In this article, we will introduce the phenomenon known as the Painlevé paradox and further discuss the associated coupled phenomena, jam and lift-off. We analyze under what conditions the Painlevé paradox can occur for a general two-body collision using a framework that can be easily used with a variety of impact laws, however, in order to visualize jam and lift-off in a numerical simulation, we choose to use a recently developed energetic impact law as it is capable of achieving a unique forward solution in time. Further, we will use this framework to derive the criteria under which the Painlevé paradox can occur in a forced double-pendulum mechanical system. First, using a graphical technique, we will show that it is possible to achieve the Painlevé paradox for relatively low coefficient of friction values, and second we will use the energetic impact law to numerically show the occurrence of the Painlevé paradox in the double-pendulum system.

References

References
1.
Brach
,
R. M.
,
2007
,
Mechanical Impact Dynamics, Rigid Body Collisions
,
Wiley
,
New York
.
2.
Keogh
,
P. S.
, and
Cole
,
M. O. T.
,
2003
, “
Rotor Vibration With Auxiliary Bearing Contact in Magnetic Bearing Systems Part 1: Synchronous Dynamics
,”
J. Mech. Eng. Sci.
,
217
(
4
), pp.
377
392
.
3.
Zhang
,
H.
,
Brogliato
,
B.
, and
Liu
,
C.
,
2014
, “
Dynamics of Planar Rocking-Blocks With Coulomb Friction and Unilateral Constraints: Comparisons Between Experimental and Numerical Data
,”
Multibody Syst. Dyn.
,
32
(
1
), pp.
1
25
.
4.
Dankowicz
,
H.
, and
Piiroinen
,
P. T.
,
2002
, “
Exploiting Discontinuities for Stabilization of Recurrent Motions
,”
Dyn. Syst.
,
17
(
4
), pp.
317
342
.
5.
Stronge
,
W. J.
,
2000
,
Impact Mechanics
,
Cambridge University Press
, Cambridge, UK.
6.
Piiroinen
,
P. T.
,
Dankowicz
,
H. J.
, and
Nordmark
,
A. B.
,
2003
, “
Breaking Symmetries and Constraints: Transitions From 2D to 3D in Passive Walkers
,”
Multibody Syst. Dyn.
,
10
(
2
), pp.
147
176
.
7.
Osorio
,
G.
,
Di Bernardo
,
M.
, and
Santini
,
S.
,
2005
, “
Chattering and Complex Behaviour of a Cam-Follower System
,”
European Nonlinear Dynamics Conference (ENOC)
, Eindhoven, The Netherlands.
8.
Shen
,
Y.
, and
Stronge
,
W. J.
,
2011
, “
Painlevé Paradox During Oblique Impact With Friction
,”
Eur. J. Mech. A/Solids
,
30
(
4
), pp.
457
467
.
9.
Burns
,
S. J.
, and
Piiroinen
,
P. T.
,
2014
, “
The Complexity of a Basic Impact Mapping for Rigid Bodies With Impact and Friction
,”
J. Regular Chaotic Dyn.
,
19
(
1
), pp.
20
36
.
10.
Painlevé
,
P.
,
1905
, “
Sur les lois de frottement de glissement
,”
C. R. Acad. Sci.
,
141
, pp.
552
564
.
11.
Brogliato
,
B.
,
2007
,
Nonsmooth Mechanics
,
Springer-Verlag
, Cham, Switzerland.
12.
Champneys
,
A. R.
, and
Varkonyi
,
P. L.
,
2016
, “
The Painlevé Paradox in Contact Mechanics
,”
IMA J. Appl. Math.
,
81
(
1
), pp.
538
588
.
13.
Leine
,
R. I.
,
Brogliato
,
B.
, and
Nijmeijer
,
H.
,
2002
, “
Periodic Motion and Bifurcations Induced by the Painlevé Paradox
,”
Eur. J. Mech. A/Solids
,
21
(
5
), pp.
869
896
.
14.
Grigoryan
,
S. S.
,
2001
, “
The Solution to the Painlevé Paradox for Dry Friction
,”
Dokl. Phys.
,
46
(
7
), pp.
499
503
(in Russian).
15.
Génot
,
F.
, and
Brogliato
,
B.
,
1999
, “
New Results on Painlevé Paradoxes
,”
Eur. J. Mech. A/Solids
,
18
(
4
), pp.
653
677
.
16.
Zhao
,
Z.
,
Liu
,
C.
,
Ma
,
W.
, and
Chen
,
B.
,
2008
, “
Experimental Investigation of the Painlevé Paradox in a Robotic System
,”
ASME J. Appl. Mech.
,
75
(
4
), p.
041006
.
17.
Liu
,
C.
,
Zhao
,
Z.
, and
Chen
,
B.
,
2007
, “
The Bouncing Motion Appearing in a Robotic System With Unilateral Constraint
,”
Nonlinear Dyn.
,
49
(
1
), pp.
217
232
.
18.
Wilms
,
E. V.
, and
Cohen
,
H.
,
1997
, “
The Occurrence of Painlevé's Paradox in the Motion of a Rotating Shaft
,”
Trans. ASME
,
64
(
4
), pp.
1008
1010
.
19.
Nordmark
,
A.
,
Dankowicz
,
H.
, and
Champneys
,
A.
,
2009
, “
Discontinuity-Induced Bifurcation in Systems With Impacts and Friction: Discontinuities in the Impact Law
,”
Int. J. Non-Linear Mech.
,
44
(
10
), pp.
1011
1023
.
20.
Nordmark
,
A.
,
Dankowicz
,
H.
, and
Champneys
,
A.
,
2011
, “
Friction-Induced Reverse Chatter in Rigid-Body Mechanisms With Impacts
,”
IMA J. Appl. Math.
,
76
(
1
), pp.
85
119
.
21.
Burns
,
S. J.
, and
Piiroinen
,
P. T.
,
2015
, “
A Hybrid Scheme for Simulation of Planar Rigid Bodies With Impacts and Friction Using Impact Mappings
,”
Int. J. Nonlinear Mech.
,
77
, pp.
312
324
.
22.
Darboux
,
G.
,
1880
, “
Etude géométrique sur les percussions et le choc des corps
,”
Bull. Sci. Math. Astron.
,
4
(
1
), pp.
126
160
.
23.
Keller
,
J. B.
,
1986
, “
Impact With Friction
,”
ASME J. Appl. Mech.
,
53
(
1
), pp.
1
4
.
24.
Nordmark
,
A. B.
, and
Piiroinen
,
P. T.
,
2009
, “
Simulation and Stability Analysis of Impacting Systems With Complete Chattering
,”
Nonlinear Dyn.
,
58
, pp.
85
106
.
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