Abstract

The smooth endochronic hysteretic Bouc–Wen model is studied from the point of view of random vibration. The sources of the
errors of the method of equivalent linearization applied to this model using the hypothesis of joint Gaussian behaviour are examined.
The method of linearization for softening hysteretic models proposed by the authors, which is based on a combination of Dirac
and Gauss densities, is developed and applied to the Bouc–Wen model under a variety of conditions. It is shown that the method
gives excellent estimations of the response statistics without increasing the computational effort required by the conventional technique.

Abstract

A new procedure for the random vibration analysis of hysteretic structures using stochastic equivalent linearization is reported. Its aim is to improve the prediction of the response obtained by conventional Gaussian linearization technique. To this purpose, mixed discrete-continuous Gaussian distributions are used taking into account the bounded nature of the non-linear restoring force. The simple but important property of the mixed distribution is its linearity, which allows the use of the previous results obtained by the Gaussian hypothesis, avoiding the need of employing non-Gaussian continuous distributions or other time-consuming techniques such as local Monte Carlo simulations. Closed-form expressions of the new linearization coefficients for the Bouc-Wen-Baber model are then provided. The relative weights of the discrete and Gaussian distributions are calculated in dependence of the degree of non-linearity in each time step. The comparison of the results with previously published ones obtained by simulation shows a good agreement, providing a substantial improvement of the method with respect to the conventional Gaussian technique with the same calculation effort.

Abstract

As the recent research by Trybicki and Sobczyk has demonstrated [1-3] the principle of maximum entropy is a powerful tool for solving stochastic differential equations. In particular, its use in connection with the moment equations generated by the Ito formula provides accurate estimations of the probability density evolution of some oscillators for which conventional methods such as the diverse closure schemes are not applicable. A major computational requirement of the method, however, lies in the need of calculating a large number of multidimensional integrals at each time step -a numerical task for which both accurate and economic algorithms are required. In this paper it is shown that conventional economic integration techniques often lead to numerical collapse of the solution, especially when dealing with higly nonlinear oscillators. A strategy that overcomes this difficulty is proposed. In essence, the integrals are reformulated in terms of multidimensional Fourier transforms, which are solved by an ad hoe FFT algorithm aimed at obtaining only one single "frequency" point. It is demonstrated that the numerical stability and the accuracy of the proposed algorithm are superior to those afforded by other integration schemes.