Lie group formalisms in flexible multibody dynamics

This course presents a recently-developed Lie group approach for the analysis of flexible multibody systems with large rotations and kinematic constraints in a parameterization-free setting. More precisely, the equations of motion are formulated as differential-algebraic equations (DAE) on a matrix Lie group and are then solved using a Lie group time integration method. Based on a nonlinear finite element approach, the lecture shows how the kinematics of a flexible multibody system, including the description of nodal frames, velocity fields, constraints, and flexible bodies, are described in the Lie group framework. The general structure of the equations of motion are derived from the Hamilton principle in a general and unifying approach. A Lie group generalized-alpha time integration scheme is proposed for DAEs on a Lie group and practical implementation issues are discussed. Finally, semi-analytical algorithms for sensitivity analysis on a Lie group are proposed for optimization purpose. The theoretical concepts and the numerical properties of the algorithms are illustrated with a number of examples.