THEORETICAL PRINCIPLES OF TORPEDO WEAPONS

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• Document Page Count: 128
• Document Creation Date: December 23, 2016
• Document Release Date: August 1, 2013
• Sequence Number: 1
• Publication Date: March 29, 1976
• Content Type: REPORT

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, 4.1.t4r
THEORETICAL PRINCIPLES OF TORPEDO WEAPONS
By
G. M. PODOBRIY, ET AL.
451
U. S. JOINT PUBLICATIONS RESEARCH SERVICE
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BIBLIOGRAPHIC DATA
SHIRT
1. Report No.3.
JPRS 67047 II
Recipient's Accession No. '
4. Tian and Subtitle
THEORETICAL PRINCIPLES OF TORPEDO WEAPONS
5. Report Date
March 1976
6.29
7.AmItor(s) G. M. Podobriy, V. S. Beloborodyy, V.V. Khalimonov,
A. I. No3OV
II. rieorformins Organization Rept.
9. Performing Organisation Name
and Address
Research Service
Road
22201
10. Pioject/Task/Work unit No.
Joint Publications
1000 North Glebe
Arlington, Virginia
11. Contract/Grant No.
12, Sponse:lap Organisation Name sad Address
As above
'
13. Type of Report & Period
Covered
14.
13. Supplementary Notes
TEORETICHESKIYE OSNOVY TORPEDNOGO ORUZHIYA, 1969, Moscow
16. Abstracts
The report contains the theoretical fundamentals of present-day torpedo
weapons: dynamics, destructive effect, proximity fuzes, gas-steam energy
sources and engines, electric energy sources and motors, guidance control
and homing systems. .
17. Rey Words and Document Analysis. 17n. Descriptors
USSR
Ordnance
Underwater Ordnance
Torpedoes
171a? ldestifiers/Open-Reded Tema
17s. C0SAT1 Pield/Group 19H
IL Avallabillty,Stotement
Unlimited Availability. Sold by NTIS
Springfield, Va. 22151
_
19.. Security Class (This
Report) Lasimp
21. Tio-.-of Pages
366
20? SscUsfir/ Class (This
Page
UNCLASSIFIED
22. Price
NT15015
USCOMM.DC 401411.1.71
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ILAItgar.?
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JPRS 67047
29 March 1976
THEORETICAL PRINCIPLES OF TORPEDO WEAPONS
Moscow TEORETICHESKIYE OSNOVY TORPEDNOGO ORUZHIYA'in Russian 1969
signed to press -7 Mar 69 pp 1.359
(Book by G.M. Podobriy, V. S. Beloborodyy, V. V. Khalimonov,
A. I. Nosov, Voyenizdat; 4,000 copies, UDC 623.946 (01)]
CONTENii PAGE
INTRODUCTION
1
CHAPTER 1. DYNAMICS OF TORPEDOES
3
General Information. Kinematic Parameters of the Torpedo
3
CHAPTER 2. DESTRUCTIVE EFFECT OF TORPEDOES
52
Warheads
52
CHAPTER 3. TORPEDO PROXIMITY FUZES
66
General Principles
66
CHAPTER 4. GAS-STEAM TORPEDO ENERGY SOURCES
106
General Description of Propulsion System. Relationship
Between Torpedo Specifications, Performance Data and
Propulsion System
106
CHAPTER 5. GAS-STEAM TORPEDO ENGINES
147
Requirements on Engines
147
CRAP ER 6. ELECTRIC TORPEDO ENERGY SOURCES AND MOTORS
186
General Description of Electric Torpedo Propulsion
Systems
186
CHAPTER 7. TORPEDO DEPTH CONTROL SYSTEMS
217
Fundamental Principles of Torpedo Guidance
217
? a [III ? USSR ? 4]
[II - USSR]
?
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CONTENTS (Continued) Pagq
CHAPTER 8. TORPEDO LATERAL GUIDANCE CONTROL SYSTEMS 267
Preliminary Remarks 267
CHAPTER 9. TORPEDO HOMING SYSTEMS 298
Functions of, and Requirements for, Homing Systems 298
-b -
oo:
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PUBLICATION DATA
Author(s)
Grigoriy Mikhavlovich Ppdobriy,
Vasiliy Sergeyevich Beloborodyy
Vladimir Viktorovich Khalimonov
Andrey /vanovich Nosov
Editor(s) S. A. Vyzvilko
Technical reviewer(s) : Ye. N.'Sleptsova
.Copies : 4,000
Printing plant
Voyenizdat .2nd Printing Plant, Leningrad
06
0 q
0 ,
0
,1
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INTRODUCTION
World War I demonstrated and World War It confirmed that the torpedo is a
potent weapon in offensive and defensive operations at sea. The effective-
ness of torpedo Utilization was determined first and foremost by the element.
of surprise in employing new models, by tactics of employment, by concealment
of attack, as well as by the difficulties involved in evading torpedo at-
tacks.
Today, in the age of nuclear energy and missile hardware, foreign navies are
continuing to devote continuous attention to the improvement of torpedo
weapons. The high level of development of modern torpedo weapons makes it
possible to fire torpedoes from surface, submerged and airborne platforms and
to employ torpedoes from considerable distances against transports, cargo
ships, surface warships, submarines and port facilities. Outstanding Russian
and Soviet engineers Ptd scientists have made a great contribution toward the
development of torpedo weapons: A. I. Shpakovskly, I. I. Nazarov, N. A.
Datskov, P. V. Bukhalo, N. N. Azarov, L. G. Goncharov, A. V. Trofimov,
Yu. A. Dobrotvorskiy, N. N. Shamarin, D. P. Skobov, and A. K. Vereshchagin.
More than 100 years has passed since gifted Russian inventorI.F. Aleksandrovekiy
proposed the first torpedo design in Russia in 1865.
While the first torpedo was an underwater unguided minsile, today's torpedo
is a complex aggregate of propUlsion, directional control, homing and war-
head detonation syytems designed on the basis of modern achievements of
science and technology.
The authors of this volume set for themselves the task of presenting the
, theoretical principles of torpedo weapons in the most comprehensible form
possible.' The authors do not examine specific torpedo models. For the sake
of greater clarity some points of theory are illustrated by examples, the
numerical values and quantities in which are hypothetical and are of a purely
illustrative nature. We have been unable to present in this volume many
elements elaborated in torpedo theory. But if this book assists the reader
in comprehending the physical principles of torpedo weapons and comprehending
hte the principles applied in designing torpedo assemblies and systems, the
1
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authors will consider their task accomplished. In working on this book the
authors utilimd Scviet and foreign published materials available to the public.
This book was written by a team of authors: Professor G. M. Podobriy, Doctor
of Technical Sciences (chapters 1, 7, 8); Docent V. S. Beloborodyy, Candidate
of Technical Sciences (Introduction, chapters 4, 5); Docent V. V. Khalimonov,
Candidate of Technical Sciences (Chapter 6); Docent A. I. Nosov, Candidate
of Technical Sciences (chapters 2, 3); Chapter 9 was written in collaboration
by G. M. Podobriy (sections 12-18) and V. V. Khalimonov (sections 1-11).
The authors would like to express their sincere gratitude to A. G. Pukhov,
I. I. Trubitsyn, A. P. Vorob'yev and S. A. Vyzvilko, who were kind enough to
inspect the manuscript and who offered a number of valuable comments, and
will, xt grateful for any comments aimed at correcting the book's deficiencies
and at improving its content.
R
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CHAPTER 1. DYNAMICS OF TORPEDOES
1.1. General Informadon. Kinematic Parameters of the Torpedo
A theoretical examination of the dynamic properties of a torpedo, its
stability of motion, controllability and maneuverability is based primarily
on an elalysis of torpedo motion equations.
The structure of differential equations of torpedo motion, as of any other
body, is determined by that system of coordinates in which this motion is
studied. Therefore coordinate systems are usually selected in such a manner
that the equations are maximally simple in form and convenient for analysis.
In torpedo dynamics one employs for the most part Cmtesian coordinate systems,
primarily right-hand systems. In a right-hand coordinate system those angles,
angular velocities and moments which are figured or which operate counter-
clockwise are considered positive.
Coordinate Systems
The following are employed in dynamics of torpedoes (Figure 1.1):
a coordinate system linked to the earth -- OXRYgZg, arbitrarily called
a fixed or stationary system; it is used in determining the parameters of the
trajectory of a torpedo's movement;
a coordinate system coupled to the torpedo -- OXYZ. Axis OX runs along
the longitudinal axis of the torpedo, axis OY runs upward and is located in
the torpedo's centerplane, while axis OZ is perpendicular to axis XY. The
origin of the coordinates is located either at center of gravity CG or at the
center of buoyancy CB. In contrast to the center of gravity, which is
defined as the point of application of the force of the torpedo's weight, the
center of gravity constitutes the point of application of its displacement
force. It is advantageous to use the center of buoyancy as origin of co-
ordinates when the CG position changes within a substantial range; in this
coordinate system it is convenient to define the components of inertial hydro-
dynamic force;
3
Il
0
0 0
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0
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?
velocity coordinate system 0X1Y1X1, where axis OX1 runs along the
velocity vector, axis 0Y1 is placed in the centerplane, while axis 021 is
perpendicular to axis X121. This coordinate system is usually employed in
determining the components of hydrodynamic force caused by viscosity of the
medium.
X
Zs ?
Figure 1.1. Fixed or Stationary (Meg%) Coupled (OXYZ) and Velocity (0X1Y121)
Coorditate Systems
Coordinates Determining a Torpedo's Position in a Fixed and Velocity Coordi-
nate System
A torpedo's motion is considered known if at any moment in time t one can
find the position in space of each point of the torpedo.
We shall designate the coordinates of point M (Figure 1.2) in a fixed co-
ordinate system by xg yg, zg, in a coupled system with x, y, and z, and
polar coordinates (point C),-- xog, yog, zog.
Xr
r
I'
tl
X
as I ale+
sin I
?ces I ele ?
r
?1n7 sin + ? am T sin A cos 4i
cosy eoe 0
*In 7 eos + + els I sin ? sln +
Z
ea / sin y + sin 7 eta I ais +
...hi T ea' I
eOsi cos ir ? sla T its I an +
Table 1.1. Direction Cosines Between the Axes of a Coupled and Fixed Coordinate
System
We shall project dashed line OABCPQM onto each of the axes OX a, OYa, 02g and,
, employing Table 1.1, we shall obtain expressions for the coor3inatgs of point M
in a stationary coordinate system [1.5]:
4
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+ x cos e cos + y (An 7 sin 4, ? cos y sin 0 cos +) +
+ 11(cm 7 sin 4, + sin y sin 0 cos 410);
YR .Yog X Sin y cos T cos I ? saln 7 cos 0; (1.1)
se vs soe? x cos 0 sin $ y (sin y cos lo 4- cos 7 sin 0 sin +) +
+ s (cos y COS 40 -?- sin T sin 0 sin +).
Any torpedo movement can be broken down into two motions: translational and
rotational [1.5].
Figure 1.2. Deriving the Relationship Between the Coordinates of Point Min
a Stationary and Coupled Coordinate System
Torpedo motion whereby only x08, Yoe, zog (current values of stationary co-
ordinates) change is called translaEional motion, while all angles remain
constant, that is, the coupled axes, displacing, remain constAutly parallel
to their original direction. /n translational motion velocities, accelera-
tions and trajectories are identical for all points of the torpedo.
Motion vhereb7 only angles change is called rotary or rotational motion of
the torpedo relative to any point (pole)*.
In conformity with expressions (1.1), the position of the torpedo in space is
determined by six synthesized coordinates: by three polar coordinates xog,
yo., zo. and by three angles 4), 8, and y, called the yaw angle, pitch angle,
ana heer angle respectively (Figure 1.3), that is, expressions (1.1) describe
the torpedo's translational and rotary motions.
The angle between axis ORg manned the torpedo's projected longitudinal axis
-onto plane RgZg is called the yaw angle.
* The origin of a moving coordinate system is called the pole. Usually the
torpedo's center of gravity is taken as the pole.
?
cro
' 00
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o
f.1
p
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Figure 1.3. Torpedo Yaw Angle 0, pitch angle 0, and heel angle y
Figure 1.4. Torpedo Angle of Attack a and Drift Angle 0
? The angle between horizontal plane. Xg2g and the torpedes longitudinal axis
ia called tiht pitch angle.
a 8
0 corresponds to control deflection down-
' ward).
At low angles of attack a and low angles of horizontal Control surface
deflection 6r the relationship between cyl and these angles can be assumed
.linear in a first approximation:
diC
c ...41 1 a AI. 154010 06 4"11
aa Ira o (1.30.
uor
Q
22
0 34
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.r
a
o
0 o
'`)
'00
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-to
41611 den- or?
0,001
0,006
?
6004
'.002
?
.1
-8
-..
-4
?
,
0,002
2
.
.
0.001
,
0,010
-
0.
pr.
4.-/o?
?
Figure 1.11. Relationship Between of Lift and Angle of Attack a and Angle
of Horizontal Control Surface Deflection dr.
OC
Partial derivatives and OC ?A are determined as the ratios
777- oar
C 00 C 00
and
a a
The value of coefficient Cm21 of a finned torpedo hull is obtained from
wind tunnel data. In tecting, the coefficients of hydrodynamic moments of
a finned and unfinned hull'are determined at various angles of attack and
control surface deflection angles. Figure 1.12 contains the results of
determining Cm21 for a torpedo. Hydrodynamic moments are given relative to
its center of gravity. As is evident from the graph, the fins and control
surface position appreciably influence the value of the hydrodynamic moment
coefficient. Cm2,
Heone mod/ iwpflyO
0,003
.
-...--?+Z
0,002
,
1
-10
-8
-6
4
6
8
,:y.(#'
,
?,,,to
,...?,
,,?????-
40--,--
?,
-0,003
Figure 1.12. Relationship Between Coefficient of Longitudinal Moment and
Horizontal Control Surface Angle of Deflection dr and Angle of Attack a
23
. 0
, ,
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At small angles of attack a and small angles of horizontal control surface
deflection 6r, coefficient Can can be approximately represented with linear
relation
ac?,
?e-ImPr 8).
(1.39)
Figure 1.13. Deriving a Formula for Converting a Torpedo's Longitudinal
Hydrodynamic Moment
Quantitiesacm and 14.1:, are determined in the same manner as
-7N-L
and
We shall note that the above also applies in equal measure to force Rzi and
moment Mil.
/n 'practical calculations the hydrodynamic moment must be converted for one
and the same torpedo as a consequence of change in its center of gravity.
Such a conversion for small angles of attack can be performed in an approx-
imate manner with formula
M,, 12 MA (1.4Q)
where Mzi -- hydrodynamic moment relative to the initial position of the
center or gravimr LATo; M'zi -- hydrodynamic moment relative to the new
. Position of the torpedo's center of gravity LAT1 (Figure 1.13); Al -- dis-
tance between LIT0 and LATi; A1>0 with a shift of 1411 toward the aft end of
the torpedo.
1.6. Aydrodynamic Heeling Moment
Moment Mx is generated with fin and control surface asymmetry. Assume the
torpedo, for example, is moving with a positive angle 6, while the rudder
is deflected by angle 69. Then lateral force Rup will act on the upper
vertical fin, RzHn on the lower vertical fin, Rzsp on the upper rudder, and
RzHp on the lower rudder (Figure 1.14). These forces generate a moment
relative to axis OX, determined by equality
24
8
a
a
0 a
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0
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0
where
itflt 111 RS MI YNII Rs .Y,1 + Rs NpYilp Rs op Yip ,
Yin, Y.11111 Yips Yir
above -indicated forces.
-- coordinates of points of application of the
Rzyin de
Rue
41e 1
Rzep ?
Rz,iptIPI
Al
. Figure 1.14. The Problem of Generation of a Torpedo's Lateral Hydrodynamic
Moment
The final expression for determining heeling moment will be
A; A 04 0 r_o 4_ PV1 0 "
???3.7?n ???2 ??-rr -r -?-r. ?
From expression 01.41)
where
acir,
"Irs
(1.41)
d se. Pin
Yon s'iM71:7371,
U. at. k + Sim
(
Pop,
di ea Sim?
5" ?Top "'" + Sup )
r. Fop , p
- distances of lines of corresponding forces
? im Pon up,
relative to axis OX, in relation to torpedo
length LT;
son, Sao ? Sip 1 Sup.
Cs on ? areas of lower and upper vertical fin, upper and
lower rudder respectively;
-- dimensionless coefficient of lateral.force-of
vertical fin;
? dimensionless coefficient of rudder lateral force.
1.7. Damping Hydrodynamic Forces and Moments
We have examined above hydrodynamic characteristics under the condition that
the torpedo moves only translationally at a constant velocity VT. If the
25
0 cf
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torpedo is involved in rotary or oscillatory motion in addition to transla-
tional, its hydrodynamic characteristics will change appreciably, since
there will be additionally generated hydrodynamic forces and. moments called
damping forces and Moments [1.2]. We shall clarify their origin with a
particular case of torpedo motion, that is, with circulation in a horizontal
plane.
As a consequence of torpedo rotation, local linear fluid velocities arise at
each point of the torpedo's hull, velocities which are normal to the torpedo's
longitudinal axis and numerically equal to the product of angular velocity w
and the distance to point x. Consequently, sum velocity V of the incident
flow will be equal to the geometric sum
171. +
Since xw is a variable quantity, the fluid flow along the torpedo will be curvi-
linear a n d local drift angles will differ from those which would occur
with rectilinear motion.
For the sake of an example Figure 1.15 contains local drift angles for two
surface elements; with a positive angular velocity local drift angles in
the aft section increase by quantity
xo)
ail w,
while in the nose section they decrease by the same amount.
Figure 1.15.
?
Change in local angles of drift is the reason for the occurrence of damping
forces and moments. These forces and moments can be determined both
theoretically and experimentally. With the theoretical method. they are deter-
mined separately for an unfinned and finned hull.
We shall first examine the method of determining damping force and moment
for a torpedo's fins. With.torpedo rotation relative to axis OZ the increase
in angle of drift AS can be determined in approximate fashion as [1.7]
'
apn
26
0
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where Con is the distance from the torpedo's center of gravity to the center
of hydrodynamic forces of the fins.
As a consequence of this an additional lateral force and an additional yaw-
ing moment arise on the fins or, as they are usually called a damping
force and a damping moment.
CoeffiCient of fin lateral force Czon can be determined with the formula
Cs en all Cs -? Cis('
where Cz is the coefficient of lateral force of a finned hull; Czk -- co-
efficient of lateral force of an unfinned hull. Assuming AO to be a small
quantity, we can write ,
ac Pv:
RrCla in 7-1021" ?11?"hafi El 4?Sr
- .' Pg:Co a Vrt?
? al A:on VT efi (1.42)
1 6C,
tit: on 1" or pax".
? where:
The fin damping moment can be computed approximately with the formula
h10 ots nog,. gs 0,V,?, , (1.43)
1ac
fA,7,3, on al "r ..12n. 024 .
For an unfinned hull damping forces
formulas of K. K. Fedyayevskiy:
RA, AA,V,?;
1,11 r= liVok
and moment are determined: with the
(1.45)
where 7r7 is a section relative (to LT) coordinate figured from the torpedo
nose; no -- relative (to LT) coordinate of the center of rotation (origin
of coordinates), figured from the torpedo nose; 7-- relative (to LT)
radius of the profile of the torpedo's meridional section at distance TT
from the torpedo's nose.
Calculation of synthesized coefficients employing formulas (1.45) is per-
formed in the following sequence.
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We determine quantity 1", for which we first compute the torpedo's fullness
coefficient, equal to the ratio of the volume of the torpedo hull Vo to the
displacement of a cylinder of equal size, that is,
From the fullness coefficient in the graph (Figure 1.16) we obtain the ratio
of radii
run
where rmak -- torpedo radius; r -- running radius of the target section in
the afterpart of the torpedo.
We then determine roltmak and relative radius
r
f an 'r ?
We compute quantity 1170 with the formula
where No. is the distance from the torpedo nose to its center of gravity.
In order to determine we determine on the drawing distance 1,1 from the
torpedo nose to the section in the after part with a radius of r. Then
/am*.
On the basis of known values of. W and F from the graph (Figure 1.16) for
force Rzo, assuming in section 0..0!-5? Czo is a linear function of angle 0,
we compute partial derivative
aC,
754'
Then we determine 14k and 140n as well as
Hyk? and Myon.
Total damping forces and damping moment are equal to:
bC
=Ig11TIAC Rion;
1? A11761 +
In like manner we can determine damping force Ry ? and moment 4 during torpedo
rotation (oscillation) in a vertical plane.
In the case of torpedo rotation on its longitudinal axis with angular
velocity wx, a damping moment also occurs, the approximate value of which
is determined with the formula
ML 66 AL NV004.1
28
0
(1.46)
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as
04
MMICHIPI
MIMI:ill'
11/MI111111
0.0 07 08 0.9
0 cm 004 ape oss
.1:09g4
0.12
0 0,02 0.04 0,00 0.01 0.10 0 002 004 ami 0,05 010 I
Figure 1.16. Graphs for Determining Lift Coefficients of an Unfinned Torpedo
Hull (C70)
where
..4:esie Clicre
r ?7 torpedo radius.
Utilizing expression (1.46), we can determine in a first approximation the
damping moment of an unfinned.hull during torpedo rotation on its longitudinal.
axis.
When a torpedo rotates on its longitudinal axis, a damping moment also arises
on the fins. For theoretical determination of the magnitude of this moment,
one can employ the formula
ACron
29
(I,47)
a
0
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4-ofe,v5i
whereby
ANIS oft 1" bac, r)oo I (113-'44
whore 1 -- is the average length of the fin plate (Figure 1.17); d -- dis-
tance between the outer edge of the fin to the axis of rotation; do -- dis-
tange from the inner edge of the fin to the axis of rotation.
In conclusion we should note that the above formulas make it possible only
approximately to determine damping characteristics. Special tests must
be performed in order to obtain more precise values. One experimentally
determines thereby rotational derivatives
ql..1f1-1.,0 and ,c4A8
OC_L'', I
josa
connected with damping forces and moments by the following relations:
PV2 V2
12; ign cp,--7-4.; M = cva?
Figure 1.17. Determining Damping Forces and Moments of Torpedo Fins
There exist two methods of determining these derivatives. One is the testing
of a model on a rotation testing unit. The model travels in a circle.
Measuring the hydrodynamic forces and moments at various angular velocities,
one can determine rotational derivatives by subsequent numerical or graphic
differentiation. 'Another method of determining these derivatives is based
on recording small model oscillations in the flow. On the basis of these
tests one determines the damping constant and subsequently rotational
derivatives proper.
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1.8. Force Effect of a Nonviscous Fluid on a Torpedo Traveling Through It
Up to the present we have been examining hydrodynamic forces caused by fluid
viscosity. As noted above, beyond the limits of the boundary layer a fluid
can be assumed ideal, nonviscous. Let us examine the force effect exerted
on a torpedo by a nonviscous fluid.
During torpedo motion certain momentum is imparted to the particles of the
.ambient fluid. If the torpedo moves at a constant velocity, this momentum
will possess a constant value. In conformity with the laws of momentum and
the moment of momentum, in this case the fluid will not exert force on the
torpedo.
With torpedo unsteady motion, the momentum of the fluid continuously changes,
and forces and moments of an inertial nature arise. Indeed, the pressure
of the external surface of the torpedo on the fluid should overcome only its
inertia; there will not be other forces, since the fluid is assumed nonvis-
cous.
We shall examine these forces and moments in greater detail [1.3]. Let us
assume, for example, that a torpedo is in unsteady translational motion
along axis OX and is rotating on that axis. The reaction force with un-
steady translational motion in the general case will not coincide with the
direction of velocity V/Ic. The value of this force will be determined by
its three projections onto the axes of the coupled coordinate system:
Rt, = ? 1.1,x; Rky = kl2f/tri RILE = AIS l'ITZ ?
Since force R1 does not pass in the general case through the origin of
coordinates, it forms a moment the projections of which onto the coordinate
axes are equal to
Alsr = 9Tx; M1,= 1Ll5 Mis =
With torpedo torpedo rotation on axis OK with angular acceleration wx, inertial
forces also arise, which can be reduced.to main vector A2. and a Rair of
forces the vector of which is equal to M2. Projections A2 and M2 onto the
coordinate axes are determined by the expressions:
R24. A414;x; R2y = /42(;).24 RaT er??
= X4e;:ex; M21 = ??????? Antj.r;M =
/n like manner one could write expressions for inertial forces and moments
for the two other axes. As a result we shall have 36 expressions of
projections of forces and moments and correspondingly 36 coefficients Aik
which, at the suggestion of N. Ye. Mukovskiy, are called apparent masses
and apparent moments of inertia respectively. One should not define ap-
parent mass thereby as a certain fluid mass moving together with the torpedo.
In actuality fluid particles possess velocity and accelerations which are
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Figure 1.18. Analysis of Coefficients of Torpedo Apparent Masses
not equal to the velocity and acceleration of the solid body. 'Therefore
apparent mass is defined as a fictitious fluid mass which, moving at the
same velocity as the torpedo, would have the same momentum which is in fact
possessed at the given moment by the fluid surrounding the torpedo.
The number of quantities Aik diminishes significantly if the body possesses
planes of symmetry. In particular, the torpedo can be assumed to be a
body which is symmetrical relative to planes XO! and XOZ (Figure 1.18).
Let us first see what Aik become zero when the torpedo moves along axis OX.
'Flow along the torpedo is symmetrical, and because of this fluid pressures P
. On elementary surfaces dgl to the right and left of plane XOY are identical.
Resultant-dRix of both these forces is directed along axis OX. Therefore
its projections onto axes OY and OZ will be equal to zero. It follows from
this that Al2Ot.w10.
Force dRix does not create moments relative to the coordinate axes, and
"therefore should be A14-A15"0,16-0.
W2
..__.__
Figure 1.19. Analysis of Coefficients of Torpedo Apparent Masses
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When a torpedo moves along axis OY, the resultant pressure Ry will lie in
the plane of symmetry XOY, but it wIll not pass through the origin of co-
ordinates. Projections of this force onto axes OX and OZ are equal to zero,
and consequently 1
-21 123.00. Force Ry generates a moment only relative to
'
axis OZ, and therefore A
-24' X25.00.
In the case of the torpedo motion along axis 02, the following four coef-
ficients become zero: A31siA3eA31elA3e0.
When the torpedo rotates on axis OX, the main vector of inertial forces
should lie in plane YOZ, while the vector of the main moment should coincide
with axis OZ (Figure 1.19). /f we ignore fin asymmetry, the torpedo can be
assumed to be a solid of revolution. In this case the main vector of in-
ertial forcee becomes zero. Consequently, RtiekeR4z .414r.444e0. It fol-
lows from this that 4114
4243X45A4604.
In the case of torpedo rotation relative to axis 02, the main vector of in-
ertial forces should be positioned symmetrically to plane XOY, and the main
moment parallel to axis 02, since plane YOZ is not a plane of symmetry.
Then R5eR53.8.445rM5e0 and A61-A63-A6,-A65-0.
When the torpedo rotates relative to axis OY, A51A52A54'A56.10.
In general courses in hydrodynamics it is proven that AileAki. /f one takes
this into account, then only eight of 36 coefficients remain for the torpedo,
namely: All, A22, A33, A4410 A559 A66, A26 and X35.
The values of apparent mass coefficients Aik are usually determined in a
coupled coordinate system with the origin at the center of buoyancy. If the
origin of coordinates is placed at any other point, at the torpedo's center
of gravity, for example, the necessity arises to recalculate the apparent
mass coefficients. This recalculation can be performed with the following
formulas [1.12j:
1u= 41; fl2
? 144 km' + Ass' y24,1
1s0 A55' 245Xas;
1.8= X661 + 424, + 246x0;
48 4' klaras ?
where Alk and Aik are apparent mass coefficients computed relative to the
center of buoyancy and relative the center of gravity respectively; we shall
note that.coefficientsI
-11.-22, A33 possess the dimension of mass, coef-
ficients A44, A65 and A66 the dimension of moment of inertia, and A26,
A35 the dimension of static moment of mass.
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For an approximate determination of coefficients of apparent mass one usually
employs the method of plane sections [1.2]. The essence of this method con-
sists in the fact that with lateral flow across an elongated body, each of
its frames is considered to be in a plane flow without longitudinal fluid
spreading. The apparent mass of the entire body in the direction of the
lateral axis is determined by adding together the apparent masses of the
separate sections. The coefficients are calculated separately thereby for
an unfinned hull and for the torpedo fins. We shall cite the formulas for
determining A229 A26 and A66, while the remaining apparent mass coefficients
are determined in analogous manner.
When an unfinned torpedo moves along axis OY
? ? as
ist sp (x)dx pVfl*,
x,
4 et up xr1 (x)ilx pVA,
(1.48)
(1.49)
where xl, x2 are the coordinates of the nose and aft add; xo -- coordinate
of the torpedo center in a coupled coordinate system; Vo -- volume of the
unfinned torpedo hull; r(x) -- variable hull radius.
In the case of torpedo rotation relative to axis OZ we have
x,
46=19 xle.01d4
Li
= X24.
(1.59)
We shall note and in a first approximation 42 and 46 can also be deter-
mined by substituting for the unfinned torpedo hull an ellipsoid of revolu-
tion the axes of which are equal to the torpedo's length and diameter.
Coefficients of apparent mass for ellipsoids of revolution have been computed
and are contained in a number of courses on hydrodynamics [1.14].
.--,
PI
44
q2
0
.4.........1..........ti
I
.'
-
2 34
5
6
7 8
8
'
Figure 1.20. Graph for Determining Correction Factor p(x) for Finiteness
of Fin Span
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With an approximate determination of fin apparent mass coefficients, the
fins can be replaced by isolated plates with an aspect ratio
1
Xillg ?
where 1 -- plate length, b plata width. Then the values of apparent
mass coefficients, for horizontal fins, for example, with movement along
axis OY, are determined with the following formulas:
(%) pb11;
ALISO
ne?"1;04m, -
where u(x) -- correction for finiteness of span, determined from the graph
(Figure 1.20); xan -- fin center coordinate in a coupled coordinate system.
In the case of torpedo rotation relative to axis OZ, the coefficients of
apparent fin mass are respectively equal to:
AgsmnpfblXidX;
(1.5M)
where xH and xi -- coordinates of fin starting and end points in a coupled
system.
Knowing the apparent mass coefficients for the hull and fins, one can
easily obtain their values for the entire torpedo by means of addition.
1.9. Principal Dynamic Properties of a Torpedo
Torpedo movement in the water consists of spatial maneuvers of various kind.
The capability and position of execution of these maneuvers are determined
chiefly by the dynamic properties of the torpedoes, particularly their '
maneuverability, controllability, and stability. These properties are in-
terlinked and are determined by the specifications, performance capabili-
ties and hydrodynamic properties of torpedoes. We shall briefly examine the
content of the fundamental dynamic properties of torpedoes.
The maneuverability of a torpedo reflects its ability to change direction,
speed and depth or, which amounts to the same thing, to change its position
in space. Following are the standard torpedo maneuvers: acceleration and
deceleration, movement along inclined trajectories, and turning in a
horizontal plane. Each of these maneuvers is evaluated by its indices,
which are the following: time and path of acceleration or deceleration,
maximum permissible angle of inclination of trajectory, range of change in
velocities during torpedo movement at various cruising depths, minimum turn-
ing radius or maximum angular velocities.
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ki
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Quantitative indices of the various maneuvers are in the final analysis
determined by the accelerations which can be imparted to the torpedo as
it moves through the water. Therefore maximum linear and angular accelera-
tions can be viewed.as-the indices of torpedo maneuverability in general.
As it moves through the water a torpedo is subjected to various kinds of
disturbing influences which disrupt the equilibrium of the forces and
moments acting on it. As a result of thib the torpedo diverges from the
originally calculated mode of mvement. Deviations of this kind also take
place as a consequence of a difference between the actual parameters and
characteristics of the torpedo and its control systems and their values .
employed in calculations. Usually all parameter deviations from computed
values are viewed as disturbing influences.
In stability theory standard rated conditions are called undisturbed motion.
Genermad motion in respect to standard (undisturbed) is called torpedo
disturbed motion. Following are characteristic undisturbed motions for
torpedoes: rectilinear motion at a specified depth, circling motion, and
inclined rectilinear or nonrectilinear motion.
We should note that undisturbed motion can be both steady-state and un-
steady. It is important that it constitutes one of a torpedo's physically
possible motions, taking place under specific given circumstances. Let us
assume, fo_ example, that a torpedo is moving on a rectilinear trajectory.
At some moment in time it diverges from undisturbed motion under the in-
fluence of external disturbing forces (Figure 1.21).
803myu4etinoe 1
Oeuwertue
N ? ....
... 1
ne603Alyuottnoe
deuncenue
Figure 1.21. Torpedo Disturbed and Undisturbed Motion
Key to figure: 1 -- disturbed motion; 2 -- undisturbed motion
If after cessation of the disturbing forces the torpedo returns to the
original conditions of motion (within the limits of allowable precision of
travel), its undisturbed motion is stable. On the other hand, if after
cessation of disturbing forces the torpedo does not return to the original
mode, its undisturbed motion is unstable.
Consequently, stability is a torpedo's ability to reestablish its original
undisturbed motion mode in all or separate kinematic parameters; depth,
trim, yaw angle, heeling angle, and velocity. The more rapidly undis-
turbed motion is reestablished, the greater the degree of stability.
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?
A torpedo's turning ability is its ability to change its direction of move-
ment. Quantitatively turning ability is defined by angular velocity or
turning radius. The greater the angular velocity or the smaller the turning
radius, the greater is the torpedo's turning ability.
Controllability is a torpedo's ability to execute commands (signals) proceed-
ing from the control system to the torpedo controls (control surfaces), and
thus to alter direction cr motion in strict conformity with a specified
program (trajectory) or to follow a signal from a homing system.
Torpedo controllability and turning ability are closely-related terms. They
are not identical, however. It may happen that a torpedo which possesses
a high degree of turning ability will control poorly and will be unable to
follow input signals which change with a specified frequency. Essentially
controllability unifies two opposite torpedo properties: turning ability and
stability.
Degree of stability of toLt:ado movement decreases with an increase in
turning ability and, on the other hand, an increase in degree of stability
results in diminished turning ability. Therefore in substantiating hydro-
dynamic design parameters (torpedo relative elongation, fin surface, control
surface area), one proceeds from the necessity of ensuring an efficient com-
bination of torpedo basic dynamic properties.
1.10. Conditions of Torpedo Rectilinear Motion in a Vertical Plane
Rectilinear motion is one of the simplest and at the same time the most
common type of motion. A torpedo executes this kind of motion on the
horizontal segment of its trajectory as well as when surfacing and diving.
Center of gravity rectilinear motion is possible under the condition that
the velocity vector does not change during the entire time of torpedo
travel, that is,
1"? const 044
?
or
14=0.- a gis amit, ?
where (his the angle between the velocity vector and the horizon.
/n order to satisfy condition (1.53) it is necessary and sufficient to en-
sure constant angle of attack a and trim angle 0 during the entire time of
torpedo motion.
Constancy of angle e is ensured by the system controlling torpedo motion
in a vertical plane. The angle of attack is determined by many factors:
negative buoyancy, torpedo hydrodynamic characteristics and speed. In
order to determine this relation it is necessary to construct an equation
of equilibrium of forces and ioments acting on the torpedo.
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Figure 1.22. Diagram of Fortes Acting on a Torpedo in the Vertical Plane
During rectilinear motion a torpedo is acted on by a number of hydrodynamic
forces and moments. If these forces and moments are mutually balanced or,
as is usually stated, the torpedo is balanced, under this and only under
this condition is 5.t possible to ensure constancy of angle of attack.
In steady-state rectilinear motion a torpedo is acted upon by the following
forces (Figure 1.22). .
Drag
? A z= C -th?
where synthesized coefficient of drag.
Lift
A
/1111A V!
?a Vt. Ti ?
R C ?or
A A 2
For small angles of attack and angles of horizontal control surface
deflection, lift can be represented by a linear relationship between these
angles ?
01 AC. pV:
=""g"."7"Telo+ le"*."?aT8ro.
The first. term of this relationship defines the torpedo's lift without
horizontal control surfaces RY1T' and the second -- lift of horizontal
control surfaces Rypo
CD,
? IDCI' ..
Rh? I. V ? Ciao es A y 11440:
OC 1 A . A.
' Ryp 74.1P. ? Ger? go 119 Vero, ?
38
d cX - ?
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where
Ayi: OC so
OiCy, g4
AP1m-1117 -- synthesized control surface lift coefficient,
characterizing their effectiveness.
-- synthesized coefficient of torpedo lift;
The torpedo's longitudinal hydrodynamic moment without taking into con-
sideration horizontal control surfaces is
AissumCmsi?t4414.
For small angles of attack the coefficient of longitudinal moment is
expressed by linear relation
Consequently,
where
?f-ts'o
?
poy
arlido Ams, 440,
-- synthesized coefficient of longi-
tudinal hydrodynamic moment.
Propeller thrust T. It is usually assumed that force T is directed along
the torpedo's longitudinal axis.
Propeller lateral force O. It arises on the propellers when the torpedo
moves at a certain angle of attack and is directed perpendicular to the
torpedo's longitudinal axis:
Che=i0leo,
where K -- synthesized coefficient of propeller lateral force.
Torpedo weight G.
Torpedo displacement force B.
The moments of forces Rypo T, %Tit B are determined according to the
general rules of mechanics.
In order to determine angles ao and aro we shall construct equations of
equilibrium of the enumerated forces and moments acting on the torpedo in
steady-state motion.
1. Equation of equilibrium of forces in projections onto axis 04
- R?,+ T Cos ao - Qr, sin ao - (at - B ) sln 0.
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?
2. Equation of equilibrium of forces in projections onto axis 0Y1
R B ?????01. TStIIao Qys tot ao Ryi,
3. Equation of equilibrium of moments of forces relative to the torpedo's
center of gravity
???? B (10)4 ao + sIn ao) ? TA ? QysCo ? CpRyp al Os
where 1 -- distanc e between center of gravity and center of buoyancy;
A lowering of center of gravity; Ca -- distance from center of gravity
to plane of propellers; Cp -- distance from center of gravity to center of
control surface pressure.
During torpedo motion the angle of attack does not exceed 5-70.
Therefore we can assume that
cos ao as 1 n sln ao as ao. ?
Terms ciosina, and Asincio are small and can be ignored. Taking these
assumptions into consideration, the equations of equilibrium of forces and
moments during torpedo rectilinear motion can be presented in the form
A 4. ,V;i1. 712 T B) sln
+ A, 4" IC) Ilae ApIlero sa B; (1.54)
? ? I