where f i is the ith vibration mode shape and q,- is the ith normal coordinate. Using the orthogonality conditions of mode shapes, Equation 22.30 can be expressed as (Balendra 1993)

where m*, c*, k*, and p* are the generalized mass, damping, stiffness, and force in the ith mode of vibration. The generalized mass and force are determined from r h m = I m(z)f2(z) dz (22.33)

Equation 22.32 consists of a set of uncoupled equations, each representing a single degree of freedom system. Using the random vibration theory (Robson 1963), the power spectrum of the response in each normal coordinate is given by

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