Chaotic behaviour of a parametrically excited damped pendulum

Leven, R.; Koch, B. P.

doi:10.1016/0375-9601(81)90167-5

Cited by 140 publications

(71 citation statements)

References 9 publications

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“…They also found that the parametric excitation promoted a sum-type combination resonance involving two modes of vibration. Analytical studies of the parametrically-excited pendulum can be traced back to the work of Leven and Koch (1981). They identified that the pendulum behaved in a chaotic way under certain parameter intervals.…”

Section: State Of the Art In Parametric Excitationmentioning

confidence: 99%

Suppressing boring bar vibrations by parametric excitation

Rezig¹,

Ouali²

2012

Adv produc engineer manag

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Abstract:Machine tool chatter causes an instability that can occur in tool-workpiece-machine system. Hence surface roughness and tool wear in metal cutting processes. This instability caused by interaction between current tool positions and surface left by previous tool passes. Many methods have been developed to minimize the effects of regenerative chatter in machine tools and to enhance the damping capability minimizing the loss in static stiffness through implementation of device uses PZT actuators or MR fluid. The possibility of suppressing selfexcited vibrations of boring process using parametric excitation is discussed. We consider a two-mass system of which the main mass is excited by a self excited force (cutting force). A single mass which acts as a dynamic absorber is attached to the main mass and, by varying the stiffness between the main mass and the absorber mass, represents a parametric excitation. Cutting experiments with an excitation current of different waveforms and diverse frequencies show that chatter can be significantly suppressed by the effect of parametric excitation.

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Section: State Of the Art In Parametric Excitationmentioning

confidence: 99%

Suppressing boring bar vibrations by parametric excitation

Rezig¹,

Ouali²

2012

Adv produc engineer manag

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“…A damped parametrically driven mathematical pendulum can be described by the following equation [17]:…”

Section: Equation Of a Parametrically Excited Damped Pendulummentioning

confidence: 99%

Construction of homoclinic and heteroclinic trajectories in mechanical systems with several equilibrium positions

Mikhlin¹,

Manucharyan²

2003

Chaos, Solitons & Fractals

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A new approach for a construction of homo and heteroclinic trajectories of some principal non-linear dynamical systems is utilized here, namely the non-linear Schrodinger equation, non-autonomous Duffing equation and the equation of a parametrically excited damped pendulum are considered. PadeÕ and quasi-PadeÕ approximants and a convergence condition used earlier in the theory of non-linear normal vibration modes made possible to solve a boundary-value problems formulated for the orbits and to determine initial amplitude values of the trajectories with admissible precision. The approach proposed here is more exact than the generally accepted one because it is not necessary to use here separatrix trajectories of the corresponding autonomous equations.

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“…This system has attracted great attention [12][13][14][15][16][17]. It presents a wide range of dynamical behaviour, such as: the stabilization of the hilltop saddle [18][19][20][21][22][23]; the occurrence of chaotic behaviour [24][25][26][27][28][29]; the observation of period-doubling cascades [30,31] and the existence of resonance regions [11,32]. Moreover, it can be used as qualitative analogue for more complex systems [31,33,34].…”

Section: Introductionmentioning

confidence: 99%

Tilted excitation implies odd periodic resonances

et al. 2016

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Our aim is to unveil how resonances of parametric systems are affected when symmetry is broken. We showed numerically and experimentally that odd resonances indeed come about when the pendulum is excited along a tilted direction. Applying the Melnikov subharmonic function, we not only determined analytically the loci of saddle-node bifurcations delimiting resonance regions in parameter space, but also explained these observations by demonstrating that, under the Melnikov method point of view, odd resonances arise due to an extra torque that appears in the asymmetric case.

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Chaotic behaviour of a parametrically excited damped pendulum

Cited by 140 publications

References 9 publications

Suppressing boring bar vibrations by parametric excitation

Suppressing boring bar vibrations by parametric excitation

Construction of homoclinic and heteroclinic trajectories in mechanical systems with several equilibrium positions

Tilted excitation implies odd periodic resonances

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