# A brief review of dynamic instability of a beam/plate in the magnetic field

## Main Article Content

## Abstract

In this review, the fundamental results concerning the dynamic instability of a beam/plate structure in the magnetic field have been presented, that have been made over the last two decades, many of which are related to the mechanical model and responses of the magneto-elastic system. The review shows the basis equation of motion of a beam/plate in an oscillating magnetic field and touches on the dynamic instability behavior of a beam/plate system which is the isotropic material and composites made of the piezoelectric/ piezomagnetic materials.

## Article Details

How to Cite

WU, Guan-Yuan.
A brief review of dynamic instability of a beam/plate in the magnetic field.

**Quarterly Physics Review**, [S.l.], v. 3, n. 3, oct. 2017. ISSN 2572-701X. Available at: <https://journals.ke-i.org/index.php/qpr/article/view/1453>. Date accessed: 22 july 2018.
Section

Review Articles

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## References

[1] F.C. Moon, Y.H. Pao, Magnetoelastic buckling of a thin plate, J. Appl. Mech. 35 (1968) 53–58.

[2] F.C. Moon, Y.H. Pao, Vibration and dynamic instability of a beam-plate in a transverse magnetic field, J. Appl. Mech. 35 (1969) 92–100.

[3] J.M. Dalrymple, M.O. Peach, G.L. Viegelahn, Magnetoelastic buckling : Theory versus experiment. Exp. Mech. 16 (1976) 26–31.

[4] J.M. Dalrymple, M.O. Peach, G.L. Viegelahn, Edge effect influence on magnetoelastic buckling of rectangular plates, J. Appl. Mech. 44 (1977) 305–310.

[5] K. Miya, K. Hara, Y. Tabata, Experimental and theoretical study on magnetoelastic of a ferromagnetic cantilevered beam-plate, J. Appl. Mech. 45 (1978) 355–360.

[6] K. Miya, T. Takagi, Y. Ando, Finite element analysis of magnetoelastic buckling of a ferromagnetic beam-plate, J. Appl. Mech. 47 (1980) 377–382.

[7] M.O. Peach, N.S. Christopherson, J.M. Dalrymple, G.L. Viegelahn., Magnetoelastic buckling: why theory and experiment disagree. Exp. Mech. 28 (1988) 65–69.

[8] N.S. Christopherson, M.O. Peach, J.M. Dalrymple, Magnetoelastic deformation Experiment: Bending of a thin plate. Exp. Mech. 29 (1989) 432–436.

[9] A. C. Eringen, Theory of electromagnetic elastic plates. Int. J. Eng. Sci. 27 (1989) 363–375.

[10] E. Pan, Exact solution for simply supported and multilayered magneto-electro-elastic plates, J. Appl. Mech. 68 (2001) 608–618.

[11] E. Pan, P.R. Heyliger, Free vibration of simply supported and multilayered magneto-electro-elastic plates, J. Sound Vib. 252 (2002) 429–442.

[12] A.B. Temnykh, R.V.E. Lovelace, Electro-mechanical resonant magnetic field sensor, Nu. In. Meth. Phy. Res. A 484 (2002) 95–101.

[13] J. Chen, H. Chen, E. Pan, P.R. Heyliger, Modal analysis of magneto-electro-elastic plates using the state-vector approach, J. Sound Vib. 304 (2007) 722–734

[14] J. Chen, E. Pan, H. Chen, Wave propagation in magneto-electro-elastic multilayered plates, In. J. Solids. Stru. 44 (2007) 1073–1085.

[15] A. Milazzo, A one-dimensional model for dynamic analysis of generally layered magneto-electro-elastic beams, J. Sound Vib. 332 (2013) 465–483.

[16] S. Priya, R. Islam, S. Dong, D. Viehland, Recent advancements in magnetoelectric particulate and laminate composites, J. Elec. 19 (2007) 149–166.

[17] C.W. Nan, M. Bichurin, S. Dong, D. Viehland, G. Srinivasan, Multi ferroic magnetoelectric composites: historical perspective, status, and future directions, J. of Appl. Phys. 103 (2008) 031101.

[18] J.S. Lee, Destabilizing effect of magnetic damping in plate strip, J. Eng. Mech. 118 (1992) 161–173.

[19] Q.S. Lu, C.W.S. To, K.L. Huang, Dynamic stability and bifurcation an alternating load and magnetic field excited magnetoelastic beam, J. Sound Vib. 181 (1995) 873–891.

[20] Y.S. Shih, G.Y. Wu, J.S. Chen, Transient vibrations of a simply-supported beam with axial loads and transverse magnetic fields, Mech. Struc. Mach. 26 (1998) 115–130.

[21] G.Y. Wu, R. Tsai, Y.S. Shih, The analysis of dynamic stability and vibration motions of a cantilever beam with axial loads and transverse magnetic fields, J. Acoust. Soc. of ROC 4 (1997) 40–55.

[22]J.S. Bae, M.K. Kwak, D.J. Inman, Vibration suppression of a cantilever beam

using eddy current damper, J. Sound Vib. 284 (2005) 805–824.

[23] C.C. Chen, M.K. Yeh, Parameter instability of a beam under electromagnetic excitation, J. Sound Vib. 240 (2001) 747-764.

[24] B. Pratiher, S.K. Dwivedy, Parametric instability of a cantilever beam with magnetic field and periodic axial load, J. Sound Vib. 305 (2007) 904-917

[25] B. Pratiher, S.K. Dwivedy, Non-linear vibration of a single link viscoelastic Cartesian manipulator, In. J. Non. Mech. 43 (2008) 683-696.

[26] B. Pratiher, S.K. Dwivedy, Non-linear dynamics of a soft magneto-elastic Cartesian manipulator, In. J. Non. Mech. 44 (2009) 757-768.

[27] Y.H. Zhou, X.J. Zheng, A general expression of magnetic force for soft ferromagnetic plates in complex magnetic fields, In. J. Eng. Sci. 35 (1997) 1405–1407.

[28] Y.H. Zhou, K. Miya, A theoretical prediction of natural frequency of a ferromagnetic beam-plate with low susceptibility in an in-plane magnetic field, J. of Appl. Mech. 65 (1998) 121–126.

[29] Y.H. Zhou, Y.W. Gao, X.J. Zheng, Q. Jiang, Buckling and post-buckling of a ferromagnetic beam-plate induced by magnetoelastic interactions, Int. J. Non. Mech. 35 (2000) 1059–1065.

[30] X.J. Zheng, X.Z. Wang, Analysis of magnetoelastic interaction of rectangular ferromagnetic plates with nonlinear magnetization, In. Solids Struc. 38 (2001) 8641–8652.

[31] G.Y. Baghdasaryan, Z.N. Danoyan, M.A. Mikilyan , Issues of dynamics of conductive plate in a longitudinal magnetic field, In. J. Solids Struc. 50 (2013) 3339–3345.

[32] X. Wang, J.S. Lee, X. Zheng, Magneto-thermo-elastic instability of ferromagnetic plates in thermal and magnetic fields, In. J. Solids Struc. 40 (2003) 6125–6142

[33] G. Y. Wu, Transient vibration analysis of a pinned beam with transverse magnetic fields and thermal loads, J. Vib. Acoust. 127 (2005) 247–253.

[34] G. Y Wu, The analysis of dynamic instability on the large amplitude vibrations of a beam with transverse magnetic fields and thermal load, J. Sound Vib. 302 (2007) 167–177.

[35] G.Y. Wu, 2013, Non-linear vibration of bimaterial magneto-elastic cantilever beam with thermal loading, In. J. Non. Mech. 55 (2013) 10–18.

[36] Z. Jia, W. Liu, Y. Zhang, F. Wang, D. Guo, A nonlinear magnetomechanical coupling model of giant magnetostrictive thin films at low magnetic fields, Sen. Act. A 128 (2006) 158–164.

[37] Anandkumar R. Annigeri, N. Ganesan, S. Swarnamani, Free vibration behaviour of multiphase and layered magneto-electro-elastic beam, J. Sound Vib. 299 (2007) 44–63.

[38] G.Y. Zhou, Q.Wang, Magnetorheological elastomer-based smart sandwich beam with nonconductive skins, Smart Mat. Struc. 14 (2005) 1001–1009.

[39] G.Y. Zhou, Q. Wang, Use of magnetorheological elastomer in an adaptive sandwich beam with conductive skins. Part I: magnetoelasticloadsin conductive skins, In. J. Solids Struc. 43(2006) 5386–5402.

[40] G.Y. Zhou, Q. Wang, Use of magnetorheological elastomer in an adaptive sandwich beam with conductive skins. PartII: dynamic properties, In. J. Solids Struc. 43(2006) 5403–5420.

[41] B. Nayak, S.K. Dwivedy, K.S.R.K. Murthy, Dynamic analysis of magnetorheological elastomer-based sandwich beam with conductive skins under various boundary conditions, J. Sound Vib. 330 (2011) 1837–1859.

[42] B. Nayak, S.K. Dwivedy , K.S.R.K. Murthy, Dynamic stability of a rotating sandwich beam with magnetorheological elastomer core, Euro. J. Mech. A/Solids 47 (2014) 143–155

[43] B. Nayak, S.K. Dwivedy, K.S.R.K. Murthy, Multi-frequency excitation of magnetorheological elastomer-based sandwich beam with conductive skins, In. J. Non. Mech. 47 (2012) 448–460.

[44] T. Kong, D.X. Li, X. Wang, Thermo-magneto-dynamic stresses and perturbation of magnetic field vector in a non-homogeneous hollow cylinder , Appl. Math. Mod. 33 (2009) 2939–2950.

[45] L. Wang, W.B. Liu, H.-L. Dai, Dynamics and instability of current-carrying microbeams in a longitudinal magnetic field, Phy. E 66 (2015) 87–92.

[46] K. Kiani, Surface effect on free transverse vibrations and dynamic instability of current-carrying nanowires in the presence of alongitudinal magnetic field, Phy. Let. A 378 (2014) 1834–1840.

[47] K. Kiani, Stability and vibrations of doubly parallel current-carrying nano

[2] F.C. Moon, Y.H. Pao, Vibration and dynamic instability of a beam-plate in a transverse magnetic field, J. Appl. Mech. 35 (1969) 92–100.

[3] J.M. Dalrymple, M.O. Peach, G.L. Viegelahn, Magnetoelastic buckling : Theory versus experiment. Exp. Mech. 16 (1976) 26–31.

[4] J.M. Dalrymple, M.O. Peach, G.L. Viegelahn, Edge effect influence on magnetoelastic buckling of rectangular plates, J. Appl. Mech. 44 (1977) 305–310.

[5] K. Miya, K. Hara, Y. Tabata, Experimental and theoretical study on magnetoelastic of a ferromagnetic cantilevered beam-plate, J. Appl. Mech. 45 (1978) 355–360.

[6] K. Miya, T. Takagi, Y. Ando, Finite element analysis of magnetoelastic buckling of a ferromagnetic beam-plate, J. Appl. Mech. 47 (1980) 377–382.

[7] M.O. Peach, N.S. Christopherson, J.M. Dalrymple, G.L. Viegelahn., Magnetoelastic buckling: why theory and experiment disagree. Exp. Mech. 28 (1988) 65–69.

[8] N.S. Christopherson, M.O. Peach, J.M. Dalrymple, Magnetoelastic deformation Experiment: Bending of a thin plate. Exp. Mech. 29 (1989) 432–436.

[9] A. C. Eringen, Theory of electromagnetic elastic plates. Int. J. Eng. Sci. 27 (1989) 363–375.

[10] E. Pan, Exact solution for simply supported and multilayered magneto-electro-elastic plates, J. Appl. Mech. 68 (2001) 608–618.

[11] E. Pan, P.R. Heyliger, Free vibration of simply supported and multilayered magneto-electro-elastic plates, J. Sound Vib. 252 (2002) 429–442.

[12] A.B. Temnykh, R.V.E. Lovelace, Electro-mechanical resonant magnetic field sensor, Nu. In. Meth. Phy. Res. A 484 (2002) 95–101.

[13] J. Chen, H. Chen, E. Pan, P.R. Heyliger, Modal analysis of magneto-electro-elastic plates using the state-vector approach, J. Sound Vib. 304 (2007) 722–734

[14] J. Chen, E. Pan, H. Chen, Wave propagation in magneto-electro-elastic multilayered plates, In. J. Solids. Stru. 44 (2007) 1073–1085.

[15] A. Milazzo, A one-dimensional model for dynamic analysis of generally layered magneto-electro-elastic beams, J. Sound Vib. 332 (2013) 465–483.

[16] S. Priya, R. Islam, S. Dong, D. Viehland, Recent advancements in magnetoelectric particulate and laminate composites, J. Elec. 19 (2007) 149–166.

[17] C.W. Nan, M. Bichurin, S. Dong, D. Viehland, G. Srinivasan, Multi ferroic magnetoelectric composites: historical perspective, status, and future directions, J. of Appl. Phys. 103 (2008) 031101.

[18] J.S. Lee, Destabilizing effect of magnetic damping in plate strip, J. Eng. Mech. 118 (1992) 161–173.

[19] Q.S. Lu, C.W.S. To, K.L. Huang, Dynamic stability and bifurcation an alternating load and magnetic field excited magnetoelastic beam, J. Sound Vib. 181 (1995) 873–891.

[20] Y.S. Shih, G.Y. Wu, J.S. Chen, Transient vibrations of a simply-supported beam with axial loads and transverse magnetic fields, Mech. Struc. Mach. 26 (1998) 115–130.

[21] G.Y. Wu, R. Tsai, Y.S. Shih, The analysis of dynamic stability and vibration motions of a cantilever beam with axial loads and transverse magnetic fields, J. Acoust. Soc. of ROC 4 (1997) 40–55.

[22]J.S. Bae, M.K. Kwak, D.J. Inman, Vibration suppression of a cantilever beam

using eddy current damper, J. Sound Vib. 284 (2005) 805–824.

[23] C.C. Chen, M.K. Yeh, Parameter instability of a beam under electromagnetic excitation, J. Sound Vib. 240 (2001) 747-764.

[24] B. Pratiher, S.K. Dwivedy, Parametric instability of a cantilever beam with magnetic field and periodic axial load, J. Sound Vib. 305 (2007) 904-917

[25] B. Pratiher, S.K. Dwivedy, Non-linear vibration of a single link viscoelastic Cartesian manipulator, In. J. Non. Mech. 43 (2008) 683-696.

[26] B. Pratiher, S.K. Dwivedy, Non-linear dynamics of a soft magneto-elastic Cartesian manipulator, In. J. Non. Mech. 44 (2009) 757-768.

[27] Y.H. Zhou, X.J. Zheng, A general expression of magnetic force for soft ferromagnetic plates in complex magnetic fields, In. J. Eng. Sci. 35 (1997) 1405–1407.

[28] Y.H. Zhou, K. Miya, A theoretical prediction of natural frequency of a ferromagnetic beam-plate with low susceptibility in an in-plane magnetic field, J. of Appl. Mech. 65 (1998) 121–126.

[29] Y.H. Zhou, Y.W. Gao, X.J. Zheng, Q. Jiang, Buckling and post-buckling of a ferromagnetic beam-plate induced by magnetoelastic interactions, Int. J. Non. Mech. 35 (2000) 1059–1065.

[30] X.J. Zheng, X.Z. Wang, Analysis of magnetoelastic interaction of rectangular ferromagnetic plates with nonlinear magnetization, In. Solids Struc. 38 (2001) 8641–8652.

[31] G.Y. Baghdasaryan, Z.N. Danoyan, M.A. Mikilyan , Issues of dynamics of conductive plate in a longitudinal magnetic field, In. J. Solids Struc. 50 (2013) 3339–3345.

[32] X. Wang, J.S. Lee, X. Zheng, Magneto-thermo-elastic instability of ferromagnetic plates in thermal and magnetic fields, In. J. Solids Struc. 40 (2003) 6125–6142

[33] G. Y. Wu, Transient vibration analysis of a pinned beam with transverse magnetic fields and thermal loads, J. Vib. Acoust. 127 (2005) 247–253.

[34] G. Y Wu, The analysis of dynamic instability on the large amplitude vibrations of a beam with transverse magnetic fields and thermal load, J. Sound Vib. 302 (2007) 167–177.

[35] G.Y. Wu, 2013, Non-linear vibration of bimaterial magneto-elastic cantilever beam with thermal loading, In. J. Non. Mech. 55 (2013) 10–18.

[36] Z. Jia, W. Liu, Y. Zhang, F. Wang, D. Guo, A nonlinear magnetomechanical coupling model of giant magnetostrictive thin films at low magnetic fields, Sen. Act. A 128 (2006) 158–164.

[37] Anandkumar R. Annigeri, N. Ganesan, S. Swarnamani, Free vibration behaviour of multiphase and layered magneto-electro-elastic beam, J. Sound Vib. 299 (2007) 44–63.

[38] G.Y. Zhou, Q.Wang, Magnetorheological elastomer-based smart sandwich beam with nonconductive skins, Smart Mat. Struc. 14 (2005) 1001–1009.

[39] G.Y. Zhou, Q. Wang, Use of magnetorheological elastomer in an adaptive sandwich beam with conductive skins. Part I: magnetoelasticloadsin conductive skins, In. J. Solids Struc. 43(2006) 5386–5402.

[40] G.Y. Zhou, Q. Wang, Use of magnetorheological elastomer in an adaptive sandwich beam with conductive skins. PartII: dynamic properties, In. J. Solids Struc. 43(2006) 5403–5420.

[41] B. Nayak, S.K. Dwivedy, K.S.R.K. Murthy, Dynamic analysis of magnetorheological elastomer-based sandwich beam with conductive skins under various boundary conditions, J. Sound Vib. 330 (2011) 1837–1859.

[42] B. Nayak, S.K. Dwivedy , K.S.R.K. Murthy, Dynamic stability of a rotating sandwich beam with magnetorheological elastomer core, Euro. J. Mech. A/Solids 47 (2014) 143–155

[43] B. Nayak, S.K. Dwivedy, K.S.R.K. Murthy, Multi-frequency excitation of magnetorheological elastomer-based sandwich beam with conductive skins, In. J. Non. Mech. 47 (2012) 448–460.

[44] T. Kong, D.X. Li, X. Wang, Thermo-magneto-dynamic stresses and perturbation of magnetic field vector in a non-homogeneous hollow cylinder , Appl. Math. Mod. 33 (2009) 2939–2950.

[45] L. Wang, W.B. Liu, H.-L. Dai, Dynamics and instability of current-carrying microbeams in a longitudinal magnetic field, Phy. E 66 (2015) 87–92.

[46] K. Kiani, Surface effect on free transverse vibrations and dynamic instability of current-carrying nanowires in the presence of alongitudinal magnetic field, Phy. Let. A 378 (2014) 1834–1840.

[47] K. Kiani, Stability and vibrations of doubly parallel current-carrying nano