多频激励下达芬系统的超谐共振

Super-harmonic resonance of Duffing-van der Pol system to multi-frequency excitation

达芬系统是自激系统的一个典型例子,研究多频激励下该系统的超谐共振在实际工程中很有现实意义。本文利用多尺度法研究了多频激励下达芬系统的超谐共振,得到了该系统的超谐共振的定常解,然后利用数值方法研究了系统参数对近似解幅频曲线的影响,并用数值仿真验证了结论的正确性。研究表明,超谐共振的幅频曲线具有叉型分岔的特性,此结果不仅可以用于分析类似系统的定常响应,而且为类似系统的设计和控制提供理论依据。

Duffing-van der Pol system is the typical example of the self-excited system. It is important to study the super-harmonic resonances of the system in the engineering . In this paper the super-harmonic resonance of Duffing-van der Pol system is researched by the Multi-Scale Method, and the steady-state response is also obtained. Then the effects of all the parameters of the system on the amplitude-frequency curve are also discussed. Finally, the numerical method is used to verify the accordance. The results show pitchfork bifurcation arises in the turning point of amplitude-frequency curve. The results can not only be used to analyze the periodic solutions of similar system, but show theoretical foundation for design and control.