Principal Resonance Bifurcation and Chaos of Rotating Annular Plates in Magnetic Fields

PIAO Jiang-min; HU Yu-da

doi:10.21656/1000-0887.370141

Volume 37 Issue 11

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PIAO Jiang-min, HU Yu-da. Principal Resonance Bifurcation and Chaos of Rotating Annular Plates in Magnetic Fields[J]. Applied Mathematics and Mechanics, 2016, 37(11): 1181-1197. doi: 10.21656/1000-0887.370141

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Principal Resonance Bifurcation and Chaos of Rotating Annular Plates in Magnetic Fields

doi: 10.21656/1000-0887.370141

PIAO Jiang-min^1,2,
HU Yu-da^1,2

¹College of Civil Engineering and Mechanics, Yanshan University, Qinhuangdao, Hebei 066004, P.R.China;²Key Laboratory of Mechanical Reliability for Heavy Equipments and Large Structures of Hebei Province, Yanshan University, Qinhuangdao, Hebei 066004, P.R.China

Funds: The National Natural Science Foundation of China(11472239)

Received Date: 2016-05-10
Rev Recd Date: 2016-06-26
Publish Date: 2016-11-15

Abstract

Abstract

The magneto-elastic principal resonance bifurcation and chaos of rotating annular plates in magnetic fields were studied. Based on the expressions of kinetic energy, strain energy and virtual work done by external forces and electromagnetic forces, the nonlinear vibration equations of a rotating annular plate in magnetic field were deduced with the Hamiltonian principle. The Galerkin method with the Bessel mode shape functions was used to achieve the ordinary differential vibration equations. The static bifurcation equations and corresponding transition sets with the physical parameters as the bifurcation control parameters were achieved by means of the method of multiple scales. Finally, the critical conditions for the break of the heteroclinic orbits were obtained under the conditions of fixed outer boundary and free inner boundary with the Mel’nikov method. Moreover, the global bifurcation diagrams under the external forces as the control parameters and other response diagrams with specified control parameters were drawn. The results show that the magnetic field deters the occurence of multi-value phenomena. With the decreasing of the external force frequency, the rotating speed and the magnetic induction, and with the increasing of the external force, the system’s heteroclinic orbits break more easily, meanwhile chaos or almost periodic motion of the system is induced.
- magneto-elastic,
- annular plate,
- bifurcation,
- chaos,
- Bessel function,
- singularity theory,
- Mel’nikov method

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References(16)

References

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