Commercial multibody system simulation (MBS) tools commonly use a redundant coordinate formulation as part of their modeling strategy. Such multibody systems subject to holonomic constraints result in second-order d-index three differential algebraic equation (DAE) systems. Due to the redundant formulation and a priori estimation of possible flexible body coordinates, the model size increases rapidly with the number of bodies. Typically, a considerable number of constraint equations (and physical degrees-of-freedom (DOF)) are not necessary for the structure's motion but are necessary for its stability like out-of-plane constraints (and DOFs) in case of pure in-plane motion. We suggest a combination of both, physical DOF and constraint DOF reduction, based on proper orthogonal decomposition (POD) using DOF-type sensitive velocity snapshot matrices. After a brief introduction to the redundant multibody system, a modified flat Galerkin projection and its application to index-reduced systems in combination with POD are presented. The POD basis is then used as an identification tool pointing out reducible constraint equations. The methods are applied to one academic and one high-dimensional practical example. Finally, it can be reported that for the numerical examples provided in this work, more than 90% of the physical DOFs and up to 60% of the constraint equations can be omitted. Detailed results of the numerical examples and a critical discussion conclude the paper.

References

1.
Craig
,
R.
,
1985
, “
A Review of Time-Domain and Frequency-Domain Component Mode Synthesis Method
,”
J. Modal Anal.
,
2
, pp.
59
72
.
2.
Craig
,
R.
,
2000
, “
Coupling of Substructures for Dynamic Analyses: An Overview
,”
Structural Dynamics, and Materials Conference and Exhibit
, pp.
1
12
.
3.
Besselink
,
B.
,
Tabak
,
U.
,
Lutowska
,
A.
,
van de Wouw
,
N.
,
Nijmeijer
,
H.
,
Rixen
,
D.
,
Hochstenbach
,
M.
, and
Schilders
,
W.
,
2013
, “
A Comparison of Model Reduction Techniques From Structural Dynamics, Numerical Mathematics and System Control
,”
J. Sound Vib.
,
332
(
19
), pp.
4403
4422
.
4.
Lehner
,
M.
,
2007
, “
Modellreduktion in elastischen Mehrkoerpersystemen
,” Ph.D. thesis, Fakultaet fuer Maschinenbau, Universitaet Stuttgart, Stuttgart, Germany.
5.
Koutsovasilis
,
P.
, and
Beitelschmidt
,
M.
,
2008
, “
Comparison of Model Reduction Techniques for Large Mechanical Systems
,”
Multibody Syst. Dyn.
,
20
(
2
), pp.
111
128
.
6.
Witteveen
,
W.
,
2012
, “
On the Modal and Non-Modal Model Reduction of Metallic Structures With Variable Boundary Conditions
,”
World J. Mech.
,
2
(
6
), pp.
311
324
.
7.
Pennestri
,
E.
, and
Valentini
,
P.
,
2007
, “
Coordinate Reduction Strategies in Multibody Dynamics: A Review
,”
Conference on Multibody System Dynamics
, pp.
1
17
.
8.
Laulusa
,
A.
, and
Bauchau
,
O.
,
2007
, “
Review of Classical Approaches for Constraint Enforcement in Multibody Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
3
(
1
), pp.
1
8
.
9.
Volkwein
,
S.
,
2008
,
Model Reduction Using Proper Orthogonal Decomposition (Lecture Notes)
, University of Constance, Constance, Germany.
10.
Ebert
,
F.
,
2010
, “
A Note on POD Model Reduction Methods for DAEs
,”
Math. Comput. Model. Dyn. Syst.
,
16
(
2
), pp.
115
131
.
11.
Ersal
,
T.
,
Fathy
,
H.
, and
Stein
,
J.
,
2009
, “
Orienting Body Coordinate Frames Using Karhunen–Loeve Expansion for More Effective Structural Simplification
,”
Simul. Modell. Pract. Theory
,
17
(
1
), pp.
197
210
.
12.
Masoudi
,
R.
,
Uchida
,
T.
, and
McPhee
,
J.
,
2015
, “
Reduction of Multibody Dynamic Models in Automotive Systems Using the Proper Orthogonal Decomposition
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
3
), p.
031007
.
13.
Chelidze
,
D.
,
2014
, “
Identifying Robust Subspaces for Dynamically Consistent Reduced-Order Models
,”
Nonlinear Dynamics
(Conference Proceedings of the Society for Experimental Mechanics Series), Vol.
2
,
G.
Kerschen
, ed.,
Springer International Publishing
, Cham, Switzerland, pp.
123
130
.
14.
Stadlmayr
,
D.
, and
Witteveen
,
W.
,
2015
, “
Model Reduction for Nonlinear Multibody Systems Based on Proper Orthogonal- & Smooth Orthogonal Decomposition
,”
Nonlinear Dynamics
(Conference Proceedings of the Society for Experimental Mechanics Series), Vol.
1
,
G.
Kerschen
, ed.,
Springer International Publishing
, Cham, Switzerland, pp.
449
458
.
15.
Heirman
,
G.
,
Bruls
,
O.
,
Sas
,
P.
, and
Desmet
,
W.
,
2008
, “
Coordinate Transformation Techniques for Efficient Model Reduction in Flexible Multibody Dynamics
,”
ISMA 2008—International Conference on Noise and Vibration Engineering
, pp.
1
16
.
16.
Heirman
,
G.
,
Bruls
,
O.
, and
Desmet
,
W.
,
2009
, “
A System-Level Model Reduction Technique for Efficient Simulation of Flexible Multibody Dynamics
,”
Multibody Dynamics ECCOMAS Conference
, pp.
1
16
.
17.
Bruls
,
O.
,
Duysinx
,
P.
, and
Golinval
,
J.-C.
,
2007
, “
The Global Modal Parameterization for Non-Linear Model-Order Reduction in Flexible Multibody Dynamics
,”
Int. J. Numer. Methods Eng.
,
69
(
5
), pp.
947
977
.
18.
Nachbagauer
,
K.
,
Oberpeilsteiner
,
S.
,
Sherif
,
K.
, and
Steiner
,
W.
,
2014
, “
The Use of the Adjoint Method for Solving Typical Optimization Problems in Multibody Dynamics
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
6
), p.
061011
.
19.
Nikravesh
,
P. E.
,
Wehage
,
R. A.
, and
Kwon
,
O. K.
,
1985
, “
Euler Parameters in Computational Kinematics and Dynamics, Part 1
,”
ASME J. Mech. Des.
,
107
(
3
), pp.
358
365
.
20.
Shabana
,
A. A.
,
2005
,
Dynamics of Multibody Systems
, 3rd ed.,
Cambridge University Press
, New York.
21.
Baumgarte
,
J.
,
1972
, “
Stabilization of Constraints and Integrals of Motion in Dynamical Systems
,”
Comput. Methods Appl. Mech. Eng.
,
1
(
1
), pp.
1
16
.
22.
Freund
,
R. W.
,
2003
, “
Model Reduction Methods Based on Krylov Subspaces
,”
Acta Numer.
,
12
, pp.
267
319
.
23.
Moore
,
B. C.
,
1981
, “
Principal Component Analysis in Linear Systems: Controllability, Observability, and Model Reduction
,”
IEEE Trans. Automatic Control
,
26
(
1
), pp.
17
32
.
24.
Chatterjee
,
A.
,
2000
, “
An Introduction to the Proper Orthogonal Decomposition
,”
Curr. Sci.
,
75
, pp.
808
817
.
25.
Kerschen
,
G.
,
Golinval
,
J.
,
Vakakis
,
A.
, and
Bergman
,
L.
,
2005
, “
The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview
,”
Nonlinear Dyn.
,
41
(
1
), pp.
147
169
.
26.
Kosambi
,
D. D.
,
1943
, “
Statistics in Function Space
,”
J. Indian Math. Soc.
,
7
, pp.
76
78
.
27.
Scilab
,
2014
, “
Scilab—Version 5.5.1 (x64)
,” http://www.scilab.org
28.
Sherif
,
K.
,
Irschik
,
H.
, and
Witteveen
,
W.
,
2012
, “
Transformation of Arbitrary Elastic Mode Shapes Into Pseudo-Free-Surface and Rigid Body Modes for Multibody Dynamic Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
7
(
2
), p.
021008
.
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