Abstract

This paper presents a new method for enhancing and extending finite element analyses of structures in the frequency domain. When a finite element model is used to predict the response of a vibrating structure to a time-harmonic excitation, the number of degrees of freedom necessary for a convergent answer scales with the frequency of excitation. At higher frequencies, inversion of the matrix equations of motion for the response vector is computationally expensive and, given a finite amount of computer resources, the response at any particular point on the structure is coarsely sampled in frequency. Direct computation of the modal vectors and natural frequencies is similarly cost prohibitive. The method presented here is intended to increase the frequency resolution of response data over a frequency band by estimating the resonant frequencies which lie in the band and their corresponding modal vectors. These estimates are obtained by inserting into Rayleigh’s Principle a trial modal vector that is a linear combination of the response vectors evaluated at frequencies inside the band. An eigenvalue problem is thereby obtained whose rank is equal to the number of sampled frequencies, which is chosen to be much smaller than the number of degrees of freedom of the structure. The method is demonstrated on a vibrating beam, where it is shown to accurately predict the displacement anywhere over a frequency band which contains four modes.

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