A Stochastic Surrogate Modelling of a NonLinear Time-Delay Mechanical System

Abstract

Nonlinear time-delay dynamic is present in a wide range of engineering problems. This is due to the modernization of structures related to the need of using lighter, more resistant and flexible materials. In mechanical systems, nonlinearities may have physical or geometric characteristics. Most of these systems may possess complex equations that demands a significant computer processing time in order to solve them. In addition, these systems may be subject to uncertainties, such as material properties, random forces, dimensional tolerances and others. The complexity and the time required to solve the equations will be increased with the addition of uncertainties to the inputs of the dynamic system model. In this case, a surrogate model based on Karhunen-Loeve decomposition or polynomial chaos of dynamic system is a viable choice to reduce the complexity and the computational time of the problem, as well as obtaining the statistical responses of the model. Surrogate modeling (also known as metamodeling) is employed to replace the original model of high complexity by a simpler model whose computation cost is reduced. In the field of uncertainty quantification, the statistical moments of a complex model can be easily obtained once a surrogate model is created. Methods like KLD (Karhunen-Loeve Expansion), which relies on the covariance function of the system and decompose the model into a set of eigenvalues and eigenvectors which represents the surrogate model, or PCE (polynomial chaos expansion), that uses a set of multivariate orthogonal polynomials to build the surrogate model are applied to represent the system output. The purpose of this paper is to build a surrogate model of a nonlinear mechanical system with time delay using PCE and KL. A comparison between the original model response will be made against the surrogate model.