Hill T L, Cammarano A, Neild S A, Wagg D J
Department of Mechanical Engineering , University of Bristol , Bristol BS8 1TR, UK.
Department of Mechanical Engineering , University of Sheffield , Sheffield S1 3JD, UK.
Proc Math Phys Eng Sci. 2015 Jan 8;471(2173):20140659. doi: 10.1098/rspa.2014.0659.
Resonance is an important phenomenon in vibrating systems and, in systems of nonlinear coupled oscillators, resonant interactions can occur between constituent parts of the system. In this paper, out-of-unison resonance is defined as a solution in which components of the response are 90° out-of-phase, in contrast to the in-unison responses that are normally considered. A well-known physical example of this is whirling, which can occur in a taut cable. Here, we use a normal form technique to obtain time-independent functions known as backbone curves. Considering a model of a cable, this approach is used to identify out-of-unison resonance and it is demonstrated that this corresponds to whirling. We then show how out-of-unison resonance can occur in other two degree-of-freedom nonlinear oscillators. Specifically, an in-line oscillator consisting of two masses connected by nonlinear springs-a type of system where out-of-unison resonance has not previously been identified-is shown to have specific parameter regions where out-of-unison resonance can occur. Finally, we demonstrate how the backbone curve analysis can be used to predict the responses of forced systems.
共振是振动系统中的一个重要现象，在非线性耦合振子系统中，系统的组成部分之间可能会发生共振相互作用。在本文中，非同步共振被定义为一种响应分量相位相差90°的解，这与通常所考虑的同步响应形成对比。一个众所周知的物理例子是在拉紧的电缆中可能发生的旋转。在这里，我们使用一种正规形技术来获得被称为主曲线的与时间无关的函数。考虑一个电缆模型，这种方法被用于识别非同步共振，并且证明这对应于旋转。然后我们展示了非同步共振如何在其他两自由度非线性振子中发生。具体来说，一个由通过非线性弹簧连接的两个质量块组成的串联振子——一种此前尚未识别出非同步共振的系统类型——被证明在特定参数区域会出现非同步共振。最后，我们展示了主曲线分析如何可用于预测受迫系统的响应。
