Stochastic seismic response of single-degree-of-freedom systems through wavelets

This paper proposes a wavelet-based approach for estimation of the stochastic response of a single-degree-of-freedom (SDOF) system which is subjected to an earthquake ground motion process modelled by its wavelet coefficients. Expressions have been derived for (i) the representation of the excitation process through wavelets, (ii) relationship between the wavelet coefficients of the excitation and response processes, and (iii) instantaneous mean-square value and power spectral density function (PSDF) of the response process. The proposed formulation considers both amplitude and frequency non-stationarities in the ground motion process, and is thus more generalized than the earlier methods based on the use of frequency-independent modulating functions. Compared with an alternative wavelet-based formulation (Basu B, Gupta VK. Seismic response of SDOF systems by wavelet modelling of non-stationary processes. J Eng Mech (ASCE) 1998;124(10):1142–50), this formulation does not require the wavelet bases to be mutually orthogonal and energy bands corresponding to different scales to be non-overlapping, for computing the instantaneous mean-square value of the response process. This is also applicable for very short excitations where the sudden application of excitation may be of crucial importance. The proposed formulation has been validated through statistical simulation in the case of an example ground motion process and oscillators of different periods.