非线性振动系统中的模态对应原理

出于系统非线性因素的存在，非线性模态具有许多与线性模态不同的性质，但非线性模态与线性模态类似，它同样反映了系统的一种同步运动关系。刘练生等根据Rosenberg非线性模态理论得到了关于非线性保守系统、非线性自治系统和非线性非自治系统的模态包含原理，在此基础上，本文提出了非线性振动系统中的模态对应原理，并给出厂模态对应原理的严格证明。该原理指出：无论非线性振动系统具有相似模态还是具有非相似模态，n个自由度的非线性振动系统至少具有n个非线性模态，且这n个非线性模态形式上对应于该非线性振动系统对应的线性振动系统的n个线性模态。算例表明该原理能够用于寻求非线性振动系统中形式上与相应线性系统的线性模态相对应的非线性模态。

Modal Corresponding Principle in the Nonlinear Vibration Systems

Although there are some different properties between nonlinear modes and linear modes because of the existence of nonlinearities in the nonlinear system, similarly, the nonlinear modes indicate a kind of synchronous motion of the system, just as the linear modes do. According to Rosenberg s nonlinear mode theory, Liu et al. (in 10]) presented the modal containing principles on the nonlinear conservative system, nonlinear autonomous system and nonlinear non-autonomous system. In this paper, the modal Corresponding Principle in the nonlinear vibration systems is proposed on the basis of the preceding works, and the certification on such principle is given as well. The principle suggests that the nonlinear vibration system with n degrees of freedom (DOF) have not less than n nonlinear modes, and the n nonlinear modes are formally corresponding to the n linear modes of the corresponding linear vibration system of the nonlinear one, no matter what the nonlinear vibration system possesses either similar modes or dissimilar ones. The calculated example is given to show that the principle proposed in this paper can be applied to seek the nonlinear modes that are formally corresponding to the linear modes of the corresponding linear vibration system of the nonlinear one.