Modified Legendre Operational Matrix of Differentiation for Solving Strongly Nonlinear Dynamical Systems

Complex vibration phenomena appear so frequently in many engineering and physical experiments, and they are well modeled using nonlinear differential equations. However, contrary to the linear models, nonlinear models are difficult to analyze analytically or numerically and particularly for long-time spans. In this paper, we propose a novel method to provide approximate analytic solutions of an important class of nonlinear differential equations that describe the underdamped, overdamped, and oscillatory motions of massspring systems subjected to external excitations. The method is based on a novel modification of the Legendre operator matrix of differentiation technique which results in solutions that are accurate not only for short-time spans but also for long-time spans as well. We provide error analysis and present several examples to demonstrate the efficiency of the proposed method.