In this paper, complex motions of a ball in the horizontal impact pair with a periodic excitation are studied analytically using the theory of discontinuous dynamical system. Analytical conditions for motion switching caused by impacts are developed, and generic mapping structures are introduced to describe different periodic and chaotic motions. Analytical prediction of complex periodic motion of the ball in the periodically shaken impact pair is completed, and the corresponding stability and bifurcation analysis are also carried out. Numerical illustrations of periodic and chaotic motions are given.
Issue Section:
Research Papers
References
1.
Hartog
, J. P. D.
, and Mikina
, S. J.
, 1932, “Forced Vibrations With Non-Linear Spring Constants
,” ASME J. Appl. Mech.
, 58
, pp.157
–164
.2.
Filippov
, A. F.
, 1964, “Differential Equations With Discontinuous Right-Hand Side
,” Am. Math. Soc. Transl., Series 2
, 42
, pp. 199
–231
.3.
Filippov
, A. F.
, 1988, Differential Equations With Discontinuous Right Hand Sides
, Kluwer Academic Publishers
, Dordrecht
.4.
Luo
, A. C. J.
, 2005, “A Theory for Non-Smooth Dynamic Systems on the Connectable Domains
,” Commun. Nonlinear Sci. Numer. Simul.
, 10
, pp. 1
–55
.5.
Luo
, A. C. J.
, 2006, Singularity and Dynamics on Discontinuous Vector Fields
, Elsevier
, Amsterdam
.6.
Luo
, A. C. J.
, 2009, Discontinuous Dynamical Systems on Time-varying Domains
, HEP-Springer
, Dordrecht
.7.
Holmes
, P. J.
, 1982, “The Dynamics of Repeated Impacts With a Sinusoidally Vibrating Table
,” J. Sound Vib.
, 84
, pp. 173
–189
.8.
Bapat
, C. N.
, and Popplewell
, N.
, 1983, “Stable Periodic Motions of an Impact-Pair
,” J. Sound Vib.
, 87
, pp. 19
–40
.9.
Shaw
, S. W.
, and Holmes
, P. J.
, 1983, “A Periodically Forced Piecewise Linear Oscillator
,” J. Sound Vib.
, 90
, pp. 129
–155
.10.
Heiman
, M. S.
, Sherman
P. J.
, and Bajaj
A. K.
, 1987, “On the Dynamics and Stability of an Inclined Impact Pair
,” J. Sound Vib.
, 114
(3
), pp. 535
–547
.11.
Heiman
, M. S.
, Bajaj
A. K.
, and Sherman
P. J.
, 1988, “Periodic Motions and Bifurcations in Dynamics of an Inclined Impact Pair
,” J. Sound Vib.
, 124
(1
), pp. 55
–78
.12.
Bapat
, C. N.
, 1988, “Impact-Pair Under Periodic Excitation
,” J. Sound Vib.
, 120
, pp. 53
–61
.13.
Nordmark
, A. B.
, 1991, “Non-Periodic Motion Caused by Grazing Incidence in an Impact Oscillator
,” J. Sound Vib.
, 145
, pp. 279
–297
.14.
di Bernardo
, M.
, Budd
, C. J.
, and Champney
, A. R.
, 2001, “Normal Form Maps for Grazing Bifurcation in n-Dimensional Piecewise-Smooth Dynamical Systems
,” Physica D
, 160
, pp. 222
–254
.15.
di Bernardo
, M.
, Kowalczyk
, P.
, and Nordmark
, A. B.
, 2002, “Bifurcation of Dynamical Systems With Sliding: Derivation of Normal Form Mappings
,” Physica D
, 170
, pp. 175
–205
.16.
Bapat
, C. N.
, 1995, “The General Motion of an Inclined Impact Damper With Friction
,” J. Sound Vib.
, 184
, pp. 417
–427
.17.
Han
, R. P. S.
, Luo
, A. C. J.
, and Deng
, W.
, 1995, “Chaotic Motion of a Horizontal Impact Pair
,” J. Sound Vib.
, 181
(2
), pp. 231
–250
18.
Luo
, A. C. J.
and Han
, R. P. S.
, 1996, “The Dynamics of a Bouncing Ball With A Sinusoidally Vibrating Table Revisited
,” Nonlinear Dyn.
, 10
, pp. 1
–18
.19.
Peterka
, F.
, 1996, “Bifurcation and Transition Phenomena in an Impact Oscillator
,” Chaos, Solitons Fractals
, 7
, pp. 1635
–1647
.20.
Blazejczyk-Okolewska
, B.
, and Peterka
, F.
, 1998, “An Investigation of the Dynamic System With Impacts
,” Chaos, Solitons Fractals
, 9
, pp. 1321
–1338
.21.
Peterka
, F.
, 2003, “Behavior of Impact Oscillator With Soft and Preloaded Stop
,” Chaos, Solitons Fractals
, 18
, pp. 79
–88
.22.
Peterka
, F.
, Kotera
, T.
, and Cipera
, S.
, 2004, “Explanation of Appearance and Characteristics of Intermittency Chaos of the Impact Oscillator
,” Chaos, Solitons Fractals
, 19
, pp. 1251
–1259
.23.
Peterka
, F.
, and Tondl
, A.
, 2004, “Phenomena of Subharmonic Motions of Oscillator With Soft Impacts
,” Chaos, Solitons Fractals
, 19
, pp. 1283
–1290
.24.
Blazejczyk-Okolewska
, B.
, and Kapitaniak
, T.
, 1995, “Dynamics of Impact Oscillator With Dry Friction
,” Chaos, Solitons Fractals
, 7
, pp. 1455
–1459
.25.
Blazejczyk-Okolewska
, B.
, and Kapitaniak
, T.
, 1998, “Co-Existing Attractors of Impact Oscillator
,” Chaos, Solitons Fractals
, 9
, pp. 1439
–1443
.26.
Blazejczyk-Okolewska
, B.
, 2000, “Study of the Impact Oscillator With Elastic Coupling of Masses
,” Chaos, Solitons Fractals
, 11
, pp. 2487
–2492
.27.
Blazejczyk-Okolewska
, B.
, and Kapitaniak
, T.
, 2003, “Influence of the Mass and Stiffness Ratio on a Periodic Motion of Two Impacting Oscillator
,” Chaos, Solitons Fractals
, 17
, pp. 1
–10
.28.
Czolczynski
, K.
, 2003, “On the Existence of a Stable Periodic Motion of Two Impacting Oscillators
,” Chaos, Solitons Fractals
, 15
, pp. 371
–379
.29.
Blazejczyk-Okolewska
, B.
, Czolczynski
, K.
, and Kapitaniak
, T.
, 2009, “Dynamics of a Two-Degree-of-Freedom Cantilever Beam With Impacts
,” Chaos, Solitons Fractals
, 40
, pp. 1991
–2006
.30.
Giusepponi
, S.
, Marchesoni
, F.
, and Borromeio
, M.
, 2004, “Randomness in the Bouncing Ball Dynamics
,” Physica A
, 351
, pp. 143
–158
.31.
Park
, J.
, Wang
, S.
, and Crocker
, M. J.
, 2009, “Mass Load Resonance of a Single Unit Impact Damper Caused by Impacts and the Resulting Kinetic Energy Influx
,” J. Sound Vib.
, 323
, pp. 877
–895
.32.
Luo
, A. C. J.
, 2002, “An Unsymmetrical Motion in a Horizontal Impact Oscillator
,” ASME J. Vibr. Acoust.
, 124
, pp. 420
–426
.33.
Luo
, A. C. J.
, and Menon
, S.
, 2004, “Global Chaos in a Periodically Forced, Linear System With a Dead-Zone Restoring Force
,” Chaos, Solitons Fractals
, 19
, pp. 1189
–1199
.34.
Menon
, S.
, and Luo
, A. C. J.
, 2005, “A Global Period-1 Motion of A Periodically Forced, Piecewise Linear System
,” Int. J. Bifurcation Chaos
, 15
, pp. 1945
–1957
.35.
Luo
, A. C. J.
, 2005, “The Mapping Dynamics of Periodic Motions for a Three-Piecewise Linear System Under a Periodic Excitation
,” J. Sound Vib.
, 283
, pp. 723
–748
.36.
Pavlovskaia
, E.
, and Wiercigroch
, M.
, 2007, “Low-Dimensional Maps for Piecewise Smooth Oscillators
,” J. Sound Vib.
, 305
, pp. 750
–771
.37.
Luo
, A. C. J.
, and Chen
, L. D.
, 2006, “Grazing Phenomena and Fragmented Strange Attractors in a Harmonically Forced, Piecewise, Linear System With Impacts
,” IMeChE Part K: J. Multibody Dyn.
, 220
, pp. 35
–51
.38.
Luo
, A. C. J.
, and Chen
, L. D.
, 2007, “Arbitrary Periodic Motions and Grazing Switching of a Forced Piecewise-Linear, Impacting Oscillator
,” ASME J. Vib. Acoust.
, 129
, pp. 276
–285
.39.
Luo
, A. C. J.
, and O’Connor
, D.
, 2009, “Mechanism of Impacting Chatter With Stick in Gear Transmission Systems
,” Int. J. Bifurcation Chaos
, 19
, pp. 2093
–2105
.40.
Luo
, A. C. J.
, and O’Connor
, D.
, 2009, “Periodic Motions and Chaos With Impacting Chatter and Stick in Gear Transmission Systems
,” Int. J. Bifurcation Chaos
, 19
, pp. 1975
–1994
.41.
Luo
, A. C. J.
, 2008, “A Theory for Flow Switchability in Discontinuous Dynamical Systems
,” Nonlinear Anal.: Hybrid Syst.
, 2
(4
), pp. 1030
–1061
.42.
Luo
, A. C. J.
, and Guo
, Y.
, 2009, “Motion Switching and Chaos of a Particle in a Generalized Fermi-Acceleration Oscillator
,” Math. Probl. Eng.
, 2009
, 40
p.43.
Luo
, A.C.J.
, and Guo
, Y.
, 2010, “Switching Mechanism and Complex Motions in an Extended Fermi-Acceleration Oscillator
,” J. Comput. Nonlinear Dyn.
, 5
(4
), pp. 1
–14
.44.
Luo
, A. C. J.
, and Guo
, Y.
, 2010, “Switchability and Bifurcation of Motions in a Double-Excited Fermi-Acceleration Oscillator
,” Proceedings of the 2010 ASME International Mechanical Engineering Congress and Exposition
, IMECE2010-39165.Copyright © 2012
by American Society of Mechanical Engineers
You do not currently have access to this content.
