基于高斯原理的Cosserat弹性杆动力学模型

Dynamical model of Cosserat elastic rod based on Gauss principle

在动力学普遍原理中,高斯最小拘束原理的特点是可通过寻求函数极值的变分方法直接得出运动规律,而无须建立动力学微分方程.Kirchhoff动力学比拟方法以刚性截面的姿态表述弹性细杆的几何形态,并发展为以弧坐标s和时间t为自变量的弹性杆分析力学.由于截面姿态的局部微小改变沿弧坐标的积累不受限制,Kirchhoff模型适合描述弹性杆的超大变形.Cosserat弹性杆模型考虑了Kirchhoff模型忽略的截面剪切变形、中心线伸缩变形和分布力等因素,是更符合实际弹性杆的动力学模型.建立了基于高斯原理的Cosserat弹性杆的分析力学模型,导出拘束函数的普遍形式,以平面运动为例进行讨论.关于弹性杆空间不可自相侵占的特殊问题,给出相应的约束条件对可能运动施加限制,以避免自相侵占情况发生.

Based on the generalized principles of dynamics, the/eature ot ~auss prmclpm o~ Jeas~ co^1~1~11,~ l~ law can be directly obtained by using the variation method of seeking the minimal value of the constraint function without establishing any dynamic differential equations. According to the Kirchhoff＇s dynamic analogy, the configuration of an elastic rod can be described by the rotation of rigid cross section of the rod along the centerline. Since the local small change of the attitude of cross section can be accumulated infinitely along the arc-coordinate, the Kirchhoff＇s model is suited to describe the super-large deformation of elastic rod. Therefore the analytical mechanics of elastic rod with arc-coordinate s and time t as double arguments has been developed. The Cosserat model of elastic rod takes into consideration the factors neglected by the Kirchhoff model, such as the shear deformation of cross section, the tensile deformation of centerline, and distributed load, so it is more suitable to modeling a real elastic rod. In this paper, the model of the Cosserat rod is established based on the Gauss principle, and the constraint function of the rod is derived in the general form. The plane motion of the rod is discussed as a special case. As regards the special problem that different p^rts of the rod in space are unable to self-invade each other, a constraint condition is derived to restrict the possible configurations in variation calculation so as to avoid the invading possibility.