Abstract

We study mathematical models of binary direct collinear collisions of convex viscoplastic bodies based on two incremental collision laws that employ the Bouc–Wen differential model of hysteresis to represent the elastoplastic behavior of the materials of the colliding bodies. These collision laws are the Bouc–Wen–Simon–Hunt–Crossley collision law (BWSHCCL) and the Bouc–Wen–Maxwell collision law (BWMCL). The BWSHCCL comprises of the Bouc–Wen model amended with a nonlinear Hertzian elastic spring element and connected in parallel to a nonlinear displacement-dependent and velocity-dependent energy dissipation element. The BWMCL comprises of the Bouc–Wen model amended with a nonlinear Hertzian elastic spring element and connected in series to a linear velocity-dependent energy dissipation element. The mathematical models of the collision process are presented in the form of finite-dimensional initial value problems (IVPs). We show that the models possess favorable analytical properties (e.g., global existence, uniqueness, and boundedness of the solutions) under suitable restrictions on the values of their parameters. Furthermore, based on the results of two model parameter identification studies, we demonstrate that good agreement can be attained between experimental data and numerical approximations of the behavior of the mathematical models across a wide range of initial relative velocities of the colliding bodies while using parameterizations of the models that are independent of the initial relative velocity.

References

1.
Flores
,
P.
,
Leine
,
R.
, and
Glocker
,
C.
,
2011
, “
Modeling and Analysis of Rigid Multibody Systems With Translational Clearance Joints Based on the Nonsmooth Dynamics Approach
,”
Multibody Dynamics: Computational Methods and Applications
(Computational Methods in Applied Sciences, Vol.
23
),
K.
Arczewski
,
W.
Blajer
,
J.
Fraczek
, and
M.
Wojtyra
, eds.,
Springer Dordrecht
,
Dordrecht, The Netherlands
, pp.
107
130
.
2.
Machado
,
M.
,
Moreira
,
P.
,
Flores
,
P.
, and
Lankarani
,
H. M.
,
2012
, “
Compliant Contact Force Models in Multibody Dynamics: Evolution of the Hertz Contact Theory
,”
Mech. Mach. Theory
,
53
, pp.
99
121
.10.1016/j.mechmachtheory.2012.02.010
3.
Panagiotopoulos
,
P. D.
,
1985
,
Inequality Problems in Mechanics and Applications: Convex and Nonconvex Energy Functions
,
Birkhäuser
,
Boston
.
4.
Pfeiffer
,
F.
, and
Glocker
,
C.
,
2004
,
Multibody Dynamics With Unilateral Contacts
(Wiley Series in Nonlinear Science),
WILEY-VCH Verlag GmbH & Co. KGaA
,
Weinheim, The Federal Republic of Germany
.
5.
Stewart
,
D. E.
,
2011
,
Dynamics With Inequalities: Impacts and Hard Constraints
,
SIAM
,
Philadelphia, PA
.
6.
Goebel
,
R.
,
Sanfelice
,
R. G.
, and
Teel
,
A. R.
,
2012
,
Hybrid Dynamical Systems: Modeling, Stability, and Robustness
,
Princeton University Press
,
Princeton, NJ
.
7.
Brogliato
,
B.
,
2016
,
Nonsmooth Mechanics: Models, Dynamics and Control
(Communications and Control Engineering), 3rd ed.,
Springer International Publishing
, Cham, Switzerland,
The Swiss Confederation
.
8.
Sanfelice
,
R. G.
,
2021
,
Hybrid Feedback Control
,
Princeton University Press
,
Princeton, NJ
.
9.
Terzopoulos
,
D.
,
Platt
,
J.
,
Barr
,
A.
, and
Fleischer
,
K.
,
1987
, “
Elastically Deformable Models
,”
Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques
, Anaheim, CA, July 27–31, pp.
205
214
.10.1145/37401.37427
10.
Platt
,
J. C.
, and
Barr
,
A. H.
,
1988
, “
Constraint Methods for Flexible Models
,”
SIGGRAPH '88: Proceedings of the 15th Annual Conference on Computer Graphics and Interactive Techniques
, Atlanta, GA, Aug. 1–5, pp.
279
288
.10.1145/378456.378524
11.
Moore
,
M.
, and
Wilhelms
,
J.
,
1988
, “
Collision Detection and Response for Computer Animation
,”
SIGGRAPH '88: Proceedings of the 15th Annual Conference on Computer Graphics and Interactive Techniques
, Atlanta, GA, Aug. 1–5, pp.
289
298
.10.1145/378456.378528
12.
Roithmayr
,
C. M.
, and
Hodges
,
D. H.
,
2016
,
Dynamics: Theory and Application of Kane's Method
,
Cambridge University Press
,
Cambridge, UK
.
13.
Stronge
,
W. J.
,
2018
,
Impact Mechanics
, 2nd ed.,
Cambridge University Press
,
Cambridge, UK
.
14.
Ruina
,
A.
, and
Pratap
,
R.
,
2019
,
Introduction to Mechanics for Engineers
,
Rudra Pratap and Andy Ruina
, Oxford, UK.
15.
Corral
,
E.
,
Moreno
,
R. G.
,
García
,
M. J. G.
, and
Castejón
,
C.
,
2021
, “
Nonlinear Phenomena of Contact in Multibody Systems Dynamics: A Review
,”
Nonlinear Dyn.
,
104
(
2
), pp.
1269
1295
.10.1007/s11071-021-06344-z
16.
Chatterjee
,
A.
,
1997
, “
Rigid Body Collisions: Some General Considerations, New Collision Laws, and Some Experimental Data
,”
Ph.D. thesis
,
Cornell University
,
Ithaca, NY
.http://ruina.tam.cornell.edu/research/topics/collision_mechanics/rigid_body_collisions.pdf
17.
Chatterjee
,
A.
, and
Ruina
,
A.
,
1998
, “
Two Interpretations of Rigidity in Rigid-Body Collisions
,”
ASME J. Appl. Mech.
,
65
(
4
), pp.
894
900
.10.1115/1.2791929
18.
Gilardi
,
G.
, and
Sharf
,
I.
,
2002
, “
Literature Survey of Contact Dynamics Modelling
,”
Mech. Mach. Theory
,
37
(
10
), pp.
1213
1239
.10.1016/S0094-114X(02)00045-9
19.
Morro
,
A.
, and
Giorgi
,
C.
,
2023
,
Mathematical Modelling of Continuum Physics
(Modeling and Simulation in Science, Engineering and Technology),
Springer Nature Switzerland AG
,
Cham, The Swiss Confederation
.
20.
Reiner
,
M.
,
1945
, “
A Classification of Rheological Properties
,”
J. Sci. Instrum.
,
22
(
7
), pp.
127
129
.10.1088/0950-7671/22/7/303
21.
Thomson
,
W.
,
1865
, “
On the Elasticity and Viscosity of Metals
,”
Proc. R. Soc. London
,
14
, pp.
289
297
.10.1098/rspl.1865.0052
22.
Meyer
,
O. E.
,
1874
, “
Zur Theorie der inneren Reibung
,”
J. Reine Angew. Math.
,
174
(
78
), pp.
130
135
.10.1515/crll.1874.78.130
23.
Voigt
,
W.
,
1890
, “
Ueber die innere Reibung der festen Körper, insbesondere der Krystalle
,”
Abh. K. Ges. Wiss. Goettingen
,
36
, pp.
3
48
.
24.
Butcher
,
E. A.
, and
Segalman
,
D. J.
,
2000
, “
Characterizing Damping and Restitution in Compliant Impacts Via Modified K-V and Higher-Order Linear Viscoelastic Models
,”
ASME J. Appl. Mech.
,
67
(
4
), pp.
831
834
.10.1115/1.1308578
25.
Maxwell
,
J. C.
,
1867
, “
On the Dynamical Theory of Gases
,”
Philos. Trans. R. Soc. London
,
157
, pp.
49
88
.10.1098/rstl.1867.0004
26.
Luding
,
S.
,
1998
, “
Collisions & Contacts Between Two Particles
,”
Physics of Dry Granular Media
(NATO Science Series E: Applied Sciences, Vol.
350
),
H. J.
Herrmann
,
J.-P.
Hovi
, and
S.
Luding
, eds.,
Springer Dordrecht, Dordrecht
,
The Netherlands
, pp.
285
304
.
27.
Kruggel-Emden
,
H.
,
Simsek
,
E.
,
Rickelt
,
S.
,
Wirtz
,
S.
, and
Scherer
,
V.
,
2007
, “
Review and Extension of Normal Force Models for the Discrete Element Method
,”
Powder Technol.
,
171
(
3
), pp.
157
173
.10.1016/j.powtec.2006.10.004
28.
Seifried
,
R.
,
Schiehlen
,
W.
, and
Eberhard
,
P.
,
2010
, “
The Role of the Coefficient of Restitution on Impact Problems in Multi-Body Dynamics
,”
Proc. Inst. Mech. Eng., Part K: J. Multi-Body Dyn.
,
224
(
3
), pp.
279
306
.10.1243/14644193JMBD239
29.
Brake
,
M. R.
,
2013
, “
The Effect of the Contact Model on the Impact-Vibration Response of Continuous and Discrete Systems
,”
J. Sound Vib.
,
332
(
15
), pp.
3849
3878
.10.1016/j.jsv.2013.02.003
30.
Khulief
,
Y. A.
,
2013
, “
Modeling of Impact in Multibody Systems: An Overview
,”
ASME J. Comput. Nonlinear Dyn.
,
8
(
2
), p.
021012
.10.1115/1.4006202
31.
Thornton
,
C.
,
Cummins
,
S. J.
, and
Cleary
,
P. W.
,
2013
, “
An Investigation of the Comparative Behaviour of Alternative Contact Force Models During Inelastic Collisions
,”
Powder Technol.
,
233
, pp.
30
46
.10.1016/j.powtec.2012.08.012
32.
Alves
,
J.
,
Peixinho
,
N.
,
da Silva
,
M. T.
,
Flores
,
P.
, and
Lankarani
,
H. M.
,
2015
, “
A Comparative Study of the Viscoelastic Constitutive Models for Frictionless Contact Interfaces in Solids
,”
Mech. Mach. Theory
,
85
, pp.
172
188
.10.1016/j.mechmachtheory.2014.11.020
33.
Ahmad
,
M.
,
Ismail
,
K. A.
, and
Mat
,
F.
,
2016
, “
Impact Models and Coefficient of Restitution: A Review
,”
ARPN J. Eng. Appl. Sci.
,
11
(
10
), pp.
6549
6555
.https://www.arpnjournals.org/jeas/research_papers/rp_2016/jeas_0516_4312.pdf
34.
Banerjee
,
A.
,
Chanda
,
A.
, and
Das
,
R.
,
2017
, “
Historical Origin and Recent Development on Normal Directional Impact Models for Rigid Body Contact Simulation: A Critical Review
,”
Arch. Comput. Methods Eng.
,
24
(
2
), pp.
397
422
.10.1007/s11831-016-9164-5
35.
Skrinjar
,
L.
,
Slavič
,
J.
, and
Boltežar
,
M.
,
2018
, “
A Review of Continuous Contact-Force Models in Multibody Dynamics
,”
Int. J. Mech. Sci.
,
145
, pp.
171
187
.10.1016/j.ijmecsci.2018.07.010
36.
Rodrigues Da Silva
,
M.
,
Marques
,
F.
,
Tavares Da Silva
,
M.
, and
Flores
,
P.
,
2022
, “
A Compendium of Contact Force Models Inspired by Hunt and Crossley's Cornerstone Work
,”
Mech. Mach. Theory
,
167
, p.
104501
.10.1016/j.mechmachtheory.2021.104501
37.
Wang
,
G.
,
Ma
,
D.
,
Liu
,
Y.
, and
Liu
,
C.
,
2022
, “
Research Progress of Contact Force Models in the Collision Mechanics of Multibody System
,”
Chin. J. Theor. Appl. Mech.
,
54
(
12
), pp.
3239
3266
.10.6052/0459-1879-22-266
38.
Wang
,
D.
,
de Boer
,
G.
,
Neville
,
A.
, and
Ghanbarzadeh
,
A.
,
2022
, “
A Review on Modelling of Viscoelastic Contact Problems
,”
Lubricants
,
10
(
12
), p.
358
.10.3390/lubricants10120358
39.
Flores
,
P.
,
Ambrósio
,
J.
, and
Lankarani
,
H. M.
,
2023
, “
Contact-Impact Events With Friction in Multibody Dynamics: Back to Basics
,”
Mech. Mach. Theory
,
184
, p.
105305
.10.1016/j.mechmachtheory.2023.105305
40.
Ding
,
S.
,
Hu
,
Y.
,
Jian
,
B.
,
Zhang
,
Y.
,
Xia
,
R.
, and
Hu
,
G.
,
2024
, “
A Review and Comparative Analysis of Normal Contact Force Models for Viscoelastic Particles
,”
Int. J. Impact Eng.
,
189
, p.
104968
.10.1016/j.ijimpeng.2024.104968
41.
Hooke
,
R.
,
1678
,
De Potentia Restitutiva, or of Spring. Explaining the Power of Springing Bodies
,
Printed for John Martyn Printer to the Royal Society
,
London
.
42.
Hertz
,
H. R.
,
1881
, “
Über die Berührung fester elastischer Körper
,”
J. Reine Angew. Math.
,
92
, pp.
156
171
.10.1515/crll.1882.92.156
43.
Hertz
,
H. R.
,
1882
, “
Über die Berührung fester elastischer Körper und über die Härte
,”
Verhandlungen des Vereins zur Beförderung des Gewerbfleißes
,
Verein zur Beförderung des Gewerbefleisses
,
Berlin, Germany
, pp.
449
463
.
44.
Goldsmith
,
W.
,
1960
,
Impact: The Theory and Physical Behaviour of Colliding Solids
,
Edward Arnold
,
London
.
45.
Dubowsky
,
S.
, and
Freudenstein
,
F.
,
1971
, “
Dynamic Analysis of Mechanical Systems With Clearances. Part 2: Dynamic Response
,”
ASME J. Eng. Ind.
,
93
(
1
), pp.
310
316
.10.1115/1.3427896
46.
Dubowsky
,
S.
, and
Freudenstein
,
F.
,
1971
, “
Dynamic Analysis of Mechanical Systems With Clearances. Part 1: Formation of Dynamic Model
,”
ASME J. Eng. Ind.
,
93
(
1
), pp.
305
309
.10.1115/1.3427895
47.
Hunt
,
K. H.
, and
Crossley
,
F. R. E.
,
1975
, “
Coefficient of Restitution Interpreted as Damping in Vibroimpact
,”
ASME J. Appl. Mech.
,
42
(
2
), pp.
440
445
.10.1115/1.3423596
48.
Johnson
,
K. L.
,
1985
,
Contact Mechanics
,
Cambridge University Press
,
Cambridge, UK
.
49.
Argatov
,
I. I.
,
2013
, “
Mathematical Modeling of Linear Viscoelastic Impact: Application to Drop Impact Testing of Articular Cartilage
,”
Tribol. Int.
,
63
, pp.
213
225
.10.1016/j.triboint.2012.09.015
50.
Poynting
,
J. H.
, and
Thomson
,
J. J.
,
1902
,
A Text-Book of Physics: Properties of Matter
,
Charles Griffin and Company
,
London
.
51.
Jeffreys
,
H.
,
1917
, “
The Viscosity of the Earth (Third Paper)
,”
Mon. Not. R. Astron. Soc.
,
77
(
5
), pp.
449
456
.10.1093/mnras/77.5.449
52.
Ishlinsky
,
A. Y.
,
1940
, “
Vibrations of a Rod in the Presence of a Linear Law of Aftereffect and Relaxation
,”
J. Appl. Math. Mech.
,
4
(
1
), pp.
79
92
.
53.
Zener
,
C.
,
1948
,
Elasticity and Anelasticity of Metals
,
University of Chicago Press
,
Chicago, IL
.
54.
Mills
,
J. K.
, and
Nguyen
,
C. V.
,
1992
, “
Robotic Manipulator Collisions: Modeling and Simulation
,”
ASME J. Dyn. Syst., Meas., Control
,
114
(
4
), pp.
650
659
.10.1115/1.2897737
55.
Veluswami
,
M. A.
,
Crossley
,
F. R. E.
, and
Horvay
,
G.
,
1975
, “
Multiple Impacts of a Ball Between Two Plates. Part 2: Mathematical Modelling
,”
ASME J. Eng. Ind.
,
97
(
3
), pp.
828
835
.10.1115/1.3438689
56.
Veluswami
,
M. A.
, and
Crossley
,
F. R. E.
,
1975
, “
Multiple Impacts of a Ball Between Two Plates. Part 1: Some Experimental Observations
,”
ASME J. Eng. Ind.
,
97
(
3
), pp.
820
827
.10.1115/1.3438688
57.
Herbert
,
R. G.
, and
McWhannell
,
D. C.
,
1977
, “
Shape and Frequency Composition of Pulses From an Impact Pair
,”
ASME J. Eng. Ind.
,
99
(
3
), pp.
513
518
.10.1115/1.3439270
58.
Lee
,
T. W.
, and
Wang
,
A. C.
,
1983
, “
On the Dynamics of Intermittent-Motion Mechanisms. Part I. Dynamic Model and Response
,”
ASME J. Mech., Transm., Autom. Des.
,
105
(
3
), pp.
534
540
.10.1115/1.3267392
59.
Lankarani
,
H. M.
, and
Nikravesh
,
P. E.
,
1990
, “
A Contact Force Model With Hysteresis Damping for Impact Analysis of Multibody Systems
,”
ASME J. Mech. Des.
,
112
(
3
), pp.
369
376
.10.1115/1.2912617
60.
Lankarani
,
H. M.
, and
Nikravesh
,
P. E.
,
1994
, “
Continuous Contact Force Models for Impact Analysis in Multibody Systems
,”
Nonlinear Dyn.
,
5
(
2
), pp.
193
207
.10.1007/BF00045676
61.
Stoianovici
,
D.
, and
Hurmuzlu
,
Y.
,
1996
, “
A Critical Study of the Applicability of Rigid-Body Collision Theory
,”
ASME J. Appl. Mech.
,
63
(
2
), pp.
307
316
.10.1115/1.2788865
62.
Marhefka
,
D. W.
, and
Orin
,
D. E.
,
1999
, “
A Compliant Contact Model With Nonlinear Damping for Simulation of Robotic Systems
,”
IEEE Trans. Syst., Man, Cybern. - Part A: Syst. Hum.
,
29
(
6
), pp.
566
572
.10.1109/3468.798060
63.
Gonthier
,
Y.
,
McPhee
,
J.
,
Lange
,
C.
, and
Piedbœuf
,
J.-C.
,
2004
, “
A Regularized Contact Model With Asymmetric Damping and Dwell-Time Dependent Friction
,”
Multibody Syst. Dyn.
,
11
(
3
), pp.
209
233
.10.1023/B:MUBO.0000029392.21648.bc
64.
Zhang
,
Y.
, and
Sharf
,
I.
,
2004
, “
Compliant Force Modelling for Impact Analysis
,”
ASME
Paper No. DETC2004-57220.10.1115/DETC2004-57220
65.
Zhiying
,
Q.
, and
Qishao
,
L.
,
2006
, “
Analysis of Impact Process Model Based on Restitution Coefficient
,”
J. Dyn. Control
,
4
(
4
), pp.
294
298
.
66.
Ye
,
K.
,
Li
,
L.
, and
Zhu
,
H.
,
2009
, “
A Note on the Hertz Contact Model With Nonlinear Damping for Pounding Simulation
,”
Earthquake Eng. Struct. Dyn.
,
38
(
9
), pp.
1135
1142
.10.1002/eqe.883
67.
Hu
,
G.
,
Hu
,
Z.
,
Jian
,
B.
,
Liu
,
L.
, and
Wan
,
H.
,
2011
, “
On the Determination of the Damping Coefficient of Non-Linear Spring-Dashpot System to Model Hertz Contact for Simulation by Discrete Element Method
,”
J. Comput.
,
6
(
5
), pp.
984
988
.
68.
Flores
,
P.
,
Machado
,
M.
,
Silva
,
M. T.
, and
Martins
,
J. M.
,
2011
, “
On the Continuous Contact Force Models for Soft Materials in Multibody Dynamics
,”
Multibody Syst. Dyn.
,
25
(
3
), pp.
357
375
.10.1007/s11044-010-9237-4
69.
Gharib
,
M.
, and
Hurmuzlu
,
Y.
,
2012
, “
A New Contact Force Model for Low Coefficient of Restitution Impact
,”
ASME J. Appl. Mech.
,
79
(
6
), p.
064506
.10.1115/1.4006494
70.
Khatiwada
,
S.
,
Chouw
,
N.
, and
Butterworth
,
J. W.
,
2014
, “
A Generic Structural Pounding Model Using Numerically Exact Displacement Proportional Damping
,”
Eng. Struct.
,
62–63
, pp.
33
41
.10.1016/j.engstruct.2014.01.016
71.
Jacobs
,
D. A.
, and
Waldron
,
K. J.
,
2015
, “
Modeling Inelastic Collisions With the Hunt–Crossley Model Using the Energetic Coefficient of Restitution
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
2
), p.
021001
.10.1115/1.4028473
72.
Hu
,
S.
, and
Guo
,
X.
,
2015
, “
A Dissipative Contact Force Model for Impact Analysis in Multibody Dynamics
,”
Multibody Syst. Dyn.
,
35
(
2
), pp.
131
151
.10.1007/s11044-015-9453-z
73.
Wang
,
X.
,
Ji
,
X.
,
Wang
,
L.
,
Wang
,
D.
, and
Han
,
B.
,
2018
, “
Modeling and Simulation of an Improved Impact Force Model for Mechanical System
,” Proceedings of the 2018 International Conference on Mathematics, Modelling, Simulation and Algorithms (
MMSA 2018
), Chengdu, People's Republic of China, Mar. 25–26, pp.
34
38
.10.2991/mmsa-18.2018.9
74.
Carvalho
,
A. S.
, and
Martins
,
J. M.
,
2019
, “
Exact Restitution and Generalizations for the Hunt–Crossley Contact Model
,”
Mech. Mach. Theory
,
139
, pp.
174
194
.10.1016/j.mechmachtheory.2019.03.028
75.
Sherif
,
H. A.
, and
Almufadi
,
F. A.
,
2020
, “
Models for Materials Damping, Loss Factor, and Coefficient of Restitution
,”
ASME J. Eng. Mater. Technol.
,
142
(
1
), p.
011006
.10.1115/1.4044281
76.
Safaeifar
,
H.
, and
Farshidianfar
,
A.
,
2020
, “
A New Model of the Contact Force for the Collision Between Two Solid Bodies
,”
Multibody Syst. Dyn.
,
50
(
3
), pp.
233
257
.10.1007/s11044-020-09732-2
77.
Yu
,
J.
,
Chu
,
J.
,
Li
,
Y.
, and
Guan
,
L.
,
2020
, “
An Improved Compliant Contact Force Model Using a Piecewise Function for Impact Analysis in Multibody Dynamics
,”
Proc. Inst. Mech. Eng., Part K: J. Multi-Body Dyn.
,
234
(
2
), pp.
424
432
.10.1177/1464419319900874
78.
Zhang
,
J.
,
Li
,
W.
,
Zhao
,
L.
, and
He
,
G.
,
2020
, “
A Continuous Contact Force Model for Impact Analysis in Multibody Dynamics
,”
Mech. Mach. Theory
,
153
, p.
103946
.10.1016/j.mechmachtheory.2020.103946
79.
Zhao
,
P.
,
Liu
,
J.
,
Li
,
Y.
, and
Wu
,
C.
,
2021
, “
A Spring-Damping Contact Force Model Considering Normal Friction for Impact Analysis
,”
Nonlinear Dyn.
,
105
(
2
), pp.
1437
1457
.10.1007/s11071-021-06660-4
80.
Wang
,
G.
,
Liu
,
C.
, and
Liu
,
Y.
,
2022
, “
Energy Dissipation Analysis for Elastoplastic Contact and Dynamic Dashpot Models
,”
Int. J. Mech. Sci.
,
221
, p.
107214
.10.1016/j.ijmecsci.2022.107214
81.
Ramaswamy
,
D. A.
,
2023
, “
Continuous Contact Force Modeling: Theoretical Formulation of Model Parameters for the Simulation of Arbitrary Compliant Impacts
,” Master's thesis,
University of Colorado
,
Boulder, CO
.
82.
Tan
,
H.
,
Li
,
L.
,
Huang
,
Q.
,
Jiang
,
Z.
,
Li
,
Q.
,
Zhang
,
Y.
, and
Yu
,
D.
,
2023
, “
Influence of Two Kinds of Clearance Joints on the Dynamics of Planar Mechanical System Based on a Modified Contact Force Model
,”
Sci. Rep.
,
13
(
1
), p.
20569
.10.1038/s41598-023-47315-1
83.
Sheikhi Azqandi
,
M.
, and
Safaeifar
,
H.
,
2024
, “
Optimal Model of the Contact Force for the Collision Between Two Solid Bodies by ICACO
,”
Int. J. Optim. Civ. Eng.
,
14
(
1
), pp.
17
35
.10.22068/ijoce.2024.14.1.573
84.
Tatara
,
Y.
, and
Moriwaki
,
N.
,
1982
, “
Study on Impact of Equivalent Two Bodies (Coefficients of Restitution of Spheres of Brass, Lead, Glass, Porcelain and Agate, and the Material Properties)
,”
Bull. JSME
,
25
(
202
), pp.
631
637
.10.1299/jsme1958.25.631
85.
Kuwabara
,
G.
, and
Kono
,
K.
,
1987
, “
Restitution Coefficient in a Collision Between Two Spheres
,”
Jpn. J. Appl. Phys.
,
26
(
8R
), pp.
1230
1233
.10.1143/JJAP.26.1230
86.
Ristow
,
G. H.
,
1992
, “
Simulating Granular Flow With Molecular Dynamics
,”
J. Phys. I
,
2
(
5
), pp.
649
662
.10.1051/jp1:1992159
87.
Tsuji
,
Y.
,
Tanaka
,
T.
, and
Ishida
,
T.
,
1992
, “
Lagrangian Numerical Simulation of Plug Flow of Cohesionless Particles in a Horizontal Pipe
,”
Powder Technol.
,
71
(
3
), pp.
239
250
.10.1016/0032-5910(92)88030-L
88.
Lee
,
J.
, and
Herrmann
,
H. J.
,
1993
, “
Angle of Repose and Angle of Marginal Stability: Molecular Dynamics of Granular Particles
,”
J. Phys. A: Math. Gen.
,
26
(
2
), pp.
373
383
.10.1088/0305-4470/26/2/021
89.
Luding
,
S.
,
Clément
,
E.
,
Blumen
,
A.
,
Rajchenbach
,
J.
, and
Duran
,
J.
,
1994
, “
Anomalous Energy Dissipation in Molecular-Dynamics Simulations of Grains: The ‘Detachment’ Effect
,”
Phys. Rev. E
,
50
(
5
), pp.
4113
4122
.10.1103/PhysRevE.50.4113
90.
Hertzsch
,
J.-M.
,
Spahn
,
F.
, and
Brilliantov
,
N. V.
,
1995
, “
On Low-Velocity Collisions of Viscoelastic Particles
,”
J. Phys. II
,
5
(
11
), pp.
1725
1738
.10.1051/jp2:1995210
91.
Shäfer
,
J.
,
Dippel
,
S.
, and
Wolf
,
D. E.
,
1996
, “
Force Schemes in Simulations of Granular Materials
,”
J. Phys. I
,
6
(
1
), pp.
5
20
.10.1051/jp1:1996129
92.
Brilliantov
,
N. V.
,
Spahn
,
F.
,
Hertzsch
,
J.-M.
, and
Pöschel
,
T.
,
1996
, “
The Collision of Particles in Granular Systems
,”
Phys. A: Stat. Mech. Appl.
,
231
(
4
), pp.
417
424
.10.1016/0378-4371(96)00099-4
93.
Brilliantov
,
N. V.
,
Spahn
,
F.
,
Hertzsch
,
J.-M.
, and
Pöschel
,
T.
,
1996
, “
Model for Collisions in Granular Gases
,”
Phys. Rev. E
,
53
(
5
), pp.
5382
5392
.10.1103/PhysRevE.53.5382
94.
Morgado
,
W. A. M.
, and
Oppenheim
,
I.
,
1997
, “
Energy Dissipation for Quasielastic Granular Particle Collisions
,”
Phys. Rev. E
,
55
(
2
), pp.
1940
1945
.10.1103/PhysRevE.55.1940
95.
Falcon
,
E.
,
Laroche
,
C.
,
Fauve
,
S.
, and
Coste
,
C.
,
1998
, “
Behavior of One Inelastic Ball Bouncing Repeatedly Off the Ground
,”
Eur. Phys. J. B - Condens. Matter Complex Syst.
,
3
(
1
), pp.
45
57
.10.1007/s100510050283
96.
Schwager
,
T.
, and
Pöschel
,
T.
,
1998
, “
Coefficient of Normal Restitution of Viscous Particles and Cooling Rate of Granular Gases
,”
Phys. Rev. E
,
57
(
1
), pp.
650
654
.10.1103/PhysRevE.57.650
97.
Jankowski
,
R.
,
2005
, “
Non-Linear Viscoelastic Modelling of Earthquake-Induced Structural Pounding
,”
Earthquake Eng. Struct. Dyn.
,
34
(
6
), pp.
595
611
.10.1002/eqe.434
98.
Jankowski
,
R.
,
2006
, “
Analytical Expression Between the Impact Damping Ratio and the Coefficient of Restitution in the Non-Linear Viscoelastic Model of Structural Pounding
,”
Earthquake Eng. Struct. Dyn.
,
35
(
4
), pp.
517
524
.10.1002/eqe.537
99.
Bordbar
,
M. H.
, and
Hyppänen
,
T.
,
2007
, “
Modeling of Binary Collision Between Multisize Viscoelastic Spheres
,”
J. Numer. Anal., Ind. Appl. Math.
,
2
(
3–4
), pp.
115
128
.https://jnaiam.org/2024/02/modeling-of-binary-collision-between-multisize-viscoelastic-spheres/
100.
Schwager
,
T.
, and
Pöschel
,
T.
,
2008
, “
Coefficient of Restitution for Viscoelastic Spheres: The Effect of Delayed Recovery
,”
Phys. Rev. E
,
78
(
5
), p.
051304
.10.1103/PhysRevE.78.051304
101.
Choi
,
J.
,
Ryu
,
H. S.
,
Kim
,
C. W.
, and
Choi
,
J. H.
,
2010
, “
An Efficient and Robust Contact Algorithm for a Compliant Contact Force Model Between Bodies of Complex Geometry
,”
Multibody Syst. Dyn.
,
23
(
1
), pp.
99
120
.10.1007/s11044-009-9173-3
102.
Mahmoud
,
S.
, and
Jankowski
,
R.
,
2011
, “
Modified Linear Viscoelastic Model of Earthquake-Induced Structural Pounding
,”
Iran. J. Sci. Technol., Trans. Civ. Environ. Eng.
,
35
(
C1
), pp.
51
62
.10.22099/ijstc.2012.656
103.
Müller
,
P.
, and
Pöschel
,
T.
,
2011
, “
Collision of Viscoelastic Spheres: Compact Expressions for the Coefficient of Normal Restitution
,”
Phys. Rev. E
,
84
(
2
), p.
021302
.10.1103/PhysRevE.84.021302
104.
Roy
,
A.
, and
Carretero
,
J. A.
,
2012
, “
A Damping Term Based on Material Properties for the Volume-Based Contact Dynamics Model
,”
Int. J. Non-Linear Mech.
,
47
(
3
), pp.
103
112
.10.1016/j.ijnonlinmec.2012.01.006
105.
Zheng
,
Q. J.
,
Zhu
,
H. P.
, and
Yu
,
A. B.
,
2012
, “
Finite Element Analysis of the Contact Forces Between a Viscoelastic Sphere and Rigid Plane
,”
Powder Technol.
,
226
, pp.
130
142
.10.1016/j.powtec.2012.04.032
106.
Alizadeh
,
E.
,
Bertrand
,
F.
, and
Chaouki
,
J.
,
2013
, “
Development of a Granular Normal Contact Force Model Based on a Non-Newtonian Liquid Filled Dashpot
,”
Powder Technol.
,
237
, pp.
202
212
.10.1016/j.powtec.2013.01.027
107.
Azad
,
M.
, and
Featherstone
,
R.
,
2014
, “
A New Nonlinear Model of Contact Normal Force
,”
IEEE Trans. Rob.
,
30
(
3
), pp.
736
739
.10.1109/TRO.2013.2293833
108.
Brilliantov
,
N. V.
,
Pimenova
,
A. V.
, and
Goldobin
,
D. S.
,
2015
, “
A Dissipative Force Between Colliding Viscoelastic Bodies: Rigorous Approach
,”
Europhys. Lett.
,
109
(
1
), p.
14005
.10.1209/0295-5075/109/14005
109.
Goldobin
,
D. S.
,
Susloparov
,
E. A.
,
Pimenova
,
A. V.
, and
Brilliantov
,
N. V.
,
2015
, “
Collision of Viscoelastic Bodies: Rigorous Derivation of Dissipative Force
,”
Eur. Phys. J. E
,
38
(
6
), p.
55
.10.1140/epje/i2015-15055-x
110.
Wang
,
W.
,
Hua
,
X.
,
Wang
,
X.
,
Chen
,
Z.
, and
Song
,
G.
,
2017
, “
Advanced Impact Force Model for Low-Speed Pounding Between Viscoelastic Materials and Steel
,”
J. Eng. Mech.
,
143
(
12
), p.
04017139
.10.1061/(ASCE)EM.1943-7889.0001372
111.
Kaviani Rad
,
H.
, and
Nejat Pishkenari
,
H.
,
2018
, “
Frictional Viscoelastic Based Model for Spherical Particles Collision
,”
Granular Matter
,
20
(
4
), p.
62
.10.1007/s10035-018-0835-9
112.
Wang
,
X.
,
Zhang
,
Y.
,
Ji
,
X.
,
Ma
,
S.
, and
Tong
,
R.
,
2019
, “
A Contact-Impact Force Model Based on Variable Recovery Coefficient
,”
J. Vib. Shock
,
38
(
5
), pp.
198
202
.https://link.oversea.cnki.net/doi/10.13465/j.cnki.jvs.2019.05.028
113.
Poursina
,
M.
, and
Nikravesh
,
P. E.
,
2020
, “
Optimal Damping Coefficient for a Class of Continuous Contact Models
,”
Multibody Syst. Dyn.
,
50
(
2
), pp.
169
188
.10.1007/s11044-020-09745-x
114.
Poursina
,
M.
, and
Nikravesh
,
P. E.
,
2020
, “
Characterization of the Optimal Damping Coefficient in the Continuous Contact Model
,”
ASME J. Comput. Nonlinear Dyn.
,
15
(
9
), p.
091005
.10.1115/1.4047136
115.
Wang
,
G.
, and
Liu
,
C.
,
2020
, “
Further Investigation on Improved Viscoelastic Contact Force Model Extended Based on Hertz's Law in Multibody System
,”
Mech. Mach. Theory
,
153
, p.
103986
.10.1016/j.mechmachtheory.2020.103986
116.
Chatterjee
,
A.
,
James
,
G.
, and
Brogliato
,
B.
,
2022
, “
Approximate Coefficient of Restitution for Nonlinear Viscoelastic Contact With External Load
,”
Granular Matter
,
24
(
4
), p.
124
.10.1007/s10035-022-01284-w
117.
Jia
,
Y.
, and
Chen
,
X.
,
2022
, “
Application of a New Conformal Contact Force Model to Nonlinear Dynamic Behavior Analysis of Parallel Robot With Spherical Clearance Joints
,”
Nonlinear Dyn.
,
108
(
3
), pp.
2161
2191
.10.1007/s11071-022-07344-3
118.
Zhang
,
J.
,
Liang
,
X.
,
Zhang
,
Z.
,
Feng
,
G.
,
Zhao
,
Q.
,
Zhao
,
L.
, and
He
,
G.
,
2022
, “
A Continuous Contact Force Model for Impact Analysis
,”
Mech. Syst. Signal Process.
,
168
, p.
108739
.10.1016/j.ymssp.2021.108739
119.
Zhang
,
J.
,
Fang
,
M.
,
Zhao
,
L.
,
Zhao
,
Q.
,
Liang
,
X.
, and
He
,
G.
,
2022
, “
A Continuous Contact Force Model for the Impact Analysis of Hard and Soft Materials
,”
Mech. Mach. Theory
,
177
, p.
105065
.10.1016/j.mechmachtheory.2022.105065
120.
Nikravesh
,
P. E.
, and
Poursina
,
M.
,
2023
, “
Determination of Effective Mass for Continuous Contact Models in Multibody Dynamics
,”
Multibody Syst. Dyn.
,
58
(
3–4
), pp.
253
273
.10.1007/s11044-022-09859-4
121.
Wang
,
S.
, and
Gao
,
P.
,
2023
, “
Development of a Contact Force Model Suited for Spherical Contact Event
,”
Actuators
,
12
(
2
), p.
89
.10.3390/act12020089
122.
Huang
,
H.-L.
,
Yao
,
G.-F.
,
Wang
,
M.
,
Hou
,
J.-H.
, and
Zhu
,
Y.-C.
,
2024
, “
A General Continuous Contact Force Model for Contact Collisions of Soft and Hard Materials
,”
Multibody Syst. Dyn.
.10.1007/s11044-024-10039-9
123.
Wang
,
G.
,
Jia
,
W.
,
Cheng
,
F.
, and
Flores
,
P.
,
2024
, “
An Enhanced Contact Force Model With Accurate Evaluation of the Energy Dissipation During Contact-Impact Events in Dynamical Systems
,”
Appl. Math. Modell.
,
135
, pp.
51
72
.10.1016/j.apm.2024.06.034
124.
Zhang
,
Y.
,
Ding
,
Y.
, and
Xu
,
G.
,
2024
, “
A Continuous Contact-Force Model for the Impact Analysis of Viscoelastic Materials With Elastic Aftereffect
,”
Multibody Syst. Dyn.
,
61
(
3
), pp.
435
451
.10.1007/s11044-023-09954-0
125.
Poursina
,
M.
, and
Nikravesh
,
P. E.
,
2025
, “
A New Model With Uniform Damping Force for Frictionless Impacts With Non-Permanent Deformation at the Time of Separation
,”
Multibody Syst. Dyn.
,
63
(
1–2
), pp.
235
253
.10.1007/s11044-024-10003-7
126.
Wang
,
Y.
,
Luo
,
Z.
,
Wang
,
H.
,
Zhou
,
X.
,
Wang
,
H.
, and
Chen
,
S.
,
2025
, “
Characteristics of Contact Force for Normal Collision of Rubber Sphere With Rigid Plate
,”
AIP Adv.
,
15
(
2
), p.
025104
.10.1063/5.0246600
127.
Khusid
,
B. M.
,
1986
, “
Collision of Polymer Particle With Rigid Barrier
,”
J. Eng. Phys.
,
51
(
6
), pp.
1387
1393
.10.1007/BF00870348
128.
Atanackovic
,
T. M.
, and
Spasic
,
D. T.
,
2004
, “
On Viscoelastic Compliant Contact-Impact Models
,”
ASME J. Appl. Mech.
,
71
(
1
), pp.
134
138
.10.1115/1.1629106
129.
Zbiciak
,
A.
, and
Kozyra
,
Z.
,
2015
, “
Dynamic Analysis of a Soft-Contact Problem Using Viscoelastic and Fractional-Elastic Rheological Models
,”
Arch. Civ. Mech. Eng.
,
15
(
1
), pp.
286
291
.10.1016/j.acme.2014.03.002
130.
Argatov
,
I. I.
,
Selyutina
,
N. S.
, and
Mishuris
,
G. S.
,
2016
, “
Impact Problem for the Quasi-Linear Viscoelastic Standard Solid Model
,”
J. Strain Anal. Eng. Des.
,
51
(
4
), pp.
294
303
.10.1177/0309324715610027
131.
Jian
,
B.
,
Hu
,
G. M.
,
Fang
,
Z. Q.
,
Zhou
,
H. J.
, and
Xia
,
R.
,
2019
, “
A Normal Contact Force Approach for Viscoelastic Spheres of the Same Material
,”
Powder Technol.
,
350
, pp.
51
61
.10.1016/j.powtec.2019.03.034
132.
Askari
,
E.
,
2021
, “
Mathematical Models for Characterizing Non-Hertzian Contacts
,”
Appl. Math. Modell.
,
90
, pp.
432
447
.10.1016/j.apm.2020.08.048
133.
Argatov
,
I.
,
2024
, “
Viscoelastic Hertzian Impact
,”
Lubricants
,
12
(
6
), p.
193
.10.3390/lubricants12060193
134.
Ding
,
S.
,
Hu
,
Y.
,
Jian
,
B.
,
Zhang
,
Y.
,
Su
,
L.
,
Xia
,
R.
, and
Hu
,
G.
,
2024
, “
Approximate Contact Force Model for Viscoelastic Materials Based on Generalized Maxwell Model
,”
Int. J. Solids Struct.
,
289
, p.
112645
.10.1016/j.ijsolstr.2024.112645
135.
Tresca
,
H. E.
,
1869
,
Mémoire sur l'écoulement des corps solides
,
Imprimerie Impériale
,
Paris, French Empire
. The reference is based on a translation that was performed using Adobe's PDF to Word conversion tool and https://stringtranslate.com.
136.
de Saint-Venant
, A. J. C. B.
,
1871
, “
Mémoire sur l'établissement des équations différentielles des mouvements intérieurs opérés dans les corps solides ductiles au delà des limites où l'élasticité pourrait les ramener à leur premier état
,”
J. Math. Pures Appl.
,
16
, pp.
308
316
.
137.
Crook
,
A. W.
,
1952
, “
A Study of Some Impacts Between Metal Bodies by a Piezo-Electric Method
,”
Proc. R. Soc. London, Ser. A: Math. Phys. Sci.
,
212
(
1110
), pp.
377
390
.10.1098/rspa.1952.0088
138.
Barnhart
,
K. E.
,
1955
, “
Transverse Impact on Elastically Supported Beams
,” Ph.D. thesis,
University of California, Berkeley
,
Berkeley, CA
.
139.
Barnhart
,
K. E.
, and
Goldsmith
,
W.
,
1957
, “
Stresses in Beams During Transverse Impact
,”
ASME J. Appl. Mech.
,
24
(
3
), pp.
440
446
.10.1115/1.4011560
140.
Walton
,
O. R.
, and
Braun
,
R. L.
,
1986
, “
Viscosity, Granular‐Temperature, and Stress Calculations for Shearing Assemblies of Inelastic, Frictional Disks
,”
J. Rheol.
,
30
(
5
), pp.
949
980
.10.1122/1.549893
141.
Iqbal
,
U.
,
Gautam Revankar
,
A.
,
Kasula
,
V.
, and
Bobji
,
M. S.
,
2025
, “
Coefficient of Restitution in a Low-Velocity Normal Impact Using Elastoplastic Contact Stiffness
,”
ASME J. Appl. Mech.
,
92
(
3
), p.
031003
.10.1115/1.4067566
142.
Biryukov
,
D. G.
, and
Kadomtsev
,
I. G.
,
2002
, “
Dynamic Elastoplastic Interaction Between an Impactor and a Spherical Shell
,”
J. Appl. Mech. Tech. Phys.
,
43
(
5
), pp.
777
781
.10.1023/A:1019860524175
143.
Andrews
,
J.
,
1930
, “
Theory of Collision of Spheres of Soft Metals
,”
London, Edinburgh, Dublin Philos. Mag. J. Sci.
,
9
(
58
), pp.
593
610
.10.1080/14786443008565033
144.
Tabor
,
D.
,
1948
, “
A Simple Theory of Static and Dynamic Hardness
,”
Proc. R. Soc. London, Ser. A: Math. Phys. Sci.
,
192
(
1029
), pp.
247
274
.10.1098/rspa.1948.0008
145.
Tabor
,
D.
,
1951
,
The Hardness of Metals
(Monographs on the Physics and Chemistry of Materials),
Oxford University Press
,
Oxford, UK
.
146.
Chang
,
W.-R.
, and
Ling
,
F. F.
,
1992
, “
Normal Impact Model of Rough Surfaces
,”
ASME J. Tribol.
,
114
(
3
), pp.
439
447
.10.1115/1.2920903
147.
Ning
,
Z.
, and
Thornton
,
C.
,
1993
, “
Elastic-Plastic Impact of Fine Particles With a Surface
,”
Proceedings of the Second International Conference on Micromechanics of Granular Media: Powders & Grains 93
, Birmingham, UK, July 12–16, pp.
33
38
.
148.
Sadd
,
M. H.
,
Tai
,
Q.
, and
Shukla
,
A.
,
1993
, “
Contact Law Effects on Wave Propagation in Particulate Materials Using Distinct Element Modeling
,”
Int. J. Non-Linear Mech.
,
28
(
2
), pp.
251
265
.10.1016/0020-7462(93)90061-O
149.
Yigit
,
A. S.
, and
Christoforou
,
A. P.
,
1994
, “
On the Impact of a Spherical Indenter and an Elastic-Plastic Transversely Isotropic Half-Space
,”
Compos. Eng.
,
4
(
11
), pp.
1143
1152
.10.1016/0961-9526(95)91288-R
150.
Yigit
,
A. S.
,
1995
, “
On the Use of an Elastic-Plastic Contact Law for the Impact of a Single Flexible Link
,”
ASME J. Dyn. Syst., Meas., Control
,
117
(
4
), pp.
527
533
.10.1115/1.2801110
151.
Thornton
,
C.
,
1997
, “
Coefficient of Restitution for Collinear Collisions of Elastic-Perfectly Plastic Spheres
,”
ASME J. Appl. Mech.
,
64
(
2
), pp.
383
386
.10.1115/1.2787319
152.
Vu-Quoc
,
L.
, and
Zhang
,
X.
,
1999
, “
An Elastoplastic Contact Force–Displacement Model in the Normal Direction: Displacement-Driven Version
,”
Proc. R. Soc. London, Ser. A: Math., Phys. Eng. Sci.
,
455
(
1991
), pp.
4013
4044
.10.1098/rspa.1999.0488
153.
Stronge
,
W.
,
2000
, “
Contact Problems for Elasto-Plastic Impact in Multi-Body Systems
,”
Impacts in Mechanical Systems
(Lecture Notes in Physics, Vol.
551
),
R.
Beig
,
J.
Ehlers
,
U.
Frisch
,
K.
Hepp
,
W.
Hillebrandt
,
D.
Imboden
, and
R. L.
Jaffe
, et al., eds.,
Springer
,
Berlin, The Federal Republic of Germany
, pp.
189
234
.
154.
Zhao
,
Y.
,
Maietta
,
D. M.
, and
Chang
,
L.
,
2000
, “
An Asperity Microcontact Model Incorporating the Transition From Elastic Deformation to Fully Plastic Flow
,”
ASME J. Tribol.
,
122
(
1
), pp.
86
93
.10.1115/1.555332
155.
Li
,
L.-Y.
,
Wu
,
C.-Y.
, and
Thornton
,
C.
,
2001
, “
A Theoretical Model for the Contact of Elastoplastic Bodies
,”
Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
,
216
(
4
), pp.
421
431
.10.1243/0954406021525214
156.
Etsion
,
I.
,
Kligerman
,
Y.
, and
Kadin
,
Y.
,
2005
, “
Unloading of an Elastic–Plastic Loaded Spherical Contact
,”
Int. J. Solids Struct.
,
42
(
13
), pp.
3716
3729
.10.1016/j.ijsolstr.2004.12.006
157.
Weir
,
G.
, and
Tallon
,
S.
,
2005
, “
The Coefficient of Restitution for Normal Incident, Low Velocity Particle Impacts
,”
Chem. Eng. Sci.
,
60
(
13
), pp.
3637
3647
.10.1016/j.ces.2005.01.040
158.
Mangwandi
,
C.
,
Cheong
,
Y. S.
,
Adams
,
M. J.
,
Hounslow
,
M. J.
, and
Salman
,
A. D.
,
2007
, “
The Coefficient of Restitution of Different Representative Types of Granules
,”
Chem. Eng. Sci.
,
62
(
1–2
), pp.
437
450
.10.1016/j.ces.2006.08.063
159.
Luding
,
S.
,
2008
, “
Cohesive, Frictional Powders: Contact Models for Tension
,”
Granular Matter
,
10
(
4
), pp.
235
246
.10.1007/s10035-008-0099-x
160.
Du
,
Y.
, and
Wang
,
S.
,
2009
, “
Energy Dissipation in Normal Elastoplastic Impact Between Two Spheres
,”
ASME J. Appl. Mech.
,
76
(
6
), p.
061010
.10.1115/1.3130801
161.
Antonyuk
,
S.
,
Heinrich
,
S.
,
Tomas
,
J.
,
Deen
,
N. G.
,
van Buijtenen
,
M. S.
, and
Kuipers
,
J. A. M.
,
2010
, “
Energy Absorption During Compression and Impact of Dry Elastic-Plastic Spherical Granules
,”
Granular Matter
,
12
(
1
), pp.
15
47
.10.1007/s10035-009-0161-3
162.
Jackson
,
R. L.
,
Green
,
I.
, and
Marghitu
,
D. B.
,
2010
, “
Predicting the Coefficient of Restitution of Impacting Elastic-Perfectly Plastic Spheres
,”
Nonlinear Dyn.
,
60
(
3
), pp.
217
229
.10.1007/s11071-009-9591-z
163.
Brake
,
M. R.
,
2012
, “
An Analytical Elastic-Perfectly Plastic Contact Model
,”
Int. J. Solids Struct.
,
49
(
22
), pp.
3129
3141
.10.1016/j.ijsolstr.2012.06.013
164.
Brake
,
M. R. W.
,
2015
, “
An Analytical Elastic Plastic Contact Model With Strain Hardening and Frictional Effects for Normal and Oblique Impacts
,”
Int. J. Solids Struct.
,
62
, pp.
104
123
.10.1016/j.ijsolstr.2015.02.018
165.
Ma
,
D.
, and
Liu
,
C.
,
2015
, “
Contact Law and Coefficient of Restitution in Elastoplastic Spheres
,”
ASME J. Appl. Mech.
,
82
(
12
), p.
121006
.10.1115/1.4031483
166.
Ghaednia
,
H.
,
Pope
,
S. A.
,
Jackson
,
R. L.
, and
Marghitu
,
D. B.
,
2016
, “
A Comprehensive Study of the Elasto-Plastic Contact of a Sphere and a Flat
,”
Tribol. Int.
,
93
(
A
), pp.
78
90
.10.1016/j.triboint.2015.09.005
167.
Mukhopadhyay
,
S.
,
Das
,
P. K.
, and
Abani
,
N.
,
2023
, “
A Theoretical Model to Predict Normal Contact Characteristics for Elasto-Plastic Collisions
,”
Granular Matter
,
25
(
2
), p.
20
.10.1007/s10035-023-01307-0
168.
Storakers
,
B.
, and
Larsson
,
J.
,
2000
, “
On Elastic Impact and Dynamic Hardness
,”
Arch. Mech.
,
52
(
4–5
), pp.
779
798
.
169.
Tomas
,
J.
,
2000
, “
Particle Adhesion Fundamentals and Bulk Powder Consolidation
,”
KONA Powder Part. J.
,
18
(
0
), pp.
157
169
.10.14356/kona.2000022
170.
Ismail
,
K. A.
, and
Stronge
,
W. J.
,
2008
, “
Impact of Viscoplastic Bodies: Dissipation and Restitution
,”
ASME J. Appl. Mech.
,
75
(
6
), p.
061011
.10.1115/1.2965371
171.
Yigit
,
A. S.
,
Christoforou
,
A. P.
, and
Majeed
,
M. A.
,
2011
, “
A Nonlinear Visco-Elastoplastic Impact Model and the Coefficient of Restitution
,”
Nonlinear Dyn.
,
66
(
4
), pp.
509
521
.10.1007/s11071-010-9929-6
172.
Burgoyne
,
H. A.
, and
Daraio
,
C.
,
2014
, “
Strain-Rate-Dependent Model for the Dynamic Compression of Elastoplastic Spheres
,”
Phys. Rev. E
,
89
(
3
), p.
032203
.10.1103/PhysRevE.89.032203
173.
Christoforou
,
A. P.
, and
Yigit
,
A. S.
,
2016
, “
Inelastic Impact and the Coefficient of Restitution
,”
J. Eng. Res.
,
4
(
4
), pp. 194–213.https://kuwaitjournals.org/jer/index.php/JER/article/view/1243
174.
Ahmad
,
M.
,
Ismail
,
K. A.
,
Mat
,
F.
, and
Stronge
,
W. J.
,
2016
, “
Improved Model for Impact of Viscoplastic Bodies
,”
Key Eng. Mater.
,
715
, pp.
180
185
.10.4028/www.scientific.net/KEM.715.180
175.
Borovin
,
G. K.
, and
Lapshin
,
V. V.
,
2019
, “
Nonlinear Visco-Elastic-Plastic Model of Impact
,”
J. Phys.: Conf. Ser.
,
1301
(
1
), p.
012004
.10.1088/1742-6596/1301/1/012004
176.
Wang
,
G.
,
Faes
,
M. G. R.
,
Cheng
,
F.
,
Shi
,
T.
, and
Gao
,
P.
,
2022
, “
Extension of Dashpot Model With Elastoplastic Deformation and Rough Surface in Impact Behavior
,”
Chaos, Solitons Fractals
,
162
, p.
112402
.10.1016/j.chaos.2022.112402
177.
Wang
,
G.
,
Ma
,
D.
,
Liu
,
C.
, and
Liu
,
Y.
,
2023
, “
Development of a Compliant Dashpot Model With Nonlinear and Linear Behaviors for the Contact of Multibody Systems
,”
Mech. Syst. Signal Process.
,
185
, p.
109785
.10.1016/j.ymssp.2022.109785
178.
Lin
,
J.
,
Wu
,
B.
,
Zhang
,
J.
,
Wei
,
Z.
,
Zhang
,
X.
, and
Dai
,
H.
,
2024
, “
Blade-Coating Rub-Impact Force Analysis Using a Dissipative Contact Force Model With Plastic Deformation
,”
J. Mech. Sci. Technol.
,
38
(
12
), pp.
6731
6745
.10.1007/s12206-024-1127-4
179.
Kikuuwe
,
R.
, and
Fujimoto
,
H.
,
2007
, “
Incorporating Geometric Algorithms in Impedance- and Admittance-Type Haptic Rendering
,”
Second Joint EuroHaptics Conference and Symposium on Haptic Interfaces for Virtual Environment and Teleoperator Systems
(
WHC'07
), Tsukuba, Japan, Mar. 22–24, pp.
249
254
.10.1109/WHC.2007.75
180.
Xiong
,
X.
,
Kikuuwe
,
R.
, and
Yamamoto
,
M.
,
2013
, “
A Differential Algebraic Method to Approximate Nonsmooth Mechanical Systems by Ordinary Differential Equations
,”
J. Appl. Math.
,
2013
, pp.
1
13
.10.1155/2013/320276
181.
Xiong
,
X.
,
Kikuuwe
,
R.
, and
Yamamoto
,
M.
,
2014
, “
A Contact Force Model With Nonlinear Compliance and Residual Indentation
,”
ASME J. Appl. Mech.
,
81
(
2
), p.
021003
.10.1115/1.4024403
182.
Boltzmann
,
L.
,
1874
, “
Theorie der elastischen Nachwirkung
,”
Sitzungsber. Kais. Akad. Wiss.. Math.-Naturwiss. Cl. Abt. 2, Math., Phys., Chem., Mech., Meteorol. Astron.
,
70
, pp.
275
306
.
183.
Schwedoff
,
T.
,
1889
, “
Recherches Expérimentales Sur la Cohésion Des Liquides
,”
J. Phys. Théor. Appl.
,
8
(
1
), pp.
341
359
.10.1051/jphystap:018890080034100
184.
Bingham
,
E. C.
,
1922
,
Fluidity and Plasticity
,
McGraw-Hill Book Company
,
New York
.
185.
Masing
,
G.
,
1926
, “
Eigenspannungen und Verfestigung Beim Messing
,”
Proceedings of the Second International Congress of Applied Mechanics
, Zürich, Switzerland, pp.
332
335
.
186.
Schofield
,
R. K.
, and
Blair
,
G. W. S.
,
1932
, “
The Relationship Between Viscosity, Elasticity and Plastic Strength of Soft Materials as Illustrated by Some Mechanical Properties of Flour Doughs, I
,”
Proc. R. Soc. London, Ser. A, Containing Pap. Math. Phys. Charact.
,
138
(
836
), pp.
707
718
.10.1098/rspa.1932.0211
187.
Ramberg
,
W.
, and
Osgood
,
W. R.
,
1943
, “
Description of Stress-Strain Curves by Three Parameters
,” National Advising Committee for Aeronautics, Washington, DC, Report No.
NACA-TN-902
.https://ntrs.nasa.gov/api/citations/19930081614/downloads/19930081614.pdf
188.
Pisarenko
,
G. S.
,
1962
, “
Vibrations of Elastic Systems Taking Account of Energy Dissipation in the Material
,” Directorate of Materials and Processes, Aeronautical Systems Division, Air Force Systems Command, Report No. WADD
60
582
.
189.
Jennings
,
P. C.
,
1963
, “
Response of Simple Yielding Structures to Earthquake Excitation
,” Ph.D. thesis,
California Institute of Technology
,
Pasadena, CA
.
190.
Rosenblueth
,
E.
, and
Herrera
,
I.
,
1964
, “
On a Kind of Hysteretic Damping
,”
J. Eng. Mech. Div.
,
90
(
4
), pp.
37
48
.10.1061/JMCEA3.0000510
191.
Iwan
,
W. D.
,
1966
, “
A Distributed-Element Model for Hysteresis and Its Steady-State Dynamic Response
,”
ASME J. Appl. Mech.
,
33
(
4
), pp.
893
900
.10.1115/1.3625199
192.
Bouc
,
R.
,
1971
, “
Modèle mathématique d'hystérésis
,”
Acustica
,
24
(
1
), pp.
16
25
.
193.
Özdemir
,
H.
,
1976
, “
Nonlinear Transient Dynamic Analysis of Yielding Structures
,” Ph.D. thesis,
University of California, Berkeley
,
Berkeley, CA
.
194.
Chen
,
W. F.
, and
Ting
,
E. C.
,
1980
, “
Constitutive Models for Concrete Structures
,”
J. Eng. Mech. Div.
,
106
(
1
), pp.
1
19
.10.1061/JMCEA3.0002559
195.
Jayakumar
,
P.
,
1987
, “
Modeling and Identification in Structural Dynamics
,” Ph.D. thesis,
California Institute of Technology
,
Pasadena, CA
.
196.
Monteiro Marques
,
M. D. P.
,
1994
, “
An Existence, Uniqueness and Regularity Study of the Dynamics of Systems With One-Dimensional Friction
,”
Eur. J. Mech. - A/Solids
,
13
(
2
), pp.
277
306
.
197.
Bastien
,
J.
,
Schatzman
,
M.
, and
Lamarque
,
C.-H.
,
2000
, “
Study of Some Rheological Models With a Finite Number of Degrees of Freedom
,”
Eur. J. Mech. - A/Solids
,
19
(
2
), pp.
277
307
.10.1016/S0997-7538(00)00163-7
198.
Charalampakis
,
A. E.
, and
Koumousis
,
V. K.
,
2009
, “
A Bouc–Wen Model Compatible With Plasticity Postulates
,”
J. Sound Vib.
,
322
(
4–5
), pp.
954
968
.10.1016/j.jsv.2008.11.017
199.
Biswas
,
S.
, and
Chatterjee
,
A.
,
2014
, “
A Reduced-Order Model From High-Dimensional Frictional Hysteresis
,”
Proc. R. Soc. A: Math., Phys. Eng. Sci.
,
470
(
2166
), p.
20130817
.10.1098/rspa.2013.0817
200.
Biswas
,
S.
, and
Chatterjee
,
A.
,
2015
, “
A Two-State Hysteresis Model From High-Dimensional Friction
,”
R. Soc. Open Sci.
,
2
(
7
), p.
150188
.10.1098/rsos.150188
201.
Maleki
,
M.
,
Ahmadian
,
H.
, and
Rajabi
,
M.
,
2023
, “
A Modified Bouc-Wen Model to Simulate Asymmetric Hysteresis Loop and Stochastic Model Updating in Frictional Contacts
,”
Int. J. Solids Struct.
,
269
, p.
112212
.10.1016/j.ijsolstr.2023.112212
202.
Maleki
,
M.
,
Ahmadian
,
H.
, and
Rajabi
,
M.
,
2024
, “
A Modified Bouc–Wen Model for Simulating Vibro-Impact Hysteresis Phenomenon and Stability Analysis in Frictional Contacts
,”
J. Vib. Eng. Technol.
,
12
(
1
), pp.
1105
1122
.10.1007/s42417-023-00896-7
203.
Bouc
,
R.
,
1967
, “
Forced Vibration of Mechanical Systems With Hysteresis
,”
Proceedings of the Fourth Conference on Nonlinear Oscillations
, Prague, Czechoslovakia, Sept. 5–9, p.
315
.
204.
Wen
,
Y.-K.
,
1976
, “
Method for Random Vibration of Hysteretic Systems
,”
J. Eng. Mech. Div.
,
102
(
2
), pp.
249
263
.10.1061/JMCEA3.0002106
205.
Sivaselvan
,
M. V.
, and
Reinhorn
,
A. M.
,
2000
, “
Hysteretic Models for Deteriorating Inelastic Structures
,”
J. Eng. Mech.
,
126
(
6
), pp.
633
640
.10.1061/(ASCE)0733-9399(2000)126:6(633)
206.
Ikhouane
,
F.
, and
Rodellar
,
J.
,
2007
,
Systems With Hysteresis: Analysis, Identification and Control Using the Bouc—Wen Model
,
Wiley
,
Chichester, UK
.
207.
Newton
,
I.
,
1729
,
The Mathematical Principles of Natural Philosophy
,
Printed for Benjamin Motte, at the Middle-Temple-Gate, in Fleet Street
,
London
(Translated by Andrew Motte).
208.
Poisson
,
S. D.
,
1842
,
A Treatise of Mechanics
, Vol.
2
,
Longman and Co
.,
London
(Translated by Henry H. Harte).
209.
Stronge
,
W. J.
,
1990
, “
Rigid Body Collisions With Friction
,”
Proc. R. Soc. London, Ser. A: Math. Phys. Sci.
,
431
(
1881
), pp.
169
181
.10.1098/rspa.1990.0125
210.
Ikhouane
,
F.
,
Mañosa
,
V.
, and
Rodellar
,
J.
,
2004
, “
Bounded and Dissipative Solutions of the Bouc-Wen Model for Hysteretic Structural Systems
,”
Proceedings of the 2004 American Control Conference,
Boston, MA, June 30–July 2, pp.
3520
3525
.10.23919/ACC.2004.1384457
211.
Ikhouane
,
F.
, and
Rodellar
,
J.
,
2005
, “
On the Hysteretic Bouc–Wen Model: Part II: Robust Parametric Identification
,”
Nonlinear Dyn.
,
42
(
1
), pp.
79
95
.10.1007/s11071-005-0070-x
212.
Ikhouane
,
F.
, and
Rodellar
,
J.
,
2005
, “
On the Hysteretic Bouc–Wen Model: Part I: Forced Limit Cycle Characterization
,”
Nonlinear Dyn.
,
42
(
1
), pp.
63
78
.10.1007/s11071-005-0069-3
213.
Ikhouane
,
F.
,
Rodellar
,
J.
, and
Hurtado
,
J. E.
,
2006
, “
Analytical Characterization of Hysteresis Loops Described by the Bouc-Wen Model
,”
Mech. Adv. Mater. Struct.
,
13
(
6
), pp.
463
472
.10.1080/15376490600862830
214.
Ikhouane
,
F.
,
Mañosa
,
V.
, and
Rodellar
,
J.
,
2007
, “
Dynamic Properties of the Hysteretic Bouc-Wen Model
,”
Syst. Control Lett.
,
56
(
3
), pp.
197
205
.10.1016/j.sysconle.2006.09.001
215.
Ikhouane
,
F.
,
Hurtado
,
J. E.
, and
Rodellar
,
J.
,
2007
, “
Variation of the Hysteresis Loop With the Bouc–Wen Model Parameters
,”
Nonlinear Dyn.
,
48
(
4
), pp.
361
380
.10.1007/s11071-006-9091-3
216.
Ma
,
F.
,
Zhang
,
H.
,
Bockstedte
,
A.
,
Foliente
,
G. C.
, and
Paevere
,
P.
,
2004
, “
Parameter Analysis of the Differential Model of Hysteresis
,”
ASME J. Appl. Mech.
,
71
(
3
), pp.
342
349
.10.1115/1.1668082
217.
Logan
,
J. D.
,
2013
,
Applied Mathematics
, 4th ed.,
Wiley
,
Hoboken, NJ
.
218.
Kharaz
,
A.
, and
Gorham
,
D.
,
2000
, “
A Study of the Restitution Coefficient in Elastic-Plastic Impact
,”
Philos. Mag. Lett.
,
80
(
8
), pp.
549
559
.10.1080/09500830050110486
219.
Rohatgi
,
A.
,
2024
, “
WebPlotDigitizer
,” accessed Nov. 10, 2024, https://automeris.io
220.
Harris
,
C. R.
,
Millman
,
K. J.
,
van der Walt
,
S. J.
,
Gommers
,
R.
,
Virtanen
,
P.
,
Cournapeau
,
D.
,
Wieser
,
E.
, et al.,
2020
, “
Array Programming With NumPy
,”
Nature
,
585
(
7825
), pp.
357
362
.10.1038/s41586-020-2649-2
221.
Virtanen
,
P.
,
Gommers
,
R.
,
Oliphant
,
T. E.
,
Haberland
,
M.
,
Reddy
,
T.
,
Cournapeau
,
D.
,
Burovski
,
E.
, et al.,
2020
, “
SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python
,”
Nat. Methods
,
17
(
3
), pp.
261
272
.10.1038/s41592-019-0686-2
222.
IEEE
,
2019
, “
IEEE Standard for Floating-Point Arithmetic
,” IEEE, New York, IEEE Standard No. 754TM-2019 (Revision of IEEE Standard No. 754-2008).
223.
Dormand
,
J. R.
, and
Prince
,
P. J.
,
1980
, “
A Family of Embedded Runge-Kutta Formulae
,”
J. Comput. Appl. Math.
,
6
(
1
), pp.
19
26
.10.1016/0771-050X(80)90013-3
224.
Prince
,
P. J.
, and
Dormand
,
J. R.
,
1981
, “
High Order Embedded Runge-Kutta Formulae
,”
J. Comput. Appl. Math.
,
7
(
1
), pp.
67
75
.10.1016/0771-050X(81)90010-3
225.
Hairer
,
E.
,
Nørsett
,
S. P.
, and
Wanner
,
G.
,
1993
,
Solving Ordinary Differential Equations I: Nonstiff Problems
(Springer Series in Computational Mathematics, Vol.
8
), 2nd ed.,
Springer
,
Berlin, The Federal Republic of Germany
.
226.
Nelder
,
J. A.
, and
Mead
,
R.
,
1965
, “
A Simplex Method for Function Minimization
,”
Comput. J.
,
7
(
4
), pp.
308
313
.10.1093/comjnl/7.4.308
227.
Cross
,
R.
,
2011
,
Physics of Baseball & Softball
,
Springer Science+Business Media
,
New York
.
228.
Cross
,
R.
,
2014
, “
Impact of Sports Balls With Striking Implements
,”
Sports Eng.
,
17
(
1
), pp.
3
22
.10.1007/s12283-013-0132-0
229.
Wolfram Research, Inc.
,
2023
, “
Mathematica, Version 13.3
,” Wolfram Research, Champaign, IL,accessed Nov. 10, 2024, https://www.wolfram.com/mathematica
230.
Mathworks
,
2023
, “
MATLAB R2023a
,” Mathworks, Natick, MA, accessed Nov. 10, 2024, https://www.mathworks.com/products/matlab.html
231.
Takeuti
,
G.
, and
Zaring
,
W. M.
,
1982
,
Introduction to Axiomatic Set Theory
(Graduate Texts in Mathematics, Vol.
1
), 2nd ed.,
Springer-Verlag New York
,
New York
.
232.
Kelley
,
J. L.
,
1955
,
General Topology
,
Van Nostrand Reinhold Company
,
New York
; Reprint, Dover Publications, Mineola, NY (2017).
233.
Morris
,
S. A.
,
2020
,
Topology Without Tears
,
Sidney A. Morris
.
234.
Baldwin
,
S. L.
,
2024
,
MATH 7500: Topology I (Lecture Notes)
,
Auburn University
,
Auburn, AL
.
235.
Bloch
,
E. D.
,
2010
,
The Real Numbers and Real Analysis
,
Springer Science+Business Media
,
New York
.
236.
Shurman
,
J.
,
2016
,
Calculus and Analysis in Euclidean Space
(Undergraduate Texts in Mathematics),
Springer International Publishing
,
Cham, The Swiss Confederation
.
237.
Ziemer
,
W. P.
, and
Torres
,
M.
,
2017
,
Modern Real Analysis
(Graduate Texts in Mathematics, Vol.
278
), 2nd ed.,
Springer International Publishing
,
Cham, The Swiss Confederation
.
238.
Chicone
,
C. C.
,
1999
,
Ordinary Differential Equations With Applications
(Texts in Applied Mathematics, Vol.
34
),
Springer-Verlag New York
,
New York
.
239.
Schaeffer
,
D. G.
, and
Cain
,
J. W.
,
2016
,
Ordinary Differential Equations: Basics and Beyond
(Texts in Applied Mathematics, Vol.
65
),
Springer Science+Business Media
,
New York
.
240.
LaSalle
,
J. P.
,
1960
, “
Some Extensions of Liapunov's Second Method
,”
IRE Trans. Circuit Theory
,
7
(
4
), pp.
520
527
.10.1109/TCT.1960.1086720
241.
Yoshizawa
,
T.
,
1966
,
Stability Theory by Liapunov's Second Method
(Publications of the Mathematical Society of Japan, Vol.
9
),
The Mathematical Society of Japan
,
Tokyo, Japan
.
242.
Yoshizawa
,
T.
,
1975
,
Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions
(Applied Mathematical Sciences, Vol.
14
),
Springer-Verlag New York
,
New York
.
243.
Lakshmikantham
,
V.
,
Leela
,
S.
, and
Martynyuk
,
A. A.
,
1990
,
Practical Stability of Nonlinear Systems
,
World Scientific Publishing
,
Singapore
.
244.
Isidori
,
A.
,
1995
,
Nonlinear Control Systems
, 3rd ed.,
Springer-Verlag London
,
London
.
245.
Sontag
,
E. D.
,
1998
,
Mathematical Control Theory: Deterministic Finite Dimensional Systems
(Texts in Applied Mathematics), 2nd ed.,
Springer Science+Business Media
,
New York
.
246.
Isidori
,
A.
,
1999
,
Nonlinear Control Systems II
,
Springer-Verlag London
,
London
.
247.
Sastry
,
S.
,
1999
,
Nonlinear Systems
(Interdisciplinary Applied Mathematics, Vol.
10
),
Springer Science+Business Media
,
New York
.
248.
Márquez
,
H.
,
2003
,
Nonlinear Control Systems: Analysis and Design
,
Wiley
,
Hoboken, NJ
.
249.
Haddad
,
W. M.
, and
Chellaboina
,
V.
,
2011
,
Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach
,
Princeton University Press
,
Princeton, NJ
.
250.
Khalil
,
H. K.
,
2015
,
Nonlinear Control
,
Pearson Education
,
Upper Saddle River, NJ
.
251.
Goebel
,
R. K.
,
2024
,
Set-Valued, Convex, and Nonsmooth Analysis in Dynamics and Control: An Introduction
,
SIAM
,
Philadelphia, PA
.
252.
Bhat
,
S. P.
, and
Bernstein
,
D. S.
,
2003
, “
Nontangency-Based Lyapunov Tests for Convergence and Stability in Systems Having a Continuum of Equilibria
,”
SIAM J. Control Optim.
,
42
(
5
), pp.
1745
1775
.10.1137/S0363012902407119
253.
Hahn
,
W.
,
1967
,
Stability of Motion
(Die Grundlehren de mathematischen Wissenschaften in Einzeldarstellungen, Vol.
138
),
Springer-Verlag New York
,
New York
.
254.
Kellett
,
C. M.
,
2014
, “
A Compendium of Comparison Function Results
,”
Math. Control, Signals, Syst.
,
26
(
3
), pp.
339
374
.10.1007/s00498-014-0128-8
You do not currently have access to this content.