free and forced vibrations of elastically connected structures

A general theory for the free and forced responses of 𝑛 elastically connected parallel structures is developed. It is shown that if the stiffness operator for an individual structure is self-adjoint with respect to an inner product defined for 𝐶𝑘[0,1], then the stiffness operator for the set of elastically connected structures is self-adjoint with respect to an inner product defined on 𝑈=𝑅𝑛×𝐶𝑘[0,1]. This leads to the definition of energy inner products defined on 𝑈. When a normal mode solution is used to develop the free response, it is shown that the natural frequencies are the square roots of the eigenvalues of an operator that is self-adjoint with respect to the energy inner product. The completeness of the eigenvectors in 𝑊 is used to develop a forced response. Special cases are considered. When the individual stiffness operators are proportional, the problem for the natural frequencies and mode shapes reduces to a matrix eigenvalue problem, and it is shown that for each spatial mode there is a set of 𝑛 intramodal mode shapes. When the structures are identical, uniform, or nonuniform, the differential equations are uncoupled through diagonalization of a coupling stiffness matrix. The most general case requires an iterative solution.