Abstract

The spatial variation of the coefficient of restitution for frictionless impacts along the length of a circular beam is investigated using a continuous impact model. The equations of motion are obtained using the finite element method, and direct time integration is used to simulate the collision on a fast time scale. For collision of a pinned beam with a fixed cylinder, the spatial variation of the coefficient of restitution, impulse magnitude, duration of collision, energetics, and the role of damping are investigated. In the absence of significant external damping, the kinematic and kinetic definitions of the coefficient of restitution provide identical results. Experiments validate the results from simulation which indicate that the coefficient of restitution is sensitive to the location of impact.

References

1.
Brogliato
,
B.
,
2016
,
Nonsmooth Mechanics: Models, Dynamics and Control. Communications and Control Engineering
,
Springer International Publishing
,
Cham, Switzerland
.
2.
Brogliato
,
B.
, and
Rio
,
A.
,
2000
, “
On the Control of Complementary-Slackness Juggling Mechanical Systems
,”
IEEE Trans. Automat. Contr.
,
45
(
2
), pp.
235
246
.
3.
Kant
,
N.
, and
Mukherjee
,
R.
,
2021
, “
Non-Prehensile Manipulation of a Devil-Stick: Planar Symmetric Juggling Using Impulsive Forces
,”
Nonlinear Dyn.
,
103
(
3
), pp.
2409
2420
.
4.
Khandelwal
,
A.
,
Kant
,
N.
, and
Mukherjee
,
R.
,
2023
, “
Nonprehensile Manipulation of a Stick Using Impulsive Forces
,”
Nonlinear Dyn.
,
111
(
1
), pp.
113
127
.
5.
Mathis
,
F. B.
, and
Mukherjee
,
R.
,
2016
, “
Apex Height Control of a Two-Mass Robot Hopping on a Rigid Foundation
,”
Mech. Mach. Theory
,
105
, pp.
44
57
.
6.
Allafi
,
A.
, and
Mukherjee
,
R.
,
2019
Apex Height Control of a Two-DOF Ankle-Knee-Hip Robot Hopping on a Rigid Foundation
,”
2019 ASME Dynamic Systems and Control Conference
,
Park City, UT
,
Oct. 8–11
.
7.
Raman.
,
C. V.
,
1920
, “
On Some Applications of Hertz’s Theory of Impact
,”
Phys. Rev.
,
15
(
4
), pp.
277
284
.
8.
Zener
,
C.
, and
Feshbach
,
H.
,
2021
, “
A Method of Calculating Energy Losses During Impact
,”
ASME J. Appl. Mech.
,
6
(
2
), pp.
A67
A70
.
9.
Bhattacharjee
,
A.
,
2019
, “
New Approximations in Vibroimpact Problems
,” Ph.D. thesis,
Department of Mechanical Engineering, Indian Institute of Technology
,
Kanpur, India
.
10.
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
.
11.
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
.
12.
Thornton
,
C.
,
1997
, “
Coefficient of Restitution for Collinear Collisions of Elastic-Perfectly Plastic Spheres
,”
ASME J. Appl. Mech.
,
64
(
2
), pp.
383
386
.
13.
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
.
14.
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
.
15.
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
.
16.
Pfeiffer
,
F.
, and
Glocker
,
C.
,
1996
,
Multibody Dynamics With Unilateral Contacts
, 1st ed.,
John Wiley & Sons, Ltd
,
Germany
.
17.
Brach
,
R. M.
,
1989
, “
Rigid Body Collisions
,”
ASME J. Appl. Mech.
,
56
(
1
), pp.
133
138
.
18.
Popov
,
V. L.
,
2017
,
Contact Mechanics and Friction
,
Springer Berlin Heidelberg
,
Berlin, Heidelberg
.
19.
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
.
20.
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
.
21.
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
.
22.
Khulief
,
Y. A.
, and
Shabana
,
A. A.
,
1986
, “
Impact Responses of Multi-body Systems With Consistent and Lumped Masses
,”
J. Sound Vib.
,
104
(
2
), pp.
187
207
.
23.
Wagg
,
D. J.
, and
Bishop
,
S. R.
,
2002
, “
Application of Non-Smooth Modeling Techniques to the Dynamics of a Flexible Impacting Beam
,”
J. Sound Vib.
,
256
(
5
), pp.
803
820
.
24.
Wagg
,
D. J.
,
2004
, “
A Note on Using the Collocation Method for Modelling the Dynamics of a Flexible Continuous Beam Subject to Impacts
,”
J. Sound Vib.
,
276
(
3
), pp.
1128
1134
.
25.
Wagg
,
D. J.
,
2007
, “
A Note on Coefficient of Restitution Models Including the Effects of Impact Induced Vibration
,”
J. Sound Vib.
,
300
(
3
), pp.
1071
1078
.
26.
Vyasarayani
,
C. P.
,
Sandhu
,
S. S.
, and
McPhee
,
J.
,
2012
, “
Nonsmooth Modeling of Vibro-Impacting Euler-Bernoulli Beam
,”
Adv. Acoust. Vibr.
,
2012
, p.
e268595
.
27.
Vyasarayani
,
C. P.
,
McPhee
,
J.
, and
Birkett
,
S.
,
2009
, “
Modeling Impacts Between a Continuous System and a Rigid Obstacle Using Coefficient of Restitution
,”
ASME J. Appl. Mech.
,
77
(
2
), p.
021008
.
28.
Yigit
,
A. S.
,
Ulsoy
,
A. G.
, and
Scott
,
R. A.
,
1990
, “
Dynamics of a Radially Rotating Beam With Impact, Part 1: Theoretical and Computational Model
,”
ASME J. Vib. Acoust.
,
112
(
1
), pp.
65
70
.
29.
Yigit
,
A. S.
,
Ulsoy
,
A. G.
, and
Scott
,
R. A.
,
1990
, “
Dynamics of a Radially Rotating Beam With Impact, Part 2: Experimental and Simulation Results
,”
ASME J. Vib. Acoust.
,
112
(
1
), pp.
71
77
.
30.
Palas
,
H.
,
Hsu
,
W. C.
, and
Shabana
,
A. A.
,
1992
, “
On the Use of Momentum Balance and the Assumed Modes Method in Transverse Impact Problems
,”
ASME J. Vib. Acoust.
,
114
(
3
), pp.
364
373
.
31.
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
.
32.
Schiehlen
,
W.
, and
Seifried
,
R.
,
2004
, “
Three Approaches for Elastodynamic Contact in Multibody Systems
,”
Multibody Sys.Dyn.
,
12
(
1
), pp.
1
16
.
33.
Schiehlen
,
W.
,
Hu
,
B.
, and
Seifried
,
R.
,
2005
, “Multiscale Methods for Multibody Systems With Impacts,”
Advances in Computational Multibody Systems
, Computational Methods in Applied Sciences,
Ambrósio
,
J. A.
, ed.,
Springer Netherlands
,
Dordrecht
, pp.
95
124
.
34.
Schiehlen
,
W.
,
Seifried
,
R.
, and
Eberhard
,
P.
,
2006
, “
Elastoplastic Phenomena in Multibody Impact Dynamics
,”
Comput. Methods Appl. Mech. Eng.
,
195
(
50–51
), pp.
6874
6890
.
35.
Seifried
,
R.
,
Schiehlen
,
W.
, and
Eberhard
,
P.
,
2005
, “
Numerical and Experimental Evaluation of the Coefficient of Restitution for Repeated Impacts
,”
Int. J. Impact Eng.
,
32
(
1
), pp.
508
524
.
36.
Seifried
,
R.
,
Minamoto
,
H.
, and
Eberhard
,
P.
,
2010
, “
Viscoplastic Effects Occurring in Impacts of Aluminum and Steel Bodies and Their Influence on the Coefficient of Restitution
,”
ASME J. Appl. Mech.
,
77
(
4
), p.
041008
.
37.
Minamoto
,
H.
,
Seifried
,
R.
,
Eberhard
,
P.
, and
Kawamura
,
S.
,
2011
, “
Analysis of Repeated Impacts on a Steel Rod With Visco-Plastic Material Behavior
,”
Eur. J. Mech. A/Solids
,
30
(
3
), pp.
336
344
.
38.
Bhattacharjee
,
A.
, and
Chatterjee
,
A.
,
2017
, “
Interplay Between Dissipation and Modal Truncation in Ball-Beam Impact
,”
ASME J. Comput. Nonlinear Dyn.
,
12
(
6
), p.
061018
.
39.
Bhattacharjee
,
A.
, and
Chatterjee
,
A.
,
2020
, “
Restitution Modeling in Vibration-Dominated Impacts Using Energy Minimization Under Outward Constraints
,”
Int. J. Mech. Sci.
,
166
, p.
105215
.
40.
Svedholm
,
C.
,
Zangeneh
,
A.
,
Pacoste
,
C.
,
François
,
S.
, and
Karoumi
,
R.
,
2016
, “
Vibration of Damped Uniform Beams With General End Conditions Under Moving Loads
,”
Eng. Struct.
,
126
, pp.
40
52
.
41.
Bathe
,
K.-J.
,
2014
,
Finite Element Procedures
, 2nd ed.,
K. J. Bathe
,
Watertown, MA
.
42.
Mohsen
,
M. F. N.
,
1982
, “
Some Details of the Galerkin Finite Element Method
,”
Appl. Math. Model.
,
6
(
3
), pp.
165
170
.
43.
Stronge
,
W. J.
,
2018
,
Impact Mechanics
, 2nd ed.,
Cambridge University Press
,
Cambridge, MA
.
You do not currently have access to this content.