Frequency Domain Analysis of Undamped Systems

Abstract

A numerical tool commonly used in digital signal processing, the exponential window method, is briefly reviewed in this paper and applied to problems in structural dynamics. This method allows one to carry out analyses of undamped structures in the frequency domain, and yields highly accurate results for both discrete and continuous systems. In essence, the solution involves: (1) Finding both the transfer function and the forward Fourier transform of the excitation for complex frequencies; (2) performing a standard inverse Fourier transformation into the time domain; and (3) removing the effect of the complex frequencies by means of an exponential factor (or window). Excellent results are obtained when this factor is chosen so that the power of the excitation and response signals at the end of the window are attenuated by some three orders of magnitude. In such case, it is found that a quiet zone (a tail of trailing zeroes) is not needed for accurate computations, and that temporal aliasing (folding) is negligible. This computational advantage is achieved at the expense of having to evaluate accurately the transfer functions at each frequency step, since interpolation schemes cannot be used in this method.