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Adomian Decomposition Method Applied to Nonlinear Normal Modes of an Inertially Coupled Conservative System

School of Mechanical Engineering, 585 Purdue Mall, Mechanical Engineering Building, Purdue University, West Lafayette, IN 47907-2088 USA

Anil K. Bajaj

School of Mechanical Engineering, 585 Purdue Mall, Mechanical Engineering Building, Purdue University, West Lafayette, IN 47907-2088 USA, bajaj{at}ecn.purdue.edu

In this work, the nonlinear normal modes (NNMs) of a conservative spring—mass—pendulum system with quadratic inertial nonlinearity are determined by applying the Adomian decomposition method. The technique is applied to the governing boundary value problem that determines the nonlinear normal modes of the two degrees-of-freedom system in configuration space. The NNMs of this system are non-similar and such that the modal curves do not pass through the origin in the configuration space. Both the low energy and the high energy cases are considered, and nonlinear normal modes near 1:2 and 1:1 internal resonances in the two linear modes of vibration of the system are developed. It is seen that the ADM polynomial approximation approach, where the model nonlinearities are approximated by polynomials in the neighborhood of center—center equilibrium, greatly reduces algebraic calculations, and improves the convergence speed and the accuracy of the results obtained. The method also captures the possible multiple nonlinear normal modes that exist near internal resonance conditions. The advantages and disadvantages in using the Adomian decomposition method to construct the nonlinear normal modes of the system are discussed.

Key Words: Nonlinear normal modes • spring mass pendulum • internal resonances • Adomian decomposition • bifurcations in normal modes

Journal of Vibration and Control, Vol. 14, No. 1-2, 107-134 (2008)
DOI: 10.1177/1077546307079401


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