非线性非保守耦合系统的模态分岔与稳定性

Modal Bifurcations and Stability Analysis of a Nonconservative Nonlinear Coupled System

用模态的方法分析了一个具有弹性耦合项的非线性耦合ＶａｎｄｅｒＰｏｌ振子系统，研究了此系统的非相似模态运动及分岔，并通过理论和数值结果对模态运动的稳定性和合成性进行了分析．研究结果表明，模态的合成能有效地模拟原系统的衰减效应，即当系统作衰减运动时，理论和数值结果之间的误差很小．然而，当系统的参数穿越某个值后，系统的模态运动方程发生Ｈｏｐｆ分岔，产生一个稳定的极限环，此时，模态的合成失效，特别表现在相位上．

This paper deals with a coupled nonlinear Van der Pol oscillator system with linear coupled elastic term,the dissimilar modes and their bifurcations have been discussed by modal analysis.Moreover,the stabilities and the superposition of the dissimilar modes have also been analyzed theoretically and numerically.It is shown that the modal superposition can effectively simulate the system's attenuation effects,the theoretical results are compared with the numerical ones,it is found that there exit very small errors between each other.However,when the parameter of the system passes through some value,the Hopf bifurcation takes place and a stable limit cycle arises in the modal equation of the system,and as a result,the modal superposition lose its efficacy,especially in their phase positions.