TY - JOUR
A2 - Liew, K. M.
AU - Kelly, S. Graham
PY - 2010
DA - 2011/01/02
TI - Free and Forced Vibrations of Elastically Connected Structures
SP - 984361
VL - 2010
AB - 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.
SN - 1687-6261
UR - https://doi.org/10.1155/2010/984361
DO - 10.1155/2010/984361
JF - Advances in Acoustics and Vibration
PB - Hindawi Publishing Corporation
KW -
ER -