Bias errors of different simulation methods for linear and nonlinear systems

Abstract

Responses of mechanical systems are often studied using numerical time-domain methods. Discrete excitation forces require a transformation of the dynamic system from continuous time into discrete time. Such a transformation introduces an aliasing error. To reduce the aliasing error, different discretization techniques are used. The bias errors introduced by some discretization techniques are studied in this paper. Algebraic expressions of the bias error obtained for some discretization methods are presented. The bias error depends on the assumption of the characteristics of the load between two subsequent time steps; here the zero-order, first-order and Lagrange second-order assumptions are studied. Different simulation methods are also studied for numerical evaluation of the derived theoretical bias errors. The discretization techniques are implemented for Runge–Kutta, the Digital Filter method and the Pseudo Force in State Space method. The study is carried out for both a linear and a nonlinear system; two numerical examples assist in validating the theory. Perfect matches between the numerically estimated bias errors and the theoretical ones are shown. The results also show that, for the nonlinear example, the fourth order Runge–Kutta method produces data that gives less accuracy in the following system identification than the Digital Filter and the used single step Pseudo Force in State Space method do.