Hong Dongxiao, Hill Thomas L, Neild Simon A
Department of Mechanical Engineering, University of Bristol, Bristol BS8 1TR, UK.
Proc Math Phys Eng Sci. 2019 Dec;475(2232):20190374. doi: 10.1098/rspa.2019.0374. Epub 2019 Dec 4.
Isolated backbone curves represent significant dynamic responses of nonlinear systems; however, as they are disconnected from the primary responses, they are challenging to predict and compute. To explore the conditions for the existence of isolated backbone curves, a generalized two-mode system, which is representative of two extensively studied examples, is used. A symmetric two-mass oscillator is initially studied and, as has been previously observed, this exhibits a perfect bifurcation between its backbone curves. As this symmetry is broken, the bifurcation splits to form an isolated backbone curve. Here, it is demonstrated that this perfect bifurcation, indicative of a symmetric structure, may be preserved when the symmetry is broken under certain conditions; these are derived analytically. With the symmetry broken, the second example-a single-mode nonlinear structure with a nonlinear tuned mass damper-is considered. The evolution of the system's backbone curves is investigated in nonlinear parameter space. It is found that this space can be divided into several regions, within which the backbone curves share similar topological features, defining the conditions for the existence of isolated backbone curves. This allows these features to be more easily accounted for, or eliminated, when designing nonlinear systems.
孤立的主干曲线代表非线性系统的显著动态响应；然而，由于它们与主要响应断开连接，因此预测和计算具有挑战性。为了探索孤立主干曲线存在的条件，使用了一个广义双模态系统，该系统代表了两个经过广泛研究的示例。首先研究了一个对称双质量振荡器，正如之前所观察到的，它在其主干曲线之间表现出完美的分岔。当这种对称性被打破时，分岔分裂形成一条孤立的主干曲线。在此，证明了在某些条件下对称性被打破时，这种表示对称结构的完美分岔可能会得以保留；这些条件是通过解析推导得出的。在对称性被打破的情况下，考虑第二个示例——一个带有非线性调谐质量阻尼器的单模态非线性结构。在非线性参数空间中研究了该系统主干曲线的演变。发现这个空间可以分为几个区域，在这些区域内主干曲线具有相似的拓扑特征，从而确定了孤立主干曲线存在的条件。这使得在设计非线性系统时能够更容易地考虑或消除这些特征。
