振动练习题-PPT幻灯片

1.AmassmissuspendedonamasslessbeamofbendingstiffnessEIthroughaspringofstiffnessk,asshownFigure1.Determinethenaturalfrequencyofoscillation.Figure1第二讲作业（3道题）1.DeterminetheparametersforanequivalentsystemanalysisofthesystemasshowninFigure3,usingy,thedownwarddisplacementoftheblockfromthesystem’sequilibriumposition,asthegeneralisedcoordinate.Hence,findthefundamentalnaturalfrequencyofthesystem.Note:,Figure23.ArigiduniformrodoflengthLandmassMasshowninFigure3isrestrainedtooscillateinaverticalplane.Alumpedmass

isattachedatthequarterlengthoftherod.Usetheenergymethodtofindthenaturalfrequencyofsystem.Figure3算例：求系统的主振型振动设在光滑水平面上有质点m，分别由三个刚度各为k的弹簧连结于三个固定点，静平衡时各弹簧无变形.试考察系统的主振型振动。解答算例：求系统的固有频率与主振型振幅比试求图示两个物体沿铅垂方向振动的固有频率及主振型的振幅比。其中，滑轮、弹簧与软绳的质量以及阻力都可以不计，且.解答作业：求系统的固有频率与振型求图示系统的固有频率与振型.解答作业：求系统的固有频率质量可以不计的刚杆,可绕点转动,中点支以弹簧,求振动系统的固有频率.解答作业：能量法求系统自由振动频率均质杆AB，质量为M，长为3l，端刚性连接一质量为

的m物体，其几何大小可略去不计.AB杆在O处用铰链连接，并用弹簧刚度系数均为K的两弹簧加以约束，如图所示.试求系统自由振动的频率解答作业：跳台问题一物体放在水平台面上（水平台面不计质量），当水平台面沿铅垂方向作频率为5Hz的简谐振动时，要使得物体不跳离水平台面，对水平台面的振幅有何限制？解答这里，作业：求系统的等效刚度求出下图所示系统的等效刚度的表达式。解答受力分析如下图Homework

Task1:Determinetheresponseofablockofmassconnectedthroughaspringofstiffnessktoabasewhenthebaseissubjecttoarectangularvelocitypulseshownasrightfigure.Task2:Findthecriticaldampingforthecompactingmachinesothatthemachinecanavoidoscillatingvibrationduringcompactingprogress,anddeterminetheresponseoftheplatformwhenasuddenpressureisapplied.Thevaluesform,kandtheforcecausedbypressureis

m=5kg;ωn=20rad/s;F0=1000;v(t)v00tt0F(t)PistonCylinderPlatformx(t)mk/2k/2cMaterialbeingcompactedArbitraryForceDuhamelintegralNoteaboveequationdoesn’tconsidertheeffectoftheinitialconditions.Non-periodicallyExcitedVibrationg(t)-ImpulseresponsefunctionxcmktHomework

Task1:Determinetheresponseofablockofmassconnectedthroughaspringofstiffnessktoabasewhenthebaseissubjecttoarectangularvelocitypulseshownasrightfigure.v(t)v00tt0Solution:Method1:Thederivativeoftheunitstepfunctionistheunitimpulsefunction.Homework

v(t)v00tt0Solution:Method2:Thederivativeoftheunitstepfunctionistheunitimpulsefunction.v(t)v00tv(t)-v00tHomework

Task2:Findthecriticaldampingforthecompactingmachinesothatthemachinecanavoidoscillatingvibrationduringcompactingprogress,anddeterminetheresponseoftheplatformwhenasuddenpressureisapplied.Thevaluesform,kandtheforcecausedbypressureis

m=5kg;ωn=20rad/s;F0=1000;F(t)PistonCylinderPlatformx(t)mk/2k/2cMaterialbeingcompactedSolution:Whenthesystemiscriticallydamped,ξ=1

WhenunitimpulseisappliedHomework

WhenunitimpulseisappliedSolutioncontinued:F(t)PistonCylinderPlatformx(t)mk/2k/2cExample–CompactingMachineMaterialbeingcompactedAcompactingmachine,modeledasaSDOFsystem,isshowninleftfigure.Theforceactingonthemassm(mincludesthemassesofthepiston,theplatformandthematerialbeingcompacted)duetoasuddenapplicationofthepressurecanbeidealizedasastepforce,shownasfigurebelow.Determinetheresponseofthesystem.

F(t)F00tExample–CompactingMachineGiven:Compactingmachinesubjectedtoastepforce.Find:Responseofthesystem

Solution:SincethecompactingmachineismodeledasaSDOFsystem,theproblemistofindtheresponseofadampedSDOFsystemsubjectedtoastepforce.BynotingF(t)=F0.ThentheresponsecanbecalculatedasExample–CompactingMachinem=5kg;ωn=20rad/s;ξ

=0.05;F0=1000;F0/kProblem:whystepforcecannotproducesteady-statevibrationfordampedsystem?Example–CompactingMachineIfdampratioiszero,thenm=5kg;ωn=20rad/s;ξ

=0.0;F0=1000;*Iftheloadisinstantaneouslyappliedtoanundampedsystem,amaximumdisplacementoftwicethestaticdisplacementisobtained.F0/kExample–CompactingMachineIfthecompactingmachineissubjectedtoconstantforceonlyaftertimet0,findthesystemresponse.F(t)F00tt0Solution:Sincethestepforceistime-delayed,substitutet-t0fortinthesolutionofpreviousexample.Ifdampratioiszero,thenExample–CompactingMachineIfthecompactingmachineissubjectedtoconstantforceonlyduringthetime0≤t≤t0,findthesystemresponse.F(t)F00tt0Solution:ThisforcefunctioncanbeconsideredasthesumofastepfunctionofmagnitudeF0beginningatt=0andasecondstepfunctionofmagnitude–F0beginningattimet=t0.ThentheresponsecanbeobtainedasF(t)F00tF(t)-F00tt00≤t≤t0t>t0Example–CompactingMachinem=5kg;ωn=20rad/s;ξ

=0.0;F0=1000;t0=1secNumericalMethodfunctionf=Rect_Fun(t,y);wn=20;m=5;k=wn^2*m;mu=0.05;c=2*mu*sqrt(k*m);F0=1000;if(t>1),F0=0;endf=[-c/m-k/m;10]*y+[F0/m;0];functionExample_Impulse(),t0=0;te=5;n=5000;x0=0.00;v0=0;y0=[v0;x0];y1=RK_2('Rect_Fun',t0,te,y0,n);t=linspace(t0,te,n+1);plot(t,y1(2,:));xlabel('Time/Sec‘)Example–CompactingMachinewithCamF(t)x(t)mk/2k/2cCamPlatformMaterialbeingcompactedFollowerMotionofCamF(t)0t1δFF(t)=δF.tExample–CompactingMachinewithCamFind:ResponseofthesystemSolution:theresponsecanbeobtainedbyevaluatingDuhamelintegralwithF(t)=δFt

m=5kg;ωn=20rad/s;ξ

=0.01;δF0=1000;TransientMotionDuetoBaseExcitationxcmktApplyingNewton’ssecondlawgivesyDenotez=x-y,thenRewrittenasHomework

Task1:Determinetheresponseofablockofmassconnectedthroughaspringofstiffnessktoabasewhenthebaseissubjecttoarectangularvelocitypulseshownasrightfigure.Task2:Findthecriticaldampingforthecompactingmachinesothatthemachinecanavoidoscillatingvibrationduringcompactingprogress,anddeterminetheresponseoftheplatformwhenasuddenpressureisapplied.Thevaluesform,kandtheforcecausedbypressureis

m=5kg;ωn=20rad/s;F0=1000;v(t)v00tt0F(t)PistonCylinderPlatformx(t)mk/2k/2cMaterialbeingcompactedNon-periodicallyExcitedVibrationApplyingNewton’ssecondlawgivesItcanberewrittenaswherexcmkF(t)Non-periodicexcitingforcehasamagnitudethatvarieswithtime;itactsforspecifiedperiodoftimeandthenstops.Theresultedsystemresponsewillbetransientassteadystateresponsewillnotbeproduced.ImpulsiveForceΔttOF(t)FFΔt=1Impulsiveforceisthesimplestnon-periodicforce,whichhasalargemagnitudeF

and

actsforaveryshortperiodoftimeΔt.IngeneralAunitimpulseisdefinedasInorderforFdttohaveafinitevalue,Ftendstoinfinity(sincedttendstozero)Ifthemassisatrestbeforeanunitimpulseisapplied,x=0;v=0fort<0,fromtheimpulse-momentumrelation,itisobtainedThustheinitialconditionsaregivenbyImpulsiveForceImpulseresponsefunctionm=10kg;ωn=10πrad/s;ξ

=0.05Ifthemagnitudeoftheimpulseisinsteadofunity,theinitialvelocityisthenThenthesystemresponseisImpulsiveForceImpulsiveForce–NumericalSimulationm=10kg;ωn=10πrad/s;ξ

=0.05;F=500N;dt=0.002secfunctionExample_Impulse(),t0=0;te=2;n=1000;x0=0.00;v0=0;y0=[v0;x0];y1=RK_2('Impulse_Fun',t0,te,y0,n);t=linspace(t0,te,n+1);plot(t,y1(2,:));xlabel('Time/Sec‘)functionf=Impulse_Fun(t,y);wn=10*pi;m=10;k=wn^2*m;mu=0.05;c=2*mu*sqrt(k*m);F0=500;if(t>0.002),F0=0;endf=[-c/m-k/m;10]*y+[F0/m;0];ΔttOF(t)FFΔt=1τIftheimpulseisappliedatanarbitrarytimet=τ,thevelocityofthemasswillchangeatt=τ.Assumingx=0andv=0untiltheimpulseisapplied,thenthedisplacementxatanysubsequenttimet,causedbythechangeinvelocityattimeτ,isgivenasImpulsiveForceImpulsiveForce-ExampleIf,afteranimpact,asecondimpactisappliedafterintervalτ,thentheimpactforcecanbeexpressedasδ(t)isdeltafunction,havingthefollowingproperties:Deltafunctionisalsocalledassamplingfunctionτissamplingperiod.ImpulsiveForce-Examplem=5kg,k=2000N/m,c=10N.s/mand.determinethesystemresponse.Answer:ThesystemresponsecausedbythefirstimpactisThesystemresponsecausedbythesecondimpactisThetotalresponseisRed–oneimpactcaseBlue–twoimpactscaseArbitraryForceTheforcecanberegardedtobemadeupofaseriesofimpulsesofvaryingmagnitude.Attimeτ,theforceF(τ)actsonthesystemforashortperiodoftimeΔτ,theimpulseactingatt=τisgivenbyF(τ)Δτ.Atanytimet,theelapsedtimesincetheimpulseist-

τ,sotheresponseofthesystemattduetothisimpulsealoneisgivenas

Thetotalresponseattimetcanbefoundbysummingalltheresponseduetotheimpulsesactingatalltimesτ.LettingΔτ→0,andreplacingthesummationbyintegration,oneobtainsArbitraryForceDuhamelintegralNoteaboveequationdoesn’tconsidertheeffectoftheinitialconditions.Example–CompactingMachinewithCamF(t)x(t)mk/2k/2cCamPlatformMaterialbeingcompactedFollowerMotionofCamF(t)0t1δFF(t)=δFtFind:ResponseofthesystemSolution:theresponsecanbeobtainedbyevaluatingDuhamelintegralwithF(t)=δFt

m=5kg;ωn=20rad/s;ξ

=0.01;δF0=1000;Example–CompactingMachinewithCamF(t)PistonCylinderPlatformx(t)mk/2k/2cExample–CompactingMachineMaterialbeingcompactedAcompactingmachine,modeledasaSDOFsystem,isshowninleftfigure.Theforceactingonthemassm(mincludesthemassesofthepiston,theplatformandthematerialbeingcompacted)duetoasuddenapplicationofthepressurecanbeidealizedasastepforce,shownasfigurebelow.Determinetheresponseofthesystem.

F(t)F00tExample–CompactingMachineGiven:Compactingmachinesubjectedtoastepforce.Find:Responseofthesystem

Solution:SincethecompactingmachineismodeledasaSDOFsystem,theproblemistofindtheresponseofadampedSDOFsystemsubjectedtoastepforce.BynotingF(t)=F0.ThentheresponsecanbecalculatedasExample–CompactingMachinem=5kg;ωn=20rad/s;ξ

=0.05;F0=1000;F0/kProblem:whystepforcecannotproducesteady-statevibrationfordampedsystem?Example–CompactingMachineIfdampratioiszero,thenm=5kg;ωn=20rad/s;ξ

=0.0;F0=1000;*Iftheloadisinstantaneouslyappliedtoanundampedsystem,amaximumdisplacementoftwicethestaticdisplacementisobtained.F0/kExample–CompactingMachineIfthecompactingmachineissubjectedtoconstantforceonlyaftertimet0,findthesystemresponse.F(t)F00tt0Solution:Sincethestepforceistime-delayed,substitutet-t0fortinthesolutionofpreviousexample.Ifdampratioiszero,thenExample–CompactingMachineIfthecompactingmachineissubjectedtoconstantforceonlyduringthetime0≤t≤t0,findthesystemresponse.F(t)F00tt0Solution:ThisforcefunctioncanbeconsideredasthesumofastepfunctionofmagnitudeF0beginningatt=0andasecondstepfunctionofmagnitude–F0