Keywords: magnetic field, axial force, Hamilton's principle, Galerkin's method, free vibration, forced vibration, natural frequency.

The development of electrical machinery, communications equipment and computers, which involve magnetic phenomena, play an important role in everyday life. In this study, the interactions among transverse magnetic fields, compressive axial forces, springs forces, and external transverse forces of the beam are investigated; the basic relation of magnetism adopted in this paper is given in [1]. Y.S. Shin, Wu and Chen [2] have studied the transient vibrations of a simply supported beam with axial loads and transverse magnetic fields. Moon and Pao [3] have presented vibration and parametric instability of a cantilevered beam-plate in a transverse magnetic field, and have provided the theoretical and experimented results. Ambartsumian [4] has written one of the reviews on magneto-elasticity. The buckling of a thin plate simply supported around its edges has been investigated by Moon [5], and magneto elastic buckling of beams and plates has been investigated by Van de Ven [6], Wallerstein and Peach [7]. Moon and Pao [8] have studied the buckling of a cantilever beam-plate in a transverse magnetic field, a mathematical model is proposed with distributed magnetic forces and torque in their study.

A beam system involving compressive axial load, transverse magnetic field, transverse force, springs force and damping effect is considered in this paper. The equation of motion is derived by the Hamilton's principle. The axial force, transverse magnetic force are assumed to be periodic functions of time, meanwhile two frequencies associated with axial force and oscillation transverse magnetic field are applied to the system. The relationship of the displacement versus time with the effect of magnetic force and axial force are determined from the modal equation by using the Runge-Kutta method.

Based on the assumption of the inextensible beam, the motion of the beam in the transverse magnetic field leads to a nonlinear damping effect that is proportional to the square of the amplitude, so the effect of the magnetic field is similar to the damping effect. Under stable situations, the more the transverse magnetic field increases, the more the displacement and the natural frequency of vibration decrease. As long as the parameters of the beam system are practical and reasonable, we can also investigate the behaviour of the micro beam system in magnetic field by the proposed approach in the present study.

1

J. Rreitz, F. Milford, R. Christy, "Foundations of Electromagnetic Theory", Addison-Wesley, 1993.

2

Y.S. Shin and G.Y. Wu, and J.S. Chen, "Transient Vibrations of a Simply-Supported Beam with Axial Loads and Transverse Magnetic Fields", J. of Mech. of Struc. & Mach., 26(2), 115-130, 1998.

3

F.C. and Moon and Y.H. Pao, "Vibration and dynamic instability of a beam-plate in a transverse magnetic field", J. Appl. Mech. 36, 92-100, 1969.

4

S.A. Ambartsumian, "Magneto-elasticity of thin plates and shells", J. Appl. Mech. Rev. 35, 1-5, 1982.

5

F.C. Moon, "The Mechanics of ferroelastic plates in a uniform magnetic field", J. Appl. Mech. 37, 153-158, 1970.

6

A.A.F. Van de Ven, "Magnetoelastic buckling of thin plates in a uniform transverse magnetic field", J. Elasticity. 8, 297-312, 1978. doi:10.1007/BF00130468

7

D.V. Wallerstein and M.O. Peach, "Magnetoelastic buckling of beams and thin plates of magnetically soft material", J. Appl. Mech. 39, 451-455, 1972.

8

F.C. Moon and Y.H. Pao, "Magnetoelastic buckling of a thin plate", J. Appl. Mech. 37, 53-58, 1968.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £135 +P&P)