Abstract

A new method of model-order reduction for the flexible multibody system which undergoes large deformation and rotation is proposed. At first, the flexible multibody system is modeled by absolute nodal coordinate formulation (ANCF), and then, the whole motion process of the system is divided into a series of quasi-static equilibrium configurations according to a given criterion. Afterward, motion equation is locally linearized based on the Taylor expansion. Therefore, the constant tangent stiffness matrix is obtained and does not need to be updated until the next configuration. Based on the locally linearized motion equation, the free-interface component mode synthesis (CMS) method is adopted to reduce the degrees-of-freedom (DOF) of the flexible multibody system molded by ANCF. The generalized-α integrator is used to solve the reduced motion equation. To verify the accuracy and efficiency of the proposed method, three examples including a free-falling pendulum, a flexible spinning beam and a deployable sail arrays are presented. Results show that the proposed method is able to reduce the computing time and maintain high accuracy.

References

1.
Shabana
,
A. A.
,
1997
, “
Flexible Multibody Dynamics: Review of Past and Recent Developments
,”
Multibody Syst. Dyn.
,
1
(
2
), pp.
189
222
.10.1023/A:1009773505418
2.
Shabana
,
A. A.
,
2005
,
Dynamics of Multibody Systems
, 3th ed.,
Cambridge University Press
,
Cambridge, UK
.
3.
Berzeri
,
M.
,
Campanelli
,
M.
, and
Shabana
,
A. A.
,
2001
, “
Definition of the Elastic Forces in the Finite Element Absolute Nodal Coordinate Formulation and the Floating Frame of Reference Formulation
,”
Multibody Syst. Dyn
,.
5
(
1
), pp.
21
54
.10.1023/A:1026465001946
4.
Dibold
,
M.
,
Gerstmayr
,
J.
, and
Irschik
,
H.
,
2009
, “
A Detailed Comparison of the Absolute Nodal Coordinate and the Floating Frame of Reference Formulation in Deformable Multibody Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
4
(
2
), p.
21006
.10.1115/1.3079825
5.
Shabana
,
A. A.
,
1996
, “
An Absolute Nodal Coordinates Formulation for the Large Rotation and Deformation Analysis of Flexible Bodies
,” University of Illinois, Chicago, IL, Report. No. MBS96-1-UIC.
6.
Shabana
,
A. A.
,
1997
, “
Definition of the Slopes and the Finite Element Absolute Nodal Coordinate Formulation
,”
Multibody Syst. Dyn.
,
1
(
3
), pp.
339
348
.10.1023/A:1009740800463
7.
Escalona
,
J. L.
,
Hussien
,
H. A.
, and
Shabana
,
A. A.
,
1998
, “
Application of the Absolute Nodal Coordinate Formulation to Multibody System Dynamics
,”
J. Sound Vib.
,
214
(
5
), pp.
833
851
.10.1006/jsvi.1998.1563
8.
Berzeri
,
M.
, and
Shabana
,
A. A.
,
2000
, “
Development of Simple Models for the Elastic Forces in the Absolute Nodal co-Ordinate Formulation
,”
J. Sound Vib.
,
235
(
4
), pp.
539
565
.10.1006/jsvi.1999.2935
9.
Shabana
,
A. A.
, and
Patel
,
M.
,
2018
, “
Coupling Between Shear and Bending in the Analysis of Beam Problems: Planar Case
,”
J. Sound Vib.
,
419
, pp.
510
525
.10.1016/j.jsv.2017.12.006
10.
Shabana
,
A. A.
, and
Yakoub
,
R. Y.
,
2001
, “
Three-Dimensional Absolute Nodal Coordinate Formulation for Beam Elements Theory
,”
J. Mech. Des.
,
123
(
4
), pp.
606
613
.10.1115/1.1410100
11.
Yakoub
,
R. Y.
, and
Shabana
,
A. A.
,
2001
, “
Three-Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Implementation and Applications
,”
J. Mech. Des.
,
123
(
4
), pp.
614
621
.10.1115/1.1410099
12.
Sugiyama
,
H.
,
Koyama
,
H.
, and
Yamashita
,
H.
,
2010
, “
Gradient Deficient Curved Beam Element Using the Absolute Nodal Coordinate Formulation
,”
ASME J. Comput. Nonlinear Dyn.
,
5
(
2
), p.
021001
.10.1115/1.4000793
13.
Mikkola
,
A. M.
, and
Shabana
,
A. A.
,
2003
, “
A Non-Incremental Finite Element Procedure for the Analysis of Large Deformation of Plates and Shells in Mechanical System Applications
,”
Multibody Syst. Dyn.
,
9
(
3
), pp.
283
309
.10.1023/A:1022950912782
14.
Nachbagauer
,
K.
, and
Gerstmayr
,
J.
,
2014
, “
Structural and Continuum Mechanics Approaches for a 3D Shear Deformable ANCF Beam Finite Element: Application to Buckling and Nonlinear Dynamic Examples
,”
ASME J. Comput. Nonlinear Dyn.
,
9
(
1
), p.
011013
.10.1115/1.4025282
15.
Dmitrochenko
,
O. N.
, and
Pogorelov
,
D.
,
2003
, “
Generalization of Plate Finite Elements for Absolute Nodal Coordinate Formulation
,”
Multibody Syst. Dyn.
,
10
(
1
), pp.
17
43
.10.1023/A:1024553708730
16.
Pappalardo
,
C. M.
,
Wallin
,
M.
, and
Shabana
,
A. A.
,
2017
, “
A New ANCF/CRBF Fully Parameterized Plate Finite Element
,”
ASME J. Comput. Nonlinear Dyn.
,
12
(
3
), p.
031008
.10.1115/1.4034492
17.
Gerstmayr
,
J.
,
Sugiyama
,
H.
, and
Mikkola
,
A.
,
2013
, “
Review on the Absolute Nodal Coordinate Formulation for Large Deformation Analysis of Multibody Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
8
(
3
), p.
031016
.10.1115/1.4023487
18.
Olshevskiy
,
A.
,
Dmitrochenko
,
O.
, and
Kim
,
C. W.
,
2014
, “
Three Dimensional Solid Brick Element Using Slopes in the Absolute Nodal Coordinate Formulation
,”
ASME J. Comput. Nonlinear Dyn.
,
9
(
2
), p.
021001
.10.1115/1.4024910
19.
Lan
,
P.
,
Wang
,
T.
, and
Yu
,
Z.
,
2019
, “
A New Planar Triangular Element Based on the Absolute Nodal Coordinate Formulation
,”
Proc IMech E, Part K
,
233
(
1
), pp.
163
173
.10.1177/1464419318771436
20.
Shabana
,
A. A.
,
2015
, “
ANCF Tire Assembly Model for Multibody System Applications
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
2
), p.
024504
.10.1115/1.4028479
21.
Eberhard
,
P.
, and
Schiehlen
,
W.
,
2006
, “
Computational Dynamics of Multibody Systems: History, Formalisms, and Applications
,”
ASME J. Comput. Nonlinear Dyn
,
1
(
1
), pp.
3
12
.10.1115/1.1961875
22.
Klerk
,
D.
,
Rixen
,
D. J.
, and
Voormeeren
,
S. N.
,
2008
, “
General Framework for Dynamic Substructuring: History, Review, and Classification of Techniques
,”
AIAA J.
,
46
(
5
), pp.
1169
1181
.10.2514/1.33274
23.
Hurty
,
W. C.
,
1965
, “
Dynamic Analysis of Structural Systems Using Component Modes
,”
AIAA J.
,
3
(
4
), pp.
678
685
.10.2514/3.2947
24.
Craig
,
R. R.
, and
Bampton
,
C. C.
,
1968
, “
Coupling of Substructures for Dynamic Analyses
,”
AIAA J.
,
6
(
7
), pp.
1313
1319
.10.2514/3.4741
25.
Hou
,
S. N.
,
1969
, “
Rev. Modal Synthesis Techniques a New Approach
,”
Shock Vib. Bull.
,
40
(
4
), pp.
25
30
.https://ntrs.nasa.gov/search.jsp?R=19700033203
26.
MacNeal
,
R. H.
,
1971
, “
A Hybrid Method of Component Mode Synthesis
,”
Comput. Struct.
,
1
(
4
), pp.
581
601
.10.1016/0045-7949(71)90031-9
27.
Rubin
,
S.
,
1975
, “
Improved Component Mode Representation for Structural Dynamic Analysis
,”
AIAA J.
,
13
(
8
), pp.
995
1006
.10.2514/3.60497
28.
Papalukopoulos
,
C.
, and
Natsiavas
,
S.
,
2007
, “
Dynamics of Large Scale Mechanical Models Using Multilevel Substructuring
,”
ASME J. Comput. Nonlinear Dyn.
,
2
(
1
), pp.
40
51
.10.1115/1.2389043
29.
Pichler
,
F.
,
Witteveen
,
W.
, and
Fischer
,
P.
,
2017
, “
Reduced-Order Modeling of Preloaded Bolted Structures in Multibody Systems by the Use of Trial Vector Derivatives
,”
ASME J. Comput. Nonlinear Dyn.
,
12
(
5
), p.
051032
.10.1115/1.4036989
30.
Ding
,
Z.
,
Li
,
L.
, and
Hu
,
Y.
,
2016
, “
A Free Interface Component Mode Synthesis Method for Viscoelastically Damped Systems
,”
J. Sound Vib.
,
365
, pp.
199
215
.10.1016/j.jsv.2015.11.040
31.
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
), pp.
146
152
.10.1115/1.4005237
32.
Chapman
,
T.
,
Avery
,
P.
,
Collins
,
P.
, and
Farhat
,
C.
,
2017
, “
Accelerated Mesh Sampling for the Hyper Reduction of Nonlinear Computational Models
,”
Int. J. Numer. Meth. Eng.
,
109
(
12
), pp.
1623
1654
.10.1002/nme.5332
33.
Brüls
,
O.
,
Duysinx
,
P.
, and
Golinval
,
J. C.
,
2007
, “
The Global Modal Parameterization for Nonlinear Model-Order Reduction in Flexible Multibody Dynamics
,”
Int. J. Numer. Meth. Eng.
,
69
(
5
), pp.
948
977
.10.1002/nme.1795
34.
Naets
,
F.
,
Tamarozzi
,
T.
,
Heirman
,
G. H. K.
, 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-z
35.
Naets
,
F.
,
Heirman
,
G. H. K.
, and
Desmet
,
W.
,
2012
, “
Subsystem Global Modal Parameterization for Efficient Simulation of Flexible Multibody Systems
,”
Int. J. Numer. Method Eng.
,
89
(
10
), pp.
1227
1248
.10.1002/nme.3284
36.
Wu
,
L.
, and
Tiso
,
P.
,
2016
, “
Nonlinear Model Order Reduction for Flexible Multibody Dynamics: A Modal Derivatives Approach
,”
Multibody Syst. Dyn.
,
36
(
4
), pp.
405
425
.10.1007/s11044-015-9476-5
37.
Wu
,
L.
,
Tiso
,
P.
,
Konstantinos
,
T.
,
Eleni
,
C.
, and
Fred
,
V. K.
,
2019
, “
A Modal Derivatives Enhanced Rubin Substructuring Method for Geometrically Nonlinear Multibody Systems
,”
Multibody Syst. Dyn.
,
45
(
1
), pp.
57
85
.10.1007/s11044-018-09644-2
38.
Weeger
,
O.
,
Wever
,
U.
, and
Simeon
,
B.
,
2014
, “
Nonlinear Frequency Response Analysis of Structural Vibrations
,”
Comput. Mech.
,
54
(
6
), pp.
1477
1495
.10.1007/s00466-014-1070-9
39.
Weeger
,
O.
,
Wever
,
U.
, and
Simeon
,
B.
,
2016
, “
On the Use of Modal Derivatives for Nonlinear Model Order Reduction
,”
Int. J. Numer. Methods Eng.
,
108
(
13
), pp.
1579
1602
.10.1002/nme.5267
40.
Holzwarth
,
P.
, and
Eberhard
,
P.
,
2015
, “
SVD-Based Improvements for Component Mode Synthesis in Elastic Multibody Systems
,”
Eur. J. Mech. A/Solids
,
49
, pp.
408
418
.10.1016/j.euromechsol.2014.08.009
41.
Gerstmayr
,
J.
, and
Ambrósio
,
J. A. C.
,
2008
, “
Component Mode Synthesis With Constant Mass and Stiffness Matrices Applied to Flexible Multibody Systems
,”
Int. J. Numer. Method Eng.
,
73
(
11
), pp.
1518
1546
.10.1002/nme.2133
42.
Humer
,
A.
,
Naets
,
F.
,
Desmet
,
W.
, and
Gerstmayr
,
J.
,
2014
, “
A Generalized Component Mode Synthesis Approach for Global Modal Parameterization in Flexible Multibody Dynamics
,”
Proceedings of the 3rd Joint International Conference on Multibody System Dynamics
,
Seventh Asian Conference on Multibody Dynamics
,
BEXCO, Busan, Korea
, June 30–July 3.https://www.researchgate.net/publication/268924032_A_generalized_component_mode_synthesis_approach_for_global_modal_parameterization_in_flexible_multibody_dynamics
43.
Pechstein
,
A.
,
Reischl
,
D.
, and
Gerstmayr
,
J.
,
2013
, “
A Generalized Component Mode Synthesis Approach for Flexible Multibody Systems With a Constant Mass Matrix
,”
ASME J. Comput. Nonlinear Dyn.
,
8
(
1
), p.
011019
.10.1115/1.4007191
44.
Ziegler
,
P.
,
Humer
,
A.
,
Pechstein
,
A.
, and
Gerstmayr
,
J.
,
2016
, “
Generalized Component Mode Synthesis for the Spatial Motion of Flexible Bodies With Large Rotations About One Axis
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
4
), p.
041018
.10.1115/1.4032160
45.
Kobayashi
,
N.
,
Wago
,
T.
, and
Sugawara
,
Y.
,
2011
, “
Reduction of System Matrices of Planar Beam in ANCF by Component Mode Synthesis Method
,”
Multibody Syst. Dyn.
,
26
(
3
), pp.
265
281
.10.1007/s11044-011-9259-6
46.
Okazawa
,
Y.
,
Kobayashi
,
N.
, and
Torisaka
,
A.
,
2014
, “
Reduction of System Matrices of Plate in Absolute Nodal Coordinate Formulation
,”
Trans. JSME.
80
(
813
), pp.
1
12
(in Japanese).10.1299/transjsme.2014dr0129
47.
Sun
,
D.
,
Chen
,
G.
, and
Sun
,
R.
,
2014
, “
Model Reduction of a Multibody System Including a Very Flexible Beam Element
,”
J. Mech. Sci. Technol.
,
28
(
8
), pp.
2963
2969
.10.1007/s12206-014-0703-4
48.
Sun
,
D.
,
Chen
,
G.
,
Shi
,
Y.
,
Wang
,
T.
, and
Sun
,
R.
,
2015
, “
Model Reduction of a Flexible Multibody System With Clearance
,”
Mech. Mach. Theory
,
85
, pp.
106
115
.10.1016/j.mechmachtheory.2014.10.013
49.
Kim
,
E.
,
Kim
,
H.
, and
Cho
,
M.
,
2017
, “
Model Order Reduction of Multibody System Dynamics Based on Stiffness Evaluation in the Absolute Nodal Coordinate Formulation
,”
Nonlinear Dyn.
,
87
(
3
), pp.
1901
1915
.10.1007/s11071-016-3161-y
50.
Kim
,
E.
, and
Cho
,
M.
,
2018
, “
Design of a Planar Multibody Dynamic System With ANCF Beam Elements Based on an Element-Wise Stiffness Evaluation Procedure
,”
Struct. Multidiscip. Optim.
,
58
(
3
), pp.
1095
1107
.10.1007/s00158-018-1954-y
51.
Luo
,
K.
,
Hu
,
H.
,
Liu
,
C.
, and
Tian
,
Q.
,
2017
, “
Model Order Reduction for Dynamic Simulation of a Flexible Multibody System Via Absolute Nodal Coordinate Formulation
,”
Comput. Methods Appl. Mech. Eng.
,
324
, pp.
573
594
.10.1016/j.cma.2017.06.029
52.
Tang
,
Y.
,
Hu
,
H.
, and
Tian
,
Q.
,
2019
, “
Model Order Reduction Based on Successively Local Linearizations for Flexible Multibody Dynamics
,”
Int. J. Numer. Method Eng.
,
3
, pp.
159
180
.10.1002/nme.6011
53.
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/s11044-007-9084-0
54.
Kobis
,
M. A.
, and
Arnold
,
M.
,
2016
, “
Convergence of Generalized-α Time Integration for Nonlinear Systems With Stiff Potential Forces
,”
Multibody Syst. Dyn.
,
37
(
1
), pp.
107
125
.10.1007/s11044-015-9495-2
55.
Omar
,
M. A.
, and
Shabana
,
A. A.
,
2001
, “
A Two-Dimensional Shear Deformable Beam for Large Rotation and Deformation Problems
,”
J. Sound Vib.
,
243
(
3
), pp.
565
576
.10.1006/jsvi.2000.3416
56.
Lan
,
P.
,
Tian
,
Q.
, and
Yu
,
Z.
,
2020
, “
A New Absolute Nodal Coordinate Formulation Beam Element With Multilayer Circular Cross Section
,”
Acta Mech. Sin.
,
36
(
1
), pp.
82
96
.10.1007/s10409-019-00897-4
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