Journal of Theoretical
and Applied Mechanics

57, 1, pp. 179-191, Warsaw 2019
DOI: 10.15632/jtam-pl.57.1.179

Magneto-elastic internal resonance of an axially moving conductive beam in the magnetic field

Jie Wang, Yuda Hu, Yu Su, Liangfei Gong, Qingnan Zhang
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.
Keywords: magneto-elastic, conductive beam, internal resonance, axially moving, multiple scales


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