A three-sub-step composite method for the analysis of rigid body rotations with Euler parameters

Abstract

This paper proposes a composite method for the analysis of rigid body rotation based on Euler parameters. The proposed method contains three sub-steps, wherein for keeping as much low-frequency information as possible the first two sub-steps adopt the trapezoidal rule, and the four-point backward interpolation formula is used in the last sub-step to flexibly control the amount of high-frequency dissipation. On this basis, in terms of the relation between Euler parameters and angular velocity, the stepping formulations of the proposed method are further modified for maximizing the accuracy of the angular velocity. For the analysis of rigid body rotation, the accuracy of the proposed method can converge to second-order, and the amount of its high-frequency dissipation can smoothly range from one (conservative scheme) to zero (annihilating scheme). Additionally, in the proposed method, the constraints at the displacement and velocity levels are strictly satisfied, and the numerical drifts at the acceleration level can be effectively eliminated. Several benchmark rigid body rotation problems show the advantages of the proposed method in stability, accuracy, dissipation, efficiency, and energy conservation.

Conserving integration of multibody systems with singular and non-constant mass matrix including quaternion-based rigid body dynamics

“…Basic textbooks on the fundamentals of quaternions can be found in [2,34] and the seminal works by Haug [23] and Nikravesh [45]. Until now, unit quaternion parametrizations of spatial rigid body rotations and their numerical integration are an active field of research [56,46,49,59,51,58,27]. The standard approach yields a singular 4 × 4 mass matrix, thus making a direct transition to the Hamiltonian setup impossible.…”