Harmonic wavelets based excitation–response relationships for linear systems: A critical perspective

An analytical approach based on generalized harmonic wavelets (GHWs) for determining the response evolutionary power spectral density (PSD) of single-degree-of-freedom linear structural systems subjected to non-stationary stochastic excitations is developed. Specifically, utilizing a periodized GHW transform, applying a Galerkin scheme, and relying on the orthogonality properties of the GHWs, an analytical relationship between wavelet coefficients of the system response and of the excitation is derived. It is shown that in comparison with a recently developed GHW based excitation–response relationship, which is based on a “locally stationary” stochastic process representation and employs localized in time monochromatic functions, the herein developed relationship exhibits enhanced accuracy for cases of relatively flexible systems and/or for systems with relatively low damping where the related impulse response function cannot be assumed to be short-lived. Pertinent numerical examples and Monte Carlo simulation data are included as well for demonstrating the reliability of the approach.