Lagrange 方程应用于连续介质力学.pdf

北京大学学报(自然科学版) 第 52 卷 第 4 期 2016 年 7 月 Acta Scientiarum Naturalium Universitatis Pekinensis, Vol. 52, No. 4 (July 2016) doi: 10.13209/j.0479-8023.2016.076 Lagrange 方程应用于连续介质力学 1 2,† 冯晓九 梁立孚 1. 常州大学环境与安全工程学院, 常州 213164; 2. 哈尔滨工程大学航天与建筑工程学院, 哈尔滨 150001; † 通信作者 , E-mail: lianglifu@hrbeu.edu.cn 摘要 如何将 Lagrange 方程应用于连续介质力学 , 一直是学术界关注的理论课题。应用变导的概念和运 算法则 , 研究 Lagrange 方程中的求导的性质 , 进而将 Lagrange 方程应用于线性弹性动力学和非线性弹 性动力学 , 并且给出相应的算例。结果表明 , 借鉴变积分学来解决将 Lagrange 方程应用于连续介质力学 的问题是可行的。 关键词 连续介质力学; Lagrange 方程 ; 变导; 线性弹性动力学 ; 非线性弹性动力学 中图分类号 O313 Lagrange Equation Applied to Continuum Mechanics 1 2,† FENG Xiaojiu , LIANG Lifu 1. School of Environmental and Safety Engineering, Changzhou University, Changzhou 213164; 2. College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001; † Corresponding author, E-mail: lianglifu@hrbeu.edu.cn Abstract How to apply the Lagrange equation to the continuous medium mechanics has been a theoretical issue of academic circles. Using variational derivative concepts and operational rules, the properties of variational derivative in Lagrange equation are studied. The Lagrange equation is applied to linear elastic dynamics and nonlinear elastic dynamics, and some corresponding numerical examples are given. The result shows that it is a feasible way to solve the problem of the application of Lagrange equation to the mechanics of continuous media by us