Dynamical model of Cosserat elastic rod based on Gauss principle

Abstract

Based on the generalized principles of dynamics, the feature of Gauss principle of least constraint is that the motion 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 parts 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.