The paper presents two novel second order conservative Lie-group geometric methods for integration of rigid body rotational dynamics. First proposed algorithm is a fully explicit scheme that exactly conserves spatial angular momentum of a free spinning body. The method is inspired by the Störmer–Verlet integration algorithm for solving ordinary differential equations (ODEs), which is also momentum conservative when dealing with ODEs in linear spaces but loses its conservative properties in a nonlinear regime, such as nonlinear SO(3) rotational group. Then, we proposed an algorithm that is an implicit integration scheme with a direct update in SO(3). The method is algorithmically designed to conserve exactly both of the two “main” motion integrals of a rotational rigid body, i.e., spatial angular momentum of a torque-free body as well as its kinetic energy. As it is shown in the paper, both methods also preserve Lagrangian top integrals of motion in a very good manner, and generally better than some of the most successful conservative schemes to which the proposed methods were compared within the presented numerical examples. The proposed schemes can be easily applied within the integration algorithms of the dynamics of general rigid body systems.

References

References
1.
Bauchau
,
O. A.
,
2011
,
Flexible Multibody Dynamics
,
Springer, Dordrecht
,
Heidelberg, London, NY
.
2.
Hairer
,
E.
,
Lubich
,
C.
, and
Wanner
,
G.
,
2006
,
Geometric Numerical Integration
,
Springer Verlag
, Berlin.
3.
Marsden
,
J. E.
, and
Ratiu
,
T.
,
1998
,
Mechanics and Symmetry
,
Springer
,
NY
.
4.
Hairer
,
E.
,
Lubich
,
C.
, and
Wanner
,
G.
,
2003
, “
Geometric Numerical Integration Illustrated by the Störmer–Verlet Method
,”
Acta Numerica
,
12
(
12
), pp.
399
450
.
5.
Leimkuhler
,
B.
, and
Reich
,
S.
,
2004
,
Simulating Hamiltonian Dynamics
,
Cambridge University
,
UK
.
6.
Reich
,
S.
,
1996
, “
Symplectic Methods for Conservative Multibody Systems
,”
Integration Algorithms and Classical Mechanics
,
American Mathematical Society, Providence
,
RI
, Vol.
10
, pp.
181
192
.
7.
McLachlan
,
R. I.
, and
Scovel
,
C.
,
1995
, “
Equivariant Constrained Symplectic Integration
,”
J. Nonlinear Sci.
,
5
(
3
), pp.
233
256
.
8.
Betsch
,
P.
, and
Steinmann
,
P.
,
2001
, “
Constrained Integration of Rigid Body Dynamics
,”
Comput. Methods Appl. Mech. Eng.
,
191
(
3
), pp.
467
488
.
9.
Müller
,
A.
,
2010
, “
Approximation of Finite Rigid Body Motions From Velocity Fields
,”
J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM)
,
90
(
6
), pp.
514
521
.
10.
Holm
,
D. D.
,
2008
,
Geometric Mechanics. Part II: Rotating, Translating and Rolling
,
Imperial College Press
,
London
.
11.
Munthe-Kaas
,
H.
,
1998
, “
Runge–Kutta Methods on Lie Groups
,”
BIT
,
38
(
1
), pp.
92
111
.
12.
Celledoni
,
E.
, and
Owren
,
B.
,
2003
, “
Lie Group Methods for Rigid Body Dynamics and Time Integration on Manifolds
,”
Comput. Methods Appl. Mech. Eng.
,
192
(
3
), pp.
421
438
.
13.
Lewis
,
D.
, and
Simo
,
J. C.
,
1994
, “
Conserving Algorithms for the Dynamics of Hamiltonian Systems on Lie Groups
,”
J. Nonlinear Sci.
,
4
(
1
), pp.
253
299
.
14.
Iserles
,
A.
,
Munthe–Kaas
,
H. Z.
,
Norsett
,
S. P.
, and
Zanna
,
A.
,
2000
, “
Lie-Group Methods
,”
Acta Numerica
,
9
, pp.
215
365
.
15.
Engø
,
K.
, and
Faltinsen
,
S.
,
1999
, “
Numerical Integration of Lie-Poisson Systems While Preserving Coadjoint Orbits and Energy
,”
University of Bergen
, Bergen, Report No. 179.
16.
Simo
,
J. C.
, and
Wong
,
K. K.
,
1991
, “
Unconditionally Stable Algorithms for Rigid Body Dynamics That Exactly Preserve Energy and Momentum
,”
Int. J. Numer. Methods Eng.
,
31
(
1
), pp.
19
52
.
17.
Austin
,
M.
,
Krishnaprasad
,
P. S.
, and
Wang
,
L. S.
,
1993
, “
Almost Lie-Poisson Integrators for the Rigid Body
,”
J. Comput. Phys.
,
107
(
1
), pp.
105
117
.
18.
Krysl
,
P.
, and
Endres
,
L.
,
2005
, “
Explicit Newmark/Verlet Algorithm for Time Integration of the Rotational Dynamics of Rigid Bodies
,”
J. Numer. Methods Eng.
,
62
(
15
), pp.
2154
2177
.
19.
Krysl
,
P.
,
2008
, “
Dynamically Equivalent Implicit Algorithms for the Integration of Rigid Body Rotations
,”
Commun. Numer. Methods Eng.
,
24
(
2
), pp.
141
156
.
20.
Bottasso
,
C. L.
, and
Borri
,
M.
,
1998
, “
Integrating Finite Rotations
,”
Comput. Methods Appl. Mech. Eng.
,
164
(
3–4
), pp.
307
331
.
21.
Müller
,
A.
, and
Terze
,
Z.
,
2014
, “
On the Choice of Configuration Space for Numerical Lie Group Integration of Constrained Rigid Body Systems
,”
J. Comput. Appl. Math.
,
262
(1), pp.
3
13
.
You do not currently have access to this content.