JPRS ID: 8280 VIBRATION DAMPING SYSTEMS IN MACHINES AND MECHANISMS

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I i6 ~FE6RUARY i979 ~ i OF 2 ~
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JPRS L/8280
16 February 1979
~
- VIBRATION DAMPING SYSTEMS IN
P1ACNINES AND MECHANISMS
~ t
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~o~r~ ~i i - -
- REPORT OOCUMENI'ATION ,t,.REaopr ?+o, t, L R~e~a~n~~~ ~ea~uon Na
- PAGE I JPRS L/8280
~.-TIII~ ~nd luDt~ll~ a. aopon o,~.
VIHRATION-DAMP~i(~ SYS~I~MS ~N MACHINES AND M~CHANISMS 16 Feb rua ~ 7 g
~
7. Auther(f) P~N~ortnln~ O~~~nlaflon R~o6 No,
M~ D~ Genkin (edi~or) ~
~ ~~rlormin~ Or~~~lt~flon N~m~ ~nd Addr~~~ 10. Pro~~et/T~~M/Work Unlt No.
Joint Publicationa Research Service
1000 North Gleb~ Road ~i. co~~n~ac~ o? n~,~ua~ No.
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cct
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= As above -
u.
14. luppl~m~nt~ry Not~~
VIBRO-IZOLIRUYUSHCHZYE UZSTEMY V MASHSNAIQi I MEKHANIZMAKH, Moscow, 1977 ~
~ -
lL AO~tn' ' f,l.zlt: 200 wo?dq
This collection is dedicated to methods of overcoming vibrat3ona ~~f machines
and mech~niams. Met~ods are examined of distributing vib~ation er~ergy along ~
structures as well as methods of acous~ic diagnosis of ball bearitigs aud
- localizing acoustic sources.
1f. Ooeum~M An~ly~l~ D~~crlpton
USSR Elastic Systems Acoustic Diagnoais
_ Engineering Rods Air-Cushions
Vibrations ,Shock Absorbers
b IOMtIMn/Op~n�tnd~0 T~rm~
~ ~
e. COlATI Mld/Crouo ],3(t ],31
- 16 Av~ll~bllity lf~l~m~nt I!. S~turltp Clats Rhis R~poR) 21. No. of PaL~s
For Ufficisl Use Only. Limited UNCLASSIFIED 11 3
NumbQr of Copies Available From JPRS~ 20.3~CwItyC:afs(TAISVip) u.?,,~.
UNCLASSIFIED
~ANi4-i]9.Ip 3~~ In~trvetlon~ on R~v~n� OriiONAL FORM 272 (~-77)
(fortn~rly NTI~7S) -
- D~D~K~~t cf Gomm~rc~
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JPR5 L/8280 ~
- ~.6 February 1.979
VIBRATION DAMPING SYSTEMS IN
MACHINES AND MECHANISNlS
Moscow VIBRO-IZOLIRUYUSHCHIYE SI5TEMY V MA5HINAKH Z MEKHANIZMAKH
" ~.n Russ3.an 1977 signed to press 1.8 May 78, pp 1-1~3 -
_ [Book edited by M.D. Genkin, Izdatel'stvo "Nauka", 2,400 copies,
116 pac~es]
CONTENTS PAGE
Active Vibroprotection Systems -
- (M. D. Genkin, V. V. Yablonakiy) 1 -
An Active Vibroprotection System With Control With Respect to Law-
I`requency and Vibrational Ferturbations
(M. D. Genkin, et al.) 11 ~
Some Problems of Vibrocompensation of. Elastic Systems
(V. A. Tikhonov, V. V. Yablonskiy) 17 -
- Wave Propagation Over Thin-Walled Rods -
- (Yu. I. Bobrovnitakiy, M. D. Gen.kin) 31
On Approximate Theories of Flexural Vibrations of Rods -
(Yu. I. Bobrovnitakiy) 47
Investigation of Space-Time Coherence of Some Types of Acoustic Fields
(A. M. Medvedkov, A. T. Shargayev) 57
A Method of Vibration Da,mping in Som~e Rod Structures
(V. M. Ry~,boy, V. V. Yablonskiy) 65
_ On the Operation of Rubber-Idetal Shock Absorbers
(R. I. Veytsman, et al.) 72 -
- a - [I - USSR - G FOUO] -
,
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CONTENTS (Contlnued) Pa~e
Eval.u~ting the Lo~xding Capc~city of an Air-Cuehion Suppor~ -
(Ft. V~ytsman) 'j~+
Problems of Acous~ic Diagnoe~s
- (Yu. S. Bobrov~aitekty, et al.) ~ 77 -
Bal]. Bearing Diagnos~.g by a Vibration Me~hod
(B. T~ 8hef`te1~, et al.) 95
' Localization of Acoustic Source~
_ (I. P. Biryukova, et al.) 99
On the Prob].em ~f Vibration Diagnoais of Technological F1aws in
Electric Machinee
(V. A~ Ava.kya.n) 104
Saftware Sy~tem for Checking and Diagnos3.ng Electric Machines _
(V. A. Mail.ya.n, et al.) 107
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PUBLICATTUN DATA
~
Engliah title ; V~TION-DAMPING SYST.EMa I1V MACHINES .
. AND MECF,ANISM,S -
Rusai~n title ; VTBRO-IZOLIRUYUSfICHIYE SISTLMY V
MASHINAHIi Z MEKHANTZMAHIi
Author (s) ,
Editor (s) . M. D. Genkin -
Publishing House , Nauka
Place of Publication . Moscow
- Date of Publication . 1977
~ Signed to press , 18 Me,y 77
~ Copies , 2,~00
COPYRIGHT � Izdatel'stvo ~~Naul~~.", 1977
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- E
~ LTex~ An examination is made of the propagation of vibration enFr~~
- through structural elements, mainly through rod s~ructures, and the
- acous~ic diagnosis of ball beaxings is considered as we11 as local~za-
tion of ac~uatic sources.
'1'tie collcction i:.~ written Por scien+ iCic and engineering-technical workers, -
AC`P1 Vf~; V 1 fifZUYRO'?'1?~"f' IUN SYSTEMS
M. I). Genkin, V. V. Y,iblonskiy
Ttiis ~,rticle examines active vibroprotection systems (AVS) as controllable
system~ for vi~ration protection and presents a classification of such -
sy~tems from different viewpoints.
A control.l.r~ble systern for vibration protection is a system for automatically
controllin~ the vibration of a mechanical ob,ject to reduce vibration to a
~ predetermined level at certain points or in a region of space, in a pre- ~
~ determined frequency band or time region for a certain class of external
~ction~. The ob~ect of control is a mechanism or attached structural unit,
the ~ource of information is the data on the vibration state of the ob,ject,
the criterion of effectiveness of control is the magnitude of the vibration or
- som~~ functional that characterizes the vibroactivity of the ob~ect in the =
- ~ l t1i11 11I1 Ql y813 .
?
l1~ r~ rule, controllable vibroprotection systems (VS) require energy input
from rsn add~tional external source. These svstems can be divided into _
three ~,roups (see the diagram). _
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C1~ssification of con~rollable anfl ~.cti.ve vibroprot;ection systems _
, (1 ~ I YnpaHnpen~tre HC ,
^ ~ -
BC BC
(AB)C c nep( ei~Fiwn~?t c nepes(e?t~~oA
- napan+erpaMii cTpy?crypoA
_ ~ ~
- ( S ) C ynpaqneujien+ ~ 7 ) ' -
C Kon~neNCauNeli no bTKJICNENNN) AAaRiHbHWO
(cTa6Nn~~sau?iei~)
8 I (9) ' ,
C ynpaeneHHea~ C ynpanneF~iieM Can~~aa~~PBH-
I1Q PHCIllHBMY f10 A~~H2~iH4eCKNM uJlOll~NOCA
I 803MyW2HH10 KUOpANH3T8M
. I
(12~
floucKOUiae I 6ecno~icxoowe
KEY: 1--Controllable VS 7--Adaptive
- 2--AVS B--With control with respect to
3--VS with variable parameters exter~nal perturbation
4--VS with variable structure 9--With dynamic coordinate control
- S--With compensation 10--Self-ad,justing
6--With deflection control 11--Searching '
(by stabilizatioii) 12--Non-searching
In the first group are the AVS. In these the actuating elements act
- directly on the ob,ject alon~ with the disturbing factors. Passive par~.meters
u:~ually remuin unaltered.
lri t:klc :~econd group are VS in which the actuating devices act on passive =
elernent~ (a mass, a spring, a damper), changing their value in some way
(continuou~ly or in steps). For instance a change takes place in the distri- -
bul,ion of unbalanced masses (autobalancing), the mass or stiffness of a
dynarnic n.ntivibrator. When there is a fairly slow change in parameters,
the ::ystem as a whole behaves like a passive system. Rapid changes that are
= comF~urable in velocity to vibrational process lead to fundamentally new
properties such as increased stability.
in ttie tt?ird group are V~ with vrsriable :~tructure; where there is a change not
- only in parameters, but in the order of activta.tion of various links.
- 2 -
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In sy~tems w3th variable parameters and vaxiable structure, energy losses
_ from the outside source are aetermined by the mechanism of action on the
p~,rameters and structure, and do not directly involve the vibrational -
proce~s it~elf.
I'r~zetically alw~,y~ the active vibroprotection is provided by the enmbined
uction of actuatin~ (active) and pass3ve elements of the system. In some
ca~~~ the.' vibroprotect3ve role af the passive elements is especially clear. -
For instance in the electrohydraulic vibropr~tection system of Ref. 1 there
is a:~priilg that is connected in series with a hydraulic cylinder, and the
AV5 with fc~rce control in the shock-~,bsorbi.ng mechanisms of Ref. 2 also
_ acts only~in combination with fairly pliable elastic elements. Some ~uthors
� [Ref. 3] call such systems hybrid in contrast to "purely active." We feel
that such a division is unsound since ther~ are no "purely active" systems
(i. e. systems that are independent of passive parameters).
- D. Karnopp [Ref. 4] uses the term "semi-acti.ve" for a system with an electro-
= hydruulic mechariism that changes thestiffness of an elastic suspr.nsion.
' Accordin~, to the classification that we propose, this is a system with
vnri~,ble parometerJ. -
In extensive use are mathematical models of AVS as systems with additional
stiff'ness, dEUnping or mass introduced by feedback with respect to the corre-
sponding kinematic parameters (displacement, velocity, acceleration) [Ref.
2, 5]. On the other he,nd the block diagrams of passive systems are depicted ~
as cont;ol systems [Ref. 6]. Obviously this gives no basis for considering
AVS a t;rpe of systems with variable parameters.
Actually the AVS, in contrast to pr~ssive systems, provides a wider choice of
reciprocal and negative inserted parameters. I~Ioreover, the region of
constancy of a given inserted parameter is limited by the passband of the
~ feedback loop. Th,~ introduction of equivalent parameters is useful for a
more ~raptiic representation of the effectiveness of the system in the
working frequency band, but is completely useless in stability analysis.
; A VS mith controZ r~ith respect to perturbation. In the theory of vibro- -
protection many versions of AVS with control with respect to perturbation
c~.n be con:;idered. The source of control is a signal proportional to the
~~erturbin~r, factor~ (force or kinematic). UsualZy in automation, perturbations
are tr.~ken as independent of the response of the system that changes in the
_ ~r~cess of regulation. Such for instance are the kinematic perturbations of
equipcnent on the base side, and also the vibrations of the shaft and bearings
of u ~prin~;-mounted rotor machine if the components of the AVS are on a
founciation or other supporting structural elements, i. e. they are "decoupled"
from the :~ource. The main goal is taken as realization of a transfer function
t~~at ensure~ "invuriunce" (independence of perturbations) of the selected
c~ynarnic coordinates. This is open-cycle c~ntrol. This requires extensive
information on the sy:~tem.
~ 3 -
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- On tt~e basis of given form of the cantinuous spectrum of random pertur-
- ba~ions and various criteria of vibroprotection, the theory of optimum
_ filter~ ia u~e~ to find the optimum ~tructure of'~the VS, which in r~, number -
= of 1.n~tanet~ must nece~a~rily contt~in active elements [Ref, 7~,
Es~entially different from this kind of control is vibrocompens~,tion w3.th
CUIltI~o1 with respect to the dynamic: coordina~l;es cf the ob~ect [Ref. 2, .
8-15]. These coordinates in a certain frequency range can be treated as
_ external perturbations of some part of ~the system. An example is provided
by the dynamic forces f transmitted throu~h elaGtic elements (shock absorbers)
~ to the foundation of machines, or from the outside to the equipment to be
siolated. The best protection for equipment to be isolated is provided
- by applying compens ating (active) forces fa to the po3nts of action of the
given perturbations; to do this, it is natural to use these perturbations
themselves as the control signal. Invariance in this case should be under-
stood as "decoupling" of the ob,ject from the source of vibr~,tions, The
idea:L transfer function of the control circuit is equal to fa/f =-1, i. e.
it is frequency-independent, which is favorable for wi.de-band vibroprotection.
- In contrast to cc,~ntrol with respect t~ external perturbations, stability
- may be disrupted i~l a system due to feedbacks through the ob~ect. Some
_ ver:.ions of. control can be partly or completely represented as control with
respect to deviation of a dynamic force [Ref. 1~+]. Stability can be improved
by using correcting circui~s, filters, or combined control (with res~ect to -
perturbation and deflection).
_ An AVS is proposed in Ref. 16 with control with respect to deformation of -
an elastic element. This coincides with control with respect to force, "
where the shock absorber is an elastic link without losses.
A V~ ulith defZeettion eontroZ. This control principle is used in the over-
whelming ma,jority of VS with electromechanical feedback that axe described in
the literature, They h~,ve certain advantages: there is no requirement for
complete information on the perturbations or (to a certain extent) on the
characteri:tics of the equipment to be isolated since a slight change in "
the latter has little effect on damping efficiency.
IZcF. 17-?_0 deal with genernl studies of feedback with respect to acceleration,
velocity ~,nd displacement in simple unidirectional mechanical circuits.
Re:sea.rch has been done [Ref. 18, 21] on the stability and other properties
- of AVS in arbitrary elastic systems and in shock-mounted ob~ects with six
de~,rec:s of' freedom.
- lief. ?2-25 exnmine multichr~,nnel systems of electromechanical feedb~,ck with
re:~}>ect to deflections as applied to ro ds, plates, shells and an acoustic
medium.
M~,ny E~a.pers have been devoted to the devQlopment and theory of individual
- dev:ice~ a~ well as to their applications. In Ref. 26, 27 a study is done on _
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electromagne~ic vibrocompensators unified into a single e7.ement to~ether
w1.i;h u vacuum tube and a feedback coil (velocity senaor). Ref. 28-36 deal
~ wi~h electrodynamic devices. Despite the great possib3lities proved for
= these dev3.ces by many authors, on1.y laboratory models have been described.
- Apparently the problem lies in the difficulties of making reli~,ble and
economic vibratora.
Matters are different when it comes to controllable hydrau].ic and pneumatic
suspensions designed for d~mping low-frequency actions and compensating
slowly changing loads. Playing a decisive role here on the one hand have
_ been the high operat.fonal reliability and the 1.arge developed forces with _
speed sufficient for quasistatic conditions (elthough according to the latest
data the frequency range has been increased to a few hundred hertz); on the
other hand there have been the extensive possibilities of control provided -
- by eleetromechanical ~ensors (there are VS w3th purely hydraulic cor,trol as
- well [Ref. 31]). Controllable hydrauli~ and pneumatic devices have been -
developed on the basis of equipment that has already been perfected, checked -
out t~nd put irrto wide u~e, The first research on electrohydraulic devices
wa:; iione in Ref. 1, tn the USSR this reseaxch has been developed in Ref. 7,
20, 31-33 and elsewhere. A model of a VS for a human operator has been -
- mo,de ~fief. 7), c~nd con~trollable suspensions are being developed for motor
vehicles [Ref. 34]. A special group is constituted by self-leveling elastic
supports for foundationless installation of machine tools, mainly pneumatic
(Barry Control Co., and in the Soviet Union an original design has been
worked out in Ref. 35).
In modern machines that ~perate on moving ob~jects it is advisable to use
pliable vibration-damping rotor suspensions in the bearings together with -
a se1P-leveling systern (stabilization of rotor position relative to the
stator and housin~). Ref. 3b, 37 describe stabilizing electromagnets placed . _
between the stator and bc:~,rings, and controlled by a signal proportional to
deflection of the rotor or change in the load on the bearings. In view of
the low time constant, the low electromagnetic stiffness of the system and
- its insensitivity to transverse displacements, it has an advantage in s peed,
particularly as compared with pneumatic devices, and combines better with
ela~tic elements (parallel connection). Calculati~ns show that an espec~.ally
al~~~reciable vibration-damping effect is realized with simul.taneous use of -
e1~ct:romafr,Ilets as controllable vibrocompensators on the vibration frequency.l
Al't;er colution of a number of engineering problems an electromagnetic system _
muy be used in rotor machines.
A VS with compensation and stabilization (particuZars of anaZz~sis). Mainly,
we use a system of ordinary differential equations or partial differential ~
ec~uations with constant coefficients and with additional terms in the second
member�. In connection with the introductian of elements of automatic control -
- circuit:; (A.mplifiers, filter~) there is a tendency to change to structural -
method , ttir~,t are common to the ob,ject and the control circuit . The Routh-
fiurwitz criterion is used to evaluate stability on~; ir. characteristic
l~ee the article by M. D. Genkin, V. G. Yelezov, M. A. Pronina and
V. V. Yablonskiy, "Active Vibroprotection System with Control with Respect
to Low-frequency and Vibrational Perturbations" in this collection. _
_ 5 _
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equ~.tion~ of up tn order 4. The Bode-Nyquis~ criterior~ [Ref. 22, 30] is most
_ widel,y u.ied in more complieated cases. In Ref. 13, 14 ari immitance criterion
i~ u3ed ttiat include:~ impedances determinc~d axperimerit~,lly by the same
- vibrocompensator;~.
_ Most; AVS ~,re made to ensure linear operation, t~nd nonlinearity i~ treated as
a spurious eradicable ef.fect. Tn systems for stabilizing quasistatic dis-
placements (including in the self-levelin~ supports mentioned above) the _
nonlinearity of the responses of' elastic and ~,ctive elements is ~onsiderable =
in operation on 1.arge displacements. In the system wi~h electromagnet of
~2ef. 37 the elasti.c elements must necessai�ily have a rigid response to ensure
~tatic stability of the ob~ect in the magnetic field. We should expect _
~ more and more extensive use of nonli.nesr methods for comprehensive evaluation -
of operability under working conditions. A number of authors have pointed
nut an improvement in stability with nonlinear operation.
_ SeZf-ad,justing AVS. 5o far these systems have remained almost unconsidered
in ttie literature, even though they are obviously needed, particularly in
control with respect to perturbation in separate frequency bands, under
conditions of varia,ble frequency and drift of the characteristics of the =
controlled ob,jeet. -
Non-searching seZf-ad,justing A VS. An exa.mple is a ~ystem of a fairly large _
numbez� n of conh,rollable vibrators tha;; compensate oscillations in a girder
structural element,l or in some region of an extended plate. For instance _
- vibrations on a given frequency ar~ minimized at a mean-square or maximum -
~ level ttir~.t i~ determined from the readings of m acceleration converters. =
Under certain conditions we can limit ourselves to compensation of the
initial vibration at n points. For this purpose we measure beforehand
the natural and reciprocal compliance (matrix l). From the vector of -
initial v~ibration Xp we use a digital computer incorpora~ted into the AVS
to calculate ~;he vector of the necessary compensating (~ctive) force -
Fa = -Y-1Xp . -
~ince the mrltrix Y can be found and inverted beforehand on z�equired fre-
quencies, r.~ comparativel,y sme11 computer unit is ~eeded for operational `
control of vibr~.tion. Non-s~arching systems provide the highest speed, which
i~ neces:~ary when there is a considerable change of' the initial vibration
i n t i rne .
fief. 2S considers selection of the points of installation of vibrocompensators
when there is a limitation on power consumption.
Searehing self-ad~justing A VS. The main peculiarity of the qualit,y function
of' i;Yie vibroprotection system for a complex ob,ject is the presence of "local
- 1See the ar~icle by V. A. Tikhonov and V. V. Yablonskiy, "Some Pz�oblems
of Vibrocompensation of Elastic Systems" in this collection.
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~Oit d~'~'~CIAI, U5~ UNLY
~^xtr~mu~" 7'tierePore method~ o� ~enuential variation of dynFUnic test force:~
~dr~ n~t wn11 c~t~ir,~~d t,o :~c:urchin~. tn R~f. ~5 ~ computer wus used to study
- t~ mr.l,hud t~im>>It,~~.ri~~~,~u? r.lui.ti~?r. tn ~~11 f'r~rrr.n in 1~rni~orti~tt th renidunl.
- v ll,r?~t,lc~ti, ri.rul uc~ r~. ;i I;n~~L~~x rn~~t,hc~rl f'~r !.?i~.~ ?truna rn~d~~.l ( rl p~t rder '1'ti~
t'dt~mer method wu~ e!'Cc~itivc f'or dtunping "~ure" normal modea with a~urplu:~
~f vibrr~tor~, The :~peed of th~ latter m~thod d~pend~ consider~bly on the
inittal point of the ~r.arch and nn the dimensinns of the ~implex. 'I'his
~~mni~x mus~ be r~duced during search to s.chi~v~ th~ re.luire~ accurucy nf
aontrol, and nf rours~: there is n,ednc~mitant reductinn of speec~, 7'he -
3peed is determined by the gpeed di` vibration mea~urement on each step, and =
~ by the setting of appropriate rated forces on the vibrators,
Considering the f~asibi~.ity of making rniniature s~1f-cnnt~ined computing
devices, we should cdnsid~r ~e1f'-ad~ustin~ AVS with n~rturbation control `
~ the most promising for ~ources of vibr~tion that are polyhn.rmonic in nttture.
No lesu important is self-ad,~ustment of control circuits with feedback to -
improve efficiency. -
Optimum controt of damping of asciZZations of eZastic ~~stems. Research on
control of o;scillating e].astic systems is bnsed on L. S. Pontryagin's maximum
principle nnd ulso on aynamic progrnmming (Bellman's principle) (Ref. 38-40]. ~
[n pnrticulur, aci ex,ur~ination was made of control of the oscillations of a
~et, a rod und a thir~ plute. Typical formulc~tion of the problem: find a
^ontrol, i, e, a;;y:,t~~rn of forces or kinematic uctions applied at certai~~
- (out:-;ide or intermediate) points of tt~e ob~ect ~o reduce its free oscillations
to u predetermined level in the shortest time. A similar problem is i'ormu-
lated for vibration dwnpers (Ref, 14). Such a control is one of the methods
of uc;tive vibroprotection as applied to nonstationary actions, and speeifi- -
cnll,y impact actions. practical realization of this control leads to AVS
witt~ o~atimum control that appreciably surpass self-ad,justing AVS in their
capubilities. Such systems have alreac~y been created for fairly slow
processe~ (e. R. dfunpin~ of gyroscope o~cillations).
Con:~iderin~; whc~t t~a~ been ~uid, let us note the ma,jor factors that are most
conducive to more extended use of AVS in technology:
l. Av~tilribility of powerful campuct actuatin~ mechanisms (hydraulic and
pneumcitic) thut provide the necessary dynamic and static forces in Lhe
- low-frecluency rnnpe (from n few hertz to tens of hertz).
2. Lc~w mu~;nitude of vibrational .forces transmitted to supports, particularly =
ir~ ::tiock -mounl:e~1 mechani:;m, of rotor type. Z'his enables the use of AVS with
control with respect to force for additional reduction of vibrations. Actu-
nLin~; elementJ CAJ a rule electromagnetic vibrators) may have small overtitl
dimen:ior~; and low ~ower consumption under these conditions.
:i. 'I't~e cai~ability for u~inf; passive vibroprotection devices (particularly
ela~tic links) to improve the conditions of operation of active elements
rsnd cnhance the total effect.
7
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/1 wlde c:hdlc~ at' mr.ttiodi~ nnd mcnn~ oi' control ([brce rlnci vibrritidn trur~c~-
ducern, Ea.mplifiers, :'iltern, correcting ~ircuit:~), and the po3~ibility of
mc~lttn~; compact ct~mputers.
Ei~~'~Ei~NC~~
1, b. Stiubert, b. ~iu~hichkr~, "7'heoretical and ~xperiment~l Ittvesti~ntion of
~~ectrohyflr~ulic Vibroprot~ctinn 5ystems," Kc~nstruirovuniye i tekhnolo~iya
ma~hinostrnyeniya [be~ign and mechnolo~ af Machine $uildin~~, No 4, -
Mo~cow, ;~1ir, 196q, -
2. M. D. Cenkin, V~ C~, Y~lezov, V, V. Yablonnkiy, "Methods of Act~ve
bemping of Vibration of Mechanisms" in: "binamika i Filtu~tikn m~shin"
( Dyne.mica ~nd Acouatics of Machin~s Mo~cow, rr~,u~~, ig7~.. _
3. b. A. His, "Inve~ti~,ation of n 1{ybrid Vibroprotectior~ System,'` ~lcspress- _
informatsiya, "Yspytatel'nyy~ pribory i stenc'~y" ~Test Devices ~.nd Stands~,
19bg, No 13.
~
4. D. Karnopp, M. J. Crosby, R. A. Harwood, "Vibration Control Using Semi-
Acti ve N"orce Generators TRANS . ASM~, 197~ , v. 96 , No 2.
5. K. V. I~rolov, "Reduction of the Amplitude of U.^.cillations of Resonant
System~ by Controlled Change of parameters," MASHINOV~DENIY~, 1g65, P1o 3.
6. S. V. Yeliseyev, "Theory ~f Ac:tive Vibroprotection Systems and its ~
Applications," author'~ nt~sti~act of doctoral di~sertation, Kiev Po1y-
_ technical Institute, 1973.
7. A. V. Sinev, B. A. Potemkin, Yu. G. S~fronov, On the Possibilities -
of Realizing Optimum Transfer Functions by Using Electroh,ydraulic
Vihroprotection Devices" in "Vliyaniye vibratsiy razlichnykh spektrov
_ na organizm cheloveka-operatora i problert~y vibrozashchity" [Influence
that Vibrations of Various Spectra E{gve on the Organism of a Human _
OF~erator, and Problems of Vibroprotection], Moscow, Nauka, 1972�
f~. M. D. Genkin, A. V. Rimskiy-Korsakov, A. M. Tselebrovskiy, V. V, Yablon- `
;;k;y, "A Shock Absorber with Automatic Contr4l" Soviet Patent No 259568,
~ ~YULLETEN' IZOBRETENIY, 1970, No 2.
9. M. D. Cenkin, V. G. Yelezov, V. V. Yablonskiy, "Vibration Dam~inp of
hiultidimen~ional Ob,~ects with the Use of Electromech~nir.al F'eedbacY,,"
Konferentsiya F>u kolebnni,yfun mekhanicheskikh sistem (tezis,y dokladov)
~Confcrence on O~cillation~ of Mechanical Systems (Abstracts of Reports)~,
F:iev, 1971.
1(). h1. D. Cenkin, V. G. Yelezov, V. V. Yublonskiy, "Some Problems of Active
Vi broprotection in a Wide F're~uency Band," Doklad,y na Vos'mo;~ Vseso,yuznoy
Akusticheskoy konferent:.~ii [Report~ to the Ei~hth 1~1.1-Union ?lcoustics
Conference), Moscow, Acoustics Institute, 1973.
8
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11~ M~ D~ (}enkin, V. C3, Ye~.ezov, V. V, Yab~.onskiy, "p~cu~.iarities n!' Som~
Arrangem~nt~ for Active Vibration Damping with Comk~ined Control.,"
in: "Akust3che~kaya din~.mike, meshin i konstruktsiy" [Acougtic Dynamics -
nf Maehines and Struc~ural ~lem~n~~~, Mo~cow, Nauk~,, 1973.
1.2. M, b. Cenkin, V, G, Y~7.ezov, V. V. Yablonskiy, "Cr3ter3a foi 'e~ec~ing
Sy~tems for Active Vibration namping of M~chanisms" in: "Akust~.che~kaya
dinemika maahin i kon~~ruk~giy," Moscow, N~uka, ~.973.
13. M, D~ Genkin, V~ (3, Y~iezov, V. V. Y~b1.~nskiy, "Use of ~'r~quency Sta- -
bility Criteria in prnblems of Active Vibratinn Urunping of Multiresonance
- Systems" 3n: "Akusticheskaya dinamika mashin i konstruktsiy," Moscow,
Nauka, 1973.
14. M. D, Genkin, V. C. Yelezov, B. St~tnikov, V. V. Yablonskiy, "Choosing
the Op~c3mum Parameters of an Active Vibr~tion bamper with Electro-
mechnnical Feedback with Respect to ~e~ative Displacement and motal
I~orce" in: "Vibroekusticheskiye protsessy v mashinakh i prisoy~dinennykh
kar?~truktsiyakh" [Vibroacoustic Processes in Machines and Attached Struc-
tural Membern], Moscow, Naul:a, 1974.
15. M. A. Cenkin, V. G. Yelezov, V. V. Yablonskiy, E. L. Fridman, "Development
of Methods af Active Vibration Demping" in: "Metody aktivnoy i passivnoy
vibroizolyatsii mashin i prisoyedinennykh konstrulctsiy" [Methods of
_ Active and Passive 'libration Damping of Machines and Attached 8tructural
Members~, Moscow, Nauka, ig75.
16. V. S. Martyshkin, I. A. Pikulev, "A method of Autometic Compensation
of F2andom Displacements of Vibratian-Damped Foundations That May be
Ylaced Under Precision Ins'cruments," Soviet Patent No 134412, BYULLETEN'
ILOBR~TENIY, 1960, No 24.
17. Sbornik "Voprosy nadezhriosti i vibratsionnoy zashchitY priborov" (Problems
of Reliability and Vibration Protection of Instrumentu), Irkutsk, IPI
~~ubli~hers, 1972.
18. :bornik "Vibratsionnaya zashchita i nadezhnost' priborov, mashin i -
mekhn,nizmov" [Vibration Protection and Reliability of Instruments,
Muchine~ nnd Mechanisms], Irkutsk, IPI publishers, 1973..
19. Sbornik "Avtomobil'nyy i bezdorozhnyy transport" ~Motor Vehicle and
Oi'f-the-Road Transportationj, Irkutsk, IPI publishers, 197~+.
~ 2U. Sbornik "Teori a akt:v kh vibrozashchitnykh sistem"
Y 1'~Y� [Theory of Active
Vibroprotection Systems], Irkutsk, IPI publishers, 197~+� ~
~1. M. Z. Kolov~kiy, "Controlling Oscillations of Mechanical S~stems,"
M/I:~fiI NOSTROYF:NI YI: ~ 1974 ~ NO 6.
- 9
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22, A. S, Kny~,z~v, B. b, '1'artglcnvskiy, "Using ~lec~tromechnnical ~'eedb~ck
_ foi� b~unp3ng P'1~xural O~cillations o~ Rdd~," AKUSmI~H~SKIY ztIUI2NAL,
1C~65 , Vol 11, No ~ ~ -
23. f3� n. '.Cartov:skiy, "A Multichannel ~lec~romechanical T'e~dback Sy~i:~m ~f _
~~neral ~i`ype ( for Vibr~.tion n~u~iping)" in: "Kolebaniya, izlucheniye i
d~mpfirovgniye upru~ikh s~ruktur" [Ogcill.ations, Ttadi~.tidn and namping _
- nF ~l~,gtic 5tructure~~, Mo~cow, Nauk~, 1973.
24. I~. n, m~rtakovskiy, "Imped~nc~ c~,nd ~tiergy Chc~xt?ct~r3gtics of a Mu1ti-
channel System for ~lectromechanical Compensation of Vibrat3ons and =
� Sound F'i~ld" in: "Kolebani.ya, izlucheniye i dempfirovaniye uprugikh
atruktur," Moscow, Nausca, i973.
25. A� Y. Vyalyshev, t3. S. Ggvrilov, C3. S. Lyubashevskiy, B. D. TartFilcov~kiy,
Yti. Ch. Choni, "Synthesis of a, Syntiem for Comp~nsation of a Vibration -
F'ield in a Limited ttegion of ~ Semi-Infinite plate," VIBROT~KHNIKA,
1g74, No 1 (22).
26. S. Korablev, "hiachine and Instrument Vibratian Exciters and Dampers -
that Contain a Self-Excited 03cillator," Authnr's abstrnct of doctoral
diss~rtation, Institute of the Sci~nce of Mr~chin~s, Moscow, 1970.
27. S. S. Korablev, V. I. St~apin, "An Electromechanical Damper of Rod ~ -
- Oscillations," in: "Mekhanika mashin" [Mechanic~ of idachines], No 35-36,
- Moscnw, Nauka, 1972.
28. Abu-~1ki1, "An Electrodynemic Vibrntion Absarber ~s a;assive or Active
bevice," Konstruirovaniye i tekhnologiya mashinostroyeniya, No 4, Mir, -
~967.
29. Kh'yudimak, "Electromechanical Vibration Suppression," U. S. Patent
- class 318-128, No 3088062, 1963.
30. T. H. Rockwell, "Investigation of Structure-Borne Vibration Dampers,"
JI1;A, 1965, v. 38, No 4.
31. F. A. Furman, "Active f{ydraulic Vibration Systems," VESTNIK MASHINO-
~Tf20Y~:NIYA, 1972, No 5�
32. A. S. Gel'mAn, K. V. F'rolov, F'. A. Furman, R. G. Voranchikt;in, "A
'Pracking Hydraulic Damper of Oscillations of Mechanical Systems" in:
"Kolebuniya i ustoychivost' pribarov i elementov sistem upra�rleniya"
[U~cillations and Stability of Instruments and Elements of Control
Systemsj, MoscoW, Nauka, 1968.
- 33� A. V. Sinev, "Construction of a Mathematical Model of a Hydraulic Power
Control System by Methods of Circuit Theory" fn: "Kolebaniya i ustoychi-
vo~t' priborov, mr~hin i elementov sistem upravleniya" [Oscillations and
- ~ i0 `
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~'OFt 0~~'ICIAL US~ ONLY
5~~bi1'i~y of Instrument~, Me~chines ~nd ~lement,p of Cnntrol Systems7,
MOgCOW~ N~ulc~, ~968,
34. A. Musarskiy, "Inves~igation of the Posgibilities of 'A~-~ive' Suspen- -
gion ~f ~ransport Vehiclea," Uchenyy~ ~apigki univergiteta, seriy~
mekhanika [Scipnt~.fic Annals of the University, M~ch~nics Series],
No 138, C~or'kiy, 197~~
35~ V. m, Shmakav, "Pneumatic Act3ve Me~ng ~f Vibra~inn Dam~ing fnr Pre-
cieion Machine Toola," Truc'~y U1'yanovskogo politekhnicheskogo 3nstitut~ -
[Proceedings of U1'yanov Polytechnicel Institute], V. 9, No l, i973.
' 36. V. G~ Yelezov, "An Electromechanical Control System for Low-Frequency
Correction of the Characterietics of an Elastic Vibration Damper,"
Doklady na Vos'moy Vseeoyuznoy Akusticheskoy konferentsi3, Moscnw,
Acou~tica Ingtitute, i973�
37~ V. G. Yelezov, "A Vibration Damping Support with an Electromechanical -
Control System" in: "Vibroakusticheskiye protsessy v mashinakh i pri-
soyedinennykh konstruk~~iyekh, Moscow, Nauka, 1974.
38. A. G. Butkovskiy, "Metody upravleniya sistemami s raspredelenny~mi
~ parametrami" [Methods of Controlling Systems with Distributed Paxameters],
Moscow, Nauka, 1975. -
39. A. I. Yegorov, G. B. Shenfel'd, "Concernin~ a Problem of Optimwr. Coritrol
- of Flexural Oscillations of a Girder," Trudy Frunzenskog~ politekhni-
cheskogo instituta (Proceedings of Frunze Polytechnical Institute],
No 45~ 1~71.
1~0. I. A. Karnovskiy, "~n the Problem of Eliminating Oscillations in
Plates," PRIKLADNAYA MEKHANIKA, 197~, v. 10, No 2.
41. H. Kvibel, "A Study of the Feasibility of Active Shock Isolation,"
INGR.-ARCH., 1968, v. 36, No 6.
AN ACTiVE VIBROPROTECTION SYSTEM WITH CONTROL WITH RESPECT TO LOW-FREQUENCY
- ANU VII3RATIONAL PERTURBATIONS
M. D. Cenkin, V. G. Yelezov, M. A. Pronina, V. V. Yablonskiy
- An examination i3 made in Ref. 1-3 of active vibroprotection systems (AVS)
with control with respect to force or strain on an elastic element that
- connects the source of oscillation to the ob~ect that is to be isolated.
These 3y:;tems contain individual control channels that include a force or
deformution transducer, an amplifier and an electromagnet that operates on
t1:e ob,~ect to be i~olated close t~ the point of application of the force.
Tt~e transfer ratio of the vibrocomperisation system has a constant value in
the vibration band and i~ close to zero in the low-frequency region. Control _
11
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~ u~
f' i~ equivalent to reducing the stiffnegs df ~he e~astic
elementn in th~ vibr~tion b~,nd, On the other hand, the
n~u ~~trxh3lizai:inn s~y~tem nf Ref. 3 han a r.on:~tunt trnnnfer
ratio ota low fr~qu~ncien n,nd doed not oper~ate in the
vibratinn b~,nd. Control in equiv~lent to incre~,sin~ thr
stiffn~ss of ela~tic ~~.~ment~ on ~.nw fr~qu~nciea. In the
- hn ~n ~a first cas~ the differ~nti~,l of oscil~a~ions i~ ~,r~ificially
inereased on re].r~tively stiff shock absorbers. Tn the
second case it becomes po~~ib7.e to use a pliable insula~ing
_ mN guspen~ion for mecha.nisms ~n moving ob~ects under conditi~ns
of varying misalignm~nts, as ~;ell as inerti~. lnads (ex-
ternal 1ow-frequenc:y perturbat3ons).
' ~H ~M -
On the be~si~ of simple compu~~tiongl exampl~s, this paper
investigates vibroprotection by simultn.neous contrnl in
~'ig. 1 the vibration and 1aw-fr~quency perturbation bands.
Dia~ram af the AVS and basie equatione. 'I'he mechanical ~y~tem (F'i~. 1)
cont~sins ma3s mN excited by external force f(source of oscillatiens), an
- elastic element with cnmpl.ex stiffnes~ k~(1 +,jan), (a~ is the loss factor),
- and also mn~s mH and an elastic element with complex stiffness kH(1 +~~H)
the model of the ob~ect to be isolated the "loed." The action of the
' controlled electromagnet with frame fastened to the source (m~ss mN) and ~
armature fastened to the ob~ect to be isolated (the rigid base) is depi~ted
in accordance with the conventional scheme [Ref. 1, 2] by two equal and
oppo~ite active fbrces f'~ applied to ma~s mN and the base.
The transfer function has the following form: -
7 = K? (P) _ ~ ~ - T~"r" ~ - -
c ~P -f' '
K/c KIK ~TKpi Z~KrMP) i
KI ~P) � T~P' 2(~ 7.cP t'- TKp~ ~ 2tK rNp + 1
for RC and LCR filters respectively.
In the frequency region (p =,~w) this function can be transformed to the
following form (for LCR filters):
~ n a, i- n~~n~ + K~K ~i - n~~n~ - aK? n�/nK
Kr( ) ~i~~~-a+~a~~~+~~n+~n~ (~""'(1fIf2K~~~..~K~~`"K
~~n~~~ ~NA~~~ .
r ~ [K~` (t - n+;n~)' + ~~n'~n~ ~ K~~ - n'i~K}'+ ~Kn'~nK J ~
(2)
rrF~ere w/wo; t~o= yk"/m"; f is the force in the elastic element (control
signal); Kp~ and KfH are the nominal coefficient~ of amplification of the
control circuit in the bands of stabilization (when i2 � 1) und vibrocompen-
, 3ution (when S~ ~ 1) respectively; f2~, S2N are the normalized natural
12
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frequencies of the ~nw-frequency gnd high-frequency LC~ fil.~ers; ac, aK ~re
the induc~~nce loss f~,c~orn of the fi~~ers, the rec3proca~s of
tiheir Q, -
~.'he f'~rc~d osci~.~atinns 3n the AVS ~hat operc~tes in ~he ob,ject (see F'3~. 1)
are deacribed by the ~oiioW~ng sygtem of equ~~.tiona:
Itt"X � k" ~~n) t~ Kr) ~Y") ~ f u~ -
Itlua u kn "i' (1-{- K~) (.1" k~~ ~ _4� j~,~~~ x~~ ~3~
- where ~/o~~"r~ ~ xoe~~W~*m~, and the transfer ratio of the con~rol circuit K f
i~ defined by expression (1)~
We evaluat~ the effectiveness of vibrat3on demping ns usual on the basis of -
the coefficient A(n)= f/f~ of transfer of force to the rigid base via an
- elaetic element in a sys~em with one degree of freedom, It is naturel to tske
the measure of effectiveness of active vibration damping as the absnlute
value of the ratio of frequency responses ~A(St)/Ag(S~)~ of the pass3ve and
active systems 3n the vibration frequency band. For a system of wide-band _
vibrocompensation [Ref. 1) when n~ 1(considerably above resonance) the
effe~tiveness is ~qual to .
~,q/AK ~.e f3K = ~ ko/k~Ke 1/(1 _ KrK),
where kp i~ the complex stiffness of the system without control ("regulax"
stiffness) that ensures resistance to low-frequency oscillations, and
~,K~ = ko (1 K~K) is the equivelent stiffness in the vibration band.
We evaluate the effectiveness of stabilization from the reduction in dis-
placements of the mass or deformations of the spring caused by the low-
frequency actions (when n� 1). The effectiveness can be expressed as
~c ~ kmn/1t 1�I' K~c? C~,
wh~~rc k i.~ the com~?lex atifFnes3 of the system without control, which is
taken as clearly ?ower than the "regular" stiffness, and therefore, generally
spenking, does not ensure stability to low-frequency perturbations; .
k,KO =/r (1 -1- K~~) is the equivalent stiffness in the low-frequency band. _
2'he effectiveness of vibration damping with an AVS having control of both
ty~~es can be expressed by the quantity
B ~s k(i + K~~)~ k(i K~K)= (1 + K~~)I(i - K~~), -
i. e. a:: the product of expressions (4) and (5). Stabilization should
ensure the required stfffness on low frequencies, i. e. the equality
I k I(i Kr~) ~ ko
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_ Vll~t~~~r.rnn~u~riui?I~~Iciti tnuiil, ~~r~i~ur~~~ t.lir. niltilrnurn r.~~iaivnlr.rit, nt,lt'I'nr,uii k(].-K~�i~) fn
I,li~~ I'~~c.~~u4~ru~y ~~ltid ul' vlt~t~uLloti tli~it, lu ~iurtn.I~ii~lt~Yc wltfi Lt~e exi~ting cnn- ~
= ~tr~inte in the low-fr~c~uency band, Agsu?ning condition (7), equation (6)
charucterizes the effectiveness of this AVS t~s compared with the "regulr~r" "
pa~~ive v3bra~ion d~.mper (wi~h gtiffn~~y kp).
UuaZit~ of th2 ~wn eontro~ syst2ma. Lct zl, z2 be the tmped~nces of
the
arms of th~ simplest L-shaped low-frenuency or high-fr~quency filter (L~'F and
HI'F'). If zl/(zl ~+z7) is the transi'er function nf ~he HF'F', then z2/(zl *z2) -
is the transfer funct3on of the LFF with the same elements but connected in ~
rever3e order~ In accordance with expresaion the equival.ent stiffness
k(1 * Kp) that appears in equat3on~ (3) can be expressefl and transformed as -
follows:
for vibrocumpensation:
' - K 1
- kaKa - k 1-- l~~K I~ r~ ~ k~1 t-- K~K z~> ~
\ I . -
f~r ::t~bilization
K
k~KU ~ k C1 /t?c z~ -h ta~ ~ It (1 1~Ic) C~ - f~~ :t t~) '
Comparing expreasions (8) and (9) we find that the AVS with compensution is '
~ equivalent to a stabilizing AVS in which the stiifness of the elastic element
i~ equal to k(1 - KfH), while the transfei� ratio in the low-frequency region
is KfH/(1 - KfH); the AVS with stabilization is equivalent to g compensating
AVS in which the stiffness of the elastic element is equal to k(1 + Kfc~+
w}~ile the transfer ratio in the vibration band is Kpc~~~ + Kfc~� ~i~ enables
' us to investigate stabilizing systems by methods presented in Ref. 1, 2, but
with consideration of the dependence of the stiffness of the equivalent
elastic element on the transfer ratio Kfc�
Both kinds of control ca;;~ also be represented in a form such as vihrocom-
pensation
k,~~ k(~ Kr~ ricZ`~' js~ ~ K~" jiKl'IK 1.~
N ~ .
= k(~ -I- K~~)(1- I~~ _ KiK =~K l f0
t � K/c ztc -M tee ~ h?e z~K -I- =s?c/ ' ( )
where xj~;, ziN are the elements of two different filters (LFF and HF'F in the
initio.l expression, HFF in the final expression). The equivalent transfer
ratio (last two terms) has a"two-step" frequency response.
Expre~~ion (10) implies directly that if the limiting frequencies axe suf- _
ficieritly separated, the controls can be treated separately, each in its
own band. Frequency separation is useful for reducing the mutual influence
of frequency distortions, and also for getting away from resonant frequencies. -
If on the other hand the limiting frequencies are equal and in this connection
ttiC element3 of filter3 zi~ and ziH are identical, the equivalent control
hu:: the r.impleat form
- 14
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- I
FOR O~FICYAL U5E ONLY
_ ' %t~nn ~1 ~~Ic) C~ ~
~
~i >
I ~d~y
. i ~la1
- i0'~ - ~ . _ - ~ ' - i0f IDj
~ ~ 6t~d1
_ _ ~~s? e _ . _ ~ __l... ,
e ~ ( si,;,'u"~ _
, i i ;c~ .o~
/OJ_._ . -
_ - t ;p~ ~
! `
/O
~ - - J. - - I ~O' r0~
~0 ^ ~ , ~ ' r ~ I ~ _
0, t 0, 6 0 ~ 0 1 i 6 ? !0 !t n ~
f~'i~;. 2 F'ig. 3
An invc~;;tif;~~tion aus mndc nf' the fc~._.ibiliLy of rEplacing the residu~ su.~~
t,,y nn imF~ro~~er intc~grnl whic}?, With accurncy to terms of no more than the
::ecorul or~ier of ~mallnc::s of the lo..s factor a is equnl ta
~ ~3
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I
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_ ~OR dF~IC~AL USL tlNt,Y
~
~ ~It~ u n~` ~-~~�X.~ e 1tl~.~...~. n r~ ~ ~ ~ol -
( ~ 1 ~ /
~ ~ ,
~r~t~ ~ r 3~ ~nl nrG t~ c
~ ~i
~ ~ -
-I� C ~ ~n, I n (n t ~ ~ 1 ~1
c ~
~ U4 (I! ~ ~ ~A ~l1 ~
~ C
~ ~ 11 N ~ ~li ( n ~ ) ~ ~
~ (n~f)~~2 n-}� )'}.c
_ ~ ~ ~ ah ~n (~i 1)! ~ c ~
4 ~ahen (?t ~ c '
~~~'~1-~ az7~'~~ b~~, c~a'-~b�~.~
~
Calcul~tiong ahoa th~t for 1~ unifdrm incre~s~ is eb~erved in th~
v~lue o~ th~ integr~~ ov~r th~ re~idu~. ~um (F'i~. 3).
L~t u~ not~ th~t d~p~nding ~n th~ nwnber di the mode t~ be enmpensatec~ -
n -
(3~ r~ gup ~j, ~r Xt ~b~~ ~ W~ W1~ d~~
mhe gr~ph for th~ depend~nce of G~/a� ~((~~t)+.... 11~ on n ig plott~d (~~e ~'ig. 3)
~d th~.t th~ product of the valu~ of th~ giv~n expreasion multiplied by Sn+1 or
I ~ n) for identica]. coorflinatps is equsl. td t~nity. Then to determine th~
numbpr of vibrators neceasary for comp~n~ntion of r harmonics With ~ given
dpgree of damping B, a segment g mu~t be 1~id dff ddwnWard from tih~ pc~int
nn the curve of (~n/an With abscinsa r, and the nearest point (on the in-
cre~~ing ~ide) aith abscissc~ equal to the integer n must be found from the
corresponding ordinate on the curve of Sn+l or I(n). On F'ig. 3 the moves are
ghoun by the points a- b- c- tl.
The resultant estimates are easily generglized to two-dimensional systems.
F'or instanee in the case oF small flexural oscillations of a thin elastic
~~late, we have in ( 3)
n m
L� ffi D~o, LK Q Ph, f" ~l) ~ S~ ~ t~^h (F - F,1 n Ir ~~k),
I~lRal
Wticr~ U i:~ cylindricnl stiffness; is the Laplace operator in dimensionl2ss
~aordir?gths ~ and h; ph i~ mass density.
'I'1~en, repeating the prer.eding considerations for the system of eigenfunctidns
XZY.~rthonormalized aith re~pect to a pair of indices, ae nrrive at an
estimate nf the form
n m
~ ~ I f(i -4-f~)n?,'y-msla~y-- }J RXty~E,~ n,~)I'
;��~Rei
z ~ But~
_ az ~
y�~ (~'~y ) ~w
- 2k
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~
~d~t 0~'~~CIAL U5~ dNLY
Let a vibratidn~l mdd~ b~ n~t up in the pl~.te witt~ r node~ r~l~,tiv~ ta
r~~drdin~t~ ~ an~ ~ n~d~~ relntive t~ conrdin~te t~. mh~n ~.i'ter determinin~
the ~,mpl3tu~e and ~,hnne di th~ fdr~e~ ~rnm the gygtem df equgtinn~ _
A MI
R
~ ~ ~ A~ly~~,t~ ~k~ ~ ~i~ ~~~W~y-- w9) a~y = 1~ ~ ~ ~ ~ ~1~ Y ~ 1~ Z~ ~ ~ . ~ Ili~
!wt R~+1
und~r ~onditi~ns thgt n yr ~nd m~g it ig ne~~gs~ry tn ~h~ck ~~tig�a~tian df
th~ in~~u~liti~~
r. `m
ar ~j ~J ~ikxly~~.~~ ~1p~
t~i ~ B~, (1~)
1 Ir~1 y~ 1 ~t'~.' J'~~~(y..~p~i
, ~~latidn C~~), ~s in the c~~~ of ~ one-dim~n~i~ngl sy~tem, i~ the c~nditibn
df minimi~ing th~ work df forc~~ f~k 1c~c~t~d ~t th~ pdint~ o� inter~~cti~n
of ~ rectan~ular nx m gri~ on g~ner~li2ed di~pl~eements ~f moc~~s af higher
order tht~r~ thoge ~h~rg~terix~d by th~ pgir df numb~rs ~~nd m.
Cantrol mad~li~g i~ith respe~t ta th~ inf~uett~e matr~x. Vibro~ampen~atian in
d 1in~ar elastic ~y~t~m assum~~ ~electi~n ~f' ~~ontrol suCh th~t it sets
up a made cloge t~ the giv~n X~, but oppnsite in sign, i. the limiting
condition -
(R'~ X�) ~ ~ X�
must be satisfied, Whpre F~ ~ {fA~ k ~ ~r 7~ . . l1~~ ~l'a ~ {X%i l = Z~ . . m}i
~X~} = tx~~,
To d~termine the required v~ctor of ~ontrolling fnrces, We Wi11 start frem
the limiting case Where X$ - X~ = 0, i. e. the case of total compens~tion of
vibrations. Then we determine F'a from the equation
A~' ~ x~. ~1y1
Nere A~{p,k; f= 1, 2, m; k= 1, 2, n} is a matrix (in the gener~.1 case
an operator - function) that in Lhe cgse of harmonic oscillatinns is obtained
u~c u r~e:.ult df formal multiplication of the elements of the influ~nce r,atrix K
by -wa; the dot~ over the acr~leration vector gre a~ymbolir desi~ngtinn
for emplitude (hereafter they sri11 be omitted).
When tt~r.ciimen~innality of vectors X~ and F~ coincides, i. e. the number of ~
F~oint:, of observution is equal to the number of points of unidirectional
discrete control (m = n), metrix equation (14) may h~ve a unique solution
for det A# 0 gnd matrix rank equal to n. Usually the number of vibro- `
iicceleration pickups in~talled to monitor the uniformity of eompensation
exceeds t1~e number of vibrators (m >n). In the general case the system of
e~~uation~ may be mutuully exclusive.
~5 -
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i,~t un fdrmul~t~ th~ problem fallowg. It i~ required to nelect ~ ve~t~r
uu~t~ th~t ~~me c;f th~e equatidnn ot' ~ygtem (14) ~r~ g~.tigfied identic~lly, nnd
bhc error r~f ~a~lEiCuc~t;inn th~ remriinit~~s r,huntidn~ i~ det~rmined by the
- ~dnditian dt' minimiznt3on nf qu~lity function on~ ~~n find ~n approxi~-
mnte ~~].ution of the ~ntir~~ gygtem d~ ~qugtinnn by the methhd nf m~~n s~utir~~.
Ac~~wning that th~ r~t~k ~f ~he m~trix A~.ttd af
th~ expundcd m~ttrix aft~r the
apprnpri~te ~~~.~~tion d~ paint~ i~ ~qun~. to n, w~ writ~ gysten~ ~f equationy
(l4) in ~he form
M
}~r ~rR,~s ~x~, (i~~ k=-~ 1,~,,,,,n}~(i,2,,,,,nc), (14~)
n
1~r ~YA ~ xf GJ (1l.~1 i r = {1, Z, ~ ~ ~ ? Iit}~~1h1~ 11~Ib ~
t ..l
5~lection of pointa {ik? the pdintg where the functinn di the resu].t~nt
di~tribution of
vibratidn vani~h~g ig ~i�bitrur,y and i~ detertr~in~d by the
requirement�~ of ~xisten~e af ~ soluti~n of equ~tion (14~.) and c~nstructian
of the best epprdxim~ti~n nf' ac~ ta ~ti.
Modelin~; W~s done with applic~tion to a found~tfon for the shock-mdunted -
ob~ect that congi~ted c~f twn identiral girc~~rs oscilluting in phase. The
initial vibration fie~.d wag assigned eith~r ~s ~n individu$1 r~r mixed ndrmal
m~d~ of fl~xural oaeill~tions ot' a girder fixed an_i supported at the ends, -
or e1~e as th~ re~ult o� action of a systen of cnncentrated force~ an the
nhock-ab~orber side. 7'he elements of
the matrix Were G~lculated it~r these
eases by approxin;ate computution nf infin3te ~ums of the form
X, (E~) tER1 As ~'s
~ ~ ! ~ 1,2,...,m;
~tR a ~ 1'1 j~~ r 1'~ - ~ ~
k~l,2,...,n,
and also in closed form in terms of Krylov functians of a complex arguemnt
a~~ x' ~F e~ ~~a t ~ ~~45 -4' Cs4 T `F' C~4U (~,t)
+ ~'~o~ xV [x - ~?)1, x' = f~' (i + j~)1"'~
~ ~oo/ W1� '
ilere the constants ClQ-~4Q were found fnr each segment by solving the corre-
~ponding boundary value problem.
A model Wa~ constructed fnr compensatior, of individual n~rmal modes of the
Nirdcrs ~upported and fixed at the ends (see ~ig. 1):
.~r � s n~ E sh ~
~ ' ~ y (4r -1- i),
r -
and t'or mtxed modes: x~ aaX,(;,,�). A preZiminnrf investigation Was made
~
of the ssurfnce of the quality funct-ion with respecL to t.he two previously
indicated criteria, preference being ~iven gmong step-r,y-step search methods
to p~rcillel change of coordinates vith a variable step
26
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~
t~0[t tll~'t~'tC~AL U5~ dNLY
~h`" ft~ ~ k ~j X? (~k)~ ~ tni ~k ~ l~X, (~a), k= f, 2, n}, . _
;3omci renu].tg c~f c6mpen~ati~n df individu~.l mdden are ~hown in f~'ig, 4-6, won
Ci~, 4 the c~xi~ of' ~b~cis~an refle~t~ the rtep-~y-~tep eh~n~~ in foree~ f~,
ancl ttte tioman r?um~r~lr~ indi~c~te: I--the methofl af influen~e coeffi~ient~;
f.t--~,tep-b~-nter ~e~,rct~. A"~u~.ly" ~tructure af tti~ ~urf~ce nf the ~uality -
i'unrtivn ia nbn~rv~d, i, c, th~r~ i~ a. ~harp minimwn dP r~ fun~tidt~ iaf m~ny
vnri~bl~n, th~ de~r~e c~f ~h~,rpn~gg di the minimum ~n the surf~.ce incre~sing
with u reduetion ~f
tih~ numb~r r ef the mdde and t~n incre~.se in the number n
- of vibr~tdrn. ~'he lgtter ~.~adg td ~~gse nf poe~rly c~nditioned mgtrix A
gnd un unnt~bl~ ~o~utidn I ClOge to th~ ~ldbal minimum of the qu~lity
i'un~tidn~
Di. Oi ~
ri g . 4 ~o
n =.f
/�f 4j
_ ?
. !l~ ~ ~
! -
/D"~
~ ' /
/O 0 : I ? if�I/~X,;I
d$ B, dB
eo fo
~0 10
10 , -
lil s Iti ~ + A ~
B / / ~ I~til IO / ~ ~ 1 ~`.Is~l �
~ 0 I ~ ,I~'~-.r"~I ~ 0 ~ .1~ (H~) = a~4~ (U). (29) -
7"he relation between forces and displacements is e:cpr~saed in terms of
linear
bendin~ stif�nesses
~=U (N~1 g�~ lN,)l
~.~c~?~y~1 air ~i~i [a~~?(N,)J ~ (3u) -
F'or ~he flunge, this relation is found from formulas (16) and (17)
r~ ?~n~ zc;?� u r,Y ~ ~n~
M~s~ ~ 11 ~t~~:~ at:) ' (31)
31
F'roc~~ding in the usual way, i. e. substituting relations (30) and (31) in
equa.litie~ (28) nnd talcing consideratian of conditions (2q), We can get the
_ uou~t~t di~persion equation of antisymmetric normal Waves in a T-rod in the
form of a zero determinant of second order
Rf' ~ + ~C;?� ~r~~ '
g~it ~~i) 2gia? = (32)
Equation (27) is the dispersion equation of longitudfnal n~rmal Waves in the
T-rod, and equation (32) is the equation for flexural-torsionul nor~nal aaves.
4. l~ox-becvn rbd and I-beam r~ith identical flanges. Rods of this type sre
shown in F'ir. 4, 5. The3e are typified by the presence of two mutu$lly
. 41
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_
~Ott d~'~~CtAL U5~ dNLY
.
perppndi~ul~r pl~?n~~ ~f mirrar gymm~try. As a c~n~equen~~ di this f~ct, in
thene rodg there ure normt~]. waves of
four modps tht~t exiat inc~epenc~~ntly of
dn~ anoth~r; longitudin~]., i~ c~ ~ymm~trie r~l~,tive tn bdth plun~~ of ~ym-
m~try; twn mod~g nf ~lexur~.1 wave~ ~hc~t ~.re ~ymm~~r3c ~~1ati~ ta ane of th~
pl~n~~ ~nd an~3gymm~tric r~l~tiiv~ ~td the other; r~n~ t~rsianal w~ve~ that ar~
chargct~rize~ by nnti~ymmetric: motion r~lative to bdth pl~nes ~t' symmetry.
Let u~ congi~~r flirgt th~ bdx-beam rnd (see ~'ig. This Cdnsists of fnur
p~irwige iflentiical strips cnnnect~d in four corne~�s. In vi~w of the ~ymmetry -
it in gufficient to con~ider the inter~ction of twn sti�ip~ in one of th~
_ cornpr ~oints, einc~ the mdtion and interaction in th~ nth~r ,~dinin~s wi11
be repeat~fl symm~trically or antisymmetricnl~.y d~pen~ing nn the mo~~ nf the
norm~.]. wav~g. Let u~ taic~ two stri~~, e. g. the upper one ~nd th~ on~ ~n th~
~~ft, that form the upper ].~ft corn~r ,jdint. Let u~ con~truct ~ cnordin~t~
sygtem for each nf th~ne ~trips so that the xl and x2 axes coincide with the
middle lines, while the yl and Y2 gxeg li~ in the pl~ne di the strips and
sre direC~ed toward the line of their intersection.
When a normril, w~.ve of any of the four mode~ enumer~ted ubove prnpagates in
_ the artgle ,~oining, forces of reaction Nith three compon~nts $rise as we~1
as ~ reaction be~nding moment with one component (alon~; the x-fucis). In the -
selected coordinate system the rel~tions between them, and r~.tso the relations
for the displacements e.nd angles of turn of the strips Will coincide in -
aceurncy with the analogous relations for an angle-iron rod (19) und (20).
However, the relation betWeen the forces and displac~ments for each of the
strips is different from (18), and Will. be expressefl in terms of nymm~tric
and gntisymmetric linear dynamic stiffnesaes. The derivation of' the necessary
formulas and the final results repeat the computations for an angle-iron rod.
The dispersion equations of the four wave modes in the box-beam rod are four
zero determinants (22) in which linear stiffnesses with symmetric or anti-
_ symmetric excitation stand in place of the lfnear dynamic stiffnesses of the
strips with asymmetric excitation. Thus in examining the longitudinal weves -
for both strips it is necessary to take symmetric linear stiffne~ses. In the
cese of torsional Waves all linear stiffnesses in equntion (22) must be taken
for antisymmetric excitation. ~ro cases when symmetric linear stiffnesses
are taken for one of the strips in equation (22) while antisymmetric stiff-
nes::e3 are taken for the other corr~spond to independent f'lexuresl Waves
in the horizontal and vertical plsnes. �
Now let us go on to the I-beam rod (see ~'ig. 4). It consists of a single
vertical ~trip (aall) and tWO horizontal double strips (flanges) that are
taker~ as identical. Thsnks to the presence of mirror symmetry, here as in
the ~~receding case We can consider only one ~oining of the strips, e. g. the
~oining of the Wall r+ith the upper flange. In the loWer ~oinin~; everything
uill take place symmetrically or antisymmetrically relative to the horizontal
plane of symmetry. If ae take txo coordinate systems one in the aell und
the other in the flange, ~ust as for a T-rod, the further derivation of dis-
persion equations here Will exactly repeat the derivation done for the T-rod.
The only difference is that for the wa11 it is necessary to take sym~netric
42
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i
~O~t O~~ICIAL U~~ dNLY
c~t~ ~ntinymm~triC ~in~ur c~yn~mi~ ~tiffne~g~~. Ag a re~ult we get tYte fdl~owin~;
aisp~rsion ~qu~tions di th~ f~ur ndrm~.L warr~ mode~ of the ~-beam rod (~or the ~
t~tilte df convenience the index "1" 3~ dmitt~~ th~ in~~x "2" is replaced by
t~ utrok~);
i'nr ldngitudinal w~v~s:
>;~t.
Cj1-~' 2Ctt Cin 0
Cnt ~in 26~~ ~ ~
or in expand~d form
~ C~ ( ~~n (~Cir) -4- ~ir (~ar~) (2Ci~)(2Bir) ~ 0, ~33~
wher~ the first term is the det~rminant of the matrix df ~inear stiffn~ss~g
nf the wa11 in th~ cQSe af symmetric excitati~n by longituflinal-~r~sverse
i'orce3;
for flexural waveg in the plane of the ual.1
Cl~ 2Clt Cin
, ~ 0,
Cnt Cnn'~' ~~tr
nr in expgnded form
~ C' ~ Cir (2Brr) Cnn ~~Crt1 ~Z~ct)(ZB~~) ~ ) -
where ~Ca) is the determinant of the matrfx of linear stiffnesses of the wall
in the case of antisymmetric longitudinal-transverse excitation for flexural
r+ave3 in the plene of the flanges
e� + ~n~ B~n ~ o
Bnf ~an'~� 2gaQ - ?
or in expanded form
~ D~ ~ -4- 6ir (2eis) ~,`~z (~Cn~) -4- (2eaa)(ZC~~) = 4, (35)
where ~F3~~ ic the determinant of the matrix nf flexural linea~r stiffnesses of
the Wnll in the case of symmetric excitstion;
�or torsionnl Wave~
~ ~
B~t 2C~~ arn
Bnt Bna'~" =~as ~ O~
or in expanded form
( B� ( Bff ~2Bs:) Baa ~~Cnn) t28as)~2Cnn) _ ~36~
~ 43
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~Oit d~~tCIAL U5C ONLY
wh~r~ ~~u) i~ ~h~ fl~t~rmin~nt d~' th~ m~trix ~f flexurn]. linenr dynnmir.
~eirrn~uESen of th~ wbll in the Ca~e c~f ~nt-,is,ymmetric ~xeitr~ti~n nn th~ ~dpeg.
~'r~neral prapert~,~~ d~' dinperciidn ~~~ur~ti~nc;. ~r~Ch rnt~t t~r th~ dinp~rnidn
~qu~tinn giv~~ n v~ue r~~ th~ cnngtant ~f prnp~gation df n n~rm~1 w~v~, ~nd
therefore det~rmin~s its d~penden~e .~n th~ lnngitudinal codre~in~te of the rod. _
mhe real rootg ~orre~p~nd t~ norm~l a~ve~ thgt propngnte over the rod withdut
- d~mping, while the imaginary ronts correspand to w~v~s that are expdn~nti~lly
damped With regpect to th~ x caordinat~, all point~ t~f the rod fluctuating in
phn~e. 7'he romplex roots corregpond tn trav~ling adves with wnplitud~s thnt
increase or d~~rec~ e~xponentially with renpect to x. Wav~~ nf th~se typ~s
have c~lreac~y been encaunt~r~d in the ~~uc~y ~f fl~xurgl anfl 1ong3tudin~].- _
. tr~?ngverge ogcill~tions of ~ thin e~.ustic gtrip [Fief. ~j. ~'hey a11 ~rise in
compn~ite thin-w~11~d rodg ~te11.
bigper~ion equatidtts (22), (~7), (~2)-(~~), e?ft~r sub~titution of the corre-
sponding expressions for 1in~ar dyr?~mic stiffne~se~ t~re ~r~ns~endental -
equations in Which the variable a= kH is ~quared ev~rywhere. This menns thst
if ~ is ~ root of the ~qur~tion, th~n -a is ~lso a root,
7'he first members of the equations are rettl quantities, and for each of them
th~ ~qua].itieg f(a) = f a = f(A) are sati~fied (the line denotes the complex `
con~ugate). In other Word~, if J1 is a ront af the equntion f(a) = 0, then
the complex con~ugate a Will also be a roat of the equgtion f(Ji) = 0.
Thu, the renl end imagin~ry roote of the d~rived dispersidn equations are
alw~ys met in pairs ta, While the complex roota occur in groups of four
~J1, iJ1.
We can easily go on and convince ourselves that since only rational attd
exponential functions nf J1 and (a2 t u2)'~ appear in the first members of
these equations, they are entire functions of first order. On the basis of
general theorems of tt~e theory of analytiral functions [Ref. 13~, suCh
functions have an infinite (even) number of zero~ that cannot have point~ of
croWding in any finite part of the complex plane. This implies that in ~gch
of the rods considered gbove there is an infinite number of normal Wnves,
and that the modulus of the constant of propagatioa increases monotonica2ly
aith $n increase in the number of the normal Wave.
Let un noW prove Lhe following statement: on ac~y fre~uency there ~xists a
!'initc number of real and imsginary roots and an infinite number of complex
root3 of the dispersion equations investi~ated here.
For the proof We con~ider the behavior of the dimensionless constant of
propuggtion 71 of Waves with hi~~}~ r.umbers whose modulus is much greater Lhan
unity, and the parameters U~, u~ and ~+2.
- To do this, We expand the first membcrs ~~f t.he dispersion equatians in series
aitt~ re~~ect to t}ie small quentities (~~;~2i.), (~i?~2i.) and Lnking only the
44
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~dEt U~~ICIAL U5~ t~NLY
i'irdt t~rmg, we invc~ti~rat~ their n~ymptoti~ beh~vid~~ mhe ~xpt~n~idt~ i3 u
cumberavme operatf~n, anrl therefore ~11 intermedinte r~~u~ts nre left out,
Ag an ~xrunpl~, we ~iv~ the exp~nsi~ns ~f th~ ~ine~.r dynt~ic dtifft~e3ses of
u strip with E1gy'tninetriG excitatidn:
fc~r t'lexural wavea .
- ~r? ~ 4~D~' (c sh ~ 2~)/~,
. ~fa = ~af ~ n~1 V) ~1 ~~C~ Ch 4~ ~ (~i~,)3)/l~i/fa~
. Qaa 4D1~ (r sh 4~ ~ 4~1/~1~FI+
(1,~ ~ ~ D~ (i v)~ (2e~ ~h /i~ - (~i~)~~/A1H~,
d, = 2c ch (4~)'; ~~7~
fdr l~n~itudinal-tr~n~ver~e waves -
c~t ~ ~~i~~Q~. ~h ~~/~i =4~ v~ HeQ, =
~~n = C�~ 2ih~Q}, (?gch hf. (~~)~~//~~a,
~nn IOhE~~ sh 4J~/(1 v) H~q,
- ICI~ 1~~I4~p~.1~~C111~~ ~~l~l~R~~~~1,2r
o, ~ ry~ ~h ~~,~~1. ~3g~
Ner~ a ~ (1 + v)/(1- v) ~ e = (3 + v)/(~. - v), f = (3 - v)/(l + v) ~ ~ = C~ - v)/f 1 + v).
7'he expc~n~ions of ~inear dynamic stiffne~ses c~f the strip for syrr~netric and
nntisymmetric excitations h~ve the sgme fdrm nnd drder with respect to
It i~ immediste~y c1~ur i'rnm thege �drmulus thgt at large ~a~ the flexural
linear stiffnesses exceed the longitudinc;l-tran~verse ~tiffnesses. tn nther
word~ for short uuves the bending of the strips is the decisive �orm of
motion, f~nd the ldngitudir,al-transverse stif.�ne~ses in the dispersion
equation~ can be disregarded in comparison with the flexural linear stiff-
nes~es.
F'or instnnce let us con~ider disper~ion ~quations (33) and (34) tor longi- _
= tudinal nnd flexurul Wavefi in ~n I-beam rvd. Substituting in these equgtions
the u~ymptotic formuln~ for linear stiffnes~es and disreggrding qu~ntities
- Cn~a und C~~a r?:: having nrder of ~mallness l~-z in comparison With B~t, ue
cnn reduce them td the form (C~Z~ + 2~t~)Btit = 0, or in exp~nded fc~rm
(e' sh 27~' -'li.') (al ch 2}, ch 2).' a~ sh 2ks1~ 2}.' ;
a~ (2}.')' cl~ 2~1 = 0, (39)
afier�e the u~~ nre con3tant~, =1, 2, 3. 7'his expr~ssion obvit~usly decomposes
- intu ~~ro iniiependent equations. ~'he first is an asymptotic dispersinn
- ecluation ~f ~ymmetric flexural Waves in the flunge thet are not influenced
by lon~itudinal-transver3e mnvements of the ~all. The second equatinn
de:;cr�ibe:; purely lon~itudinal Wgves in the uall-flange system under candition
thr~t no bending of Lhe flanp,e3 occurs.
After carrying out the same computations on equgtions (35) and (36) for
flexurul s~nves of the second ty~?e and torsional Waves of an I-beam rod, ~e can
45
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~dEt ~~~IC~lIt, t1~~ dNt~Y
~,~t th~ �ollaaing ~~ymptati~ ~quution:
bl sh ~Jl Ch 2~?` b~ 5h ~i.' ~h ~t= bs st~ 7~ --3 U, (dn) .
`~'11ic~ ~xpren~ion d~~~rib~g C~mbfned flexur~l. ns~illr~tiong (symmet~i~ ~.nd a.nti-
_ gymmetri~) dt' g Wul~ ~nd t'l~nge~ aithdut the p~rtiripation of lon~itudin~.l-
trt~nc~ver~e unve~ in thi~ prr~ce~s, ~qu~tia~~ c~f ty~es (39) nnfl t4~) gre nlsn
obt~in~d f~r ~n ~-b~~m r~d.
_ ~'ar the r~ngl~-irdn rad ~n ~na1d ~u~ procedure 1~~~s t~ tu~ in~ependent
~qur~tion~ : C~~ * C~~ ~ d~nd I~t~g ~ ~+~t ~ n~ = 0 Wt~ich, gfter ~ubstitu~in~ ~h~m
in ~37) ~a C~~), r~r~ r~~u~e~ ta equ~tion~ like (~9) ~nd cb~). On~ c~f th~se
~qu~tic~n~ d~~~ribe$ pureiy langitudin~l a~ve~ in the ~ng1e-irdn ro~ Nher~
b~n~i ng daes~ n~t o~~ur ~ 7'he ~~~~nd re~.ates t~i purely flexur~ t+aves th~t
nr~ not influenced by l~ngitu~.n~].-transver~e di~placements of the strins.
- ~c~u~tion~ of the ~~m~ kinfl gr~ ~btgfn~d for a Channel-irdn ~nd btix-be~,rn roc~. -
F'rom ttii~ ue cun s~~ thut in the cgse of
1~rg~ ~,1~ dispersion equ~tiz~ns (~2),
- (~'t). (32)-(36) redu~e tb simpl~ ~qugti~ns trith ~~ner~ f~rm that C~h be
repr~~~nted ag a line~r combination of qugntities sh(2a Ch(2a �~a~)~
a2ch 27? and a~sh ~ut nuch cqugtions huve ~nly cbmplex root~. We cun
- convince ~urselve~ nf this by repeeting the deriv~tion r~f the f'ormulas for
th~ re~t~ that W~ done in e~ef. 2 fbr simpler equntions of this t;~pe (setting
~+in, substituting ~xp n/~ for sh n nnd ~h n, disregarding all terms
tt~nt vanigh as n-~~). It turn~ out that niter suCh trsnsf~rr,~atit~ns the
- equ~ti~an hu~ a aolutic~n only Wh~n and expr~ sre qu~ntities ef the sgme
order. This impli~s that all its roots are complex. 7'he imaginary parts r~
are proportional tn the numbpr of Lhe raot (tsking them in the arder af
incrcasing ~bsolute value), Whi1e the real parts are proportion~l to the
- logarithm of th~ imagin~ry parts.
7'hus the constants of propagation of normal uave~ c~f high numbers, b~ginning
aith ~ome J~n, are complex quanLiti~R. But since there ~r~ no point~ of
cro+~dinp, af roots in a Finite section of plane a, only a finite number of ai
cgn be ~ituated within a circle a0, pl Qi~rce of Random Perturbations in an Oscillatory System," MASHINOVEDENIYE, -
1)'f 2, No 6 . _
~
93
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b~. K, W, Qoff, "An Applicut;ion df ~nrrel~tion Techniquen to Some Acuust3c -
Measurem~ntn," J. ACOUSm, 50C. AMEfi~CI1, i9y5, v, 27, No 2,
47~ A. K~ Navikov, "~pplicat3nn of the Method of Mutua~. Spec~ra td Some
Acou~tic Me~surementg" in: "13or'ba s sttumami i vibra~::ty~rni" CCon- _
L�rolltng Noises and Vibratiinns~, Mo~cow, Stroyizd~t, ~.g66.
48. nzh, Bendat, A, Prisol., "Tzmereniye i~n~.liz s~.uct:aynykh protse~sov"
[M~a~ur~men~ an~ Analy~in of' Random Procease~], Moscow, Mir, i974, ~
4g. A. K, Novikov, "Korrelyatsinnnyye izm~reniya v korab~l'noy akustik~"
[Correlation Analyais in Ship Acoustics], Leningrad, Sudostroyeniye, ig7i.
50. N. A. Rubichev, "A Method of Experimental netermination of the Contri-
bution of Individual Source~ to Overall No3.se" in: "~Cibernetichesknya -
diagnostika mekhanicheskikh aietem po vibroakusticheskim protsessrun,"
Kaunas, Kaunas Polytechnical Institute, 1972.
51. F. Ya. Balitskiy, M. D. Genkin, A. G. Sokolnva, "A Correlation Technique
for Locelizing ~tatistically Dependent Sources of Noise" in: "Bor'ba, s
shumami i vibratsiyami metodami stroitel'noy akustiki" [Controlling
Noises e.nd Vibration~ b;r the Methods of Structural Acoustics], Moscow,
Strnyizdat, 1966. ,
52. F� Ya. Balitskiy, Yu. I. Bobrovnitskiy, A. G. Sokolova, "Measurement of
Vibrations in the Presence of Correlation Interference" in: "Analiz i -
vosproizvedeniye vibrat$iy" [~nalysis and Reproductifln of Vibrations],
part 2, Lenii~grad, Leningrad House of Scientific atid Technic~al. Fropagunda,
1967.
53. M. D. Ge;ikin, V. I. Sergeyev, A. G. Sokolova, "Application of a Corre-
lation Technique to Calculation of the Dynamic Characteristics of
Mechanical Systems Under Conditions of Irarmal Operation" in: "Vibro-
akusticheskaya aktivnost' mekhanizmov s zubchatymi peredachami," Moscow,
Nauka, 1971.
54. D. G. Levchenko, "Problems of Separating Signals that Differ in Fluctu-
= ations of Autocorrelation" in: "Metod,y predstavleniya i apparaturnyy
analiz sluchaynykh protsessov i poley," section 1, Leningra~, All-Union
~cientific Research Institute of Electrical Measuring Devices, 197~.
55� D. G. Le�rchenko, "On the Problem of Measurements in an Inaccessible
Region" in: "Metody predstavleniya i apparaturnyy analiz sluchaynykh
protsessov i poley," section 1, Novosibirsk, State Scien~ific Research
institute of Measures and Measuring Instruments, 1968.
56. C. J. Dodds, J. D. Robson, "Partial Coherence in Multivariate Random -
!'rocesses," J. SOUND AND VIBRATION, 1975, v. 42, No 2.
94
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~
~
~
_ ~ ~OR n~~ICIAL US~ ONLY "
57. Ya. Y~,. ~
Gel'f~,ndbeyn, "Me~tddy kiberne~icheskoy di~,gnostiki dinamiche;~kikh
_ ~istem" [Methods of Cybernetic biagnosis of I?ynamic Systems], ~ii~a,
zinatne, ~967,
y8. I'. C. Chang, A. Lin, C. A. 5e~nr, K, S. Su, "netermitta~tion of the I'ulse
Wave V~lncity by F'iltered Cross-Corre~.~tion mechnique," J, I3IOM~;Ct{ANICS,
- i97i~, ~ v, 4, No 6.
$ALL B~AI~ING DIAC}NOSIS BY A VIB~ATION METHOb -
T. Sheftel', V. A. Gushchict, G. K. Lipskiy, A. A,. 5hant~syn, F. F'. Yudin
, _
- 5pectrograms of ba11 bearing vibration show a background of a continuous
sFectrum u~,rzinat which characteristic discrete component3 are observed to
var,; in ~~ro~jnrtion to the apeed of rotation. T~ese spectiral components
ari~e 'bec~use of deviations of the races from circular3.ty in a pattern c~lled
wavinesn. To evaluate the vibroactivity of waviness, a spectral method is
u~ed i;hat enables the derivation of ane,lytical relations between the param-
- eter~ of the harmonics of wa,viness and the parameters of the resultant
vibr~tion: frequency and amplitude. On this bas3s, solutions are f'ound for
the following problems in ball bearing di~gnosis;
by a~f=:,ration method, without disassembling the bearing, the technical
state of the outer and inner rings is determined, w:~~ch is characterized
by the presence of harmonic components with high vil,roactivity in the
spectra of waviness of the races;
Before a Ue~ring has been assembled, the harmonic components of waviness on
the ball races of the inner and outer rings are established (by a vibration -
method), and a prediction is made concerning the corresponding frequencies -
and amplitudes of vibration of the bearing assembled from these rings;
f~r each harmonic component of waviness of the outer or inner ring, the
admisaible value of its amplitude is determined as a function of vibration
requirements.
The information on the technfcal state of the ba11 races of the rings can
be used to select rings for assembling 1ow-noise ball bearings; to determine
ttie de~;rec of suitability of a bearing under working conditions; to calculate
~ the vibration of bearing units that arises as a consequence of kinematic
perturbation due to waviness of the ball races of the bearing.
7'he ~pectrul method of evaluating waviness is to use a Fourier series to
repre~ent the deviations 6pk(�k) of the ball race from circularity
~ -
. ~Pk ~~Qk~ = Up Qki COS ~!(Pk T aki)~
t:~L
where i= 2n/ai is the number of the harmonic that shows how many waves of
_ a~;iven angu:lar pitch ai fit into the nominal circle; aki is the amplitude of
~
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FOFt O~FICT.4I, US~ ONLX
, the i-th harmonic of the k-th ring; a~3 i~ the pha~e ~hif't; ~.p i~ the aver~,~e '
value of �rhe function ~pk(~k) for the perind; ~k is ~he polar angl.e (for the .
~ inner ring k= l, for the outer rin~ k= 2) .
- C~, l~tn -
_ o,f .~.j r . . . _ . . . .1
- o�~
- Qo~ ~ .
- o~f - - _
a
1 d /J t0 !J ?0 ~D JO ~O JO a0.~0 /OD /PO i
Fig. 1 =
The set of Funplitudes e~ki of the Fourier series form~ a spectrum nf waviness _
of the b~.11 race for each ring of ~he ba].1 bearing. $y u3ing the electronic 4
equipment described in Ref. 1, 2 this spectrum is automatically recorded (as a
probe runs over the surface of the rotating part) in the form of a diagram .
(Fig. 1) on which the numbers uf the harmonics are plotted along the axis of _
absc3ssas, while the corresponding amplitudes are plotted along the axis of -
ordinates. In this way, the harmonics of even very high numbers can be
- fixed with amplitudes of hundredthes of a micrometrF.
The vibration frequencies in Hertz of the harmonic component~ of wavinesa are
calculated by the formula
� fkr = ~n~/60 (k - 1, 2), (1)
where
n1= n(1 - t/2(1 r/R)); n2 = n/2(1 r/R). ~
Here nl and n2 are the speeds (in rpm) of the inner (ni) and outer (nZ)
rings relative to the separator); R is the radius of the ba11 race of the
inner ring; r is the radius of the ball; n is the shaft speed.
2'he vib.ration amplitudes with respect to acceleration of ttie harmonic com-
ponenti3 of waviness are determined by the expression
Wk,=~ 4naf k~ak~~ (2)
which :ts transformed to
1~Ikl- ak;i', k= 1, 2. (3)
Since the vibration acceleration in this expression is directly proportional _
to the quantity in parentheses, even very small amplitude components aki of
the spectrum of wQViness the,t could have been disregarded at low numbers of
the liarmonics can cause considerable vibration acceleratiuns if they are
related to harmonics of sufficiently high numbers. Therefore the discrete -
components of the spectrum of waviness should be evaluated from the standpoint �
of vibroactivity as a funetion of the products akii2 rather than with respecL
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~OR OFFICIAL US~ ONLY
_ ~o the value~ of the ~mplit~,udes aki of these componen~s. To illustrate t.his,
F'ig, 1 showti the ~pect~,rogram of wavinesn of the ball race of the inner rinp;
= of one of the b~,11 bearing~, and we ~ive here the vatues ~f the quantitie~
ak~ nnd ~,kii2 for a number of h~,rmonics of the spectrum . `
, i 2 3 ~.8 59 ~.20 -
~ki+ um o.i4o o,n55 o.i7o 0.052 0.050
akii2~ um 0.56 0.50 55.0 175.0 720.0
As we can see, the harmon3cs of low numbers (i = 2, 3) that ~xe associated
with a flat spot have low vibroactivity with respect to acceleration, while
the 120-th haxmonic of waviness, despite low ampl3tude, has very high vibro-
activity.
- Thu~ the most dangerous from the standpoint of vibro~,etivity are the harmonics
of waviness of higher numbers. ~
In machine building, low-noise ball bearings are often required to have a
spectrum of vibration recorded with respect to acceleration at a given
rotational velocity that is bounded throughout the entire frequency range
by ~ome line, e. g. a sloping straight line with equation
j ' _
~max = F ~f~~ ~4~ _
This requ3rement also imposes limitations on the amplitudes of harmonics of
different numbers. After substituting the value of in expression (2),
we get a formula for calculating the limiting admissible amplitude of the -
i-th harmonic of wavines~
akr = F~fkr)l4n''fk~� ~5)
Relations (1), (2) and (5) enable solution of all the problems formulated
above for ball bearing diagnosis. To get the solutions, these relations
must be ~~epresented as tables or nomograms plotted for a given bearing size,
predetermined rotational velocity and vibration requirements. ~
As an example, Fig. 2 shows a nomogram plotted for ball bearing No 306 at
a rotational velocity of n= 1500 rpm.
' Straight lines 1 and 2 on the nomogrsm establ'�h the relation between thP
numbers of harmonics of waviness of the inner ,l) and outer (2) rings and
the frequency ~ of the stimulated vibration. For instance the 27-th harmonic
oF waviness of the outer ring causes bearing vibration on a frequency of
2j5 Hz, while the same harmonic of waviness on the inner ring causes vibration
with a frequency of 420 Hz.
The parallel straight lines 3 on the nomogram relate the amplitudes of the =
- harmonics of waviness, the vibration frequency (number of the harm~nic) and
the vibration level. For instance the 27-th harmonic of waviness of the -
outer ring with an amplitude of 0.1 um on a frequency of 255 Hz causes
bet~ring vibration of 55 dB. The reverse relation also holds: in the vibration
_ ti
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a, um I, .d~3
90
� - Bd
- _ _ ~ 60
_ ~ ~i - -
/A ~ I ~ ~ I ~ -
~ i '
1, 0 ~ - 1!~
_ ~ ~ I
O, / - - 0 =
P~s ~ I c
~
- D,4/
_ o,od~
I.f ?0 0 ~O d0 ~0 /OD /SO IODZJC?00 f00 ~04 /040 /60D 104D
rk
HZ
l
?
_ ~ _
~ s
~
_ ~
i0
~J
to
IJ � - - -
' ?O
fp .___r....__~
JO - �
60
~0 � /
o~v
90 ,
i
Fig. 2
_ ~pectrum of the ball bearin~ the 27-th harmonic of waviness of the outer ~
ring with amplitude of 0.1 um corresponds to a level af 55 dB on a frequency
of 255 Hz if the vibration frequency changes in proportion to the speed of -
rotntion.
Strni~?ht line 4 of the nomogram that reflects the requirements wi~h respect
to vibration imposed on the bearing sh'ows the relation bel;k~een ti:e -
vibration frequency (number of the Y,armonic)'and the lin~iting admissible ~
amplitude of the harmonic of wa~iness. For instance in order for the vibra-
tion from the 27-th harmonic of waviness of the outer ring on a frequency
of 255 Hz to stay below the admissible level, the amplitude of this harmonic
mu:~t not exceed 0.11 um.
~ 98
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i
~
~ ~'ti", ~OR O~F'ICZAL U5E ONLY
- ~'hus the deyr.ribed nomogrem enables solution of
different problems in the
diagnosia,,of a ball bearing. The accuracy of the qol.ution dEpends on ~he -
resolu~ion'`~of the ~na].yzer and the scales of the nomogram.
~~F~RENCES ~
1. B, T, Sheftel', "~valuation of the Waviness of I;o113ng Surfaces by ~he .
_ Method of Spec~ral Analysis," STANKI T INSTRUMENT, 1970, No 7.
2. ~3. T. Sheftel', G. K. Lipskiy, "Investigation of Radial V3brations of a .
Ball Bearing due to Errors of Surface Shape by ~he Method of Spectral
Analy~i3," PODSHIPNIKOVAYA PROMXSHLENNOST', 1969, No 1.
LOCALl'LAmION OI~ ACOUSTIC SOURCES
I. P. f3iryukova, A. M. Medvedkov, V. V. Naumova `
Determination of the contribution that individual mechanisms operating as a `
group make to the overall acoustic field is usually handled by appr.opriate -
analy~is of synchronous multichannel recordings of signals taken from micro-
phonec set up close to the mechanisms as well as at the investiaated point
of the field. The signal taken from a given microph~ne is determined not
only by the physical parameters of the acoust3c field that i.s formed by the _
mechanism closest to the microphone, but also by the parameters of the acoustic
fields that are induced simultaneously by the working mechanisms. In this
case the signals that are usually taken as steac~y-state random processes of
- second order are statistically related or partly coherent.
Coherence in this case is understood in the sense of the definition presented
- in Ref. 1.
m~ ~ ~ ~ ~
mr
+ 1'r.
+
� ~t
+ a~
_ !!/f
t ~'�f
~6
~ '~6
Let us introduce the following notation:
wi(t) component of the random vector of unobservable signals that corre-
~F~onds to the i-th source (mechanism);
99
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x3(t) componen~ of the random vector of observable signals that corre-
sponds to the i-th microphone; _
y(t) ob~ervc~ble random eignal of the mierophone set up a~: the given
point of the acoustiC field;
~ gi~(t) impulse transfer funetion of the linear system that connects th,e
i-th and ,~-th sourcea; _
~ h(t) impulse transfer funetion of the linear system tha~ connecty the
- i-th source to the inv~stigated point of the acoustic field; -
t time.
In these definitions, an unobservab].e signal is under:~tood as the microphone
signal determined by the corresponding parameters of the acoustic field that
is formed by a given mechanism in autonomous operat3on, while an observable
_ si~nal i~ understood as e~ signal corresponding to the superposition of
nc~uatic fields that are formed by the combined operation of inechanisms _
[Ref. 2]. -
The Fourier transfot~ms of the signals and the impulse transfer functions will
be denoted by the corresponding capital letters.
A schematic of the formation of observable signals in the case of N sta-
tist3cally independent sources for N= 6 is shown in the diagram on the
- preceding page, where the time dependence of the signals is not ii~dicated.
Symmetric dynamic systems are consiciered for which Gi~(f) = C~i(f), and it
is alco assumed that Gii(f)=1, where f is frequency. This enables us to _
get an exact solution of the ~roblem in the frequency region. We will look
for the solution on the basis of the example of two acoustic sources; the -
ca5e N>2 can be examined in an analogous scheme.
In accordance~with the diagram we write
a _
_ ~I (t) = ~1 S 81r (t) ~r (t - t) dY, ~1~
_ a -
y (t) = ~il S i ~ - Y) dY. � (2)
After carrying out Fourier transformation on both members of the equations,
we get
fXll f i G~2lr W1l
L Xs J L i J L u's J' ~3)
fW l
~ S' = f ~f i, Ha~ L~,s J. . ' .~4~
100 .
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- I~'OR OrFICIAL USL ONLY
- , ~~.r�i,
- l:n L�hene~annlytic~l exprenntona ~nd the one~ to follow, freqtitency dependence f
lu n~L indlcated 1or t,hc n~{ce of hrevity.
- LeL u~s aci; u~~ the r.llrceL produet ol' the two-component vector (X*], the complex =
cnn,~ug~,te to vector [X], with its Nermit3an con,jugate [X~')~', and ~hen carry
out the operation of
mathernatical. expec~a~ion for both members of the re- _
sultant equr~lity and inte~ra~e :~t in the vicinity of ~he investigated fre- -
quency, r~nd as a resu].t [Ref. 1] we get
r Sx~x~ Sxix~ 1 Gtt ~ Stuitei ~ 1 G~a ~
~SX~x, Sx.x.~=ca.~ 1~ c ~ S~,~..~cGz~ 1~~ ~5> _
where SxiX~ is the spectral density of the observable signal x~(t); SXiX~ is -
_ tlie mutual spectral density of the observable signals xi(t) and x~(t); SWiWi
i~ the spectral density of the unobservable signal wi(t); Gi.1 is the frequency
response of the linear system that connects the i-th $nd ,~-t~i sources.
- l~y r~nal.ogouc calculations we cari get the following expression for the spectral -
_ density of signal y(t):
_ . sdU - , Hl N,,� ( S ~?u'' s U , ] [ Hz, ~ _
u~,u ,
To study equation (5) we introduce a complete orthogonal basis system of
Pauli spin matrices
- ao=l(I i]' ai=[~_~], ~x=[~ ~J, G~_ I~t 0~.
~
_ The al@;ebraic properties of the Pauli spin matrices are determined by the
followitig relations [Ref. 3]:
_ a�~~ = - a~ca = ivy,
where a, R, y represent a cyclic permutation in (1)-(3),
(a~)' vo, f- 0, l, 2, 3, (6) -
- a~Qo = ao~~ = 0, l , 2, 3), . (7)
`SP ~`'~31~ Z3~1 f= 1 r 2, 3~~ _
wkiere :;p is the Spur of the matrix; Qi~ is the Kronecker syr,ibol.
In m~.trix equation (5) the unknown quantities are the elements of the matrix
of spectral densities of the unobservable signals [SW.W~] and of the matrix ~
of frec~uency responses [Gi~ ~ . 1 =
Let u~ expand the matrices that appear in equation (5) with respect to the
,
complete system of Pauli spin matrices. With consideration of equalities
_ (6)-(~3), the matrix of spectral densities of observable signals is written as
101
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APPROVED FOR RELEASE: 2007/02/48: CIA-RDP82-44850R000100024425-1
I~'OR OF'F'ICIAL USE ONLY
~ ~
Isxix~a 1 ~~r3r~ (9)
y wU
- where ~he S~ = Sp(o~ISx~Xl1) are St~,okes parameters.
It~troducing the notat3on
Gl~ = al !a�
= for the matrix of i'requency reaponse~ we get the expression ~
. ~
[ Gr~~ = 2 ~ ~r~r, ~ (10) -
r-o
where ~o ~ 1 ~ ~i = Q~ !'9 ~ al ~ ~aSi ~a = 0.
- The matrix of spectral densities of observable signal.s can be presented as
a
~ (5~,~~~1= z ~ 5~~~, (11)
_ , ~a~ ~
where ~a ~ S~~ - 0. _
Substituting expressions (9)-(11) in equation (6), we get
Fi Sr~t = ~'i ~~Qi ~'i ~ia~ ~ ~kak� (12) _
" 1~0 1~0 1=0 kvU
By using relations (6), (7), let us simplify the second member of equation ~
(12), and then by equating the coefficients before spin matrices with identi-
' cal 3ndices standing in the first and second members, we r~duce this equation
to the following system
So I1 -I- (ai -I- a~)1 = So, Sl [1- (ai -I- aa)1 = Si~
' (13)
2tt~So - S~~ 2a~S1 = S9. . -
Usin~ the representation of the complex frequency response G12 in trigono-
metric farm
= Gi~ = al taz = ~ Gl: ( cos cp i ~ Gia ~ sin
we gPt the following equations for ~etermining the elements of frequency -
response from system (13) when al ~ 0 and a2 ~ 0: _
t6 ~P = SoS~ (1 - ~ Gi~ ~2)/,SiSz (1 -I- I Gla ~a)+ (14) _
I~lzl'+2AIu1~I~+i =o,
where ' -
A = (SiSa - SoS3 - 2SiSo)/(Sis~ -f- Soss'),
102
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APPROVED FOR RELEASE: 2007/02/48: CIA-RDP82-44850R000100024425-1
,
,
FOR OFrZCIAL U5E ONLY
It can be simply shown that A 9At38f~.@S the 3.nequality A