We study a ball-beam impact in detail; and in particular, we study the interplay between dissipation and modal truncation. With Hertzian contact between a solid ball and an Euler–Bernoulli beam model, we find using detailed numerical simulations that many (well above 60) modes are needed before convergence occurs; that contact dissipation (either viscous or hysteretic) has only a slight effect; and that contact location plays a significant role. However, and more interestingly, we find that as little as 2% modal damping speeds up convergence of the net interaction so that only about 25 modes are needed. We offer a qualitative explanation for this effect in terms of the many subimpacts that occur in the overall single macroscopic impact. In particular, we find that in cases where the overall interaction time is long enough to damp out high modes yet short enough to leave lower modes undissipated, modal truncation at about 25 modes gives good results. In contrast, if modal damping is absent so that higher mode vibrations persist throughout the interaction, final outcomes are less regular and many more modes are needed. The regime of impact interactions studied here occurs for reasonable parameter ranges, e.g., for a 3–4 cm steel ball dropped at speeds of 0.1–1.0 m/s on a meter-long steel beam of net mass 1 kg. We are unaware of any prior similarly detailed numerical study which clearly offers the one summarizing idea that we obtain here.

References

References
1.
Ahn
,
J.
, and
Stewart
,
D. E.
,
2006
, “
Existence of Solutions for a Class of Impact Problems Without Viscosity
,”
SIAM J. Math. Anal.
,
38
(
1
), pp.
37
63
.
2.
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
.
3.
Yigit
,
A. S.
,
Ulsoy
,
A. G.
, and
Scott
,
R. A.
,
1990
, “
Dynamics of a Radially Rotating Beam With Impact—Part 1: Experimental and Simulation Results
,”
ASME J. Vib. Acoust.
,
112
(
1
), pp.
65
70
.
4.
Yigit
,
A. S.
, and
Christoforou
,
A. P.
,
1998
, “
The Efficacy of the Momentum Balance Method in Transverse Impact Problems
,”
ASME J. Vib. Acoust.
,
120
(
1
), pp.
47
53
.
5.
Christoforou
,
A. P.
, and
Yigit
,
A. S.
,
1998
, “
Effect of Flexibility on Low Velocity Impact Response
,”
J. Sound Vib.
,
217
(
3
), pp.
563
578
.
6.
Goldsmith
,
W.
,
1960
,
Impact: The Theory and Physical Behavior of Colliding Solids
,
Edward Arnold
,
London
.
7.
Stronge
,
W. J.
,
1990
, “
Rigid Body Collisions With Friction
,”
Proc. R. Soc. London, Ser. A
,
431
(
1881
), pp.
169
181
.
8.
Chatterjee
,
A.
, and
Ruina
,
A.
,
1998
, “
Two Interpretations of Rigidity in Rigid-Body Collisions
,”
ASME J. Appl. Mech.
,
65
(
4
), pp.
894
900
.
9.
Rakshit
,
S.
, and
Chatterjee
,
A.
,
2015
, “
Scalar Generalization of Newtonian Restitution for Simultaneous Impact
,”
Int. J. Mech. Sci.
,
103
, pp.
141
157
.
10.
Hurmuzlu
,
Y.
,
1998
, “
An Energy-Based Coefficient of Restitution for Planar Impacts of Slender Bars With Massive External Surfaces
,”
ASME J. Appl. Mech.
,
65
(
4
), pp.
952
962
.
11.
Gilardi
,
G.
, and
Sharf
,
I.
,
2002
, “
Literature Survey of Contact Dynamics Modeling
,”
Mech. Mach. Theory
,
37
(
10
), pp.
1213
1239
.
12.
Khulief
,
Y. A.
,
2013
, “
Modeling of Impact in Multibody Systems: An Overview
,”
ASME J. Comput. Nonlinear Dyn.
,
8
(
2
), p.
021012
.
13.
Lee
,
Y.
,
Hamilton
,
J. F.
, and
Sullivan
,
J. W.
,
1983
, “
The Lumped Parameter Method for Elastic Impact Problems
,”
ASME J. Appl. Mech.
,
50
(
4a
), pp.
823
827
.
14.
Goyal
,
S.
,
Pinson
,
E. N.
, and
Sinden
,
F. W.
,
1994
, “
Simulation of Dynamics of Interacting Rigid Bodies Including Friction—I: General Problem and Contact Model
,”
Eng. Comput.
,
10
(
3
), pp.
162
174
.
15.
Chatterjee
,
A.
, and
Ruina
,
A.
,
1998
, “
A New Algebraic Rigid-Body Collision Law Based on Impulse Space Considerations
,”
ASME J. Appl. Mech.
,
65
(
4
), pp.
939
951
.
16.
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
.
17.
Yigit
,
A. S.
,
Ulsoy
,
A. G.
, and
Scott
,
R. A.
,
1990
, “
Spring-Dashpot Models for the Dynamics of a Radially Rotating Beam With Impact
,”
J. Sound Vib.
,
142
(
3
), pp.
515
525
.
18.
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
.
19.
Pashah
,
S.
,
Massenzio
,
M.
, and
Jacquelin
,
E.
,
2008
, “
Prediction of Structural Response for Low Velocity Impact
,”
Int. J. Impact Eng.
,
35
(
2
), pp.
119
132
.
20.
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
.
21.
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
.
22.
Qi
,
X.
, and
Yin
,
X.
,
2016
, “
Experimental Studying Multi-Impact Phenomena Exhibited During the Collision of a Sphere Onto a Steel Beam
,”
Adv. Mech. Eng.
,
8
(
9
), pp.
1
16
.
23.
Singh
,
S. J.
, and
Chatterjee
,
A.
,
2004
, “
Nonintrusive Measurement of Contact Forces During Vibration Dominated Impacts
,”
ASME J. Dyn. Syst. Meas. Control
,
126
(
3
), pp.
489
497
.
24.
Lazan
,
B. J.
,
1968
,
Damping of Materials and Members in Structural Mechanics
,
Pergamon Press
,
Oxford, UK
.
25.
Biswas
,
S.
,
Jana
,
P.
, and
Chatterjee
,
A.
,
2016
, “
Hysteretic Damping in an Elastic Body With Frictional Microcracks
,”
Int. J. Mech. Sci.
,
108–109
, pp.
61
71
.
26.
Chatterjee
,
A.
,
2004
, “
The Short-Time Impulse Response of Euler–Bernoulli Beams
,”
ASME J. Appl. Mech.
,
71
(
2
), pp.
208
218
.
You do not currently have access to this content.