Effects of static loads on the nonlinear vibration of circular plates

Effects of static loads on the nonlinear vibration of circular plates

Xu, Pengpeng; Wellens, Peter

DOI

10.1016/j.jsv.2021.116111

Publication date

2021

Document Version

Final published version

Published in

Journal of Sound and Vibration

Citation (APA)

Xu, P., & Wellens, P. (2021). Effects of static loads on the nonlinear vibration of circular plates. Journal of

Sound and Vibration, 504, [116111]. https://doi.org/10.1016/j.jsv.2021.116111

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Effects

of

static

loads

on

the

nonlinear

vibration

of

circular

plates

Pengpeng

Xu,

Peter

Wellens

∗

Department of Maritime and Transport Technology, Delft University of Technology, Mekelweg 2, Delft 2628 CD, the Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 1 December 2020 Revised 4 March 2021 Accepted 28 March 2021 Available online 3 April 2021

Keywords:

Nonlinear plate vibration Static transverse load Helmholtz–Duffing equation Stiffening

Asymmetry Softening

a

b

s

t

r

a

c

t

Maritime structures in water always experience a mean hydrostatic pressure. This paper investigates the nonlinear vibration of a clamped circular thin plate subjected to a non- zero mean load. A set of coupled Helmholtz–Duffing equations is obtained by decomposing the static and dynamic deflections and employing a Galerkin procedure. The static deflec- tion is parameterized in the linear and quadratic coefficients of the dynamic equations. The effects of the static load on the dynamics, i.e. stiffening, asymmetry and softening, are investigated by means of the numerical solution of the coupled multi-mode system. An analytical solution of the single-mode vibration near primary resonance is derived. The analytical solution provides a theoretical explanation and quick quantification of the in- fluence of the static load on the dynamics. The numerical and analytical results compare well, especially for lower values of the static deflection, confirming the effectiveness of the analytical approach. The proposed analysis method for plate vibration can be applied to other structures such as beams, membranes and combination forms.

© 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Maritime structures in waterare always exposed to a meanload in the formof the hydrostaticpressure. Thin plate structuressubjectedtocomplexloadsinwaterarelikelytoundergoamoderate-to-largestaticdeflectionwithanamplitude of the sameorder asthe platethickness. Thecorresponding dynamicbehavior differs fromthat of a platenot statically loaded.

There are numerousprior studies on the nonlinear dynamicsof plates withstatic in-plane pre-loads. An early study on nonlinearvibratorypropertiesofcircularplateswithinitial deflectionisattributedto Yamakietal.[1],whichmodels thenonlinear dynamicsofclamped circularplateswithoutdampingthat areinitiallystretched andcompressedinthe ra-dialdirection.Naturalfrequenciesoffreevibrationsandthefrequency-responseintermsofharmonics,sub-harmonics,and super-harmonicsareformulatedanalyticallyandsolvednumerically.TheresultisvalidatedagainstanexperimentinYamaki etal.[2].Later,themethodologywasalsoadaptedandappliedtorectangularplates[3,4].

Hui [5,6] investigatethe influenceof imperfections onthe dynamicsof plates.The imperfectionis modeled asa per-turbation,andthe asymptoticsolutionclearlyshowsthe physicalpropertiesofthe primaryvibrations. EslamiandKandil

∗_{Corresponding author.}

E-mail address: p.r.wellens@tudelft.nl (P. Wellens).

https://doi.org/10.1016/j.jsv.2021.116111

0022-460X/© 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

[7] extend a similaranalysis to orthogonal panels, ofwhich the primary and secondary modes were found and studied.

[8,9] investigatetheinteractionofmultiplemodesofarectangularplatevibratingunderin-planepre-loads.Their numeri-calresultsshowbranchinginthefrequency-imperfectamplitude-curve.Quasi-degeneracyphenomenathatmodalfrequency curvesavoidcrossingandcoalesceareobservedinthesecondaryimperfectionmode.Rotaryinertiaeffectsarediscussedin ChenandShabana[10].ThethermaleffectisstudiedbyAmabiliandCarra [11].Directnumericalsolutionsandnumerical simulationsarestudiedinHui[12],Singhetal.[13],Amabili[14],Touzé etal.[15],ZareiandKhosravifard[16].Thestability ofthebucklingpathandvibrationsofin-planepre-compressedcircularplatesareanalyzedinZhouetal.[17],where non-linearityistackledbyexpandingcoefficientsandsolvingtheeigenvalueproblemofasetofalgebraicequations.Guojunand Huijian [18],DuandMa [19] investigatethe effectsofstaticloadsonthevibrations ofcircularsandwichplates,inwhich theeigenvalueproblemissolvedbyamodifiediterationmethod.

Another research angleis tostudy shallowspherical shells orplatesbeing curvedbecausethe curvature canbe seen asstaticdeformation.Over theyears,manystudieshavecontributedtothemodelingofnonlinear shellsandpanelswith curvatures, pre-loads, andimperfections [20–27].A comprehensiveintroduction tothe nonlinear vibrations ofplates and shellscanbefoundinAmabili[28].

Few studies haveconcentrated on platevibrations undertransverse static load. [29] analyzes the effect oftransverse deadloadsinrectangularplatedynamics.Itshowsthatthedeadloadsincreasethenaturalfrequencies.Amabili[30] com-pares numerical andexperimental results for large-amplitudevibrations of a circular cylindricalpanel. The results show the asymmetry inthe vibratoryamplitudein thepositive andnegative direction.[31] investigatethe nonlinearvibration characteristics ofa rotatingplateunderstaticloads ina magneticfield. Anordinary differential equation isadapted toa Duffing-likeequationby introducingperturbationsinthenonlinearterms.Amulti-time-scalemethoddeliversthe approx-imated analyticalsolution that clearlyshowsthe frequencyshift andtheresponse asymmetry.The numerical resultsare onlyvalidfortheprimarymodeofvibration,andtheperturbationsusedtotransformtheequationcouldnotbeinterpreted physically.Wangetal.[32] studythevibroacousticbehaviorofarectangularplatewithimperfectionssubjectedtoastatic point load.Morerecently,the changesofnonlinearfrequencies andmodesofbeam[33,34] andshell[35,36] duetolarge deformationsinnonlinearvibrationsareinvestigatedwiththeCarreraUnifiedFormulation[37,38].

TheHelmholtz–Duffingequationiscloselyrelatedtothenatureofnonlinearplatevibrationssubjectedtocomplex trans-verse loads (see Section ). The quadratic stiffness makes this category of nonlinear differential equations special. Chen et al.[39] extendstheelliptic Lindstedt-Poincaré (ELP) methodtosolve stronglyquadratic nonlinearoscillators with per-turbeddampingandstiffness.Hu[40],41]solvefreevibrationsofconservativefreeoscillatorswithweaklyquadraticterms with the harmonic balance method (HBM) andan auxiliary sign function in positive andnegative direction. Leung and Guo [42] deploys the homotopy perturbationtechniqueintroduced inHe [43],whichalso assumestwo half-solutions,to solve Helmholtz–Duffing equations. Comparedto numerical results, the homotopy perturbation generates smaller errors than[40,41].Yeasminetal.[44] developsasinglesolutiontofreeoscillatorswithmoderate-to-strongnonlinearity,wherea modifiedharmonicbalancemethodreducesthealgebraiccomplexity.Exactsolutions basedonJacobiellipticfunctionsare givenbyZhu[45] fortheHelmholtzequation,and[46] fortheHelmholtz–Duffingequation.

Most studies mentioned above are about the free Helmholtz or Helmholtz–Duffing oscillator. The forced vibration of quadraticandmixednonlinearDuffingequationshasreceived increasedattentioninrecentyears.Jiangetal.[47] demon-stratethesofteningeffectofnonlinearquadraticsystemsandcomparesdifferentanalyticalapproaches.GussoandPimentel

[48] improvesthe approximation tothe analytical solutionofthe forcedHelmholtz–Duffingequation. The approximation canbefurtherimprovedbyaone-stepcorrection,asproposedinZhouetal.[49].

Little quantitative information is available on changes instructural dynamics due to uniformlydistributed transverse staticloadsanditsrelationwiththeHelmholtz–Duffingequation,whichmotivatestheauthorstorevisit theforced vibra-tionofedge-clampedcircularplates.Otherpopularcases,in-planepre-loadsandimperfectionsforexample,areoutofthe scope ofthisresearch. Stiffening,asymmetry andsofteningareinvestigatedby decomposingthetotaldisplacementinthe equationsofmotionandisolatingthestaticdeflectionbeforeincludingitseffectinthesystemofdynamicequations.The systemissolvednumerically,andparametersarevariedsystematicallytodemonstratetheinfluenceofthestaticdeflection onthedynamicproperties.AHBM-basedanalyticalsolutionisthenderivedtogiveimmediateinsightintothemechanism behind, andaquickquantificationoftheeffectofstaticdeflectiononthedynamicsoftheplate.The analyticalsolutionis comparedtothenumericalresults.

2. Problemstatement

2.1. Governingequations

Consider an isotropic homogeneous circular platesubjected to a uniformly distributed transverse load, whose lateral section issketchedinFig.1.The loadhasa non-zerostaticpartanda dynamicpart.Thevibration willoccuraround the meandeflectionthatresultsfromthestaticload.Table1 givestheparametersrelevanttotheplate’smechanics.

WefollowvonKármánnonlinearplatetheorythattacklesthegeometricalnonlinearityassociatedwithmoderate deflec-tion.ThebendingstiffnessoftheplateiscalculatedasD= Eδ3

12

(

1−ν2

)

.Thedynamicpartofthein-planedeflectionisneglected becausethein-planetransientchangesaresmallcomparedtothetransverseones.Inthisaxisymmetriccase,thegoverning

Fig. 1. Lateral view of a clamped circular plate under a uniform transverse load. The total load is decomposed into a static and a dynamic part. The plate vibrates around the mean deflection.

Table 1

Symbols for the plate model.

Symbol Description Value Unit

ρ Density 7.78 kg m −2

E Young’s modulus 210 GPa

ν Poisson ratio 0.31

δ Thickness 0.001 m

R Radius 0.1 m

D Bending stiffness 18.44 Pa

c Damping coefficient 0.02

0 First linear EF 1572.7 rad s −1

r Radial coordinate m

t Time coordinate s

u In-plane displacement m

w Transverse displacement m

equationsare givenintermsofthe platedisplacement fieldinpolarcoordinates [50,51].The compatibilityequationthen reads u ,rr+1_r u ,r−_r u 2+w ,rw ,rr+ 1−

ν

2r w 2 ,r=0, (1)

wherecommasinsubscriptsdenotepartialdifferentiationinspacewhiledotsaboveparametersdenotepartial differentia-tionintime.

The transverse load isuniformly distributed witha staticand a time-varyingpart q

(

t

)

=qs+qdcos

(

ω

t

)

, with

ω

the frequencyofoscillation.Theequilibriumequationisgivenby

D

∇

2

_∇

2_{w +}

_ρ

_{w ¨+2c}

_ρ

_

0w ˙ − [q s+q dcos

(

ω

t

)

] − E

δ

1−

ν

2



u ,rw ,rr+3₂w ,r2w ,rr+

ν

_r uw ,rr+1+_r

ν

u ,rw ,r+u ,rrw ,r+₂1_r w 3,r



=0. (2)

Here,thebi-Laplacianoperatorinpolarcoordinatesisdefinedas

∇

2

_∇

2₌ ∂4 ∂r4 +2r ∂

3 ∂r3−_r12 ∂

2 ∂r2 +_r13_∂∂r.

Theboundaryconditionsoftherigidlyclampedplateareimposedattheplate’sedgeas

u

(

r, t

)

=0, w

(

r, t

)

=0, w ,r

(

r, t

)

=0, (3)

andatthecenteras

|

u

(

0, t

)

|

< ∞,

|

w

(

0, t

)

|

< ∞. (4) Thevariablesarerewrittentonon-dimensionalformforgenerality.Thenewindependentvariablesare

ξ

and

τ

andthe newdependentU

(

ξ

_,

τ

)

andW

(

ξ

_,

τ

)

.

ξ

= r R , W = w

δ

, U = uR

δ

2,



=

ω



0,

τ

=



0t, (5) inwhich



0isthefirsteigenfrequency(EF)oftheplatecalculatedby



0=

λ

2 1 R 2



D

ρ

, (7)

and

λ

1 (see[52])isthefirst(n=1)rootofthesystem

J0

(

R

λ

n

)

I1

(

R

λ

n

)

+I0

(

R

λ

n

)

J1

(

R

λ

n

)

=0, (8)

whereJandIstandforBesselfunctionsofthefirstandsecondkind.Thenon-dimensionalformofEqs.(1)–(4) nowreads

D

δ∇

2 ξ

∇

ξ2W +

ρ

R 4

δ



20W ¨ +2cR 4

δρ



0W ˙ − R4[q s+q dcos

(



τ

)

] − E

δ

4 1−

ν

2



U _,ξW _,ξξ+3 2W 2 ,ξW ,ξξ+

ν

_ξ

UW ,ξξ+1+

_ξ

ν

U ,ξW ,ξ+U ,ξξW ,ξ+₂1

_ξ

W ,3ξ



=0, (10) U

(

1,

τ

)

=0, W

(

1,

τ

)

=0, W _,ξ

(

1,

τ

)

=0, (11) U

(

0,

τ

)

=∞, W

(

0,

τ

)

=∞. (12) 2.2. Proposedapproach

The staticcomponentoftheexternal loadcausesa staticdeflection.Such astaticloadandits resultantdeflection will change thedynamicpropertiesinthe nonlinearmodel.Increasingthe staticdeflectioncancause stiffeningandsoftening oftheplate.Alsothevibrationscausestiffeningandsoftening,dependingonwhetherthemotionphaseworkstowards in-creasingordecreasingthedeflection,respectively.Thiseffectwillresultinanasymmetricdynamicsolutionofthevibration amplitudes.

There is no direct analytical solution to the dynamicsystem describedby Eqs. (9)–(12) that can isolate the effect of the staticdeflectionon thedynamics oftheplate. Therefore,we propose to modifythe equation.The staticdeflection is calculatedfirstandthencombinedwiththe-stillunknown-dynamicdisplacement.Thedynamicdisplacementoscillates around themeanposition thatis thestaticdeflection.Thecombinationofstaticdeflectionanddynamicdeflectionisthe totaldeflection.Thesubstitutionofthetotaldeflectionintothesystemfortheplaterevealshowtheplatedynamicschange whenthestaticdeflectionchanges.TheproposedapproachyieldsasetofHelmholtz–Duffingequationswithquadraticterms thatdependonthestaticdeflection,seeSection3.3.

2.3. Initialdeflection

Overdecades,academiaproposed differentanalyticalexpressionsofthestaticdeflectionsubjectedtoatransverseload. Examples include solutions by assuming deflection shapesand/or slopes [53], the perturbation method[54], the energy minimization approach [55,56], the fracture mechanics approach [57,58]. In this work, authors choose the solution give by Zhang [59],which buildson Timoshenko’sassumption. The reasonforthischoice is that (a).the assumedtransverse shapeisidenticaltothefirstbasisfunctionthatisgoingtobeappliedinourdynamicanalysisEq.(21) and(b).thein-plane displacementsufficesthecompatibilityrelationshipEq.(1).Inabsenceofin-planepre-strain,thetransversedeflectionWs

(

r

)

duetotheuniformlydistributedtransverseloadqs isgivenby:

W s

(

ξ

)

=W ˆs



1−

ξ

2



2_{, (13)}

wheretheamplitudeofstaticdeformationWˆsisreadilysolvedasafunctionofthetransversestaticloadqs ˆ

W s+



0. 4118+0. 25

ν

− 0. 16088

ν

2



W ˆs3=

q sR 4

64D

δ

. (14)

Thecurve inFig.2 showsthedependenceofdimensionalwˆs anddimensionlessWˆs onqs withininaloadrangeunder laboratoryconditions,fortheplateparametersgiveninTable1.Intheremainderofthearticlethestaticloadqsisnolonger used;insteadwewillusethestaticdeflectionamplitudeWˆsintheequationstorefertotheeffectofthestaticload.

Thecorrespondingin-planestaticdeflectioncanbesolvedthroughsubstitutingEqs.(13) into(9),

U s

(

ξ

)

=W ˆs2



_ν

_{− 7} 6

ξ

7₊10− 2

ν

3

ξ

5₊

₍

_ν

_{− 3}

₎

_ξ

3₊5− 3

ν

6

ξ



. (15) 3. Dynamicmodel 3.1. Decompositionofdeflection

Thetotaldeflectioniscomposedofthestaticandthedynamicdeflection:

U T

(

ξ

,

τ

)

=U

(

ξ

,

τ

)

+U s

(

ξ

)

, (16)

W T

(

ξ

,

τ

)

=W

(

ξ

,

τ

)

+W s

(

ξ

)

, (17)

wheresubscripts

(

·

)

Tdenote“total”.Termswithoutsubscriptarethedynamicin-planeandtransversedeformationtosolve for.

Fig. 2. Amplitude of static deflection as a function of the transverse static load.

Withthetotaldeformation,Eqs.(9) and(10) become

U T,ξξ+U

_ξ

T,ξ −U

_ξ

T2+W T,ξW T,ξξ+1₂−

_ξ

ν



W T,ξ



2=0, (18) L

(

U T, W T

)

= D

δ∇

_ξ2

∇

_ξ2W T+

ρ

R 4

δ



20W ¨T+cR 4

δ



0W ˙T− R4q dcos

(



τ

)

− E

δ

4 1−

ν

2



U T,ξW T,ξξ+3₂W T2,ξW T,ξξ+

ν

_ξ

UW T,ξξ+1+

_ξ

ν

U T,ξW T,ξ+U T,ξξW T,ξ+₂1

_ξ

W T3,ξ



=0. (19) NoteinEq.(19) thattheentireequationontheright-handsideofthefirstequalsignisbeingreferredtoasanoperation

LofUTandWT.

3.2. Separationofvariables

Theunknownsareseparatedintotime-dependentandspace-dependentpartsandtruncatedtoN

W

(

ξ

,

τ

)

=

N

n=1

[

φ

n

(

τ

)

η

n

(

ξ

)

], (20)

withtime-dependentpart

φ

nandspace-dependent

η

n.Inaddition,thetransversedeflectionisexpressedintermsof

poly-nomialbasisfunctions(BF)[50]

η

n

(

ξ

)

=



1−

ξ

2



2

_ξ

2n−2_{, n =1, . . . , N. (21)}

The exactexpressionsofeigenmodes(EM)arenotappliedinthepresentworkbecausethepurposeofthispaperisto investigate theplatevibration wherethe time-dependentbehavior isthe keything to solvefor; polynomialBFs simplify solving thespatialdifferential equationsandperformingthe Galerkinprocedure.Aspecific EMcanbe constructedby as-semblingthechosenbasisfunctionswithcoefficientsthatcanbecomputedfromaL2 _{projection[60].Fig.3 illustratesEMs} thatarecomposedofbasisfunctionsuptoN=3and4.

Thein-planedeformationisobtainedfromsubstitutingEqs.(20) into(18) andsolvingthesecond-orderdifferential equa-tion.

3.3. Dynamicequations

Substitution ofEq.(20) andtheexpressionforU intoEq.(19) yieldsapartialdifferentialequation intime

τ

andspace

ξ

.ThestandardGalerkinprocedureisapplied.Weobtain

1 0

Fig. 3. Examples of basis functions (BF) and eigenmodes (EM). (a) Polynomial BF; (b) Exact EM; (c) comparison between exact EM and EM shapes con- structed by three or four basis functions.

PerformingtheintegrationgeneratesasetofN coupledsecond-orderordinarydifferentialequationswithrespectto

τ

N k=1 c 1nk

φ

¨k+ N k=1 c 2nk

φ

˙k+ N k=1 c 3nk

φ

k+ N k,i=1 c 4nki

φ

k

φ

i + N k,i, j=1

c 5nki j

φ

k

φ

i

φ

j+c 6nq dcos

(



τ

)

+c 7n=0 for n =1, . . . , N. (23)

Thedynamicsystem(23) canbecategorizedasasetofcoupled,forcedHelmholtz–Duffingoscillators.Coefficientsc are

determinedwiththeGalerkinprocedure.Thec_1nkstandfortheinertialcoefficients,c_2nkforthedamping,c_3nkforthelinear stiffness, c4nki forthequadraticstiffnessandc5nki j forthecubicstiffness.Thec6n arethecoefficientsofthedynamicload.

Coefficients c7n are constantscontainingWˆs andWˆs3 originatingfromthe staticdeflection.Thistermisbalanced withthe initialstatictransverseloadqsinEq.(2).Itisworth emphasizingthattheasymmetryofthedynamicsystemisnotcaused byc7.Instead,thequadraticdynamictermswillresultinconstanttermsthatleadtoasymmetryinthesolution,whichwill bedemonstratedinthesectionsfollowingup.

4. Numericalsolutiontothecoupledmulti-modesysteminalargerangeoffrequency

Fig.3 ccomparesexacteigenmodesandtheapproximationsconstructedbythecombinationofbasis functions.It indi-catesthat usingN=3givesagoodapproximationofthefirstandsecond eigenmodebutan unsatisfactoryapproximation of the third.Higher modes and/or higheraccuracy can be obtainedwhen morepolynomial basis functionsare included. Becausefourtermsgiveaonlyamarginalimprovementoftherepresentationofthefirsttwoeigenmodesattheexpenseof athree-foldincreaseincomputationaleffort,hereafterallcomputationsareperformedwithN=3inpolynomialexpression

(20).

Thenumericalcomputationisperformedwithdimensionlessvariables,whicharebasedonthedimensionalparameters statedinTable1.We havevaried thestaticdeflectionWs; andforeachstaticdeflection,we havevaried thefrequency



of thedynamic load. Thenumerical integrationand thefrequencyvariation are carried out withMatCont [61]; that isa MATLAB®-basedpackagethatutilizescontinuationtechniques[62] andtheode45solverforthenumericaltimeintegration. Onlysteady-stateresultsinthepresenceofdampingarereported.

4.1. Trajectoriesinthestate-spacerepresentation

Fig.4 illustratesthetrajectoriesofthesteady-stateofthedynamicvariable

φ

nwithdifferentstaticdeflectionsunderthe

externalexcitation.Theexcitationamplitudeisfixedatq_d=500andthestaticdeflectionvariedasWˆs=0.0,0.1and0.2in differentrows.Theexcitationfrequencyis



= 1.125correspondingtothefoldregioninFigs.5 and6.Inthisregion,there arethreesolutions,twostableandoneunstable.Thefirsttwocolumnsofthefigureareforthestatespacerepresentationof thefirstandsecondBFinthepolynomialofthedynamicdeflectionW.Thethirdcolumnshows3Dplotsthatcomparethe dynamicdisplacementamplitudesofthefirstthreeBFs.Itisworth notingthatthetrajectoriesappeartocrossthemselves in the projection towards lower dimensionsshownin the second columnof Fig.4, butthat they are not crossed in the six-dimensionalphaseportrait



φ

1,

φ

˙1,

φ

2,

φ

˙2,

φ

3,

φ

˙3



.

Fig.4 hasevidenceoftheeffectsofstiffeningandofasymmetryonthedynamicresponseamplitudeinduced bystatic deflections.Withoutstaticdeflection,aharmonicexcitationgeneratescentro-symmetrictrajectoriesinstatespace.Stiffening becomesapparentfromtheextremitiesofthecurvesbecomingsmallerforlargervaluesofthestaticdeflection.Asymmetry is apparent fromthecentroids ofthe curvesshiftingalong the horizontalaxis,particularlyforthepositive position half-plane ofthe second BFbeingdifferentin theopposingpositionhalf-plane. Later,we willalsoseethat, besidesstiffening andasymmetry,thestaticdeflectioncausessoftening-softeningisnotshownstraightforwardlybythesetrajectories.

Fig. 4. Comparison of trajectory projections in the fold region ( = 1 . 125 ) with different static deflection ˆ Ws = 0 . 0 , 0.1 and 0.2. The transverse dynamic

load is fixed, q d = 500 .

4.2. Frequency-responsecurvesforbasisfunctionswithdifferentstaticdeflections

Fig.5 showsthefrequencyresponse (FR)ofthefirstthree BFsinthree differentrowsfordifferentvaluesofthestatic deflectionineachplot.The amplitudesofBF1,BF2,andBF3becomelargenearthefirst,second,andthirdeigenfrequency, respectively. ThequadratictermsinthesystemofDuffingequationsEq.(23) lead toasymmetryinthedynamicsolutions. Therefore,itisnecessarytoshowboththemaximumpositivedeflectionoftheplateandthemaximumnegativedeflection. The negativedeflectionamplitudesare showninthesecond columnofFig.5,in whichparticularlythenegativevalue of theleft-mostpeakofBF1isdifferentfromthepositivevalue.Also,note thatthesignofBF2isalwaysopposite toBF1and BF3duetotheconstructionoftheEM.EnlargementsoftheareasinFig.5 indicatedbyAtoFareshowninFig.6.

4.2.1. Stiffening,asymmetry,andsofteningeffectsofstaticdeflections

TheaforementionedthreechangesinvibratorybehaviorduetoWˆsaremoreclearlyvisualizedinFig.6b,c,f:thestiffened effect increases the eigenfrequencies and shifts the FR curves towards higher frequencies; the asymmetry decreases the positiveamplitudesbutincreasesthenegative;thesofteningeffectadjuststheoriginallyrightward-bendingFR curves,and bendsthemleftward.TheincreasedstiffnessalsobecomesapparentfromthelowerFRamplitudesforhighervaluesofthe staticdeflection.Allthreeeffectsareproportionaltothestaticdeflection.

Fig. 5. Frequency-response curves of the first three basis functions (BF) with different static deflections. The transverse dynamic load is fixed, q d = 500 .

Fig. 6. Enlargements of frequency-response curves with different static deflections ˆ Ws in Fig. 5 with specific frequency ranges for basis function 1 (a, b and

c) and basis function 2 (d, e and f). The dynamic transverse load is fixed, q d = 500 .

4.2.2. Sub-andsuper-harmonics

As an importantrepresentativeof nonlinearity,sub-harmonics arevisiblein thenumericalresults.Fig. 6a,e show the 1

2



sub-harmonics,whichareduetothequadratictermsofBF1andBF2forthefirstandsecondeigenfrequency.Interesting illustrationsareFig.6b,d,inwhichthe 1₃



sub-harmonics ofBF2 forthesecondaryresonanceareclosetotheharmonics oftheprimaryresonance.Thatthesetwopeaksapproacheach otherforincreasing staticdeflectionisevidentatWˆ =0.8;

Fig. 7. FR curves of the first and second eigenmode obtained from the numerical results. The dynamic transverse load is fixed, q d = 500 . These eigenmode-

based results are computed from the BF-based results shown in Fig. 5 .

the peaksmerge whenWˆs=0.9,which amplifiesthe responseamplitudes. Thepresent workdoesnot detect the super-harmonics,thecounterpartofsub-harmonics,becauseoftherelativelyhighdampingratio.

4.3. Transfertoeigenmodesfrombasisfunctions

The analysis so far was based on BFs (see Eq. (21)). The results in terms of eigenmodes (EM) can be calculated by transferringtheBF-basedfrequencyresponse(FR)

φ

totheEM-basedFR

φ

˜ utilizingtheL2projection.TheEMamplitudeis denotedas

˜.

Athree-termcombinationofpolynomial BFonlygivessatisfactoryaccuracyup tothesecondEM(seeSection 3.2)and the secondEF(see [1]).Therefore,we analyzethefirstandsecond eigenmodes andleave thethird outofthe discussion.

Fig.7 showsFR curvesbasedonthe EMs.Threeinfluencesofstaticdeflection-stiffening,asymmetryandsoftening-are visibleinasimilarwayasinFigs.5 and6.

Backbones(BB)oftheFR,representingtheasymptotesofthetheoreticalundampedeigenfrequencies,depictedinFig.8, demonstratethestaticloadeffectsinaclearoverview.BackbonesareobtainedbylinearlyinterpolatingthecuspsoftheFR withvaryingdynamicloadamplitudes,q_d=10(or20)to500.Thewell-orderedrankingofBBsdemonstratesthestiffening effect;thedifferenceinamplitudesinpositive-andnegative-directionisevidenceoftheasymmetricaleffect;thechangein bend-directionofEM1showsthesofteningeffect.TheeffectofthestaticloadonstiffeningisstronginbothEM1andEM2; theasymmetryissignificantinEM1andinsignificantbutstillobservableinEM2;softeninghardlyaffectsEM2.

5. Analyticalsolutiontothesingle-modesystemneartheprimaryresonance

Qualitative analysis provides insight into the mechanisms behind the effect of a static load on the dynamics of the circularplate. Hence welookintoan approximationto theanalyticsolution ofasingleeigenmode(EM)ofamulti-mode

Fig. 8. Backbones of EM-based FR obtained from the numerical results. Both static deflection ˆ Ws and dynamic transverse load q d are varied. For a certain

backbone with fixed ˆ Ws , the load amplitudes vary among q d = 500 , 400, 300, 200, 100, 20 (or 10). Table 2

Coefficients of the first and second EM. Coefficient EM1 EM2

σ 0.04 0.04

γ 1.520274439 0.257259273

α 1.537908112 0.4013065576

β 0.502206616 0.6613560788

χ 8.856630555 ×10 −5 _{4.254542778 ×10} −6

system inEq.(23) nearits eigenfrequency (EF). The response of one EMwill be dominantand other EMs are negligible whentheloadisexertednearoneoftheEFs.Usingthreepolynomialterms,Eq.(20) isrewrittento

˜ W

(

ξ

,

τ

)

=

φ

(

τ

)



z 1



1−

ξ

2



2_+z 2



1−

ξ

2



2

ξ

2_+z 3



1−

ξ

2



2

ξ

4



_{. (24)}

Thetildeaboveavariableindicatestime-dependentvariablesassociatedwithanEM.znarecoefficientsofbasisfunctions

toconstructaparticularEM,whichcanbeexplicitlycalculatedbytheL2_{projection.SubstitutionofEqs.(24) into(18) yields} thein-planeEMU˜

(

ξ

_,

τ

)

.

SubstitutionofW˜

(

ξ

,

τ

)

andU˜

(

ξ

,

τ

)

intoEq.(19) givesasingleordinarydifferentialequtionabout

φ

˜

(

τ

)

inthesameform ofEq.(23).TheforcedHelmholtz–Duffingequationisobtainedbynormalisingthesingledynamicequation andneglecting constantterms. ¨

φ

+

σ

_φ

˙ ₊



_1+W ˆ2 s

γ



φ

+_W ˆ_s

_αφ

2₊

_βφ

3₌

_χ

_q dcos

(



τ

)

, (25)

The unity partinthe lineartermiscalculated from λ21 λ2 1

(see Eq.(7)),which isthe firstlinear EFwithoutstatic defor-mation,anditsEM.Coefficients

σ

,

γ

,

α

,

β

and

χ

areexplicitlycalculatedwithmaterialparametersstatedinTable1 and coefficientszn foraparticularEMW˜.ThestaticdeflectionWˆsduetostaticloadqs existsinthelinearandquadraticterm. Table2 liststheparametervaluesofEq.(25) forthefirstandsecondEM.

Valuesof

γ

and

α

-valuesareassociatedwithWˆs -ofthefirstEMareaboutsixandfourtimesthatofthesecondEM, respectively.TheeffectsofstaticdeflectiononthefirstEMaremoreconspicuousthanthoseonthesecond.Whatistested butnotshownisthatthecubiccoefficient

β

isactuallymuchlargerthan

α

foralltheotherEMfromthethirdEMon.This indicates thattheforcedvibrationofhigherEMsisstronglycubicsothatthequadraticisnegligible.Hence weinvestigate theinfluenceofthestaticloadaroundprimaryresonancetodemonstrateandestimatethestaticinfluenceonthedynamics oftheplate.

TruncatingtheFourierexpansiontothefirstorder,thesolutionofEq.(25) isassumedas

φ

(

τ

)

= A 0+a 1cos

(



τ

)

+b 1sin

(



τ

)

and A 1=



a 2

1+b 21, (26)

wherethetotalamplitudeis

Fig. 9. Comparison of approximations of Eq. (37) (Sim.), Eq. (33) (Full Cub.) and Eq. (35) (Full Qua.).

Fig. 10. FRs and BBs of HBM. with different ˆ Ws . Eq. (38) calculates FRs with different dynamic load q d . Only FRs with q d = 500 are shown for reasons of

clarity. Interpolating the cusps of FRs generates the BBs. Correcting FRs and BBs in amplitudes with the asymmetry Eq. (35) yields curves in the positive and negative direction.

Fig. 11. Backbones of EM-based FR of analytical approximations. Both static deflection ˆ Ws and dynamic transverse load q d are varied. For a certain backbone

with fixed ˆ Ws , the load amplitudes vary from q d = 500 to 10.

Fig. 13. Comparison of BBs of analytical approximation (HBM) and numerical method.

Proceeding withtheHBM,we substituteEqs.(26) into(25) andequatethesummationofconstantsandcoefficientsof theprimaryharmonictozero.Theresultingequationsare

β

A 30+W ˆs

α

A 20+



3

β



a 2 1+b 21



2 +



1+_W ˆ2 s

γ



A 0+ ˆ W s

α



a 2₁+b 2₁



2 =0 (28) 2W ˆs

α

A 0a 1+ 3

β

a 1 4



a 2 1+b 21



− a1



2+



1+W ˆ2 s

γ



a 1+

σ

b 1



+3

β

A 20a 1=

χ

q d (29) 2W ˆs

α

A 0b 1+ 3

β

b 1 4



a 2 1+b 21



− b1



2+



1+W ˆ2 s

γ



b 1−

σ

a 1



+3

β

A 20b 1=0. (30) 5.1. Stiffening

ThelinearcoefficientinEq.(25) determinesthelinearEF,whichiscalculatedas



s0=



1+W ˆ2

s

γ

, (31)

in which



s0 standsforthefirstEFaffected bythe staticdeflection.Thereforethestaticdeflectionstiffens theplateand raisestheEFregardlessofthedeformingdirection.Incaseofsmallstaticdeflection,theEFincreaseswiththestatic deflec-tionsquaredbecauseEq.(31) canbeexpandedas



s0=1+

γ

2W ˆ 2 s +O



_ˆ W 4 s



. (32) 5.2. Asymmetry

SubstitutionofEq.(31) into(28) leavesusacubicequationaboutA0,

β

A 3 0+W ˆs

α

A 20+



_3βA2 1 2 +



2s0



A 0+ ˆ WsαA21 2 =0. (33)

Solving Eq.(33) andeliminating the imaginary parts fromthe three roots generates the solution of A0. However, its solution is nontrivial because of the cubic nature of the equation. Alternatively, we neglect the highestpower term in

Eq.(33) andobtainaquadraticequation: ˆ W s

α

A 20+



_3βA2 1 2 +



2s0



A 0+ ˆ WsαA21 2 =0. (34)

SolvingEq.(34) andchoosingthesolutionwithphysicalmeaningfromthetwogives

A 0=− 3A 12

β

+2



02 4W ˆs

α

+



9A 14

β

2− 8W ˆs2

α

2A 12+12A 12

β



02+4



04 4W ˆs

α

. (35)

2



_s0+3

β

A ₁

The minussignindicates thattheasymmetry isopposite tothedirectionofstaticdeflection.It impliesthat, subjected to a staticload in positive direction, the plate’s dynamicbending is larger in negativedirection and smallerin positive direction. Eq.(37) isidenticaltotheapproximation obtainedbyGussoandPimentel[48] whereboth cubicandquadratic termsofEq.(33) areneglected.Itisalsointeresting tosimplifyEq.(37) furthertoA0=−

ˆ WsαA2 1 21+Wˆ2 sγ  incaseof

β

 1.This approximationisidenticaltothesolutionobtainedwiththemulti-time-scalemethodinBenedettiniandRega[63].

Fig.9 comparesthesimplifiedapproximationgivenbyEq.(37),labelledas“Sim.”,thefullsolutiontothecubicequation

Eq.(33),labelledas“FullCub.”,andtheapproximationgivenbyEq.(35),labelledas“FullQua.”.Thecomparisonshowsthat thesimplifiedapproximationisvalidforsmallstaticdeflections.AsWˆs increasestothemoderate-to-largerangeWˆs,A11, thesimplifiedapproximationisstillacceptablewithamaximumerrorof7.42%atWˆs=1andA1=1.Thecomparisonalso confirmsthatthequadraticapproximationofEq.(35) isaccurateuptoWˆs=1withanerrorlessthan0.1%.

5.3. Softening

Cubicstiffnessisthenatureofgeometricnonlinearhardeninginplatemodels.QuadraticnonlinearityintheHelmholtz– Duffingequationcausessoftening[47,64].Thisimpliesthatthesofteningeffectofthestaticdeflectionisacorrectiontothe hardeningeffect.Thecorrectionmakesthebackbones(BB)bendtowardslower frequenciesanditsstrengthisproportional tothestaticdeflection,seeEq.(25).

SummingthesquaresofEqs.(29) and(30) andsubstitutingEq.(37) leadstoanimplicitfunctionF





,A1; ˆWs,qd



=0 A 2 1





4₊

_

4 s0



+9

β

2_A 6 1 16 − 3A 4 1

β





2₋

_

2 s0



2 + 4A 4 1

α

2W s2





2₋

_

2 s0



3A 2 1

β

+2



2s0 +



2_A 2 1



σ

2_{− 2}

_

2 s0



+ 4A 6 1W s4

α

4



3A 2 1

β

+2



2s0



2− 3A 6 1W s2

α

2

β

3A 2 1

β

+2



2s0 =

χ

2_q 2 d. (38)

Eq. (38) generates FR curves in which Wˆs and qd are parameters. Here we substitute the simplified expression Eq. (37) rather than the quadratic Eq. (35) or the cubic solution of Eq. (33) because they complicate the expression of

Eq.(38) whileprovidingonlyaminorimprovementinaccuracy(seeFig.9).

Fig.10 illustratestheaverage,positive,andnegativeFRandBBswithdifferentstaticdefection.Fig.11 depictsBBsofmore staticloadcases.Inthemoderaterange,max

(

A

)

≈ 1,thecubicstiffnesscausesrightwardbending(towardshigher frequen-cies),whilethequadraticstiffnessresultingfromthestaticloadbendsthecurvesleftwards(towardslowerfrequencies).This softening correction increaseswiththe staticload. The coexistenceofleft-bendingandright-bendingismostpronounced around Wˆs=0.7.Then thesoftening becomesdominant overhardening. Figs.10 and11 alsoillustrate the stiffening(see Section5.1)andtheasymmetry(seeSection5.2).

5.4. Comparisonwithnumericalresultsanddiscussion

NumericalresultsfromSection4.3 areusedtoverifytheapproximationderivedhere.Figs.12 and13 comparethe numer-icalandanalyticalFR andBB underdifferentstaticdeflections.Fromthecomparisonwefindthatthetwo approachesare ingoodagreementfortheoveralltrendofcurves.TheagreementverifiesnotonlythehardeningeffectofthevonKármán platemodelbutalsothestiffening,asymmetry,andsofteningeffectsduetothestaticdeflections(loads).Whenconsidered inmoredetail,theanalyticalapproximation,comparedwithnumericalresults,overestimatesstiffeningandasymmetry.As the staticdeflectionincreases,thedifferencesbetweencurvesfromthe numericalandtheanalytical approximationgrow accordingly.Thedifferencesareduetotheapproximationsused andthefactthat asingleeigenmodeisconsidered inthe analyticalapproximation.Theaccuracyforasingle-modeapproximationtothecompletesystemcanbeimprovedby includ-ingmoretermsintheFourierseries[48] andcorrectingthesolutionwithmultiplesteps[49].

Thedifferenceinasymmetrybetweenthenumericalresultsandtheanalyticalapproximationareattributedtothe over-estimationofA1.TheasymmetryA0becomeslargeraccordingly,seeEqs.(35) and(37).

Thedifferenceinleftwardorrightwardbending,i.e.hardeningandsoftening,isbecauseoftheeffectofstiffening decre-ments (the rootsofthebackbones)andoverestimatedamplitudes(the cuspsofthe backbones).Itcan enhanceorreduce the softeningorhardeningcurvesundercertain conditions.In general,both thehardeningandsofteningoftheanalytical

approximation are stronger than that of the analytical approximation for larger static deflections. Thisdifference is also mainlybecauseoftheoverestimationofA1.

Theanalyticalapproximationprovidesagoodexplanationandthefirstestimationofthechangesindynamicbehaviorof theplateduetostaticdeflection.

6. Conclusion

This paper theoretically investigates the effects of a static transverse load on the nonlinear dynamic behavior of a clamped circularplate. Forgenerality, the variables andthegoverning equationshavebeen madedimensionless. Decom-posingthetotaldeflectionintoa staticdeflectionandadynamicdeflection,andparametrising thestaticdeflectioninthe dynamic equations lead to a set of coupled, forced Helmholtz–Duffingequations. The linear and quadratic terms inthe Helmholtz–Duffingequationsareaffectedbythestaticdeflection.

Numerical calculations have been performed and an analytical model has been derived to capture the effect of a moderate-to-largestaticloadhasonthetransversevibrations.Numericalresults,aswellasanalyticalresults,illustratethe frequencyresponsecharacteristicsfordifferentvaluesofthestaticloadandshowitseffectintheformofstiffening, asym-metry, andsoftening. The numericalapproach forthemulti-degree offreedom Galerkinsystemis validfora wide range ofexcitationfrequencies andmultipleaxisymmetricEM.Stiffening,asymmetry andsofteningduetothestaticloadhavea significanteffectonthefirstEManditsfrequency.Itisalsofoundthat,forhigherEM,thestiffeningeffectisconsiderable, whilesofteninglesssoandthattheasymmetryisnegligible.

The analytical approach utilizes harmonic balancing to solve a single EM vibration near the primary resonance. The analyticsolutionprovidesamathematicalexplanationandafastapproximationoftheaforementionedstiffening,asymmetry andsofteningasaresultofthestaticdeformation.Numericalandanalyticalresultsareingoodagreementforthefrequency response in arange offrequencies andforthe backbonecurvesfor lowervalues ofthe staticdeflection.The verification of theargumentisaddressedby the comparableresultsofnumericalcalculation andtheoreticalanalysis.The differences between the numerical method and the analytical approximation grow larger as the static deflection increases, butthe trendsremainthesame.

The analytical approximation isfound tobe a good predictorforthe effectofa staticloadon thenonlinear dynamic behavior of a clamped circularplate. In differentcases,the relation betweendeflection andstaticloading isnot unique. Thepresentanalysishasthepotentialofextensiontothosedifferentcases.Thereisalsoaprospecttoapplytheproposed methodinothertypicalmodelslikeabeam,string,andmembrane.

DeclarationofCompetingInterest

The authors declare that they have no knowncompeting financial interests or personal relationshipsthat could have appearedtoinfluencetheworkreportedinthispaper.

CRediTauthorshipcontributionstatement

PengpengXu:Conceptualization, Methodology,Software,Validation, Formalanalysis,Investigation,Data curation, Writ-ing - original draft, Writing - review & editing, Visualization. Peter Wellens: Conceptualization, Methodology, Software, Validation,Formalanalysis,Investigation,Datacuration,Writing-originaldraft,Writing-review&editing,Supervision.

Acknowledgement

ThisworkisfinanciallysupportedbytheprogramofChinaScholarshipCouncil (CSC)withprojectno.201707720039.

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