DYNAMIC ANALYSIS USINGMODE SUPERPOSITIONThe Mode Shapes Used To Uncouple TheDynamic Equilibrium Equations Need Not BeThe Exact Free-Vibration Mode Shapes13.1 EQUATIONS TO BE SOLVEDThe dynamic force equilibrium Equation (12.4) can be rewritten in the followingform as a set of N second order differential equations:MuCuKuFf g(t) (t)(t)(t)(t)jjj+ + = = =1J∑(13.1)All possible types of time-dependent loading, including wind, wave and seismic, canbe represented by a sum of “J” space vectors jf, which are not a function of time,and J time functions g(t)j, where J cannot be greater than the number ofdisplacements N.The number of dynamic degrees-of-freedom is equal to the number of lumpedmasses in the system. Many publications advocate the elimination of all masslessdisplacements by static condensation prior to the solution of Equation (13.1). Thestatic condensation method reduces the number of dynamic equilibrium equations tosolve; however, it can significantly increase the density and the bandwidth of thecondensed stiffness matrix. In building type structures, in which each diaphragm

STATIC AND DYNAMIC ANALYSIS2has only three lumped masses, this approach is effective and is automatically used inbuilding analysis programs.For the dynamic solution of arbitrary structural systems, however, the elimination ofthe massless displacement is, in general, not numerically efficient. Therefore, themodern versions of the SAP program do not use static condensation in order toretain the sparseness of the stiffness matrix.13.2 TRANSFORMATION TO MODAL EQUATIONSThe fundamental mathematical method that is used to solve Equation (13.1) is theseparation of variables. This approach assumes the solution can be expressed in thefollowing form:uY(t)(t)= Φ(13.2a)Where Φis an “N by L” matrix containing L spatial vectors which are not a function oftime, and Y(t)is a vector containing L functions of time.From Equation (13.2a) it follows thatuYuY(t)(t) (t)(t)= and = ΦΦ(13.2b) and (13.2c)Prior to solution, we require that the space functions satisfy the following mass andstiffness orthogonality conditions:TT2M= IandK= ΦΦΦΦΩ(13.3)where Iis a diagonal unit matrix and 2Ωis a diagonal matrix which may or may notcontain the free vibration frequencies. It should be noted that the fundamentals ofmathematics place no restrictions on these vectors, other than the orthogonalityproperties. In this book all space function vectors are normalized so that the GeneralizedMassφφnTnM=1.After substitution of Equations (13.2) into Equation (13.1) and the pre-multiplication byTΦ, the following matrix of L equations are produced:

MODE SUPERPOSITION METHOD3IYdYp g(t) (t)(t)jjj+ + = 2=1JΩ∑(13.4)where jjpf= TΦand are defined as the modal participation factors for time functionj. The term njpis associated with the n th mode.