Rapid calculation of selected fourier spectrum ordinates

Abstract

The use of Fourier spectrum techniques in earthquake engineering has grown rapidly in recent years because of the economy of programs using the Fast Fourier Transform (FFT) and the widespread use of Fourier techniques in other fields of engineering and science. Typically, the standard FFT programs take 2N equally spaced data points in the time domain as input and produce as output 2N-1 Fourier amplitude spectrum ordinates equally spaced in the frequency domain from 0 cps to the maximum frequency permitted by the digitization interval. By appropriate choice of filters, sampling interval and length of record, the FFT approach can be adapted to most purposes, but there is occasionally a need to calculate a few spectrum points in narrow frequency bands or to analyze, over selected frequency bands, records of longer duration than can be accommodated conveniently by standard FFT programs. The technique presented below permits such calculations to be made rapidly and accurately. In addition, the method helps in the interpretation of Fourier spectra used in earthquake engineering because it is developed from the point of view of elementary vibration theory. The first part of the text reviews the relation between the response of an undamped, single degree-of- freedom oscillator subjected to the same accelerogram. This review shows that the calculation of the Fourier amplitude and phase spectrum ordinates is equivalent to finding the potential and kinetic energies of an undamped oscillator at the end of the excitation. The analysis is then extended to include an associated free vibration problem useful in the interpretation of Fourier spectra. The next portion of the study shows that these final response values can be calculated rapidly and accurately by reducing the accelerogram, regardless of length, to an equivalent excitation with a duration of one natural period, and by further reduction to two excitations - one for displacement and one for velocity - of only one-quarter period duration. The response of the oscillator to the shortened excitations can then be calculated by standard methods. The next section is devoted to the development of a subroutine for calculating ordinates of Fourier amplitude spectra by this approach, and to the presentation of examples of its use. The study concludes with a discussion of possible applications and extensions of the method.