Random Vibration of Coulomb Oscillators Subjected to Support Motion With a Non-Gaussian Moment Closure Method*

Abstract

The response of one-degree-of-freedom systems with frictional devices (Coulomb oscillators) undergoing Gaussian support motion (earthquake) is investigated by adopting a moment equation approach in the context of Itoˆ's calculus. Such equations contain expectations of the signum function of the velocity. In order to evaluate them, the joint probability density function of the variables is expanded in a truncated series of modified Hermite polynomials, which allows the computation of the response moments. The truncation is equivalent to neglecting the Hermite moments of the variables 381 382 FATICA AND FLORIS beyond a given order. Starting from the response moments so calculated, approximate marginal and joint PDF, mean upcrossing rate functions, and the largest value distributions can be constructed. Conversely, the estimates of the response correlations require the evaluation of a first-order differential system, which also is written by using Itoˆ's rule. The results of the applications compare well with the simulation, and show that the assumption of Gaussian response is unacceptable and non-conservative.

Mean square stability of a second-order parametric linear system excited by a colored Gaussian noise

“…Thus, it is necessary to close it. Several methods have been proposed to perform the closure: see [26,39,[46][47][48][49][50][51][52]. The cumulant neglect closure method is chosen here.…”

“…In order to circumvent the difficulties of the problem, other studies [15,21,23,31,33,34,38] make the assumption of weak excitation. This assumption allows using the stochastic averaging method [51,52] as in [15,21,23], which simplifies the problem. However, in many instances the assumption of weakly excitation is not valid.…”

“…Because of parametric colored excitation, the moment equations form an infinite hierarchy: in other words, the moment equations that are written for computing the moments of an order s contain moments of order larger than s. Thus, the hierarchy must be closed. For doing it, various closure methods have been proposed: [43][44][45][46][47][48][49][50][51][52]. For the problem under examination Bolotin [10,55] uses the cumulant neglect closure method; Wedig [26] makes a stochastic state transformation, and truncates an infinite determinant.…”

Equivalent nonlinear potential systems: Review of previous assumptions

“…When the nonlinear system has a polynomial nonlinearity, the moment equations form an infinite hierarchy even when the FPK equation associated with the dynamic system admits an analytical solution: in other words, the ME that are written for computing the moments of an order s contain moments of order larger than s. In such a way, only approximate values of the moments can be obtained by using some closure scheme, which may be a severe limitation. Many closure schemes have been advanced, which in most cases have the common feature of expressing the hierarchical moments by means of the lower order moments; other criteria use approximate values of the hierarchical moments [47][48][49][50][51][52][53][54][55][56][57][58][59].…”