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ARTICLE

Constrained Optimization Multi-dimensional Harmonic Balance Method for Quasi-periodic Motions of Nonlinear Systems

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1 Chinese Aeronautical Establishment, Beijing 100012, China. Email: ht0819@163.com

Computer Modeling in Engineering & Sciences 2013, 95(3), 207-234. https://doi.org/10.3970/cmes.2013.095.207

Abstract

The constrained optimization multi-dimensional harmonic balance method for calculating the quasi-periodic solutions of nonlinear systems is presented. The problem of determining the worst quasi-periodic response is transformed into a nonlinear optimization problem with nonlinear equality constraints. The general nonlinear equality constraints are built using a set of nonlinear algebraic equations which is derived using the multi-dimensional harmonic balance method. The Multi- Start algorithm is adopted to solve the resulting constrained maximization problem. Finally, the validity of the proposed method is demonstrated with a Duffing oscillator and numerical case studies for problems with uncertainties are performed on a nonlinear two-degree of freedom with non-regular nonlinearities. It is illustrated that the proposed approach can be used to find the worst resonant response and the upper and lower response bounds of quasi-periodic solution and is also able to quantify the combined influences of structural uncertainties and non-regular nonlinearities on the nonlinear quasi-periodic vibrations of nonlinear systems.

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