In this paper some existing codes for the determination of joint reactions in multibody mechanisms are first reviewed. The codes relate to the DAE (differential-algebraic equation) dynamics formulations in absolute coordinates and in relative joint coordinates, and to the ODE (ordinary differential equation) formulations obtained by applying the coordinate partitioning method to these both coordinate types. On this background a novel efficient approach to the determination of joint reactions is presented, naturally associated with the reduced-dimension formulations of mechanism dynamics. By introducing open-constraint coordinates to specify the prohibited relative motions in the joints, pseudoinverse matrices to the constraint Jacobian matrices are derived in an automatic way. The involvement of the pseudo-inverses leads to schemes in which the joint reactions are obtained directly in resolved forms—no matrix inversion is needed as it is required in the classical codes. This makes the developed schemes especially well suited for both symbolic manipulators and computer implementations. Illustrative examples are provided.

Issue Section:
Technical Papers
Keywords:
dynamics, differential equations, Jacobian matrices, mechanical engineering
Topics:
Differential equations, Dimensions, Dynamics (Mechanics), Equations of motion, Jacobian matrices, Open loop systems, Computers, Algebra
1.
Kane, T. R., and Levinson, D. A., 1985, Dynamics: Theory and Applications, McGraw-Hill, New York.
2.
Nikravesh, P. E., 1988, Computer-Aided Analysis of Mechanical Systems, Prentice-Hall, Englewood Cliffs, NJ.
3.
Garcı´a de Jalo´n, J., and Bayo, E., 1994, Kinematic and Dynamic Simulation of Multibody Systems: the Real-Time Challenge, Springer, New York.
4.
Schiehlen
,
W.
,
1997
, “
Multibody System Dynamics: Roots and Perspectives
,”
Multibody Syst. Dyn.
,
1
(
2
), pp.
149
–
188
.
5.
Nikravesh
,
P. E.
,
1990
, “
Systematic Reduction of Multibody Equations of Motion to a Minimal Set
,”
Int. J. Non-Linear Mech.
,
25
(
2/3
), pp.
143
–
151
.
6.
Langer
,
F. D.
,
Hemami
,
H.
, and
Brown
,
D. B.
,
1987
, “
Constraint Forces in Holonomic Mechanical Systems
,”
Comput. Methods Appl. Mech. Eng.
,
62
, pp.
255
–
274
.
7.
Blajer
,
W.
,
1977
, “
A Geometric Unification of Constrained System Dynamics
,”
Multibody Syst. Dyn.
,
1
(
1
), pp.
3
–
21
.
8.
Blajer
,
W.
, 2001, “A Geometrical Interpretation and Uniform Matrix Formulation of Multibody System Dynamics,” ZAMM, 81(4), pp. 247–259.
9.
Eich-Soellner, E., and Fu¨hrer, C., 1998, Numerical Methods in Multibody Dynamics, Teubner, Stuttgart.
10.
Ascher, U. M., and Petzold, L. R., 1998, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia.
11.
Wehage
,
R. A.
, and
Haug
,
E. J.
,
1982
, “
Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems
,”
ASME J. Mech. Des.
,
104
(
1
), pp.
247
–
255
.
12.
Wampler
,
C.
,
Buffinton
,
K.
, and
Shu-hui
,
J.
,
1985
, “
Formulation of Equations of Motion for Systems Subject to Constraints
,”
ASME J. Appl. Mech.
,
52
(
2
), pp.
465
–
470
.
13.
Blajer
,
W.
,
Schiehlen
,
W.
, and
Schirm
,
W.
,
1994
, “
A Projective Criterion to the Coordinate Partitioning Method for Multibody Dynamics
,”
Arch. Appl. Mech.
,
64
(
2
), pp.
86
–
98
.
14.
Lesser
,
M.
,
1992
, “
A Geometrical Interpretation of Kane’s Equations
,”
Proc. R. Soc. London, Ser. A
,
A436
, pp.
69
–
87
.
15.
Djerassi
,
S.
,
1997
, “
Determination of Noncontributing Forces and Noncontributing Impulses in Three-Phase Motions
,”
ASME J. Appl. Mech.
,
64
(
3
), pp.
582
–
589
.
16.
Yamaguchi, G. T., 2001, Dynamic Modeling of Musculoskeletal Motion: A Vectorized Approach for Biomechanical Analysis in Three Dimensions, Kluwer, Dordrecht.
17.
Campbell, S. L., and Meyer, C. D., Jr., 1979, Generalized Inverses of Linear Transformations, Pitman, London.
18.
Blajer
,
W.
,
2002
, “
Elimination of Constraint Violation and Accuracy Aspects in Numerical Simulation of Multibody Systems
,”
Multibody Syst. Dyn.
,
7
(
3
), pp.
265
–
284
.
19.
Udwadia, F. E., and Kalaba, R. E., 1996, Analytical Dynamics: A New Approach, Cambridge University Press, New York.
20.
Singh
,
R. P.
, and
Likins
,
P. W.
,
1985
, “
Singular Value Decomposition for Constrained Dynamical Systems
,”
ASME J. Appl. Mech.
,
52
, pp.
943
–
948
.
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