多自由度内共振系统非线性模态的分岔特性

利用多尺度法构造了一个立方非线性1:3内共振系统的内共振非线性模态(NonlinearNormal Modes associated with internal resonance)．研究表明,内共振非线性系统除存在单模态运动外还存在耦合模态运动．耦合内共振模态具有分岔特性．利用奇异性理论对模态分岔方程进行分析发现此类系统的模态存在叉形点分岔和滞后点分岔这两种典型的分岔模式．

BIFURCATION OF NONLINEAR NORMAL MODES OF MULTI-DEGREE-OF-FREEDOM SYSTEMS WITH INTERNAL RESONANCE

The method of multiple scales is applied for constructing nonlinear normal modesof a three-degree-of-freedom system which is discretized from a two-link flexible arm connectedby a nonlinear torsional spring. The discrete system is with cubic nonlinearity and 1:3 internalresonance between the second and the third modes. The approximate solution for the nonlinearnormal modes associated with internal resonance are presented. The NNMs determined here tendto the linear modes as the nonlinearity vanishes, which is significant for one to construct nonlinearnormal modes. Greatly different from results of those nonlinear systems without internal resonance,it is found that the nonlinear normal modes involved in internal resonance include both coupled anduncoupled kinds. The bifurcation analysis of the coupled NNM of the system considered is givenby means of the singularity theory The pitchfork and hysteresis bifurcation are simultaneouslyfound. Therefore, the number of nonlineax normal modes arising from the internal resonance mayexceed the number of linear modes, in contrast with the case of no internal resonance, wherethey are equal. Curves displaying variation of the coupling extent of the coupled NNM with theinternal-resonance-deturing parameter are proposed for six cases.