In this paper, periodic motion in an oscillator moving on a periodically oscillating belt with dry friction is investigated. The conditions of stick and nonstick motions for such an oscillator are obtained in the relative motion frame, and the grazing and stick (or sliding) bifurcations are presented as well. The periodic motions are predicted analytically and numerically, and the analytical prediction is based on the appropriate mapping structures. The eigenvalue analysis of such periodic motions is carried out. The periodic motions are illustrated through the displacement, velocity, and force responses in the absolute and relative frames. This investigation provides an efficient method to predict periodic motions of such an oscillator involving dry friction. The significance of this investigation lies in controlling motion of such a friction-induced oscillator in industry.
Issue Section:
Research Papers
1.
Luo
, A. C. J.
, 2005, “A Theory for Non-Smooth Dynamical Systems on Connectable Domains
,” Commun. Nonlinear Sci. Numer. Simul.
1007-5704, 10
, pp. 1
–55
.2.
Filippov
, A. F.
, 1964, “Differential Equations With Discontinuous Right-Hand Side
,” Am. Math. Soc. Transl.
0065-9290, Series 2, 42
, pp. 199
–231
.3.
Filippov
, A. F.
, 1988, Differential Equations with Discontinuous Righthand Sides
, Kluwer
, Dordrecht.4.
Luo
, A. C. J.
, and Gegg
, B. C.
, 2005, “On the Mechanism of Stick and Non-Stick Periodic Motion in a Forced Oscillator Including Dry-Friction
,” ASME J. Vibr. Acoust.
0739-3717, 128
, pp. 97
–105
.5.
Luo
, A. C. J.
, and Gegg
, B. C.
, 2005, “Grazing Phenomena in a Periodically Forced, Friction-Induced, Linear Oscillator
,” Commun. Nonlinear Sci. Numer. Simul.
(in press).6.
Den Hartog
, J. P.
, 1931, “Forced Vibrations with Coulomb and Viscous Damping
,” Trans. ASME
0097-6822, 53
, pp. 107
–115
.7.
Levitan
, E. S.
, 1960, “Forced Oscillation of a Spring-Mass System Having Combined Coulomb and Viscous Damping
,” J. Acoust. Soc. Am.
0001-4966, 32
, pp. 1265
–1269
.8.
Hundal
, M. S.
, 1979, “Response of a Base Excited System with Coulomb and Viscous Friction
,” J. Sound Vib.
0022-460X, 64
, pp. 371
–378
.9.
Shaw
, S. W.
, 1986, “On the Dynamic Response of a System with Dry-Friction
,” J. Sound Vib.
0022-460X, 108
, pp. 305
–325
.10.
Feeny
, B. F
, and Moon
, F. C.
, 1994, “Chaos in a Forced Dry-Friction Oscillator: Experiments and Numerical Modeling
,” J. Sound Vib.
0022-460X, 170
, pp. 303
–323
.11.
Hinrichs
, N.
, Oestreich
, M.
, and Popp
, K.
, 1997, “Dynamics of Oscillators with Impact and Friction
,” Chaos, Solitons Fractals
0960-0779, 8
(4
), pp. 535
–558
.12.
Hinrichs
, N.
, Oestreich
, M.
, and Popp
, K.
, 1998, “On the Modeling of Friction Oscillators
,” J. Sound Vib.
0022-460X, 216
(3
), pp. 435
–459
.13.
Natsiavas
, S.
, 1998, “Stability of Piecewise Linear Oscillators with Viscous and Dry Friction Damping
,” J. Sound Vib.
0022-460X, 217
, pp. 507
–522
.14.
Leine
, R. I.
, Van Campen
, D. H.
, De Kraker
, A.
, and Van Den Steen
, L.
, 1998, “Stick-Slip Vibrations Induced by Alternate Friction Models
,” Nonlinear Dyn.
0924-090X, 16
, pp. 41
–54
.15.
Virgin
, L. N.
, and Begley
, C. J.
, 1999, “Grazing Bifurcation and Basins of Attraction in an Impact-Friction Oscillator
,” Physica D
0167-2789, 130
, pp. 43
–57
.16.
Ko
, P. L.
, Taponat
, M.-C.
, and Pfaifer
, R.
, 2001, “Friction-Induced Vibration-With and Without External Disturbance
,” Tribol. Int.
0301-679X, 34
, pp. 7
–24
.17.
Andreaus
, U.
, and Casini
, P.
, 2002, “Friction Oscillator Excited by Moving Base and Colliding With a Rigid or Deformable Obstacle
,” Int. J. Non-Linear Mech.
0020-7462, 37
, pp. 117
–133
.18.
Thomsen
, J. J.
, and Fidlin
, A.
, 2003, “Analytical Approximations for Stick-Slip Vibration Amplitudes
,” Int. J. Non-Linear Mech.
0020-7462, 38
, pp. 389
–403
.19.
Kim
, W. J.
, and Perkins
, N. C.
, 2003, “Harmonic Balance/Galerkin Method for Non-Smooth Dynamical System
,” J. Sound Vib.
0022-460X, 261
, pp. 213
–224
.20.
Utkin
, V. I.
, 1981, Sliding Regimes in Optimization and Control Problems
, Nauka
, Moscow.21.
Aubin
, J.-P.
, and Cellina
, A.
, 1984, Differential Inclusions: Set-Valued Maps and Viability Theory
, Springer-Verlag
, Berlin.22.
Luo
, A. C. J.
, 2004, “Imaginary, Sink and Source Flows in the Vicinity of the Separatrix of Non-Smooth Dynamic System
,” J. Sound Vib.
0022-460X, 285
(1–2
), pp. 443
–456
.23.
Luo
, A. C. J.
, and Zwiegart
, Jr, P.
, 2005, “Existence and Analytical Predictions of Periodic Motions in a Periodically Forced, Nonlinear Friction Oscillator, Journal of Sound and Vibration
” (in revision).24.
Luo
, A. C. J.
, and Zwiegant
, P.
, Jr., 2005, “Analytical Dynamics of a Piecewise Linear Friction Oscillator Under a Periodic Excitation
,” Proc. ASME IMECE, ASME, New York, ASME Paper No. IMECE2005-80108.25.
Luo
, A. C. J.
, 2006, Singularity and Dynamics on Discontinuous Vector Fields
, Amsterdam
, Elsevier.
This content is only available via PDF.
You do not currently have access to this content.
