This paper presents an equation of motion for numerical simulation of constrained mechanical systems with holonomic and nonholonomic constraints. In order to avoid the error accumulation typically experienced in such simulations, the standard equation of motion is enhanced with embedded force and impulse terms which perform continuous constraint and energy correction along the numerical solution. To avoid interference between the kinematic constraint correction and the energy correction terms, both are derived by taking the geometry of the constrained dynamics rigorously into account. In this light, enforcement of the (ideal) holonomic and nonholonomic kinematic constraints are performed using ideal forces and impulses, while the energy conservation law is considered as a nonideal nonlinear nonholonomic constraint on the simulated motion, and as such it is enforced with nonideal forces. As derived, the equation can be directly discretized and integrated with an explicit ODE solver avoiding the need for expensive implicit integration and iterative constraint stabilization. Application of the proposed equation is demonstrated on a representative example. A more elaborate discussion of practical implementation is presented in Part II of this work.

References

References
1.
Lagrange
,
J. L.
, 1787,
Mecanique Analytique
,
Mme Ve Courcier
,
Paris
.
2.
Gear
,
C. W.
, 988, “
Differential-Algebraic Equation Index Transformations
,”
SIAM J. Sci. Stat. Comput.
,
9
, pp.
39
47
.
3.
Gauss
,
C. F.
, 1829, “
Über ein neues allgemeines Grundgesetz der Mechanik
,”
J. Reine Angew. Math.
,
4
, pp.
232
235
. Available at: http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN236006339http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN236006339
4.
Maggi
,
G. A.
, 1896,
Principii della Teoria Mathematica del Movimento dei Corpi: Corso di Meccanica Razionale
,
Ulrico Hoepli
,
Milano
.
5.
Gibbs
,
J. W.
, 1879, “
On the Fundamental Formulae of Dynamics
,”
Am. J. Math.
,
2
(
1
), pp.
49
64
.
6.
Appell
,
P.
, 1899, “
Sur une forme generale des equations de la dynamique
,”
C. R. Acad. Sci., Paris
,
129
, pp.
459
460
.
7.
Hamel
,
G.
, 1949,
Theoretische Mechanik
,
Springer
,
Berlin
.
8.
Kane
,
T. R.
, and
Levinson
,
D. A.
, 1985,
Dynamics: Theory and Applications
,
McGraw-Hill
,
New York
.
9.
Udwadia
,
F. E.
, and
Kalaba
,
R. E.
, 1992, “
A New Perspective on Constrained Motion
,”
Proc. R. Soc. London, Ser. A
,
349
(
1906
), pp.
407
410
.
10.
Udwadia
,
F. E.
, and
Kalaba
,
R. E.
, 1996,
Analytical Dynamics: A New Approach
,
Cambridge University Press
,
Cambridge
.
11.
Udwadia
,
F. E.
, and
Kalaba
,
R. E.
, 2002, “
On the Foundations of Analytical Dynamics
,”
Int. J. Non-Linear Mech.
,
37
(
6
), pp.
1079
1090
.
12.
Pars
,
L. A.
, 1965,
A Treatise on Analytical Dynamics
,
Wiley
,
New York
.
13.
Neimark
,
J. I.
, and
Fufaev
,
N. A.
, 1972,
Dynamics of Nonholonomic Systems
,
AMS
,
Providence
.
14.
Goldstein
,
H.
, 1980,
Classical Mechanics
,
Addison-Wesley
,
Reading, MA
.
15.
Arnold
,
V. I.
, 1989,
Mathematical Methods of Classical Mechanics
,
Springer
,
New York
.
16.
Lurie
,
A. I.
, 2002,
Analytical Mechanics
,
Springer
,
New York
.
17.
Papastavridis
,
J. G.
, 2002,
Analytical Mechanics: A Comprehensive Treatise on the Dynamics of Constrained Systems; for Engineers, Physicists, and Mathematicians
,
Oxford University
,
New York
.
18.
Vujanović
,
B. D.
, and
Atanacković
,
T. M.
, 2003,
An Introduction to Modern Variational Techniques in Mechanics and Engineering
,
Birkhäuser
,
Boston
.
19.
Schiehlen
,
W.
, 1997, “
Multibody System Dynamics: Roots and Perspectives
,”
Multibody Syst. Dyn.
,
1
(
2
), pp.
149
188
.
20.
Shabana
,
A. A.
, 1997, “
Flexible Multibody Dynamics: Review of Past and Recent Developments
,”
Multibody Syst. Dyn.
,
1
(
2
), pp.
189
222
.
21.
Brogliato
,
B.
,
ten Dam
,
A. A.
,
Paoli
,
L.
,
Génot
,
F.
, and
Abadie
,
M.
, 2002, “
Numerical Simulation of Finite Dimensional Multibody Nonsmooth Mechanical Systems
,”
Appl. Mech. Rev.
,
55
(
2
), pp.
107
150
.
22.
Eberhard
,
P.
, and
Schiehlen
,
W.
, 2006, “
Computational Dynamics of Multibody Systems: History, Formalisms, and Applications
,”
J. Comput. Nonlinear Dyn.
,
1
(
1
), pp.
3
12
.
23.
Laulusa
,
A.
, and
Bauchau
,
O. A.
, 2008, “
Review of Classical Approaches for Constraint Enforcement in Multibody Systems
,”
J. Comput. Nonlinear Dyn.
,
3
(
1
), p.
011004
.
24.
Bauchau
,
O. A.
, and
Laulusa
,
A.
, 2008, “
Review of Contemporary Approaches for Constraint Enforcement in Multibody Systems
,”
J. Comput. Nonlinear Dyn.
,
3
(
1
), p.
011005
.
25.
Wehage
,
R. A.
, and
Haug
,
E. J.
, 1982, “
Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems
,”
J. Mech. Des.
,
104
(
1
), pp.
247
255
.
26.
Gear
,
C. W.
,
Leimkuhler
,
B.
, and
Gupta
,
G. K.
, 1985, “
Automatic Integration of Euler-Lagrange Equations with Constraints
,”
J. Comput. Appl. Math.
,
12–13
, pp.
77
90
.
27.
Führer
,
C.
, and
Leimkuhler
,
B.
, 1991, “
Numerical Solution of Differential-Algebraic Equations for Constrained Mechanical Motion
,”
Numer. Math.
,
59
(
1
), pp.
55
69
.
28.
Eich
,
E.
, 1993, “
Convergence Results for a Coordinate Projection Method Applied to Mechanical Systems with Algebraic Constraints
,”
SIAM J. Numer. Anal.
,
30
(
5
), pp.
1467
1482
.
29.
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
.
30.
Cuadrado
,
J.
,
Cardenal
,
J.
, and
Bayoj
,
E.
, 1997, “
Modeling and Solution Methods for Efficient Real Time Simulation of Multibody Dynamics
,”
Multibody Syst. Dyn.
,
1
(
3
), pp.
259
280
.
31.
Yun
,
X.
, and
Sarkar
,
N.
, 1998, “
Unified Formulation of Robotic Systems with Holonomic and Nonholonomic Constraints
,”
IEEE Trans. Rob. Autom.
,
14
(
4
), pp.
640
650
.
32.
Chen
,
S.
,
Hansen
,
J. M.
, and
Tortorelli
,
D. A.
, 2000, “
Unconditionally Energy Stable Implicit Time Integration: Application to Multibody System Analysis and Design
,”
Int. J. Numer. Methods. Eng.
,
48
(
6
), pp.
791
822
. Available at http://onlinelibrary.wiley.com/doi/10.1002/(SICI)1097-0207(20000630)48:6%3C791::AID-NME859%3E3.0.CO;2-Z/abstracthttp://onlinelibrary.wiley.com/doi/10.1002/(SICI)1097-0207(20000630)48:6%3C791::AID-NME859%3E3.0.CO;2-Z/abstract
33.
Hairer
,
E.
, and
Wanner
,
G.
, 2002,
Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems
,
2nd ed.
,
Springer
,
Berlin
.
34.
Blajer
,
W.
, 2002, “
Elimination of Constraint Violation and Accuracy Aspects in Numerical Simulation of Multibody Systems
,”
Multibody Syst. Dyn.
,
7
(
3
), pp.
265
284
.
35.
Kövecses
,
J.
,
Piedbœuf
,
J.-C.
, and
Lange
,
C.
, 2003, “
Dynamics Modeling and Simulation of Constrained Robotic Systems
,”
IEEE/ASME Trans. Mechatron.
,
8
(
2
), pp.
165
177
.
36.
Arnold
,
M.
,
Fuchs
,
A.
, and
Führer
,
C.
, 2006, “
Efficient Corrector Iteration for DAE Time Integration in Multibody Dynamics
,”
Comp. Methods Appl. Mech. Eng.
,
195
(
50–51
), pp.
6958
6973
.
37.
Petzold
,
L.
, 1982, “
A description of DASSL: A Differential/Aalgebraic System Solver
,” Sandia National Laboratory Report SAND82-8637, pp.
4
7
.
38.
Baumgarte
,
J.
, 1972, “
Stabilization of Constraints and Integrals of Motion in Dynamical Systems
,”
Comp. Methods Appl. Mech. Eng.
,
1
(
1
), pp.
1
16
.
39.
Baumgarte
,
J.
, 1983, “
A New Method of Stabilization for Holonomic Constraints
,”
J. Appl. Mech.
,
50
(
4a
), pp.
869
870
.
40.
Gear
,
C. W.
, 2006, “
Towards Explicit Methods for Differential Algebraic Equations
,”
BIT Numer. Math.
,
46
(
3
), pp.
505
514
.
41.
ten Dam
,
A. A.
, 1992, “
Stable Numerical Integration of Dynamical Systems Subject to Equality State-Space Constraints
,”
J. Eng. Math.
,
26
(
2
), pp.
315
337
.
42.
Ascher
,
U. M.
,
Chin
,
H.
, and
Reich
,
S.
, 1994, “
Stabilization of DAEs and Invariant Manifolds
,”
Numer. Math.
,
67
(
2
), pp.
131
149
.
43.
Burgermeister
,
B.
,
Arnold
,
M.
, and
Esterl
,
B.
, 2006, “
DAE Time Integration for Real-Time Applications in Multi-Body Dynamics
,”
Z. Angew.e Math. Mech.
,
86
(
10
), pp.
759
771
.
44.
Braun
,
D. J.
, and
Goldfarb
,
M.
, 2009, “
Eliminating Constraint Drift in the Numerical Simulation of Constrained Dynamical Systems
,”
Comp. Methods Appl. Mech. Eng.
,
198
(
37–40
), pp.
3151
3160
.
45.
Ben-Israel
,
A.
, and
Greville
,
T. N. E.
, 2003,
Generalized Inverse: Theory and Applications
,
Springer
,
New York
.
46.
Kövecses
,
J.
, and
Piedbœuf
,
J. C.
, 2003, “
A Novel Approach for the Dynamic Analysis and Simulation of Constrained Mechanical Systems
,”
ASME Design Engineering Technical Conferences, 19th Biennial Conference on Mechanical Vibrations and Noise
, Chicago, Illinois, Paper no. DETC2003/VIB-48318, pp.
143
152
.
47.
d’Alembert
,
J.
, 1743,
Traite de Dynamique
, Paris.
48.
Parczewski
,
J.
, and
Blajer
,
W.
, 1989, “
On Realization of Program Constraints: Part I - Theory
,”
J. Appl. Mech.
,
56
(
3
), pp.
676
679
.
49.
Glocker
,
C.
, 2001,
Set-Valued Force Laws: Dynamics of Non-Smooth Systems
,
Springer
,
Berlin
.
50.
Kövecses
,
J.
, 2008, “
Dynamics of Mechanical Systems and the Generalized Free-Body Diagram - Part I: General Formulation
,”
J. Appl. Mech.
,
75
(
6
), p.
061012
.
51.
Vujanović
,
B. D.
, and
Jones
,
S. E.
, 1989,
Variational Methods in Nonconservative Phenomena
,
Academic
,
New York
.
52.
Yoon
,
S.
,
Howe
,
R. M.
, and
Greenwood
,
T. D.
, 1994, “
Geometric Elimination of Constraint Violations in Numerical Simulation of Lagrangian Equations
,”
J. Mech. Des.
,
116
(
4
), pp.
1058
1064
.
53.
Betsch
,
P.
, 2006, “
Energy-Consistent Numerical Integration of Mechanical Systems With Mixed Holonomic and Nonholonomic Constraints
,”
Comp. Methods Appl. Mech. Eng.
,
195
(
50–51
), pp.
7020
7035
.
54.
Mei
,
F.
, 2000, “
Nonnolonomic Mechanics
,”
Appl. Mech. Rev.
,
53
(
11
), pp.
283
305
.
55.
Braun
,
D.
, and
Goldfarb
,
M.
, 2010, “
Simulation of Constrained Mechanical Systems - Part II: Explicit Numerical Integration
,”
J. Appl. Mech.
You do not currently have access to this content.