Abstract

The aim of this work is to extend the finite element multibody dynamics approach to problems involving frictional contacts and impacts. The nonsmooth generalized-α (NSGA) scheme is adopted, which imposes bilateral and unilateral constraints both at position and velocity levels avoiding drift phenomena. This scheme can be implemented in a general purpose simulation code with limited modifications of pre-existing elements. The study of the woodpecker toy dynamics sets up a good example to show the capabilities of the NSGA scheme within the context of a general finite element framework. This example has already been studied by many authors who generally adopted a model with a minimal set of coordinates and small rotations. It is shown that good results are obtained using a general purpose finite element code for multibody dynamics, in which the equations of motion are assembled automatically and large rotations are easily taken into account. In addition, comparing results between different models of the woodpecker toy, the importance of modeling large rotations and the horizontal displacement of the woodpecker's sleeve is emphasized.

References

1.
Géradin
,
M.
, and
Cardona
,
A.
,
2001
,
Flexible Multibody Dynamics: A Finite Element Approach
,
John Wiley & Sons Inc
,
New York
.
2.
Laursen
,
T.
,
2003
,
Computational Contact and Impact Mechanics
,
Springer-Verlag
,
Berlin
.
3.
Negrut
,
D.
,
Serban
,
R.
, and
Tasora
,
A.
,
2018
, “
Posing Multibody Dynamics With Friction and Contact as a Differential Complementarity Problem
,”
ASME J. Comput. Nonlinear Dyn.
,
13
(
1
), p.
10
.10.1115/1.4037415
4.
Tasora
,
A.
, and
Anitescu
,
M.
,
2010
, “
A Convex Complementarity Approach for Simulating Large Granular Flows
,”
ASME J. Comput. Nonlinear Dyn.
,
5
(
3
), p.
5
.10.1115/1.4001371
5.
Mylapilli
,
H.
, and
Jain
,
A.
,
2017
, “
Complementarity Techniques for Minimal Coordinate Contact Dynamics
,”
ASME J. Comput. Nonlinear Dyn.
,
12
(
2
), p.
12
.10.1115/1.4033520
6.
Trinkle
,
J. C.
,
2003
, “
Formulation of Multibody Dynamics as Complementarity Problems
,”
ASME
Paper No: DETC2003/VIB-48342.10.1115/DETC2003/VIB-48342
7.
Acary
,
V.
,
2013
, “
Projected Event-Capturing Time-Stepping Schemes for Nonsmooth Mechanical Systems With Unilateral Contact and Coulomb's Friction
,”
Comput. Methods Appl. Mech. Eng.
,
256
, pp.
224
250
.10.1016/j.cma.2012.12.012
8.
Glocker
,
C.
,
2006
, “
An Introduction to Impacts
,”
Nonsmooth Mechanics of Solids
, J. Haslinger and G. E. Stavroulakis, eds.,
Springer
,
Vienna
, pp.
45
101
.
9.
Pfeiffer
,
F.
,
1984
, “
Mechanische Systeme Mit Unstetigen Übergängen
,”
Ing. Arch.
,
54
(
3
), pp.
232
240
.10.1007/BF00555662
10.
Glocker
,
C.
, and
Pfeiffer
,
F.
,
1995
, “
Multiple Impacts With Friction in Rigid Multibody Systems
,”
Nonlinear Dyn.
,
7
(
4
), pp.
471
497
.10.1007/BF00121109
11.
Pfeiffer
,
F.
, and
Glocker
,
C.
,
2000
, “
Contacts in Multibody Systems
,”
J. Appl. Math. Mech.
,
64
(
5
), pp.
773
782
.10.1016/S0021-8928(00)00107-6
12.
Leine
,
R. I.
,
Van Campen
,
D.
, and
Glocker
,
C.
,
2003
, “
Nonlinear Dynamics and Modeling of Various Wooden Toys With Impacts and Friction
,”
J. Vib. Control
,
1261
, pp.
58
59
.10.1177/107754603030741
13.
Leine
,
R. I.
, and
Van Campen
,
D. H.
,
2006
, “
Bifurcation Phenomena in Non-Smooth Dynamical Systems
,”
Eur. J. Mech., A/Solids
,
25
(
4
), pp.
595
616
.10.1016/j.euromechsol.2006.04.004
14.
Glocker
,
C.
, and
Studer
,
C.
,
2005
, “
Formulation and Preparation for Numerical Evaluation of Linear Complementarity Systems in Dynamics
,”
Multibody Syst. Dyn.
,
13
(
4
), pp.
447
463
.10.1007/s11044-005-2519-6
15.
Slavic
,
J.
, and
Boltezar
,
M.
,
2006
, “
Non-Linearity and Non-Smoothness in Multi-Body Dynamics: Application to Woodpecker Toy
,”
Proc. Inst. Mech. Eng., Part C
,
220
(
3
), pp.
285
296
.10.1243/095440605X31562
16.
Charles
,
A.
,
Casenave
,
F.
, and
Glocker
,
C.
,
2018
, “
A Catching-Up Algorithm for Multibody Dynamics With Impacts and Dry Friction
,”
Comput. Methods Appl. Mech. Eng.
,
334
, pp.
208
237
.10.1016/j.cma.2018.01.054
17.
Peng
,
H.
,
Song
,
N.
, and
Kan
,
Z.
,
2020
, “
A Nonsmooth Contact Dynamic Algorithm Based on the Symplectic Method for Multibody System Analysis With Unilateral Constraints
,”
Multibody Syst. Dyn.
,
49
(
2
), pp.
119
153
.10.1007/s11044-019-09719-8
18.
Peng
,
H.
,
Song
,
N.
, and
Kan
,
Z.
,
2020
, “
A Novel Nonsmooth Dynamics Method for Multibody Systems With Friction and Impact Based on the Symplectic Discrete Format
,”
Int. J. Numer. Methods Eng.
,
121
(
7
), pp.
1530
1557
.10.1002/nme.6278
19.
Dubois
,
F.
,
Jean
,
M.
,
Renouf
,
M.
,
Mozul
,
R.
,
Martin
,
A.
, and
Bagneris
,
M.
,
2011
, “
LMGC90
,”
Proceedings of the 10eme Colloque National en Calcul Des Structures (CSMA)
, Giens, France. May 9–13, pp.
1
8
.
20.
Acary
,
V.
, and
Brogliato
,
B.
,
2008
,
Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics
,
Springer-Verlag
,
Berlin
.
21.
Heyn
,
T.
,
Mazhar
,
H.
,
Pazouki
,
A.
,
Melanz
,
D.
,
Seidl
,
A.
,
Madsen
,
J.
,
Bartholomew
,
A.
,
Negrut
,
D.
,
Lamb
,
D.
, and
Tasora
,
A.
,
2013
, “
Chrono: A Parallel Physics Library for Rigid-Body, Flexible-Body, and Fluid Dynamics
,”
Mech. Sci.
,
4
, pp.
49
64
.10.5194/ms-4-49-2013
22.
Lacoursière
,
C.
,
2007
, “
Ghosts and Machines: Regularized Variational Methods for Interactive Simulations of Multibodies With Dry Frictional Contacts
,” Ph.D. thesis,
Umeå University, Computing Science, Umeå
,
Sweden
.
23.
Cardona
,
A.
,
Klapka
,
I.
, and
Géradin
,
M.
,
1994
, “
Design of a New Finite Element Programming Environment
,”
Eng. Comput.
,
11
(
4
), pp.
365
381
.10.1108/02644409410799344
24.
Cosimo
,
A.
,
Galvez
,
J.
,
Cavalieri
,
F. J.
,
Cardona
,
A.
, and
Brüls
,
O.
,
2020
, “
A Robust Nonsmooth Generalized-α Scheme for Flexible Systems With Impacts
,”
Multibody Syst. Dyn.
,
48
(
2
), pp.
127
149
.10.1007/s11044-019-09692-2
25.
Brüls
,
O.
,
Acary
,
V.
, and
Cardona
,
A.
,
2014
, “
Simultaneous Enforcement of Constraints at Position and Velocity Levels in the Nonsmooth Generalized-α Scheme
,”
Comput. Methods Appl. Mech. Eng.
,
281
, pp.
131
161
.10.1016/j.cma.2014.07.025
26.
Brüls
,
O.
,
Cardona
,
A.
, and
Arnold
,
M.
,
2012
, “
Lie Group Generalized-α Time Integration of Constrained Flexible Multibody Systems
,”
Mech. Mach. Theory
,
48
, pp.
121
137
.10.1016/j.mechmachtheory.2011.07.017
27.
Studer
,
C.
,
2009
, “
Numerics of Unilateral Contacts and Friction: Modeling and Numerical Time Integration in Non-Smooth Dynamics
,”
Lecture Notes in Applied and Computational Mechanics
, Vol.
47
,
Springer
,
Berling
.
28.
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
.10.1016/0377-0427(85)90008-1
29.
Galvez
,
J.
,
Cavalieri
,
F. J.
,
Cosimo
,
A.
,
Brüls
,
O.
, and
Cardona
,
A.
,
2020
, “
A Nonsmooth Frictional Contact Formulation for Multibody System Dynamics
,”
Int. J. Numer. Methods Eng.
,
121
(
16
), pp.
3584
3609
.10.1002/nme.6371
30.
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
31.
Chung
,
J.
, and
Hulbert
,
G.
,
1993
, “
Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method
,”
ASME J. Appl. Mech.
,
60
(
2
), pp.
371
375
.10.1115/1.2900803
32.
Alart
,
P.
, and
Curnier
,
A.
,
1991
, “
A Mixed Formulation for Frictional Contact Problems Prone to Newton Like Solution Methods
,”
Comput. Methods Appl. Mech. Eng.
,
92
(
3
), pp.
353
375
.10.1016/0045-7825(91)90022-X
33.
Zander
,
R.
,
2008
, “
Flexible Multi-Body Systems With Set-Valued Force Laws
,” Ph.D. thesis,
Technische Universität München
,
Munich, Germany
.
34.
Piatkowski
,
T.
,
2014
, “
Dahl and LuGre Dynamic Friction Models—The Analysis of Selected Properties
,”
Mech. Mach. Theory
,
73
, pp.
91
100
.10.1016/j.mechmachtheory.2013.10.009
35.
Uchida
,
T. K.
,
Sherman
,
M. A.
, and
Delp
,
S. L.
,
2015
, “
Making a Meaningful Impact: Modelling Simultaneous Frictional Collisions in Spatial Multibody Systems
,”
Proc. R. Soc. A: Math., Phys. Eng. Sci.
,
471
(
2177
), p.
20140859
.10.1098/rspa.2014.0859
36.
Pfeiffer
,
F.
, and
Glocker
,
C.
,
1996
,
Multibody Dynamics With Unilateral Contacts
,
Wiley-VCH Verlag GmbH
,
Weinheim, Germany
.
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