含阻尼器的刚体的稳定性与长时间行为
Stability and long-time behaviour of a rigid body containing a damper
AI总结：
本文分析了由薄润滑剂隔开的两个刚体耦合系统的稳定性与长时间行为，证明了仅当旋转轴为外刚体最大转动惯量主轴时永久旋转指数稳定，且所有解均收敛至平衡构型。
摘要：
本文对由两个刚体组成的耦合系统的稳定性和长时间行为进行了全面分析，这两个刚体由一层薄润滑剂隔开。我们证明了整个系统的永久旋转（此时两固体处于相对静止状态）是指数稳定的，当且仅当旋转轴为外刚体最大转动惯量所对应的主惯性轴。所有其他平衡点均是通常双曲的，因而不稳定。此外，我们证明了无论选择何种初始条件，控制方程的所有解均收敛至一个平衡构型。文中还给出了上述结果的数值证据，以及确保稳定构型可达性的条件。在稳定性分析中，我们使用了具有中心流形的动力系统的线性化原理。长时间行为的刻画则是通过对控制该耦合系统运动的局部耗散方程组进行仔细分析而获得的。
英文摘要：
This paper provides a comprehensive analysis of stability and long-time behaviour of a coupled system constituted by two rigid bodies separated by a thin layer of lubricant. We show that permanent rotations of the whole system, with the solids at relative rest, are exponentially stable if and only if the axis of rotation is the principal axis of inertia corresponding to the largest moment of inertia of the outer body. All other equilibria are normally hyperbolic, and hence unstable. In addition, we show that all solutions to the governing equations converge to an equilibrium configuration, no matter which initial conditions are chosen. Numerical evidence of the above results as well as conditions ensuring attainability of the stable configurations are also presented. For the stability analysis we use a linearization principle for dynamical systems possessing a center manifold. The characterization of the long-time behaviour is obtained by a careful analysis of the partially dissipative system of equations governing the motion of the coupled system.
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