Slip along a frictional contact between elastic bodies can be stable or unstable, leading to stick-slip motion. Frictional slip can also be associated with vibrations. The condition for these vibrations and their characteristics remains poorly understood. To address this issue, which is relevant to engineering and earth science, we carry out a linear stability analysis of a spring-and-slider system obeying rate and state friction. We first identify the solution space for the linearized equation and define the conditions for different slip modes from the real and imaginary parts of the solution. We then derive asymptotic equations for all boundaries between overdamped stable sliding, inertial/non-inertial underdamped oscillation, stick-slip, and harmonic vibration. Finally, we verified the conditions with numerical simulations. Our work provides rigorous criteria regarding the conditions for the various frictional slip modes and the emergence of vibrations. It can help design appropriate approaches for suppressing undesired vibrations in mechanical systems and investigate the mechanisms generating vibrations (tremor) associated with fault slip in nature.

沿着弹性体之间的摩擦接触滑动可以是稳定的也可以是不稳定的，从而导致粘滑运动。摩擦滑移也可能与振动有关。这些振动的条件及其特性仍然知之甚少。为了解决这个与工程和地球科学相关的问题，我们对服从速率和状态摩擦的弹簧和滑块系统进行了线性稳定性分析。我们首先确定线性方程的解空间，并根据解的实部和虚部定义不同滑移模式的条件。然后，我们推导出过阻尼稳定滑动、惯性/非惯性欠阻尼振荡、粘滑和谐波振动之间的所有边界的渐近方程。最后，我们通过数值模拟验证了这些条件。我们的工作为各种摩擦滑移模式和振动出现的条件提供了严格的标准。它可以帮助设计适当的方法来抑制机械系统中的不需要的振动，并研究与自然界中的断层滑动相关的振动（震颤）产生机制。