Piecewise-linearized methods for single degree-of-freedom problems

A piecewise linearization method based on the Taylor series expansion of the nonlinearities and forcing with respect to time, displacement and velocity for the study of smooth single degree-of-freedom problems, is presented. The method provides piecewise analytical solutions which are smooth everywhere, is second-order accurate in time and yields explicit finite difference formulae for the displacement and velocity. The method is applied to nine single degree-of-freedom problems and its accuracy is assessed in terms of the displacement, velocity and energy as functions of the time step, and its results are compared with those of piecewise linearization methods that use Taylor series expansion of the forcing and nonlinearities with respect to time. It is shown that, for nonlinear problems with unknown free frequency and damping, the linearization method presented here is more accurate and robust than linearization techniques based on Taylor series expansions with respect to time. For linear problems with oscillatory forcing, linearization methods that employ fourth-order expansions in time are more accurate than the linearization method proposed here provided that the time step is sufficiently small.