Piecewise-linearized methods for oscillators with fractional-power nonlinearities

Abstract

A piecewise linearization method based on the Taylor series expansion of nonlinear ordinary differential equation with respect to time, displacement and velocity is developed for the study of one-degree-of-freedom nonlinear oscillators with smooth and fractional-power nonlinearities. The method provides smooth solutions and explicit, nonstandard finite difference expressions for the displacement and velocity, and is exact for constant coefficients, linear ordinary differential equations with linear time-dependent forcings. The method is applied to ten oscillators with fractional-power nonlinearities and its results are compared with those of harmonic balance techniques, Ritz procedure and numerical solutions based on nonstandard finite difference methods, in terms of displacement, velocity, energy and angular frequency, when available. It is shown that the linearization method presented here provides slightly more (less) accurate results than those of nonstandard methods for fractional powers greater (smaller) than one, and both techniques are more accurate than those that freeze the nonlinearities at the previous time step or linearize the nonlinear terms with respect to only the displacement or the velocity. For the examples and time steps considered in this paper, the linearization method has been found to be more accurate than the harmonic balance and Ritz procedures.