超细长弹性杆动力学的Gauss原理

研究基于Gauss 变分的超细长弹性杆动力学建模的分析力学方法.分别在弧坐标和时间的广义加速度空间定义虚位移，给出了非完整约束加在虚位移上的限制方程；建立了弹性杆动力学的Gauss原理，由此导出Kirchhoff方程、Lagrange方程、Nielsen方程以及Appell方程；对于受有非完整约束的弹性杆，导出了带乘子的Lagrange方程；建立了弹性杆截面动力学的Gauss最小拘束原理并说明其物理意义. 关键词： 超细长弹性杆动力学 分析力学 Gauss变分 最小拘束原理

Gauss principle for a super-thin elastic rod dynamics

The analytical formulation of dynamics of a super-thin elastic rod was studied on the basis of Gauss variation. The definition of virtual displacement in the generalized acceleration space with respect to the arc-coordinate and the time were given with expression of Gauss variation，respectively. The nonholonomic constraint equations of the virtual displacements expressed by Gauss variation were given. The Gauss’s principle of the dynamics of a super-thin elastic rod was established, from which the Kirchhoff equation, the Lagrange equation, the Nielsen equation and the Appell equation of the rod can be derived. The Lagrange equation with indeterminate multipliers was obtained for the case when the rod is subjected to the nonholonomic constraints. The Gauss's principle of least compulsion of a super-thin elastic rod was proved and the compulsion function has a minimum for the actual motion and its physical meaning was indicated.