The competing solutions of a planar pendulum parametrically excited by the vertical motion of the pivot are investigated in terms of both attractor robustness and basin integrity. Two different measures are considered to highlight how the integrity of the system is modified by changing the amplitude of the excitation. Various attractors, both in-well and out-of-well, are considered, and the integrity profiles of each of them are determined. A detailed discussion of the interaction and mutual erosion of the various attractors is performed by clarifying the role of the two complementary measures, and the most relevant characteristics of the erosion profiles are interpreted in terms of the underlying topological mechanisms involving local or global bifurcations.
1.
Thompson
, J. M. T.
, 1989, “Chaotic Phenomena Triggering the Escape From a Potential Well
,” Proc. R. Soc. London, Ser. A
1364-5021, 421
, pp. 195
–225
.2.
Soliman
, M. S.
, and Thompson
, J. M. T.
, 1989, “Integrity Measures Quantifying the Erosion of Smooth and Fractal Basins of Attraction
,” J. Sound Vib.
, 135
, pp. 453
–475
. 0022-460X3.
Soliman
, M. S.
, and Thompson
, J. M. T.
, 1992, “Global Dynamics Underlying Sharp Basin Erosion in Nonlinear Driven Oscillators
,” Phys. Rev. A
1050-2947, 45
, pp. 3425
–3431
.4.
Lansbury
, A. N.
, Thompson
, J. M. T.
, and Stewart
, H. B.
, 1992, “Basin Erosion in the Twin-Well Duffing Oscillator: Two Distinct Bifurcation Scenarios
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274, 2
, pp. 505
–532
.5.
Gan
, C.
, Lu
, Q.
, and Huang
, K.
, 1998, “Nonstationary Effects on Safe Basins of a Forced Softening Duffing Oscillator
,” Acta Mech. Sin.
, 11
, pp. 253
–260
. 0567-77186.
de Freitas
, M. S. T.
, Viana
, R. L.
, and Grebogi
, C.
, 2003, “Erosion of the Safe Basin for the Transversal Oscillations of a Suspension Bridge
,” Chaos, Solitons Fractals
0960-0779, 18
, pp. 829
–841
.7.
Wu
, X.
, Tao
, L.
, and Li
, Y.
, 2004, “The Safe Basin Erosion of a Ship in Waves With Single Degree of Freedom
,” Proceedings. of the 15th Australasian Fluid Mechanics Conference
, Sydney, Australia
, Dec. 13–17.8.
Lenci
, S.
, and Rega
, G.
, 2006, “Control of Pull-in Dynamics in a Nonlinear Thermoelastic Electrically Actuated Microbeam
,” J. Micromech. Microeng.
0960-1317, 16
, pp. 390
–401
.9.
Goncalves
, P. B.
, Silva
, F. M. A.
, and Del Prado
, Z. J. G. N.
, 2007, “Global Stability Analysis of Parametrically Excited Cylindrical Shells Through the Evolution of Basin Boundaries
,” Nonlinear Dyn.
, 50
, pp. 121
–145
. 0924-090X10.
Rega
, G.
, and Lenci
, S.
, 2005, “Identifying, Evaluating, and Controlling Dynamical Integrity Measures in Nonlinear Mechanical Oscillators
,” Nonlinear Anal.: Real World Appl.
, 63
, pp. 902
–914
. 0362-546X11.
Xu
, J.
, Lu
, Q.
, and Huang
, K.
, 1996, “Controlling Erosion of Safe Basin in Nonlinear Parametrically Excited Systems
,” Acta Mech. Sin.
, 12
, pp. 281
–288
. 0567-771812.
Rega
, G.
, and Lenci
, S.
, 2008, “Dynamical Integrity and Control of Nonlinear Mechanical Oscillators
,” J. Vib. Control
1077-5463, 14
, pp. 159
–179
.13.
Butikov
, E.
, 1999, “The Rigid Pendulum–An Antique But Evergreen Physical Model
,” Eur. J. Phys.
0143-0807, 20
, pp. 429
–441
.14.
Blekhman
, I. I.
, 2000, Vibrational Mechanics
, World Scientific
, Singapore
.15.
Szemplinska-Stupnicka
, W.
, Tyrkiel
, E.
, and Zubrzycki
, A.
, 2000, “The Global Bifurcations That Lead to Transient Tumbling Chaos in a Parametrically Driven Pendulum
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274, 10
, pp. 2161
–2175
.16.
Xu
, X.
, Pavlovskaia
, E.
, Wiercigroch
, M.
, Romeo
, R.
, and Lenci
, S.
, 2007, “Dynamic Interactions Between Parametric Pendulum and Electrodynamical Shaker
,” Z. Angew. Math. Mech.
0044-2267, 87
, pp. 172
–186
.17.
Bishop
, S. R.
, and Clifford
, M. J.
, 1996, “Zones of Chaotic Behavior in the Parametrically Excited Pendulum
,” J. Sound Vib.
0022-460X, 189
, pp. 142
–147
.18.
Koch
, B. P.
, and Leven
, R. W.
, 1985, “Subharmonic and Homoclinic Bifurcations in a Parametrically Forced Pendulum
,” Physica D
0167-2789, 16
, pp. 1
–13
.Copyright © 2008
by American Society of Mechanical Engineers
You do not currently have access to this content.
