Magnetoelastic combined resonance and stability analysis of a ferromagnetic circular plate in alternating magnetic field
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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, Poincar′e 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.
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, Poincar′e 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.
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, Poincar′e 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.
引文
[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)
[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)