基于广义谐波平衡法,求解了强非线性杜芬振子自由振动和简谐激励下受迫振动的周期-m解,并与数值解进行了比较,从而讨论非线性项的系数以及激励参数对系统周期解的影响.对自由振动而言,倍周期响应的周期是派生系统固有周期的整倍数;对受迫振动而言,倍周期响应的周期是外激励周期的整倍数.结果表明,为使近似解析谐波解与数值解比较接近,系统的非线性越强,所需的谐波项数越多;所设倍周期分岔解的周期越大,所需的项数也越多.
Based on the generalized harmonic balance method, the period-m responses of free and forced Duffing oscillators are constructed and compared with numerical solutions from which the effects of nonlinear term and excitation parameters can be observed. For free vibrations, the response periods are assumed to be integer times of the natural period of the corresponding linear system and the excitation period respectively. It is shown that the number of harmonic terms should be large enough for both strongly nonlinear systems and period-doubling bifurcation solutions with large period so that the approximate analytic solutions are comparative to numerical results.
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