拉格朗日动力学方程在弹性体方程推导中的应用及其教学

APPLICATION AND TEACHING OF LAGRANGIAN DYNAMICS EQUATIONS IN DERIVATION OF ELASTIC BODY EQUATION

为了避免牛顿力学中对微元体做受力分析的复杂过程,本文采用拉格朗日力学分别推导了弹簧质量系统、轴向杆、弯曲梁、弯曲板、三维连续体5种弹性体的拉格朗日量及其等价的动力学微分方程表达式。在弹性力学课程教学中,可以让学生从能量的视角建立控制方程,避免对微元体进行复杂的受力分析。在有限单元法课程教学中,可以让学生直接对拉格朗日量进行变分与离散。本文是对现有弹性体的拉格朗日量和微分方程的总结,有利于学生形成较为系统的知识体系。

To avoid the complex process of force analysis on infinitesimal elements in Newtonian mechanics,this paper uses Lagrangian mechanics to derive the Lagrangians and their equivalent dynamical differential equations for five types of elastic bodies:mass-spring systems,axial rods,bending beams,bending plates,and three-dimensional continua.In the teaching of elasticity mechanics courses,students are encouraged to establish control equations from the perspective of energy and avoid the complex analysis of forces for infinitesimal body.In finite element method courses,students can directly perform variations and discretizations on the Lagrangians.This paper summarizes the Lagrangians and differential equations for existed elastic bodies to form students a more systematic knowledge framework.