Damped Fourier Spectrum and Response Spectra by F. E. Udwadia and M. D. Trifunac

This paper describes the physical relationships that exist between the Fourier transform and the response spectrum of a strong-motion accelerogram. By developing the new concept of the "Damped Fourier Spectrum" (D.F.S.), we show that the velocity and displacement of the damped oscillator can be represented by a linear combination of the real and imaginary parts of the D.F.S. and by the initial conditions. The D.F.S. represents a new way of "smoothing" the classical Fourier Transform by using a physically based filter. INTRODUCTION The computational economy afforded by the Fast Fourier Transform algorithm (Tukey, 1967) has made Fourier analysis the eondicio sine qua non in the processing of strong-motion data (e.g. Udwadia and Trifunac, 1973b; Trifunac, 1972). To take full advantage of this economy, it is now necessary to develop new methods that are capable of extracting the maximum possible information from the complex Fourier transform for use in vibration analysis. In this paper, we show that the Damped Fourier Spectrum bears the same relationship to the damped velocity spectrum, as the classical Fourier transform does to the undamped velocity spectrum (Kawasumi, 1956; Rubin, 1961; Hudson, 1962; Jennings, 1972). THE DAMPED FOURIER SPECTRUM The governing equation of relative response x(t) of a damped linear oscillator subjected to an absolute ground acceleration -~( t ) is + 2~On~ + COn2X = ~(t), (1) where ~ is the percentage of critical damping and ~o, = (k/m) 1⁄2 is the natural frequency (Figure 1). Using the transformation we get y = x exp (o~,~t) (2) y-k09a2y = 2"(0 exp (~Odflt), (3) where ~o~ = ~o,(1 ~2 )1⁄2 is the damped natural frequency of the oscillator and fl = 4/(1 ~ 2 ) 1⁄2 Defining equation (3) becomes r/d*(O~d, t) = •(t) + ic%y(t), (dqd*/dt)io~dr/a* = ~(t) exp (~Odflt) whose solution is r/a*(~%, t) = exp (io~dt)[Sto ~(z) exp (rndflZi~odz)dz +r/~o]" Here r/~o is the value of r/a* at t = 0. 1775 (4)