Wuhan University of TechnologyChapter12Analysis of dynamic response usingsuperposition12-1
12-1 Wuhan University of Technology Chapter 12 Analysis of dynamic response using superposition
Wuhan University of TechnologyContents12.1 Normal coordinates12.2 Uncoupled equations of motion-undamped12.3 Uncoupled equations of motion-viscous damping12.4 Response analysis by mode disp. Superposition12.5 Construction of proportional viscous damping matrices12.6 Response analysis using coupled equations of motion12.7 Relationship between time- and frequency - domaintransferfunctions12-2
12-2 Wuhan University of Technology 12.1 Normal coordinates 12.2 Uncoupled equations of motion-undamped 12.3 Uncoupled equations of motion-viscous damping 12.4 Response analysis by mode disp. Superposition 12.5 Construction of proportional viscous damping matrices 12.6 Response analysis using coupled equations of motion 12.7 Relationship between time- and frequency - domain transfer functions Contents
Wuhan University of Technology12.1 Normal coordinatesIntheprecedingdiscussionofanarbitraryNDOFlinearsystem,thedisplacedposition was defined by the N components in the vector v. However, for thepurpose of dynamicresponse analysis,it is often advantageousto expressthisposition in terms of the freevibration mode shapesTheseshapesconstituteN independentdisplacementpatterns,theamplitudesof which mayserveas generalized coordinates toexpress anyset ofdisplacements.Themodeshapesthusservethe samepurposeasthetrigonometricfunctionsin a Fourier series, and they are used for the same reasons; because: (1) theypossess orthogonality properties and (2) they are efficient in the sense that theyusually can describe all N displacements with sufficient accuracy employing onlyafewshapes.12-3
12-3 Wuhan University of Technology 12.1 Normal coordinates In the preceding discussion of an arbitrary NDOF linear system, the displaced position was defined by the N components in the vector v. However, for the purpose of dynamicresponse analysis, it is often advantageous to express this position in terms of the freevibration mode shapes. These shapes constitute N independent displacement patterns, the amplitudes of which may serve as generalized coordinates to express any set of displacements. The mode shapes thus serve the same purpose as the trigonometric functions in a Fourier series, and they are used for the same reasons; because: (1) they possess orthogonality properties and (2) they are efficient in the sense that they usually can describe all N displacements with sufficient accuracy employing only a few shapes
Wuhan University of Technology12.1 Normal coordinatesD1V1300U2U21-U23V22-++U31U3U33-V32nV=DYU,=O,YU2=02Y2U,=O,Y3FIGURE12-1Representingdeflectionsassumofmodalcomponents12-4
12-4 Wuhan University of Technology 12.1 Normal coordinates FIGURE 12-1 Representing deflections as sum of modal components
Wuhan University of Technology12.1 Normal coordinatesVn=OnYnNV=0iYi+2Y2+...+Y=0nYnn=1V=ΦY$Tmv=oTmoiYi+Tm2Y2+...+OTmoY12-5
12-5 Wuhan University of Technology 12.1 Normal coordinates
Wuhan University of Technology12.1 Normal coordinatesTmpm = 0m+n, mv=OTmon Yn&,mvn= 1,2, ...,NYn=OTmonTm(t)Yn(t) =STmon12-6
12-6 Wuhan University of Technology 12.1 Normal coordinates
WuhanUniversityof Technology-12.2 Uncoupled equations of motion-undampedThe orthogonality properties of the normal modes will now be used to simplifytheequationsofmotionoftheMDOFsystem.Ingeneralformtheseequationsare given by Eq. (913) [or its equivalent Eq. (919) if axial forces are present);fortheundampedsystemtheybecomemv(t) +kv(t) =p(t)V-UYm 重Y(t) + k 重Y(t) =p(t)12-7
12-7 Wuhan University of Technology 12.2 Uncoupled equations of motion - undamped The orthogonality properties of the normal modes will now be used to simplify the equations of motion of the MDOF system. In general form these equations are given by Eq. (913) [or its equivalent Eq. (919) if axial forces are present]; for the undamped system they become
Wuhan University of Technology?12.2 Uncoupled equations of motion- undampedT m重Y(t) +T k重Y(t) =T p(t)OTmon Yn(t) +oTkon Yn(t) =ΦTp(t)Nownewsymbolswillbedefinedasfollows:Mn =ΦTmonKn=ΦTkonPn(t) =p(t)12-8
12-8 Wuhan University of Technology 12.2 Uncoupled equations of motion - undamped Now new symbols will be defined as follows:
Wuhan University of Technology12.2 Uncoupled equations of motion- undampedMn Yn(t) + Kn Yn(t) = Pn(t)kon=wempnTkon=wnoTmonKn=wrMn12-9
12-9 Wuhan University of Technology 12.2 Uncoupled equations of motion - undamped
Wuhan University of Technology12.2 Uncoupled eguations of motion-undampedTheproceduredescribedabovecanbeusedtoobtainanindependentSDOFequation for each mode of vibration of the undamped structure.Thustheuseofthenormal coordinatesservestotransformtheequationsofmotion from a set of N simultaneous differential equations, which are coupledbytheoffdiagonaltermsinthemassandstiffnessmatrices,toasetofNindependentnormalcoordinateeguations.Thedynamicresponsethereforecanbeobtainedbysolvingseparatelyforthe response of eachnormal (modal)coordinateand then superposing these byEg.(123)toobtaintheresponseintheoriginalgeometriccoordinates.This procedure is called the modesuperposition method, or more preciselythemodedisplacementsuperpositionmethod.12-10
12-10 Wuhan University of Technology 12.2 Uncoupled equations of motion - undamped The procedure described above can be used to obtain an independent SDOF equation for each mode of vibration of the undamped structure. Thus the use of the normal coordinates serves to transform the equations of motion from a set of N simultaneous differential equations, which are coupled by the offdiagonal terms in the mass and stiffness matrices, to a set of N independent normalcoordinate equations. The dynamic response therefore can be obtained by solving separately for the response of each normal (modal) coordinate and then superposing these by Eq. (123) to obtain the response in the original geometric coordinates. This procedure is called the modesuperposition method, or more precisely the mode displacement superposition method