In this paper, the exact solution of a linearlized equation is employed as initial conditions for the next timestep in the numerical integration of the end displacement and velocity. This exact solution is calculated by means of the Duhamel integration. The system equations are satisfied continuously and not discretely as done traditionally. The essential difference of the present method from other works is that the performance of dynamics systems can be traced continuously. Comparisons between the proposed method with traditional techniques are presented. Examples investigated include the large amplitude nonlinear vibration of a simple pendulum of a conservation system, the period of vibration and chaos in the forced vibration of the Duffing oscillator and forced vibration of the van der Pol oscillator of a non-conservation system. The results obtained indicate that the accuracy of the proposed method supersede that of the traditional techniques.

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