When redundant constraints are present in a rigid body mechanism, only selected (if any at all) joint reactions can be determined uniquely, whereas others cannot. Analytic criteria and numerical methods of finding joints with uniquely solvable reactions are available. In this paper, the problem of joint reactions solvability is examined from the point of view of selected numerical methods frequently used for handling redundant constraints in practical simulations. Three different approaches are investigated in the paper: elimination of redundant constraints; pseudoinverse-based calculations; and the augmented Lagrangian formulation. Each method is briefly summarized; the discussion is focused on techniques of handling redundant constraints and on joint reactions calculation. In the case of multibody systems with redundant constraints, the rigid body equations of motion are insufficient to calculate some or all joint reactions. Thus, purely mathematical operations are performed in order to find the reaction solution. In each investigated method, the redundant constraints are treated differently, which—in the case of joints with nonunique reactions—leads to different reaction solutions. As a consequence, reactions reflecting the redundancy handling method rather than physics of the system are calculated. A simple example of each method usage is presented, and calculated joint reactions are examined. The paper points out the origins of nonuniqueness of constraint reactions in each examined approach. Moreover, it is shown that one and the same method may lead to different reaction solutions, provided that input data are prepared differently. Finally, it is demonstrated that—in case of joints with solvable reactions—the obtained solutions are unique, regardless of the method used for redundant constraints handling.

References

1.
Garcia de Jalon
,
J.
, and
Bayo
,
E.
,
1994
,
Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge
,
Springer-Verlag
,
Berlin
.
2.
Haug
,
E. J.
,
1989
,
Computer Aided Kinematics and Dynamics of Mechanical Systems
,
Allyn and Bacon
,
Boston
.
3.
Nikravesh
,
P. E.
,
1988
,
Computer-Aided Analysis of Mechanical Systems
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
4.
Shabana
,
A. A.
,
2010
,
Computational Dynamics
, 3rd ed.,
Wiley
,
New York
.
5.
Frączek
,
J.
, and
Wojtyra
,
M.
,
2011
, “
On the Unique Solvability of a Direct Dynamics Problem for Mechanisms With Redundant Constraints and Coulomb Friction in Joints
,”
Mech. Mach. Theory
,
46
(
3
),
pp.
312
334
.10.1016/j.mechmachtheory.2010.11.003
6.
Wojtyra
,
M.
,
2005
, “
Joint Reaction Forces in Multibody Systems With Redundant Constraints
,”
Multibody Syst. Dyn.
,
14
(
1
),
pp.
23
46
.10.1007/s11044-005-5967-0
7.
Wojtyra
,
M.
,
2009
, “
Joint Reactions in Rigid Body Mechanisms With Dependent Constraints
,”
Mech. Mach. Theory
,
44
(
12
),
pp.
2265
2278
.10.1016/j.mechmachtheory.2009.07.008
8.
Blajer
,
W.
,
2004
, “
On the Determination of Joint Reactions in Multibody Mechanisms
,”
ASME J. Mech. Des.
,
126
(
2
),
pp.
341
350
.10.1115/1.1667944
9.
Udwadia
,
F. E.
, and
Kalaba
,
R. E.
,
1996
,
Analytical Dynamics: A New Approach
,
Cambridge University Press
,
Cambridge, England
.
10.
Bayo
,
E.
,
Garcia de Jalon
,
J.
, and
Serna
,
M. A.
,
1988
, “
A Modified Lagrangian Formulation for the Dynamic Analysis of Constrained Mechanical Systems
,”
Comput. Methods Appl. Mech. Eng.
,
71
(
2
),
pp.
183
195
.10.1016/0045-7825(88)90085-0
11.
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/BF01833296
12.
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)90018-7
13.
Müller
,
A.
,
2006
, “
A Conservative Elimination Procedure for Permanently Redundant Closure Constraints in MBS-Models With Relative Coordinates
,”
Multibody Syst. Dyn.
,
16
(
4
),
pp.
309
330
.10.1007/s11044-006-9028-0
14.
Müller
,
A.
,
2011
, “
Semialgebraic Regularization of Kinematic Loop Constraints in Multibody System Models
,”
ASME J. Comput. Nonlinear Dyn.
,
6
(
4
),
p.
041010
.10.1115/1.4002998
15.
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
.10.1115/1.3256318
16.
de Falco
,
D.
,
Pennestri
,
E.
, and
Vita
,
L.
,
2009
, “
An Investigation of the Influence of Pseudoinverse Matrix Calculations on Multibody Dynamics Simulations by Means of the Udwadia-Kalaba Formulation
,”
J. Aerosp. Eng.
,
22
(
4
),
pp.
365
372
.10.1061/(ASCE)0893-1321(2009)22:4(365)
17.
Neto
,
M. A.
, and
Ambrósio
,
J.
,
2003
, “
Stabilization Methods for the Integration of DAE in the Presence of Redundant Constraints
,”
Multibody Syst. Dyn.
,
10
(
1
),
pp.
81
105
.10.1023/A:1024567523268
18.
Lawson
,
C. L.
, and
Hanson
,
R. J.
,
1995
,
Solving Least Squares Problem
,
SIAM
,
Philadelphia
.
19.
Garcia de Jalon
,
J.
, and
Gutiérrez-López
,
M. D.
,
2012
, “
Multibody Dynamics With Redundant Constraints and Singular Mass Matrix
,”
Proc. of the 2nd Joint International Conference on Multibody System Dynamics
,
Stuttgart
,
Germany
,
May
29
June
1
.
20.
Kövecses
,
J.
, and
Piedbœuf
,
J.-C.
,
2003
, “
A Novel Approach for the Dynamic Analysis and Simulation of Constrained Mechanical Systems
,“
Proc. of DETC’03 ASME Design Engineering Technical Conferences
,
Chicago
,
IL
,
Sept.
2–6
.
21.
Doty
,
K. L.
,
Melchiorri
,
C.
, and
Bonivento
,
C.
,
1993
, “
A Theory of Generalized Inverses Applied to Robotics
,”
Int. J. Robot. Res.
,
12
(
1
),
pp.
1
19
.10.1177/027836499301200101
22.
Nocedal
,
J.
, and
Wright
,
S. J.
,
2006
,
Numerical Optimization
, 2nd ed.,
Springer
,
New York
.
23.
Ruzzeh
,
B.
, and
Kövecses
,
J.
,
2011
, “
A Penalty Formulation for Dynamics Analysis of Redundant Mechanical Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
6
(
2
),
p.
021008
.10.1115/1.4002510
24.
Dormand
,
J. R.
, and
Prince
,
P. J.
,
1980
, “
A Family of Embedded Runge-Kutta Formulae
,”
J. Comput. Appl. Math.
,
6
(
1
),
pp.
19
26
.10.1016/0771-050X(80)90013-3
25.
Wojtyra
,
M.
, and
Frączek
,
J.
,
2011
, “
Joint Reactions in Overconstrained Rigid or Flexible Body Mechanisms
,”
Proc. of the ECCOMAS Thematic Conference Multibody Dynamics
2011
,
Brussels
,
Belgium
,
July
4–7
.
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