Abstract
The nonaxisymmetric magnetoelastic nonlinear coupling free vibration study is performed for a conductive thin annular plate in the nonuniform toroidal magnetic field generated by a long straight current carrying wire in this article. From the electromagnetic theory, expressions for the magnetic field, electromagnetic force and torque acting on the plate are deduced. According to Hamilton principle, nonaxisymmetric magnetoelastic nonlinear vibration equation is derived. The displacement functions for plate under three different boundary conditions are solved, which is combined with Galerkin integral method for derivation of nondimensional coupling nonlinear differential equations. The method of multiple scales is introduced to solve the coupling equations and achieve the second-approximation analytical solution, and then, expressions for the first three mode nondimensional natural frequencies of plate are obtained. In numerical examples, diagrams of electromagnetic characteristics and the first three frequencies under magnetic field and modal coupling effect are presented, which shows the influence of different parameters, e.g., current intensity, plate size and time on natural frequencies and electromagnetic forces. The variation of system singularity stability is discussed, and the obtained analytical results are also validated. The results indicate that current, plate size and time parameters have obvious influence on natural frequencies, which also shows quite different variations under different boundaries. Additionally, initial conditions have significant effects on natural frequencies, which becomes more complicated under modal coupling effect. In nonaxisymmetric vibration case, electromagnetic forces show complicated changing rules along radial and circumferential directions. Furthermore, system equilibrium point will be changed by the induced nonuniform magnetic field.
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The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Funding
This work was supported by the National Natural Science Foundation of China (Grant No. 11472239 and Grant No. 12172321) and Hebei Provincial Natural Science Foundation of China (Grant No. A2020203007).
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Appendix
Appendix
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(1)
Expressions of \(E_{n1}\), \(E_{n2}\), \(E_{n3}\) and \(E_{n4}\) under C-F boundary
\(15R_{a}^{6} \mu - 63R_{a}^{4} R_{b}^{2} \mu + 72R_{a}^{3} R_{b}^{3} \mu - 25R_{a}^{2} R_{b}^{4} \mu + R_{b}^{6} \mu ))\).
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(2)
Expressions of \(E_{n1}\), \(E_{n2}\), \(E_{n3}\) and \(E_{n4}\) under C–C boundary
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Expressions of \(E_{n1}\), \(E_{n2}\), \(E_{n3}\) and \(E_{n4}\) under S–S boundary
$$ E_{01} = - (60R_{a}^{6} + 180R_{a}^{5} R_{b} - 180R_{a}^{4} R_{b}^{2} - 600R_{a}^{3} R_{b}^{3} - $$$$ 660R_{a}^{2} R_{b}^{4} + 372R_{a} R_{b}^{5} + 60R_{b}^{6} + 36R_{a}^{6} \mu + 98R_{a}^{5} R_{b} \mu - $$$$ 20R_{a}^{4} R_{b}^{2} \mu - 150R_{a}^{3} R_{b}^{3} \mu - 166R_{a}^{2} R_{b}^{4} \mu + 156R_{a} R_{b}^{5} \mu + $$$$ 46R_{b}^{6} \mu + 3R_{a}^{6} \mu^{2} + 8R_{a}^{5} R_{b} \mu^{2} - 2R_{a}^{4} R_{b}^{2} \mu^{2} - $$$$ 12R_{a}^{3} R_{b}^{3} \mu^{2} - 13R_{a}^{2} R_{b}^{4} \mu^{2} + 12R_{a} R_{b}^{5} \mu^{2} + 4R_{b}^{6} \mu^{2} )/ $$$$ R_{b}^{2} (40R_{a}^{4} + 80R_{a}^{3} R_{b} - 312R_{a}^{2} R_{b}^{2} + 80R_{a} R_{b}^{3} + 40R_{b}^{4} + $$$$ 14R_{a}^{4} \mu + 28R_{a}^{3} R_{b} \mu - 84R_{a}^{2} R_{b}^{2} \mu + 28R_{a} R_{b}^{3} \mu + $$$$ 14R_{b}^{4} \mu + R_{a}^{4} \mu^{2} + 2R_{a}^{3} R_{b} \mu^{2} - 6R_{a}^{2} R_{b}^{2} \mu^{2} + $$$$ 2R_{a} R_{b}^{3} \mu^{2} + R_{b}^{4} \mu^{2} )), $$$$ E_{02} = (20R_{a}^{7} + 60R_{a}^{6} R_{b} + 100R_{a}^{5} R_{b}^{2} - 420R_{a}^{4} R_{b}^{3} - $$$$ 1060R_{a}^{3} R_{b}^{4} + 244R_{a}^{2} R_{b}^{5} + 204R_{a} R_{b}^{6} - 60R_{b}^{7} + $$$$ + 22R_{a}^{7} \mu + 56R_{a}^{6} R_{b} \mu + 90R_{a}^{5} R_{b}^{2} \mu - 96R_{a}^{4} R_{b}^{3} \mu - $$$$ 306R_{a}^{3} R_{b}^{4} \mu \, + 132R_{a}^{2} R_{b}^{5} \mu + 98R_{a} R_{b}^{6} \mu + 4R_{a}^{7} \mu + $$$$ 2R_{a}^{7} \mu^{2} + 5R_{a}^{6} R_{b} \mu^{2} + 8R_{a}^{5} R_{b}^{2} \mu^{2} + 9R_{a}^{4} R_{b}^{3} \mu^{2} - $$$$ 26R_{a}^{3} R_{b}^{4} \mu^{2} + + 11R_{a}^{2} R_{b}^{5} \mu^{2} + 8R_{a} R_{b}^{6} \mu^{2} + R_{b}^{7} \mu^{2} )/ $$$$ (R_{b}^{3} \cdot (40R_{a}^{4} + 80R_{a}^{3} R_{b} - 312R_{a}^{2} R_{b}^{2} + 80R_{a} R_{b}^{3} + $$$$ 40R_{b}^{4} + 14R_{a}^{4} \mu + 28R_{a}^{3} R_{b} \mu - 84R_{a}^{2} R_{b}^{2} \mu + 28R_{a} R_{b}^{3} \mu + $$$$ 14R_{b}^{4} \mu + R_{a}^{4} \mu^{2} + 2R_{a}^{3} R_{b} \mu^{2} - 6R_{a}^{2} R_{b}^{2} \mu^{2} + $$$$ 2R_{a} R_{b}^{3} \mu^{2} + R_{b}^{4} \mu^{2} )), $$$$ E_{03} = - (48R_{a}^{7} - 24R_{a}^{6} R_{b} - 264R_{a}^{5} R_{b}^{2} - 192R_{a}^{4} R_{b}^{3} - $$$$ 240R_{a}^{3} R_{b}^{4} + 72R_{a} R_{b}^{6} + 54R_{a}^{7} \mu + 30R_{a}^{6} R_{b} \mu - $$$$ 74R_{a}^{5} R_{b}^{2} \mu - 48R_{a}^{4} R_{b}^{3} \mu - 46R_{a}^{3} R_{b}^{4} \mu + 58R_{a}^{2} R_{b}^{5} \mu + $$$$ 26R_{a} R_{b}^{6} \mu + 6R_{a}^{7} \mu^{2} + 3R_{a}^{6} R_{b} \mu^{2} - 8R_{a}^{5} R_{b}^{2} \mu^{2} - $$$$ 6R_{a}^{4} R_{b}^{3} \mu^{2} - 4R_{a}^{3} R_{b}^{4} \mu^{2} + 7R_{a}^{2} R_{b}^{5} \mu^{2} + 2R_{a} R_{b}^{6} \mu^{2} )/ $$$$ R_{b}^{3} \, \left( {40R_{a}^{4} + 80R_{a}^{3} R_{b} - } \right.312R_{a}^{2} R_{b}^{2} + 80R_{a} R_{b}^{3} + $$$$ 40R_{b}^{4} + 14R_{a}^{4} \mu + 28R_{a}^{3} R_{b} \mu - 84R_{a}^{2} R_{b}^{2} \mu + $$$$ 28R_{a} R_{b}^{3} \mu + 14R_{b}^{4} \mu + R_{a}^{4} \mu^{2} + 2R_{a}^{3} R_{b} \mu^{2} - $$$$ 6R_{a}^{2} R_{b}^{2} \mu^{2} + 2R_{a} R_{b}^{3} \mu^{2} + R_{b}^{4} \mu^{2} )), $$$$ E_{04} = (28R_{a}^{7} - 24R_{a}^{6} R_{b} - 184R_{a}^{5} R_{b}^{2} - 72R_{a}^{4} R_{b}^{3} - $$$$ 20R_{a}^{3} R_{b}^{4} + 32R_{a}^{2} R_{b}^{5} + 32R_{a}^{7} \mu - 66R_{a}^{5} R_{b}^{2} \mu + $$$$ 10R_{a}^{6} R_{b} \mu - 14R_{a}^{4} R_{b}^{3} \mu + 26R_{a}^{3} R_{b}^{4} \mu + 12R_{a}^{2} R_{b}^{5} \mu + $$$$ 4R_{a}^{7} \mu^{2} + R_{a}^{6} R_{b} \mu^{2} - 8R_{a}^{5} R_{b}^{2} \mu^{2} - 2R_{a}^{4} R_{b}^{3} \mu^{2} + $$$$ 4R_{a}^{3} R_{b}^{4} \mu^{2} + R_{a}^{2} R_{b}^{5} \mu^{2} )/(R_{b}^{3} (40R_{a}^{4} + 80R_{a}^{3} R_{b} - $$$$ 312R_{a}^{2} R_{b}^{2} + 80R_{a} R_{b}^{3} + 40R_{b}^{4} + 14R_{a}^{4} \mu + 28R_{a}^{3} R_{b} \mu $$$$ - 84R_{a}^{2} R_{b}^{2} \mu + 28R_{a} R_{b}^{3} \mu + 14R_{b}^{4} \mu + R_{a}^{4} \mu^{2} + $$$$ 2R_{a}^{3} R_{b} \mu^{2} - 6R_{a}^{2} R_{b}^{2} \mu^{2} + 2R_{a} R_{b}^{3} \mu^{2} + R_{b}^{4} \mu^{2} )). $$$$ E_{01} = E_{11} = E_{21} ,\;E_{02} = E_{12} = E_{22} ,\;E_{03} = E_{13} = E_{23} ,\;E_{04} = E_{14} = E_{24} . $$
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Hu, Y., Xu, H. Nonaxisymmetric magnetoelastic coupling natural vibration analysis of annular plates in an induced nonuniform magnetic field. Nonlinear Dyn 109, 657–687 (2022). https://doi.org/10.1007/s11071-022-07475-7
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DOI: https://doi.org/10.1007/s11071-022-07475-7