Cosserat弹性杆动力学普遍定理的守恒量问题

根据Cosserat弹性杆的动力学普遍定理,讨论其守恒量问题. 因弹性杆的动力学方程是以截面为对象,并且是以弧坐标和时间为双自变量,其守恒量必定是以积分的形式给出,分别存在关于弧坐标或时间守恒的问题. 根据弹性杆的动量和动量矩方程,导出其动量守恒和动量矩守恒的存在条件及其表达,并讨论了关于沿中心线弧坐标的守恒问题;再分别根据弹性杆关于时间和弧坐标的能量方程导出了各自的关于时间和弧坐标的守恒量存在条件及其表达, 结果包括了弹性杆的机械能守恒以及平衡时的应变能积分;守恒问题给出了例子. 积分形式的守恒量对于弹性杆动力学的理论分析和数值计算都具有实际意义. 关键词： 守恒量 Cosserat弹性杆 动力学普遍定理 双自变量

The conserved quantities in general theorems ofthe Cosserat elastic rod dynamics

Conserved quantities of the Cosserat elastic rod dynamics are studied according to the general theorems of dynamics. The rod dynamical equation takes the cross section of the rod as its objective of study and is expressed by two independent variables, the arc coordinate of the rod and the time, so the conserved quantities are written in the integral forms and there exist the arc coordinate conservation and the time conservation. The existence conditions and the formulas of conservations of momentum and moment of momentum are derived from the theorem of momentum and the theorem of moment of momentum respectively, which contain two cases of conserved quanties, one is the time and the other is arc coordinate. Also existence conditions and formulas of conservations of energy about time and are coordinate, which contain mechanical energy conservation, are derived from energy equations about the time and arc coordinate of the rod respectively. All of conservative motions of the rod are explained by examples. The conserved quantities in the integral form are of practical significance in both theoretical and numerical analysis for the Cosserat elastic rod dynamics.