摘要
高斯原理给出了通过求函数极值、从可能运动中鉴别出真实运动的规则,它可以使得多体系统动力学问题不需通过求解微分(代数)方程,而是采用求解最小值的优化方法来解决,从而提供了一种适用于优化算法的建模思路,因此,如何定义恰当的高斯拘束函数是动力学优化方法得以实现的前提.对于理想系统而言,约束对系统的作用可以通过约束方程来体现,故高斯拘束可表达为系统质点加速度的函数,系统的动力学问题因此可以描述为目标函数为高斯拘束函数、优化变量为质点加速度的约束最优化问题;当系统中需要考虑干摩擦等非理想因素时,部分相互作用不能被所定义的约束方程所涵盖而需要采用额外的物理规律来描述,这种相互作用破坏了原有的针对理想系统的高斯拘束函数的极值特性.基于变分类的高斯原理,推导并证明了目标函数以理想约束力所表达的非理想系统的极值原理,针对目前文献中用于非理想系统的高斯原理进行了讨论,指出其实际为文中的极值原理在非理想约束力与理想约束力无明显关联时的一种特殊表达形式,当非理想约束力与理想约束力有明显的函数关系(如库仑摩擦定律中滑动摩擦力与法向约束力间的线性关系)时,该形式失效;同时根据文中的极值原理,得到了考虑库仑摩擦时非理想的多体系统动力学问题的优化模型.例子中分析了优化模型及相应的线性互补性模型的关系,分析发现在满足刚体滑动问题的唯一性条件下二者互为充分必要条件,从而证明了文中优化模型的可靠性;并采用优化计算方法进行了动力学模拟,模拟结果显示了将高斯原理与优化算法相结合的可行性及有效性.
The Gauss’ s principle gives the rules to identify the real motion from the possible motion by finding the extreme value of the function, which can make the dynamic problem of the multi-body system not need to solve the differential(algebra) equation, but adopt the optimization method of solving the minimum value, therefore, how to define the appropriate Gaussian constraint function is the prerequisite for the realization of dynamic optimization method. For the ideal system, the effect of constraints on the system can be reflected by the constraint equation and so the Gaussian constraint can be expressed as a function of the particle acceleration of the system, then the dynamic problem of the system can be described as the constrained optimization problem with the objective function as the Gaussian constraint function and the optimization variable as the particle acceleration. When the non-ideal factors such as dry friction need to be taken into account in the system, the partial interaction can not be covered by the defined constraint equation and needs to be described by additional physical laws. This sort of interaction destroys the extreme value characteristics of the original Gaussian restraint function of the system. Based on Gauss’ s principle of the variable classification, the extreme value principle of non-ideal system is derived and proved, whose objective function is expressed by ideal constraint forces. The Gauss’ s principle for non-ideal system in the existing literature is discussed and it is pointed out that it is an expression of the extreme value principle given when there is no obvious function correlation between the nonideal constraint force and the ideal constraint force. But when they have obvious function relation(such as the linear relationship between the sliding friction force and the normal constraint force in the Coulomb friction law), this form will fail. And according to the extreme value principle given, the dynamic optimization model of contact problem for multi-body system
作者
姚文莉
刘彦平
杨流松
Yao Wenli;Liu Yanping;Yang Liusong(Qingdao Key Laboratory for Geomechanics and Offshore Underground Engineering,School of Science,Qingdao University of Technology,Qingdao 266520,China;College of Pipeline and Civil Engineering,China University of Petroleum,Qingdao 266580,China)
出处
《力学学报》
EI
CSCD
北大核心
2020年第4期945-953,共9页
Chinese Journal of Theoretical and Applied Mechanics
基金
国家自然科学基金资助项目(11272167)。
关键词
非理想
多体系统
高斯原理
优化模型
non-ideal
multibody
Gauss’s principle
optimization model
