The number of dynamic degrees of freedom is equal to

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The number of dynamic degrees of freedom is equal to the number of lumped masses in the system. Some publications advocate the elimination of all mass- less displacements by static condensation prior to the solution of Eq. ( 5.5 ). The static 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, the elimination of the massless displacement is, in general, not numerically efficient. Therefore, the method of static condensation should be avoided and other methods can be used. The fundamental mathematical method which is used to solve Eq. ( 5.5 ) is separation of variables. This approach assumes that the solution can be expressed in the following form: u ð t Þ ¼ F Y ð t Þ (5 : 6a) where F is an ‘‘ N by L ’’ matrix containing L spatial vectors which are not a function of time and Y( t ) is a vector containing L functions of time. From Eq. ( 5.6a ) it follows that _ u ð t Þ ¼ F _ Y ð t Þ (5 : 6b) 5.3 Solution of the Dynamic Equilibrium Equations 307
and u ð t Þ ¼ F Y ð t Þ (5 : 6c) Prior to solution, it is necessary that the space functions satisfy the following mass and stiffness orthogonality conditions: F T 00 M F ¼ 1 and F T 00 K F ¼ O 2 (5 : 7) T 00 =Transpose where I is a diagonal unit matrix and O 2 is a diagonal matrix which may or may not contain the free vibration frequencies. It should be noted that the funda- mentals of mathematics place no restrictions on these vectors, other than the orthogonality properties. Here, all space function vectors are normalized so that the generalized mass f T 00 n M f n ¼ 1. After substitution of Eq. ( 5.6 ) into Eq. ( 5.5 ) and the premultiplication by F T 00 the following matrix of L equations are produced: I Y ð t Þ þ d _ Y ð t Þ þ O 2 ¼ X J j ¼ 1 P j g ð t Þ j (5 : 8) where P j ¼ F T 00 f ± j and are defined as the modal participation factors for time equation j . The term P nj is associated with the n th mode. For all structures the ‘‘square’’ matrix [ d ] is not diagonal; however, in order to uncouple the modal equations it is necessary to assume that there is no coupling between the modes. Hence, it is assumed to be diagonal with the modal damping forms defined by d nn ¼ 2 x n o n (5 : 9) where z n is defined as the ratio of the damping in mode n to the critical damping of the mode. A typical uncoupled modal equation, for linear structural systems, is of the following form: y ð t Þ n þ 2 x n o n _ y ð t Þ n þ o 2 n y ð t Þ n ¼ X J j ¼ 1 P nj g ð t Þ j (5 : 10) For three-dimensional seismic motion, this equation can be written as y ð t Þ n þ 2 x n o n _ y ð t Þ n þ o 2 n y ð t Þ n ¼ P nx u ð t Þ gx þ P ny u ð t Þ gy þ P nz u ð t Þ gz (5 : 11) where the three mode participation factors are defined by P ni ¼ f n M i in which i is equal to x , y or z .
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