Dynamics of slender beam based on geometrically exact Kirchhoff beam theory formulated on SO(3) group

A geometrically exact Kirchhoff beam formulation (GEKBF) established on the Lie group SO(3) for simulating the dynamics of a slender beam is proposed. The kinematic description, dynamic equilibrium equations and their linearization are derived in this framework. Then, a second-order interpolation function for the torsion angle is introduced to improve the convergence of the element. Next, the rotation vector parameterization is developed to reduce the geometric nonlinearity caused by the rotation of the rigid body. In addition, the influence of the reference frame is considered, and the derivation of the elastic force and Jacobian matrix is simplified using the spatial-parallel transport. The semi-discrete equations of motion are in the form of a second-order ordinary differential equation on Lie group, which are solved using the Lie group generalized α-method. Finally, the accuracy of the proposed formulation is verified using several numerical examples.