Equivalent linearization of a randomly excited yielding oscillator

Text includes handwritten formulasEquivalent linearization of bilinear hysteretic systems subjected to a white noise-excitation is attempted by using 2nd and 3rd order linear systems. The bilinear hysteretic systems considered have the slope ratio between the initial and the reduced stiffness of a = 1/2 (moderately nonlinear case) and a = 1/21 (nearly elasto-plastic case). The technique is to match both energy dissipation per unit of time and average frequency between the original system and its equivalent linear system in stationary motion. In the 2nd order linearization these criteria are essentially the same as the requirements from the Krylov-Bogoluibov method. However, special attention is given to the estimation of the hysteretic energy dissipation per unit of time, resulting in improved predictions of stationary levels of root-mean-square displacement and velocity response. Satisfying the above matching criteria does not require explicit specification of the parameters in the equivalent linear system. In this investigation several linearization matching the above criteria are considered. These include: the usual 2nd order linear system, a model with two uncorrelated 2nd order modes whose undamped natural frequencies correspond to the initial stiffness and the reduced stiffness of the bilinear hysteretic system, a 3rd order linear system which has the same stiffness arrangement as the bilinear hysteretic system model but replaces the Coulomb friction slider in the original system by a viscous damper, and a model with two uncorrelated 3rd order modes which have the same root-mean-square displacement. A severe test of the equivalence to the original system is executed by comparing the response power spectral densities. After getting the specified equivalent linear systems, their transient root-mean-square responses are compared with the experimental results. Tor this response analyses, the Rice method is applied for the 2nd order linear systems and the Markov process approach is taken for the 3rd order linear systems. As a result, a correlation between the stationary response power spectral density matching and the transient root-meansquare response matching is found. As a whole, the two-mode 3rd order linear system proves to be the best linearization among those considered herein