The objective of this paper is to show, through the planar rocking block example, that kinetic angles play a fundamental role in multiple impact with friction. Even in the presence of Coulomb friction, a critical kinetic angle θcr is exhibited that allows one to split the blocks into two main classes: slender blocks with a kinetic angle larger than θcr, and flat blocks with a kinetic angle smaller than θcr. The value of θcr varies with the friction value, but it is independent of the restitution coefficient (normal dissipation). Numerical results are obtained using a multiple impact law recently introduced by the authors. Some comparisons between numerical and experimental results that validate the used model and numerical scheme are presented. However, this paper is mainly based on numerical simulations.

References

References
1.
Andreaus
,
U.
, and
Casini
,
P.
,
1999
, “
On the Rocking-Uplifting Motion of a Rigid Block in Free and Forced Motion: Influence of Sliding and Bouncing
,”
Acta. Mech.
,
138
, pp.
219
241
.10.1007/BF01291846
2.
Housner
,
G. W.
,
1963
, “
The Behaviour of Inverted Pendulum Structures During Earthquakes
,”
Bull. Seismol. Soc. Am.
,
53
(
2
), pp.
403
417
.
3.
Lipscombe
,
P. R.
, and
Pellegrino
,
S.
,
1993
, “
Free Rocking of Prismatic Blocks
,”
J. Eng. Mech.
,
119
(
7
), pp.
1387
1410
.10.1061/(ASCE)0733-9399(1993)119:7(1387)
4.
Pena
,
F.
,
Lourenco
,
P. B.
, and
Campos-Costa
,
A.
,
2008
, “
Experimental Dynamic Behavior of Free-Standing Multi-Block Structures Under Seismic Loadings
,”
J. Earthquake Eng.
,
12
, pp.
953
979
.10.1080/13632460801890513
5.
Prieto
,
F.
, and
Lourenco
,
P. B.
,
2005
, “
On the Rocking Behavior of Rigid Objects
,”
Meccanica
,
40
, pp.
121
133
.10.1007/s11012-005-5875-7
6.
Shi
,
B.
,
Anooshenhpoor
,
A.
,
Zeng
,
Y.
, and
Brune
,
J. N.
,
1996
, “
Rocking and Overturning of Precariously Balanced Rocks by Earthquakes
,”
Bull. Seismol. Soc. Am.
,
86
(
5
), pp.
1364
1371
.
7.
Tso
,
W. K.
, and
Wong
,
C. M.
,
1989
, “
Steady State Rocking Response of Rigid Blocks. Part 1: Analysis
,”
Earthquake Eng. Struct. Dyn.
,
18
(
1
), pp.
89
106
.10.1002/eqe.4290180109
8.
Tso
,
W. K.
, and
Wong
,
C. M.
,
1989
, “
Steady State Rocking Response of Rigid Blocks. Part 2: Experiment
,”
Earthquake Eng. Struct. Dyn.
,
18
(
1
), pp.
107
120
.10.1002/eqe.4290180109
9.
Yilmaz
,
C.
,
Gharib
,
M.
, and
Hurmuzlu
,
Y.
,
2009
, “
Solving Frictionless Rocking Block Problem With Multiple Impacts
,”
Proc. R. Soc. London, Ser. A
,
465
, pp.
3323
3339
.10.1098/rspa.2009.0273
10.
Yim
,
C. S.
,
Chopra
,
A. K.
, and
Penzien
,
J.
,
1980
, “
Rocking Response of Rigid Blocks to Earthquakes
,”
Earthquake Eng. Struct. Dyn.
,
8
(
6
), pp.
565
587
.10.1002/eqe.4290080606
11.
Heidenreich
,
B.
,
2004
, “
Small and Half-Scale Experimental Studies of Rockfall Impacts on Sandy Slopes
,”
Ph.D. thesis
,
Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
.
12.
Brach
,
R. M.
,
1991
,
Mechanical Impact Dynamics
,
John Wiley
,
New-York
.
13.
Brogliato
,
B.
,
1999
,
Nonsmooth Mechanics
,
2nd ed.
,
Springer
,
London
.
14.
Brogliato
,
B.
,
Zhang
,
H.
, and
Liu
,
C.
,
2012
, “
Analysis of a Generalized Kinematic Impact Law for Multibody-Multicontact Systems, With Application to the Planar Rocking Block and Chains of Balls
,”
Multibody Syst. Dyn.
,
27
, pp.
351
382
.10.1007/s11044-012-9301-3
15.
Chatterjee
,
A.
, and
Ruina
,
A.
,
1998
, “
A New Algebraic Rigid-Body Collision Law Based on Impulse Space Considerations
,”
ASME J. Appl. Mech.
,
65
, pp.
939
951
.10.1115/1.2791938
16.
Djerassi
,
S.
,
2009
, “
Collision With Friction; Part A: Newton's Hypothesis
,”
Multibody Syst. Dyn.
,
29
, pp.
37
54
.10.1007/s11044-008-9126-2
17.
Glocker
,
C.
,
2004
, “
Concepts for Modeling Impacts Without Friction
,”
Acta Mech.
,
168
, pp.
1
19
.10.1007/s00707-004-0076-3
18.
Modarres Najafabadi
,
S. A.
,
Kövecses
,
J.
, and
Angeles
,
J.
,
2008
, “
Generalization of the Energetic Coefficient of Restitution for Contacts in Multibody Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
3
, p.
041008
.10.1115/1.2960477
19.
Modarres Najafabadi
,
S. A.
,
Kövecses
,
J.
, and
Angeles
,
J.
,
2008
, “
Impacts in Multibody Systems: Modeling and Experiments
,”
Multibody Syst. Dyn.
,
20
, pp.
163
176
.10.1007/s11044-008-9117-3
20.
Modarres Najafabadi
,
S. A.
,
Kövecses
,
J.
, and
Angeles
,
J.
,
2007
, “
Energy Analysis and Decoupling in Three-Dimensional Impacts of Multibody Systems
,”
ASME J. Appl. Mech.
,
74
, pp.
845
851
.10.1115/1.2712226
21.
Lubarda
,
V.
,
2010
, “
The Bounds on the Coefficients of Restitution for the Frictional Impact of Rigid Pendulum Against a Fixed Surface
,”
ASME J. Appl. Mech.
,
77
(
1
), pp.
1
7
.10.1115/1.3172198
22.
Payr
,
M.
, and
Glocker
,
C.
,
2005
, “
Experimental Treatment of Multiple Contact Collisions
,”
Proceedings of the Euromech Conference ENOC
,
Eindhoven
,
August 7–12
, pp.
450
459
.
23.
Stronge
,
W. J.
,
2000
,
Impact Mechanics
,
Cambridge University Press
,
Cambridge, UK
.
24.
Leine
,
R.
, and
van de Wouw
,
N.
,
2008
, “
Stability Properties of Equilibrium Sets of Non-Linear Mechanical Systems With Dry Friction and Impact
,”
Nonlinear Dyn.
,
51
, pp.
551
583
.10.1007/s11071-007-9244-z
25.
Liu
,
C.
,
Zhao
,
Z.
, and
Brogliato
,
B.
,
2008
, “
Frictionless Multiple Impacts in Multibody Systems: Part I. Theoretical Framework
,”
Proc. R. Soc. London, Ser. A
,
464
(
2100
), pp.
3193
3211
.10.1098/rspa.2008.0078
26.
Liu
,
C.
,
Zhao
,
Z.
, and
Brogliato
,
B.
,
2008
, “
Energy Dissipation and Dispersion Effects in a Granular Media
,”
Phys. Rev. E
,
78
(
3
), p.
031307
.10.1103/PhysRevE.78.031307
27.
Zhao
,
Z.
,
Liu
,
C.
, and
Brogliato
,
B.
,
2009
, “
Planar Dynamics of a Rigid Body System With Frictional Impacts. II. Qualitative Analysis and Numerical Simulations
,”
Proc. R. Soc. London, Ser. A
,
465
(
2107
), pp.
2267
2292
.10.1098/rspa.2008.0520
28.
Liu
,
C.
,
Zhao
,
Z.
, and
Brogliato
,
B.
,
2009
, “
Frictionless Multiple Impacts in Multibody Systems: Part II. Numerical Algorithm and Simulation Results
,”
Proc. R. Soc. London, Ser. A
,
465
(
2101
), pp.
1
23
.10.1098/rspa.2008.0079
29.
Nguyen
,
N. S.
, and
Brogliato
,
B.
,
2012
, “
Shock Dynamics in Granular Chains: Numerical Simulations and Comparisons With Experimental Tests
,”
Granular Matter
,
14
(
3
), pp.
341
362
.10.1007/s10035-012-0338-z
30.
Falcon
,
E.
,
Laroche
,
C.
,
Fauve
,
S.
, and
Coste
,
S.
,
1998
, “
Collision of a 1-D Column of Beads With a Wall
,”
Eur. Phys. J. B
,
5
(
1
), pp.
111
131
.10.1007/s100510050424
31.
Nakagawa
,
M.
,
Agui
,
J. H.
,
Wu
,
D. T.
, and
Extramiana
,
D. V.
,
2003
, “
Impulse Dispersion in a Tapered Granular Chain
,”
Granular Matter
,
4
, pp.
167
174
.10.1007/s10035-002-0119-1
32.
Santibanez
,
F.
,
Munoz
,
R.
,
Caussarieu
,
A.
,
Job
,
S.
, and
Melo
,
F.
,
2011
, “
Experimental Evidence of Solitary Wave Interaction in Hertzian Chains
,”
Phys. Rev. E
,
84
, p.
026604
.10.1103/PhysRevE.84.026604
33.
Ceanga
,
V.
, and
Hurmuzlu
,
Y.
,
2001
, “
A New Look at an Old Problem: Newton's Cradle
,”
ASME J. Appl. Mech.
,
68
(
4
), pp.
575
583
.10.1115/1.1344902
34.
Dorbolo
,
S.
,
Volfson
,
D.
,
Tsimring
,
L.
, and
Kudrolli
,
A.
,
2005
, “
Dynamics of a Bouncing Dimer
,”
Phys. Rev. Lett.
,
95
(
4
), pp.
1
4
.10.1103/PhysRevLett.95.044101
35.
Zhang
,
H.
, and
Brogliato
,
B.
,
2011
, “
The Planar Rocking Block: Analysis of Kinematic Restitution Laws, and a New Rigid-Body Impact Model With Friction
,” INRIA Research Report No. RR-7580.
36.
Liu
,
C.
,
Zhao
,
Z.
, and
Brogliato
,
B.
,
2008
, “
Variable Structure Dynamics in a Bouncing Dimer
,” INRIA Research Report No. 6718.
37.
Acary
,
V.
, and
Brogliato
,
B.
,
2008
,
Numerical Simulation for Nonsmooth Dynamical Systems (Lecture Notes in Applied and Computational Mechanics)
, Vol.
35
,
Springer Verlag
,
Heidelberg, Germany
.
38.
Heemels
,
W. P. M. H.
,
Schumacher
,
J. M.
, and
Weiland
,
S.
,
2000
, “
Linear Complementarity Systems
,”
SIAM J. Appl. Math.
,
60
, pp.
1234
1269
.10.1137/S0036139997325199
39.
Paoli
,
L.
,
2005
, “
Continuous Dependence on Data for Vibro-Impact Problems
,”
Math. Models Meth. Appl. Sci.
,
15
(
1
), pp.
1
41
.10.1142/S0218202505003903
40.
Stronge
,
W. J.
,
2000
,
Chain Reaction From Impact on Coaxial Multibody Systems
,”
ASME J. Appl. Mech.
,
67
, pp.
632
635
. 10.1115/1.1309541
41.
“Siconos: A Software for Modeling and Simulation of Nonsmooth Dynamical Systems,” 2012, http://siconos.gforge.inria.fr/
42.
Aslam
,
M.
,
Godden
,
W. G.
, and
Scalise
,
D. T.
,
1980
, “
Earthquake Rocking Response of Rigid Bodies
,”
ASCE J. Struct. Div.
,
106
(
2
), pp.
377
392
.
43.
Priestley
,
M. J. N.
,
Evenson
,
R. J.
, and
Carr
,
A. J.
,
1978
, “
Seismic Response Analysis of Structures Free to Rock on Their Foundations
,”
Bull. New Zealand Seismol. Soc. Earthquake Eng.
,
11
(
3
), pp.
141
150
.
44.
ElGawady
,
M. A.
,
Ma
,
Q.
,
Butterworth
,
J. W.
, and
Ingham
,
J.
,
2011
, “
Effects of Interface Material on the Performance of Free Rocking Blocks
,”
Earthquake Eng. Struct. Dyn.
,
40
(
4
), pp.
375
392
.10.1002/eqe.1025
45.
Chatzis
,
M. N.
, and
Smyth
,
A. W.
,
2012
, “
Modeling of the 3D Rocking Problem
,”
Int. J. Non-Linear Mech.
,
47
(
4
), pp.
85
98
.10.1016/j.ijnonlinmec.2012.02.004
You do not currently have access to this content.