Three Dimensional Nonlinear Dynamics of Slender Structures: Cosserat Rod   Element Approach
  In this paper, the modelling strategy of a Cosserat rod element (CRE) is addressed systematically for 3-dimensional dynamical analysis of slender structures. We employ the exact nonlinear kinematic relationships in the sense of Cosserat theory, and adopt the Bernoulli hypothesis. For the sake of simplicity, the Kirchoff constitutive relations are adopted to provide an adequate description of elastic properties in terms of a few elastic moduli. A deformed configuration of the rod is described by the displacement vector of the deformed centroid curves and an orthonormal moving frame, rigidly attached to the cross-section of the rod. The position of the moving frame relative to the inertial frame is specified by the rotation matrix, parametrized by a rotational vector. The approximate solutions of the nonlinear partial differential equations of motion in quasi-static sense are chosen as the shape functions with up to third order nonlinear terms of generic nodal displacements. Based on the Lagrangian constructed by the Cosserat kinetic energy and strain energy expressions, the principle of virtual work is employed to derive the ordinary differential equations of motion with third order nonlinear generic nodal displacements. A simple example is presented to illustrate the use of the formulation developed here to obtain the lower order nonlinear ordinary differential equations of motion of a given structure. The corresponding nonlinear dynamical responses of the structures have been presented through numerical simulations by Matlab software. 
D. Q. Cao, Dongsheng Liu, Charles H.-T. Wang