In this paper, a spring-mass system with impacts and frictions is formulated by the impulsive differential system. An energy-like Lyapunov function and an auxiliary step function are constructed to analyze the finite-time stability of such impact system with a time-varying external force and sliding friction as well as air resistance. We establish the sufficient conditions of finite-time stability for three cases of the spring-mass system, and present numerical simulations for each case to verify the validity of the theoretical results.

中文翻译：

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具有单边冲击约束和摩擦的弹簧质量系统的有限时间稳定性

在本文中，通过脉冲微分系统制定了具有冲击和摩擦的弹簧质量系统。构造了类能量Lyapunov函数和辅助阶跃函数，分析了这种具有时变外力和滑动摩擦以及空气阻力的冲击系统的有限时间稳定性。我们为弹簧质量系统的三种情况建立了有限时间稳定性的充分条件，并对每种情况进行了数值模拟，以验证理论结果的有效性。