The eigen vector of any mode can be determined from the singular matrix in Equation 3.59 after zero determinant [1, pp. 187-194].

3.4.4 Response Analysis Formulation

The dynamic deflection of a member at any point x and time t can be represented as the summation of modal components as y (x, t )=£ Yi (x) qi (t )

where Y;(x) is the shape function of the member, q,(t) are generalized coordinates (similar to xj in the lumped-mass model), and i = 1,2,..., n, the number of normal modes.

The uncoupled motion equation for the kth mode is

Jo mYkdx

Then, Equation 3.62 becomes

Mkpror ^ p2 pnm MM

where NM is the total number of members of a structure and p v^NM p qk = TTZäAk(t) or qk - ,^-M " Ak(t) (3.65)

Note that Fk is a general expression, which can represent dynamic force or ground excitation. Ak(t) can be simplified not to include the effect of damping on frequency, then

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