A simple method for exponentially modulated random excitation

Abstract

The spectral method in stationary random process is extended to exponentially modulated stationary and non-stationary random processes in a straightforward manner by replacing the frequency parameter iω by α=β+iω where exp (βt) is the modulating function. Therefore, the transfer function is now H(α) instead of H(iω), and the formulation follows the conventional method. Integration formulae for the response spectra for the root mean square (r.m.s.) response, velocity and acceleration are presented. The results are compared to approximate formulae, for resonance conditions. For the case of a constant modulating function (β=0), the usual approximate formula for the r.m.s. displacement is more accurate when the natural frequency ω₁ is less than that, ω₀, at the peak of the excitation spectral density curve. For the approximation, the behaviour of the acceleration is just the reverse of that of the displacement, and the influence of the damping is not as significant as that of ω₀−ω₁, as is usually the case. The method is suitable for applications in earthquake engineering and blasting analysis.