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

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.

[1]  Peter Mora,et al.  Implementation of Particle-scale Rotation in the 3-D Lattice Solid Model , 2006 .

[2]  H. Goldstein,et al.  Classical Mechanics , 1951, Mathematical Gazette.

[3]  Benedict Leimkuhler,et al.  A symplectic method for rigid-body molecular simulation , 1997 .

[4]  J. Kuipers.  Quaternions and Rotation Sequences , 1998 .

[5]  D. C. Rapaport.  The Art of Molecular Dynamics Simulation: Basic molecular dynamics , 2004 .

[6]  J. Banavar,et al.  Computer Simulation of Liquids , 1988 .

[7]  D. Allen-Booth,et al.  Classical Mechanics 2nd edn , 1974 .

[8]  Sohail Murad,et al.  Singularity free algorithm for molecular dynamics simulation of rigid polyatomics , 1977 .

[9]  John-Paul Latham,et al.  3D dynamics of discrete element systems comprising irregular discrete elements—integration solution for finite rotations in 3D , 2003 .

[10]  P. Cundall,et al.  A discrete numerical model for granular assemblies , 1979 .

[11]  Thomas F. Miller,et al.  Symplectic quaternion scheme for biophysical molecular dynamics , 2002 .

[12]  O. Bauchau,et al.  The Vectorial Parameterization of Rotation , 2003 .

[13]  Denis J. Evans,et al.  On the representatation of orientation space , 1977 .

[14]  P. Cundall,et al.  A bonded-particle model for rock , 2004 .

[15]  David Swigon,et al.  Theory of sequence-dependent DNA elasticity , 2003 .

[16]  M. Oda,et al.  Rolling Resistance at Contacts in Simulation of Shear Band Development by DEM , 1998 .

[17]  David Fincham,et al.  Leapfrog Rotational Algorithms , 1992 .

[18]  Peter Mora,et al.  Simulation of the frictional stick-slip instability , 1994 .

[19]  B. Leimkuhler,et al.  Symplectic splitting methods for rigid body molecular dynamics , 1997 .

[20]  M. Thachuk,et al.  Mobilities of NO+ drifting in helium: A molecular dynamics study , 1999 .

[21]  Hai-Sui Yu,et al.  A novel discrete model for granular material incorporating rolling resistance , 2005 .

[22]  S. Buss.  Accurate and efficient simulation of rigid-body rotations , 2000 .

[23]  Numerical integration of the equations of motion for rigid polyatomics : The matrix method , 1998, physics/9901026.

[24]  Igor P. Omelyan.  On the numerical integration of motion for rigid polyatomics: the modified quaternion approach , 1998 .

[25]  Igor P. Omelyan.  Algorithm for numerical integration of the rigid-body equations of motion , 1998 .

[26]  Peter Mora,et al.  The Lattice Solid Model to Simulate the Physics of Rocks and Earthquakes: Incorporation of Granular Friction , 1999 .

[27]  P. Mora,et al.  Modeling Wing Crack Extension: Implications for the Ingredients of Discrete Element Model , 2008 .