Magnetoelastic combined resonance and stability analysis of a ferromagnetic circular plate in alternating magnetic field

doi:10.1007/s10483-019-2496-7

Applied Mathematics and Mechanics (English Edition) ›› 2019, Vol. 40 ›› Issue (7): 925-942.doi: https://doi.org/10.1007/s10483-019-2496-7

Magnetoelastic combined resonance and stability analysis of a ferromagnetic circular plate in alternating magnetic field

Yuda HU^1,2, Bingbing MA^1,2

1. School of Civil Engineering and Mechanics, Yanshan University, Qinhuangdao 066004, Hebei Province, China;
2. Hebei Key Laboratory of Mechanical Reliability for Heavyquipment and Large Structures, Yanshan University, Qinhuangdao 066004, Hebei Province, China

收稿日期:2018-10-31 修回日期:2018-12-17 出版日期:2019-07-01 发布日期:2019-07-01
通讯作者: Yuda HU E-mail:huyuda03@163.com
基金资助:
Project supported by the National Natural Science Foundation of China (No. 11472239)

Magnetoelastic combined resonance and stability analysis of a ferromagnetic circular plate in alternating magnetic field

Yuda HU^1,2, Bingbing MA^1,2

1. School of Civil Engineering and Mechanics, Yanshan University, Qinhuangdao 066004, Hebei Province, China;
2. Hebei Key Laboratory of Mechanical Reliability for Heavyquipment and Large Structures, Yanshan University, Qinhuangdao 066004, Hebei Province, China

Received:2018-10-31 Revised:2018-12-17 Online:2019-07-01 Published:2019-07-01
Contact: Yuda HU E-mail:huyuda03@163.com
Supported by:
Project supported by the National Natural Science Foundation of China (No. 11472239)

摘要/Abstract

摘要： The nonlinear combined resonance problem of a ferromagnetic circular plate in a transverse alternating magnetic field is investigated. On the basis of the deformation potential energy, the strain potential energy, and the kinetic energy of the circular plate, the Hamilton principle is used to induce the magnetoelastic coupling transverse vibration dynamical equation of the ferromagnetic circular plate. Based on the basic electromagnetic theory, the expressions of the magnet force and the Lorenz force of the circular plate are presented. A displacement function satisfying clamped-edge combined with the Galerkin method is used to derive the Duffing vibration differential equation of the circular plate. The amplitude-frequency response equations of the system under various combined resonance forms are obtained by means of the multi-scale method, and the stability of the steady-state solutions is analyzed according to the Lyapunov theory. Through examples, the amplitude-frequency characteristic curves with different parameters, the amplitude of resonance varying with magnetic field intensity and excitation force, and the time-course response diagram, phase diagram, Poincaré diagram of the system vibration are plotted, respectively. The effects of different parameters on the amplitude and stability of the system are discussed. The results show that the electromagnetic parameters have a significant effect on the multi-valued attribute and stability of the resonance solutions, and the system may exhibit complex nonlinear dynamical behavior including multi-period and quasi-periodic motion.

关键词: detonation wave, singular perturbation method, ratio of specific heat, combined resonance, magnetoelasticity, ferromagnetic circular plate, alternating magnetic field, multiscale method

Abstract: The nonlinear combined resonance problem of a ferromagnetic circular plate in a transverse alternating magnetic field is investigated. On the basis of the deformation potential energy, the strain potential energy, and the kinetic energy of the circular plate, the Hamilton principle is used to induce the magnetoelastic coupling transverse vibration dynamical equation of the ferromagnetic circular plate. Based on the basic electromagnetic theory, the expressions of the magnet force and the Lorenz force of the circular plate are presented. A displacement function satisfying clamped-edge combined with the Galerkin method is used to derive the Duffing vibration differential equation of the circular plate. The amplitude-frequency response equations of the system under various combined resonance forms are obtained by means of the multi-scale method, and the stability of the steady-state solutions is analyzed according to the Lyapunov theory. Through examples, the amplitude-frequency characteristic curves with different parameters, the amplitude of resonance varying with magnetic field intensity and excitation force, and the time-course response diagram, phase diagram, Poincaré diagram of the system vibration are plotted, respectively. The effects of different parameters on the amplitude and stability of the system are discussed. The results show that the electromagnetic parameters have a significant effect on the multi-valued attribute and stability of the resonance solutions, and the system may exhibit complex nonlinear dynamical behavior including multi-period and quasi-periodic motion.

Key words: detonation wave, singular perturbation method, ratio of specific heat, ferromagnetic circular plate, multiscale method, magnetoelasticity, combined resonance, alternating magnetic field

中图分类号:

Yuda HU, Bingbing MA. Magnetoelastic combined resonance and stability analysis of a ferromagnetic circular plate in alternating magnetic field[J]. Applied Mathematics and Mechanics (English Edition), 2019, 40(7): 925-942.

参考文献

[1] HASANYAN, D. J. and PILIPOSYAN, G. T. Modelling and stability of magnetosoft ferromagnetic plates in a magnetic field. Proceedings of the Royal Society A, 457, 2063-2077(2001)
[2] ZHOU, Y. H. and MIYA, K. A theoretical prediction of in crease of natural frequency to ferromagnetic plates under in plane magnetic fields. Journal of Sound and Vibration, 222(1), 49-64(1999)
[3] DONG, C. Y. Vibration of electro-elastic versus magneto-elastic circular/annular plates using the Chebyshev-Ritz method. Journal of Sound and Vibration, 317(1/2), 219-235(2008)
[4] BAGDOEV, A. G. and VARDANYAN, A. V. Analytical and numerical studies of free vibration frequencies of ferromagnetic plates with arbitrary electric conductance in a transverse magnetic field in the 3D setting. Mechanics of Solids, 43(5), 808-814(2008)
[5] LIANG, W., SOH, A. K., and HU, R. L. Vibration analysis of a ferromagnetic plate subjected to an inclined magnetic field. International Journal of Mechanical Siciences, 4(49), 440-446(2007)
[6] DING, H., HUANG, L. L., MAO, X. Y., and CHEN, L. Q. Primary resonance of traveling viscoelastic beam under internal resonance. Applied Mathematics and Mechanics (English Edition), 38(1), 1-14(2017) https://doi.org/10.1007/s10483-016-2152-6
[7] LU, Q. S., TO, W., and HUANG, K. L. Dynamic stability and bifurcation of an alternating load and magnetic field excited magnetoelastic beam. Journal of Sound and Vibration, 181(5), 873-891(1995)
[8] WANG, X. Z. and LEE, J. S. Dynamic stability of ferromagnetic beam-plates with magnetoelastic interaction and magnetic damping in transverse magnetic fields. Journal of Engineering Mechanics, 132(4), 422-428(2006)
[9] WANG, X. Z. and LEE, J. S. Dynamic stability of ferromagnetic plate under transverse magnetic field and in-plane periodic compression. International Journal of Mechanical Sciences, 48(8), 889-898(2006)
[10] HU, Y. D. and WANG, T. Nonlinear free vibration of a rotating circular plate under the static load in magnetic field. Nonlinear Dynamics, 85(3), 1825-1835(2016)
[11] HU, Y. D., LI, Z., DU, G. J., and WANG, Y. N. Magneto-elastic combination resonance of rotating circular plate with varying speed under alternating load. International Journal of Structural Stability & Dynamics, 4, 1850032(2018)
[12] HU, Y. D., PIAO, J. M., and LI, W. Q. Magneto-elastic dynamics and bifurcation of rotating annular plate. Chinese Physics B, 26(9), 269-279(2017)
[13] HU, Y. D. and LI, W. Q. Study on primary resonance and bifurcation of a conductive circular plate rotating in air-magnetic fields. Nonlinear Dynamics, 93(2), 671-687(2018)
[14] ZHOU, Y. F. and WANG, Z. M. Transverse vibration characteristics of axially moving viscoelastic plate. Applied Mathematics and Mechanics (English Edition), 28(2), 209-218(2007) https://doi.org/10.1007/s10483-007-0209-1
[15] HU, Y. D., HU, P., and ZHANG, J. Z. Strongly nonlinear sub-harmonic resonance and chaotic motion of axially moving thin plate in magnetic field. Journal of Computational & Nonlinear Dynamics, 10(2), 021010(2015)
[16] HORIGUCHI, K. and SHINDO, Y. Bending tests and magneto-elastic analysis of ferritic stainless steel plate in a magnetic field. Transactions of the Japan Society of Mechanical Engineers, 64(621), 1296-1301(1998)
[17] CHEN, J. Y., DING, H. J., and HOU, P. F. Analytical solutions of simply supported magnetoelectroelastic circular plate under uniform loads. Journal of Zhejiang University-Science A (Applied Physics and Engineering), 4(5), 560-564(2003)
[18] YANG, W., PAN, H., ZHENG, D., and CAI, Q. An energy method for analyzing magnetoelastic buckling and bending of ferromagnetic plates in static magnetic fields. Journal of Applied Mechanics, 66(4), 913-917(1999)
[19] LI, Y. S., REN, J. H., and FENG, W. J. Bending of sinusoidal functionally graded piezoelectric plate under an in-plane magnetic field. Applied Mathematical Modelling, 47, 63-75(2017)
[20] ZHOU, Y. H., ZHENG, X. J., and MIYA, K. Magnetoelastic bending and snapping of ferromagnetic plates in oblique magnetic fields. Fusion Engineering & Design, 30(4), 325-337(1995)
[21] HARIK, I. E. and ZHENG, X. J. FE-FD model for magneto-elastic buckling of ferromagnetic plates. Computers and Structures, 61(6), 1115-1123(1996)
[22] ZHENG, X. J. and WANG, X. Z. Analysis of magnetoelastic interaction of rectangular ferromagnetic plates with nonlinear magnetization. International Journal of Solids and Structures, 38(48), 8641-8652(2001)
[23] HASANYAN, D. J., LIBRESCU, L., and AMBUR, D. R. Buckling and postbuckling of magnetoelastic flat plates carrying an electric current. International Journal of Solids & Structures, 43(16), 4971-4996(2006)
[24] ZHOU, Y. H. and ZHENG, X. J. Electromagnetic Solid Mechanics (in Chinese), Science Press, Beijing (1999)

Magnetoelastic combined resonance and stability analysis of a ferromagnetic circular plate in alternating magnetic field

Magnetoelastic combined resonance and stability analysis of a ferromagnetic circular plate in alternating magnetic field

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 14

编辑推荐

Metrics

本文评价

[1]	Chun WANG, Gaoxiang XIANG, Zonglin JIANG. Theoretical approach to one-dimensional detonation instability[J]. Applied Mathematics and Mechanics (English Edition), 2016, 37(9): 1231-1238.
[2]	谢春梅冯民富. New nonconforming finite element method for solving transient Naiver-Stokes equations[J]. Applied Mathematics and Mechanics (English Edition), 2014, 35(2): 237-258.
[3]	张燕林平司新辉. Perturbation solutions for asymmetric laminar flow in porous channel with expanding and contracting walls[J]. Applied Mathematics and Mechanics (English Edition), 2014, 35(2): 203-220.
[4]	刘玉锦盛万成. Generalized Riemann problem for gas dynamic combustion[J]. Applied Mathematics and Mechanics (English Edition), 2011, 32(8): 1079-1090.
[5]	李军陈予恕. Combined resonance of low pressure cylinder-generator rotor system with bending-torsion coupling[J]. Applied Mathematics and Mechanics (English Edition), 2011, 32(8): 957-972.
[6]	丁亮韩波刘家琦. A wavelet multiscale method for inversion of Maxwell equations[J]. Applied Mathematics and Mechanics (English Edition), 2009, 30(8): 1035-1044.
[7]	郭永辉;田宙;郝保田. IMPLICIT TVD SCHEMES APPLIED TO GAS-DROPLET DETONATION CALCULATION[J]. Applied Mathematics and Mechanics (English Edition), 2000, 21(6): 725-732.
[8]	周又和;王省哲;郑晓静. MAGNETOELASTIC BENDING AND STABILITY OF SOFT FERROMAGNETIC RECTANGULAR PLATES[J]. Applied Mathematics and Mechanics (English Edition), 1998, 19(7): 669-676.
[9]	程昌钧;杨骁. VNONLINEAR STABILITY ANALYSIS OF A CLAMPED ROD CARRYING ELECTRIC CURRENT[J]. Applied Mathematics and Mechanics (English Edition), 1997, 18(9): 825-834.
[10]	袁镒吾. INTERPOLATION PERTURBATION METHOD FORSOLVING NONLINEAR PROBLEMS[J]. Applied Mathematics and Mechanics (English Edition), 1997, 18(11): 1115-1123.
[11]	袁镒吾;刘又文. M.E.SHVEZ ITERATIVE METHOD AND ITS APPLICATION IN MECHANICS[J]. Applied Mathematics and Mechanics (English Edition), 1996, 17(7): 699-704.
[12]	袁镒吾. INTERPOLATION PERTURBATION METHOD FOR SOLVING THE BOUNDARY LAYER TYPE PROBLEMS[J]. Applied Mathematics and Mechanics (English Edition), 1996, 17(1): 91-98.
[13]	苟兴华;张发祥. MOTION EQUATIONS OF MULTILAYERED ELASTIC ELECTROCONDUCTIVE PLATES IN A MAGNETIC FIELD[J]. Applied Mathematics and Mechanics (English Edition), 1994, 15(1): 29-36.
[14]	袁镒吾. AN APPROXIMATE ANALYTICAL SOLUTION OF THE CYLINDRICAL DETONATION WAVE GENERATED BY THE LINEAR EXPLOSION[J]. Applied Mathematics and Mechanics (English Edition), 1993, 14(5): 437-441.