This article uses physical arguments to derive variational integration schemes for dissipative mechanical systems. These integration algorithms find utility in the solution of the equations of motion and optimal control problems for these systems. Engineers usually represent dissipation effects using phenomenological devices such as “dampers.” In this article, we replace these dampers with a lossless transmission-line in order that the equations of motion are derivable from a variational principle. The associated system Lagrangian can then be discretized and used to develop low-order variational integration schemes that inherit the advantageous features of their conservative counterparts. The properties of a lossless spring-inerter based transmission system are analyzed in detail, with the resulting variational integration schemes shown to have excellent numerical properties. The article concludes with the analysis of a dissipative variant of the classical Kepler central force problem.

中文翻译：

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耗散系统的变分积分器


本文使用物理论证来推导耗散机械系统的变分积分方案。这些积分算法可用于解决这些系统的运动方程和最优控制问题。工程师通常使用“阻尼器”等唯象装置来表示耗散效应。在本文中，我们用无损传输线代替这些阻尼器，以便可以从变分原理推导出运动方程。然后可以将相关系统拉格朗日离散化并用于开发低阶变分积分方案，该方案继承了保守对应方案的优势特征。详细分析了基于无损弹簧惯性器的传输系统的特性，所得变分积分方案显示出优异的数值特性。本文最后分析了经典开普勒中心力问题的耗散变体。