This preview shows page 340 - 342 out of 739 pages.
The number of dynamic degrees of freedom is equal to the number of lumpedmasses in the system. Some publications advocate the elimination of all mass-less displacements by static condensation prior to the solution of Eq. (5.5). Thestatic condensation method reduces the number of dynamic equilibrium equa-tions to solve; however, it can significantly increase the density and the band-width of the condensed stiffness matrix.For the dynamic solution of arbitrary structural systems, however, theelimination of the massless displacement is, in general, not numerically efficient.Therefore, the method of static condensation should be avoided and othermethods can be used.The fundamental mathematical method which is used to solve Eq. (5.5) isseparation of variables. This approach assumes that the solution can beexpressed in the following form:uðtÞ ¼F€YðtÞ(5:6a)whereFis an ‘‘NbyL’’ matrix containingLspatial vectors which are not afunction of time and Y(t) is a vector containingLfunctions of time.From Eq. (5.6a) it follows that_uðtÞ ¼F_YðtÞ(5:6b)5.3Solution of the Dynamic Equilibrium Equations307
and€uðtÞ ¼F€YðtÞ(5:6c)Prior to solution, it is necessary that the space functions satisfy the followingmass and stiffness orthogonality conditions:FT00MF¼1andFT00KF¼O2(5:7)T00=TransposewhereIis a diagonal unit matrix andO2is a diagonal matrix which may or maynot contain the free vibration frequencies. It should be noted that the funda-mentals of mathematics place no restrictions on these vectors, other than theorthogonality properties. Here, all space function vectors are normalized sothat thegeneralized massfT00nMfn¼1.After substitution of Eq. (5.6) into Eq. (5.5) and the premultiplication byFT00the following matrix ofLequations are produced:I€YðtÞ þd_YðtÞ þO2¼XJj¼1PjgðtÞj(5:8)wherePj¼FT00f±jand are defined as the modal participation factors for timeequationj. The termPnjis associated with thenth mode.For all structures the ‘‘square’’ matrix [d] is not diagonal; however, in orderto uncouple the modal equations it is necessary to assume that there is nocoupling between the modes. Hence, it is assumed to be diagonal with themodal damping forms defined bydnn¼2xnon(5:9)whereznis defined as the ratio of the damping in modento the critical dampingof the mode.A typical uncoupled modal equation, for linear structural systems, is of thefollowing form:€yðtÞnþ2xnon_yðtÞnþo2nyðtÞn¼XJj¼1PnjgðtÞj(5:10)For three-dimensional seismic motion, this equation can be written as€yðtÞnþ2xnon_yðtÞnþo2nyðtÞn¼Pnx€uðtÞgxþPny€uðtÞgyþPnz€uðtÞgz(5:11)where the threemode participation factorsare defined byPni¼fnMiin whichiis equal tox,yorz.