Acta Geotechnica

, Volume 4, Issue 2, pp 117–127

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

• Yucang Wang
Research Paper

## 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

## Coordinate systems

OXYZ

the space-fixed system

O1X1Y1Z1

the body-fixed frame of particle 1

O2X2Y2Z2

the body-fixed frame of particle 2

O2X2Y2Z2

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 2Y 2Z 2′ and X 2 Y 2 Z 2

O2X1Y1Z1

another auxiliary frame. The relative rotation from X 2 Y 2 Z 2 to X 1 Y 1 Z 1 makes X 2Y 2Z 2′ rotate to X 1Y 1Z 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

fr

normal force between two particles

fs1, fs2

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

Δur

relative displacement in normal direction

Δus1, Δus2

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

r10, r20

initial position of particle 1 and particle 2, measured in XYZ

r1, r2

current positions of particle 1 and particle 2, measured in XYZ

r0

initial position of particle 1 relative to particle 2, measured in X 2 Y 2 Z 2 (or XYZ)

rc

current position of particle 1 relative to particle 2, measured in X 2 Y 2 Z 2

rf

current position of particle 1 relative to particle 2, measured in XYZ

Δr

translational displacement of particle 1 relative to particle 2, measured in X 2 Y 2 Z 2

fst

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

fsr

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 2Y 2Z 2

$${\varvec{\tau}}^{\prime}_{\varvec{t}}$$

twisting torque cause by the rotation of particle 1 relative to particle 2, measured in X 2Y 2Z 2

fs

shear force caused by the rotation of particle 1 relative to particle 2, measured in X 2Y 2Z 2

$${\varvec{\tau}}^{\prime}_{\varvec{s}}$$

torque generated by f s , measured in X 2Y 2Z 2

## Matrices

Ω

matrix of angular velocity

$${\dot{\varvec{Q}}}$$

matrix of quaternion derivative

Q0 (q)

matrix linking Ω and $${\dot{Q}}$$

## Quaternions

p0, q0

initial orientations of particle 1 and particle 2, expressed in XYZ

p, q

current orientations of particle 1 and particle 2, expressed in XYZ

go

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 2Y 2Z 2′ to X 1Y 1Z 1′, expressed in X 2Y 2Z 2

h

rotation from X 2 Y 2 Z 2 to X 2Y 2Z 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

Kr

normal stiffness

Ks, Ks1, Ks2

shear stiffness

Kt

twisting (torsional) stiffness

KbKb1, Kb2

bending stiffness

M

mass of a rigid body or particle

Ixx, Iyy, Izz

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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