1957 molloy

13
842 J. S. NISBET AND J. N. BRENNAN i(01= + The shock spectrum of one such pulse s shown n Fig. 10. The machine esonance as been chosen o correspond ith the second usp n the shock pectrum of the "one minus cosine" ulse used or the rise and decay orms. It is apparent hat though he arge peak n magnifica - tion factor has been removed at the machine resonant frequency, he shock spectrum s radically different from that of the rectangu lar ulse at all but the lowest frequencies. 5. CONCLUSION It is apparent hat care mustbeexercised n approxi- mating the wave form of a shock machine by an idealizedpulse. The discontinu ities in the acceleration, or time derivative of the acceleration, which are not present in the test pulse an esult n marked differences etween theoretical nd practical shock pectra. If ringing s prese nt n a pulse, he possibility f its exciting esonant modes n the test structure should be investigated. A pulse of this type may cause ittle excitation of resonant modes of frequency igher than the ringing requency. THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 29, NUMBER 7 JULY, 1957 Use of Four-Pole Parameters in Vibration Calculations* C. T. Mo•.Lo¾ Lockheed ircraft Corporation, Burbank,California (Received January 16, 1957) Linear elastic systems which have a single nput point and a single output point can be characteriz ed y a pair of si mple inear equations, namely, Finout auFoutput q-am• ou put, Viaput- a2•Foutput -a'• Vouti•ut• where he F's are forces, he V's are velocities, nd the a's are the four-pole arameters or the system. he use of four-pole aram- eters permits combinations f mechanica l lements o be handled as a single entity, and, as the parameters belong only to the elastic system for which they are defined, they do not depend upon what precedes hat particular system or follows it in a mechanical network. This permits the statement of results in a rather general orm. These parameters ave been used or many years n the analysis f electrical ircuits, nd t is the purpose f this paper to show their utility in the field of mechanical vibra- tions. In particular, the four-pole parameters or the ba sic me- chanical lements f mass, pring, and resist ance, ill be obtained. Also, a escription f mechanical ources ill be given. The rules for the connection f four poles nd a method of experimentally measuring our-pole arameters ill he presented. few specific problems llustrating he use of four poles will be set forth, namely: a generalized ibration solation problem, he response of an elastically mounted mass on an electrodynamic shake table,shock excitation nalysis, nd some distributed arameter systems i.e., helical springs and rubber n shear). 1. INTRODUCTION •OR any ears, lectrical ngineers ave een sing block diagrams. hese diagrams have proved convenient n the discussion f complicated electrical systems. argely because hey pe rmitted the encom- passing of a rather complicated rrangement of com- ponents nto a single entity which could be represented as a so-called black box." It was then relatively easy to connect the various black boxes together to form the complete system. In order to use such a technique efficiently, the engineers characterize d he various black boxes by equations whichdescribed heir performance, nd, in addition, gave meth ods or determining he charac- teristics of these boxes by measurements ade at their accessible terminals. When these black boxes had a, * Presented by invitation at the Los Angeles meeting of the Acoustical Society of America, November 16, 1956. single pair of input terminals and a single pair of output terminals hey were called four poles or four terminal networks. Over the years an extensive heory of these electrical four poles has grown up in the literature, •4 and they are still a subject of research s evidenced y the fact that several papers were published on four pole theory n 1956. The success of the black box idea in electrical circuit theory has encouraged arious writers to try a similar thing for special problems n acoustics .• and electro- mechanical ystems2 However, he appearance f these • L. A. Pipes, Phil. Mag. 30, 370 (1940). • M. B. Reed, Electrical Network Synthesis Prentice Hall, Inc., New York, 1955), Chap. 2. a L. C. Peterson, Bell System Tech. J. 27, 593 (1948). * S. Darlington, J. Math. Phys. 18, 257 (1939). s W. P. Ma son, Bell System Tech. J. 6, 258 (1927). a L. C. Peterson and B. P. Bogert, J. Acoust. Soc. Am. 22, 369 (1950). ? W. R. MacLean, J. Acoust. Soc. Am. 12, 140 (19 40).

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842 J. S. NISBET AND J. N. BRENNAN

i(01=

+

The shockspectrumof one suchpulse s shown nFig. 10. The machine esonance as been chosen ocorrespondith the second usp n the shock pectrumof the "one minuscosine" ulseused or the riseanddecay orms.

It is apparent hat though he argepeak n magnifica-tion factor has been removed at the machine resonant

frequency, he shock spectrum s radically differentfrom that of the rectangular ulseat all but the lowestfrequencies.

5. CONCLUSION

It is apparent hat caremust be exercisedn approxi-mating the wave form of a shock machine by an

idealized pulse.The discontinuities in the acceleration, or time

derivative of the acceleration,which are not presentin the testpulse an esult n markeddifferencesetweentheoretical nd practicalshock pectra.

If ringing s present n a pulse, he possibility f itsexciting esonantmodes n the test structureshouldbeinvestigated.A pulse of this type may cause ittleexcitationof resonantmodesof frequency igher thanthe ringing requency.

THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 29, NUMBER 7 JULY, 1957

Use of Four-Pole Parameters in Vibration Calculations*

C. T. Mo•.Lo¾

Lockheed ircraft Corporation,Burbank, California

(ReceivedJanuary 16, 1957)

Linear elastic systemswhich have a single nput point and asingleoutput point can be characterized y a pair of simple inearequations,namely,

Finout auFoutputq-am• ou put,Viaput- a2•Foutput -a'• Vouti•ut•

where he F's are forces, he V's are velocities, nd the a's are thefour-pole arametersor the system. he useof four-pole aram-eterspermitscombinations f mechanical lements o be handledas a singleentity, and, as the parametersbelongonly to theelastic system for which they are defined, they do not dependupon what precedes hat particular system or follows it in amechanicalnetwork. This permits the statement of results in a

rather general orm. Theseparameters ave been used or manyyears n the analysis f electrical ircuits, nd t is the purpose fthis paper to show their utility in the field of mechanicalvibra-

tions. In particular, the four-poleparameters or the basic me-chanical lements f mass, pring,and resistance, ill be obtained.Also,a description f mechanicalources ill be given.The rulesfor the connectionf four poles nd a methodof experimentallymeasuringour-pole arameters ill he presented. few specificproblems llustrating he use of four poles will be set forth,namely: a generalized ibration solationproblem, he responseof an elastically mounted mass on an electrodynamicshaketable, shockexcitation nalysis, nd somedistributed arametersystems i.e., helical springsand rubber n shear).

1. INTRODUCTION

•OR anyears,lectricalngineersaveeensingblockdiagrams. hesediagramshaveprovedconvenient n the discussion f complicatedelectrical

systems. argely because hey permitted the encom-passingof a rather complicated rrangementof com-ponents nto a singleentity whichcouldbe representedas a so-called black box." It was then relatively easyto connect the various black boxes together to formthe completesystem.

In order to use such a technique efficiently, theengineerscharacterized he various black boxes byequationswhich describedheir performance, nd, inaddition, gave methods or determining he charac-

teristicsof theseboxesby measurements ade at theiraccessible terminals. When these black boxes had a,

* Presentedby invitation at the Los Angelesmeetingof theAcousticalSocietyof America, November 16, 1956.

single pair of input terminals and a single pair ofoutput terminals hey were called four polesor four

terminalnetworks.Over the yearsan extensiveheoryof these electrical four poles has grown up in theliterature,•4 and they are still a subjectof research sevidenced y the fact that severalpaperswerepublishedon four pole theory n 1956.

The success of the black box idea in electrical circuit

theory has encouraged ariouswriters to try a similarthing for specialproblems n acoustics.• and electro-mechanical ystems2However, he appearance f these

• L. A. Pipes,Phil. Mag. 30, 370 (1940).• M. B. Reed,ElectricalNetworkSynthesisPrenticeHall, Inc.,

New York, 1955), Chap. 2.aL. C. Peterson,Bell SystemTech. J. 27, 593 (1948).

* S. Darlington,J. Math. Phys. 18, 257 (1939).s W. P. Mason, Bell SystemTech. J. 6, 258 (1927).a L. C. Petersonand B. P. Bogert, J. Acoust. Soc.Am. 22, 369

(1950).?W. R. MacLean, J. Acoust.Soc.Am. 12, 140 (1940).

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FOUR-POLE PARAMETERS IN VIBRATION CALCULATIONS 843

isolatedeffortsdoesnot seem o have brought orth, atleast o the writer'sknowledge, ny attempt to developfour pole heorysystematicallys a working echniquein fields other than the electrical field.

For the past severalyears, the writer has beenworking ariousproblemsn sound nd vibrationby

the four-pole echnique nd has found it very useful.It is the purpose f this paper o presenta brief surveyof some fundamentals nd some applicationsof thistheory.As the subject s quite extensive, nly a verybrief treatment of a few items is feasible. Much more

could be said about each topic and many additionalones ouldbe presented. owever, t is hoped hat eventhe limited samplegiven here will indicatesomeof theareaswhere this theory has proved usefuland perhapsencouragehe reader o try it himself.

A few examplesof subjectseffectively treated byfour-poleheorybut whichhavebeenomitted rom the

present aperare mechanicalilters,vibrationpickups,generalpropertiesof four-poleparameters, nd theirrelation to other parameters.Some of the advantagesof the four pole technique re listedbelow:

(1) It is an essentiallysimple idea and for thisreasons helpful n providing point of view.

(2) It is applicable o a very largeclassof systems,for example, coustical ystems,mechanical ibrationproblems,passiveelectrical networks,and activeelectrical networks.

(3) It is easy o handlemathematically.(4) All of the pertinentproperties f a system anbeexpressedn termsof four poleparameters.

(5) Four-poleparametersbelong o the particularsystem or which hey are defined nd do not dependupon what precedes r what follows hat system.Forthis reason hey allow the packaging f the parts of asystem nd the handling f these ackagesn a mannersimilar to the black box techniqueso commonand souseful n electrical ircuit analysis.

(6) They are usefuln obtaining quivalent lectricalcircuits or dynamical ystems. hey are particularly

useful in obtainingequivalent electricalcircuits fordistributedparametersystems.

In the followingdiscussiont is proposed irst todevelop he four-pole arametersor simplemechanicalelementsand mechanicalsources.Next, rules will be

given or the combinationf four poles or two im-portant ypesof connections.shortdiscussionn themeasurement f four-pole parameters ollows.Thefour-pole echniques then applied o a few specificproblems, amely: he generalizedibration solation

problem, he problemof the response f a four poleon an electrodynamic hake table, an outline of themethod of applying four-pole parameters to shock

excitation roblems, nd finally to the discussionf afew distributedparameter ystemsn which our polesare a useful analytical tool. Throughout this paper

theauthorhasemployed iagrams hich re essentiallypictorial and which he hopesare clear.The drawingshave beenmadeon an ad hocbasis,much as one defines

convenient ymbolsn a mathematical nalysis. heyare thereforenot comparableo the mechanical ircuitsor equivalentelectricalcircuitswhichare describedn

the literature.Figures17, 18, 19, and 21 are exceptionsto the above remark. These are equivalentcircuits nthe usual meaning of the term and furthermore areso labeled. Readers interested in symbolisms orrepresentingmechanical ircuitsor in the subjectofequivalent circuits are referred to the literature,especially he excellentpaper by F. A. Firestone.•-

2. FOUR-POLE PARAMETERS FOR MECHANICAL

ELEMENTS

(A) Four-Pole Parameter Definitions

The term four-poleparameterss the nameappliedto the four coefficients ,,, ix12, 21, ff29 n the pair ofequations hownbelow:

F•=otnF2+ot•2V2 (1)

Vt=et2tF2q-a*_*_V•. (2)

Theseequations re the performance quations f themechanicalystem hownn Fig. 1. The elastic ystemcan be any combination f lumped, inear,mechanicalelements uchas masses, prings, nd resistances.tcan alsobe combinationsf linear,distributedparam-

eter systems, uchas beams, lates,diaphragms,tc.The elastic ystemmusthave two identifiable onnec-tion points (1) and (2) whichare called he input and.output points.At the input point thereexists n inputforce (F•) and a velocity (V•). The input force andvelocityare produced y connectionf point (1) tothat portionof the completemechanicalystemwhichprecedest. At theoutput oint 2) there xists force(F2) and a velocity V•) which esult rom the appli-cationof (F•) and (V•) at point (1) and the reaction ftheportion f the mechanicalystemollowinghe our

pole. t should e noted hat Eqs.(1) and (2) are thecanonicalorm adopted n this paper or the four poleequations. he input quantities • and Vt are on theleft handsideand the outputquantitiesFa, and V2 areon the right-hand ide.Otherarrangementsre possiblebut it is believed that this is the most convenient.

The unitsemployedn Eqs. 1) and (2) maybeany:self-consistentet, such as those shown n Table I.

• 0 Elaslic 0 •,

I:•,V• System F . V ß Oire•tian

F[o, l, Four-poleparameternotation.

• F. A. Firestone, . Acoust.Soc.Am. 28, 1117 (1956).1 The author has discussedhis subjectwith Floyd A, Firestone

who has ndicated hat he prefers he mechanical nd acousticalsymbolsset forth in his paper to the pic•corialr•presentationemployed n this paper.

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844 C.T. MOLLOY

TABLE I.

Name of unit in which quantity is expressed

System System System SystemOuan[ity No. [ No. 2 No. 3 No. 4

Length Feet Feet Centimeters MetersMass Pounds Slugs Grams KilogramsTime Seconds Seconds Seconds Seconds

Force PoundMs Pounds Dynes NewtonsVelocity Feet per Feet per Centimeters per Meters per

second second second second

Other units may be employed, ut if they are, caremustbe exercisedo insure hat theproper djustmentsare made.

The dimensions f the four-poleparametersdependupon the dimensions f the other variables n the four

poleequation nd heycanbedeterminedy inspection.In the case f Eqs. (1) and (2) where he othervariablesare forces nd velocitiest is readilyseen hat (ml) and

(azz) are dimensionless,an) has the dimensionsofforcedivided by velocity, .e., (mass/time),andhas the dimensionsf velocitydividedby force, .e.,(time/mass). his is equivalento assertinghathas the dimensions f a mechanicalmpedance nd(a2•) has the dimensions f a mechanical dmittance.

We shallnowproceedo obtain he four-pole aram-eters for a singlemass,a singlespring,and a singleresistor.The generalprocedure or determining hefour pole parameters f a systemordinarilyrequiresthat three stepsbe taken, namely:

(1) Write the performancequationsor the givensystem.

(2) Solve he performancequationsubject o theboundarycondition hat at the input the force andvelocityare respectivelyt and Vt and at the outputthe forceand velocityare Fz and V•.

(3) Cast the solutions hus found into the canonical

form n Eqs. (1) and (2).

(B) Four-Pole Parameters for a Mass

Since he mass s regarded s a rigid body,we have

V•= V•, (3)

and by Newton's aw

Ft-- F, = mdV•/dt = mdV•/dt. (4)

Restricting ourselves o sinusoidal ime variation forthe motion and adopting he conventional omplexnumber representationwe have

Fl=Fm'ei•t; Fz=F•o'e•; Vt=

V2= V•0e ", (5)

Direction

Fro. 2. Four-poleparametersor a mass.

whereFro,Feo, tc., may be complexunctions f mass,stiffness,resistance,and frequency, but are timeindependent. t is to be noted that no phaseanglesarenecessaryn the (e t) term since he (F•0), etc., termsare allowed to be complex. t is to be understood hatdisplacementst each nput and output point are to bemeasured rom the respective quilibriumpositionsof

thesepoints.The equilibriumposition s definedas theposition occupiedby the point when no sinusoidalexcitations applied o the point.Equations 3) and (4)become

F• = F•+ (mwi)V, (6)

Vt=O.F•+V•, (7)

and the four-pole arametersor a mass seeFig. 2) are

an=l; a•=rmoi; a2x=0; a•=l. (8)

(C) Four-Pole Parameters for a Massless Spring

The force applied at the input point (1) of thesprings thesame sthe orcewhich he spring eliversat its outputpoint (2). Also he magnitude f the forceat either end is the springconstant k) multipliedbythe "stretch" of the spring. Keeping in mind ourcomplexnotation and also that velocity is the time

I 2

FI.g - • F•.V;• + 0irection

Fla. 3. Four-Doleparameters or a massless pring.

derivativeof displacement e have

F• = F•,

[V• V•I

(9)

(10)

which can be rewritten

Fi=Fo•+0-V2, (11)

Vl=--F•+ Va. (12)k

Hence he four-poleparametersor a spring seeFig. 3)are

an=l; a,,=0; a•=--; a=•=l. (13)k

(D) Four-Pole Parameters for a Resistor

As in the caseof the spring, he input force F, isequal to the output force Fz. Also for the kind ofdamping-consideredere, namely, viscousdamping,such as that providedby a dashpot, he force F• isequal to the resistance r) times the relative velocity

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FOUR-POLE PARAMETERS IN VIBRATION CALCULATIONS

rI F2, 2 + Direction

FiG. 4. Four-poleparametersor a resistor.

of point (1) with respecto point (2). We have henFt = F2 (14)

m•= r(V•- V2), (15)

and rearranging:

F•= F2+O' V•

1

V• = -F•+ V2.?.

pole parameters f the elasticportion of the sourceare known. nput force Fs) known:

F=(•)Fs-xan•V.1input velocity Vs) known:

F= -- V.s- -- V. (20)Ot21t \ Or21

Theseequations escribehe source n terms of the(16) quantitieshichre ssumednownnd reobtain

by directapplication f Eqs. (1) and (2) to the system

(17) shownn Fig.6.

The four-pole arametersor a resistor seeFig. 4) are

au=l; a•=0; ao.•=l/r; a2o.=l. (18)

The precedingconstitutea few simple examplesoffour-poleparameters.We now proceed o apply someof these deas o the problemof describingmechanicalsources.

(E) Mechanical Sources

Devices which can supply vibratory mechanical

energy are called mechanicalsources.They may bephysicallyquite complicatedbut in general theybehave n a rather simplemanner;namely,as if theyare constructed from two components, an idealgenerator, nd an elasticsystem.The ideal generatorserves he purposeof exertinga prescribed inusoidalforceor velocity (amplitude,and phase)at prescribedfrequencies n the input point of the elasticstructureassociated with it. The mechanical source drives its

load through the output point of the elastic system.It is to be noted that the term elastic system s usedhere n the general ense escribedn Sec.2A.

It is possible o employ different parameters ndescribing he performanceof mechanicalsources.The choice s based upon convenience,.e., what isknown or easily ascertainable bout a source.Thesymbols hichare employedo represent mechanicalsource re shown n Fig. 5.

The performance quationsare given below for asourcewhen either the force or velocity at the inputto its elastic ystems knownand n addition, he four-

SYMBOL OR DEAL

MECHANICAL EN'

SYMBOLEORELASTIC SYMBOLFOR COMPLETE

SYSTEMOF SOURCE MECHANICALSOURCE

FIO. 5. Symbol for ideal mechanicalgenerator;symbol forelasticsystemof source;symbol or completemechanical ource.

FiG. 6. Mechanical source.

+ Direetlon

In the Eq. (19) it is necessaryhat the dealgeneratoralways exert the force Fs at the input to the elasticsystem, egardless f the load which s coupled o theoutput of the elasticsystem.This is a physicalcharac-teristicof the idealgenerator nd is an additionalpieceof information over and above the numerical value of

the force (Fs), whichmust be known f Eq. (19) is toproperly describe the mechanical source.Suchgenerators will be called "Ideal Constant-ForcGenerators."

The samesituationwhichobtains or Eq. (19) alsoholds or Eq. (20). In this case,however, he generatormaintainsa constantsinusoidal elocity (Vs) at theinput pointof the elastic tructure, gain, ndependentlyof the load attached to the output. This type ofgeneratorwill be called an "Ideal Constant-VelocitGenerator."

Frequentlysituations risewhere t is not feasible ophysically separate the mechanicalsource nto itscomponentsnd for thesecasest is desirable o havea means f describinghe sourcen termsof quantitieswhichcan be measured t the only accessibleunction,namely, he output point. It is possibleo do this bymeasuring he mechanicalmpedanceooking nto thesource i.e., looking rom the output point toward theinput point) and alsomeasuring ither the "BlockedForce" or the "Free Velocity" of the source.TheBlocked Force is the sinusoidal force which would be

measured at point (2) in Fig. 6 if this junction isrestrained o hat its velocity s zero.The Free Velocityis the velocitywhichwouldbe measured t point (2)of Fig. 6 if this point werecompletely nrestrained othat the forceexertedat this point is equal to zero. tis to be noted that the Free Velocity is the velocitywhich is measuredby a vibration pickup at a point

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846 C. T. MOLLOY

when no load exists at this point. This, of course,assumeshat the pickup tselfproduces egligibleoad.

In describing mechanical sources two distinctsituationsarise: first, when the source s to be described

in terms of experimentallymeasuredquantities,andsecond,when the nature of the ideal generator isknown (i.e., constant orce or constantvelocity type)

and the elasticstructure s also ully described.n thefirst caseabove t is not necessaryo know the type ofideal generator,whereas n the secondcase this in-formation must be taken into account n the analysis.This will be evident rom the following.Let us considerfirst a mechanical ource omprising n Ideal Constant-Force Generatorcoupled o an elasticsystemsuchasthat shown n Fig. 6. Let us further arrange hat theoutputpoint (2) be restrainedrommoving.Under hiscondition he input force and velocity and also theoutput forceand velocityof the sourcewill have thevalues:

F,=Fs (This condition s what makes it an IdealConstant-ForceGenerator.)

V•=V,o• (The particular value of Vs under theassumed ondition.)

F=Fo• [The output force becomes he BlockedForce Fo½).•

V=0 (No motion permitted at the output point.)

Now for any input F,, V•, and any output F, V, the

four-poleelastic systemwill obey the equationsF•=ot,iF-l-ct•2V (21)

V, = a2•F+a2•_V, (22)

and for the special aluesabove heseequations ecome

F,=auFo, (23)

If Eq. (23) is substitutedn (21) and rearranged,wehave

F= (., v. (25)

This is the performance quation or an Ideal Constant-Force Generator coupled to a four-pole in terms of the

blocked orce which it can generateand the four-poleparametersof the elasticsystem.

We now consider xactly the same "constant force"mechanical ourcedescribed bove, only this time weshall permit the output point to move in an unre-strainedmanner. The values of the input and outputforces and velocities will then be:

F,= F, (Ideal Constant-ForceGenerator.)

l/•=V**• (The particular value of V, under theassumed onditions.)

F= 0 (No forceexertedat output.)

V= V,• [Output velocity becomes Free Velocity"

Inserting hese n (21) and (22):

F ,=oqaV • (26)

(27)

Putting (26) in (21) and rearranging, e obtain

F = (•2t12/.11)0• -. (.12/rvll)V. (28)

This is the performance quationof an Ideal Constant-ForceGenerator oupled o a four pole, n termsof thefree velocitywhich t deliversand the four-poleparam-eters of the elastic structure. f the above procedureis repeated or an "Ideal Constant-Velocity enerator"coupled o a four-poleelastic system, the pertinentquantitieshave the valuesshownbelow.

OutputPoint RestrainedF, = F,o• (The particularvalueof F, under he assumed

conditions.)

V•= V, (Ideal Constant-Velocity enerator.)

F=Fo• (The output orcebecomeshe blockedorce.)

V=0 (No motionpermittedat the output point.)

OutputPoint Free to I/love

F•=F,,• (The particular value of F, under the assumed

conditions.)

V,= V, (Ideal Constant-VelocityGenerator.)

F= 0 (No forceexertedat the output.)

V = V,• (Output Velocitybecomeshe free velocity.)

Inserting hesedata into Eqs. (21) and (22) the twoalternative descriptions f an Ideal Constant-VelocityGenerator oupled o an elastic our pole,givenbelow,are obtained.

(X22

F=Fo•----V (29)

(3o)•X21 0/21

The set of Eqs. (25), (28), (29), and (30), all describemechanical sources but all involve the four-poleparametersof the elastic system.We now show howthese parameters may be eliminated and all four

equationseplaced y twoequations. he ratio (a•2/otn)is the mechanical mpedanceof the elastic structure ofthe sourcewhen measuredat the output point (2),

looking oward the input point (1) when point (1) isunrestraiued i.e., has zero force or zero impedance).Similarly the ratio ((:I/22/13/21)s the mechanicalmpedance

at point (2) when point (1) is completely estrained

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FOUR-POLE PARAMETERS IN VIBRATION CALCULATIONS 847

(i.e., has zero velocityor infinite impedance). heseassertionswill not be proved here but they arereadily deducible from Eqs. (1) and (2). If welabel he firstof thesempedancesZ,) Free mpedanceand the second Zoo)Blocked mpedance hen Eqs.(25), (28), (29), and (30) become

F= o,-Z.V"Constantorceype,"F=Z,,V.--Z,•V]

(31)

(32)

F= oc--ZocV (33)F= o,,•-ZocV.IConstantelocityype."34)Now if we consider that an Ideal Constant-Force

Type Generator would automatically terminate theelastic our pole n a zero mpedance nd also hat anIdeal Constant-Velocity-Typeeneratorwouldneces-

sarily terminate its elastic four pole in an infiniteimpedance, hen if we measure the mechanical m-pedance f the mechanical ource he measured aluewill be eitherZ, or Zo•depending ponwhich ype ofsource s being measured.Therefore, f we denote thismeasured mpedanceby (Z,) it would replace (Zoo)in (31) and (32) or it wouldreplaceZo• in Eqs. (33)and (34). The experimentwouldyield then the singlepair of equations

(35)

(36)

for both typesof generators. quations 35) and (36)describe the mechanical source in terms of the measured

quantities Fo• and Z8 (see Fig. 7) or the measured

4-DIr•ctlo•

Fro. 7. Mechanicalsource haracterized y the pair of quantities(Fo•and Zs) or by the pair (Vo• and Z,).

quantities • and Z,. Thus n the casewhereexperi-mental measurements re made, it is not necessaryoknow whether the source is a Constant Force or a

ConstantVelocity Type.

3. CONNECTION OF FOUR POLES

(A) Tandem Connection

Two fourpoles resaid o be in tandem onnectionwhen he output rom he first s preciselyhe input tothe second. his is shown n Fig. 8 for (n) four poles ntandem.

The analysis f this type of connections efficientlyhandled y matrix techniques.he structure esultingfrom the tandem connectionof four poles is simplyanother our pole. Equations 1) and (2), i.e., thecanonicalequations for a single four pole can be

written in the matrix form thus:

F•]=[an)a• [)] FqVtJLa•o).o)Jx[vJ (37where the superscriptsignify that the parametersbelong o the four polenumbered1). If the outputoffour pole (1) is the input of four pole (2) then anequationsimilar o (23) can be written for it thus:

LV•J Last a) a•(•)J Va

If thematr•

[F•]s elatedoheutrom q. 38)henput

[V•J-La•t) • )]a•t)a2•)]VaJ'This process an obviouslybe generali•d for (•)

four poles n tandem.The parametersor this compositefour pole are givenby the matr• productof the matricesof the component our poles.This is shownbelow:

V•]= [a• n

xlan(•'t•(•)l...[an{"'•(")lx[F•[a••)a,2ø)Jt (")a• •)] V•J' (•)When the indicted multiplications re •rformed,

the four-pole equations for the compositesystembecome

V, ta•_] a•J LV,•41

It should be noted that each matrix belongs o asingle four pole and that any change• made in that

four poleaffectsonly ts own matrix. This is a consider-

I • I I 2 + Dlra;tion

Fro. 9. Spring and mass n tandem.

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848 C. T. MOLLOY

able conveniencen certain problems.As a simpleillustrativeexampleof the useof the matrix techniquethe four-pole parametersof a spring and mass intandem (see Fig. 9) are calculated.

1 0 1 • 1

1] 0 = - 1-= (42)Spring Mass Combined pring

matrix matrix and mass matrLx

Four-pole quationsor system re

Ft = F2+ (m•0 V2 (43)

(44)

(B) Parallel Connection

When our poles re connectedo hat--O) all theirinput unctionsmovewith the samevelocity, 2) alltheir output junctionsmove with the samevelocity,(3) the input force to the compositeour pole is thesum of the input forcesof the individual our poles,(4) the output force from the composite our pole isthe sum of the output forces of the individual four

poles,--the four poles are said to be connected nparallel.This is shown n Fig. 10.

The four-poleparametersor the composite ystemof(n) four polesare givenby the formulas 45). In thegeneralcase heseare somewhat omplicated ut formany importantspecial aseshey yield simple esults.

AC'

an=A/B;B

1

a2•------; a22=C/BB

(4S)

A=E --l-I •.0t2•

C=12 -- ß

Note.--In deriving he above equations,use has beenmade of the relation

Otll1121which is a consequencef the fact that the system

obeys he reciprocity rinciple.Fx=aixF•+at•V2

h. vl* Direction

Ricjidf, ssless RigidMosslesslarBar Constrained Constrained o Move

to Move n Axial in •laJ Direction.

Direction. Translationnly.Translation nly.

Fro. 10. Parallel connection f four poles.

Four-pole quationsor parallelconnectio•.--A impleexampleof the applicationof the foregoing ormulas

for parallelconnections shownbelow.The four-poleparametersof combination pring and resistor (seeFig. 11) are

au =1 a•x=l/(r+k/•oi)

a,2=0 a•= 1. (46)

5. v, •r v•o o

œ

Fro. 11. Springand resistor n parallel.

4. MEASUREMENT OF FOUR-POLE PARAMETERS

The four-pole parametersof a structure can bemeasured xperimentally. he measurementsequiredare the "Free" and "Blocked"mechanical npedancesat the input and output junctions. The Blockedmechanicalhnpedance t the input (Z•o•) is the me-chanical mpedance t the input junction when theoutput junction s not permittedto move. The otherimpedances re defined in a similar manner. The

formulas elating he impedancesnd the four-poleparametersare shownbelow:

1

+EZ2o.

Choosesign of radical so oa, has positivereal part.

a2•=Z•o•-a•, (47)

Oll2=Zltw. Z2ae.ol21.

5. VIBRATION ISOLATION PROBLEM

We shallnow nvestigateheproblem f determiningthe effectof inserting four polebetween massand a

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FOUR-POLE PARAMETERS IN VIBRATION CALCULATIONS 849

mechanical ource f vibration.This is a generalizationof the ordinaryvibration solation roblemwhere t iscommonpractice to regard the mechanical ourceashaving nfinite mechanicalmpedance nd to considerthe four polewhich s to perform he isolationunctionas a springor possibly spring-resistorombination.

In the case o be analyzed here the mechanicalsourcehas an arbitrary internal impedance Zo•) and aprescribed ree velocity (V,c). Also the elasticsysteminserted between the mass and the source need not be

a simplespringor spring-resistorombination ut maybe a very complicated ombination f linearmechanicalelements--a mechanical filter for example. Thisillustrates neof the advantages f the four-polepointof view, namely, hat the equations ritten in termsoffour pole parameters pply to a wide classof systemsand it is only necessary o insert the appropriate

four-poleparameters o obtain results or a particularsystem. Figure 12 shows the mass-sourceprior to

UNISOLATEOYSTEM

ß MechanicalourceIsolatingourj• •i• po,e -Ihas•lISOLAI•D SYSTEM

Fro. 12. Unisolated ystem; solated ystem.

isolation nd alsoafter the isolating our polehas beeninserted between the mass and the source.

By a straightforwardpplication f the mechanicalsource quation, he four-pole quationsor a mass nd

the four-poleequationsor the isolator, he velocityof the massbeforeand after isolationcan be computedand the results re shownn Eqs. (48) and (49).

\ ZocVnu.i(Velocity f mass eforesolation) (48)

(Zoc)= (Velocityof mass fter isolation). (49)

if we consider that the effectiveness of isolation is

measured y the ratio of velocitybefore solation othat after isolationand if we call this the insertion atio,then the functions below define the effectiveness of the

mountingsystem.

V0

Insertion ratio=•=-- (50)V•

,= (51)(Zoc+mwi)

Insertion oss=/5=20Log•0]•l decibels). (52)

6. ELECTRODYNAMIC SHAKE TABLE

Figure13 shows n idealized iagramof an electro-dynamicshake able. Figure 14 shows lockdiagramsuitable or four pole epresentation. e note hat thetwo input quantities, ppliedvoltage,and current,areelectricalwhile the two output quantities, orce, andvelocity,are mechanical. hesevariables re relatedbya pairof linearequationsndcanbeput n thecanonicalfour-poleorm as shownn Eqs. (53).

Z •Z,,,an=Ze/K; a•=lO-TK+

K

1

a=•=--; ao.o.=Z,•/KK

E = AppliedVoltage

I = Input Current(Amperes),

F=Output Force dynes),

(53)•

F,V

-Magnet

Input

Fzo. 13. Diagram of electrodynamic hake able.

it Mixed units have beenemployedn Eq. (53). If either cgsorinks units were employed hroughout, he factor 10 * would bereplacedby unity.

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850 C. T. MOLLOY

• I• F,VOutput

Input

SHAKE ABLEWITH NO EXTERNALOA9

+ DIrectlo?

I "I,j jo.tp.tInput

+ Direction

Fro. 14. Shake able driving a four-pole ead.

V=Output Velocity (cm/sec),

K = ForceFactor (dynes/amperes)

Flux Density Gauss)X oilLength cm)

10

ZE= BlockedElectrical mpedance ohms),

Z,•= OpenCircuitMechanicalmpedance,(mechanical hms).

The shakerand the load are simply wo four polesin tandem nd theproblem f determininghe motionat the outputof the oad s bestdoneby useof matrixnotation sshownn the ollowingetof equations.

• LO/21

• [•11where

The

•11]X]F]hakerquations.•1 [F•fi221X•Loadquations.

(54)

(55)

a2•l L•t

•q•]F•'•22XLt]

Combinedequa-tion for shaker

and load,

72•3•221=ta2122] /•2t 22]'

specific orm of the parameters •i are given

TABLE II.

Pure mass Spring supported mass

belowandwereobtained y performinghe indicatedmultiplicationf Eq. (56).

(57)

Equation 58) showshe four-pole quationsor thecombination, haker,and load.

(ss)I=T2xF•+'•22Vt.

Putting heoutput orce f the oadequal o zero, heresponsef he oad o the nput oltageE) isobtained.

F•=0 and Vx=E/T•2 velocityf the oad). (59)

Also hecurrentn thedriving oil sgiven y

(60)These esults requitesimplewhenexpressedn termsof four-polearameters.heyarealsoquitegeneralnthat the formulas nclude the effectsof the source nd

alsopermit he load o be any four-pole-typelasticstructure. or examplef the oadwerea puremass r aspring upported ass,hequantities•. •) wouldhavethevaluesn TableI. Perhapshenotablehing boutthis analysis s the ease with which the results were

obtainedndalso heirwidegenerality.

7. SHOCK EXCITATION ANALYSIS

The four-poleparameter echniques useful ndealingwith variousproblems f shock xcitedelasticsystems.he theoryof transient nalysiss an extensivesubject which is well treated in the literature and it is

not proposedo discussts details ere.The objectiveis merely o show owsome f these roblemsanbeformulatedn terms f four-poleheory.n doinghis,the pointof viewof Fourier ntegral nalysiss setforth by Campbelland Foster in their book "Fourier

IntegralsorPractical pplications"has een dopted.

The analysis resentedereapplies nly to systemswhich reat restat the imeof applicationf theshock.If thesystemsnot nitially t rest, hen t isnecessary

• G. Campbell nd R. Foster, ourierntegralsor PracticalApplicationD. VanNostrand ompany,nc.,NewYork,19S0).

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FOUR-POLE PARAMETERS IN VIBRATION Ca, LCULATIONS 851

to prescribenitialconditionsoreachof thecoordinatesrequired to describe he system and to solve thedifferentialequationsof the systemsubject to theseinitial conditions.However, the "Initial-Rest" case sof sufficient mportance o warrant its presentation.The procedureor calculating he response f a system

initially at rest to a nonperiodic xcitation nvolves hefollowing hree steps.

Stepnumber Operation

(1) Determine he response f the system o anapplied, single frequency, sinusoidalexcitation.

(2) Analyze the shockexcitation unction intoits sinusoidal componentsby means ofFourier ntegral.

(3) From he results f steps 1) and 2) determinethe response f the system o eachsinusoidalcomponentof the shock and add all theresponsesy use of a Fourier integral. Theresult is the desired esponse f the systemto the applied shock.

It is in step (1) that the four-pole echnique s useful.Since the four-poleequationsare relationsbetween

Vl{Ua• v2(U0 F2=O

Input Output

Fro. 15. Four-pole ubject o a velocity hock.

sinusoidalnput forces nd velocitiesnd sinusoidaloutput orces ndvelocities,he responsesequiredorthe Fourier ntegral formulation f the problemcanusually e readdirectly rom he four-polequations.This is illustrated n the following xample.Consideran arbitrary our-pole ystem t whosenput there sapplied velocity hockseeFig. 15)vt(/) andat whoseoutputno restrainingorce s applied.For sinusoidal

excitation he four-poleequations reFx-- anF2+ ax2V2

(61)

SinceFo.=0, the sinusoidalesponseunction equiredfor step 1) of the ransient nalysiss

%= g,/ot2•. (62)

Theanalysisf the nputvelocity hocksaccomplishedas follows:

v•(/)==J_•(63)

where

V,(f)=f_•v,(g)e-6Now the resultantvelocityz,.o(t)an alsobe written as aFourier integral thus:

v.(t)=_•ø_•v,(g)ea•'/"(6S)

=but V•) and g:•) are related hrough he four-poleequations,.e., Eq. (62) andwehave

v•(/)=•V•(f)ea'•df,6or in terms of the original vel•ity shock we have

vx(t)= •.ei"i(t-o)dfdg. (67)

Of coume,many other examples ouldbe cit• suchas a force nput or an accelerationnput or a displace-ment output. These wouldall be handled n a manners•ilar to that employedn the previous xample.

8. DISTRIBUTED PARAMETER SYSTEMS

(A) Helical Springs

The mechanical ystemswhich we have discussedup to nowhave all beenof the "LumpedParameter"type, .e.,systems hose erformanceould edescribedby oneor moreordinarydifferential quations.t willnow be shown hat four-pole arameter echniques realso applicable to certain distributed parametersystems. he particularsystems hosenor illustration

are all described y a one-dimensional ave equation.The first case which will be considered is that of a

helicalspringon whichelasticwaves re propagating.The equationsor the system f Fig. 16 are

O• 1 02//=0 (68)

Ox d at

o•F= --kl.-- (69)

Ox

where =displacernen• t point (•e)and time (t), c•speed f wavepropagation--l(k/m)«,=lengthof un-

Fro. 16. Helical spring.

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852 C. T. MOLLOY

Fro. 17. Equivalent electrical"T" for mechanical ystemwhosefour-poleparametersare eti. .

stretched pring,k= springstiffness onstant, nd m=mass of spring. Solution of the equationsassumingsinusoidal ime variation and subject to the boundaryconditions a, V• at the input and F2, V2 at the outputyields he four-pole quations:

+i(krn)'sin[(•-)o•]V.o70)

1]-=(krn)tin -•oF•

An equivalent electrical"T" network can always befound or any systemwhose our-poleparameters re

known, and which obey the reciprocity relation ofEq. (45a).The equivalencempliedhere s simply hatthe input and output currents n the electricalcircuitare numericallyequal to the correspondingnput andoutput velocities nd likewise or forcesand voltages.The relationship between the impedancesof thebranches f the electricalcircuit of Fig. 17 to its four-poleparametersre given n the equation et (72).

Z•= (ore- 1)/a2•

Z2= (a•.•- 1)/a•l (72)

Zs= 1/ot2•.

When these formulasare applied to a helical spring,the exactequivalentcircuitof Fig. 18 is obtained.

-I 0tin) 1m

O O

Input 0uDut

FIo. 18. Exact equivalent circuit for a helical spring.

The above equivalentcircuit is good for the wholefrequencyrange for which the differential equationdescribing he spring motion is valid. It can be em-ployed or the study of wave propagationn springs.If we restrict ourselves o the lower frequencies,namely, to situationswhere the argumentsof thetranscendentalunctions re small,say less han one-

tenth, the approximate quivalent ircuitof Fig. 19 isobtained.This approximations obtainedby expanding

• L-•

itr••• 0 CircuitalidwhenE

rio. 19. Approximate quivalentcircuit for helicalspring.

the transcendentalmpedanceunctionsn a Maclaurin's

series nd retaining he first term.The two precedingcircuits are typical and similar

circuitscan be obtained or other distributedparmetersystems.For example, the four-pole parametersandequivalentcircuit or a uniformbar in whichcompres-sionalwaves re traveling anbe obtained y replacingthe spring stiffness k) in the precedingby theequivalent tiffness f the bar.

(B) Rubber in Shear

The equations or the propagationof plane shear

wavesare identicalwith those or the propagation fwaveson a helicalspring.The only differenceies n thedefinition of the symbols.The diagram applicable othe shear case is given in Fig. 20. The performance

Fro. 20. Diagram for shearwaves.

equations re numbers 73) and (74).

0%1 1•-•- )

0x • c2 0t •

F= - SUr-,Ox

(73)

(74)

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FOUR-POLE PARAMETERS IN VIBRATION CALCULATIONS 853

wheren= displacementn shearplaneat distancex)and ime (t), c= speed f propagationf shearwaves=(G/p) , F=total shear orceon surface S), G--shearmodulus, = density f material,and = lengthofsample.If we follow the sameprocedure s was employed orthe helical spring the first approximatecircuit of Fig.

21 is obtained. f we assume hat the particular elas-tomerbehaves s f it hasa complex hearmodulus henthe second ircuit of Fig. 21 results.

9. CONCLUSION

The easewith which resultscan be achievedby thefour-pole echnique s well as the generalityof thoseresults make it a valuable addition to the collection of

methods vailable or solving ibrationproblems.

ACKNOWLEDGMENTS

The author wishes o thank his many friendswithwhomhe had the privilege ver he yearsof discussingvariousaspects f four-poleparameter heory. Thesediscussions ere most helpful in aiding the author toachieve larity in his own thinking.He is particularlyindebted to Mr. F. Mintz and Dr. S. Rubin of the

Input

o

l,=m/2

• o

Tc: o.u,(a)

L=mI2 L=m12

Input Ou  SG"

o o

Fro. 21. (a) Equivalent circuit for plane shear wave. Propa-

gation n an elastomer ith a realshearmodulus;b) equivalentcircuit or planeshearwavepropagationn an elastomer ith acomplex hearmodulus G-G'+iG").

LockheedAircraft Corporation, nd to Dr. F. A.Firestoneor having ead the originalmanuscript ndfor havingprovidedmany constructiveuggestionsorthis paper.