free and forced vibrations of elastically connected structures
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ID: 134480
2010
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Abstract
A general theory for the free and forced responses of ๐ 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 ๐ถ๐[0,1], then the stiffness operator for the set of elastically connected structures is self-adjoint with respect to an inner product defined on ๐=๐
๐ร๐ถ๐[0,1]. This leads to the definition of energy inner products defined on ๐. 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 ๐ 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 ๐ 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.
| Reference Key |
kelly2010advancesfree
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| Authors | ;S. Graham Kelly |
| Journal | malaria journal |
| Year | 2010 |
| DOI |
10.1155/2010/984361
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