Journal of Sound and Vibration "0885# 087"2#\ 250267

TRANSIENT RESPONSE OF STRUCTURES ONVISCOELASTIC OR ELASTOPLASTIC MOUNTS:

PREDICTION AND EXPERIMENT

K[ GJIKA\ R[ DUFOUR AND G[ FERRARIS

Laboratoire de Me canique des Structures\ ESA CNRS 4995\ INSA de Lyon\19\ Avenue Albert Einstein\ 58510 Villeurbanne\ France

"Received 1 August 0884\ and in _nal form 3 June 0885#

This paper deals with the response of structures on non!linear elastomer or all!metalmounts under impact excitations[ The mounts are modelled with the tangent andinstantaneous sti}nesses deduced by using experimental load!de~ection loops\ and withmodal damping factors[ The _nite element method and a dynamic condensation are usedto predict _rst the response of a beam on elastomer or all!metal mounts under hard andsoft impact forces\ and then the response of a rigid body on elastomer mounts under baseexcitation[ Associated experimental tests enable the non!linear model to be validated[

7 0885 Academic Press Limited

0[ INTRODUCTION

The viscoelastic behaviour of elastomer mounts and the elastoplastic behaviour of dry!friction mounts are commonly used for the passive isolation of fragile on!boardequipment[ In recent years much e}ort has been spent on the dynamic modelling of thesuspension and on the prediction of the dynamic behaviour of the structure!suspensionassembly[

Elastomer and metal mounts have a mostly non!linear behaviour[ The non!linearitiesarising from material behaviour\ geometrical design and dry!friction depend on de~ection\temperature and forcing frequency\ especially for the elastomer mounts 02[ As there isan interaction between these parameters\ the dynamic behaviour of such mounts iscomplex and a general model taking into account the type of excitation is di.cult toestablish[

The harmonic response of the system can be deduced from the superposition of linearvibration and large elastic deformation[ Padovan 3 has established a theory based on thesuperposition of the independent static and dynamic modulus\ while Nashif et al[ 1\who have extended the non!linear static equation of Mooney!Rivlin to dynamics\combined them[ They considered that the complex modulus approach is often moreconvenient for modelling the linear behaviour of the elastomer material than thegeneralized standard or derivative models[ They carried out characterization experimentsusing the harmonic response of a single!degree!of!freedom system "SDOF#[ This type ofsystem has been used in reference 4 to characterize elastomer mounts according toparameters such as overheating or external temperature^ then the harmonic response ofa ~exible beam on the frequency dependent mount was predicted and it compared wellwith the experimental result[ In reference 5\ a method involving an experimental and

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K[ GJIKA ET AL[251

numerical treatment was used to establish the harmonic and impact responses of a slenderstructure containing frequency dependent viscoelastic materials[

The hysteretic model is often used in the presence of dry!friction or of elastoplasticbehaviour[ The original di}erential model proposed by Bouc 6\ generalized by Wen 7\is commonly used for non!linear random dynamic analysis 800\ but its adaptation tothe prediction of steady state response is di.cult and has some disadvantages[ The tracemethod is based on the polynomial approximation of the hysteretic loop[ In the case ofthe bilinear system under stationary random excitation\ Badrakhan 01 extended the tracemethod\ keeping the non!linear sti}ness and approximating the hysteretic damping by anequivalent viscous damping[ Using an identical hypothesis\ Ko et al[ 02 establishedan empirical model with the amplitude displacement dependent parameter of a wirecable subjected to harmonic excitation[ The identi_cation of the hysteretic loop isperformed with an alternative use of the Fast Fourier Transform and of the Chebychevpolynomials[

The restoring force surface method is used for the non!parametric identi_cation ofnon!linear systems[ This method\ developed by Masri and Caughey 03\ establishes thesurface of the restoring force of the mount versus the phase plane\ the de~ection\ velocityand acceleration being known each time[ Worden 04\ 05 proposed a technique to measureonly one signal and then to obtain the two others using an SDOF system[ The integrationis better than the di}erentiation[ Especial attention must be paid to determine theintegration constants and the type of excitation[

There are di}erent analytical and numerical methods to solve non!linear dynamicsystems containing a low number of degrees of freedom\ particularly dynamic stationaryanalysis methods[ In the frequency domain the incremental method of Lau et al[ 06\ 07\integrated to the harmonic balance\ Timoshenko et al[ 08\ the IHB method\ isoften employed to predict the stationary response of strongly non!linear structuresunder arbitrary excitation[ There is also the Galerkin!Newton!Raphson "GNR# methoddeveloped by Ferri and Dowell 19 which is adapted to multi!harmonic excitation[ Speci_ctechniques are necessary if the response is di.cult to predict in the frequency domain[Cameron and Grif_n 10 developed a method based on frequency:time domain alternance"AFT# which is applied to an SDOF system containing a non!linear elastic:perfectly!plasticmodel under multi!frequency excitation[ Such an alternance using the Galerkin!Levenberg!Markardt approach was presented by Wong et al[ 11 to analyze the multi!harmonicstationary response of a system with amplitude displacement dependent parameters[ Astep!by!step incremental linearization technique "SILFD#\ performed at each integrationstep in the frequency domain through an FFT algorithm\ was presented by Venancio!Filhoand Claret 12 for an SDOF system with displacement dependent sti}ness and frequencydependent viscous damping\ subjected to an arbitrary excitation[

Current computation facilities permit the easy use of the _nite element method "FE# topredict the linear or non!linear response of a structure modelled with several thousandsof DOF[ In addition\ the reduction methods such as the modal or sub!structure methodsdecrease computer time consumption[ The pseudo!modal method can be used to predictthe frequency response of a system with frequency dependent characteristics 13[ Thismethod has been extended to the impact response of a ski with the updating of the modalsti}ness matrix 5[ Setio et al[ 14\ 15 introduced the non!linear mode and modalsuperposition method to predict the steady state response of a multi!DOF system withdisplacement dependent sti}ness under harmonic excitation[

In this paper\ the experimental quasi!static force de~ection curves enable the calculationof the tangent and instantaneous sti}nesses of elastomer and all!metal mounts which areused to compute the static equilibrium position while only instantaneous sti}ness is

STRUCTURES ON NON!LINEAR MOUNTS 252

Figure 0[ Tested elastomer mount[

employed to predict the impact response[ An FE model\ a dynamic condensation and theNewmark method are used to predict the time response of structures with localizednon!linearities due to the two kinds of mounts\ under force or base excitations[Experimental investigations carried out on a rigid structure and on a ~exible structurevalidate the predicted response[

1[ MODELLING OF THE NON!LINEAR MOUNTS

The elastomer mount\ shown in Figure 0\ has a viscoelastic behaviour while the all!metalmount\ shown in Figure 1\ has an elastoplastic behaviour[ They have overheating\ externaltemperature\ frequency and de~ection dependent behaviour 2[ Under a steady statetemperature condition\ it has been shown that the de~ection amplitude has a predominante}ect\ especially for the all!metal mount which is\ in addition\ almost independent offorcing frequency[ In order to predict the static equilibrium position under preload andthe transient response of the structure\ the quasi!static load!de~ection curves areinvestigated to evaluate the sti}ness of the mounts[ The classical experimental test consistsof picking!up the _fth load!de~ection loop\ the _rst four loops allowing the overheatingof the mount[ Axial and transverse tests are performed[ A linear curve _tting of theload!de~ection curves de_nes the tangent sti}ness kT and the instantaneous sti}ness k[The procedure is illustrated in Figure 2[

Figure 1[ Tested all!metal mount[

K[ GJIKA ET AL[253

Figure 2 "a# Load!de~ection loop of a mount[ "b# Load!de~ection curve[ "c# Linear curve _tting[

Let x be the de~ection in the applied force direction[ The restoring force Fr of themount between the de~ections at points i and j is "a list of symbols is given in theAppendix#

Fr ax b[ "0#

At any point "Fr \ x# of segment ij let the instantaneous sti}ness k be

kFr :x a b:x "1#

and the tangent sti}ness kT be

kT k"1k:1x#x a[ "2#

The constants a and b are de_ned as

a"kjxj kixi #:"xj xi #\ b"ki kj #xixj :"xj xi # "3\ 4#

with

ki "Fr #i :xi and kj "Fr #j :xj [ "5#

The quasi!static load!de~ection curve\ its approximation and the instantaneous sti}nessk of the elastomer mount are shown in Figures 36\ and those of the all!metal mount inFigures 700[ Figures 3\ 4 and Figures 7\ 8 concern the axial behaviour while the transversebehaviour is illustrated by Figures 5\ 6 and 09\ 00[ It should be noted that the transverseloops shown in Figures 5 and 09 are performed with two mounts[

STRUCTURES ON NON!LINEAR MOUNTS 254

Figure 3 Elastomer mount] measured load!de~ection loop[ "a# Traction[ "b# Compression[

2[ DYNAMIC CONDENSATION

The _nite element model of an industrial structure uses a large number of DOF[The updating of non!linear parameters at each step of the response requires techniquessuch as dynamic condensation to reduce the number of equations and the computertime consumption[ In the case of a linear structure on non!linear mounts\ the reductionis performed on the DOF describing the structure while all the DOF of the suspensionare kept[ The Craig and Bampton method 16 is well adapted to this problem[ With noexternal force\ the static equilibrium of the linear structure\ schematically representedin Figure 01\ is governed by a set of N equations which can be separated into twosub!systems\

$ KstriiKstrci Kstric Kstrcc %6"xi#"xc#76"9#"9#7\ "6#

Figure 4[ Elastomer mount] axial linear curve _tting "**#\ and instantaneous sti}ness "=====#[

K[ GJIKA ET AL[255

Figure 5[ Elastomer mount] measured transverse load!de~ection loop of two identical elastomer mounts[

Figure 6[ Elastomer mount] transverse linear curve _tting "**#\ and instantaneous sti}ness "=====#[

where "xc# and "xi# are respectively the interface and interior nodal displacement vectors[System "6# expresses the interior DOF as a function of the interface DOF which in factcorrespond to the connection of the mounts]

"xi#Kstrii 0Kstric "xc#[ "7#

The matrix Fstrc of the static displacement shapes permits a change of the basis\

6"xi#"xc#7Fstrc "xc#\ "8#

Figure 7[ All!metal mount] measured traction!compression load!de~ection loop[

STRUCTURES ON NON!LINEAR MOUNTS 256

Figure 8[ All!metal mount] axial linear curve _tting "**#\ and instantaneous sti}ness "=====#[

with

Fstrc $Kstrii 0Kstric I %\ "09#I being the identity matrix[ The _rst lower dynamic modes of the linear structure builtat the mount connections\ computed with

Mstrii "x i#Kstrii "xi# "9#\ "00#

yield the matrix Fstrn which is extended to the whole DOF of the system]

Fstrn $Fstri 9 %[ "01#The reduction matrix FCB can be established\

FCB Fstrn \ Fstrc $Fstri 9 Kstrii 0Kstric I %\ "02#

Figure 09[ All!metal mount] measured transverse load!de~ection loop[

K[ GJIKA ET AL[257

Figure 00[ All!metal mount] transverse linear curve _tting "**#\ and instantaneous sti}ness "=====#[

to permit the change of basis

6"xi#"xc#7FstrCB "xCB#\ "03#to phrase the structure|s reduced mass and sti}ness matrices]

mstrCB FCB TMstrFCB \ kstrCB FCB TKstrFCB [ "04\ 05#

In addition\ let msusCB and ksusCB "xCB # be the reduced mass matrix and the reduced variablesti}ness matrix\ respectively\ so that]

msusCB FCB TMsusFCB \ ksusCB "xCB # FCB TKsus"xCB #FCB [ "06\ 07#

Finally\ the dynamic characterization of the structure!suspension system expressed in theCB base is

"mstrCB msusCB #"x CB#"kstrCB ksusCB "xCB ##"xCB# "9#[ "08#

The static equilibrium is computed with the iterative process of the Newton!Raphsonmethod using the tangent ksusTCB and instantaneous ksusCB sti}nesses of the suspension\

"kstrCB ksusTCB "xiCB ##"DxCB# "FS#"kstrCB ksusCB "xiCB #"xiCB#\ "19#

Figure 01[ Schema of the structure with localized non!linearities[

STRUCTURES ON NON!LINEAR MOUNTS 258

with "DxCB# the displacement increment at the actual iteration\ "xiCB# the displacement atthe previous iteration\ and "FS# the applied static force[

3[ PREDICTION OF THE NON!LINEAR TIME RESPONSE

Here the structure!suspension system is subjected to constant concentrated "FC#or density forces and to di}erent forces "F"t## which are commonly time dependent[The FE model uses the classical beam\ shell\ and solid elements for the structure\ anda two node variable sti}ness element with three translations per node for the mount[Half of the mount mass is distributed at each node[ It is possible to introduce additionalmass and sti}ness matrices[

It is convenient to reduce the computation basis once again\ by using the modalreduction method in particular to introduce modal damping factors[ The modal basis Fp deduced from the conservative system "08# at the static equilibrium position permits achange of base]

"xCB#Fp "p#[ "10#

Consequently\ equations "08#\ in the presence of forces\ become

m"p #c"p #"kstr ksus"p##"p# " fc# " f"t##\ "11#

with "p# the modal parameters\ m Fp T"mstrCB msusCB #Fp the modal massmatrix\ kstr Fp TkstrCB Fp the modal sti}ness matrix of the structure\ ksus"p# Fp TksusCB "xCB #Fp the modal sti}ness matrix of the suspension\ c the constantmodal damping matrix\ " fc#Fp TFCB T"Fc#\ and " f"t## Fp TFCB T"F"t## the modalforce vectors[

In the modal basis\ the modal damping factors are assumed to be the diagonalcoef_cients of matrix c\

cj\ j zmj\ jkj\ j :Qj \ "12#

where mj\ j and kj\ j are the computed modal mass and sti}ness parameters while theQ!factor Qj is evaluated by using an experimental sine wave excitation[ The non!lineartime response of the structure!suspension system subjected to di}erent types of excitationis computed by using the Newmark method[ At each time step Dt\ the Newmark methodsolves the linear system

"3m:Dt1 1c:Dtkstr ksus"tDt##"p"tDt##

" fc# " f"tDt##"3m:Dt1 1c:Dt#"p"t##

"3m:Dtc#"p "t##m"p "t##\ "13#

which\ with the relationships

"p "tDt##"p "t##"1:Dt#""p"tDt## "p"t###\ "14#

"p "tDt##"p "t##"1:Dt#""p "tDt## "p "t###\ "15#

predicts "p"tDt# from "p"t##\ "p "t##\ "p "t##[If the system is assumed to be at rest at the static equilibrium position\ the iterative

process can be initialized[ The acceleration is computed by using the system

m"p "t9## " fc# " f"t9##"kstr ksus"p"t9###"p"t9##\ "16#

K[ GJIKA ET AL[269

and the DOF of the structure by using the relationship

6xixc7FCB Fp "p#[ "17#The Newmark method used\ known as the average acceleration method\ is unconditionallystable in the case of linear systems 17[ This implicit integration scheme employed fornon!linear analysis 17 is one of the most ef_cient methods 18[ An initial dynamiccomputation gives the highest frequencies of the system and the time step can be chosen[The sti}ness ksus"t# takes the place of ksus"tDt# in equation "13#\ in order to avoidan additional iterative process inside a time step[ The small variation of the suspensionsti}ness inside small time!step justi_es this substitution[

3[0[ FORCE EXCITATIONThe impact force applied to DOF l at the time ti9 can be described by a half!sine series]

F"t# sn

i0

ai sin Vi "t ti9#[ "18#

The unit component of the vector "Fl# expressed in the basis of the structure\

"Fl#T "9 = = = 9 0 9 = = = 9#\ "29#

corresponds to the DOF l[ The two successive basis transformations give the vector " fl#so that

" fl#Fp TFCB T"Fl#\ "20#

which\ in the case of the simultaneous force applied at m!DOF\ enables expression ofthe force vector " f"t## as

" f"t## sm

l0" fl#F"t#[ "21#

Thus the force vector is rapidly built at each time step[ Relationship "21# can be extendedfor any type of time dependent force F"t#[

3[1[ BASE EXCITATIONHere the base is rigid and the acceleration vector "G# contains only translation

components[ Equation "13# remains valid if vectors "p#\ "p #\ "p # are considered withregard to the moving base^ i[e[\ as relative quantities[ Let the force due to the baseacceleration be

"g"t##m"g"t##\ "22#

where the acceleration "g"t##\ de_ned by

"G"t##FCB Fp "g"t##\ "23#

is calculated by solving the linear system]

Fp TFCB TFCB Fp "g"t##Fp TFCB T"G"t##[ "24#

STRUCTURES ON NON!LINEAR MOUNTS 260

Figure 02[ Experimental set!up of the force excitation test[

4[ APPLICATIONS

4[0[ FORCE EXCITATIONA 9=264 m long steel beam\ shown in Figure 02\ is built!in at one of its ends and

connected onto the elastomer or all!metal mount at LP 9=224 m[ Its rectangular crosssection is 9=93 m wide and 9=993 m high[ A proximity probe\ with a linear measurementrange of 91 mm is located at LD 9=956 m\ while a piezo!electric force transducer is stuckonto the beam at LF 9=259 m[ The soft impact is performed on the force transducer witha tennis ball and the hard impact with an hammer[

The structure is modelled with 01 _nite beam elements\ the force transducer with anadditional mass element and the mount with a variable sti}ness element[ The damping isintroduced by the Q!factors measured with a sine wave excitation of the beam!mountsystem[ The _rst six modes are kept to de_ne the CB basis[

Figure 03[ Measured soft impact[

K[ GJIKA ET AL[261

Figure 04[ "a# Vertical displacement response of the beam] prediction "**#^ experiment "=====#^ "b# predictedvertical displacement response of the beam at the mount location "**#\ and instantaneous sti}ness "=====#[

Figure 05[ Measured soft impact[

4[0[0[ Beam:elastomer mount systemThe signal of the soft impact performed\ displayed in Figure 03\ can be considered as

a half sine series[ The predicted response of the beam is compared in Figure 04"a# to themeasured displacement given by the proximity probe[ Figure 04"b# shows the predicted

STRUCTURES ON NON!LINEAR MOUNTS 262

displacement of the beam at the mount location and the non!linear mount sti}ness versusthe time of the analysis[ The non!linear zone is reached[

4[0[1[ Beam:all!metal mount systemIn the impact and response graphs the previous presentation is used[ Figures 05\ 06"a#

and 06"b# are for the soft impact analysis\ and Figures 07\ 08"a# and 08"b# for the hardimpact[ As previously\ the predicted and measured displacement of the beam compare well[

4[1[ BASE EXCITATIONThe tested system used in reference 29 is schematically represented in Figure 19[

The 9=0399=0199=979 m parallelepiped structure weighing 09=14 kg is non!linearlysupported by four vertical elastomer mounts[

A signal data processing station controls the base acceleration produced by usinga 79 kN electrodynamic shaker[ The vertical base acceleration\ shown in Figure 10"a#\picked!up with a piezo!electric transducer\ is approximated by a half sine series\ displayed

Figure 06[ "a# Vertical displacement response of the beam] prediction "**#^ experiment "=====#^ "b# predictedvertical displacement response of the beam at the mount location "**#\ and instantaneous sti}ness "=====#[

K[ GJIKA ET AL[263

Figure 07[ Measured hard impact[

Figure 08[ "a# Vertical displacement response of the beam] prediction "**#^ experiment "=====#^ "b# predictedvertical displacement response of the beam at the mount location "**#\ and instantaneous stiffness "=====#[

STRUCTURES ON NON!LINEAR MOUNTS 264

Figure 19[ Schematic of the parallelepiped structure on four elastomer mounts[ Values "m#] a9=039^b9=019\ c9=979^ H9=997[

in Figure 10"b#[ The piezo!electric accelerometer located at the top centre of the structuregives the vertical response[ By using measured sine wave transmissibilities in the X!\Y! and Z!directions\ the _rst six viscous damping factors are evaluated at approximately9=914[

Four solid parabolic brick elements model the structure\ and one variable sti}nesselement models each mount[ Only the _rst six modes are kept in the basis[

The gravity e}ect computed with the Newton!Raphson method establishes the staticequilibrium position[ The comparison of the predicted and measured absolute accelerationresponses is shown in Figure 11"a#[ The relative displacement and the non!linear mountsti}ness versus the analysis time is displayed in Figure 11"b#[

5[ CONCLUSIONS

Techniques for predicting the transient response of linear structures on two typesof non!linear suspensions have been developed[ They are based on the experimentalmodelling of the mounts\ on a reduction FE model of the structures and on experimentalvalidations[ The modelling of the mount has been carried out mainly on non!linearsti}ness\ because\ the e}ect of non!linear sti}ness is more important than the e}ectof damping[ The damping is taken into account by using the modal damping factor[In future studies it will be convenient to distinguish the mount damping from the structuraldamping[

Figure 10[ "a# Measured base acceleration^ "b# half sine series approximation of the base acceleration[

K[ GJIKA ET AL[265

Figure 11[ "a# Absolute vertical acceleration response of the structure on four elastomer mounts] prediction"**#^ experiment "=====#^ "b# predicted relative vertical displacement response of the structure on four elastomermounts "**#\ and instantaneous sti}ness "=====#[

The experimental characterizations of elastomer and all!metal mounts permit thede_nition of the quasi!static load de~ection curves and the establishment of a mount _niteelement with account taken of the instantaneous and tangent sti}nesses[ In the completeFE model\ the structure is condensed to the interface DOF while all the DOF of thesuspension are kept[ The FE method with the Craig and Bampton reduction is welladapted to the time response of linear structures with localized non!linearities[

The predicted and measured transient responses of rigid and ~exible structures on thetwo types of mounts correspond well[ Thus\ it has been shown that the instantaneoussti}ness and modal damping factors\ both deduced from simple experiments\ ef_cientlymodel the non!linear behaviour of the two types of mounts under soft or hard impactexcitation[ Under these force or base excitations\ the role of the damping is of secondaryimportance[

ACKNOWLEDGMENTS

The authors would like to thank Vibrachoc S[A[ and the French Ministry of Researchand Technology for providing technical and _nancial support[

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APPENDIX] LIST OF SYMBOLS

x suspension de~ectionk suspension instantaneous

stiffnesskT suspension tangent stiffnessFr restoring forcef natural frequency\ HzV excitation frequency\ rad:s"xc# interface nodal displacement

vector"xi# interior nodal displacement

vectorI identity matrix"9# vector of 9|s9 matrix of 9|sFstrc matrix of static displacement

shapesFstrn matrix of _xed interface mode

shapesMstr structure mass matrixMsus suspension mass matrixKstr structure stiffness matrixKsus"x# suspension stiffness matrix"Fc# external constant excitation

force

"F"t## external time function excitationforce vector

"G"t## base acceleration vectorCB Craig!Bampton representationxCB Craig!Bampton displacement

vectorFCB reduction matrixmCB reduced mass matrixkCB reduced stiffness matrix"p#\ "p #\ "p # modal co!ordinate\ velocity and

acceleration vectorFp mode shape matrix"m# modal mass matrix"k# modal stiffness matrix"c# modal damping matrix" fc# modal external constant exci!

tation force vector" f"t## modal external time function

excitation force vector"g"t## modal base acceleration vectort timeDt numerical time stepQj quality factor of the jth mode