Subharmonic resonance bifurcation and chaos of the simple pendulum system with vertical excitation  and horizontal constraint

The homoclinic bifurcation and chaos in a system of weakly coupled simple pendulum and harmonic oscillator subject to light dampings and weakly external and (or) parametric excitation of bounded noise is studied. The random Melnikov process is derived and mean-square criteria is used to determine the threshold amplitude of the bounded noise for the onset of chaos in the system. The threshold amplitude is also determined by vanishing the numerically calculated maximal Lyapunov exponent. The threshold amplitudes are further confirmed by using the Poincaré maps, which indicate the path from periodic motion to chaos or from random motion to random chaos in the system as the amplitude of bounded noise increases.

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Resonance, bifurcation and chaos control in electrostatic transducers with two external periodic forces

Physica A Statistical Mechanics and its Applications ◽

10.1016/j.physa.2003.10.056 ◽

2004 ◽

Vol 333 ◽

pp. 87-105 ◽

Cited By ~ 6

Author(s):

F.M. Moukam Kakmeni ◽

S. Bowong ◽

C. Tchawoua ◽

E. Kaptouom

Keyword(s):

Chaos Control ◽

Bifurcation And Chaos ◽

Electrostatic Transducers ◽

Resonance Bifurcation

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Homoclinic bifurcation and chaos in simple pendulum under bounded noise excitation

Chaos Solitons & Fractals ◽

10.1016/j.chaos.2003.08.010 ◽

2004 ◽

Vol 20 (3) ◽

pp. 593-607 ◽

Cited By ~ 19

Author(s):

Z.H. Liu ◽

W.Q. Zhu

Keyword(s):

Homoclinic Bifurcation ◽

Bounded Noise ◽

Bifurcation And Chaos ◽

Simple Pendulum

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Some extended results of “subharmonic resonance bifurcation theory of nonlinear Mathieu equation”

Applied Mathematics and Mechanics ◽

10.1007/bf02015206 ◽

1990 ◽

Vol 11 (3) ◽

pp. 255-261 ◽

Cited By ~ 1

Author(s):

Chen Yu-shu ◽

Zhan Kai-jun

Keyword(s):

Bifurcation Theory ◽

Mathieu Equation ◽

Subharmonic Resonance ◽

Resonance Bifurcation

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Extended Results of Subharmonic Resonance Bifurcation of Nonlinear Mathieu Equation and Some Experimental Results

Nonlinear Dynamics in Engineering Systems ◽

10.1007/978-3-642-83578-0_6 ◽

1990 ◽

pp. 43-50 ◽

Cited By ~ 1

Author(s):

Yu Shu Chen ◽

Kaijun Zhan ◽

W. F. Langford

Keyword(s):

Mathieu Equation ◽

Subharmonic Resonance ◽

Experimental Results ◽

Resonance Bifurcation

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A Hybrid Command-Shaping Control System for Highly Accelerated Double-Pendulum Gantry Cranes

Volume 4: 7th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B and C ◽

10.1115/detc2009-87501 ◽

2009 ◽

Cited By ~ 7

Author(s):

Ziyad N. Masoud ◽

Khalid A. Alhazza ◽

Majed A. Majeed ◽

Eiyad A. Abu-Nada

Keyword(s):

Control Systems ◽

Degrees Of Freedom ◽

Gantry Crane ◽

System Control ◽

Double Pendulum ◽

Pendulum System ◽

Command Shaping ◽

Simple Pendulum ◽

Time Optimal ◽

Gantry Cranes

A gantry cranes is generally modeled as a simple-pendulum with a point mass attached to the end of a massless rigid link. Numerous control systems have been developed to reduce payload oscillations in order to improve safety and positioning accuracy of crane operations. However, large-size payloads transforms the crane model from a simple-pendulum system to a double-pendulum system. Control systems that consider only one mode of oscillations of a double-pendulum may excite large oscillations in the other mode. In multi-degrees-of-freedom systems, command-shaping controllers designed for the first mode may eliminate oscillations of higher modes provided that their frequencies are odd integer multiples of the first mode frequency. In this work, a hybrid command-shaping controller is designed to generate acceleration commands to suppress travel and residual oscillations of a highly accelerated double-pendulum gantry crane. It is shown that the suggested hybrid command-shaper is capable of minimizing oscillations of both modes of a scaled experimental double-pendulum model of a gantry crane. Results show that the hybrid command-shaper produces a reduction of 95% in residual oscillations in both modes of the double-pendulum over the time-optimal rigid-body commands.

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Stabilization of 1/3-Order Subharmonic Resonance Using an Autoparametric Vibration Absorber

Journal of Vibration and Acoustics ◽

10.1115/1.2893981 ◽

1999 ◽

Vol 121 (3) ◽

pp. 309-315 ◽

Cited By ~ 26

Author(s):

Hiroshi Yabuno ◽

Yasuhiro Endo ◽

Nobuharu Aoshima

Keyword(s):

Magnetic Force ◽

Frequency Component ◽

Harmonic Excitation ◽

Subharmonic Resonance ◽

Vibration Absorber ◽

Spring Stiffness ◽

Stabilization Method ◽

Pendulum System ◽

Main System ◽

Subharmonic Frequency

The paper proposes a stabilization method for 1/3-Order subharmonic resonance with an autoparametric vibration absorber. A main system with nonlinear spring stiffness and harmonic excitation, i.e., subjected to a sinusoidally changed magnetic force, is introduced as a model which produces 1/3-Order subharmonic resonance. A damped pendulum system, whose natural frequency is in the neighborhood of one-half of the main system, is attached to the main system as an absorber, in order to induce 1:2 internal resonance. The 1/3-Order subharmonic resonance which occurs in the case of locked pendulum is avoided due to energy transfer between the main system and the absorber, and due to energy dissipation by the absorber. It is also theoretically shown that a stable nontrivial steady state with respect to the 1/3-Order subharmonic frequency component is changed into an unstable one due to the absorber. Experimental results show the validity of the autoparametric vibration absorber for the 1/3-Order subharmonic resonance.

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