NATURAL FREQUENCIES OF REPETITIVE STRUCTURES

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Abstract
SUMMARY An algorithm has recently been presented (1) for determining the natural frequencies of any linearly elastic structure if its dynamic stiffness matrix K(a>) corresponding to any finite set of displacements D is known. It does not assume that K(o>) is a linear function of coa and enables one to determine how many natural frequencies lie below any given frequency, without determining them. With this information infallible iterative methods for finding natural frequencies in a small number of iterations can be developed. Such a method, the “repetition method”, is presented in this paper. It is applicable to structures which consist entirely, or in part, of a number, N, of identical sub-structures identically connected together to form a chain or ring. Theoretical and practical comparisons are made, for chain-type structures, with banded stiffness matrix and transfer matrix methods. These show that for N > 12 the repetition method is the fastest. For large N the time saving can be considerable; for N > 48, for example, the repetition method never takes more than about a third as long as the banded stifEness matrix method, if the time taken to set up the matrices which are operated on, which is common to both methods, is ignored. Furthermore the results obtained show that the repetition method, like the banded stiffness matrix method, but unlike the transfer matrix method, is well-conditioned for all the problems solved.

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