This paper proposes a family of Lie group time integrators for the simulation of flexible multibody systems. The method provides an elegant solution to the rotation parametrization problem. As an extension of the classical generalized-$α$ method for dynamic systems, it can deal with constrained equations of motion. Second-order accuracy is demonstrated in the unconstrained case. The performance is illustrated on several critical benchmarks of rigid body systems with high rotation speeds, and second-order accuracy is evidenced in all of them, even for constrained cases. The remarkable simplicity of the new algorithms opens some interesting perspectives for real-time applications, model-based control, and optimization of multibody systems.

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