Wuhan University of TechnologyChapter19Analysis of dynamicresponse19-1
19-1 Wuhan University of Technology Chapter 19 Analysis of dynamic response
Wuhan University of TechnologyLContents19.1 Normal coordinates19.2 Uncoupled flexural equations of motion:undamped case19.3 Uncoupled flexural equations of motion:damped case19.4 Uncoupled axial equations of motion:undamped case19-2
19-2 Wuhan University of Technology 19.1 Normal coordinates 19.2 Uncoupled flexural equations of motion: undamped case 19.3 Uncoupled flexural equations of motion: damped case 19.4 Uncoupled axial equations of motion: undamped case Contents
Wuhan University of Technology19.1 Normal coordinatesThemodesuperpositionanalysisofadistributedparametersystemisentirelyequivalenttothatofadiscretecoordinatesystemoncethemodeshapesandfrequencieshavebeendetermined,becauseinbothcasestheamplitudesofthemodalresponsecomponentsareusedasgeneralizedcoordinatesindefiningtheresponseofthestructure.Inprincipleaninfinitenumberofthesecoordinatesareavailableforadistributedparametersystemsinceithasaninfinitenumberof modesofvibrationbutinpracticeonlythosemodalcomponentsneedbeconsideredwhichprovidesignificantcontributionstotheresponse.Thustheproblemisactuallyconvertedintoadiscreteparameterforminwhichonlyalimitednumberofmodal(normal)coordinatesisusedtodescribetheresponse.19-3
19-3 Wuhan University of Technology 19.1 Normal coordinates The modesuperposition analysis of a distributedparameter system is entirely equivalent to that of a discretecoordinate system once the mode shapes and frequencies have been determined, because in both cases the amplitudes of the modalresponse components are used as generalized coordinates in defining the response of the structure. In principle an infinite number of these coordinates are available for a distributedparameter system since it has an infinite number of modes of vibration, but in practice only those modal components need be considered which provide significant contributions to the response. Thus the problem is actually converted into a discreteparameter form in which only a limited number of modal (normal) coordinates is used to describe the response
WuhanUniversityof Technology福19.1 Normal coordinatesThe essential operation of the modesuperposition analysis is the transformationfromthegeometricdisplacementcoordinatestothemodalamplitudeornormalcoordinates.Foraonedimensionalsystem,thistransformationisexpressedas8v(a,t) =d;(r) Y;(t)i=1whichissimplyastatementthatanyphysicallypermissibledisplacementpatterncanbemadeup by superposingappropriate amplitudesof thevibrationmodeshapesforthestructure.19-4
19-4 Wuhan University of Technology 19.1 Normal coordinates The essential operation of the modesuperposition analysis is the transformation from the geometric displacement coordinates to the modalamplitude or normal coordinates. For a onedimensional system, this transformation is expressed as which is simply a statement that any physically permissible displacement pattern can be made up by superposing appropriate amplitudes of the vibration mode shapes for the structure
Wuhan UniversityofTechnology19.1 Normal coordinatesU(x, t)=中, (x)Y, (t)+中,(x) Y,(t)+中3(x)Y,(t)9+etc.FIGURE19-1Arbitrarybeamdisplacementsrepresentedbynormalcoordinates.19-5
19-5 Wuhan University of Technology 19.1 Normal coordinates FIGURE 19-1 Arbitrary beam displacements represented by normal coordinates
Wuhan University of Technology19.1 Normal coordinatesThemodal componentscontained inanygivenshape,suchasthetopcurve ofFig.191,canbeevaluatedbyapplyingtheorthogonalityconditionson(a) m(a) v(r,t) da =Y;(t)di(r)m(a)on(c)da=1pn(r) m(r) dr= Yn(t)where only one term remains of the infiniteseriesonthe far right hand sidebyvirtue of theorthogonality condition.Hencethe expression canbesolved directlyfortheoneremainingamplitudetermJb on(a) m(a) v(r,t) deYn(t) :Je on(r)2 m(r) dr19-6
19-6 Wuhan University of Technology 19.1 Normal coordinates where only one term remains of the infinite series on the far right hand side by virtue of the orthogonality condition. Hence the expression can be solved directly for the one remaining amplitude term The modal components contained in any given shape, such as the top curve of Fig. 191, can be evaluated by applying the orthogonality conditions
WuhanUniversityofTechnology19.2 Uncoupled flexural equations of motionundampedcaseThe two orthogonality conditions [Eqs. (1834) and (1839) or (1840)] provide themeansfordecouplingtheequationsofmotionforthedistributedparametersysteminthesamewaythatdecouplingwasaccomplishedforthediscreteparameter system.Introducing Eq.(191) into the equation of motion [Eq.(177)]02a2v(r,t)a2v(r,t)EI(Cmrp(r,t)0r20r2Ot2m(a) 0(a)()2[E(a)Y;(t) = p(r,t)dar219-7
19-7 Wuhan University of Technology 19.2 Uncoupled flexural equations of motion: undamped case The two orthogonality conditions [Eqs. (1834) and (1839) or (1840)] provide the means for decoupling the equations of motion for the distributedparameter system in the same way that decoupling was accomplished for the discrete- parameter system. Introducing Eq. (191) into the equation of motion [Eq. (177)]
WuhanUniversity of Technology19.2 Uncoupled flexural eguations of motionundamped cased2dbi(r)Y(t)EI(r)m(r)o;(r)on(r)da+Yi(t)On(r0drdr210i=1--Lon(r)p(r,t)drd2d?on(r)Y,(t)m(r) on(r)2 da + Yn(t)EI(a)daron(a)dr2da2Lon() p(r,t) dr1on(a)2m(r) dadrda219-8
19-8 Wuhan University of Technology 19.2 Uncoupled flexural equations of motion: undamped case
Wuhan Universityof Technology19.2 Uncoupled flexural eguations of motionundampedcaseRecognizingthattheintegral ontherighthand sideofthisequation isthegeneralized mass of the nth mode [Eqs. (814)]Mn:Φn(r)2 m(r) daMn Yn(t) +w, Mn Yn(t) = Pn(t)TPn(t) =n(a) p(r,t) drAnequationofthetypeofEq.(1910)canbeestablishedforeachvibrationmode of the structure, using Eqs. (199) and (1911) to evaluate its generalizedmassandloading,respectively.Itshouldbenotedthattheseexpressionsarethedistributedparameterequivalentsofthematrixexpressionspreviouslyderivedforthediscreteparametersystems.19-9
19-9 Wuhan University of Technology 19.2 Uncoupled flexural equations of motion: undamped case Recognizing that the integral on the right hand side of this equation is the generalized mass of the nth mode [Eqs. (814)] An equation of the type of Eq. (1910) can be established for each vibration mode of the structure, using Eqs. (199) and (1911) to evaluate its generalized mass and loading, respectively. It should be noted that these expressions are the distributedparameter equivalents of the matrix expressions previously derived for the discreteparameter systems
Wuhan Universityof Technology19.3 Uncoupled flexural eguations of motiondamped caseTodeterminetheeffectof thenormalcoordinatetransformation[Eq.(191)]onthe damped equation of motion, substitute Eq. (191) into Eq. (1713) to getXo0d22 ) () (+ ( (+[n ] Y(t)dr2i=1i=12[E(n) 0]Y;(t) = p(a,t)drMultiplying by n (α), integrating, and applying the two orthogonalityrelationships together with the definitions of generalized mass and generalizedloadingleadstodo;(a)aiEI(r)Mn Yn(t) +Yi(t)[n(a)/e(m) 0() +dadr2+wzMnYn(t)=Pn(t)19-10
19-10 Wuhan University of Technology 19.3 Uncoupled flexural equations of motion: damped case To determine the effect of the normalcoordinate transformation [Eq. (191)] on the damped equation of motion, substitute Eq. (191) into Eq. (1713) to get Multiplying by , integrating, and applying the two orthogonality relationships together with the definitions of generalized mass and generalized loading leads to