Quantifying periodic, multi-periodic, hidden and unstable regimes of a magnetic pendulum via semi-analytical, numerical and experimental methods

Skurativskyi, Sergii; Polczyński, Krystian; Wojna, Mateusz; Awrejcewicz, Jan

doi:10.1016/j.jsv.2021.116710

Cited by 13 publications

(3 citation statements)

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“…In both cases, the primary bifurcation from the trivial state could be shifted to smaller B and changed from sub-to supercritical by increasing C. In these cases with large C, the former primary bifurcations seem to occur as secondary bifurcations at roughly the same B values as before. Chaotic motion is present in all the cases in the subharmonic instability range but largely absent for those in the harmonic range except for the lowest C. It may well appear for B > 3, which could be explored further as well as a variation of the parameter D. The periodic orbits obtained in the bifurcation diagrams could also be continued numerically into the unstable parameter range by a shooting method as done in [14]. This could highlight common structures of the solution branches between different values of C further.…”

Section: Discussionmentioning

confidence: 99%

“…Numerical solu-tions reveal numerous interesting bifurcation phenomena in such systems at subharmonic frequency ratios [13]. Moreover, magnetic pendulums that are excited by time-dependent fields were recently studied experimentally [14,15]. The corresponding mathematical models, which can reproduce the fully nonlinear behavior successfully, include additional terms for a damping element with elasticity and friction.…”

Section: Introductionmentioning

confidence: 99%

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Parametric instability of a vertically driven magnetic pendulum with eddy-current braking by a flat plate

2022

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The vertically driven pendulum is one of the classical systems where parametric instability occurs. We study its behavior with an additional electromagnetic interaction caused by eddy currents in a nearby thick conducting plate that are induced when the bob is a magnetic dipole. The known analytical expressions of the induced electromagnetic force and torque acting on the dipole are valid in the quasistatic limit, i.e., when magnetic diffusivity of the plate is sufficiently high to ensure an equilibrium between magnetic field advection and diffusion. The equation of motion of the vertically driven pendulum is derived assuming that its magnetic dipole moment is aligned with the axis of rotation and that the conducting plate is horizontal. The vertical position of the pendulum remains an equilibrium with the electromagnetic interaction. Conditions for instability of this equilibrium are derived analytically by the harmonic balance method for the subharmonic and harmonic resonances in the limit of weak electromagnetic interaction. The analytical stability boundaries agree with the results of numerical Floquet analysis for these conditions but differ substantially when the electromagnetic interaction is strong. The numerical analysis demonstrates that the area of harmonic instability can become doubly connected. Bifurcation diagrams obtained numerically show the co-existence of stable periodic orbits in such conditions. For moderately strong driving, chaotic motions can be maintained for the subharmonic instability.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Parametric instability of a vertically driven magnetic pendulum with eddy-current braking by a flat plate

2022

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“…Indeed, one can find numerous papers, books, and video clips concerning their non-linear and chaotic dynamics. To exemplify it, let us mention the paradigm models such as the spring pendulum [2][3][4][5][6], the magnetic pendulum [7][8][9], the double and the triple pendulums [10][11][12][13][14][15][16][17][18][19], the coupled pendulums [20][21][22][23][24] and the swinging Atwood machine [25][26][27][28][29]. These models have been extensively studied by researchers both theoretically and practically [29][30][31][32][33][34][35].…”

Section: Introductionmentioning

confidence: 99%