Free and Forced Vibrations of Elastically Connected Structures

Abstract EN

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.