线弹性动力学中的最小势能原理(含最小余能原理)  

THE LEAST POTENTIAL PRINCIPLE AND THE LEAST REMAINING PRINCIPLE IN LINEAR ELASTODYNAMICS

作　　者：唐松花[1] 罗迎社[1] 周筑宝[2] 

机构地区：[1]湘潭大学基础力学与材料工程研究所,湘潭411105 [2]中南大学土木建筑学院,长沙410075

出　　处：《动力学与控制学报》2005年第1期34-38,共5页Journal of Dynamics and Control

摘　　要：线弹性静力学中有最小势能原理和最小余能原理,但只适用于物体或结构在给定约束条件下处于稳定平衡状态的情况,而在一般情况下动力学问题不可能存在稳定平衡状态,因此在动力学领域中是否存在最小势能原理值得认真考虑.本文对动力学问题中存在最小势能原理的可能性进行了探讨,并以摆脱了“平衡态”和“稳定态”的限制的最小功耗原理为理论基础,导出了线弹性动力学中的最小势能原理和最小余能原理.给出了计算实例,结果正确.因此在线弹性动力学中存在瞬时意义下的最小势能原理和最小余能原理.但其含义与静力学中的最小势能原理和最小余能原理并不相同.其主要区别在于:动力学中的原理适用于不稳定过程之任一瞬时,其“最小”是指“当时(即该瞬时)所有可能值的最小”.而静力学中的最小势能原理则只适用于稳定平衡状态,其“最小”是指系统从不稳定最后达到稳定平衡的整个过程中所有“真实值中的最小”.即前者是“当时的最小”,后者则是“全过程中的最小”.这两类变分原理可成为线弹性动力学中各种变分直接解法的理论基础.In elastic-static mechanics there are the least potential principle and the least remaining principle, which is only applicable to the situation of the stable and equilibrium state. But generally speaking there are no stable and equilibrium state in dynamic problems, so it is worthwhile considering carefully whether there is the least potential principle in the dynamic field. This paper studied the possibility of the least potential principle existing in dynamic problems, and derived the least potential principle and the least remaining principle based on the least work consumption principle, which get rid of the limitations of “equilibrium' and “stable state'. The practical calculating examples were proposed and the results were correct. So in linear elastodynamics there also exist the least potential principle and the least remaining principle in instantaneous sense, which have different physical meaning. The physical meaning of the former is to take “the minimum of all probable value meantime' at any moment in the dynamical process,and the latter is to take “the minimum' in the whole dynamical process. That is to say, the former is “the minimum at that time' and the latter is “the minimum in the whole process'. These two variational principles may become the theoretical foundation for all sorts of variational direct solving methods in linear elastodynamics.

关 键 词：最小势能原理 最小余能原理 弹性动力学 动力学问题 平衡状态 理论基础 弹性静力学 不稳定过程 约束条件 最小功耗 计算实例 稳定平衡 直接解法 变分原理 适用 可能性 稳定态 平衡态 瞬时 真实值 全过程 物体 

分 类 号：O313[理学—一般力学与力学基础]