Data on the flexural vibration of thin plate with elastically restrained edges: Finite element method and wave based method simulations

ContentslistsavailableatScienceDirect

Data

in

Brief

journalhomepage:www.elsevier.com/locate/dib

Data

Article

Data

on

the

flexural

vibration

of

thin

plate

with

elastically

restrained

edges:

Finite

element

method

and

wave

based

method

simulations

Ling

Liu

a ,b

_,

_Roberto

_Corradi

b

_,

_Francesco

_Ripamonti

b

_,

_Zhushi

_Rao

a ,∗

a State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

b Department of Mechanical Engineering, Politecnico di Milano, Milano 20156, Italy

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 20 May 2020 Revised 8 June 2020 Accepted 11 June 2020 Available online 29 June 2020 Keywords:

Wave based method Elastically restrained edge Damping connection

Non-uniform boundary conditions Kirchhoff plate

Rectangular plate Pentagonal plate Plate bending problem

a

b

s

t

r

a

c

t

The data here reported refer to the numerical examples shownintheresearcharticle“Wavebasedmethodfor flex-uralvibrationofthinplatewithgeneralelasticallyrestrained edges” (Liu et al.,2020 [1]).Within theexamples, only the datasetsregardingtheplateswithelasticorelastic-damping supportsareprovided.Thedatasetscontaintherawdata di-rectly obtained from the forced vibration simulations. The simulationsarecarriedoutusingtwomethods:thefinite el-ementmethodrealized inANSYSMechanicalAPDLand the proposed wavebasedmethod(Liu etal., 2020[1]), imple-mented inaMATLABcode.The dataobtainedfrom ANSYS servesasreference fortheresponseoftheplateunder dif-ferent boundaryconditions. For each frequency, the trans-versedisplacements ofthe plateattwo pre-selectedpoints arelistedinthespreadsheet(e.g.MSExcel).Whendamping ispresent, theyareseparated intorealpart and imaginary part.Thispartofdatacanbeusedasreferencewhenother novel methods are developed. The datasets obtained from MATLABincludethecontributionfactorsaswellasthewave functions.Basedonthem,onecanobtainthedisplacement asacomplexnumberatanypointoftheplateafterasimple

DOI of original article: 10.1016/j.jsv.2020.115468

∗ _{Corresponding author.}

E-mail address: zsrao@sjtu.edu.cn (Z. Rao). https://doi.org/10.1016/j.dib.2020.105883

2352-3409/© 2020 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

postprocessing.Postprocessingcodestoobtainthefrequency response function for auser-given point and the displace-mentfieldatauser-givenfrequencyarealsoprovided.This part ofdatapresentsmuchmoreinformationthanthe pre-viouspartaswellasthecorrespondingresultsintherelated researcharticle.Itmakesitpossibletoseetheresponsesat otherpointsorotherfrequenciesthatarenotconsideredin the researcharticle, withoutrepeating thetime-consuming simulations.Moreover,ifsomeonewantstofurtherimprove thewavebasedmethod,thispartofdatawillbehelpful, ei-therforanalysingthelimitationsoftheproposedmethodor formoredirectcomparisons.Anyresearchrelatedtothe flex-uralvibrationofplatecanalsoconsiderthedataprovidedin thisarticle.

© 2020 The Authors. Published by Elsevier Inc. ThisisanopenaccessarticleundertheCCBYlicense. (http://creativecommons.org/licenses/by/4.0/)

SpecificationsTable

Subject Mechanical Engineering

Specific subject area Structural vibration Type of data Table (excel files)

Dataset (mat files)

Postprocessing codes (m files)

How data were acquired Data in the excel files were obtained through the simulations in ANSYS Mechanical APDL 17.2, using Finite Element Method (FEM). Datasets in the mat files were obtained from the numerical computation implemented in MATLAB, using the modified Wave Based Method (WBM) introduced in the associated research article [1] .

Data format Raw

Parameters for data collection In the numerical simulations of the plate, the important parameters include geometric parameters, material parameters, excitation parameters, boundary conditions, measuring points and quantity, and modelling parameters. While the modelling parameters refer to element type and size in FEM, they are wave functions and truncation factors in WBM. Description of data collection The datasets were collected from five numerical examples. They

respectively analyse the rectangular plate with uniformly elastic supports (E-E-E-E), with uniformly elastic-damping supports (ED-ED-ED-ED), with non-uniformly elastic supports (E1-E2-E3-E4), with elastic-damping partially supports (PS), and the irregular pentagonal plate with non-uniformly elastic-damping supports (IR). The E-E-E-E example considers five cases regarding different spring constants. The ED-ED-ED-ED example has two cases for different damping coefficients.

Data source location Institution: Department of Mechanical Engineering, Politecnico di Milano City/Town/Region: Milan

Country: Italy Data accessibility With the article

Related research article L. Liu, R. Corradi, F. Ripamonti, Z. Rao, Wave based method for flexural vibration of thin plate with general elastically restrained edges, J. Sound Vib. In Press.

ValueoftheData

• Thedataisusefulforinvestigatingtheflexuralvibrationofthinplates,developingnew nu-mericalmethodsforstructuraldynamicproblemsandimprovingthewavebasedmethod.

• Thedataisusefulforengineersandresearcherswhoneedtocontroloranalysethevibration ofa structure likeplate, andalsohelpful forthose whowork on thenumericalprediction tools.

• Whendevelopingother methodsforplatebendingproblem, onecanusethedatafor com-parisonorasreference.

• Thedatacanbe furtheranalysedtogetmoreinsightintheflexuralvibrationofthinplates. Forexample,itispossibletoinvestigatetheinfluenceofboundaryconditionsbyadditional comparisonamongdatasets.

• TheresearchersinterestedintheWBMmightfindthepostprocessingcodesuseful.Working onthe WBMforplatebending problems,theycan directlygetthe wave functionsandthe solutionsofcontributionfactors.

1. DataDescription

Thedatapresentedinthisarticlewereobtainedfromthenumericalsimulationsoftwoplates withgeneralelasticallyrestrainededges.ThetwoplatesareshowninFig. 1 .Theyaremadeof thesamematerialwithdensity

ρ

₌2700kg_/m3_{,Young’s modulusE=70× 10}9_N/m2_,and

Pois-son’s ratio

ν

=0.3.Theyalsohavethesamethicknessh=0.002m.Theedges oftheplatesare restrainedbyspringsanddampers.Thetransversedisplacementofedge



iissubjecttothe

lin-ear springwith stiffnesskwi,andthe lineardamperwithdamping coefficient cwi.Meanwhile,

therotationaldisplacementofedge



iissubjecttotherotationalspringwithstiffnessk_θi,and

therotationaldamperwiththedampingcoefficientc_θi.Allthestiffnesses anddamping

coeffi-cientsaredefinedper unitlength,andtherotationalspringsanddampersact inthedirection aligned withthe bendingmoment. Forthe two plates,the simulations evaluate their flexural vibrationwhen theyareunderthe steady-stateforceexcitation ejωt _{atpoint F(0.235m,0.14m).}

Theamplitudeoftheforceis1Nand

ω

isthecircularfrequency.

AsshowninFig. 1 (a),thefirstplateisrectangularwiththedimensionsa_{× b=}0_.75m_{× 0.}5m. Datasets includethe resultsof thisplate underfour typesof boundary conditions(BCs) with respecttothesupportsalongedges,i.e.,theuniformlyelasticsupports(E–E–E–E),theuniformly elastic-dampingsupports(ED-ED-ED-ED),thenon-uniformlyelasticsupports(E1–E2–E3–E4)and the elastic-dampingpartiallysupports(PS).Among them,the E-E-E-Etype containsfivecases, withrespect tothe differentcombinationsof translationalstiffnesskw androtational stiffness

k_θ.TheED-ED-ED-EDtypehastwocases,correspondingtothetwolevelsofdamping,cw/kw=

c_θ/k_θ=0.001andcw/kw=c_θ/k_θ=0.01.

The second plateispentagonal, andits geometryis showninFig. 1 (b),determinedby the coordinates of its vertexes.As is shown, thepentagonal platecan be enclosed by the rectan-gular with dimensionsLx× Ly=0.75m× 0.5m.Regarding thisplate, only the datasets forone

typeofBCsareprovided.Inthiscase,theplateiswiththenon-uniformlyelastic-damping

sup-Fig. 1. The plates with general elastically restrained edges: (a) the rectangular plate for numerical examples E–E–E–E, ED–ED–ED–ED, E1–E2–E3–E4 and PS; (b) the irregularly-shaped pentagonal plate for numerical example IR.

Table 1

Summary of the simulation cases corresponding to the datasets collected from ANSYS FEM. In these excel files are the tables for the transverse displacements of the points w 1 and w 2 (shown in Fig. 1 ) within the given frequency range at

the intervals of 1 Hz.

Excel filename Sheet Tyre of BCs Stiffness

Damping

cw / k w = c θ/ k θ Frequency (Hz)

ANSYS_EEEE Kw100Ka1000 E–E–E–E kw a 3 /D = 100 , k θa/D =

10 0 0

0 1–10 0 0

Kw10Ka10 0 0 E–E–E–E kw a 3 /D = 10 , k θa/D = 10 0 0 0 20 0–50 0

Kw10 0 0Ka10 0 0 E–E–E–E kw a 3 /D = 10 0 0 , k θa/D =

10 0 0

0 20 0–50 0 Kw100Ka10 E–E–E–E kw a 3 /D = 100 , k θa/D = 10 0 20 0–50 0

Kw100Ka100 E–E–E–E kw a 3 /D = 100 , k θa/D = 100 0 20 0–50 0

ANSYS_EDEDEDED C0 0 01 ED–ED–ED– ED kw a 3 /D = 100 , k θa/D = 10 0 0 0.001 20 0–50 0 C001 ED–ED–ED– ED kw a 3 /D = 100 , k θa/D = 10 0 0 0.01 20 0–50 0 ANSYS_E1E2E3E4 C = 0 E1–E2–E3– E4 kw1 a 3 /D = k θ1 a/D = 10 kw2 a 3 /D = k θ2 a/D = 100 kw3 a 3 /D = k θ3 a/D = 10 0 0 kw4 a 3 /D = k θ4 a/D = 10 0 0 0 0 20 0–50 0 ANSYS_PS C0 0 01 PS kw a 3 /D = 100 , k θa/D = 10 0 0 ∗ 0.001 ∗ _{20 0–50 0} ANSYS_IR C0 0 01 IR (ED1– ED2–ED3– ED4–ED5) kw1 L 3x /D = 10 4 , k θ1 L x /D = 10 2 kw2 L 3x /D = 10 3 , k θ2 L x /D = 10 4 kw3 L 3x /D = 10 4 , k θ3 L x /D = 10 3 kw4 L 3x /D = 10 3 , k θ4 L x /D = 10 2 kw5 L 3x /D = 10 2 , k θ5 L x /D = 10 3 0.001 20 0–50 0

∗_{For the PS type, the stiffness and damping parameters are only for the supported parts, i.e., edges from a /4 to 3 a /4}

in x direction and from b /4 to 3 b /4 in y direction. Otherwise, the values are zero.

ports(ED1-ED2–ED3–ED4–ED5).Since it is the only casefor an irregularly-shaped plate, it is representedinshortusing“IR” andthecorrespondingdatasetswerenamedwith“IR”.

For all the simulations, datasets are provided separately for the two different numerical methods.DatacollectedfromANSYSFEMarethetransversedisplacementsofthepointsw1and

w2 (seeFig. 1 )ateachcomputedfrequency.Inno-dampingcases,onlytherealpartisprovided.

In thedamping cases, the realpart andimaginary part are separated into two columns.The correspondingcases,aswellastheavailablefrequencyrangeforeachexcelfile,arespecifiedin

Table 1 .Thestiffnessesaregivenastheirnon-dimensionalforms andthedampingcoefficients arerelatedtothestiffnesses.ThenotationscanrefertoFig. 1 ,andDisthebendingstiffnessof theplate,whichisgivenbyD=Eh3_/[12

₍

₁₋

_ν

2

₎

_].

Data obtained from MATLAB WBM are the contribution factors of wave functions, which are the primary solutions of the wave based model. The datasets are saved in the MATLAB workspace(matfiles). Thecorresponding casesofthesematfiles,aswell astheavailable fre-quencyrangearesummarizedinTable 2 .SincethetruncationfactorT,whichareusedtolimit thenumber ofwave functions, is notalways thesame forall cases,its value is alsolisted in

Table 2 foreachcase. Forthecaseofthepentagonal plate(IR),twodatasetsare provided.The dataset“PDD_IR_T_2” wasobtainedwithT=2,and“PDD_IR_T_2” wasobtainedwithT=4.The latter one, with larger value ofT, used more wave functionsin the simulation andprovides moreaccurateresults.EachmatfileinTable 2 isthedatasetforasinglesimulation.Itcontains the information that is necessary for postprocessing and specific to the corresponding simu-lation.There are threevariables. One isthe structurearray “PDD”,which containstwo fields: “PDD.freq” is for the frequency and “PDD.coe” is for the contribution factors.For each frequency,

Table 2

Summary of the simulation cases corresponding to the datasets collected from MATLAB WBM. In these mat files are the wave function contribution factors for each computed frequency.

Mat filename Tyre of BCs Stiffness

Damping

cw / k w = c θ/ k θ Frequency (Hz) T

PDD_Kw100_Ka1000 E–E–E–E kw a 3 /D = 100 , k θa/D =

10 0 0

0 1–10 0 0 2

PDD_Kw10_Ka10 0 0 E–E–E–E kw a 3 /D = 10 , k θa/D = 10 0 0 0 20 0–50 0 2

PDD_Kw10 0 0_Ka10 0 0 E–E–E–E kw a 3 /D = 10 0 0 , k θa/D =

10 0 0 0 20 0–50 0 2

PDD_Kw100_Ka10 E–E–E–E kw a 3 /D = 100 , k θa/D = 10 0 20 0–50 0 2

PDD_Kw100_Ka100 E–E–E–E kw a 3 /D = 100 , k θa/D = 100 0 20 0–50 0 2

PDD_ED0 0 01 ED–ED–ED– ED kw a 3 /D = 100 , k θa/D = 10 0 0 0.001 20 0–50 0 2 PDD_ED001 ED–ED–ED– ED kw a 3 /D = 100 , k θa/D = 10 0 0 0.01 20 0–50 0 2 PDD_E1E2E3E4 E1–E2–E3– E4 kw1 a 3 /D = k θ1 a/D = 10 kw2 a 3 /D = k θ2 a/D = 100 kw3 a 3 /D = k θ3 a/D = 10 0 0 kw4 a 3 /D = k θ4 a/D = 10 , 0 0 0 0 20 0–50 0 2 PDD_PS_T_4 PS kw a 3 /D = 100 , k θa/D = 10 0 0 ∗ 0.001 ∗ _{20 0–50 0 4}

PDD_IR_T_2, PDD_IR_T_4 IR (ED1– ED2–ED3– ED4–ED5) kw1 L 3x /D = 10 4 , k θ1 L x /D = 10 2 kw2 L 3x /D = 10 3 , k θ2 L x /D = 10 4 kw3 L 3x /D = 10 4 , k θ3 L x /D = 10 3 kw4 L 3x /D = 10 3 , k θ4 L x /D = 10 2 kw5 L 3x /D = 10 2 , k θ5 L x /D = 10 3 0.001 20 0–50 0 2 or 4

∗ _{For the PS type, the stiffness and damping parameters are only for the supported parts, i.e., edges from a /4 to 3 a /4}

in x direction and from b /4 to 3 b /4 in y direction. Otherwise, the values are zero.

thecontributionfactorsareexpressedasavectorinthe“PDD.coe”.Theothertwovariablesare theflagfordamping“D_flag” and thetruncationfactorT.Ifdampingisconsidered intheBCs, “D_flag” is equalto 1,otherwise“D_flag” is equalto 0.Thetruncationfactor Tis aparameter thatisusedtolimitthenumberofwavefunctionsbasedonthetruncationrule[1 ,2 ]:

nS₁

π

Lx ≈

nS₂

π

Ly ≥ T

k_b. (1)

where, kb is the plate bending wavenumber and kb= 4



ρ

h

ω

2_{/D, ns}

1

π

/Lx

(

ns1∈N

)

and

ns2

π

/Ly

(

ns2∈N

)

arethelargestwavenumbersoftheconsideredwavefunctions.Inotherwords,

thewavefunctionswiththewavenumberskxs₁=s1

π

/Lx (s1=0,1,2,...,ns₁)andkys₂=s2

π

/Ly

(s2=0,1,2,...,ns₂) are used for the wave based modelling. Then, the total number of wave

functionsisgivenbyns=4

(

ns1+1

)

+4

(

ns2+1

)

,whichalsoindicatesnsunknowncontribution

factorsofthewavefunctions.Therefore,thetruncationfactorTinthedatasetrecordsthewave functionsthatareusedintheparticularsimulation.

FortheWBM,postprocessingisnecessarytogettheplatedisplacement.Hence, postprocess-ing codesareprovided. Withthesecodes andthe matfiles, moreresultsthanthose shownin the associated research articlecan be obtained. There are five MATLAB codes (mfiles) inthe supplementaryfiles:

• “Postprocessing_FRF.m” is to obtain the Frequency Response Function (receptance) at any pointoftheplates;

• “Postprocessing_Field.m” is to get the displacement field for the rectangular plate; • “Postprocessing_Field_IR.m” is to get the displacement field for the pentagonal plate;

• “CoeBaseFuncX.m” and“CoeBaseFuncY.m” arethefunctionsusedtoconstructthewave func-tions in the postprocessing m files, so that the contributionfactors can combinewith the wavefunctionstoobtainthedisplacements.

Thereisalsoanadditionalmatfile(map.mat)containingthecolourinformationforthe con-tourplotsinthe“Postprocessing_Field.m” and “Postprocessing_Field.m”.

2. Experimentaldesign,materialsandmethods

Numericalsimulations were carried out on two thinplates that satisfy theKirchhoff plate theory [3] . The two plates have different shapes and plane sizes, as shown in Fig. 1 , but thesame thicknessh=0.002m. Bothplatesare madeof aluminium.The materialparameters

ρ

=2700kg_/m3_{,E=70× 10}9_N/m2_and

_ν

_{=0.3.Asteadystateexcitationthroughaunitnormal}

forceactingonpointF(0.235m,0.14m)isconsidered.The translationalandrotational displace-mentsoftheplatealongedgeswererestrainedwithelasticityanddamping.Therestraintswere modelled usingtranslational and rotationalsprings and dampers. Translationaland rotational springconstantswere respectivelygivenbykw andk_θ,andthecorresponding damping

coeffi-cientsweregivenbycw andc_θ.Theboundaryconditionsweredeterminedbytheirvaluesand,

fordifferentsimulations, thevalues are showninTables 1 and2 .The numericalmethods for simulationsaretheFEMachievedviaANSYSMechanicalAPDL17.2andtheWBMimplemented inMATLAB.

ForthesimulationsinANSYS,thetwoplateswere modelledbySHELL63,thefour-node lin-earshellelementbasedontheKirchhoff-Lovetheory,withsixdegreesoffreedomateachnode. Theunuseddegreesoffreedom, i.e.thetranslationsinxandydirectionsandtherotationinz

direction,weresettozero.Elementsizewas0.0025m,fortherectangularplate,and0.0024m, forthe pentagonal plate. Alongthe edges ofthe plates,the generic elasticallyrestraints were simulatedusingCOMBIN14, thenodal-based spring-damperelement. EachCOMBIN14 element combinedone fixed node and one node on the plate edge, witha single degree of freedom, eithertranslationinthezdirectionorrotationalong theedge.Sincetheedgepressureis allo-catedtothe elementnodes,the continuousstiffnessanddamping alongthe edgeshould also beallocatedtonodes.Forthelengthlebetweentwonodes,thestiffnessanddampingallocated

tothetwonodesweregivenby

k+=k−= kle

2,c+=c−=

cle

2, (2)

where‘+’denotestheleftsideofthelengthleand‘–’denotestherightside,kandcare

respec-tivelythestiffnessanddampingperunit length.Eachofthenodesalong therestrainededges areconnectedtooneCOMBIN14elementfortranslationandoneCOMBIN14elementforrotation. AfterapplyingtheunitforceatthepointF(0.235m,0.14m),harmonicanalysiswasperformedat theintervalsof1HzwithinthefrequencyrangelistedinTable 1 ,foreach corresponding simu-lationcase.Afterthesimulations,thedisplacementatpointsw1(0.48m,0.31m)andw2(0.525m,

0.14m)intermsofitsrealpartandimaginarypartwerelistedandsavedtothecorresponding spreadsheet(excelfiles).

Forthesimulations inMATLAB, theyfollowed theWBMforflexural vibrationofthinplate with general elastically restrained edges proposed in the associated research article [1] . The methodismodifiedfromtheconventionalWBM[4] forplatewithclassicalboundaryconditions. Afterdefining thegeometry,material properties,excitation force andboundary conditions,the wave basedmodelwasbuiltforeach computingfrequency.The mainpointsin themodelling processarethedefinitionofwave functions,theparticularsolutionoftheforceandthe imple-mentationoftheweightedresidualformulation.Meanwhile,thetruncationruleEq. (1) isused tolimitthedegreesoffreedomofthemodel,whichisequaltothenumberofwavefunctions. BasedonEq. (1) ,morewavefunctionswillbeusedifthetruncationfactorTisincreased.Since thewave based modelconverges towards theexact solutionwith theincrease of thedegrees offreedom[3] ,thetruncationfactorTisalsorelatedtothecomputationaccuracy.Inpractice,

T=2 was the start point andthen T was increasedto check the convergence. For the inte-grationsinthe weightedresidual formulation,Gauss-Legendre quadraturewasapplied,and51 Guass pointswere usedforeachedge.Theset-up modelisa linearalgebraicequation system. Solutionoftheequationsystemisa vectorofthewave functioncontributionfactors.Withthe contributionfactorsstoredinthedatabase,themodellingandsolutionprocesseswererepeated fornextfrequency.After finishingthesolutions ofallthefrequencies forone case,thedataset wassavedtothecorrespondingmatfile,asshowninTable 2 .

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompetingfinancialinterestsorpersonal rela-tionshipswhichhave,orcouldbeperceivedtohave,influencedtheworkreportedinthisarticle.

Acknowledgments

Theauthors gratefullyacknowledge thefinancialsupportfromtheprogramofChina Schol- arship Council (No.201806230718 ). The research shown in thispaper wascarried out atthe PolimiSoundandVibrationLaboratory(PSVL)ofPolitecnicodiMilano.

Supplementarymaterials

Supplementary material associatedwiththisarticle canbe found, inthe onlineversion, at doi:10.1016/j.dib.2020.105883 .

References

[1] L. Liu , R. Corradi , F. Ripamonti , Z. Rao , Wave based method for flexural vibration of thin plate with general elastically restrained edges, J. Sound Vib. (2020) .

[2] E. Deckers, O. Atak, L. Coox, R. D’Amico, H. Devriendt, S. Jonckheere, K. Koo, B. Pluymers, D. Vandepitte, W. Desmet, The wave based method: an overview of 15 years of research, Wave Motion 51 (2014) 550–565, doi: 10.1016/j. wavemoti.2013.12.003 .

[3] A. Nilsson, B. Liu, Vibro-Acoustics, Volume 1, 2nd ed., Science Press, Beijing, 2015, doi: 10.1007/9783662478073 . [4] W. Desmet, A Wave Based Prediction Technique for Coupled Vibro-Acoustic Analysis, KULeuven, division PMA, Ph.D.