# A new algorithm to model the dynamics of 3-D bonded rigid bodies with rotations

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## Abstract

In this paper we propose a new algorithm to simulate the dynamics of 3-D interacting rigid bodies. Six degrees of freedom are introduced to describe a single 3-D body or particle, and six relative motions and interactions are permitted between bonded bodies. We develop a new decomposition technique for 3-D rotation and pay particular attention to the fact that an arbitrary relative rotation between two coordinate systems or two rigid bodies can not be decomposed into three mutually independent rotations around three orthogonal axes. However, it can be decomposed into two rotations, one pure axial rotation around the line between the centers of two bodies, and another rotation on a specified plane controlled by another parameter. These two rotations, corresponding to the relative axial twisting and bending in our model, are sequence-independent. Therefore all interactions due to the relative translational and rotational motions between linked bodies can be uniquely determined using such a two-step decomposition technique. A complete algorithm for one such simulation is presented. Compared with existing methods, this algorithm is physically more reliable and has greater numerical accuracy.

## Keywords

Bonded rigid-bodies Decomposition of 3-D finite rotations Quaternion## List of symbols

## Coordinate systems

*O*−*XYZ*the space-fixed system

*O*_{1}−*X*_{1}*Y*_{1}*Z*_{1}the body-fixed frame of particle 1

*O*_{2}−*X*_{2}*Y*_{2}*Z*_{2}the body-fixed frame of particle 2

*O*_{2}−*X*_{2}′*Y*_{2}′*Z*_{2}′an auxiliary body-fixed frame of particle 2, obtained by directly rotating

*X*_{2}*Y*_{2}*Z*_{2}at T = 0 such that its*Z*_{2}′-axis is pointing to particle 1. There is no relative rotation between*X*_{2}′*Y*_{2}′*Z*_{2}′ and*X*_{2}*Y*_{2}*Z*_{2}*O*_{2}−*X*_{1}′*Y*_{1}′*Z*_{1}′another auxiliary frame. The relative rotation from

*X*_{2}*Y*_{2}*Z*_{2}to*X*_{1}*Y*_{1}*Z*_{1}makes*X*_{2}′*Y*_{2}′*Z*_{2}′ rotate to*X*_{1}′*Y*_{1}′*Z*_{1}′

## Vectors

**f**total force acting on the particle, measured in the space-fixed system

*XYZ*- \({\varvec{\tau}}^{\varvec{b}}\)
total torque acting on the particle expressed in body-fixed frame

**f**_{r}normal force between two particles

**f**_{s1},**f**_{s2}shear forces between two particles

- \({\varvec{\tau}}_{\varvec{t}}\)
torque cause by twisting or torsion between two particles

- \({\varvec{\tau}}_{\varvec{{b1}}} , {\varvec{\tau}}_{\varvec{{b2}}}\)
torques cause by relative bending between two particles

- \(\Updelta {\varvec{\alpha}}_{\varvec{t}}\)
relative angular displacement caused by twisting motion

- \(\Updelta {\varvec{\alpha}}_{\varvec{{b1}}}, \Updelta \alpha_{b2}\)
relative angular displacements caused by bending motion

- Δ
**u**_{r} relative displacement in normal direction

- Δ
**u**_{s1}, Δ**u**_{s2} relative displacements in tangent directions

**r**position vector of a particle, measured in

*XYZ*- \({\varvec{\omega}}^{\varvec{b}}\)
angular velocities measured in the body-fixed frame

**r**_{10},**r**_{20}initial position of particle 1 and particle 2, measured in

*XYZ***r**_{1},**r**_{2}current positions of particle 1 and particle 2, measured in

*XYZ***r**_{0}initial position of particle 1 relative to particle 2, measured in

*X*_{2}*Y*_{2}*Z*_{2}(or*XYZ*)**r**_{c}current position of particle 1 relative to particle 2, measured in

*X*_{2}*Y*_{2}*Z*_{2}**r**_{f}current position of particle 1 relative to particle 2, measured in

*XYZ*- Δ
**r** translational displacement of particle 1 relative to particle 2, measured in

*X*_{2}*Y*_{2}*Z*_{2}**f**_{s}^{t}shear force caused by translational motion of particle 1 relative to particle 2, measured in

*X*_{2}*Y*_{2}*Z*_{2}- \({\varvec{\tau}}_{\varvec{s}}^{\varvec{t}}\)
torque generated by

*f*_{ s }^{ t }, measured in*X*_{2}*Y*_{2}*Z*_{2}**f**_{s}^{r}shear force caused by the rotation of particle 1 relative to particle 2, measured in

*X*_{2}*Y*_{2}*Z*_{2}- \({\varvec{\tau}}_{\varvec{s}}^{\varvec{r}}\)
torque generated by

*f*_{ s }^{ r }, measured in*X*_{2}*Y*_{2}*Z*_{2}- \({\varvec{\tau}}_{\varvec{b}}^{\varvec{r}}\)
bending torque cause by the rotation of particle 1 relative to particle 2, measured in

*X*_{2}*Y*_{2}*Z*_{2}- \({\varvec{\tau}}_{\varvec{t}}^{\varvec{r}}\)
twisting torque cause by the rotation of particle 1 relative to particle 2, measured in

*X*_{2}*Y*_{2}*Z*_{2}- \({\varvec{\tau}}^{\prime}_{\varvec{b}}\)
bending torque cause by the rotation of particle 1 relative to particle 2, measured in

*X*_{2}′*Y*_{2}′*Z*_{2}′- \({\varvec{\tau}}^{\prime}_{\varvec{t}}\)
twisting torque cause by the rotation of particle 1 relative to particle 2, measured in

*X*_{2}′*Y*_{2}′*Z*_{2}′**f**_{s}′shear force caused by the rotation of particle 1 relative to particle 2, measured in

*X*_{2}′*Y*_{2}′*Z*_{2}′- \({\varvec{\tau}}^{\prime}_{\varvec{s}}\)
torque generated by

*f*_{ s }, measured in*X*_{2}′*Y*_{2}′*Z*_{2}′

## Matrices

**Ω**matrix of angular velocity

- \({\dot{\varvec{Q}}}\)
matrix of quaternion derivative

**Q**_{0}(*q*)matrix linking Ω and \({\dot{Q}}\)

## Quaternions

*p*^{0},*q*^{0}initial orientations of particle 1 and particle 2, expressed in

*XYZ**p*,*q*current orientations of particle 1 and particle 2, expressed in

*XYZ**g*^{o}rotation from

*X*_{2}*Y*_{2}*Z*_{2}to*X*_{1}*Y*_{1}*Z*_{1}, expressed in*X*_{2}*Y*_{2}*Z*_{2}*g*rotation from

*X*_{2}′*Y*_{2}′*Z*_{2}′ to*X*_{1}′*Y*_{1}′*Z*_{1}′, expressed in*X*_{2}′*Y*_{2}′*Z*_{2}′*h*rotation from

*X*_{2}*Y*_{2}*Z*_{2}to*X*_{2}′*Y*_{2}′*Z*_{2}′, expressed in*X*_{2}*Y*_{2}*Z*_{2}

## Others

- ψ
twisting angle between two particles

- θ
bending angle between two particles

- φ
orientation angle determining the plane on which bending occurs

*K*_{r}normal stiffness

*K*_{s},*K*_{s1},*K*_{s2}shear stiffness

*K*_{t}twisting (torsional) stiffness

*K*_{b}*K*_{b1},*K*_{b2}bending stiffness

*M*mass of a rigid body or particle

*I*_{xx},*I*_{yy},*I*_{zz}three principle moments of inertia in the body-fixed frame

## Notes

### Acknowledgments

Funding support is gratefully acknowledged by the Australian Computational Earth Systems Simulator Major National Research Facility, The University of Queensland and SGI. The ACcESS MNRF is funded by the Australian Commonwealth Government and participating institutions (University of Queensland, Monash U, Melbourne U., VPAC, RMIT) and the Victorian State Government. The author would like to thank Dr. Louise Kettle, Mr. William Hancock and Dr. Junfang Zhang for their valuable suggestions to improve the manuscript.

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