This study addresses a fundamental problem of linear friction-induced vibration (FiV), the intriguing role of damping on the instability of FiV in the form of mode-coupling. Two well-known classic 2-degree-of-freedom (DoF) slider-belt models are revisited, and a 4-DoF slider-belt model considering different damping configurations in the slider and the belt is investigated. Since proportional damping is very rare in reality, non-proportional damping is given much attention in this paper.

Firstly, the analytical solutions of bifurcation boundary of the critical coefficients of friction of the classic 2-DoF models are derived, which show the subtle role of damping clearly and theoretically explain and clear misunderstandings and confusions on the phenomenon about the drop of the stability boundary from that of the undamped system caused by adding non-proportional damping. Secondly, a further increase of non-proportional damping is found to improve the stability and even bring higher degree of stability than increasing proportional damping (which is a new finding). Thirdly, the influences of the different damping distributions among the system components on the stability of the 4-DoF slider-belt model are explored. The evolution of the bifurcation boundary indicates the importance of the ratio of the horizontal to the vertical damping in the slider and in the belt. For some non-proportional damping values, the bifurcation boundary fluctuates, which could exhibit local optimal values of the ratio of damping in different directions of a system component and can be exploited in design against FiV. These findings are valid for general linear second-order dynamic systems with an asymmetric stiffness matrix.

这项研究解决了线性摩擦引起的振动（FiV）的一个基本问题，即阻尼以模态耦合的形式对FiV的不稳定性产生了有趣的作用。回顾了两个著名的经典2自由度（DoF）滑块带模型，并研究了考虑滑块和皮带中不同阻尼配置的4-DoF滑块带模型。由于比例阻尼在实际中是非常罕见的，因此非比例阻尼在本文中得到了很多关注。

首先，推导了经典2-DoF模型的临界摩擦系数分叉边界的解析解，它们清楚地显示了阻尼的微妙作用，从理论上解释并消除了关于稳定边界下降现象的误解和混淆。这是由于非比例阻尼所导致的无阻尼系统的问题。其次，发现非比例阻尼的进一步增加比增加比例阻尼（这是一个新发现）可以提高稳定性，甚至带来更高的稳定性。第三，探讨了系统各部件之间不同的阻尼分布对4-DoF滑带模型稳定性的影响。分叉边界的演变表明滑块和皮带中水平阻尼与垂直阻尼之比的重要性。对于某些非比例阻尼值，分叉边界会波动，这可能会在系统组件的不同方向上表现出局部最优的阻尼比值，并且可以在针对FiV的设计中加以利用。这些发现对于具有不对称刚度矩阵的一般线性二阶动力系统是有效的。