ARTICLE
Magneto-elastic internal resonance of an axially moving conductive beam in the magnetic field
Jie Wang 1, 2
,  
Yuda Hu 1
,  
Yu Su 2
,  
Liangfei Gong 3
,  
 
 
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1
Key Laboratory of Mechanical Reliability for Heavy Equipments and Large Structures of Hebei Province, Yanshan University, Qinhuangdao, P.R. China
2
Department of Mechanics, School of Aerospace Engineering, Beijing Institute of Technology, Beijing, P.R. China
3
State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing, P.R. China
Publish date: 2019-01-20
Submission date: 2018-03-10
Acceptance date: 2018-09-02
 
Journal of Theoretical and Applied Mechanics 2019;57(1):179–191
KEYWORDS
ABSTRACT
The Hamiltonian principle is applied to the nonlinear vibration equation of an axially moving conductive beam in the magnetic field with consideration of the axial velocity, axial tension, electromagnetic coupling effect and complex boundary conditions. Nonlinear vibration characteristics of the free vibrating beam under 1:3 internal resonances are studied based on our approach. For beams with one end fixed and the other simply supported, the nonlinear vibration equation is dispersed by the Galerkin method, and the vibration equations are solved by the multiple-scales method. As a result, the coupled relations between the first-order and second-order vibration modes are obtained in the internal resonance system. Firstly, the influence of initial conditions, axial velocity and the external magnetic field strength on the vibration modes is analysed in detail. Secondly, direct numerical calculation on the vibration equations is carried out in order to evaluate the accuracy of the perturbation approach. It is found that through numerical calculations, in the undamped system, the vibration modes are more sensitive to the initial value of vibration amplitude. The amplitude changes of the first-order and second-order modes resulting from the increase of the initial amplitude value of the vibration modes respectively are very special, and present a “reversal behaviour”. Lastly, in the damped system, the vibration modes exhibit a trend of coupling attenuation with time. Its decay rate increases when the applied magnetic field strength becomes stronger.
 
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