Sixty years of stochastic linearization technique

Abstract

Stochastic linearization technique is a versatile method of solving nonlinear stochastic boundary value problems. It allows obtaining estimates of the response of the system when exact solution is unavailable; in contrast to the perturbation technique, its realization does not demand smallness of the parameter; on the other hand, unlike the Monte Carlo simulation it does not involve extensive computational cost. Although its accuracy may be not very high, this is remedied by the fact that the stochastic excitation itself need not be known quite precisely. Although it was advanced about six decades ago, during which several hundreds of papers were written, its foundations, as exposed in many monographs, appear to be still attracting investigators in stochastic dynamics. This study considers the methodological and pedagogical aspects of its exposition.

A comprehensive review on dual approach to the vibration analysis: Some dual techniques and application

“…In the theory of random vibration, the stochastic equivalent linearization method is a popular method since it preserves some essential properties of the original nonlinear system. The method has been described in numerous review articles [34,35], and was summarized in the monographs by Roberts and Spanos [29] and Socha [36]. The essential of the equivalent linearization method is how to find the linearization coefficients for a given nonlinear system.…”

“…The essential of the equivalent linearization method is how to find the linearization coefficients for a given nonlinear system. In the literature, several criteria of equivalent linearization have been suggested to define the linearization coefficients where the original version is the conventional criterion [29,[34][35][36] that minimizes the mean-square of equation error. Despite the advantages, the main disadvantage of this criterion is that its accuracy decreases as the nonlinearity is increasing, in many cases it results in unacceptable errors.…”

“…If substituting the local averaging operator determined in (35) into the mean-square error criterion (1) for the equivalent replacement of A(ωt) by kB(ωt) one gets the local mean-square error criterion The behavior of the equivalent replacement coefficient k(r) depending on the local value r needs to be investigated. The question of how r can be chosen is open and minimal values of k(r) are ones of interest.…”

Efficient computation of optimal open-loop controls for stochastic systems

“…This kind of method for probabilistic analysis of nonlinear stochastic dynamical systems is called statistical (or stochastic) linearization. We direct the reader to the recent survey (Elishakoff and Crandall, 2017) and to the monographs (Roberts and Spanos, 2003;Socha, 2008) for details.…”

Harmonic linearisation of aerodynamic loads in a frequency‐domain model of a floating wind turbine

“…An alternative method of linearisation is a form of 'equivalent linearisation', which rather than looking at the nonlinear system's behaviour locally around one operating point aims to minimise the expected error of the linearisation over the range of operating conditions being considered. 30 Within this general approach, 'harmonic linearisation' and 'stochastic linearisation' apply to deterministic and stochastic situations, respectively. Stochastic linearisation has long been applied to modelling the nonlinear viscous drag forces on floating structures 31 and was applied to fixed wind turbines by Savenije and Peeringa 32 and to floating wind turbines by Hall et al, 11 Lupton 33 and more recently by Lemmer et al 34 to tailor the linearisation of the hydrodynamic loads to different sea states.…”