Non-Linear Oscillations of a Rotor in Active Magnetic Bearings

Ji, Jinchen; Hansen, Colin H.

doi:10.1006/jsvi.2000.3257

Cited by 94 publications

(52 citation statements)

References 5 publications

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“…The fixed points of the averaged Equation (32) correspond to the phase-locked periodic solutions of system (13). The stability of the solutions can be determined by investigating the characteristic equations.…”

Section: Stability Analysis Of Steady State Solutionsmentioning

confidence: 99%

“…Equation (7) does not have Z 2 ⊕ Z 2 symmetry. Other papers [10][11][12][13] do not consider the influence of the weight of the rotor on the electromagnetic force resultants.…”

Section: Nonlinear Electromagnetic Forces Of Amb Systemmentioning

confidence: 99%

“…They indicated that the steady state solutions lose their stability by either the saddle node bifurcation or the Hopf bifurcation. Furthermore, Ji and Handen [13] investigated nonlinear oscillations, saddle-node and Hopf bifurcations of a rotor-AMB system in the case of primary and internal resonances.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing with Quadratic and Cubic Terms and Time-Varying Stiffness

Zhang

Zhan

2005

Nonlinear Dyn

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In this paper, we use the asymptotic perturbation method to investigate nonlinear oscillations and chaotic dynamics in a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time varying in a periodic form. Because of considering the weight of the rotor, the formulation on the electromagnetic force resultants includes the quadratic and cubic nonlinearities. The resulting dimensionless equations of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions are a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found that there exist period-3, period-4, period-6, period-7, period-8, quasiperiodic and chaotic modulated amplitude oscillations in the rotor-AMB system with the time-varying stiffness. It is seen from the numerical results that there are the phenomena of the multiple solutions and the soft-spring type and the hardening-spring type in nonlinear frequency-response curves for the rotor-AMB system. The parametric excitation, or the time-varying stiffness produced by the PD controller is considered to be a controlling force which can control the chaotic response in the rotor-AMB system to a period n motion.

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Section: Stability Analysis Of Steady State Solutionsmentioning

confidence: 99%

“…Equation (7) does not have Z 2 ⊕ Z 2 symmetry. Other papers [10][11][12][13] do not consider the influence of the weight of the rotor on the electromagnetic force resultants.…”

Section: Nonlinear Electromagnetic Forces Of Amb Systemmentioning

confidence: 99%

See 1 more Smart Citation

Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing with Quadratic and Cubic Terms and Time-Varying Stiffness

Zhang

Zhan

2005

Nonlinear Dyn

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“…which describes the relationship among the parameters for Hopf bifurcation which is expected to lead to amplitude modulation of steady state response [16,17] . Therefore it is possible to avoid Hopf bifurcation for those amplitude values satisfying m 2 = 0 if the following inequality is held:…”

Section: Stability and Bifurcation Of Periodic Responsesmentioning

confidence: 99%

Response of parametrically excited Duffing-van der Pol oscillator with delayed feedback

et al. 2006

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Abstract:The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance. It is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node (SN) or Hopf bifurcation (HB) as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.

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“…For magnetic bearing systems, much work has been done on the subject of non-linear modelling, stability and bifurcation [8][9][10][11][12][13][14]. In this paper, the influence of inevitable time delays on the linear stability of trivial equilibrium will be studied, and the critical length of time delay will be determined.…”

Section: Introductionmentioning

confidence: 99%