Bifurcation analysis of a double pendulum with internal resonance

毕勤胜,; 陈予恕,; Bi, Qinsheng; Chen, Yushu

doi:10.1007/bf02459003

Cited by 3 publications

(1 citation statement)

References 10 publications

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“…Many researchers have studied similar systems. Studies of bifurcations in the non-linear dynamics of double pendula can be found in [4][5][6]. In 2001, Bridges and Georgiou [7] studied a transverse rotating double pendulum, in which the axes of the two pivots are not parallel, so that the pendula do not swing in the same plane.…”

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confidence: 99%

On the Equilibria and Bifurcations of a Rotating Double Pendulum

Tot¹,

Lewis²

2022

Preprint

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The double pendulum, a simple system of classical mechanics, is widely studied as an example of, and testbed for, chaotic dynamics. In [1], Maiti et al. study a generalization of the simple double pendulum with equal point-masses at equal lengths, to a rotating double pendulum, fixed to a coordinate system uniformly rotating about the vertical. In this paper, we study a considerable generalization of the double pendulum, constructed from physical pendula, and ask what equilibrium configurations exist for the system across a comparatively large parameters space, as well as what bifurcations occur in those equilibria.Elimination algorithms are employed to reduce systems of polynomial equations, which allows for equilibria to be visualized, and also to demonstrate which models within the parameter space exhibit bifurcation. We find the DixonEDF algorithm for the Dixon resultant [2], written in the computer algebra system (CAS) Fermat [3], to be capable to complete the computation for the challenging system of equations that represents bifurcation, while attempts with other algorithms were terminated after several hours. I. INTRODUCTIONThis work is concerning a system of classical physics, namely a rotating double pendulum (RDP).Here, 'rotating' is used to indicate a double pendulum that is made to rotate about a vertical axis with a constant angular velocity ω a . A double pendulum has two joints or pivots, and we take the vertical axis of this rotation to pass through the inner, stationary pivot. Additionally, our consideration is of a physical double pendulum, constructed from two 3-dimensional rigid bodies.Thus the physics depends on many dimensional parameters: masses, lengths, ω a , the strength of gravity, and principal moments of inertia -13 in total. However, only 6 dimensionless parameters are required to describe the dynamics of the system.

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confidence: 99%

On the Equilibria and Bifurcations of a Rotating Double Pendulum

Tot¹,

Lewis²

2022

Preprint

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Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems

陈衍茂

刘济科

2008

Appl. Math. Mech.-Engl. Ed.

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The Hopf bifurcations of an airfoil flutter system with a cubic nonlinearity are investigated, with the flow speed as the bifurcation parameter. The center manifold theory and complex normal form method are used to obtain the bifurcation equation. Interestingly, for a certain linear pitching stiffness the Hopf bifurcation is both supercritical and subcritical. It is found, mathematically, this is caused by the fact that one coefficient in the bifurcation equation does not contain the first power of the bifurcation parameter. The solutions of the bifurcation equation are validated by the equivalent linearization method and incremental harmonic balance method.

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Bifurcation and dynamic response analysis of rotating blade excited by upstream vortices

Chen

Wang

Wiercigroch

et al. 2016

Appl. Math. Mech.-Engl. Ed.

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A reduced model simulating vortex-induced vibrations (VIVs) for turbine blades is proposed and analyzed. The rotating blade is modeled as a uniform cantilever beam while a van der Pol oscillator is used to represent the time-varying characteristics of the vortex shedding, which interacts with equations of motion for a blade to simulate the fluid-structure interaction. The action for the structural motion on the fluid is considered as a linear inertia coupling. Nonlinear characteristics for the dynamic responses are investigated with the multiple scale method and the modulation equations are derived. The transition set consisting of the bifurcation and hysteresis sets is constructed by using the singularity theory and the effects of system parameters including the van der Pol damping and the coupling parameter on the equilibrium solutions are analyzed. Frequency-response curves are obtained and the stabilities are determined by using the Routh-Hurwitz criterion. The phenomena including the saddle-node and Hopf bifurcations are found to occur under certain parameter values. A direct numerical method is used to analyze the dynamic characteristics for the original system and verify the validity of the multiple scale method. The results indicate that the new coupled model can be useful to explain the rich dynamic response characteristics including possible bifurcation phenomena in the VIVs.

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Bifurcation analysis of a double pendulum with internal resonance

Cited by 3 publications

References 10 publications

On the Equilibria and Bifurcations of a Rotating Double Pendulum

On the Equilibria and Bifurcations of a Rotating Double Pendulum

Supercritical as well as subcritical Hopf bifurcation in nonlinear flutter systems

Bifurcation and dynamic response analysis of rotating blade excited by upstream vortices

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