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 C-k [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 = R-n x C-k [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 non uniform, the differential equations are uncoupled through diagonalization of a coupling stiffness matrix. The most general case requires an iterative solution.