研究两自由度强非线性振动系统的规范形方法

Study on the strongly nonlinear oscillation systems with two degrees of freedom by normal form method

传统的规范形理论常用于研究弱非线性振动问题,对于非线性项不再是小量的强非线性振动系统则并不适用。为进一步拓展这一理论的适用范围,基于研究单自由度强非线振动问题的待定瞬时固有频率法,提出了可用来求解两自由度强非线性振动系统的改进规范形方法。首先引入了复数形式的一阶方程并且利用新的未知瞬态基频替换系统原有的固有频率,再依照规范形理论计算了一类两自由度强非线性Du ffing-V an der Po l振子的5阶传统规范形。最后求解平均方程获得了此类系统的瞬时频率、振幅以及相应的稳态渐近解。通过对比算例中本文方法、原有规范形理论及数值仿真的结果,证明了改进的规范形理论对于多自由度强非线性振动问题的适用性。

Conventional normal form theory is generally used to study the weakly nonlinear oscillation system, thus it confronts the limitation of dealing with the strongly nonlinear oscillation system, because the nonlinear terms aren＇ t small. To expand the validity of the normal form theory a refined normal form theory based on the approach of undecided instantaneous fundamental frequency method was applied to study strongly nonlinear system with two degrees of freedom in this paper. The first order equations were written in a complex form and the former fundamental frequencies were substituted by the new instantaneous fundamental frequencies, and then the conventional normal form of the strongly nonlinear Dulling-Van der Pol system up to the fifth order could be obtained. Finally the new introduced frequencies, amplitude of vibration, and relevant asymptotic solutions were computed by solving the average equations. The computation examples verified the validity of the refined theory in such multi-degrees of freedom which coincided very well with the solutions of numerical integration which demonstrated the adaptability of the proposed method for strongly nonlinear oscillation systems.