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Research Papers

# Intrinsic Time Integration Procedures for Rigid Body Dynamics

[+] Author and Article Information
Olivier A. Bauchau

Professor, Fellow of ASME
e-mail: olivier.bauchau@sjtu.edu.cn

Hao Xin, Shiyu Dong, Zhiheng Li

Undergraduate student

Shilei Han

Graduate student
e-mail: hanl@sjtu.edu.cn
University of Michigan-Shanghai
Jiao Tong University Joint Institute,
Shanghai, 200240, P. R. C.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 25, 2011; final manuscript received February 26, 2012; published online June 14, 2012. Assoc. Editor: Dan Negrut.

J. Comput. Nonlinear Dynam 8(1), 011006 (Jun 14, 2012) (9 pages) Paper No: CND-11-1158; doi: 10.1115/1.4006252 History: Received September 25, 2011; Revised February 26, 2012

## Abstract

The treatment of rotations in rigid body and Cosserat solids dynamics is challenging. In most cases, at some point in the formulation, a parameterization of rotation is introduced and the intrinsic nature of the equations of motions is lost. Typically, this step considerably complicates the form of the equations and increases the order of the nonlinearities. Clearly, it is desirable to bypass parameterization of rotation, leaving the equations of motion in their original, intrinsic form. This has prompted the development of rotationless and intrinsic formulations. This paper focuses on the latter approach. The most famous example of intrinsic formulation is probably Euler’s second law for the motion of a rigid body rotating about an inertial point. This equation involves angular velocities solely, with algebraic nonlinearities of the second-order at most. Unfortunately, this intrinsic equation also suffers serious drawbacks: the angular velocity of the body is computed, but not its orientation, the body is “unaware” of its inertial orientation. This paper presents an alternative approach to the problem by proposing discrete statements of the rotation kinematic compatibility equation, which provide solutions for both rotation tensor and angular velocity without relying on a parameterization of rotation. The formulation is also generalized using the motion formalism, leading to very simple discretized equations of motion.

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## Figures

Fig. 1

Schematic of the discretization of the rotation tensor at times ti and tf

Fig. 2

Schematic of the discretization of the motion tensor at times ti and tf

Fig. 3

The displacement error measure, eD, versus time step size for the solution of the motion compatibility equation

Fig. 4

The rotation error measure, eR, versus time step size for the solution of the motion compatibility equation

Fig. 5

The displacement error measure, eD, versus time step size for the energy/momentum preserving scheme

Fig. 6

The rotation error measure, eR, versus time step size for the energy/momentum preserving scheme

Fig. 7

The displacement error measure, eD, versus time step size for the generalized-α scheme

Fig. 8

The rotation error measure, eR, versus time step size for the generalized-α scheme

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