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

摘要

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.