Abstract
In this work, stabilized explicit integrators for local parametrization are introduced to calculate the dynamics of constrained multi-rigid-body systems, including those based on the orthogonal Runge–Kutta–Chebyshev (RKC) method and the extrapolated stabilized explicit Runge–Kutta (ESERK) method. Both of these methods have large stability regions at the negative real axis, and this property makes them suitable to settle the introduction of the stabilization parameter for a constraint equation. The local vectorial rotation parameters are adopted to describe rotations in each rigid body, and a stabilization technique is developed to transform the differential-algebraic equations (DAE) into a set of first-order ordinary differential equations (ODEs) that can be computed efficiently. Several benchmarks are calculated and the results are compared to those by the generalized-α integrator and ADAMS models, verifying their effectiveness in nonstiff problems.
Issue Section:
Research Papers
References
1.
Bauchau
,
O. A.
, and
Trainelli
,
L.
, 2003
, “
The Vectorial Parameterization of Rotation
,” Nonlinear Dyn.
,
32
(1
), pp. 71
–92
.10.1023/A:10242654015762.
Robinson
,
A. C.
, 1958
, “
On the Use of Quaternions in Simulation of Rigid-Body Motion
,” WADC, OH, Report No. 58–17.3.
Verbin
,
D.
,
Lappas
,
V.
, and
Ben-Asher
,
J. Z.
, 2011
, “
Time-Efficient Angular Steering Laws for Rigid Satellite
,” J. Guid. Control Dyn.
,
34
(3
), pp. 878
–892
.10.2514/1.481544.
Pittelkau
,
M. E.
, 2003
, “
Rotation Vector in Attitude Estimation
,” J. Guid. Control Dyn.
,
26
(6
), pp. 855
–860
.10.2514/2.69295.
Stuelpnagel
,
J.
, 1964
, “
On the Parametrization of the Three-Dimensional Rotation Group
,” SIAM Rev.
,
6
(4
), pp. 422
–430
.10.1137/10060936.
Singla
,
P.
,
Mortari
,
D.
, and
Junkins
,
J. L.
, 2005
, “
How to Avoid Singularity When Using Euler Angles?
,” Adv. Astronaut. Sci.
,
119
(1
), pp. 1409
–1426
.https://www.researchgate.net/publication/2897248717.
Mortari
,
D.
,
Angelucci
,
M.
, and
Markley
,
F. L.
, 2000
, “
Singularity and Attitude Estimation
,” Adv. Astronaut. Sci.
,
105
(1
), pp. 479
–496
.https://www.researchgate.net/publication/2541997408.
Cardona
,
A.
, and
Geradin
,
M.
, 1989
, “
Time Integration of the Equations of Motion in Mechanism Analysis
,” Comput. Struct.
,
33
(3
), pp. 801
–820
.10.1016/0045-7949(89)90255-19.
Geradin
,
M.
, 2001
, Flexible Multibody Dynamics: A Finite Element Approach
,
Wiley, New York.10.
Brüls
,
O.
, and
Cardona
,
A.
, 2010
, “
On the Use of Lie Group Time Integrators in Multibody Dynamics
,” ASME J. Comput. Nonlinear Dyn.
,
5
(3
), p. 031002
.10.1115/1.400137011.
Brüls
,
O.
,
Cardona
,
A.
, and
Arnold
,
M.
, 2012
, “
Lie Group Generalized-α Time Integration of Constrained Flexible Multibody Systems
,” Mech. Mach. Theory
,
48
(1
), pp. 121
–137
.10.1016/j.mechmachtheory.2011.07.01712.
Demoures
,
F.
,
Gay-Balmaz
,
F.
,
Leyendecker
,
S.
,
Ober-Blöbaum
,
S.
,
Ratiu
,
T. S.
, and
Weinand
,
Y.
, 2015
, “
Discrete Variational Lie Group Formulation of Geometrically Exact Beam Dynamics
,” Numer. Math.
,
130
(1
), pp. 73
–123
.10.1007/s00211-014-0659-413.
Ding
,
J.
, and
Pan
,
Z.
, 2018
, “
The Lie Group Euler Methods of Multibody System Dynamics With Holonomic Constraints
,” Adv. Mech. Eng.
,
10
(4
), pp. 1
–10
.10.1177/168781401876415414.
Terze
,
Z.
,
Müller
,
A.
, and
Zlatar
,
D.
, 2015
, “
Lie-Group Integration Method for Constrained Multibody Systems in State Space
,” Multibody Syst. Dyn.
,
34
(3
), pp. 275
–305
.10.1007/s11044-014-9439-215.
Bottasso
,
C. L.
, and
Borri
,
M.
, 1998
, “
Integrating Finite Rotations
,” Comput. Methods Appl. Mech. Eng.
,
164
(3–4
), pp. 307
–331
.10.1016/S0045-7825(98)00031-016.
Munthe-Kaas
,
H.
, 1998
, “
Runge-Kutta Methods on Lie Groups
,” BIT
,
38
(1
), pp. 92
–111
.10.1007/BF0251091917.
Holzinger
,
S.
, and
Gerstmayr
,
J.
, 2021
, “
Time Integration of Rigid Bodies Modelled With Three Rotation Parameters
,” Multibody Syst. Dyn.
,
53
(4
), pp. 345
–378
.10.1007/s11044-021-09778-w18.
Brasey
,
V.
, and
Hairer
,
E.
, 1993
, “
Symmetrized Half-Explicit Methods for Constrained Mechanical Systems
,” Appl. Numer. Math.
, 13(1), pp. 23
–31
.10.1016/0168-9274(93)90128-E19.
Arnold
,
M.
, 1998
, “
Half-Explicit Runge-Kutta Methods With Explicit Stages for Differential-Algebraic Systems of Index 2
,” BIT
,
38
(3
), pp. 415
–438
.10.1007/BF0251025220.
Linh
,
V. H.
, and
Truong
,
N. D.
, 2020
, “
On Convergence of Continuous Half-Explicit Runge-Kutta Methods for a Class of Delay Differential-Algebraic Equations
,” Numer. Algorithms
,
85
(1
), pp. 277
–203
.10.1007/s11075-019-00813-821.
Prentice
,
J. S.
, 2011
, “
Stepsize Selection in Explicit Runge-Kutta Methods for Moderately Stiff Problems
,” Appl. Math.
,
02
(6
), pp. 711
–717
.10.4236/am.2011.2609422.
Stasio
,
J. D.
,
Dureisseix
,
D.
,
Georges
,
G.
,
Gravouil
,
A.
, and
Homolle
,
T.
, 2021
, “
An Explicit Time-Integrator With Singular Mass for Non-Smooth Dynamics
,” Comput. Mech.
,
68
(1
), pp. 97
–112
.10.1007/s00466-021-02021-523.
Bae
,
D. S.
,
Lee
,
J. K.
,
Cho
,
H. J.
, and
Yae
,
H.
, 2000
, “
An Explicit Integration Method for Realtime Simulation of Multibody Vehicle Models
,” Comput. Methods Appl. Mech. Eng.
,
187
(1–2
), pp. 337
–350
.10.1016/S0045-7825(99)00138-324.
Naets
,
F.
,
Tamarozzi
,
T.
,
Heirman
,
G.
, and
Desmet
,
W.
, 2012
, “
Real-Time Flexible Multibody Simulation With Global Modal Parameterization
,” Multibody Syst. Dyn.
,
27
(3
), pp. 267
–284
.10.1007/s11044-011-9298-z25.
Verwer
,
J. G.
,
Hundsdorfer
,
W. H.
, and
Sommeijer
,
B. P.
, 1990
, “
Convergence Properties of the Runge-Kutta-Chebyshev Method
,” Numer. Math.
,
57
(1
), pp. 157
–178
.10.1007/BF0138640526.
Martín-Vaquero
,
J.
, and
Kleefeld
,
B.
, 2016
, “
Extrapolated Stabilized Explicit Runge-Kutta Methods
,” J. Comput. Phys.
,
326
(1
), pp. 141
–155
.10.1016/j.jcp.2016.08.04227.
Medovikov
,
A. A.
, 1998
, “
High Order Explicit Methods for Parabolic Equations
,” BIT
,
38
(2
), pp. 372
–390
.10.1007/BF0251237328.
Abdulle
,
A.
, and
Medovikov
,
A. A.
, 2001
, “
Second Order Chebyshev Methods Based on Orthogonal Polynomials
,” Numer. Math.
,
90
(1
), pp. 1
–18
.10.1007/s00211010029229.
Abdulle
,
A.
, 2002
, “
Fourth Order Chebyshev Methods With Recurrence Relation
,” SIAM J. Sci. Comput.
,
23
(6
), pp. 2041
–2054
.10.1137/S106482750037954930.
Ren
,
H.
, and
Zhou
,
P.
, 2020
, “
A Fast Explicit Integrator for Numerical Simulation of Multibody System Dynamics
,” Multibody Dynamics 2019
,
A.
Kecskeméthy
and
F.
Geu Flores
, eds.,
Springer
,
Cham
, Switzerland, pp. 348
–355
.31.
Shabana
,
A. A.
, 2005
, Dynamics of Multibody Systems
, 3rd ed.,
Cambridge University Press
,
Cambridge, UK
.32.
Bahar
,
L. Y.
, 1970
, “
Direct Determination of Finite Rotation Operators
,” J. Franklin Inst.
,
289
(5
), pp. 401
–404
.10.1016/0016-0032(70)90208-533.
Argyris
,
J.
, 1982
, “
An Excursion Into Large Rotations
,” Comput. Methods Appl. Mech. Eng.
,
32
(1–3
), pp. 85
–155
.10.1016/0045-7825(82)90069-X34.
Sonneville
,
V.
, and
Brüls
,
O.
, 2014
, “
Sensitivity Analysis for Multibody Systems Formulated on a Lie Group
,” Multibody Syst. Dyn.
,
31
(1
), pp. 47
–67
.10.1007/s11044-013-9345-z35.
Yu
,
W.
, and
Pan
,
Z.
, 2015
, “
Dynamical Equations of Multibody Systems on Lie Groups
,” Adv. Mech. Eng.
,
7
(3
), pp. 1
–9
.10.1177/168781401557595936.
Ren
,
H.
, and
Yang
,
K.
, 2021
, “
A Referenced Nodal Coordinate Formulation
,” Multibody Syst. Dyn.
,
51
(3
), pp. 305
–342
.10.1007/s11044-020-09750-037.
Gear
,
C.
,
Leimkuhler
,
B.
, and
Gupta
,
G.
, 1985
, “
Automatic Integration of Euler-Lagrange Equations With Constraints
,” J. Comput. Appl. Math.
,
12–13
, pp. 77
–90
.10.1016/0377-0427(85)90008-138.
Baumgarte
,
J.
, 1972
, “
Stabilization of Constraints and Integrals of Motion in Dynamical Systems
,” Comput. Methods Appl. Mech. Eng.
,
1
(1
), pp. 1
–16
.10.1016/0045-7825(72)90018-739.
Gear
,
C. W.
, 2006
, “
Towards Explicit Methods for Differential Algebraic Equations
,” BIT
,
46
(3
), pp. 505
–514
.10.1007/s10543-006-0068-x40.
Bauchau
,
O. A.
, 2003
, “
A Self-Stabilized Algorithm for Enforcing Constraints in Multibody Systems
,” Int. J. Solids Struct.
,
40
(13–14
), pp. 3253
–3271
.10.1016/S0020-7683(03)00159-841.
Bayo
,
E.
, and
Ledesma
,
R.
, 1996
, “
Augmented Lagrangian and Mass-Orthogonal Projection Methods for Constrained Multibody Dynamics
,” Nonlinear Dyn.
,
9
(1–2
), pp. 113
–130
.10.1007/BF0183329642.
Braun
,
D. J.
, and
Goldfarb
,
M.
, 2009
, “
Eliminating Constraint Drift in the Numerical Simulation of Constrained Dynamical Systems
,” Comput. Methods Appl. Mech. Eng.
,
198
(37–40
), pp. 3151
–3160
.10.1016/j.cma.2009.05.01343.
Hairer
,
E.
, and
Wanner
,
G.
, 1996
, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems
, 2nd ed.,
Springer-Verlag
,
Berlin Heidelberg, Germany
.44.
45.
Van Der Houwen
,
P. J.
, and
Sommeijer
,
B. P.
, 1980
, “
On the Internal Stability of Explicit M-Stage Runge-Kutta Methods for Large M-Values
,” ZAMM Z. Angew. Math. Mech.
,
60
(10
), pp. 479
–485
.10.1002/zamm.1980060100546.
Kleefeld
,
B.
, and
Martín-Vaquero
,
J.
, 2016
, “
SERK2v3: Solving Mildly Stiff Nonlinear Partial Differential Equations
,” J. Comput. Appl. Math.
,
299
, pp. 194
–206
.10.1016/j.cam.2015.11.04547.
O'Sullivan
,
S.
, 2015
, “
A Class of High-Order Runge-Kutta-Chebyshev Stability Polynomials
,” J. Comput. Phys.
,
300
, pp. 665
–678
.10.1016/j.jcp.2015.07.05048.
Hairer
,
E.
,
Nørsett
,
S. P.
, and
Wanner
,
G.
, 1987
, Solving Ordinary Differential Equations I: Nonstiff Problems
, 2nd ed.,
Springer-Verlag
,
Berlin Heidelberg, Germany
.49.
Shampine
,
L. F.
, 2018
, Numerical Solution of Ordinary Differential Equations
,
Chapman & Hall
,
NewYork
.50.
Gustafsson
,
K.
, 1994
, “
Control-Theoretic Techniques for Stepsize Selection in Implicit Runge-Kutta Methods
,” ACM T. Math. Software
,
20
(4
), pp. 496
–517
.10.1145/198429.19843751.
Watts
,
H. A.
, 1984
, “
Step Size Control in Ordinary Differential Equation Solvers
,” Sandia National Labs., Albuquerque, NM, Report No. SAND-83-2262
.https://www.osti.gov/biblio/537105852.
Sommeijer
,
B.
,
Shampine
,
L.
, and
Verwer
,
J.
, 1998
, “
RKC: An Explicit Solver for Parabolic PDEs
,” J. Comput. Appl. Math.
,
88
(2
), pp. 315
–326
.10.1016/S0377-0427(97)00219-753.
Chung
,
J.
, and
Hulbert
,
G. M.
, 1993
, “
A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method
,” ASME J. Appl. Mech.
,
60
(2
), pp. 371
–375
.10.1115/1.290080354.
Arnold
,
M.
, and
Brüls
,
O.
, 2007
, “
Convergence of the Generalized-α Scheme for Constrained Mechanical Systems
,” Multibody Syst. Dyn.
,
18
(2
), pp. 185
–202
.10.1007/s11044-007-9084-055.
Bricard
,
R.
, 1927
, Leçons de Cinématique. II: Cinématique Appliquée
,
Gauthier-Villars
, Paris, France.Copyright © 2022 by ASME
You do not currently have access to this content.
