooFing 
structures are used extensively on ships for acoustic covers for machines
and screens, control centers, air ducts and, importantly, bulkheads, decks ~
and partitions that separate rooms with noise sources from quiet rooms.
The basic types of soundproofing ship structures are shown in Figure 10.2.
: Figure 10.1. Paths of propagation of noise from one room
to another.
 In doublewall structures a liner can be attached to the hull with the aid
of acoustic or soundinsulating bridges. Acoustic bridges are ordinary
parts, used for fastening a liner to the hull. Soundinsulating bridges
are special fasteners, which substanti.ally reduce the transmission of sound
~ from the hull to the liner and thus improve the soundproofing of a double
= wall structure.
10.2. Basic Principles of Acoustic Insulation
Th3 term "acoustic insulation" is used for describing an acoustic structure,
a physical process, and also for numerical description of the process.
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�
3 R
.
,
f
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c
~
ll
A
/
1
f
c,j
c,
i
:
Figure 10.2, Types of soundproofing ship structures:
1 hull structure; 2 vibrationdamping material;
3 soundabsorbing material; 4 liner; 5 acoustie
bridge; 6 soundinsulating bridge.
Acoustic insulation is analyzed numerically with the aid of transmission
coefficient T(the term "penetration coefficient" is also used), which is
equal to the ratio of the flux of acoustic energy passing through an
examined cross section of a bulkhead to the flux of acoustic energy striking
that cross section. The inverse transmission coefficient is acoustic
insulation capacity r.
Transmission coefficient T is related to scatter coefficient d and reflec
tion coefficient e by the equation that expresses the law of conservation
of energy, specifically 8+ E+ T= 1.
If a is the absorption coefficient, then a + c = 1. The acoustic insula
tion coefficient (the term "acoustic insulation" is most often used) R, in
dB, is
R= 10 tg r =  lO lgt =101g (a  S).
(10.2.1)
The International Standardization Organization recommends the use of the
terms penetration coefficient and acoustic insulation coefficient [19].
It is important to distinguish between acoustic insulation R and the
acoustic pressure drop OL = L1  L2. For a soundproofing partition,
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separating two reverberative rooms in the absence of bypasses, this
difference in ship structures may be 510 dB.
Acoustic insulation in consideration of bypasses is sometimes called the
actual acoustic insulation [5].
In addition to the above interpretation, according to which acoustic
~ insulation does not depend on the coordinates of the points in space in
front of and behind a partition, the concept of local acoustic insulation
is also utilized. In this case the acoustic flux behind a partit;.on in
the Fraunhofer zone of open or of enclosed space is examined in the
presence of and in the absence of the partition. The fact that acoustic
insulation depends on the point of observation (its localization in space)
necessitates the determination in thiscase of the spatial spectra that
occur when sound passes through a partition [14]. _
tiVhen plane acoustic waves strike the interface between two semiinfinite
media (Figure 10.3), in which longitudinal waves propagate, acoustic
insulation R, in dB, is
R = 201 I j/ ZZ
~ ~ Z, Y Zl)6' (10.2.2)
and the normal acoustic resistances (impedances) are
'L c1 an d Z pzCa
1 = p'cos~?1 ? cos (10. 2. 3)
where,3 1 is the angle of incidence of acoustic waves; 6 2 is the angle of
transmission of acoustic waves. The other values with the subscript 1
refer to the first medium, and the values with the subscript 2 to the
second.
Impedance Mismatch Principle. It follows from formula (10.2.2) that when
impedances Z1 and Z2 are equal acoustic insulation is equal to zero. To
increase acoustic insulation it is necessary to increase the difference
(mismatch) of the impedances of the first and second media. This situation
is called the impedance mismatch principle. To achieve good acoustic
insulation it is necessary to have a substantial mismatch of impedances,
~ and the amount of acoustic insulation does not change when the acoustic
waves propagate in the opposite direction. The impedance mismatch princi
ple is applied to any partition, medium and type of acoustic wave.
Determination of Acoustic Insulation by Tnput Impedance. The ratio of the
 acoustic pressure to the normal component of vibration velocity, taken on
the interface, from which sound propagates from the opposite side of the
interface, is called input impedance. The formula for the input impedance
of the jth layer of thickness s, in which just longitudinal waves propa
gate, acquires the form [4]
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P?1
~
"
Figure 10.3. Transmission of acoustic waves through inter
face between two media.
Key: 1. lst medium 2. 2nd medium
[ex=in] Z(j) _ Z(i+i)  iZ; tgh;si
nx  Zi  iZ(i+ t) t; ia;s; ' (10. 2. 4)
The layers are calculated from the side of the medium that the acoustic
 waves enter behind the partition. A soundproofing partition consisting of
n layers, R in dB, is determined by the formula
j11 Z Z(j_;_ i)
_ R _ 20 Ig + ,U7(iii) err:Xi11) 5(r;i) . (10.2. 5)
 j_1 Zc~~~1) uX
Determination of acousti. insulation by input impedance is the most general
way of calculating the acoustic insulation of infinite (or of largearea,
in practice) partitions.
Law of Mass. The input impedance of an extremely thin single layer with
air on both sides, and with a large input impedance is, according to
formula (10.2.4),
nC
Z�x cos i! ` iwrn, (10 . 2. 6)
where 0 is the angle of transmission of acoustic waves through the layer;
m is the surface mass of the layer.
Substituting this expression in (10.2.5) we obtain acoustic insulation R,
in dB, relative to the partition according to the "law of mass" for any
.angle of incidence of acoustic waves:
/c ,nt cos )2] R
10 Iq I{ 12~~ (10. 2. 7)
~ 292
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It follows from the formula t hat in addition to the mass of the partition,
the angle of incidence of acoustic waves also exerts an important
influence on acoustic insulat ion. This phenomenon is ca11Ed the camponent
effect [16]. In the case of tangential incidence of acoustic waves the
 acoustic insulation is equal to zero, and in the case of normal incidence
it reaches the maximum value.
In diffusive incidence of sound acoustic insulation R, in dB, is
12=201g(nc1201g~ lOlgl~[1~I p(10.2.8) `
This formula may also be writ ten in the form
R=ai 2g (I I as) Ia31, (rn+ a.i)Iaa�' (10.2.9)
The values ala5 are correcte d on the basis of experimental data and then
formulas are derived for engineering calculations. That, in particular, is
how all the formulas presente d in this chapter were derived.
For ship bulkheads, sides, partitions, etc., made of steel, duraluminum and
plywood with thicknesses of from 1 to 12 mm, and with rigidity ribs extend
ing in one direction at a hei ght of not more than 30 times the thickness of
the plate, and with frame sp aces of from 500 to 1,000 mm, formula (10.2.9)
acquires the form
R = 14,5 []g (jin d 100) 2], (10. 2. 10) 
where f is the mean geometri c thirdoctave frequency band, in Hz; m is the
surface mass of the partition, in kg/m2.
Acoustic insulation is determined by the law of mass by just one partition
parameter, namely the surface mass, and it increases monotonically with
frequency. The formula of th e law of mass conservation is valid for thin
barriers with a thickness of less than 1/6 the bending wavelength in them,
and for the 2fr < f< O.Sfcr frequency range, where fr is the first
resonance frequency of the partition and fcr is its critical frequency.
There should be a nearly diffusive acoustic field on both sides of such a
partition, and there should b e no bypasses through which the sound can
penetrate.
The law of mass conservation is the fundamental law that determines the
acoustic insulation of thin p artitions. In cases when it is valid either
materials with high density must be used to increase acoustic insulation,
or the thickness of a partiti on must be increased. Steel has the highest
density of materials used on ships, but lead is considered to be a better
material for acoustic insulat i on. Thin tinplated lead foils should be
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used on ships when especially effective and reliable noise suppression is ~
required [8].
10.3. Acoustic Insulation of SingleWall Structures
Physical Principles. b9ost ship hull structures are thinwall construc c
tions. In such structures sound propagates primarily in the form of
bending waves, which are readily excited by acoustic air waves and which C
readily radiate acoustic energy into surrounding space.
Let us examine the basic principles of acoustic insulation of singlewall
structur,;s in application to bending waves, for the case of the simul
taneous propagation of these waves in an endless plate. The impedance of
such a plate for induced bending waves, which is expressed as the ratio of
the difference between the acoustic pressures on both sides of the plate to
its vibration velocity [16], is
[H=b]
er (1F tn) Wa s;o+a ~
Z, = f r (Om 
L c4 (1  oa) '
(io.3.i)
where 0 is the angle of incidence of the acoustic waves on the plate; c is
the speed of sound in air; J is the moment of inertia of a unit width of
the plate.
In addition to induced bending waves, caused by the incidence of acoustic
waves on the plate, free bending waves can also propagate in and on the
plate. The velocity of these waves is
4
c,~ = 1~2nf Ef~ (10. 3. 2)
m(1
and wavelength is
[n=1o] 7~� = V1.8 fs , (10. 3. 3)
where s is the thickness of the plate; clo is the velocity of longitudinal
acoustic waves in the plate.
Formula (10.3.2) does not take into account internal losses, but for ship
construction materials they arP negligible and therefore have little effect
on the velocity of free bending waves.
If in formula (10.3.1) the expression enclosed in braces is equal to zero,
a plate with small internal losses will not offer resistance to external
acoustic pressure. In this case acoustic insulation is nil on the matching
wave frequency:
c2 m (1  Qz)
fo  2n sin2il 1 . E~ (10. 3.4)
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,
The following relation exists on that frequency between lengths X of
acoustic waves in air and ab of free bending waves in the plate:
[H=b]
2, = R� sin v.
(10.3.5)
The lowest matching wave frequency is called the critical frequency:
[Kp=cr, � n=1o] f 12p(1o') rv 0 cL
Kp  2ns ef ,55 S~n
(10.3.6)
On frequencies below critical the matching wave phenomenon cannot occur and
acoustic insulation for this reason cannot have a minimum. On higher fre
quencies the bending wave velocity becomes asymptotic to the Rayleigh
surface wave velocity as frequency increases.
The domain of the laws that are inherent to bending waves is limited to a
thin plate, which should satisfy the relation
s < 0,05 ~ .
(10.3.7)
The frequency above which, according to bending wavetheory., sound insulation
does not exist, can be calculated by the formula
fa ~ 0,05 c�
s
/l~~ 10 IG ~ L 1111 71(1 EF QZ) sin pcos ~ l~
J
cos" = ~~Ef J a ^ (10.3.9)
2pc / [Cl (I _U2) sin ~?1}.
1
 According to this formula acoustic insulation on frequencies below the
critical frequency obeys the law of mass conservation. On frequencies
equal to the critical frequency and higher acoustic insulation is determined
 by the internal losses in the plate and by the bending rigidity of the
plate. This has been confirmed experimentally.
Calculation of Acoustic Insulation. A singlewall ship partition is
 examined in engineering calculation of acoustic insulation as a thin plate
witji large dimensions, which always has rigidity ribs. The range of appli
~ cation of the graphoanalytical method presented below for calculating
acoustic insulation of singlewall partitions has the following constraints.
The minimum linear dimension Z of a room (height, width or length) on both
sides of the structure intended for acoustic insulation, s}:ould be
295
(io.3.s)
The acoustic insulation of an infinite plate (i.e., a very large plate in
practice) Rp, in dB, is determined by the formula
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Z > 8a
 cr'
where acr is the bending wavelength on the critical frequency.
(io.3.l0)
The second constraint pertains to the bottom and top boundaries of the
calculated frequency range. The bottom boundary of the calculated fre _
quency range of acoustic insulation should be approximately one octave
higher than the first resonance frequency of the singlewall partition. It
is determined to an accuracy satisfactory for practice by the formula 
fbot C/Z' (10.3.11) ,
where Z is the minimum linear dimension of the partition, in m.
 The top boundary of the calculated frequency range is determined by formula
(10.3.8).
The acoustic insulation of singlewall structures (Figure 10.4) is
calculated as follows [2].
R,a6
45
41
3(
21
15f00 115 160 200 2.`0 315 400 500 630 gOG vCD 1250 %500 2000 7500 J150 %000
025fP OSf,,P fKp 2 Kp 1,'!u
~
6
1~~i
~

i
_

'
Z.
~
3
Figure 10.4. Calculation of acoustic insulation of steel
bulkhead. Seven millimeter thick plate with welded No.
12 bulbplate rigidity ribs; 1 theory; 2 experi
ment. [86=dB; kp=cr; Fu=Hz]
Key: 1. 4 dB per octave 2. 8 dB per octave
1. The critical frequency of a given material for a singlewall plate
structure of a given thickness is calculated by formula (10.3.6).
2. Four values of the abscissa:
off, as shown in Figure 10.4, o
thirdoctave frequency bands f,
logarithmic scale, and acoustic
ordinate axis) .
0.25fcr, O.Sfcr' fcr
n the coordinate grid
in Hz, are plotted oi
insulation R, in dB,
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and 2fcr, are marked
(the mean geometric
1 the abscissa axis in
is plotted on the
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3. For the abovementioned four abscissas four ordinates are marked off
in accordance with the data in Table 10.1.
Table 10.1. Ordinates for Plotting Calculation Cui�ve of
Acousti c Insulation
e j T
nq 0,1,_ J) 30yH0H30lIAL(HH, AG, xa 43CT0T8X
Aiereptan xortttpyKqs+e xcerb,
KNxl 0.251KP I 0.5fxP I fKP I 2fKP
4) CTanb
5) TuTax
61 AAlOA(fllf1iEB0dtBTF[NC8610
~ cnnana
7) CTexnonnarrNx
8) ~aaepa
9) CTexno cE~~NxaTxce
!jp) CTexno opraHHtecxce
7800
35
37
30
39
4500
31
33
26
,35
28CO
29
31
23
' 31
1700
28
31
28
34
800
26
29
26
32
2500
32
34 '
27
34
1200
34
36
30
38
Key: 1. Construction material 6. Aluminummagnesium alloys
2. Density, kg/m3 7. Glassreinforced plastic
3. Acoustic insulation, in 8. Plywood
dB, on frequencies 9. Silicate glass
4. Steel 10. Organic glass
S. Titanium [kp=cr]
4. The four points thus obtained (see Figure 10.4) are connected with
straight lines, and then a straight line is drawn from the first point
toward lower frequencies downward with an inclination of 4 dB per octave,
and a straight line is drawn from the fourth point toward higher fre
quencies, upward with an inclination of 8 dB per octave.
The limiting deviatiolis of the calculated values of acoustic insulation
frQm the experimental values do not exceed 5 dB with an estimated
reliability of 0.95. The errors will be smaller if the rigidity ribs go in
the same direction at distances more than 0.5 m apart, and their height
_ should not exceed 30 times the plate thickness. We note that the calcu
lated values of acoustic insulation left of point 1 correspond to the law
 of conservation of mass, and consequently to formula (10.2.10).
 Acoustic Insulation of SingleSide Partition with Soundproofing. One
effective way to improve ttie acoustic insulation of thin singlewall parti
tions is to use lightweight soundabsorbing materials, covering an entire
insulated partition surface in a uniform layer. This produces a twolayer
structure consisting of the singlewall structure with acoustic insulation
R1 and soundproofing layer, which increases acoustic insulation by AR1.
_ The acoustic insulation of a singlewall partition with soundproofing is
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R = R1+aR1. (10. 3.12)
where Rl is calculated by the abo>>edescribed graphoanalytic method.
The additional acoustic insulation of a singlewall partition with sound
proofing, applied snugly on the structure to a thickness of 20100 mm, can
be calculated by the formulas
AR1= 8.7 19 so forsP > i
and aR, = 0 for so < 1, (10 . 3.13)
where s is the thickness of the soundproofing material, in mm; R is the
attenuation coefficient for that material, in 1/cm.
Attenuation coefficient R for basic soundproofing materials used in ship
building are listed in Table 10.2. Lining singlewall bulkheads with
soundproofing is used extensively in rooms with noise sources (the engine
compartments of ships, ventilation rooms, etc.), in soundproofing covers of
machinery, and in control centers.
Acoustic Insulation of Single{Vall Partition with Holes. Holes can have an
appreciable effect on acoustic insulation and must always be taken into
account during the design and manufacture of soundproofing structures.
The acoustic insulation of four plates made of 3 mm thick duraluminum is
plotted in Figure 10.5 as a function of frequency. The plates differ in
that the first (curve 1) has a 1 mm wide 1,000 mm long slit, which divides
it into two equal halves; the second plate (curve 2) has 16 9 mm diameter
holes, uniformly distributed on the surface of the plate; the thixd plate
(curve 3) has a singlz hold right in the middle, with a diameter of 36 mm;
the fourth plate is solid and has no holes (curve 4). The total area of
the holes in all the plates with holes is identical.
Q,~6
f 2 d 4
30
~
20
500 f000 ?000 4000 8Gq7 F,r4
Figure 10.5. Acoustic insulation of 3 mm thick duraluminum
panels with 1,000 mm2 hole area as function of frequency:
1 plate with one slit; 2 plate with 16 holes; 3
plate with one hole; 4 plate with no holes.
298
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3)
4)
5)
6)
7)
8)
9)
Table 10.2. Attenuation Coefficient S, in 1/cm
1~ I 2) 48aotd oKraettdx nonoc, 1'u
o., ~ ..w
barepuen I
63
I 125 I
250 `
I
500
I 1000
I 2000
I 4000
I 8000
1(.A703IIYK0fi30.7A11}10H
0,014
0,020
0,035
0,080
0,142
0,198
0,220
0,220
Hbl~t 1118TCpIi27 ATM1
floponnacr n0axypera
0,026
0,074
0,122
0,190
0,300
0,370
0,420
0,500
HOB61H 37aC?I14H613~
C3\[038T}'X 2101qN {f
�
Ten:1oEiaonauFtoxamii
0,016
0,041
0,066
0,082
0,091
0,120
0,130
0,130
n+aTepxas rtapxH BT4
Ten.101130n3Muj10xawi+
0,019
0,033
0,053
0,062
0,081
0,140
0,170
0,170
rtarepxa:t Mapxa
BT4C
TCiIJ10H307A1j11011HblIi
0,012
0,028
0,044
0,058
0,082
0,190
0,200
0,200
ntarepEim rtapxx
ATI4MCC
flnxrbl Ha mraneabxoro
0,014
0,038
0,061
0,083
0,105
0,132
0,156
0,176
crexnoBOnoxxa
Clniirbt noapxerrxfle
0,018
0,061
0,104
0,150
0,180
0,320
0,450
0,470
xmHepaaoearxbie fta
cpeHOnbxoii ceasxe
,
Key: 1.
Soundproofing material
2.
Frequencies of octave bands,
in Hz
3.
ATM1 heatinsulation
material
4.
Elastic selfattenuating poly
urethane poroplast
5.
VT4 heat insulation
6.
VT4S heat insulation
7. ti1IMSS heat insulation
8. Staple glass fiber panels
9. Semirigid phenolbonded rock
wool panels
As can be seen in the figure, even small holes and the slit can reduce the
acoustic insulation in a wide frequency range by approximately 10 dB. It
also follows from Figure 10.5 that one large round hold and a group of small
round holes with the same area reduces the acoustic insulation of a panel by
about the same amount, whereas a slit with the same area reduces the
acoustic insulation of a panel to a much greater extent in a considerable
frequency range (below the critical frequency).
Thus, slots and holes reduce acoustic insulation basically on frequencies
below critical, and a slot is worse in that frequency range than holes.
Slots anrl holes can have a negligib le influence on acoustic insulation on
the critical frequency. On frequencies above critical slots and holes again
exert a negative effect on acoustic insulation, and the shape of holes has
no effect on the deterioration of acoustic insulation; theirarea is more
important.
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Slots and holes in ships can greatly detract from acoustic insulation of
doors and windows in soundproof control centers, in ward rooms, etc. T}ie
elimination of these weak points in acoustic insulation is the prerequisite
for the successful control of noise.
~ 10.4. Acoustic Insulation of DoubletiVall Structures
Physical Principles. It has long been known in the practice of noise con
trol that the acoustic insulation of doublewall partitions can be sub
stantially more effective than of a singlewall structure of equal mass.
That is why m2ny soundp roof partitions on modern ships are based on the use
of doublewall structures with layers of air and soundproofing materials
between the walls. These structures are made by lining ship rooms. The
liner is installed some distance from the main hull structure (bulkhead,
si.de, 3eck, etc. ) .
If the space between the panels of a doublewall structure is filled with a
soft porous material and its absolute characteristic impedance is close to
that of air (tivhich for modern soundabsorbing materials is, for practical
purposes, the case), then the acoustic insulating capacity of the double
wall structure for acoustic waves with normal incidence is
r I \1 + 2Zo / \1 + 2Za / edO (10.4.1)
 Z1117112 . eAP (cos 2kd  i sin 2kd) I r,
`370
where Zil and Zi2 are the impedances of the first and second panels.
An analysis of the .formula shows that the specific way to improve the
acoustic insulation of a doublewall structure is to increase distance d
between the panels and attenuation constant B of the soundproofing layer.
lVhen the values of ds are large enough the effects of acoustic insulation
of the panels and of soundproofing between the p anels (in dB) are indepen
dent of each other and the acoustic insulation R, in dB, of the doublewall
structure is maximum when the mass is minimum:
3
'R= {Ogtlp� (10.4. 2)
lolg 1f 2Z'o I"a' lolg I 1 2Zo I
If there is little absorption of sound between the panels the acoustic
insulation of t}ie doublewall structure on the lowest frequencies, when
, kd.� 1, is equivalent to the acoustic insulation of a singlewall
structure with the same total surface mass as two panels.
In the lowfrequency range, when cos kd ~ 1 and sin kd kd, the acoustic
insulation is minimum. A doublewall structure behaves in this case like a
system with concentrated parameters in the masselasticitymass system.
300
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'I'he panels function as mass here, and air functions as the elastic element
between them. Por ship structures, however, the attainment of this mini
mum usually does not matter much.
The acoustic insulating properties of a doublewall structure that are
important in practice are manifested on medium and high frequencies. It is .
specifically on these frequencies that the acoustic insulation of a double
wall structure can suUstantially exceed the acoustic insulation of a
singlewall structure of the same mass. T'he acoustic insulation in the
indicated frequency range has several successive maxima and minima, and the 
minima are attributed to the resonances of the layer between the panels,
and the maxima to antiresonances. The extent of the minima of acoustic
insulation on the frequency axis is negligible in comparison with the
extent of increased acoustic insulation. The installation of even a little
soundproofing material between the panels sharply increases the minima of
acoustic insulation, which greatly improves the acoustic insul.ation of a
doublewall structure on medium and high frequencies.
Experimental data that show how much a soundproofing material, installed
between panels, affects the acoustic iiisulation of a doublewall structure,
are plotted in T'igure 10.6. The first panel i.s made of 4 mm thick duralum
inum, and the second of 1 mm thick steel; the distance between the panels
is 60 mm. Curve 1 shows the acoustic insulation of a doublewall structure
when all the space between the panels is filled with ultrafine glass fiber.
Curve 2 characterizes the acoustic insulation of the same structure, but
tivith onehalf the air space between the panels filled with ultrafine glass
fiber (30 mm thick soundproofing layer). Curve 3 corresponds to the case
when there is no soundproofing between the panels and there is only a 60 mm
 thick layer of air. It is noteworthy that the first and second panels do
not have acoustic bridges.
80
0
500 6d0 600 1000 1250 1600 2000 250D 3150 4000
,P, ru
Figure 10.6. Acoustic insulation of doublewall structure
with soundproofing. [a6=dB; FL~=Hz]
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The increase of acoustic insulation due to soundproofing increases with
frequency and reaches 2025 dB in the indicated case, which, naturally, is _
= not the limit. It is important in practice that the main increase of
acoustic insulation come from filling just onehalf the air space between
the panels with soundproofing.
Acoustic Bridges. The sound insulation of doublewall partitions depends
to a great extent on the structural connection between the panels, which is
made up of acoustic or soundinsulating bridges. The effect of acoustic
and soundproofing bridges on acoustic insulation is illustrated in Figure
10.7, where test data on the structure illustrated in Figure 10.6, but
without soundproofing, are plotted. The acoustic bridges, as can be seen
by the data, can reduce acoustic insulation by 1015 dB in an extremely
wide frequency range.
;d6
70
65
6a
9
55
~
SD
2
k5
40
35
d0
500 6d0 600 1000 1150 1600 2000 2500 J150 4000 5000
. IV; ru
Figure'10.7. Acoustic insulation of doublewall structure
with acoustic bridges: 1 acoustic insulation without
acoustic bridges; 2 acoustic insulation with acoustic
bridges; 3 acoustic insulation with soundproofing
bridges.
Three types of soundproofing bridges inertial, elastic and combined, may
be used in ship structures for reducing this harmful effect for acoustic
 insulation.
Inertial bridges should be used for doublewall partitions, in which the
critical frequencies of the first and second panels exceed 5,0008,000 Hz
(the top limit is preferable). The acoustic insulation properties of an
inertial bridge are determined by its mass, which usually should satisfy
the requirement
M: S? 1f P,El
(10.4.3)
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where sl is the thickness of the hull structure or liner, in m; pI is the
density of the material of which the hull structure or liner is made, in
kg/m3; E1 is the dynamic Young's modulus of the material from which the
hull structure or linear is made, in Pa.
The first boundary frequency (see below) also enters in formula (10.4.3).
It is calculated by the formula
r
fn = rd mrx ma
(10.4.4)
where ml is the surface mass of the first panel, in kg/m2; m2 is the sur
face mass of the second panel, in kg/m2; d is the distar.ce between the
panels, in m.
Mass M is calculated separately for the hull structure and for the liner,
and then the larger value is taken. An inertial bridge should be made of
steel, bronze or other solid material in the form of a cylinder or ball.
Elastic bridges should be used for massive and rigid doublewall ship
partitions, in which the critical frequencies of the first and second panels
lie below 3,0005,000 Hz. The acoustic properties of an elastic bridge are
determined entirely by its rigidity, which should satisfy the following
requirement:
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Closed Booth. The effectiveness of a closed booth in terms of air noise
AI,b, in dB, is calculated for any type of source by the formula
n St
ALK 101g `~1 160. 1Rl~p Sn ~ (10 . 6.13)
where Si is the area of the ith wall of the booth, in m2; Ri is the
acoustic insulation of the ith wall, in dB; aav is the average sound
absorption coefficient in the booth; Sc is the total enclosure are of the
booth, in m 2 ; n is the number of enclosures exposed to noise.
Data on the effectiveness of local sound reduction devices (shields and
booths) are presented in Table 10.9 for directional and nondirectional
sources: turbochargers, engines, etc. The data are based on measurements
(see Figure 10.14 for designations).
10.7. Acoustic Insulation of Ship Machinery
The problem of reducing noise on ships to established standards can be
solved most economically in many cases by using special soundproofing
devices (SPD), which form a closed soundproofing shield around the source.
The basic parts of soundinsulating systems are the frame and wall panels.
The structure also includes hardware, with which the panels are installed,
fastened to the frame and connections are sealed. SPD also must have
325
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Table 10.9. Effectiveness of Sliields and Booths, dB
Loca1 noise reduction
Structural design and
re
uencies of octave ban
ds, Hz
device and installa
measurement condi
250
500
1,000
2;.000
4,000
8,000
tion features
tions
.
Plane shield in open
1. Shield dimensions
space 0.5 m from
1 x 1 m, material
directional noise
plywood (8 = 10 mm)
source: (a = 0.2 m)
d= 0.5 m
3
2
15
21
33
31
d= 1.5 m
0
1
10
20
29
27
2. Shield dimensions
0.5 x 0.3 m, mate
rial plywood
.
(S = 10 mm)
d=O.Sm
0
0
3
7
13
16
d= 1 m
0
0
3
6
12
7
Plane shield in open
Shield dimensions
space 0.5 m from
1 x 1 m, material
nondirectional noise
plywood (S = 10 mm)
source
d= 0.5 m
0
3
10
14
12
14
d= 1 m
0
2
10
,
12
10
12
Plane shield in
Shield dimensions
engine room of cargo
1 X 1.5 m, mate
motor ship 1.5 m
rial steel (S =
from engine
= 2 mm), room
constant Q =
= 50300 m2
d= 0.7 m
0
2
3
4
7
6
Plane shield with
Shield dimensions
soundproofing in
1 x 1.2 m, mate
engine room of motor
rial steel (S =
tugboat over engine
= 1.5 mm), sound
(r = 0.4 m)
proofing staple
glass fiber (8 =
= 50 mm)
d= 0.3 m
2
5
6
6
6
5
Engine noise reduc
tion
1
2
2
3
2
4
Plane shield in room
Shield dimensions
0.5 m from direc
1 x 1 m, material
tional noise source
plywood (6 = 4 mm),
(a = 0.2 m)
soundproofing
staple glass fiber
(d = 50 mm), room
constant Q = 3070
m : shield without
soundproofing": (d =
0.5 .m).
0
2
2
11
21
23
shield,with;sound
roofing
0
2
10
14
24
26
326
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Table 10.9 (continued)
Loca1 noise reduction
Structural design and
Fre
uencies of octave ban
ds, Hz
device and installa
measurement condi
tion features
tions
250
500
,000
,000
4,000
8,000
Plane shield with
Shield dimensions
soundproofing
1 x 1 m, material
installed in engine
steel (d = 1.5 mm),
room of cargo motor
soundproofing
ship 0.5 m from
staple glass fiber
turbocharger of
(8 = 40 mm); Qn back
engine
ground of inechanical
engine noise (d = lm)
0
0
5
11
12
14
Enclosure with sound
Shield dimensions
proofing, installed
2 x 1 x 0.7 m,
in room
material plywood
(S = 4 mm), sound
proofing VT3
(d = 30 mm), room
constant Q = 3070 m2
r= 0.8 m
7
8
9
9
10
12
in center of enclo
sure
r= 2.3 m
4
4
5
S
6
7
Boiler partition
Partition dimensions
(open at top),
3 x 3 x 2.5 m,
installed in engine
material steel
room of cargo motor
(d = 3 mm); at cen
ship 1.5 m from
ter of partition
3
4
5
5
4
6
engine
Soundproof SEU con
Dimensions of booth
trol center, perma
5 x 3 x 2.5 m, con
nently installed in
struction of booth:
engine room of motor
steel (d = 4 mm),
tugboat (engines and
foam plastic FF
diesel generators on
(8 = 40 mm), ply
shock absorbers)
wood (S = 4 mm),
plastic (S = 1.5 mm),
double glazed win
dows (d = 5 mm), in
center of booth
25
27
29
32
34
33
Soundproof control
Booth dimensions
booth installed in
1.8 x 1.3 x 2 m,
engine room of
construction of
floating crane on i
booth enclosure
AKSSM shock t
steel (d = 5 mm);
absorbers l
foam plastic FF
l
(S = 40 mm), rock
wool (d = 50 mm),
327
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Table 10.9 (continued)
Local noise reducti on
Structural design and
Frequencies o
f octave bands, Hz
device and installa
tion features
measurement condi
tions
250
500
1,000
2,000
4,000
8,000
steel (8 = 2 mm),
double glazed windows
(d = 4 mm), in cen
ter of booth
30
32
34
36
38
45
Comment. The measured effectiveness of soundproofing systems is zero on
frequencies of 63125 Hz, and the effectiveness of soundproof booths is
2023 dB.
devices that prevent an unacceptable increase of temperature and pressure
and dangerous concentration of oil vapors in the space under state rooms,
and also serve for venting vapors and for repairing leaks in systems. The
effective acoustic insulation of passages through SPD walls for engine con
 trol levers and mounts, and of fasteners that tie the entire sound
insulating system t o the foundation, is extremely important.
_ However, the main s oundiRSUlating elements are the panels, which along
with the frame form the walls of the device (here the frame plays a
secondary role). The frames of all soundproofing devices are multihinged
_ rolled steel rod frames. Transverse connections with longitudinal connec
tions and with the b ase frame'are made detachable.
SPD are made in the shape of shallow boxes and consist of a main sicin,
body, soundproofing material and perforated liner. A typical SPD wall
panel is illustrated in Figure 10.19. The main skin functions as the
soundinsulating element. It is made of thin sheet metal and reinforced
_ plastic and is desi gned as a"sandwich" structure (alternating layers of
metal and polymer). The panel body forms the side walls and serves to
increase the rigidity and to protect the soundproofing material along the
edges of the panel. The metal parts of the panel are argon arcwelded and
spot welded on cement. This kind of connectior, assures not only the
necessary rigidity, but also air tightness, without producing great thermal
deformations.
The soundabsorbing elements of the panel are made in the form of packages.
Their shape and length and width dimensions match the panel cells into ahich
they are inserted. The packages shotild not be less than 50 mm thick. A
material based on ultrafine glass or rock wool is recommended as th.e sound
proofing. The packages are waterproofed with films, which should jiot be
more than 25 micron thick. A perforated aluminummagn?sium alloy liner, not
thicker than 1 mm, is placed on the inside of the pane?.. The perforations
should cover 25% of the total liner area. The liner holds the soundproofing
material in the panel and protects it from mechanical damage.
328
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Figure 10.19. Typical soundproofing device wall panel:
 1 panel body; 2 main skin; 3 soundproofing material;
 4 handle; S 1ock; 6 gasket; 7 interior perfor
ated liner; 8 protective film.
Panel to frame connections may be sealed by the most popular and easiest
 technique with elastic soft porous gaskets with closed pores. The space
~ between the engine and soundproofing device walls may be ventilated through
a system of special channels. Ventilation by natural convection usually is
not enough to assure the necessary air exchange. Therefore a combined
_ system, in which two lowpressure fans are used in addition to natural
ventilation, are often utilized in soundproofing devices. Air is circulated
into and from the space %nder rooms through special mufflers. The design
of such a muffler is shjwn in Figure 10.20. As can be seen in the figure,
the panels with the ventilation channels are in the form of thinwall
boxes, but the wa11s are higher than the panels of the SPD itself, and one
solid and two perforated liners are additionally installed on the inside of
each wall, para11e1 to the outside liner. The height of the ventilation
channel is 25 mm and width is 210 mm. The slotlike cross section of the
channel and its great length provide adequate sound insulation.
Special seals (Figure 10.21) are installed to prevent sound from penetrating
from the space under cabins through pipe and cable holes. Small pipes are
combined into bundles, each of which is drawn through one seal.
It is also important to take measures to limit the shifting of soundproofing
 devices during rolling and vibration. This is best done by installing
 fasteners that hold SPD to the foundation of vibrationdamping elements.
~ The foundation for SPD, as is shown in Figure 10.22, may be: the ship foun
dation (Figure 10.22a); engine block (Figure 10.22b); intermediate frame
_ (Figure 10.22c). However, because the required vibration insulation of a
fastener increases with the level of vibration of the foundation, preference
should be given to fastening to the ship foundation. 
Measurements of the acoustic effect of the noise reduction of SPD for a ship
compressor disclosed (Figure 10.23) that SPD reduces noise in a wide fre
quency range by an average of 17 dB (curve 1). This applies to a11 versions
of testing (different suspensions, perforated or solid liner, soundproofed
panels and panels without soundproofing), except one, when the compressor
draws air from the space under the cabins; in this case the effectiveness of
a system decreases on low frequencies by about 10 dB (,Figure 10.23, curve 2).
On medium and high frequencies the noise reduction effect is the same.
329
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Figure 10.20. Noise muffler (wall Figure 10.21. Soundproof gasket for
panel of ventilation channel): passage of pipe through wall of hood:
1 frame; 2 outside branch; 1 main liner; 2 rubber diaphragm;
3 perforated liner; 4 main 3 pipe; 4 seal.
skin; 5 gasket; 6 soundproof
ing material; 7 protective film;
8 inside liner.
a~ b) C)
Figure 10.22. Diagrams showing how soundproof hood is
fastened: a to ship foundation; b to engine block;
c to intermediate frame. .
330
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Other important results of ineasurements of the effectiveness of SPD are
presented in Figure 10.24. Curve 1 corresponds to the case when external
pipes are not insulated and the walls are not sealed. The use of SPD
reduced noise by 213 dB in the medium and highfrequency ranges in the
absence of pipes and connections (Figure 10.24, curve 2). Sealing slots
and holes in places where the panels are next to the block, on tne floor
and around locks increasethe effectiveness of SPD by another 58 dB on
high frequencies (Figure 10.24, curve 3). The described measures sub
stantially increased the acoustic insulation of SPD, but the maximum pos
sible acoustic insulation for a given wall structure still could not be
achieved (Figure 10.24, curve 4). In real SPD structures, obviously, if
acoustic insulation is incorporated in the design of the machine itself,
all kinds of inechanical noise can be reduced to acceptable levels. The
acoustic insulation of machinepy saves more materials and space in the hull
of a ship than the acoustic insulation of rooms. The combining of machinery
into modules substantially reduces the cost of acoustic insulation and
improves its reliability and effectiveness.
Q
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Figure 10.23. Effectiveness of unsealed soundproofing 
system: 1 air drawn from outside of soundproofing
device through muffler; 2 air drawn from basement
space of_ soundproofing device. [aE~=dB; Fu=Hz]
331
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R, 86
4
i0
3
2
)5
9
20
O
~63 125 250 500 1000 2000 4000 8000
r r.u
Figure 10.24. Effectiveness of sealed soundproofing system:
1 before sealing and insulation of pipes; 2 after
insulation of pipes; 3 after additional sealing of con
nections; 4 maximum effectiveness, achieved under
laboratory conditions. [86=0; fL~=Hz]
10.8. Materials for Marine Acoustic Insulation and Soundproofing Structuresl
The materials used for reducing noise on ships must meet a whole set of
requirements, including efficiency, manufacturability, performance reli
ability, ergonomic suitability and other indices. The general requirements
are incombustibility (at least flame retardant) and nontoxicity. The use
of materials that do not meet these requirements is prohobited or strictly
limited.
bfaterials for SoundInsulating Structures. The materials that are used in
soundinsulating structures are divided by purpose and acoustic effect
into main and auxiliary. Main materials include sheet materials, used for
making hull structures and for finishing rooms: metals, reinforced concrete,
reinforced cement, asbosilit, plywood, laminated plastics, glass, etc.
Auxiliary materials include decking and vibrationabsorbing carpets, heat
insulation, air space fillers fox doublewall soundproofing structures,
vibrationinsulating spacers, etc.
1This section was written by V. M. Frolkova with G. S. Beregova and T. A.
Firkovich.
334
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The nomenclature, classification and characteristics of basic materials,
approved by the USSR Ministry of Health for use in ship rooms, are listed
in Table 10.10, where the average internal loss coefficients and elasti
city moduli of materials are given for the 100 to 2,000 Hz frequency range
at 20�C. The combustibility index K of the materials according to GOST
1708871 is indicated and combustibility groups (J) according to the Rules
of the USSR Register are explained.
Materials for Soundproofing Structures. The materials used in soundproof
ing structures are classified by purpose and acoustic effect as basic
soundabsorbing, and auxiliary materials. Basic materials are indicated in
Table 10.11. The best in terms of properties (efficiency, incombustibility,
nontoxicity, weight and manufacturability) are mats made of superfine glass
and basalt fiber.
Tab1e 10.12. Acoustic Characteristics
of ATM1 Heat and Acoustic Insula
tion (Density 10 kg/m3)
1)
K~3441W+eHr
31soat,onoe
liactora
Pacnpo�
conpo
39yKE,
CTpaHCHHN
T3:[17CIII10
I'u
Ym = am f
�m = tiV'nt'}"
F !lVtnt
250
3.2{iG,O
2,4011,0
315
4,8+i9,0
2,20i0,75
400
6,0}113.5
2,0010,55
500
8,0}i18,0
1.65i0,48
630
10,0+01,0
1,80 i0,40
800
12, 0r i 2G, 0
1, 75 i 0, 38
1000
14,2}i32.0
1.70i0,35
1250
16, I}i38,0
1,62i0,31
1600
18,0+i44,0
1,60i0,30
2000
19.8{i51,0
1,58i0,29
2500
20,5{i60,0
1,5010,27
3150
21,0{i69.0.
1,50i0.25
9000
22,0+i74,0
1,50i0,25
6000
22,0{i80,0
1,500,25
6300
22,0i82.0
1,50i0,25
8000
22,0{i84,0
1,5010,25
Table 10.13. Acoustic Characteristics
of Ultrasuperfine Basalt Fiber Cloth
and Grade BZM Products (Density
1720 kg/m3)
l)
llacTOra
BDyKB.
ru
Ko3"itwieeT
pacnpo�
CTpHfifllNA
Vm  am 'f
+ ipm. 1/M
3onHOn0e
conpo
7HBIICHH@
Wm � 1Grm f
I 1Wtm
250
8,0
i18,0
3,72 11.00
315
9,0
l19,0
3.75 i1,05
400
10,0
120,0
3.70t(, 11
500
11,0
i22,0
3,6311,20
630
14,0
t25,0
3,56f1,31
800
17,0
i29,0
3.4211.47
1000
25,0
i33,0
3.1211.55
1250
29,0
139.0
2,64i1,54
1600
32,0
i46.0
2,43i1,46
2000
34,0+
154,0
2,27ll,25
_ 2500
35,0}
i63,0
2,14lI,10
3150
36,0}
l7(,0
2,03i0,97
4000
37.0
l78,0
1,94i0,86
5000
38,0 t
i85.0
I,gSip,yg
6300
38,0
193,0
1,77i0,70
8000
' 38,0{
i100,0
1,72i0,65
Key: 1. Frequency of sound, Hz 3. Wave impedance
2. Propagation coefficient
The acoustic characteristics of the basic soundabsorbing materials are
listed in Tables 10.1210.14. Auxiliary materials include films for lining
soundabsorbing materials (Tab.le 10.15).
The physical mecnanism of the absorption of sound by soundabsorbing
materials is examined in Chapter 11.
10.9. Measurement of Acoustic Insulation
The acoustic insulation of ship bulkheads can be measured with sufficient
accuracy and reliability with the aid of the system illustrated in
336
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Table 10.14. Acoustic Characteristics
Figure 10.25. Both acoustic
of Soundproofing DischargeFree Rock
measurement chambers of the instru
Wool Panels (Density 130 kg/m3)
ment (or one of them) are mounted

on wheels, so that complex
1) 01,,6, ,Hu,�:,a 3)8
structures of any length can be
tIacrora Pacnpo� conpo
3ti}~KO, crpaitetitcn rHenci~xe
mggSUT@Cl U lacin them between
Y P g
ru Vrn = am + wnt  Wrm'+'
chambers. By virtue of the
+`P"`.'ik + i1wlm
mobility of the chambers it is
sso 17,0+il3,0 ~ 11804119.1
possible to use them independently
315 18,01cis,0 1,75il,80
of each other with real noise
aoo 18,0+t20,0 1,72n,69
500 19,0104,e 1,69n,�Ea
sources. The boundary conditions
cao 20,0fi46,0 1.65rl,23
RUO 21,0}i34,0 1,60il, 14
in the chambers for testing an
1000 22,0}i39,0 1,50i1,00
1250 24,0+r;3.0 1,400,e7
object can be regulated and
1600 28,0fi48,0 1.30i0,78
2000 31,0}iF4.0 1,24i0.70
adjusted with a special mechanism.
2500 37,0Fi60,0 I,16i0,65
3150 91,0{i65,0 1,15i0,62
aooo 45,0+;7i.0 1.13co,58
A schematic diagram of an instru
5000 48,0+176,0 1,12i0,55
6300 50,0}i8=.0 1,10i0,54
ment'for measuring acoustic insula
8000 52.0}07,0 1,09i0,50
t1011 1S ShOWIl in Figure 10.26.
Each chamber may be both a high
level, and a lowlevel chamber.
Key: 1. Frequency of sound, Hz
The acoustic system produces high
_ 2. Propagation coefficient
intensity noise through loud
3. Wave impedance
speakers; there is also a mechanical
noise generator. The entire
metering process is controlled from a
remote control panel.
With an acoustic insulation meter it i
s possible to check a variety of ship
structures: soundinsula�ting panels, windows, portholes, etc. The acoustic
 ansulation of large structures should
be measured with this instrument on
models.
To determine the experimental values of acoustic insulation it is neces
sary to measure the energy levels of acoustic pressure L1 in the highlevel
chamber, L2 in the lowlevel chamber, and reverberation time T. The R
values of acoustic insulation, in dB, are determined for a given frequency
band by the formula
R=L1La+ 1019 0,16V
(10.9.1)
where S is the area of the opening between the chambers, in m2; V is the
volume uf zhe l.owlevel chamber, in W.
Instruments for measuring acoustic insulation are described in detail in
the literature [9], and the determination of the acoustic insulation
measurement precision is examined in [2].
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6 5 4 d 2
, Figure 10.25. Diagram of instrument for measuring acoustic
insulation: 1, 7 acoustic measurement chambers; 2
electric motor; 3, 6 set screw and lock nut; 4 test
structure; 5 end switch; 8 insulators.
1~ rb 1)
Figure 10.26. Schematic diagram of acoustic insulation
meter: 1 acoustic radiator; 2, 3 microphone with
preamplifier; 4 mechanical noise generator; 5
power amplifier; 6 acoustic filters; 7 white
noise generator; 8 autotransformer; 9 microphone
switch; 10 spectrometer; 11 level recorder;
12 stabilizer.
Key: 1. Line
340
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10.10. Personal Acoustic Insulation Gearl
There are three types of acoustic insulation, or antinoise devices:
internal or endaural earplugs, inserts and tampons which are inserted
in the outer ear passage and do not require fastening; mixed or "half
plugs," worn over the entrance to the outer auditory passage and requiring
fastening with a headband; outer wear or extraural earmuffs and helmets.
Plugs, inserts or
exceeding 100 dB;
should be worn at
exceeds 125 dB it
endaural devices.
tampons are usually adequate at a total noise level not
they reduce noise to the acceptable level. Earmuffs
a total noise level of 100 to 125 dB. If the noise
is recommended that a helmet be worn in addition to
Researchers have been devoting attention in recent years to ultrafine
polymer fiber, used extensively in Petryanov filter cloth (FPP15), and
made by electrostatic spinning of perchlorovinyl resin. Petryanov cloth
was first used for personal noise safety purposes in the form of noise
proof tampons by A. I. Vozhzhova, V. V. Perelygin, L. S. Basova, et al.
These personal safety devices passed 'extensive longterm tests under actual
sailing conditions and produced positive results. The fabric has a fiber
diameter of 1 to 1.5 micron. It is nonhygroscopic and retains its struc
ture at temperatures up to +60�C. The noiseproof tampon is made in the
form of a ball, onehalf of which is drawn to a point to facilitate
removal from the ear. Petryanov cloth has residual electrical resistance
and sticks to the fingers. Therefore the tampon, to impart to it the
tampon shape which it must retain, is coated with a thin (5 to 10 micron)
elastic film consisting of 2% alcohol solution of polyvinyl butyral. By
virtue of the elasticity of the film the tampon can be squeezed in the
fingers to insert it into the outer ear passage, where the tampon swells
and completely fi11s it (this is the essential condition for effective
protection against noise). The tampons can be washed and are intended for
repeated usage, and therefore a special pencil case with a:,ap is provided
so that they can be stored in a shirt pocket.
It is recommended that Pe�tryanov cloth be used in the form of disposable
earplugs of the "berusha" type (designed by I. V. PetryanovSokolov, I. K.
Razumov, L. N. Shkarinov, et al) for protecting large contingents of
workers for an entire work shift. Berushas are pads measuring 4 x 4 cm,
weighing 150 � 10 mg, cut from FPSh soundproofing material. The material
consists of ultrafine polymer fibers, reinforced with coarser fibers (from
3 to 100 micron in diameter) of the same polymer. To use the earplugs the
worker folds the pad on the diagonal, rolls it into a cone and, inserting
the point into the ear passage, packs in the plug with his finger.
The effectiveness of endaural personal safety devices is given in Table
10.16, and the effectiveness of earmuffs*is given in Table 10.17. The most
1This ssction was written by A. I. Vozhzhova.
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effective and easiest to use are earmuffs with liquid or viscous filler, in
particular G. I. Petrova's outfit. It consists of a metal headband and
muffs in the form of aluminum cups with flanges, covered with polyvinyl
chloride pads, filled with glycerine. F. A. Kulikov's earmuffs are of
similar design. They consist of inetal cups with snugly fitting pads, also
filled with glycerine. Earmuffs developed in East Germany, Hungarian ear
boxes and earmuffs developed by VTsNIIOT [AllUnion Scientific Research
Institute of Work Safety] VTsSPS [A11Union Centrai TradeUnion Council]
belong to the same group.
Table 10.16. Noiseproofing Effectiveness of Endaur.al Safety Devices
Txna 137Y.T01C, raunoitoa
)Cpexic
e ocna6neFti+e
wyMa, A
G, xa
aaerorax, i'u
u aKnaaraweit

125 I
250 I
500 I
1000
(2000 I
4000 I
6000
 3) AHTn~oEita Oponat;c (I'AP)

20
23
26
35
38

4)1�ieonpexoueic oxiopo,iiiLie
31
31
31
34
37
41
41
R75.7hH z, = 29
5) HcoupeEloBbte aT~'axit ~SS:1
29
29
34
36
37
31
41
v  30
6) QI:IIN.1NT0f1bIC Dryaxt Tuna
15
20
22
25
32
30
35
V51 R
7) unime npooxff (Iie;irpESA)
25
30
25
25
35
40
25
g) 7'aN11101M 113 vnb1PaT011soro
8
10
15
22
25
32

no.,ohn3 *nni5 .
9) [ir:.ia;tE,itwt jm *1I1II
15
III
18
24
26
36
31
lO)~j'eC1UB3tlf:sc I'OCT 15762.70
10
12
15
17
25
30
30
Key: 1. Type of plug, tampon, insert 7. Hungarian earplugs
2. Average noise attenuation, 8. FPP15 ultrafine fiber
dB, on frequencies, Hz tampons
3. Qropaks antiphones (GDR) 9. FPSh inserts
4. Uniform neoprene plugs v= 29 10. GOST 1576270 requirements
5. MSA neoprene plugs v= 30
_ 6. Type v51R vinylite plugs
Liquid or viscousfilled earmuffs with a snugly fitting ring should be
used in extremely high noise levels. Dry porous elasticfilled or air
filled earmuffs should be used in other cases.
Under actual sailing conditions personal safety gear is selected in
_ accordance with the noise situation and nature of the work beirg done. For
instance, during occasiona] stays in extremely noisy rooms, and when
standing watches in tnem that do not require movement of the head and body,
G63 outfits with Z63 muffs (ordinary and in the commullications version,
i.e., equipped with telephones and laryngophones) shoixld be used. When
doing work requiring movement of the head and body the type ShSh1 helmets,
equipped with the same Z63 and ShShII earmuffs, and also equipped with
headsets and laryngophones, i.e., communications helmets, should be used.
i
342
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Table 10.17. Effectiveness of Soundproofing Earmuffs
J I2)crC11ICC GC7d6:1Q11N0 I11}'\18. AG. Ila 48CTOT.3%r rIt
Tnnw fiayniiuuos
it cTpana�Fi3roronuTenb
125 I 250 I 500 I 1000 I 2000 I 9000 I 8000
3) I'aptiEin'P WVAf038IWNT11L1}I TIf
20
22
Qa
27
27
37

iia riii i (cccF)
4)nr0T,1�o1t,~',%niwe naytmiFtxFt
14
15
20
20
30
34
40
KAena.Wuiihon rima CI[i2K
(CCCP)
5)ITponM0111ys1111tile xaywHttxtt
6
7
10
IS
29
26
19
Tttna E3a1/63 (I'IIP)
6)3aiunTUwc (j)}~r~npbl (BIIP)
15
20
30
30
40
35
30
7)I7poTHoowyMfit.ic xaynu1nr.u
7
11
14
22
35
47
38
II3611P7TP.1tittoro ;[ciirrnuA
'
noncrp)'r.itFM B. C. Uaii
It1uia (CCCP)
.
8)Ilporni3oiu%'a111b1e iiayiuniixt
5
6
16
25
29
27
23
�rEina EM (KaimRa)
9)?!aN'tuuuxtt T11na A1SA (CifIr1)
7
9
12
18
32
98
35
10)11PnM0:1111fcrinbiii 3BYHUBOIi
17
20
32
33
37
45
42
111)OTCi
>Iflt'OlJtd
~
~ .
~
0
U
IJIION I
OS
~ ~d
t/1
I
w
f
IICf '{IL'L�ttl
C N
~ ' .
`r
44
ocru ttianodU
~ O 0
~
W
W N 4H
0
04
Y
R1
Q)
~b
~
� U[
~
fd FI V1
~
o
�rl ~ �rl
,a
x
oaca
.
'1 N M
~
~
~
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 and with a 30 mm thick capron fiber filler, were installed on the ceiling
and bulkheads of the engine room of a motor ship, on an area of approxi
mately 20% of the total area of all the surfaces. The resulting average
sound absorption coefficient increased by a factor of 1.52 in the entire
frequency range, which yielded a reduction of the airborne noise level in
the engine compartment by 34 dB. The noise in the dining room, located
right behind the engine room, also decreased by 810 dB due, in addition
to the sound absorption effect, also to some increase of the acoustic
insulation of the deck (Figure 11,4).
Figure 11.3. Soundabsorbing pyra
mid: 1 perforated lining; 2
soundproofing material; 3 rod;
4 spring.
r,
aN
i
�CF
O,i
0,1
L j,
IJ2 63 115 150 SDO f000 2000 40008000
Fru
C% L
f; ru
D~
_ Figure 11.4. Average sound absorption coefficient (a) and
airborne noise levels in engine room (b) and in dining
room (c) of motor ship "Sputnik": 1 before installation
of pyramids; 2 after installation of pyramids.
[cp=av; f~=Hz]
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L
120 f
f10
f00
90d? 63 175 250 500 1000 2000 4000 80
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Resonance soundproofing structures, like flat and space structures, are
not yet used extensively on ships due to problems in manufacture and rather
large sizes.
Soundproofi_ng structures are also used on ships in swimming pools, movie
halls, etc., for providing the best conditions for enjoying music and
speech; in air ducts, ventilation shafts and exhaust stacks for reducing
propagating noises; in noise mufflers; for lining bulkheads and shields
located near noise sources and on the inner walls of the hoods of machinery
and mechanisms for improving their soundinsulating properties.
Soundabsorbing materials are also used extensively for improving the
effectiveness of acoustic insulation structures.
The soundabsorbing structures and materials used on ships must be
sufficiently fireproof and vibrationresistant and must not emit dust and
toxic substances above the maximum permissible concentration [1, 11, 15].
11.2. {Vave Parameters. Porous Panel. Resonant Sound Absorbers
The soundabsorbing properties of materials and structures are character
ized by reflection coefficient S, equal to the amplitude ratio of reflected
 and incident waves; R is a function of the acoustic impedance Za of the
interface, which is equal to the ratio of the acoustic pressure tio the
normal component of vibration velocity ~n [17]:
Za =~1 = Raf iXa,
where Ra is the active, and Xa the reactive components of acoustic
impedance.
In normal incidence
ZQ _ L I a, Zl _ Zu ~ .
Za  T, (11.2.2)
Zu '
The sound absorption coefficient (SAC) is
v=1Ip I'; (11.2.2a)
421
a  ,
(R1 + 1);:.+Xi
In these formulas Z1 = R1 + iXl; Z= pc is the specific acoustic impedance
of air, pc = 410 Pa�m/s.
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The wave parameters of an acoustic medium are the propagation coefficient
ym and wave impedance tiVm. In a medium with losses Wm = Wmr + iWmi; ym
= am + iRm, where am is the attenuation coefficient; Sm is the wave
number. Wm and ym describe the acoustic behavior of a medium. They are
related to density pm and to the bulk elasticity modulus Km for a harmonic
process by the equations [18]:
Ynt = w wrn Vl(rnpm � (11 . 2. 3)
In porous materials porosity P must be taken into account, i.e., the ratio
of pore volume to total volume. For a porous medium Km = Km/PK, where K is
the ratio of specific heats for constant pressure and volume.
For a harmonic process and a wave propagating in a porous material
i WPnt6 = YmP; Pne = n~ = prnr iPmi; (11.2.4)
1
Km = Km 1. tlt uii ` Px IKmr f' iKm't),
where m is a structural constant; 6c is the impedance constant.
Expression (11.2.4) is valid for the average air pressure in a fibrous
material [2]:
` r~tp c
}'nt iw 7, ( /p  i 1 P3G (Kmi `f` lKrn1)li
~ ~ /
. (11.2.5)
lVm = y~ ( pP  i ail 1Wrnr + iKmi) (px)x ,
~
For K it is customary to use K = 1 for an adiabatic, and K = 1.4 for an
isothermal process in a material. From (11.2.5) we have
Pmr  iPnsi = �m'?m .1 ;
iw
Kmr iKm! = j~m m i
Pmr _ ltt ' 1
P p = (AmWmi  umWm!) Pck '
I(I)Pmt I= Qc rmlbmr PrnWml I;
Km! _ (l~mrPm 'I' Wmium) k i
(Xin + 0in
Kmi ITmlAm) k .
~un1~n~
 351
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(11.2.6)
(11.2.7)
(11.2.8)
(11.2.9)
(11.2.10)
(11.2.11)
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Porous Panel. A rigid panel with holes, called perforations, is used in
soundabsorbing structures. The inertial impedance of one hole with area
s0 and diameter d[17] is
z
iXo = icD:b1'  icap K= i(DPso (t F2t'),
(11. 2.12)
where M' is conjugate mass; K is conductance.
The inertial impedance per unit area of a perforated panel, standardized
to Z, is
CK ,
. `Xl _ i o)
(11.2.13)
where K= (f 1�2r,) ; t' is the correction factor for the end of a hole.
The active acoustic impedance of a hole attributed to the viscosity of the
air flowing through it is
Ro = 32� (i ; 21') (11. 2.14)
d
for d~8 10
Ro = 2(1 21') 1~2P~� ' d�
(11.2.15)
_ The specific dimensionless acoustic impedance R1 of a panel is
Ri= oP~ � (11.2.16)
Formulas are given in Table 11.2 for calculating K, t' of holes of dif
ferent shape in cells of different configuration: for a square hole with
side a in a square cell with side al; for a round hole with diameter d in a
circular cell with diameter dl and a square cell with side al. The value t'
can be found in Figure 11.5, where 2t' _ ~I(~); I(~) is the ordinate of
the point with abscissa C, where a/al, d/dl (curves 1, 2) and d/al
(curve 3). For C < 0.4 [20]
21' = l/,
so 0,96 (i 1,25g), (11. 2 .17)
Resonant acoustic absorbers (RAA) are a perforated panel, installed at dis
tance Z1 from a rigid wall. Losses in RAA are caused either by the friction
of air in the holes, or by friction in a cloth or screen type of material,
inserted in the holes.
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a)
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FOR OFFICIAL USE ONLY
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~
so
0,8
o,s
2
0�4
a,2
u u,z qff qs gB Z;
Figure 11.5. End correction factor
for holes of different shape:
~ = d/dl; ~ = a/al (1, 2) ; ~ _
= d/ al (3) ; d, di diameters of
hole and cell; a, al sides of
square holes and cell.
The parameters of RAA are the following: s(the area of a cell, in which
_ there is one hole), and also the values s0, Z1, R1, Qc , for which the sound
absorption coefficient will be greater than or equal to prescribed value al
in the frequency range flf2, calculated by the formulas
`a' iz / . . (11. 2 .18)
1; ,
4nf1 Ifl  ai
2ai .
_ [Rl= ai 7
s 2cV1 a;
K  +'a1(is  fi) R' (11. 2 .19)
Far a specified panel thickness t and diameter d, al, the distance between
holes in a square cell with s= ai, is determined (for s/k > 4) by the
formula [2]: 
1,05d2 Yrl, lds }3,53Bd O,SSd)
ai  2(1 d 0,85d) ' (11. 2. 20)
The specific impedance of a materi al, inserted in holes, is
saZ (2  a;) _ nd2Z (2
(11.2.21)
a~ sat 9alai .
If the losses in th e holes are caused only by the viscosity of air, diameter
d is not assigned, but is determined by the expression
da _ 321t Ifl ai
(fz  fl)(2  al) ' (11.2.22)
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11.3. Laminated Soundproofing Structures. Space Absorbers. Membrane
Acoustic Absorbers
Laminated Acoustic Absorbers. Laminated absorbers consist of layers of
different materials (sometimes with a panel). The simplest structure is a
layer of material, applied on a wall with impedance Z2. In normal inci
dence the input impedance of the structure is
Zi = Wm1Z12 ch y! Wmi Sh 71 _ Wm cth (Yl
Z1,2 Sh YL Nrni Ch Y! Z 4')where Z is the thickness of the layer; ~ is the phase shift contributed by
~ impedance Z1,2; Z2, Wm, Z' are normal to Z, which is denoted by the sub
script 1; Wm/Z = iyml '
For Z12 ~ (a rigid wall) [10]:
Zl,. 0'cl Knti W/It! hmi
~ 3Z + w/'/, F t( c3P w12
_ For materials with large cs , for example a`l > K"'~ � Rl ;t:~.a,`' . For bulk
c 3Z w1Z ' aZ
materials Z �Kmi ~j~. FoT a layer, applied at distance Z1 from a rigid
wall (for Km  Kmr)' the input impedance is
Y] ' vc (Pxl  li) wm (Pxl f It) c
[cn=1a Zicn _ 3ZPr, + t [ c3P'r. w (Pr.! }11) (11. 3. 2)
_ Formulas (11.3.1a) and (11.3.2) are valid to a frequency that is 3040%
, higher than the resonance frequency of the layer (determined by the condi
tion Xilay 0)'
On low frequencies soundproofing structures in the form of a layer are
ineffective. Absorption on these frequencies is improved by using a per
forated panel, applied snugly or loosely to a sheet on the incidence side.
In the case of loose contact the impedance [2] is
[k=C ] Zi K= Zi cn I i c so (21' t). (11. 3. 3)
In snug contact
ziK = ~icn i s ~Ii ~ [(ui {1].
o .
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The maximum acoustic absorption coefficient
(AAC) a
is achieved when
max
 f= fres'
If the resonance frequency f
is given and it i
s required to
res
determine
(PKZ + Z1) and s/K, then
[Pea=res]
nt (Px[ !,)2 s (Pxl f!i)

Co

(st~=:c)
K
~tn~J= '
pes
(11.3.4a)
_ The best

practical values must be selected
from combinat.ions
of the values
of (PKZ +
Z1) and s/K. If the AAC must not
only be maximum
on frequency
f= fres'
but also have a certain value ama
x  al~ the parameters of a
_
structure
(with a snug fit) are calculated
by the follawing
formulas [2]:
sPy _ nc 1) R� �
L (�t
(Pxl f li) =
[ Q
/ki Z cu l
I)= l
)
'
~ ~11. 3.5~
l
2
61
3P=xrtf
~
pea.l
o
Ri = (2 al) _ V (..al)~ al
(Pv! l,) ais .
= 3P2r 2K '
(11
3
6)
.
.
s r2
(Pxl 1=1,)
crc
Q; Q
1
K 3P=;c
7 ,
i 
(11. 3. 7)
Cell distance al for given t and d are determined by expression (11.2.20).
Example of Calculation. For a structure with fres  400 Hz find al = 1.
 Determine Z, Z1, al and R1, if m= 2, P= 0.98, 61 = 0.2.
The panel has d= 5�103 m, t= 2�103 m. The formulas yield R1 =
= 1.1(PKZ + Z1) = 0.13 m; Z= 0.05 m; Z1 = 6�102 m, al = 1.8�102 m.
The values of a are: for f= 200 Hz a= 0.83; for f= 400 Hz a= 1; for
f= 1,0001,200 Hz a= 0.85; for f= 2,000 Hz a= 0.5.
It is recommended that one more layer of material be applied on the
acoustic incidence side of the partel to improve the absorption character
istics. If the frequency, for which ai on the boundary of the range of
AAC is smaller than desired, is f', then the thickness of the top layer may
be calculated by the formulas [5]1
c{ n~ 21 n T$o
~ /
12 2,cfi Vm:ti
So = arctg 2 [X  WmrI~R 14~mr)2 f  Wm[)a,
where R, X are components of the acoustic impedance of the bottom layer
(next to the panel), but expressed in fractions of Wm (of the wave impedance
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of the material of which the top layer is made), and taken on frequency
fi, and VYmr and Wmi are the active and inertial wave impedances of the
material of the top layer, but expressed in fractior.s of Z.
If on the abovedescribed structure is applied a layer of the same material
with Zz = 2�102 m(basc3 on calculation data), it is possible to reach the
value a= 0.8 for f= 200 Hz, a= 0.99 for f= 2,000 Hz and a= 0.9 for f=
= 1,600 Hz.
Influence of Film on Acoustic Absorption of Porous b4aterial. This
_ influence is determined basically by the inertia of the film, i.e., by
surface mass b9S. Films with MS < 0.020 kg/m2 have no influence on the
acoustic absorption coefficient. The values MS = 0.0200.070 kg/m2 are
taken into account by adding to the inertial impedance of the material the
 through impedance of the film X 2 :
~ [f1=s ] Xz = (ilA4n; Xi2 = MDG) ~
Z
where MS is the surface mass of the film, kg/m2.
Consequently X1c = X1 + X12.
Given in Table 11.3 are the values of X2 of films, used for lining fiber
materials. Films with M> 0.070 k 2
S_ g/m should not be used, since they sub
_ stantially reduce the acoustic absorption coefficient on high frequencies.
Space Acoustic Absorbers. Space absorbers (SA) are designed as separate
bodies (spheres, cubes, cones, etc.), made of soundabsorbing materials,
sometimes with a perforated cover. The working area of the absorber varies
due to diffraction. This variable area is called the effective area. The
ratio of effective area Qa to the actual area of a space absorber expresses
the change of the number of units of absorption due to the use of a
material not in the form of a film, but in the form of a space absorber.
This value, called the conditional acoustic absorption coefficient a~, for
an absorber in the form of a sphere with radius r, covered with a film of
material with impedance R1 + iXi = Z1, which should be locall in normal
_ incidence, is calculated by the formula [3, 4]
[Y=c; s ~ (2n 1)R1 sin (S~  b')
n=a] k~
~ (11.3.9)
11 n=0 Un 2 [Xl CGS (Sn  Sn,) R1 Slft )rt (Xi~'R;)~
n
1Impedance that does not depend on the angle of incidence is called local
impedance.
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63
7)
8)
9)
Table 11.3. Reactive Component X2 of Through Impedance of
Fi lms
1)
Tun nneIfKH
2).
s s
~
4) t12c7orz>i okraoiMIx nonoc, I'u

5~
0
"
Fc
a
v
o 11 ~
G~`s
63
125
250
I 500
I 1000
I 2000
I 4000
I 8000
flo.titt1tNtti'lrtas fl,Ni
40� 10"
0,057
0,05
0,11
0,21
0,43
0,85
1,70
3
41
6
82
Ti+na Capau (I(pe
30� 10'6
0,019
0,05
0,09
0,18
0,37
0,74
1,48
,
2
96
,
5
92
z a.ioti)
1
,
,
CreKa0rha~1b ntap
50� lU'6
0,050
0,05
0,09
0,19
0,37
0,75
1,99
2
99
5
98
r:it CTdD
,
,
II0.jE13rJ+.TC1ITCp4
25106
0,035
0,03
0,06
0,13
0,26
0,52
1,05
2,09
4
19
raAarnast
1 la1113rEi.7CFJ0UA
50�10'1;
25�10~
0,070
0,023
0,07
0
02
0,13
!1
04
0,26
0
08
0,52
17
0
1,05
0
34
2,09
0
69
4,19
1
37
,
8,37
2
50�10'a
0,0461
,
0,04
,
0,08
,
0,17
,
0,34
,
0,69
,
1,37
,
2,75
,75
5,50
Key: 1. Type of film
2. Thickness of film, m2
3. Surface mass, kg/m2
4. Frequencies of octave bands,
in Hz
5. Polyimide PM
6. Saran (Krekhalon)
7. Grade STF glass fiber cloth
8. Polyethylene terephthalate
9. Polyethylene
where Da(kr), Da(kr), 8a(kr), Sa(kr) are the amplitudes and phases of the
spherical Bessel functions and their derivatives, k= w/c.
Formula (11.3.9) is also applicable for ca.lculating ac of a cube with side
al = r/0.64, but the resulting values should be reduced by a factor of 1.25.
The calculated values of ac as a function of kr for R1 and X1, taken as
parameters, are nomograms. One such nomogram is presented in Figure 11.6,
where it is seen that the character of the dependence of ac on kr varies
for reactive impedances of different signs: when X1 < 0 ac is maximum, and
when X1 > 0 all the curves decline gradually as kr increases.
To maximize absorption in the lowfrequency range, i.e., where kr < 1, it
_ is necessary that R1 and X1 have the values
hr
1 (kr)' '
(hr)a (11. 3.10)
Ri 1 + (hr)a ' �
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ay
(0
0,6
0,8
Q6
~6
Op
~
1,0
f,0
I L
Q2
 �w n7 /,u e,l( cu 0,u 3,Ir Jn 4,7 4,b; kr
Figure 11.6. Nomogram for determining acoustic absorption
= coefficient on basis of known components X1 and R1 of
local impedance of surface of space absorber, or for
determining X1 and R1 for given AAC. [y=c]
The conditions of maximum absorption for kr 1 are plotted in general form
in Figure 11.7. A set of n space absorbers will be most effective when the
absorbers are placed at distances apart, such that areas Q will not over
a
lap. Then the total n.umber of absorption units, contributed by the n space
absorbers, will be Qa = n7rr2ac m2.
Membrane Acoustic Absorbers. These are absorbers whose front wall has
pliancy; they include a resonant acoustic absorber (RAA) with a pliant front wall. For such RAA the resonance frec{uency depends on the ratio
to 1 /w2 between the resonance frequencies, where wl is the frequency of the
cavity and w2 is the frequency of the wall.
The calculation is presented for a cell with area s. Let wl =41%M1; _
w2 2 2; S1 = pc2/sZl is the elasticity of the cavity resonator (air
space); SZ = e2/s2; e2 is the specific elasticity of the cell; M1 = ml/s2;
 ml = s0 p(t + 2t') is the mass of the air in the hole plus the connected
mass; M2 = m2/s2 is the mass of the cell.
When Nil/MZ = u< 1 and wl/w2< 1[16]
a "
c0ul~wi[1I/1EEt~
' ~uz ~ wa [ 1 [t  (2  }t)] wi 1 /1 f' u~~
~l .1 L w~
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There turn out to be two resonance frequencies, and w01 < wl' w02 > w 2'
If wl > w2 there again are two resonance frequencies:
~o~~Wi fli): >W i: w2,.;.wzCl (11.3.12)
19hen W1 = w2 there is one resonance frequency:
� k, /
�M0 R: 12 ' (11. 3.13)
a)
X, �
~A
O,B
0,6
0,4
. a?
~
/
.
0 Of5 0,7 0,9 f,f 1,3 115 1,7 1,9 2,1 ?,d 2,5 2,7kr
b) 
Rr
f,0
qB
0,6
O,li
4?
i
i
I
^
p ~
/
~
�
/
~
iL�3
,
_e .
/L
�
0 0,5 0,7 0,9 l,f f,3 1,5 l,'I l,9 2,1 2,J 2p 2,5 kr `
Figure 11.7. Reactive (a) and active (b) components of
local surface impedance of spherical space absorber (as
functions of kr), corresponding to perfect absorption of
spherical wave of given order (n = 0, 1, 1, 3, 4,
The absorption characteris*ic of a front wall witli~little damping of sound
may show a valley on that frequency.
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If the wall has no holes it is a membrane absorber, the resonance frequency
of which is
w = c J/~pi
(11.3.14)
.
11.4. Soundproofing Structures with Oblique and Diffusive Incidence _
All the calculations in 911.2 and �11.3 pertain to the case of normal
incidence of sound on structures. In practice it is possible to experience
oblique incidence, or nearly diffusive incidence, when all angles of
incidence are equiprobable, and the acoustic energy density in the room has
a uniform distribution.
However, if the field in a room was also diffuse before the installation of
a structure, the diffusivity usually will remain after the installation due
to acoustic absorption. In practice it is necessary to deal with condi
tions that are nearly diffuse. In oblique incidence of sound the impedance
is standardized in fractions of Z/cos 6(denoted by the symbol and
therefore the factor cos 6 appears. If the normal component of vibration
velocity Cp depends on incidence angle A, the impedance is called nonlocal.
For resonant acoustic absorbers, in which the air space is separated into
compartments with a width smaller than X,
Z5 =1?1cos0{i rws ks0_ctg t 1 cos0
L J
If there are no compartments, then the impedance [8] is
7.u R1cos0}1 r wscks0 _ctg(w! cs0 )I
(
11.4.2)
~ Here al < 0.2X, which means that the connected mass is independent of 6.
In the case of resonant acoustic absorbers without compartments the acous
tic absorption coefficient is maximum on the very same frequency for all A,
but amax will be different. In the latter case the maximum AAC will fall
on different frequencies for different A.
 The impedance of a layer applied on a rigid wall [21] is
zu =(Wmi cos 0/Z cos 111) cth (yl cos (11. 4. 3)
 where cos ka sin20,
_ ~V = l V, , ~ is the complex angle of refraction.
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For small y Z [7]
m
Zo~tk mpsACl ( I  I
,
!h=( sin=0  m  iin ~ 3
~
(11.4.4)
where SZ = w/W0 ; w0 is the characteristic frequency, equal to ojo = ctc P/mp.
When S2 < 1
i cps 0 NM_'___E__1+ cc! Q p sin 0 cos 9
ZD a 2 i
c
�c, M (11.4. 5)
3Z cos e = iYl cos o+ Ro,
_ where Y1 is the local reactive component of the impedance of the film.
Consequently, when SZ < 1 only the reactive component of the acoustic
impedance is local.
The impedance of a very thick film (Z is
Zl~m CI  a ~
Zw 1 i_ sln2(i Plx ' (11.4.6)
V C 63 mx )
� cosp(1(11.4.6a)
0 1 (bulk materials)
Zo = cos 01~m � 
sinaOxP (11.4.8)
m
The impedance of a layer, installed at distance Z1 from a rigid wall [15],
is
~[I%m, Cos O r~
w Wml cos 0 cos Sh (Ynil cos V)  t ctg k!i cos 0 sh (yml cos t( )
zo I. (11.4.9)
cos ~ ri ctg (kll cos 0) sii (y,,,l cos 1)) d cos U~~~
~ cos ~ (1'"�1 cos lp)
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If the angular function Ze or a is known it is possible to calculate the
diffuse acoustic absorption coefficient:
for plane absorbers
n/~
[A=d] uA = 2 I' uo sin 0 cos0d0, (11.4.10)
b
for space absorbers
lY=c]
uA = 9ay.
(11.4.11)
11.5. Acoustic Absorption in High Acoustic Pressure Levelsl
A level at which an amplitude fun ction of impedance and of tne acoustic
absorption coefficient of sound absorbing materials appears, in addition to
the frequency dependence, is call ed a high acoustic pressure level (HPL).
Impedance exhibits greatest nonlinearity in perforated panels, starting
 with HPL 100 dB, i.e., at level s that are frequent on ships. The
dependence of impedance and the a coustic absorption coefficient on HPL can
be determined experimentally or c alculated by empirical formulas.
 Data from measurements of linear impedance and of stationary flux impedance
_ Qc as a function of velocity [22] are used for calculating the impedance of
a thin film of porous material in the nonlinear range. In the linear range
acoustic and hydrodynamic impedances R and 6c are virtually identical. At
amplitudes of vibration velocity (for Qc at flux velocities), exceeding
several meters per second, the va lues of R and Qc begin to increase with
velocity, and a onetoone relati onship exists between them.
The components of the impedance o f a thin film of porous material with crc _
= 5 in the linear range, are plot ted in Figure 11.8 as functions of HPL.
The calculated and measured value s of the active component of impedance
agree quite well. The reactive part of acoustic impedance depends less on
velocity and stays in the linear mode longer.
The specific impedance of a single hole in a panel with HPL R= R0 + kp~,
where the nonlinear additive is proportional to the amplitude of the vibra
tion velocity in the hole, is t(k z 1).
When ~ crit the additive become s many times greater than Ro and entirely
determines the active losses in t h e hole. The value R= pk is the limiting
I 1This section was written by I. V. Lebedeva.
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f,
0"
0)
D,~
0,2
0
;vo
~
~o
~i
0
17U 140 P d6
PR,oN r,06
Figure 11.8. Comparison of experi Figure 11.9. Boundary acoustic pressure
mental and theoretical (calculated) levels of nonlinear impedance as func
components of impedance of thin tions of perforation coefficient n.
porous film: 6c = S rel, f= 1,250 [HpHT=crit]
Hz, Z1 = a/4. [86=dB]
value for the specific impedance of a hole of any diameter with HPL. In
consideration of the transformation of impedance from a hole to a cell in a
panel, this circumstance can be used for achieving high acoustic absorption
coefficients with largediameter holes in highintensity fields. Thin
panels exhibit stronger nonlinear effects.
Nonlineaz effects are strongest when tcrit > 10 m/s. In practice it is
better to use not tcrit' but the boundary HPL p crit, in Pa, of the incident
sound, which for the resonance frequency is estimated through the empirical
formula [19]:
[HPNT=CTit ] Pxpxr = 62 f 2090 n{ 140 tj2. (11. 5.1)
The boundary values of HPL (above which the nonlinear term prevails),
calculated by this formula, are given in Figure 11.9 as functions of per
foration coefficient n. Modern technology creates levels that exceed p
crit,
even for n= 400 (Pcrit  153.2 dB). The deviation of these values of
impedance from linear begins at substantially lower velocities > 1 m/s)
and at correspondingly smaller HPL.
Nonlinearity R can be expressed by the socalled "nonlinear end correction
factor" A nl'
[ Fin=n 1] R a l~s~c~p .(t 21' E Al~,)� (11. 5. 2)
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For holes with the parameters 0.5 cm < d< 2 cm, ri = 0.5100 one may use
the empirical dependence of the relative value Anl/d on HPL (in decibels):
log ([1,,n/d) =  1,685 + o,o is5p.
(11.5.3)
The reactive component of the acoustic impedance of a hole with HPL exceed
ing Pcrit is assumed to decrease to onehalf of the linear value. In con
sideration of this, and by determining R in accordance with (11.5.2), it is
possible to calculate the frequency characteristic of absorption for dif
ferent HPL values. As the impedance of a hole in the case of ordinary per
foration (n = 1120) increases amax increases in the range of HPL values
from 120 to approximately 140 dB, and then it begins to decrease, and the
resonance absorption curve expands. The resonance frequency increases as
the reactive component of impedance decreases.
h,
~
r,'    1
Cco sev 1:c~? 1.9116
0, 5
.
p15
~
='t M
i'Q
01.
X r.
~v i[u 7.'U 74U T50
n17Rg, ae
Figure 11.10. Frequency character Figure 11.11. Amplitude function of
istic of absorption coefficient of acoustic absorption coefficient of
panel with hole d= 0.2 cm for holes with different diameter (f =
_ different acoustic pressure levels. = 1,000 Hz, Z1 = X/4).
The results of the calculation are plotted in Figure 11.10; they coincide
satisfactorily with the experimental data for a hole with the parameters
t= 0.16 cm; d= 0.2 cm; n= 1.450; Z1 = 4.8 cm.
The sound absorption coefficient of holes with d= 0.15, 0.25, 0.5, 0.75 cm
in a cell with a diameter of 2.35 cm for ZI = a/4, in normal incidence, is
plotted in Figure 11.11 as a function of HPL (the measurements were done
with an interferometer, f= 1,000 Hz) [13]. The sound absorption
coefficient quickly reaches its maximum value, which is low, as R increases
in a panel with a smalldiameter hole. Largediameter holes are more
effective.
Nonlinear phenomena in H�L substantially alter the absorption characteristic
and often increase acoustic absorption, which should be taken into consider
ation during the design af acoustic absorbers for deadening highintensity
noise.
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11.6. hleasurement of Acoustic Absorptionl
Acoustic Interferometry. This method is used for measuring the acoustic
_ absorption coefficient (AAC) and impedance in normal instruments. A speci
men is placed in one end of a hollow cylindrical or square tube with rigid
walls. The diameter of the specimen is the same as the cross section of
the tube. A radiator, driven by an acoustic frequency generator, is placed
in the other end of the tuUe. The receiver is a probe (a hollow metal tube
34 mm in diameter), inserted in the tube and connected to a capsule con
 taining a microphone. Potential from the microphone is applied to a
metering amplifier with a filter (to eliminate harmonics). The acoustic
waves, striking the specimen, are reflected an d produce standing waves in
the tube.
The probe may introduce distortions, and therefore the socalled correction
factor for the probe must be determined. To do this a plate is placed over
the end of the tube and the coordinate of the first minimum, which in this
case should be theoretically at a distance of A1 =X/4, is marked. The
difference between the calculated and measured values of O1 is the correc
tion far_tor for the probe. The frequency dependence of this correction
factor must be determined and taken into account during the measurements.
The top frequency boundary, up
fb = 2�106/dl for a cylindrical
square interferometer with side
which measurements can be taken
= c/4Z.
to which measurements may be conducted, is
interferometer with diameter dl, and for a
ai it is fb = c/2a1. The frequency at
in an interferometer with length Z is f=
To measure the AAC it is necessary to measure the pressure at the maximum
pmax and minimum Pmin of the standing wave:
4d Pmax
u d~~ 1 ' d pmin ~ (11.6.1)
To measure impedance it is also necessary, in addition to d, to determine
A 1 and calculate phase shift 26:
26 _2sc (,1t  X!q) .
K/2 1 (11.6.2)
The components of impedance are calculated by the formulas
(da 1) sin 28
Y~ (dj i )  (r13  1) cus 26
'
Rl b 2d (11. 6. 3)
(d 1) (d=  1) cos 26
'
lI. V. Lebedeva contributed to this section.
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X1 and R1 can also be determined on the basis of the impedance diagram [17].
Measurement of Wave Parameters Wm and ym. This method is based on measure
ment of impedance Z1 of a film with thickness Z of the analyzed material,
placed on a rigid wall, and of impedance ZI of a film, applied over the
wall at distance Z1 = a/4:
147,n _ lY~ml '
Z =VZiZi :
1 I (1 R)' Xa
Y~n = ~ { ~ ln (1 _ R) a + Xa
T arctg I nrc 2} R X 1 1~ arctg [ nrc  1 XR
where R and X are the real and imaginary components of Z1/Z1I.
The values of Wm and ym can also be determined by measuring impedance Zi of
layer Z on a rigid wall, and of layer. 2Z  Z1 (also on a rigid wall). If
Zi/Z1 = u+ iv, then the expressions
Pm= ~ 1 rccos I+l~ll[(u1)zuZIZI(u1)l fval2 .
41 (u1)2 }u2 9, (11.6.6)
am= 1 arcch 1+1~121(�l)2 }val"+ I(�I)'fvIA1'
41 (u 1)2 u3 (11. 6 . 7)
are used for determining am and Sm (ym = am + ism). The radical is deter
mined by the conditions am > 0, a m > 0. The wave impedance is
 
wmZ ctlt (am ipm) ! _ Wmr iWml . (11. 6 . 8)
The values of Km and pm are calculated by (11.2.7, 8), whence
Pmr= pP IPmlI= ~ . (11.6.9)
The value ac = Ipmiwl on some frequency may coincide with the impedance of
the stationary flux, determined as the ratio of the pressure drop to the
space velocity.
Reverberation Technique. The reverberation technique is used for measuring
the acoustic absorption coefficient in a diffuse acoustic field, generated
in reverberation chambers (RC), which are large irregular rooms with smooth
walls, which reflect sound well. RC must have a separate foundation or a
floating floor for insulation from external noise and vibrations. The
recommended volume for RC is 200 � 20 m3; structures that are used in prac
tice are tested in them.
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 An RC is a threedimensional vibrating system, in which a certain spectrum
of natural frequencies is excited by an acoustic signal. Width B of the
resonance curves is determined by the losses in RC, which occur both on
boundary surfaces, and in the air mass. The absorption of RC walls is
characterized by the average acoustic absorption coefficient a= EaiSi/ESi
(ai is the AAC of a surface with area Siand the absorption in the space
is determined by the absorption coefficient m in air.
The acoustic characteristic of losses in a room is the standard reverbera
 tion time T the segment of time during which the intensity of sound,
after the source is turned off, falls to onemillionth (i.e., by 60 dB),
_ and the pressure to onethousandth of the initial values. In RC with a<
< 0.1 T, in seconds, is determined by Sebin's formula
T 0,16V ,
as 4mV (11. 6 .10)
Eyring's formula is more exact for a> 0.1:
T _ 0,16V
 s ln (1 a) 4rriV
Due to the frequency dependence of a and m the reverberation time decreases
monotonically as frequency increases. According to GOST To in a vacant
chamber with a volume of 200 m3 should be T0 > 5 s on low frequencies
(starting at 100 Hz) and T0 > 2 s on 4 kHz.
The field in RC is assumed to be diffuse if the natural vibration fre
quencies are so close that the frequency distance between them is Af < B/3.
This condition limits the effective range of RC on the lowfrequency end:
fb = 2,000v/T/V, where V, in m3, is the volume of RC.
The highfrequency end is limited by the strengthening of attenuation in
air, as a result of which the direct sound from the source begins to pre
dominate.
Acoustic absorption in a diffuse field is determined by measuring the
reverberation time T0 in an empty chamber and the reverberation time T in
the chamber with the specimen. Then the acoustic absorption coefficient of
the analyzed material with area S1 is 
[A= ] o,isv > >
d aA~ ~ + Si ~ ~ f. ~ ,o }  25 (m  mo)1. (11. 6 .12)
J
The first term takes into account the absorption of surface of RC shielded
by the specimen (usually a� a, and for calculation purposes it may be
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ignored). The last term is related to the change of absorption in air due
to changes of temperature and hi,imidity during the time of ineasurements.
T0 and T should be measured under t'tte identical conditions. Then
O,1Gl! l 1
ua Sl ( 1  7.0 (11. 6.13)
The measured values of a for a specimen of limited size, due to the so
called "boundary effect," caused by diffraction on the boundary of the
specimen, depend on the area of the specimen and may exceed unity [12].
For an RC with a volume of 200 m3 it is recommended that specimens with an
area of 1012 m2 be used in the form of a rectar.gle with a side ratio of
0.7:1. The specimen is mounted flush against the floor or its sides are
covered with wood strips to eliminate absorption by the side surfaces.
Diffusers in the form of randomly oriented, easily folded sheets of rigid
material, commensurable in size with wavelength, are susper.ded in RC to
maintain diffusivity after the soundabsorbing specimen is set up. A
homogeneous field in the stationary mode and linear decline of the acoustic
pressure level in decibels, during the time of reverberation, are indirect
indications of diffusivity.
The frequency dependence of the acoustic absorption coefficient should be
measured with thirdoctave bands of white noise with the measured values
 of ad expressed in terms of the mean geometric frequencies of the band.
The values of T, ay�iraged in time (three tests at one point) and in space
(the microphone is set up in tizree positions) should be substituted into
(11.6.13). To obtain statistical experimental data the test points should
be separated by distance Z>X/2 from the walls, absorbing specimen, sound
source and each other.
The recordings, made with a recording instrument, are processed in the
pressure drop range of from 5 to 35 dB from the stationary level. The
 reverberation time is determined on the basis of the slope of the rever
 beration curves with a special protractor, included in the test instrument
kit. Recordings with a nonlinear curve are not used in the calculation.
The values of a may also be measured in reverberation chambers in the
 stationary mode, using the socalled "method of intensities." The energy
density, or the mean square acoustic pressure in the empty chamber and
in the chamber with the specimen , proportional to the energy density,
at the same source output power is determined for this purpose. After
 determini P
bieasurements can be conducted in the stationary mode not only with third
 octave, but also with narrower noise bands, with multitone and discrete
frequencies. In these cases the acoustic field in the reverberation
chamber is described, respectively, by the average measured value, by the
limits of deviation from the average (for Pxample of standard dispersion
62), and by the fiducial probability of these limits Piri/i; i2 >l2 _/i12; i3>Y312; ~ (12.2.2)
'a Js 11 /3; i5 13; L''~
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The impedances of the elements of this system for a harmonic vibration
process arel
Zi icomi; Z. Z3
Iw
'L, C_ ; Z,p X f iY� (12.2.3)
I 
We will determine below the values that characterize the acoustic effect of
vibration damping. From expressions (12.2.2) in consideration of I1, I25
1 3 according to (12.2.1) and impedances according to (12.2.3), it follows
that:
the difference of the vibration velocities in the first sh,ck absorption
stage is
Yi ~i
y9 i3
and in the second stage
y3 i `a ~
ys S
the differences of the vibration velocities and of the forces on the entire _
shock absorption system are
.
n f . n Fo ~ Fo .
b ys ~s , F=
the difference of the vibration forces transmitted from the mounts of a
machine to the foundation is
rlp n  Fo  %WmtJi , Fo  %omlii
[n=mo) z'bi5 �
To determine the vibration damping of a twostage shock absorber it is
necessary also to examine a system, corresponding to a rigid mount of a
machine (without an extra frame, Figures 12.5c, d). The current ir in it, ~
corresponding to the vibration velocity of the foundation with a rigid
mount, is _
U
[m=r; O=f] ym = Zi i 4 .
The total vibration damping of the two stages of the shock absorbers is
[BH=VD] BH = YK _ ~ FF~ .y U~ = i?
1Here and in the ensuing analysis (until the end of the chapter) V__1 is
denoted not through i, but through j. This is done in order to avoid con
fusion with current i[see, for instance, formula (12.2.2)].
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Calc.ulatioii by blatrix Method (Using Primarily Fundamentals of nPole
Tlleory). The fundamentals of the acoustic calculation of tlie effect of
installing shock absorbers using the tool of inechanical quadrupoles were
presented in the preceding section. The coefficients of the quadrupoles,
corresponding to complex vibrationdamping elements, are determined
basically experimentally. Considerable possibilities for analysis are
available also if the coefficients of the equivalent vibrationdamping
elements of c{uadrupoles are found directly from the known physical constants
of the material of the elements and from their geometric dimensions. This
kind of representation is perfectly valid for shock absorbers with the
simplest vibrationdamping elements (elastic pads and KAS [not further
identified] absorbers) and for certain nonbearing structures.
For this kind of homogeneous mechanic element with distributed parameters
(including dissipative) and with harmonic longitudinal vibrations, the
coefficients of the equivalent quadrupole, entering in formula (1.7.1) and
its analogies are (see, for example, [6]) :
A~~ch I(a +lk) 11 D;
B= pc ( l~ j 2~ Ssh [(u rjh) lJ pcSsh ~I/~) ~l;
~
(12.2.4)
C 2
PcS sh [(a + ik) PcS Sh [(a + ik) 1],
Here Z and S are the length and cross section area of a given element of a
vibration conductor; k is the wave number; a is the dissipation constant,
a= ~ 2; c is the speed of sound in the material ri < 0.3 is the loss
coefficient.
For the homogeneous part of any vibration cenductor with longitudinal
symmetry the diagonal terms of the matrix of the coefficients (A and D) are
equal to each other.
In the case of extremely small losses, which can be the case in a metallic
vibration conductor, a~ 0, and the coefficients of the elementary
mechanical quadrupole with distributed constants are
A= cli jl;l = cos kt = U;
BpcSsli jkl 1PcS sin k1; ( (12. 2. 5)
C = ~Ssh jk!  PS sin kf. ~
These coefficients for lowfrequency vibrations are given in matrices
 (1.7.8) and (1.7.9). In consideration of these expressions and of the
_ dependence for seriesconnected (mechanically) elements, expressed by
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matrix (1.7.3), we may write for the system illustrated in Figure 12.5a
the equation (the symbols are indicated in the figure):
[Fol 1 jc�rrii 1 0 1 jcunM I 0 F:b
10=f ] Ji o t ico 1 o i /c1 I i (12.2.6)
Cl C., ~ h
From this and analogous equations we can derive two algebraic equations by
multiplying the matrices using the "row by column" rule [3].
iVith the relation Ff = Zfyf we can determine from equation (12.2.6) the
differences of the vibration forces on a structure. The analogous expres
sion for a rigid mount (Figure 12.5c) will be
Fa l'O jcuntlI I F~p,;~
[r~=r] 14j  I y ,p;KJ ~ (12. 2. 7)
whence, in combination with the eQuality Ffr = Zfyfr and the results of
calculation by formula (12.2.6), we can determine the vibration damping of
a structure, which, in dB, is
20 !p
F~b
For bending vibrations the equation of an elementary quadrupole will acquire
the form (the symbols are given in Figure 1.24)
 
yuti~~
 S (k,{1) ! l 1 
T(k111) . U(k,, a V(kuI)
k~t kiil3 k1I3
 
Jtn!(u
~ilw
k�l' (knl) S (kul) i T (kut) ~ U (kn!)
~Pzn/w
[H=b]
_
k�B h~e
x
~ (12.2.8)
;Yi,
ki !3U (F�1) k,,BV (k,~l) S (k�!) I T (knl)
Ms
k�
_kHBT (knl) k;,BU (k�!) kl,V (k�l) S (k�1) _
 F~ 
where kb is the wave number for bending vibrations; B= EJ is the bending
rigidity; S(kbZ), T(kbZ), U(kbZ), V(kbZ) are Krylov functions [4]. The
matrices in equation (12.2.8) are obtained from matrices presented in the
literature [4] by means of certain transformations (introduction of kb and
B and utilization of y and $ instead of y and fl .
Calculation of Parameters of Vibration Process in Branched ShockAbsorbing
System by Matrix Procedure. Let several units j8], corresponding to
machines installed on an nstage shockabsorbing system, be installed on a
common shockabsorbing frame (Figure 12.6a). When the top of one unit is
acted upon by vibration force F0 the transformation matrix of a unit will be
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n,
lb1i=t = n (!Ht),
t=i
where Mi are the matrices of the elements of the unit. To determine the
reaction of the other units to the frame it is necessary to find their
input impedances on the frame side. The transformation matrix of the jth
unit on the frame side is equal to the product of the matrices of the ele
ments from bottom to top:
]Z(,tif i )
~
[6n=u] r=,l I C~11 D6,IJ
1 f ~Gnl BGn1
where n3 . is the number of elements in the jth branch. To determine the
input impedance of the jth unit on the frame side it is necessary to
determine it from the frame and to apply to it an arbitrary vibration force
Fin. Here the equation
Fnx ~ L
[BX=1ri; BbIX=OUt ~
~
 A6ni BGn/ J L0 J
Yex C5.1; DSn, yadx
is valid, where yin and Yout are the vibration velocities at the input and
output of the unit. According to the last.equation the input impedance of
the jth unit is 
Zp _ Fex
ex~  Jex
a) D
.1'1 Fo ' J r J=k b) .
9o I I
is1 ~ I i1
I i=1 I
M~ ~ I
i=n, ~ L=n~ I i=raK
I 17pZ6sj
� i j'=1 ~
I
~ I � 1 r~ ~ ~
~ Zq, FIP ~
4
ZIP F47
yIp .
Figure 12.6. Calculation of effect of branched shock
absorbing systems. [O=f; ex=in; p=fr]
It can be shown [8] that the input impedances of branches on the frame side
_ enter in the calculation system as dipoles, i.e., their summary (except the
first branch) transformation matr�ix is
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k ' k
~1 Zx86'i
j_! ~r U~!
1 i
[0,
0 (
The overall matrix equation for vibrations in a structure will acquire tlte
form
 k
J n, 1 1 BG.nj 1 F4,
[Pfr] li~ (:11i) ;t 0 ! 1 D6n, X A1p X~(A1i,) X Ffi ~ (12. 2. 9)
i ' 1 ZT
I
where Mfr is the transformation matrix of the frame, and Mil is the same
for the j'th vibrationdamping element under the frame (the number of
elements is Z, see Figure 12.6a).
Thus, by introducing the input impedances of extra branches it is possible
to reduce the total transformation matrix of a branched structure tb the
product of a series of matrices, i.e., to a form characteristic of non
branched, chainlike structures.
From (12.2.9) we can derive an equation for'determining the effect of
vibration damping of one or several vibrationdamping elements in a
branched structure. In the structure illustrated in Figure 12.6a let us
determine the combined effect of the vibration damping of two elements: of
some element in a branch, where force FD is applied (to this element we
assign the subscript s), and the tth element in the assembly, located
under the frame. It follows from equation (12.2.9) that
[m=r; O=f]
1�'o S1 n,
 n (Ali) X e,; n(,tirt) x I
yom r=1 r=s+l
0
ri
xMn x II (,til,.) xE x
k
Y B6ai
, D6,; x
i=
1
(12.2.10)
l F(~*
~ (m; X Fepm ,
4
_ where unit matrices E indicate the locations of the absolished elements;
the subscript "r11 in the expression for the vibration velocity at the input
and vibration force at the output of the structure refers to a rigid connec
tion between the elements, contiguous with the absolished elements.
By taking Ffr from (12.2.10) and Ff from (12.2.9), we can determine the
total vibration effect of the sth and tth elements under the given condi
tions.
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If the vibration force acts not in one branch, but in m branches, then, by
writing for each of the branches an equation of the type (12.2.9) and ~
determining from them forces Ffl on the foundation, we can find the total
averagE+ vibration force FfE on the foundation using the expression
rn
i=1
For a simpler structure, consisting of two branches (Figure 12.11b), acted
upon by relatively lowfrequency vibration, equation (12.2.9) is simplified:
l r1 jwrt91l 1 0 Zo~= [I0 jca;Ylp 1 0 F~
[ex=in; J Lo 1 J1 . l[1o 1 ~ t ] i(t) F~ . (12. 2.11)
12p=fr] C l 3 1 Z4,
The symbols of the parameters of the elements, which on these frequencies
represent bunched mass and rigidity, are given in Figure 12.6.
The input impedance of the second branch "from the bottom" Zin2 can be
found from the equation
LCz DzJ W ~1 L~ 1wM=J'
C2
whence
r B2 1
Ztix~ _  D. = 1~M2 w 21
t _ (1 (12.2. 12)
\ ~oz /
where w02 is the partial circular frequency of the second branch:
Woa = ~ Mz Using equation (12.2.11) we can determine the difference of vibration
forces F0 /Ff, and expressing the terms of the last matrix in this equation
in the form yfZf and yf, we can determine the difference of vibration
velocities y0 /yf in the analogous shock absorber system.
Substituting, as before, rigidity matrices C1 or C2 (or both together) with
unit matrices, we can determine from the equation thus derived the vibra
tion force (or velocity) on the foundation with these shock absorbers
eliminated (i.e., when the corresponding elements M1 and Mfr are rigidly
fastened). Then, by comparing the vibration parameters on the rigidly
fastened foundation and with shock absorbers, we can determine their
vibration damping. To determine the effect of shock absorption with some
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inertial element excluded we replace tlle matrix in (12.2.11), corresponding
to that inertial element, with the unit matrix.
Calculation of Acoustic Effect of Installation of Shock Absorber Using
Orthogonal Polynomials. This method is an extension of the quadrupole
technique in application to chain structures consisting of n series
connected identical component complexes (assemblies), each of which, in
turn, may consist of m vibration elements (in the special case n or m may
be equal to unity). Orthogonal polynomials of the Hegenbauer type of a
complex argument are used for this purpose. The identity of recursive
relations, demonstrated by Parody [2] for the described polynomials and for
_ passive linear electric (and consequently mechanical) quadrupoles, is the
foundation of this approach. The method is examined in the literature [7]
in application to complex vibrationdamping systems, and the expressions
that follow are taken from the cited work.
In the general case the difference of the vibration forces and vibration
velocities for a structure of n identical assemblies (Figure 12.7a) (the
symbols are in the figure) is
Fn = tiCnI (x)  Cri2 (x); (12 . 2 . 13)
yo = tzC ~I (X)  C ~2 (X), (12 . 2 . 14)
where yn
f1=Aa+ L' + t:=CaZ(b +Da+,
Z'~, (12.2.15)
Aa, Ba, Ca, Da are the coefficients of the mechanical quadrupoles that are
equivalent to each of the identical assemblies of elements. The Hegenbauer
polynomials entering in formulas (12.2.13) and (12.2.14), of somewhat lower
orders (the number of assemblies is usually small in vibration technology)
for the generating function, equal to unity, are expressed as
Ci (x) 1; Cl (.e) = 4x  1;
_ rl(X)2X; Cj(X)HX3 4X'. } (I2.2.16)
The argument of the polynomials is
. x= 2 (AaTnu)� (I2.2.17)
From equations (12.2.13) and (12.2.14), in consideration of the trivial
relation Fn = Zfyn, we can derive the input mechanical impedance of a
structure, closed on impedance Z f :
t
Zax = Fn _ Z~ 11C,ti (x)  Cin2 ("L) .
yo t2C,ti (�r)  C t_z (x) (12 . 2. 18 )
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a) Fo ZBr b)
Yo Fc ~v .
FrP Z re D N= 2131 ~ ZM
Zn
nD " 32 N '
i i �T
Ysl ~ I n ; n=~
c)
39
N3 32
31
Zh
ZR
Zok
Fn Zo Fn Zpyn FnZ~ yn
Figure 12.7. Calculation of multiple assembly chain vibra
tion damping systems using orthogonal polynomials.
[ex=in; $=f; 3=E]
We also present expressions for the vibration force, vibration velocity and
mechanical impedance at the input of the pth assembly from the end of a
structure closed on impedance Zf:
P  [t1CP1 ('C)  Cp_2 F,:  tlCi1(X)  Ci2 (X) Fo~ 12. 2. 19
t1C�_1(x)  Cn_2 (x) ~ )
Jn_P = 112CP1 (a)  Cp: (.C)J yn = t zCPI (�Y) Cp2 (X) (12 . 2 . 20)
t_Crii (x)  Cn2 (x) yo'
Znp Fn'p _ Zo t tCPI (X)  CP2 (X~ (12)
.2.21
ynP t zCP1 (x)  CA2 (x)
The application of some of the expressions derived above for determining the
differences of vibration on complex (multistage) structures and their
vibration damping Ais examined below.
ThreeStage Sho;.k Absorber on Foundation with Arbitrary Impedance. In this
case there will be three seriesconnected identical assemblies (n = 3),
. each consisting of two elements E1 and E2, N= 2(Figure 12.7b).
From expressions (12.2.13)(12.2.17) for n= 3 we find the differences of
the vibration forces and velocities on the structure in the form
nF Fn (Aa Z~� ) [(Aa + Da)`  1]. (Aa Da); (12 . 2. 22)
ny  yn  (CQZ'b 2 Da) [(Aa DQ)2  1J  (Aa + Da)� (12. 2. 23)
In the last expressions the coefficients of the compound quadrupoles,
equivalent to each of the assemblies, are determined by multiplying the
matrices of two components of the assemblies of elementary quadrupoles
[see (1.7.3)]:
(Aa BQl fA;,~ B~~l f~'1R Bn
[np ] LCa DaJ  LCM A5, J LCn An ~
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where the subscript "m" indicates the parameters of inetal structures (t}le
machine and intermediate frames), and the subscript "p" indicates the
parameters of elastic pads.
In consideration of the expressions for the coefficients of quadrupoles
with losses [formula (12.2.4)], the parameters of compound quadrupoles in
this case are written as
Aa = ch (Yl)~ ch (YI)n ~ Z Sh (YI)K sh (VI)n; 1
Ba = Zn ch (1'I)n, Sh (Yl)n Za, sh (}+l),t ch (Yj)n;
Ca Znt Sh (Y!)x, ch (Yl)n Z~ cli (yl)~~ Sh (1'1)n; (12. 2. 24)
D� Z" sh (Y sh (Y!) ch 1 + (l)Rt ch (Y1)n
Zni .
The following symbols are used in the last expressions for brevity:
_ Zm _ (PCS)a, (I + 1 2M ) i Zn = (PCS)n ~I f j 2n 1 ;
� 1
W ~ (12.2.25)
+ ! W
YM= 2 1~ ; Yn=~ 2 ~ .
M x n n
TwoStage Shock Absorber with Deadened Niounts on Foundation with Finite
Impedance. In this case n= 2, N= 3(Figure 12.7c). The differences of
the vibration forces and vibration velocities, according to expressions
(12.2.13)(12.2.17), are expressed as
, o
nF = Fn  (Aa Z~ ) (Aa + Da)  ~ ~ (12 . 2 . 26)
(CaZ41 + D.) (`~a ~ Da)1. (12 . 2 . 2 7)
yn
The coefficients Aa, Ba, Ca, Da are determined in this case by the products
of three matrices, corresponding to the elements E1, E2, E3 [7].
The vibration damping of certain elements of a vibration conductor is
determined by comparing the vibration forces (or velocities) on the founda
tion in the absence of.a given element and with it installed. By using
Hegenbauer polynomials it is possible to find tb.e vibration damping for two
cases.
1. Identical elements (for example, vibrationdamping pads) are absolished
immediately in all assemblies of the structure. When some element is
excluded from all assemblies of a structure the true vibration damping of
the elements excluded from the calculation is expressed as the ratio
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Fn Jn
Fn
[B{~=VD] BI~i = =`v (12. 2. 28)
N_t Nnl ~
where N is the number of elements in the assembly.
When the number of assemblies is n= 2 and the number of elements in them
is N= 3(see Figure 12.7c) the true vibration damping of two stages of
shock absorbers is
Fn (,1a 13n17,,~) (An T Du)  (
f,fi r BN "2, A3 A'=3 A'=:i A'~3 .V_:3
LW1~ 1'r1 (aa 13u'Zq,) (A a'7 Va) 1 (I2.2.29)
n=?, N=? A'= 2 N=^_ N=2 N=S .
where the coefficients with the subscript N= 3 are taken from (12.2.24).
 2. Several identical assemblies (or one) are abolished simultaneously. By
comparing the vibration force on the foundation with the total number of
assemblies (for example, see Figure 12.7b) and on the foundation with the
number of assemblies minus one, we find the vibration damping of the
excluded assembly in the form
F� (A~ BU Z \ /A' D 1 ( 1~ U
I311 n_'g. _ `V? :~'=2 11,~ an'~r_a~ ~ l~r ��~'~r.� 1
Fn / A. $a; L~) / Aa i Ua ~  1 ' (12 . 2 . 30)
n2, lY=2 lA~_3 A'_~, h'_ 2
where the coefficients Aa, Ba, Ca, Da are taken from the literature [7].
Vibration Damping with Diffuse Vibration Field in VibrationDamped Object
(VibrationActive Mechanism) and in Foundation. The vibration field in
metal structures on medium and high acoustic frequencies can be diffuse
under certain conditions. For the derivation in which we are interested we
will use the successive approximation procedure. When the machine is
rigidly fastened to the foundation the average energy density wm+f in this
combined space (Figure 12.8a) is
E
y�+.V~a' (12.2.31)
where E is the energy of the vibration source (assumed to be constant); V
and Vf are the volumes of the mechanism and foundation. m
Through VD' we denote the "original" vibration damping effectiveness of a
shockabsorbing mount, characterized approximately by the ratio of vibra
tion energy E, radiated in the mechanism, to energy E', traiismitted through
the shock absorbers into the foundation (Figure 12.8b):
BH' = E/E'. (12 . 2. 32)
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a)
� E
VM
mM
Vo w(n+,P)
mq,
b)
Figure 12.8. Determination of influence of foundation
parameters on vibrationdamping effectiveness of shock
absorber.
The energy density in the foundation in the presence of shock absorbers is
w4, = E'lvc~� (12 . 2 . 33)
From the above expressions we find the shockabsorbing effectiveness of the
shock absorbers under these conditions in the form
BN =w++d' _ E V,p _ BH, y,p
. r~~ E' VM I V(b Vm V(b (12 . 2. 34)
If the average densities of the mechanism and foundation are equal, then,
as follows from the last equation, in a diffuse acoustic field
BN = BH' m4' (12 . 2. 35)
mM m,p '
where mm and mf are the masses of the mechanism and foundation, respectively.
= As can be seen, the vibrationdamping effectiveness of the shockabsorbing
mount increases with the ratio of the mass of the foundation to the mass of
the machine. When mf/mm > 2 the loss of acoustic insulation does not
exceed 3 dB in comparison with the case of an infinitely massive foundation.
_ Difference of Vibration on Shock Absorbers as Function of Mass Ratio of
Foundation and Machine. The vibration energy density in a vibrationdamped
machine is
EE' I E
VM E',CE V� '
The difference IIe of the vibration energy on the shock absorbers is equal
to the ratio of wm to wf [see (12.2.33)]:
EVq, , V~p BN, m~
[3=e] n3= vMEr = B~ VM lpm=pcb mM .
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From (12.2.36) and (12.2.35) we find the ratio of the difference of the
vibration energy to vibration damping:
B 3 nM ' (12 . 2. 37)
Thus, the larger the ratio mf/mm the more the difference of the vibration
of the shock absorbers exceeds their vibration damping, even though the
latter also increases [see formula (12.2.35)].
In twostage shock absorbers on a foundation with an arbitrary impedance
the relationship between the difference of vibration and vibration damping
is more complicated, and the differen ce between the gradient and vibration
damping is greater than in singlestage shock absorbers [9].
 12.3. Energy Analysis of Effectiveness of ShockAbsorbing Mounts
It is often difficult to give an exact description of vibrationdamping
systems in terms of kinematic parameters. In such cases it is helpful to
use energy parameters jllJ.
To analyze the effectiveness of shock absorbers (of shockabsorbing
structures) we will use the following energy parameters: 
vibration power, absorbed by the vibrationdamping system, Na;
energy absorption coefficient IIaw, determined as the ratio of powers Naf
and Nf, where Naf is the vibration power absorbed in the shock absorber
foundation system:
Na~
nacv = w ~
energy vibrationdamping coefficient Bw = N~/Nf, determined as the ratio
of power Nf, radiated into the foundation in the absence of shock absorp
tion, to power Nf, radiated into the foundation in the presence of shock
absorption.
Case of Unidirectional Vibrations of Machine with One Point of Contact
with Shock Absorber. The shock absorber absorbs the power
2
N. = Na4,  N(b = I 9y~ I (Re IZa xx (Z~ + ZQ K3) (Za xx d Z4J`J
[xx=id; I Zq) F Taxx I"
/ l 3 7
K3=s cJ  Re Z~p ( Za Xx l7Q %X  Za,K3) t~
~ 1  I Z IF(I3L Z~ Ig {[~C (7aM IZ8M + Gdl IZ 
I
~zBM4) (7'2N + T'd)>� J I Z8h11~1 I2 Re Zd11,
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Where Zaid' z asc' Zf are the impedances of the shock absorber (in the idle
and short circuit modes) and foundation; Zan, Zamf are the input and
transient impedances of the shock absorbers.
The energy absorption coefficient of a shock absorber [11] is
TI _ Re [Za xr (7.~ + Z. K3) (TQ !CX L Z4,)']
I Za xr (7a rr  7. x3) I Ile Z4,
_ Re 1Zanr I Za,a j ZtbIZ  ZaM4) (Zani + Z(~),]
~ Zati+(b 1= RC Z4.
The vibration energy damping coefficient of the shock absorber is _
B (Z~ Z. xa)2 1 Z~ (Zo Z. xx) T Za xx (Zo Z.
xs~ ~
vr = Z _
I 0 TZ4,12 r LZQSY (ZQXXZRK3~~
= I (ZM 7aM)(28M + .Z4))  zBYfp IZ � �
.
I Z. T Zo 121 Z3M4l 12
It is often desirable to determine the energy vibration parameters on the
basis of the impedances of the shock absorber and vibration velocities of
the machine and foundation. This can be done, since the gradient on a
 shock absorber is a function of Zf and the influence of Zf on the vibration
energy flux in the foundation, and also the amount of absorption of energy
in the shock absorbers can be taken into account [11]:
1V4, = 2 Re (Q~qj) 2 I qb I2 Re Zb  2 Re (ZaM~99~,)
� ~ .
  2 [Re Ze,, + Re,(ZeM,,II
1
Na~ = 2 Re (Q9') = 2 I 4 12 Re Ze,~ f 2 Re (Ze~q'9~) = .
' 2 I 9 Ia I Re Ze� Re ( ZeM~ I
aq
Iy 111 QQ Iz
1
Na  2 I 9b Ia [I na'v I' Re Zee~
Re (Z81AfpII' 121 9 12 f Re Ze~ I n. I9a0 J
aq
Re (ZeM~,no' ) J
Re Zaw Q
s .
I apl
Re Ze,b f Re ~Za,(bIIQ'v) Re (Z8..bIIQv
The energy absorption coefficient of a shock absorber is
I QQ 12 Re ZaM Re (ZeM~II.)
n�u, Ii Re Ze~ + Re (Ze,,xIIai) '
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_ In a symmetric shock absorber
I TIav. Iz Re Z T Re (Z A�. )
aM aw~ av
na"'  Re Zem Re (ZeN4, IIO '
_ Na= 2 I 9~ 1'(ReZaw(InaQl�1) ImZaxb Im17QQj.
The effect of vibration damping of vibration energy by shock absorbers is
attributed to the establishment of conditions for the vibration force that
acts in a source that are unfavorable for the excitation of vibrations in
foundation structures. If a machine rigidly fastened to a foundation is
acted upon by the force Qf
H ZO
[H=s] Q~=Q zx+z0'
_ the ratio between the vibration powers, absorbed in the machine N and
radiated in the foundation N f is m
NM Re Zu
N~  ReZo ' (12. 3.1)
Let us examine vibrations in the frequency range up to wave resonances
inside a shock absorber. In this frequency range the input impedance of a
shock absorber is equal to the transient impedance and is elastic in nature:
Sa (1 f1~la)
Za  iW ,
where na is the loss coefficient.
Then
ZaZ,p
Z�`~ Za f  Zqr
and ReZ~I ZQI~~ReZQIZwI2
ReZQ~= [Zalz~IZ~bIaF'2(KeZaReZ(pImZQImZd,) '
With shock absorbers (in the examined frequency range) the foundation is
acted upon by the force
QQ~b = QH  ZMZ+ 4 �
Usually Za < Zf, and then
� ZQ
ZQ4,Za and Qa4,ZF, ZQ '
In this connection, and also because internal vibration forces on fre
quencies above the frequency of free vibrations of a machine on shock
absorbers Zm > Za is expended basically on the excitation of vibrations of
the machine itself
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Nat � ReZ,, (12.3.2) 
Ala4, Re ZQ �
For example, when the loss coefficient in the material of a shock absorber 
is small (pa + 0) and Za < Z f
Re ZQ4, Re Z4, IZu F Re Zkb ~
14 I' ~ III I_ (12.3.3)
a
where IIa is the gradient of vibration on the shock absorber.
The expression for Nm/Naf acquires the form
A'a, Re Z,t TI .
NT  Re Z4, I v ~ (12 . 3. 4)
_ A comparison of (12.3.1) with (12.3.4) shows that with a shock absorber the
energy absorbed in the machine in relation to the energy radiated increases
by a factor of IIIa12 in comparison with the case of a machine without a
shock absorber. The reason for this is that in the case when a machine is
installed on shock absorbers only the 1/111 Q .l2th fraction of the active
impedance of the foundation is connected to the machine [see expression
(12.3.3)]. The ratio between the powers absorbed in a machine in the
absence of shock absorption Nm and with it Nm is 
IVM 41 i Z~ 2
I ZM T Za IWhen Za < Zm < Z f
u 2
VNZ�I'
i.e., the energy absorbed in the machine increases sharply.
Let us examine three characteristic cases.
1. The case when Zf � Zm � Za and na � nf� Then ReZaf � ReZa and
z Z. z
rI,~, , u
Re Z ~ B. ~ 1 � 
D I Zu IIZ I ZaI' (12.3.5)
The relation between the coefficients of absorption and vibration damping in
terms of energy is
B,v _ Re Z,p 1Z� Iz
17a Re ZQ I Z,y Ia '
In the case when there is no active impedance in the shock absorbers (fl _
12 a
= 0) Re Ta,p ReZ,p ~ Z~ 12 and, as can be seen by (12.3.5), llaW = 1. This is
obvious, since in the absence of losses in a shock absorber all the energy
radiated by the machine is transmitted into the foundation.
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Then
z
BI:J I Z.,1
 I1(t[:1 1 Z. I '
In the case at hand the vibration power radiated into the shock absorber
foundation system is proportional to the active impedance of the shock _
absorber:
1QIIz
[$=f; H=s] Na~ IZM12 ReZa.:ReZa.
Under these conditions the use of a shock absorber with a large r1a leads to
some increase of Naf, but the power absorbed by the shock absorber
increases simultaneously:
Ro~N ReZa n.I2NRe7Q (ImZ,b)a
, ReZb ~ 9 ReZd' (ImZQ), (I }1a) As a result the power radiated into the foundation is
N(~ _ ~ua~ L , (1 .~Q)� (12.3. 6)
Expression (12.3.6) shows, for example, that when the coefficient ri a
increases from 0.15 to 0.3 the power radiated into the foundation increases
, by 7%.
2. The case when Z m � Z f� Z a. In this case the excitation of the shock
_ The expressions for coefficients IIaW and BW are
n Re Z�~p N ~ Re Z~ ~ 11v I2 . RRe e ZZ~a p I~a
B IZ ~or ZQ < Z~and~lQ~ ~l~);
I Za '~"Zcb ~
 I Za Ie ~ q ~z;
n~, " Re ZQ for ~1Q } 0 n~'nw v
~ ~
absorberfoundation system is kinematic in nature. The vibration velocity
of the foundation changes in proportion to the vibration gradient on the
shock absorbers. Therefore the vibration damping coefficient in terms of
vibration Irelocity Ba is equal to the vibration gradient: Ba ~ IIa.
3. The case when Zf � Zm and '.f � Za. Then
Re Z� I Z~U I~ . I Z., f Za Iz
ReZ,p jZul' ' B"'~ IZaI= .
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Case of Unidirectional Vibrations of MachineShock AbsorberFoundation
System with Multipoint Contact. The vibration power radiat' into the
foundation is calculated in this case by the formula
m m nt
N`p 2 Ej'qhj ReZ";` f Re (T'~k9d, q ~fi
~~=1 rt=l k=t
k#.rt
wnerekZfn and Zfk are the point and transient impedances of the foundation;
qf' qf are the vibration velocities of the foundation at points n and k,
i.e., in contrast to the previous case, additional information is needed
about the transient impedance of the foundation and the degree of inter
action of the vibration velocities at different points of the foundation
[11].
The energy flux through each shock absorber, and then through all the shock
absorbers, can be calculated on the basis of data on the impedance of the
shock absorbers and on the vibration velocities at their inputs and out
puts:
m ni 
A'~ Re(Qp9V)~ 2~ Iq~l2(ReZa~,Re(Za~~,IlQQ)];
. n=J rt=1
n~!'
 ni ~ m Pke !Zan n
N ~ 2 Re (Q"9u�) Jq" j2 Re ZeM ` ~ a�Q ~ 

_ n=1 n=1 2 I na9l
m
2 I~,I2[InaRI2ReZantfRe(Zati~IIQn 11 
n1 Q l Q'1 J
If the shockabsorbing mount consists of identical symmetric shock
absorbers, which provide the average vibration gradient .0
. y �rl
Cd >
i~
~ (1)
d) b0 �
~ N Tl ~ r1
t1 tb N �ri
b0 w Cd 4
�rl 4I ta N ~
P. o +j
3 aa
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If the transmission of vibrations through shockabsorbing hangers must be
reduced on relatively low acousti c frequencies the most effective way is to
fasten the bearing plate with a reinforcing rib, so that it will be rigidly
connected between the space framing. It must be remembered in this case
that the transmission of vibrations to a bearing structure can increase
considerably in the highfrequency range.
By installing concentrated masses in the form of heavy plates where shock
absorbing hangers are fastened it is possible to greatly increase their
vibrati on damping effectiveness on frequencies }iigher than the lowest free
vibration frequency of the bearing structure.
The installation of concentrated mass and simultaneous fastening of the
bearing structure with a reinforcing rib reduce the transmission of vibra
tions through a vibrationdamping hanger in a wide frequency range. The
positive increment of vibration damping of a hanger on high frec{uencies is
a little smaller than in the case when just one concentrated mass is
installed, due to the negative effect of the reinforcing rib.
12.7. Measurement of Vibration Damping of ShockAbsorbing Mounts
The vibrationdamping properties of shockabsorbing mounts are determined
experimentally by measuring vibration levels, dynamic forces and vibration
powers.
Vibration sensors for measurements should be installed on the heads of
bolts, which fasten vibrationdamping elements to a machine or foundation
(or to a pipe).
The vibration gradient is usually measured on each shock absorber and
branch pipe, both in terms of the total vibration level in the low and
medium frequency ranges, and on individual frequencies in in individual
frequency bands.
The first measurement establishes the "condition" of a shock absorber (for
example, if the rubber pad is broken the total level gradient, according
to experience, approaches zero). The second measurement provides a
detailed evaluation of the effectiveness of a shockabsorbing mount.
The components of viUrations in three mutually perpendicular directions
are measured. The spectra of the gradient are plotted for each of the
directions of vibration.
The information obtained is processed by one of two methods. One: the
vibration "gradient field" on a shockabsorbing mount is plotted and tho
vibration gradients on all shock absorbers are indicated on the curves
(Figure 12.23). Two: the difference of the levels of the mean square
vibrations around the periphery of a vibrationdamping mount is determined
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a) 7Ry , d5
60 ~
2
40
20
0
101 10 s f,!'u
c na9,,a6
60 ~
2
%,0
20 22
0
70p 103 r,ru
b) na � a6
60
40 l
20
0
10? 10~ r
ru
d)
nQ~, a6
40
30
20
0
ia2 103 f ru
Figure 12.23. Frequency characteristics of gradients on
shockabsorbing mount of diesel engine: a gradient
IIaQ1 of vibration on x axis; b gradient IIaa2 of vibra
tion on y axis; c gradient IIaQ3 of vibration on z
axis (1, 2 maximum and minimum gradients); d
gradient of vibration energy. [86=dB; fu=Hz]
 Figure 12.24. Schematic diagram of instrument for measur
ing mean square (around the perimeter of a mount) vibra
tion level: 1 vibration receiver with preamplifier;
2 heterodyne filter; 3 control generator; 4
level recorder; 5 digital coding system; 6 per
forator.
on the flange of the machine and on the flange of the foundation for each
direction or vibration. This method has physical meaning in relation co
_ the determination of the vibrationdamping properties of a mount as a whole,
since t;he mean square vibration levels are proportional to the flux of
vibration energy through the analyzed cross section of the shockabsorbing
_ mount.
The frequenr.y characteristic of the energy gradients on a shockabsorbing
mount i.s givan in Figure 12.23.
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Figure 12.25. Schematic diagram of instrument for measur
ing combined vibration gradient: 1 vibration receiver
with preamplifier; 2 filters; 3 control generator;
4 oscillograph; 5, 7 level recorders; 6 corre
lator.
During stepbystep measurement of vibrations at different points the mean
square vibration at the different points is determined with the aid of an
instrument, a block diagram of which is illustrated in Figure 12.24.
Simpler meters, assembled from standard instruments, may be used. Founda
tion vibrations thus measuxed, with and without a shockabsorbing mount,
are used for determining the vibration damping of a mount in terms of the
kinematic parameter [5, 9].
The energy parameters for analyzing the vibration properties of a shock
absorber can be determined, as was mentioned above, by direct measurement
of radiated power [11]. The powers radiated through the machineshock
absorber and shock absorberfoundation cross sections can also be deter
mined on the basis of ineasurement data on the combined vibration gradients.
The combined gradients are measured with an instrument, a schematic diagram
of which is shown in Figure 12.25.
EIBLIOGRAPHY
1. Belyakovskiy, N. G., "Konstruktivnaya Amortizatsiya Mekhanizr~ov,
Priborov i Apparatury na Sudakh" [Structural Vibration Damping of
Machines, Instruments and Equipment on Ships], Leningrad,
 Sudostroyeniye, 1965.
2. Brillyuen, L. and M. Parody, "Rasprostraneniye Voln v Periodicheskikh
Strukturakh" [Wave Propagation in Periodic Structures], Moscow, IL,
1959.
3. Yefimov, N. V., "Kvadratichnyye Formy i Matritsy" [Quadratic Forms and
Matrices], Mos cow, Nauka, 1967.
4. Ivovich, V. A., "Perekhodnyye Matritsy v Dinamike Upr.ugikh Sistem"
[Conversion Matrices in Dynamics of Elastic Systems], Moscow,
Mashinostroyeniye, 1969.
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5. I1'kov, V. K. and V. I. Popkov, "Vibrations of Complex Active
Mechanical Systems," "Akusticheskaya Dinamika Mashin i Konstruktsiy"
[Acoustic Dynamics of Machines and Structures], Moscow, Nauka, 1973.
6. Klyukin, I. I., "Bor'ba s Shumom i Zvukovoy Vibratsiyey na Sudakh"
[Control of Noise and Acoustic Vibration on Ships], Leningrad,
Sudostroyeniye, 1971.
7. Klyukin, I. I., "Transmission and Insulation of Longitudinal Waves in
Chain Structures of Inertial and Elastically Dissipative Elements,"
TRUDY LKI [Proceedings of Leningrad Shipbuilding Institute], 1972,
No 77, pp 39.
8. Klyukin, I. I., "Propagation and Insulation of Vibrations in Branched
Niechanical Structures with Open and Closed Branch Ends," TRUDY LKI,
1975, No 97, pp 39.
9. Klyukin, I. I., "On Criteria of Vibration Damping and Relations
between Them," AKUSTICHESKIY ZHURNAL AN SSSR [Acoustics Journal of the
USSR Academy of Sciences], 1975, No 5, pp 747750.
10. Naydenko, 0. K. and P. P. Petrov, "Amortizatsiya Sudovykh Dvigateley i
Mekhanizmov" [Vibration Damping of Ship Engines and Machinery],
Leningrad, Sudpromgiz, 1962.
11. Popkov, V. I., "Vibroakusticheskaya Diagnostika i Umen'sheniye Vibro
aktivnosti Sudovykh Mekhanizmov" [Vibroacoustic Diagnostics and Reduc
tion of Vibration Activity of Marine Machinery], Leningrad,
Sudostroyeniye, 1974.
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CHAPTER 13. VIBRATION DAMPING OF SHIP HULL STRUCTURES
13.1. Basic Principles and Laws of Vibration Damping for Bending Waves
The ability of barriers to insulate hull structures from propagating
elastic waves of vibration energy from vibration sources is defined as
vibration damping.
Vibration damping VD, in dB, is the attenuation of the energy of vibrations
of a structure after insulating barriers are installed between it and vibra
tion sources:
 [BH=VD] BN = lO lg (w2,)  101; (rC'2:). (13. 1. 1)
where w21, w22 are the vibration energy densities of the structure behind
a barrier before and after its installation.
Both additional structural elements (vibrationimpeding masses VIM, elastic
pads, etc.), and all inhomogeneities of the shell of a hull structure
(stiffening rihs, changes of inertial and rigidity parameter5, etc.) are
defined as ba�rr:iers.
' Vibration amplitude transmission coefficient T through a barrier on an
_ infinite plate (rod) is used extensively for analyzing the vibrationdamping
capacity of bar.riers:
BIi =i0 Ig [I T la m` 1,.
/Til (13.1.2)
where ml, m2 are the masses of a plate (rod) before and after a barrier.
In the case of a diffuse field of incideni waves energy techniques are
utilized, and vibration damging is given by the formula
BH = lOlg (T=)o� (13.1. 3)
where e is the average vibration energy transmission coefficient in
terms of incidence angle.
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Vibration level gradients AL on a barrier are usually found as a direct
result of ineasurements:
AL = 10 Ig (ECh2)  lO lg (W.~,), (13. 1. 4)
These gradients may differ considerably from the attainable vibration
damping, depending on the resonance properties and vibration absorption in
plates, divided by a barrier.
Vibration damping as a physical process is the result of the reflection of
elastic vibration waves, propagating in a hull structure, from an inhomo
geneity (barrier). The expressions for reflection coefficient R and trans
mission coefficient T of traveling bending waves on a plate with an
arbitrary symmetric barrier are
. R = [I  ta]' [t  tA]',
T=t  I i a~,_~~t~J~,
_ where
= KX~h"a(Fa)s~~ .
[nn=p1] A ClDK~n (t a) +C3g,D ,
CID (I f Q) h 11.1  K~
S ;
. (1  I c3!n
B_ t i(Ia)~
CuI1XK11nD(t
1 ~CCDK'Kn)
a  C2K I (DX17.1)
~
a=(1  u)Ky/Kpl =(1  u) sin2 A; 8 is the angle of incidence of a wave;
Kpl, D are the wave number of the bending waves and bending stiffness of
~ the plate, and KX = KP1 cos 6, Ky = KP1 sin 6, K' = KPlfl + sin2 A, Coef
ficients Ci entering in these expressions depend on the type of barrier
and express its shear and bending compliance and dynamic rigidities in ~
relation to moments and forces:
[cp=av] Co = nZ ; Cl = A`P ; C2 = _LVf ; C3 _ AQ > (13.1.6)
2Qcp 2!1?,p lrp,p 2ZcP
where the symbol A denotes a jump (difference), and the subscript "av" is
the average value of the corresponding physical parameter on both sides of
a barrier. For the most characteristic barriers these coefficients are:
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vibrationimpeding mass (VIM) on a plate
Co = 0. C, 0,5 [GKfiy  Vlpco`],
(13.1.7)
Ci = 0, C3 0,5 (l3Ky  psw`
where GK and B are the torsional and bending stiffnesses of VIM, and Ip is
the polar moment of inertia of the cross section of VIM;
elastic pad in joi;it of plates
_ Co = Lv (Gvsy)1, Cz = 0,
C1_L,,(r�Y1y)`,` c'==o,
where GYsY and EyIY are the shear and bending stiffnesses of the pad, and
2Ly is the thickness of the pad;
hinged barrier in plate joint
Co =21_2[GKKyplPw']I, C20,
C3 = 0,5 [l3Ky  Psru21 I, C oo, (13.1.9)
where 2L is the distance between the axes of the hinges.
In a plane bending wave onehalf the vibration energy flux is produced by
the transmission of moments and angles of rotation of the cross sections of
the plate parallel to the leading edge of the wave, and the other half is
. the result of the transmission of shear forces and displacements in the
described cross sections. Limiting just one of these characteristics of a
bending wave of a barrier (for instance, making the moment in a hinge equal
to zero) does not counteract the transmission of vibration energy through
the plate at the expense of the other pair of characteristics, which do not
depend on the constrained characteristic (force and displacement in the
case of a hinge). Therefore the vibration damping of the simplest barriers
(a hinge, bearings, movable clamp) is only 3 dB.
When bending waves propagate through a barrier at oblique angles the
equality of the four characteristics (displacement, rotation angle, moment
and force) that describe the transmission of bending wave energy is vio
lated. In this case placing a constraint on displacements (a bearing) or
on shear forces (conventional cross section) provides greater vibration
damping than the other two methods the use of a hinge or moving clamp.
A direct way to achieve total vibration damping is to set two independent
characteristics of the bending wave equal to zero at the same time. _
Examples of perfect barriers are: a hinged bearing (M = 0, z= 0) ; a fixed
bearing = 0, z= 0) ; complete cross section (M = 0, Q= 0) ; fixed cross
section = 0, Q = 0). "
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The vibrationdamping properties of specific barriers are determirr?. by how
well some barrier realizes the constraints imposed on the reci,:. ;,;:ating
force and rotary moment characteristics of the vibrations of a plate.
13.2. Vibration Damping of VibrationImpeding Masses and Stiffening Ribs
A vibrationimpeding mass (VIM) is a metal beam, usually of square cross
section, installed in the joints of the plates of ship structures for the
purpose of preventing the transmission through them of acoustic vibrations
due to a change of impedance. VIM is most often installed in Tshaped and
straightline joints of plates.
A stiffening rib (SR) with an elongate cross section may be viewed from the
 standpoint of vibration damping as a VIM, if the height of the cross sec
tion of the SR is less than 1/6 of the bending wavelengtn. This require
ment is satisfied if the inequality K b H < 1 is valid, where Kb is the wave
number of bending vibrations in a plate with thickness t, equal to the
thickness of SR, and H is the height of SR.
Vibration Damping of VIM in Straight Plate Joint. VIM is installed in a
~ straight joint ia the manner illustrated in Figure 13.1. Its vibration 
_ damping is calculated by formula (13.1.3). The transmission coefficient
, of the diffusive iield of bending waves in a plate is
[nn=p1 j (.I.=) N mnn l�H. M+ ,\K. M(1 T�N.IlAa~2~I\N nn /2~
H=b; K=t]  m K2 . (13.2.1) 
b H. nn
where mm = abpm; mpl = hplppl' pm' ppl are, respectively, the density of
VIM and of the plate on which it is installed; Kb.pl is the wave number of
bending vibrations of a plate; Kb.m, Kt,m are the wave numbers of bending
and torsional vibrations of VIM: r= 5b is the radius of inertia of the
VIbt cross section relative to the line of intersection of the planes of
symmetry of the plate and VIM.
The values a, b and hpl are explained in Figure 13.1. The value Kb.m is
determined for the bending vibrations of VIM perpendicular to the plate.
The values a and b are selected from the condition
[ H=bot ] a 6> 0,213
1/ IOt.,n (13 . 2. 2)
f. ' 
where fbot is the bottom frequency of the range in which VIM is supposed to
perform effectively (VD > 5 dB), Hz; tpl is expressed in meters.
Vibration Damping of VIM in TJoint of Plates. The vibration damping of
VIM in this kind of joint (see Figure 13.1) in relation to bending waves,
, 441
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'b
D >
)
hnn
C)
a
 h =
.
Figure 13.1. Structures connecting plate and vibration
impeding mass: a straight joint; b Tjoint; c
stiffening rib on plate. [nn=p1]
incident on VIM from a vertical plate, is determined by formula (13.1.3).
The value entering in this furmula is in this case
[H=t; N=b;
nn=p 1 ]
where mpl = Ppltpl'
2
(7.2> N 81Cx. et ~ 1+ K e. nna~2 ~ mnn
4 2 ~
3Ka. nnmM rii
ru+ +d(13.2.3)
r az ( a )2]1/2
is the radius of inertia of the cross section of VIM relative to the line
of intersection of the pla.nes of symmetry of the plate.
The values Kt.m, Kb.pl' hpl and ppl in expression (13.2.3) are the same as
in formula (13.2.1), and Kb.pi, hpl and ppl refer to a vertical plate. If
calculation by formula (13.1.3) yields VD < 0 dB it is recommended that
VD = 0 dB be used.
Formula (13.2.3) explains possibilities for increasing the vibration damping
of VIM. This can be done, in particular, b.y increasing rm. It is for this
very purpose that a spacer with height d is placed in a joint. The values
of d can be arbitrarily large, but not greater than 1/6 of the bending
wavelength in the spacer. This requirement is observed when the inequality
[n=sp; e=top]
fn 3.Cfe
d2 ~ 103
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(13.2.4)
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is satisfied, where tsp is the thickness of the spacer; ftop is the top
frequency of the band in which 1/IM is required to perform effectively, Hz;
tsp and d are expressed in meters. When this inequality is not satisfied
the installation of a vibrationdamping VIM will not correspond to the case
of installation in a linear plate joint.
The desirability of using a spacer is determined by the inequality
d > 0.3a,
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(13.2.5)
which corresponds to a 3 dB and larger increment of vibration damping of
VIM. The cross section of VIM is selected from the condition
[H=bot]
r 1013
G= b~ 103 nn ,
iH '
(13.2.b)
here fbot is the bottom frequency of the range in which the VIM is required
to perform effectively (VD > 5 dB), Hz, and tpl is expressed in meters.
Vibration Damping of Stiffening Ribs. The vibration damping of a stiffen
ing rib (see Figure 13.1) is determined by formula (13.1.3). The trans
mission coefficient of bending wave energy in this formula, on fre
quencies above
 [n=10] fo= 0'~3"t ,
(13.2.7)
(where clo is the velocity of longitudinal waves in SR, and t and H are
explained in Figure 13.1), is
= 0.2. (13.2.8)
~ Thus the vibration damping of SR on frequencies f> f0 is 7 dB. On fre
quencies f< f0 the coefficient , in accordance with the above explana
tion, is determined for SR by formula (13.2.1). On frequencies higher than
fl, determined by formula
fi= 4H'
(13.2.9)
vibration damping of a stiffening rib diminishes gradually to zero on fre
quency f2 = 2f1. On frequencies above f2 vibration damping of stiffening
ribs is infinitesimal; they behave a though they are "disconnected" from
the hull and exert only a negligible influence on the propagation of bending
waves.
Vibration Damping of VIM and Stiffening Ribs Fastened Elastically to Hull.
The elastic fastening of VIb1to the hull can be used for effective
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vibration damping in ship structures. Here spacers made of vibration
absorbing materials are placed between the VIM and hull (Figure 13.2).
The vibration damping of this kind of VIbi is manifested in the frequency
range
!3'
. 4
�kt : .ti~r: : ti.:
/ / .
4
Figure 14.5. Structure of hard vibrationabsorbing coating
(left) and nature of its deformation (right) : a hard
coating; b hard coating with liner. 1 damped plate;
2 vibrationabsorbing material (plastic) ; 3 liner;
A deformation of vibrationabsorbing material.
10 I
iQ 5
~2
0,1 1,0 90,G 970.0
a2
2
113
0
.
1
Figure 14.6. Reduced loss coeffi
cient n/n2 of plate faced with
hard vibrationabsorbing coating
as function of thickness ratio of
coating and plate a2 = t2/tl for
different ratios ~2 = E2/E1.
Loss modulus rIE of materials for hard coatings usu.ally depends to a great
extent on temperature. This relationship is plctted in Figure 14.7 for
Antivibrit2 material [16]. As can be seen, its acoustic effectiveness is
high in the temperature range to approximately 30�C ar.d is maximum near
+20�C.
The physicochemical properties of special and other materials that can be
used in shipbuilding for hard vibrati on absorbing coatings are listed in
Table 14.1. Special materials obvioualy produce the best results. Anti
vibrit2 vibrationabsorbing mastic, as calculation by formula (14.2.1)
shows, gives a lined plate a loss coefficient of about 0.1 for the ratio
a2 = 1.5. When linoleum is used as a hard vibrationabsorbing coating the
loss coefficient in the plate wil]'. be more than an orlPr of magnitude
lower.
PKhV1 foam plastic is used as aliner for a hard coating. The data are:
C= 4�106 N/m2, p= 104 kg/m3, n= 0.04.
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8)
1�3
13)
Table 14.1. Characteristics of Materials for Hard Vibration
Absorbing hlaterials
3)
!

;
4)
5)
6)
7)
2)
U
a
Alarepna~
~
I511
\I
a
T
G~ h
U
~
~ a
cpifa
a
a7c
,
G~H
~
~
F.'t't
uCam
~+H
0p
K.
a
tt�
oF
OL
9
IR&
~W
CW
C.p
.11�HOley,t nxa
).T�TCno�

0,03
1,18
054
0
,�Hesam
1.Macruxa

0,016
4
,
0
064
, > [ ~_I~_ J [ y(f) I ~ 
where y is the vibration velocity of the frame; yf is the vibration velo 
city of the foundation; Zf is its impedance; Ff is the vibration force on _
it; A1, B1, C1, D1 are coefficients of the mechanical quadrupole corre
sponding to a frame (the same terms with the subscript 2 refer to shock
absorbers).
It may be assumed on the basis of published data [19] that up to certain
frequencies a frame is a concentrated mass mfr (including the part of the
mass of a machine that rests on one mount). Equation (14.7.1) is written
in the form
rj~~ I I jcontp . 1 0 r JpTp I
[
P=fr; co , (14 . 7. 2)
_ am=sa] ~ " 1 C;i,t
where csa is the stiffness of the shock absorber; w is the circular fre
quency of vibrations.
a) Fo
1, B1, cy, Dt
, 92, C2,D2
b)
Fo
Y� I.
t Z6sQ
Z,P ~,p ~PQ
Yip . . yipd
Figure 14.20. Determination of vibrationdamping effect
of local antivibrator with one degree of freedom on frame
(flange) of machine: a system without antivibrator;
b system with antivibrator, [p=fr; $=f; ex=in; am=sa]
When a simple shock absorber with input impedance Z. is installed on a
frame the equation in.a
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[BX=1ri; = ( 0
� (ro (t 7,,,..:~ I t jo~nrc (IY'h'Z'hl (14.7.3)
p=fr� $=f ~~pa J
am=sa ~ 0 0 I co 1 1. !
~ 1~I
is valid, where the subscript "a" indicates the existence of a shock
 absorber in the system (see Figure 14.20a, b).
The input impedance of a shock absorber with mass m, natural frequency
wOa and loss coefficient t1a in its elastic element ais [10]
ZBx. :1 (14 . 7 . 4)
From equations (14.7.1)(14.7.3) in considerati.on of expression (14.7.4) it
is possible to find the effectiveness of the vibration absorption of a
shock absorber with one degree of freedom on a frame (flange) of a damped 
machine L1L, in dB, as
nL 20 i6 I`! I  20 19 I`'" I =
ya N* a
20 hl " V ,11= i~ N2 (14. 7. 5)
n I l/tp'i/Il4,'__ (t0 / lA,t) I l,ld I L( fa
where Yf.a' Ff.a are the vibration velocity and force on the foundation
with a shock absorber with mass m:
a
~ n1 nt, \To G~cmib
hl im,h [j  (fo )2J+ nip J ~11' + 1( fot3 /2 _ 1]21
_ [ C,o, I  ,,aJ ;
f0 ~'~,rl?M1~ is the natural frequency of mass mfr on a shock absorber
n
with elasticity csa; f0a is the natural frequency of the shock absorber.
The results of calculation ot the effectiveness of the vibration damping of
a foundation (and frame) with a rubberized metal shock absorber are pre
sented in Figure 14.21. The raw data for the calculation are: mass of the
 frame (and part of the machine) per shock absorber and antivibrator 50 kg;
mass of foundation mf = 100 kg; natural frtquency of shockabsorbing mount
_ f0 = 13 Hz; stifness of each shock absorber Csa = 350,000 N/m; loss
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1
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coefficient of antivibrator rj a = 0.1; the other data on the antivibrator
are given in the key.
As can be seen in Figure 14.21, the antivibrator exerts much of its effect
primarily on its own natural frequency. But in addition to this maximum
there is an additional maximum on lower frequencies. An effect can also
be observed between them, i.e., the antivibrator behaves as a more or less
wideband device. The effect of vibration damping increases with the mass
 of the antivibrator.
Such sharp peaks should not be expected in the curves of the acoustic
effectiveness of a real antivibrator, as in the theoretical curves. The
flattening of the curves of vibration damping of a real antivibrator is
attributed both to the existence of losses in the shock absorbers and
foundation that are not taken into account in the given calculation, and to
the fact that a real antivibrator is a system with six degrees of freedom,
the effects in which will be superimposed one on the other.
Experimental Data. To determine the acoustic characteristics of anti
vibrators one may use a test stand, such as the one schematically depicted
in Figure 14.22. It is a plate (12 mm thick), installed on shock absorbers.
The shock absorbers are fastened through a welded intermediate foundation
to rails in the concrete bed of the test stand. The antivibrator is
installed on a plate over t}ie shock absorbers; the plate is driven by a
vibrator, which develops a force with constant frequency (using a compres
sion circuit). Additional weights, simulating machines of different
weights, may be placed on the plate if needed.
 A sample of the spectrograms that were recorded is presented in Figure
14.23. As can be seen by the figure, the effect of the vibration damping
of an antivibrator (the difference of the ordinates of the curves) is
manifested not only on the natural vertical frequency of the antivibrator
(85 Hz), but also on frequencies above and below that frequency due, in
particular, to the absorption of vibration in other degrees of freedom of
the antivibrator.
The characteristics of antivibrators, in which the elastic elements are
the familiar AKSS shock absorbers, are presented in Table 14.2. Data on
the impedance and weight of the antivibrators may be used for selecting
the necessary degree of vibration damping in bearing structures with dif
ferent mechanical impedances.
Multielement Antivibrator. An antivibrator with three independent absorb
ing elements, the frequencies of which are separated relative to each
other, is illustrated in Figure 14.24. The input impedance of this kind of
antivibrator, consisting of n elements, is
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tn
c
_t
~O
~
~ d
'O
~
n J
Cd a
N
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I
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4 fI x
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> z o
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cd O
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N
cd 1
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cd II
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7
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(Zb
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r
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Figure 14.22. Experimental setup for studying vibration
damping effect of antivibrators: 1 shock absorber;
2 steel plate simulating frame of machine; 3 test
antivibrator; 4 electronic circuit for measuring and
analyzing vibrations; 5 vibration meter; 6 force
transmitter; 7 vibrator; 8 excitation circ�it;
 9 foundation ot stand; 10 intermediate (mounting)
foundation.
L, g6
v0
25
20
?5
10
5
D
9
2

o
p
~o 60 70 900 950 200 no f, rq
rigure 14.23. Vibration spectra of test foundation under
_ vibrationdamped plate: 1 plate without antivibrator;
2 antivibrators installed on plate under shock
absorbers (fOa = 86 Hz). [g6=dB; fL4=Hz]
,t
,
[BX=1.II] Znx. a= Zox. n tt (14. 7.6)
i=1
where Z. in.ai is the input impedance of .)ach of the elements on a given
frequency in a given direction.
, Experimental studies of a test specimen of an antivibrator disclosed that '
the vibration of a 2 mm thick plate is attenuated by (45)(1518) dB in
the 1003,000 Hz range. The attenuation in the bottom of that frequency
range is attributed basically to the inertial reaction of the rather
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Table 14.2. Characteristics of Antivibrators with AKSS
Shock Absorbers
'
4)
 5)
6
?
ita
i va
lcrnra
Q a~, ~p~r�
0611la:1 An
I18.`.11(VCCiC35t
'113 CC J 811TN'
L,.. r,rdr.,a
dllilt'i:l',; :
J.'dVj~TilJiITU~),l
ItC~)til
K:l:ILIIlA\
Ita Uiilll
'{'CiTKOC'fb.
111161!.ITOr:I.
i017,l 1:
i
KO:II'(j:1
:111'tiS'
f1
115P JT0P
~I V\1) .10V
Kf'
Pf:iVll:~w.,'.
(}~�C:.'i) ~'i ~
I11171. ( !l
AKCG25II
25
1
0,6
31
3.2
:1KCC60I1
1
1,0
89
9,2
r1 KCGGO{ I
1
1,0
10
14,0
AKCG25M
'
I
l,l
11
15,4
AhCCG011
50
2
2,0
20
28,0
AKCC60M
1
2,2
22
30,8
11hCC40011
1
5,7
53 
74,3
A 1CCG220M
1
9,8
100
I40,0
:11