2dof Forced vibration of two degrees of freedom system

Twodegreeoffreedomsystems •Equationsofmotionforforcedvibration •Freevibrationanalysisofanundampedsystem Introduction Introduction h dd d dbh • Systemst h atrequiretwoin d epen d entcoor d inatesto d escri b et h eir motionarecalledtwodegreeoffreedomsystems. f Nb masseach ofmotion of s y stemin the s y stem theof typespossible ofnumber masses ofNumber freedom of degrees o f N um b er  y y Introduction Introduction h f d f fd f h • T h erearetwoequations f oratwo d egreeo f  f ree d omsystem,one f oreac h  mass(preciselyoneforeachdegreeoffreedom). • Theyaregenerallyintheformofcoupleddifferentialequations‐thatis, eachequationinvolvesallthecoordinates. • Ifaharmonicsolutionisassumedforeachcoordinate,theequationsof motionleadtoafre q uenc y e q uationthat g ivestwonaturalfre q uenciesof qyq g q thesystem. Introduction Introduction If i it bl iitil it ti th t ib t t f th • If weg i vesu it a bl e i n iti a l exc it a ti on, th esys t emv ib ra t esa t oneo f  th ese naturalfrequencies.Duringfreevibrationatoneofthenatural frequencies,theamplitudesofthetwodegreesoffreedom(coordinates) are related in a specified manner and the configuration is called a normal are  related  in  a  specified  manner  and  the  configuration  is  called  a  normal  mode,principlemode,ornaturalmodeofvibration. • Thusatwodegreeoffreedomsystemhastwonormalmodesofvibration correspondingtotwonaturalfrequencies. • Ifwegiveanarbitraryinitialex citationtothesystem,theresultingfree vibrationwillbeasuperpositionofthetwonormalmodesofvibration. However , ifthes y stemvibratesundertheactionofanexternalharmonic , y force,theresultingforcedharmonicvibrationtak esplaceatthefrequency oftheappliedforce. Introduction Introduction d f h h h f h f f • Asisevi d ent f romt h esystemss h ownint h e f igures,t h econ f igurationo f a systemcanbespecifiedbyasetofindependentcoordinatessuchas length,angleorsomeotherphysicalparameters.Anysuchsetof coordinatesiscalledgeneralizedcoordinates. • Although the equations of motion of a two degree of freedom system are • Although  the  equations  of  motion  of  a  two  degree  of  freedom  system  are  generallycoupledsothateachequationinvolvesallcoordinates,itis alwayspossibletofindaparticularsetofcoordinatessuchthateach i f i i l di Th i f i equat i ono f mot i onconta i nson l yonecoor di nate. Th eequat i onso f mot i on arethenuncoupled andcanbesolvedindependentlyofeachother.Such asetofcoordinates,whichleadstoanuncoupledsystemofequations,is calledprinciplecopordinates. Equationsofmotionforforced vibration d l ddd f fd • Consi d eraviscous l y d ampe d two d egreeo f  f ree d omspring‐masssystem showninthefigure. • Themotionofthesystemiscompletelydescribedbythecoordinatesx1(t) andx 2(t),whichdefinethepositionsofthemassesm1 andm2 atanytimet from the respective equilibrium positions from  the  respective  equilibrium  positions . Equationsofmotionforforced vibration h l f d h d l • T h eexterna l  f orcesF1 an d F2 actont h emassesm1 an d m2,respective l y. Thefreebodydiagramsofthemassesareshowninthefigure. • Thea pp licationofNewton’ssecondlawofmotiontoeachofthemasses pp givestheequationofmotion: Equationsofmotionforforced vibration b h h f l h • Itcan b eseent h att h e f irstequationcontainstermsinvo l vingx2,w h ereas thesecondequationcontainstermsinvolvingx 1.Hence,theyrepresenta systemoftwocoupledsecond‐orderdifferentialequations.Wecan thereforeexpectthatthemotionofthem1 willinfluencethemotionof m 2,andvicaver sa. Equationsofmotionforforced vibration h b f • T h eequationscan b ewritteninmatrix f ormas: where [m] [c] and [k] are mass damping and stiffness matrices where  [m] , [c]  and  [k]  are  mass , damping  and  stiffness  matrices , respectivelyandx(t)andF(t)arecalledthedisplacementandforce vectors,respectively. whicharegivenby: Equationsofmotionforforced vibration b h h [][] d [k] ll h • Itcan b eseent h att h ematrices [ m ] , [ c ] an d  [k] area ll 2x2matricesw h ose elementsaretheknownmasses,dampingcoefficienst,andstiffnessofthe system,respectively. • Further,thesematricescanbeseentobesymmetric,sothat: Freevibrationanalysisofanundampedsystem • For the free vibration analysis of the system shown in the figure we set For  the  free  vibration  analysis  of  the  system  shown  in  the  figure , we  set  F 1(t)=F2(t)=0.Further,ifthedampingisdisregarded,c1=c2=c3=0,andthe equationsofmotionreduceto: [...]... conditions Free vibration analysis of an  undamped system Free vibration analysis of an  undamped system Free vibration analysis of an  undamped system Frequencies of a mass‐spring system Frequencies of a mass spring system Example: Find the natural frequencies and  l d h lf d mode shapes of a spring mass system , which  is constrained to move in the vertical  direction Solution: The equations of motion are given ... phase. In this case, the midpoint of the middle spring remains stationary  for all time. Such a point is called a node.  Frequencies of a mass‐spring system Frequencies of a mass spring system • Using equations the motion (general solution) of the system can be expressed as:  Forced vibration analysis Forced vibration analysis • The equation of motion of a general two degree of freedom system under  Th ti f ti f lt d ff d t d external forces can be written as:... inverse of the impedance matrix is given by: • Therefore, the solutions are: • By substituting these into the below equation, the solutions can be  By substituting these into the below equation the solutions can be obtained • Multi‐degree of freedom systems •Modeling of continuous systems as multidegree of freedom systems •Eigenvalue problem Multidegree of freedom systems Multidegree of freedom systems... equations. The analysis of a multidegree of freedom system on the  ti Th l i f ltid ff d t th other hand, requires the solution of a set of ordinary differential  equations, which is relatively simple. Hence, for simplicity of analysis, continuous systems are often approximated as  analysis continuous systems are often approximated as multidegree of freedom systems • For a system having n degrees of freedom,  there are n associated ... Frequencies of a mass‐spring system Frequencies of a mass spring system • The solution to the above equation gives the natural frequencies: Frequencies of a mass‐spring system Frequencies of a mass spring system • From the amplitude ratios are given by: Frequencies of a mass‐spring system Frequencies of a mass spring system • From • The natural modes are given by Frequencies of a mass‐spring system Frequencies... natural frequencies, each associated with its own mode shape.  Multidegree of freedom systems Multidegree of freedom systems • Different methods can be used to approximate a continuous system as a  Different methods can be used to approximate a continuous system as a multidegree of freedom system.  A simple method involves replacing the  distributed mass or inertia of the system by a finite number of lumped masses or  rigid bodies.  •... Linear coordinates are used to describe the motion of the lumped masses. Such  models are called lumped parameter of lumped mass or discrete mass systems.  • The minimum number of coordinates necessary to describe the motion of the  lumped masses and rigid bodies defines the number of degrees of freedom of the  system.  Naturally, the larger the number of lumped masses used in the model, the  higher the accuracy of the resulting analysis. ... As stated before, most engineering systems are continuous and  t i i t contin o s d have an infinite number of degrees of freedom.  The vibration analysis of continuous systems requires the solution of partial  differential equations, which is quite difficult.  differential equations which is quite difficult • In fact, analytical solutions do not exist for many partial differential  equations. The analysis of a multidegree of freedom system on the ... Frequencies of a mass‐spring system Frequencies of a mass spring system • The natural modes are h l d given by: Frequencies of a mass‐spring system Frequencies of a mass spring system • It can be seen that when the system vibrates in its first mode, the  b h h h b f d h amplitudes of the two masses remain the same. This implies that the  length of the middle spring remains constant. Thus the motions of the  mass 1 and mass 2 are in phase. ... because solution of this equation yields the frequencies of the  characteristic values of the system.  The roots of the above equation are  g given by: y Free vibration analysis of an  undamped system • This shows that it is possible for the system to have a nontrivial harmonic  Thi h th t it i ibl f th t t h t i i lh i solution of the form  when =1 and =2 given by: We shall denote the values of X1 and X2 . Two degree of freedom systems •Equations of motionfor forced vibration •Free vibration analysis of anundamped system Introduction Introduction h dd d dbh • Systemst h atrequire two in d epen d entcoor d inatesto d escri b et h eir motionarecalled two degree of freedom systems. f Nb masseach. Systemst h atrequire two in d epen d entcoor d inatesto d escri b et h eir motionarecalled two degree of freedom systems. f Nb masseach ofmotion of s y stemin the s y stem theof typespossible ofnumber masses ofNumber freedom of degrees o f N um b er  y y Introduction Introduction h. Asisevi d ent f romt h esystemss h ownint h e f igures,t h econ f igurationo f a system canbespecifiedbyaset of independentcoordinatessuchas length,angleorsomeotherphysicalparameters.Anysuchset of coordinatesiscalledgeneralizedcoordinates. • Although the equations of motion of a two degree of freedom system are • Although  the  equations  of  motion  of  a  two  degree  of  freedom  system  are  generallycoupledsothateachequationinvolvesallcoordinates,itis alwayspossibletofindaparticularset of coordinatessuchthateach i