# Interdisciplinary Applied Mathematics

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N x N matrix whose columns are the mode shape vectors. The generalized inertia matrix MG and the generalized stiffness matrix KG are defined as

MG = ST MS and KG = ST KS,

where both Mg and KG are diagonal. Substituting x = Sq in equation (17.17) and premultiplying by ST, we get

MGq + KGq = ST F(Sq, t).

Since Mg and KG are diagonal matrices, the coupling of the q’s is through the nonlinear force term. In practice, only a few modes are sufficient to describe the deformation. So the N equations are reduced to a smaller number of m equations, where m is the number of mode shapes considered.

If damping properties are to be included, the damping force is added as an additional term on the right-hand side of the equation, or a new set of geometric basis functions is generated by including the space external to the structure where the damping is present. If an additional force term is added, it contains a dependence on velocity. Another approach to account for damping is to assume Rayleigh damping, in which case the linear modes obtained from M and K would be sufficient to capture the full behavior. However, in general, to include the effect of damping, one may have to solve the quadratic eigenvalue problem (A2M + AD + K)s = 0 for the desired modal matrix S.

If the structure undergoes large-amplitude deformation, then in an ideal case, the stiffness matrix needs to be recomputed as the amplitude changes. An alternative approach is to retain the original stiffness matrix and add an extra force term to account for the large-amplitude effects, such as stress stiffening of the structure. It is convenient to express the right-hand side in terms of modal coordinates instead of the meshed coordinates. The energy method (Senturia, 1998b) can be used for this purpose, and this procedure is summarized below:

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