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