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