zbMATH — the first resource for mathematics

Experimental versus theoretical robustness of rotating solutions in a parametrically excited pendulum: a dynamical integrity perspective. (English) Zbl 1218.37116
From an operative point of view the dynamical integrity of a dynamical system consists in studying the properties of the safe basin (to be properly defined and measured) and their evolution with a varying driving parameter. In this paper the safe basin is the basin of attraction of the clockwise rotating solution of a pendulum and the considered measure of integrity is the integrity factor, which is the normalized radius of the largest circle entirely belonging to the safe basin, and which has been shown to be appropriate for rotations. A main part of this work is Section 5, where the dynamical integrity of the parametric pendulum is discussed in detail and used to justify the experimental observations.

37N05 Dynamical systems in classical and celestial mechanics
70E17 Motion of a rigid body with a fixed point
70K20 Stability for nonlinear problems in mechanics
Full Text: DOI
[1] M. Wiercigroch, A new concept of energy extraction from waves via parametric pendulor, UK Patent Application, Pending, 2010.
[2] ()
[3] Marlin, J.A., Periodic motions of coupled simple pendulums with periodic disturbances, Int. J. non-linear mech., 3, 439-447, (1968) · Zbl 0169.55605
[4] Chacon, R.; Martinez, P.J.; Martinez, J.A.; Lenci, S., Chaos suppression and desynchronization phenomena in periodically coupled pendulums subjected to localized heterogeneous forces, Chaos solitons fractals, 42, 2342-2350, (2009)
[5] Chacón, R.; Marcheggiani, L., Controlling spatiotemporal chaos in chains of dissipative Kapitza pendula, Phys. rev. E, 82, 016201, (2010)
[6] Koch, B.P.; Leven, R.W., Subharmonic and homoclinic bifurcations in a parametrically forced pendulum, Physica D, 16, 1-13, (1985) · Zbl 0585.70022
[7] Butikov, E., The rigid pendulum—an antique but evergreen physical model, European J. phys., 20, 429-441, (1999)
[8] Szemplinska-Stupnicka, W.; Tyrkiel, E.; Zubrzycki, A., The global bifurcations that lead to transient tumbling chaos in a parametrically driven pendulum, Int. J. bifurcation chaos, 10, 2161-2175, (2000) · Zbl 0965.70036
[9] Garira, W.; Bishop, S.R., Rotating solutions of the parametrically excited pendulum, J. sound vibration, 263, 233-239, (2003) · Zbl 1237.34080
[10] Xu, X.; Wiercigroch, M.; Cartmell, M.P., Rotating orbits of a parametrically-excited pendulum, Chaos solitons fractals, 23, 1537-1548, (2005) · Zbl 1135.70313
[11] Xu, X.; Wiercigroch, M., Approximate analytical solutions for oscillatory and rotational motion of a parametric pendulum, Nonlinear dynam., 47, 311-320, (2007) · Zbl 1180.70033
[12] Lenci, S.; Pavlovskaia, E.; Rega, G.; Wiercigroch, M., Rotating solutions and stability of parametric pendulum by perturbation method, J. sound vibration, 310, 243-259, (2008)
[13] Horton, B.; Wiercigroch, M.; Xu, X., Transient tumbling chaos and damping identification for parametric pendulum, Philos. trans. R. soc. lond. ser. A math. phys. eng. sci., 366, 767-784, (2008) · Zbl 1153.70305
[14] Lenci, S.; Rega, G., Competing dynamic solutions in a parametrically excited pendulum: attractor robustness and basin integrity, ASME J. comput. nonlin. dyn., 3, 041010, (2008)
[15] Zhu, Q.; Ishitobi, M., Experimental study of chaos in a driven triple pendulum, J. sound vibration, 227, 230-238, (1999)
[16] de Paula, A.S.; Savi, M.A.; Pereira-Pinto, F.H.I., Chaos and transient chaos in an experimental nonlinear pendulum, J. sound vibration, 294, 585-595, (2006)
[17] Blackburn, J.A.; Zhou-jing, Y.; Vik, S.; Smith, H.J.T.; Nerenberg, M.A.H., Experimental study of chaos in a driven pendulum, Physica D, 26, 385-395, (1987) · Zbl 0613.58023
[18] Awrejcewicz, J.; Supeł, B.; Lamarque, C.-H.; Kudra, G.; Wasilewski, G.; Olejnik, P., Numerical and experimental study of regular and chaotic motion of triple physical pendulum, Int. J. bif. chaos, 18, 2883-2915, (2008) · Zbl 1165.70307
[19] Xu, X.; Pavlovskaia, E.; Wiercigroch, M.; Romeo, F.; Lenci, S., Dynamic interactions between parametric pendulum and electro-dynamical shaker, ZAMM Z. angew. math. mech., 87, 172-186, (2007) · Zbl 1342.70060
[20] M. Wiercigroch, Private communication, 2007.
[21] S. Lenci, M. Brocchini, C. Lorenzoni, Experimental rotations of a pendulum on water waves (2010) (submitted for publication).
[22] Thompson, J.M.T., Chaotic behavior triggering the escape from a potential well, Proc. R. soc. lond. ser. A, 421, 195-225, (1989) · Zbl 0674.70035
[23] Soliman, M.S.; Thompson, J.M.T., Integrity measures quantifying the erosion of smooth and fractal basins of attraction, J. sound vibration, 135, 453-475, (1989) · Zbl 1235.70106
[24] Rega, G.; Lenci, S., Identifying, evaluating, and controlling dynamical integrity measures in nonlinear mechanical oscillators, Nonlinear anal. TMA, 63, 902-914, (2005) · Zbl 1153.70307
[25] Rega, G.; Lenci, S., Dynamical integrity and control of nonlinear mechanical oscillators, J. vib. control, 14, 159-179, (2008) · Zbl 1229.70085
[26] Goncalves, P.B.; Silva, F.M.A.; Rega, G.; Lenci, S., Global dynamics and integrity of a two-d.o.f. model of a parametrically excited cylindrical shell, Nonlinear dynam., 63, 61-82, (2011) · Zbl 1215.74049
[27] Sanjuan, M.A.F., Subharmonic bifurcations in a pendulum parametrically excited by a non-harmonic perturbation, Chaos solitons fractals, 9, 995-1003, (1998) · Zbl 0990.70017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.