forced harmonic response analysis of nonlinear structures

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  • 7/27/2019 Forced Harmonic Response Analysis of Nonlinear Structures


    AIAA J O U R N A LVol.31, No. 7,July 1993

    Forced Harmonic Response Analysis of Nonlinear StructuresUsing Describing FunctionsOmer Tanrikulu*Tilbitak-Sage,Ankara, TurkeyBayindirKurantand H. Nevzat OzgiivenJMiddle East Technical University, Ankara 06531, TurkeyandMehmet ImregunImperial College of ScienceTechnology andM edicine, London SW 7 2BX, England, United Kingdom

    Th edynamic response of multiple-degree-of-freedom nonlinear structuresisusually determinedbynumericalintegration of the equations of motion, an approach whichis computationally very expensive for steady-stateresponse analysisof large structures.Inthis paper,analternative semianalytical quasilinear method basedon thedescribing functionformulation isproposedfor the harmonic response analysiso f structures with symmetricalnonlinearities. T he equations of motion are converted to a set of nonlinear algebraic equations and the solutionisobtainediteratively. Thelinearandnonlinear partsof thestructureare dealt with separately,theformerbeingrepresented by theconstant linear receptance matrix[a],and thelatterby thegeneralized quasilinear matrix[A ]which isupdatedateach iteration.Aspecial technique that reducesthe computation time significantly whenthenonlinearities are localized is used with success to analyze large structures. Th e proposed method is fullycompatible w ith standard modal analysis procedures. Several examples dealing with cubic stiffness, piecewiselinearstiffness,and coulomb friction typeof nonlinearitiesare presentedin thecaseo f aten-degree-of-freedomstructure.

    IntroductionEXPERIMENTAL and analytical modal analysis tech-niques are routinely used to examine the dynamic re-sponse of structures.However, standard proceduresa rebasedon the assumption o f linearity and can fail to give accurateresults when there is significant nonlinear behavior. Previousinvestigations on nonlinearities were mainly focused on thedetection an d identification of nonlinear characteristics ofstructures from measured frequency response data.112 Thelack of analytical work can be attributed to the fact thatnonlinear system theory is not very well developed. Most ofthe existing nonlinear analysis methods are complicated andincompatible with the state-of-the-art modal analysis tech-niques. Moreover, they arerestrictedto certain types of non-linearities, and tosystems with veryfewdegreesof freedom .13 '14Th e most commonly used method fo r determining the re-sponse of nonlinear structures is the numerical integration ofthe equations of motion.


    However, to obtain accurateresults,the time stepo fintegration mustb e asmall fractionoftheperiod that corresponds to the highest natural frequency ofinterest. This makes numerical integration procedures verycostly forsteady -state response analysis, especiallyin thecaseof lightly damped, stiff structures with a large number ofdegrees of freedom. Hence, time domain techniques are notsuitable for theparametric clesignstudyo f structuresinwhicha large number of case studies have to be perfo rmed to deter-mine th e optimal structural properties.

    Received April 24, 1992; revision received Oct. 27, 1992;acceptedfor publication Oct. 27, 1992. Copyright 1993by the AmericanInstitute o fA eronautics an dAstronautics, Inc. A ll rights reserved.*Research Engineer, P. K. 119, Bahcelievler.tResearch Assistant, Department o f Mechanical Engineering.JProfessor, Department o f Mechanical Engineering.Lecturer, Department of Mechanical Engineering.

    In many branches of structural dynamics, attention ha sbeen focused on alternative, albeit approximate, frequencydomain methods fo r determiningth e steady-state response o fstructures, particularly to periodic external forcing. The start-ing point inallof - these quasilinear methods is the nonlinearordinarydifferential equations o fmotion,obtainedthroughaLagrangian procedure suchas the finiteelement method. Theresponse of the structure is assumed to be periodic and isexpressed as a Fourier series. The nonlinear differential equa-tions of motion are converted to nonlinear algebraic equa-tions, and thesolution is obtained iteratively. Ozgiiven,18 Bu-dak,1 9and Budak and Ozgiiven20 '21 used this approach for thefundamental harmonic response analysis o f structures withsymmetrical polynomial type nonlinearities. They used com-plex algebra with special rules for nonlinear operations toquasilinearize the internal nonlinear forces and to formulatethem in matr ix form. A substructuring technique previouslydeveloped for the analysis of nonproportionally dampedstructures was usedto increase the speed o f calculations fo rcases inwhich nonlinearities were confinedto a fewdegreeso ffreedom. Watanabe an d Sato22 '23 investigated the effects o fclearance type nonlinearities on the harmonic response ofstructures. They u sed the describing functio n technique for thequasilinearization of the differential equations of motion anda bisection methodto determinethe response. They extendedth e receptance coupling method to nonlinear structures byusing quasilinear receptance matrices. Their results were ex -perimentally verifiedby Murakami an dSato,24whotested anL-shaped beam with a clearance type nonlinearity. Bowdenand Dugundji25 also used describing functions to examine theeffects of local nonlinearities on the dynamics of large spacestructures.

    This paper isbased on theworko f Tanrikulu26

    an d focuseson thedevelopment o f a technique for the fundamental har-monic response analysis of structures with general symmetri-cal nonlinearities. The describing functionapproach is usedfo r th e quasilinearization o f nonlinearities, and the methodproposedb yOzgiiven18isusedtoincreasethespeedo f compu-tations when the nonlinearities arelocal.1313

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    Consider a nonlinear structure that isvibratingdue to har-monic external forcing. If the structure is discretized to ndegrees of freedom, then the matrix differential equation o fmotion can be written as[N ] = (1)

    [M], [H], and [K ]represent thelinear mass, structural damp-ing, and stiffness matrices, respectively, an d / is the unitimaginary number . Here, [x ] is the vector o f generalizeddisplacements and the dot denotesdifferentiation withrespectto time. The {/} and { T V } represent the external forcing andtheinternal nonlinear forces,respectively.T he A r t h element of{ T V } can be expressed as a series of the form

    nx r _ yt / f j ^

    wherenkj represents the force of the nonlinear element actingbetween theco ordinates k a n d y fo r (k & j)9 and between thegroundand the coordinatek for(k-j . Notethatnkj ~njk (3)

    wherenkj can be anyarbitrary function of theintercoordinatedisplacementykj and its timede rivatives


    ykj=xk if k =j



    Th eexternal harmo nicforcing w ith theangular frequency wcan be written as

    (6)where {F } is the vector of complex external forcing ampli-tudes,Im()denotes the imaginarypart,and thegenericangle\l / is defined as the product of o > and time t

    \l / = (7)If it is assumed that the structure vibrates periodically inresponse to the external harmonic forcing, then the displace-ment can be expressed as a Fourier series of the form

    = E (8)Th eresponse of the structureconsists of the bias term {x}0,the fundamental harmonic term {x} i,and the superharmonicterms ( x }r (r = 2, 3, 4, . . .). [X]r is thevector of complexdisplacement amplitudes of therthharmonic response comp o-nent. Thetermswith even coefficients are due tononlinearitieswith asymmetrical characteristics. If interest is confined tosymmetrical nonlinearities only, and superharmonic termsthat are small compared with the fundamental harmonic areneglected, thenEq. (8) simplifies to

    (9)wherethe subscript1 indicating the fundamental harmonic isdropped fo rconvenience.Th e internal nonlinear force nkj canbeapproximateda s aharmonic function o ftimefo rharmonicdisplacement

    Akj is the complex amplitude of the fundamental harmoniccomponent of nkj and it can be determined by using thefollowing Fourier integral:H)

    Th e nkj ca n also be expressed by using the harmonic inputdescribing function vkj which can be defined as the optimumequivalent linear complex stiffness representation of nkj fo rharmonicykj. It is awell-known fact that vkj is simplyequaltothe ratio ofAy an d Ykj (Refs. 27 and 28)

    n kj e(12)(13)(14)(15)

    In evaluating the integral in Eq. (11) and (13), the followingform ofykj should be used to expressnkj:

    ykj = Ykj si (16)where Ykj an d kj are thereal amplitudeand thephaseo fykj,respectively. Aswillbe seenlater,use of describing functionsin theanalysisisveryimp ortant sinceit allows theexpressionof internal nonlinear forces inmatrix form . Using Eq. (2) and(15), the vector [N ] can be written as

    { T V } (17)where {G } is the vector o f complex amplitudes o f internalharmonic nonlinear forces. The fcth element of { G } can beexpressed as





    IfEqs. (6), (9), an d (17)representing theharmonicexternalforcing, displacement, and internal nonlinear forces are in-serted incomplexform intoth em atrixdifferential equationo fmotion (1), o ne obtains

    where [a] is the receptance matrix of the linear part of thestructure which is defined as(21)

    and it can alsobeo btained bymodal superposition as will bediscussed later. Using Eqs. (17) and (18) vector { G } can bewritten as{G}=[A]m (22)

    [A ] is defined as the generalized quasilinear matrix, and itselements can beobtained by using the following:n

    k k= V kk + E v (23)(24)

    Using Eq. (22), Eq. (20) can be simplified to(10) (25)

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    where the coefficient matrix of {F } can be identified as theresponse-level-dependentquasilinearreceptance matrix of thestructure[0]= (26)

    Linear receptance matrix elementajk isdefined as the har-monic displacement amplitude of the y'th coordinate whenonlyasingleharmonicforceo funitam plitude is applied to thekthcoordinate.The quasilinear receptance Qjk cannotbe de-fined inthisway,sincefor a givennonlinearstructurediffer-en t Qjk values can be determined fo r different forcing levelsand configurations bysolvingE q. (25).A t agiven [F ] and co ,th e linearpart of the structurewith th e receptance matrix [a]is modified by the generalized quasilinear matrix [A]whichdependson [X ] .Theresultingequivalent linear structurehasthequasilinear receptance matrix [0] which is valid for thespecified frequency, forcing level, an d forcing configuration.[0]shouldbeconsideredas aresponse-level-dependentopera-to r between {X } and [F ] fo r forced harmonicmotion.In most cases the linearreceptance matrix [a]issym metricsince the linearmass, damping, an d stiffness matrices of thestructure are symmetric. The same is true for thequasilinearreceptance matrix[0] sincethegeneralizedquasilinear matrix[A ] is symmetric. At a first glance this might be viewed as aflawin thequasilineartheory sincenonlinearitiesareknowntobe one of thesources of asymmetry in transfer functionchar-acteristics of structures. Consider a hypothetical experiment inwhichone tries toobtainthe receptance matrix of ann-degree-of-freedom structurewith symmetrical nonlinearitiesas if it islinear, byapplyinga harmonicforceo funit magnitudeto eachcoordinate,respectively, and measuring the response atothercoordinates. This experiment can be simulated bysolvingEq.(25) n times, once for each forcingcase. This will result in ndifferent [0] matrices. In each [0] matrix only the columncorresponding to theexcitation coordinate wou l d be of rele-vance in terms of thesimulation considered.The desired re-ceptance matrix, which can be obtained by collating thesecolumns, will not be symmetric in general. Symmetricity o f[0] willbe discussed further in theCaseStudiessection.

    Solution TechniqueTh e quasilinear theory jus t presented converts a set of nnonlinear ordinary differential equations (1 ) into a set of nnonlinear complex algebraicequations(25).The fundamentalharmonic response of a structurewith sym metricalno nlineari-tiesunder externalharmonicforcing can now beobtained bysolving Eq. (25) iteratively. The following simple iterationscheme can be used at a particular frequency co :

    (27){ X } i + i is the complex displacement amplitudevector at the(/ +l)thiterationstep,while[0]/is the quasilinear receptancematrix at the /th iteration step that is determined by using( X ) i . The iterations can be continued until th e percentagedisplacement erro r

    (28)dropsbelow acertain value. The number o fiterationscan bereduced considerably if instead of [X}/ + i one uses the aver-aged displacement [ X*}/+ i to determine [0] /+ i

    (29)Inpracticalcases, response information is required for a cer-tainconstant forcing {F } over a frequency range with lowerlimit c o / and upper limit c o w . Structures with certain types o fnonlinearities, such as cubic stiffness, can exhibit multipleresponse behavior in certainfrequency ranges. The magnitudeo f th e response observed depends on whether the frequency

    range of interest is swept by increasing or decreasing theexcitation frequency. This is the well-known j u m pphenome-non. Thesemu ltiplesolutionscan bedeterminedby thecarefulselection of the initial iteration [X}{ at each frequency.1) Low -to-high-frequency sweep c o / > c o w ) : At frequency c o = c o / ) response of the linearpart of the structure isused as[X ] i. At other frequencies c o /< co

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    ~ r m -vV -u ,

    m ~U x2

    mL -vw- -vw-F i g . 1 Ten-degree-of-freedom systemwithlocalnon linearity, k - 50 k N / m m = 1 kg, y=0.01).




    COULOMB DAMPING) P I E C E W I SE L I N E A R S T I F F N E S S )F i g . 2 Symmetrical nonlinearities considered in thiswork.





    1009.4 9.6 9.8 1 0 . 0 1 0 . 2 1 0 . 4 1 0 . 6 1 0 . 8Frequency H z )400q


    0^300C DQC D(/)O



    1 00 - 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 M 1 1 1 1 1 M 1 1 n 1 1 1 1 1 1 1 1 1 1 1 1 1 19.4 9.6 9.8 1 0 . 0 1 0 . 2 1 0 . 4 1 0 . 6 1 0 . 8Frequency H z )F i g . 3 Linear and quasilinear receptances an and On for cubicstiffness typeo fnonlinearity csn=1 X 1 07 N /m 3 F\ =175 N):arrowsshow the direction of frequency sweep.

    all elements of [A],thefinalreceptance matrix [9] willcontainthe required quasilinear receptances which ar eobtained with-o u t havingto invert th e (n x n) matrix on the right-hand sideof Eq . 2 6 ) . Altho ugh this procedu re, in general, is not muchdifferent thanmatrixinversionfor afullypopu latedmatrixasfar as the number o f operations isconcerned, in this case itincreases theo verall speedo fcomputations considerably sinceno computat ions ar e required fo r zero elements o f [ A ] , amatrix which isusually sparseinstructu ral dynamics applica-tions.In most cases the nonlinearities arelocalized to a fewcoor-dinates only. This can be used to reduce the computationaleffort further by rearranging the differential equations o fmotion and writing [A ] in the follo wing partitioned form:

    [A ] = [[An] [0]]L [0] [ 0 ] J 3 6 )where the submatrix [ A n ] has an order o f m which is thenumber ofcoordinates affected bynonlinearities. When onlythe A r t h column o f [An] isconsidered, Eq . 3 3 ) becomes

    mQ = a p j - E ap s A s k e k j ,=

    ( p =l,2,. . . , ;j = 1 , 2 ,. . . , ) 3 7 )and for p=k)

    1+ E oiks A sks =l(7=1 , 2 , ... ,n) 3 8 )

    After calculating th e 0^^ (j =1, 2, . . . , ) which include theA:thco lumn of the matrix [An] , therem aining elements of thesubmatrices [0n] and [012]can be found from Eq. 3 7 ) for( p =l,2, . . . ,k-1,k +1, . . . , m;j=1, 2, . . . ,n).Repeatingthisproceduremtimes A := 1,2, . . . ,m)yieldsthefinal valuesof the upper(mxn)portion of [0].If the linearmass, damping, an d stiffness matrices are symmetric, [0] isalso symmetric a n d , therefore,[02i]=[Bi2]r 09)

    [022] can be determined by finding its diagonal an d uppertriangular elements only by using5=1 k=\

    +l, . . . ,n;j=p9 4 0 )

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    -* A ea

    9.4 9.6 9.8 10.0 10.2 10.4 10.6 10.8Frequency (Hz)

    In general, thelinear structuraldampingmatrix [ H ] can bewritten as the sum of proportional an d nonproportionaldamping matrices [H ]p and [H ]n[ H ] = [H ]p + [H ]n

    [ H ] p is usuallyexpressed as[H \p =


    (43)where 17 is the structural damping proportionality constant.[H ]n can beincorporated into [ A] and the linear receptancematrix [a] can be determined through the following modalsummation:


    where[$]is themodalmatrixandc o r(r= 1,2, . . . ,n)are thenatural frequencies of the undampedlinearpart of the struc-ture. Matrix inversion can be totally avoided by using thisapproach.400q


    C DQC Oo



    1009.4 9.6 X9.8 10.0 10.2 10.4Frequency (Hz) 10.6 10.8Fig.4 Linear receptances :1 2 and an and quasilinear receptances612 with F2= 75 N and 621 with Fi =75 N): cubic stiffness type ofnonlinearity, cs n= 1 x107N/m3); arrows show thedirection of fre-quency sweep.

    Hence,when [0] issymm etric, [On]a nd [612]are recom putedm times, thediagonal and upper triangular elements of [822]are computed once, and no computations are needed for theremaining elements at the /th iteration step of Eq. (27). Atypical [A ] matrix with local nonlinearities and the corre-sponding matrixthatshowsthe num ber of recomp utations foreach element of [0] is shownnext.A l w 0 mm 00 0







    m m... o

    . . . o. . . o. . . o

    m mm m1 1

    ... o

    m m m m.. . i i

    1 1 10 1 1 0 1



    CD -65^OC O




    859.4 9.6 9.8 10.0 10.2 10.4 10.6 10.8Frequency (Hz)400q


    0^3000>QC DC OD




    1009.4 9.6 9.8 10.0 10.2 10.4 10.6 10.8Frequency (Hz)

    (41)Fig.5 Linear and qu asilinear receptancesaisand 615 forpiecewiselinear stiffness type of nonlinearity [ fci)n=1kN/m, 2)11= 25 kN/m ,5n=0.01m,F5= 5N].

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    1318 TANRIKULU ETAL.: ANALYSIS OF NONLINEARSTRUCTURESCaseStudiesIn this section, th e forced fundamental harmonic responsecharacteristics of a ten-degree-of-freedom structureareinves-tigated (Fig. 1). All calculations were performed in doubleprecisionusing an HP Vectra386/25 personal computer. Cu-bic stiffness, piecewiselinearstiffness, andcou lomb dampingtype of nonlinearities wereconsidered(Fig.2). Mathematicalexpressions fo r nkj and vkj of these nonlinearities are given intheAppendix.In allcases,asinglenonlinearelementis placedbetween th e first coordinate and the ground. Responsedataarepresentedaroundthe firstresonance ( fn i = 10.1Hz) whereasingleharmo nic analysis hasbeen showntohave high accu-racy.InFig. 3,thelinearandquasilinear receptanceso and0nare shown fo r cubic stiffness type o f nonlinearity(csu = 1 x 107N/m3,F t =175 N). Multiplesolutionsthat de -pend on the direction of frequency sweep can be easily seen.Th e response was determined at 100 frequency points usingboth direct matrix inversion and the method proposed byOzgiiven. There were no appreciable differences between theresults. However, the calculation time was 871 CP U seconds




    851.......9.4 9.6 9.8 100 10 2 104 106 10 8Frequency (Hz)400n





    \ e 5

    100~ M nn1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1M1 1 1 1 1 1 1 1 1 1 1 1 1 1 m mi T[ m m |9.4 9.6 9.8 10 0 10 2 104 10 6 10 8Frequency (Hz)Fig. 6 Linear and quasilinear receptancesais and 615 for piecewiselinear stiffness typeof nonlinearity[ *i)n= l k N / m , A:2)n= 25kN/m,6n=0.01 m, Fs= 20 N]: arrows show thedirection of frequencysweep.







    4 9.6 9.8 10 0 10 2 104 10 6 10 8Frequency (Hz)




    100-9.4 9.6 9.8 100 10 2 104Frequency (Hz) 10.6 108Fig.7 L inear and quasilinear receptances x is and 615 for piecewiselinear stiffness type of nonlinearity [ /ti)n = l kN/m, 2)11=25kN/m , SnO.Olm,/5= 100N].

    withmatrix inversion and 287 CPU seconds using Ozgiiven'smethod.Figure 4 isrelated to the earlier discussion on the symmetryof th e quasilinear receptance matrix [0]. 012 was obtainedwith (F2 =75 N) whereas02i wasobtained with (F i =75 N) .Whereas O L \ 2 anda2iare identical to eachother,012and02iare completely different, althought heforcing levelswerekeptthe same. Multiple solutions occur for012 although no suchtrend is observed for021In Figs. 5-7 the linear and quasilinear receptances M I S and015areshownforpiecewiselinear stiffnesstype ofnonlinear-ity [(i)n= 1kN/m,(k2)n = 25 kN/m,6n= 0.01 m] forthreedifferent forcing levels.At the low forcing levelo f (F5 = 5 N) ,the displacement of the first coordinateX\ issmallerthan 6 1 1 ,and,hence,0i5is the receptance of a linear structurethatca nbe obtained th rough a modification of the original linear

    structure by thestiffness (^i)n (Fig.5) .Whenth eforcing levelis increased to ( F5 = 20 N), both (&i)n and (k2)\ \ becomeeffective and nonlinear behavior with multiple solutions isobtained (Fig. 6). At high forcing levels (F5 = 100 N),Xl ismuch higherthan 6 1 1an d (k2)n becomesdominant.Hence,0isbecomes the receptance of the original linear structure modi-fied effectively by the stiffness (k2)n (Fig. 7).

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    T3 -7CH




    -95-9.4 9.6 9.8 10 0 10 2 10 4 10 6 10 8Frequency (Hz)400q






    9.4 9.6 9.8 10 0 10 2 10 4 10 6 10 8Frequency (Hz)Fig.8 L inear and quasilinearreceptances ctis and 615 for coulombdamping type of nonlinearity an 5 N, Fs= 6 N, and Fs = 8 N9:arrow shows the direction of increasing external forcing level.

    InFig. 8,thelinearand quasilinearreceptancesa15 and9i5are shown for coulomb dampingtypeof nonlinearity (0n = 5N) for twoforcing levels ( F5 = 6and 8 N). As the forcing levelis increased, the quasilinear receptance approachesthe linearreceptance. This is due to the fact that internal linear forcesare am plitude dependentand increasewithincreasingexternalforcing whilethe coulomb damping forces stayconstant.Conclusion

    1 ) In this paper a semianalytical, frequency domainmethodhas beenpresentedthatcan be used for thepredictionof the fundamentalharmonic responseofmultiple-degree-of-freedom structures with symmetrical nonlinearities. Themethodcan beusedas an alternative tonum ericalintegrationprocedureswhichare computationallyveryexpensive. Its dis-tinct advantage over other nonlinear analysis methods is itsability to deal withlarge structurescontaining a widevarietyof nonlinearities with minimal computational effort.2) In most realistic cases, nonlinearities are localized to af e w degrees of freedom and this can be used to reduce thecomputing time through the application o f a special tech-

    nique. Significant savings can be obtained even fo r structureswith a relatively small number of degrees of freedom.3) The internal nonlinear forces are expressed in matrixform fo r harmonic motion using describing functions whichmakesit possible to define thequasilinear receptance matrix.It is important to carry the receptance matrix concept tononlinear analysis, since this provides compatibility with thestandard linear modal analysis procedures. The differencesbetweenthelinearand quasilinear receptancesare discussed indetail.4) Typical nonlinear frequency response phenomena suchasjump behavior can be fully simulated using this method.5) Themethodcan be used to dealwith awidevariety ofproblems in structural dynamics: investigation of the effectsofnonlinearities on linearmodal analysis andm odel updatingprocedures;structural coupling and modification analysisus -ing quasilinear receptances; detection and identification ofstructuralnonlinearities;rotordynamics; geardynamics; dy-namicsof largespacestructures; and dynamics ofbladeddiskswheredryfriction is used to improve the dynamicresponse.6) Although this paper discusses the fundamental har-monicresponseof structureswith symmetricalnonlinearities,thesameapproach can beextendedto the multiplefrequencyanalysisof structureswithnolimitationon thetypeo f nonlin-earities. These extensions will beaddressed in a forthcomingpaper.

    Appendix:HarmonicInput DescribingFunctionsMathematical expressions and the corresponding harmonicinput describing functions for theno nlinearities consideredinthis work are listed as follows.Cubic stiffness:n=csy

    v= 3 /4 cs~Y2Piecewise linear stiffness:

    n=kiy, \y\

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