New approach to beating phenomenon in coupled nonlinear oscillatory chains
New approach to beating phenomenon in coupled nonlinear oscillatory chains
复制标题
DOI：
10.1007/s00419-006-0081-1
复制
发表时间：
2007-03
影响因子：
2.8
通讯作者：
L. Manevitch
中科院分区：
文献类型：
--
作者：
L. Manevitch
本文提出了一种描述弱耦合振子间强能量交换的非线性振动的解析方法。介绍了与全能量转移相对应的极限相轨迹的基本概念。在某种意义上，这是替代非线性正常模式（NNM）的特点是能量守恒。许多著名的近似基于NNM的结果是有效的弱能量交换的情况下，所提出的方法可以用于描述弱耦合振子或振荡链之间的强能量交换的非线性过程。这样的描述在形式上类似于振动冲击过程的描述，并且当处理具有密集能量传递的其他过程的描述时，可以被认为是产生这样的过程。随后，我们考虑两个弱耦合保守，阻尼和强迫非线性振动器以及系统的两个弱耦合振动链。结果表明，当非线性参数增大时，系统发生两次动态跃迁。众所周知，第一个转变对应于一个合作非线性简正模（NNM）的分叉，并导致其不稳定并出现两个稳定简正模，其中一个振子具有主导激励。第二次跃迁的特征是极限相轨迹（LPT）的不稳定性，对应于振子之间的完全能量转移。由于这样的过渡，完整的能量转移被打破，并且在振荡器之一上的能量局部化（而最初集中在它上）变得可能而不是跳动。这种运动与最密集的能量转移原来是作为重要的NNMs的解释动态转换和描述密切相关的制度。
We propose an adequate analytical approach to the description of nonlinear vibration with strong energy exchange between weakly coupled oscillators. The fundamental notion of the limiting phase trajectory (LPT) corresponding to full energy transfer is introduced. In a certain sense this is alternative to the nonlinear normal mode (NNM) characterized by conservation of energy. Many well-known approximations based on NNMs turn out to be valid for the case of weak energy exchange, and the proposed approach can be used for the description of nonlinear processes with strong energy exchange between weakly coupled oscillators or oscillatory chains. Such a description is formally similar to that of a vibro-impact process and can be considered as generating such a process when dealing with a description of other processes with intensive energy transfer. We consider subsequently two weakly coupled conservative, damped and forced nonlinear oscillators as well as a system of two weakly coupled oscillatory chains. It is shown that two dynamic transitions occur in the system under consideration when the nonlinearity parameter increases. The first, well-known, transition corresponds to the bifurcation of one of the cooperative nonlinear normal modes (NNMs) and leads to its instability and the appearance of two stable normal modes with predominant excitation of one of oscillators. The second transition is characterized by the instability of the the limiting phase trajectory (LPT), corresponding to complete energy transfer between the oscillators. As a result of such a transition the complete energy transfer is broken, and energy localization on one of oscillators (whereas it was initially concentrated on it) becomes possible instead of beating. This motion with the most intensive energy transfer turns out to be as important as NNMs for the explanation of dynamic transformations and the description of closely related regimes.