2

which has the following orthogonality characteristics:

{X>T[K]{X}v = 0, {X>t[M]{X}v = 0, for u^ v (3.31a)

In Equation 3.30, p is the eigenvalue (also angular frequency) and X is the eigenvector (also normal modes). u and v in Equation 3.31 represent uth and vth modes associated with frequencies pu and pv, respectively. Note that normal modes (or mode shapes) have significant meaning in displacement response, which in fact, results from a combination of the modes of a system. Let {F}u and {F}vbe modal displacements corresponding to uth and vth modes, respectively, such that

When the modal vectors are collected in a single square matrix of order n, corresponding to n modes, the resulting matrix is called modal matrix, [F]. Using [F] in Equation 3.31 yields Equation 3.33

Note that Equations 3.31 and 3.33 are derived on the basis that [M] = [M]T, [K] = [K]T, and pu ^ pv. The orthogonality condition for the unsymmetrical case as well as for zero and repeating eigenvalues is available [1, pp. 98-106].

3.3.3 Response Analysis and Relevant Parameters 3.3.3.1 Response Analysis and Participation Factor

This section covers both undamped and damped vibration analyses. Mathematical formulations are first established in general form with viscous damping and then simplified to the undamped case. Consider the shear building shown in Figure 3.12 subjected to applied force F(t) and viscous damping force expressed as cxo Using the free-body diagrams, the motion equations may be written as

M1€1 + k1x1 + c1xc1 — k2x2 + k2 x1 — c2x2 + c2x1 = F1(t) (3.34a)

M2€2 + k2x2 — k2x1 + c2xi2 — c2;fc 1 = F2(t) (3.34b)

and in matrix notation as

kl 2

0 0

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