Mathematical modeling and numerical computation for the vibration of a magnetostrictive actuator

Mathematical modeling to analyze the vibration problem of a Terfenol-D actuator is presented, using Hamilton's variational principle. The total kinetic, strain and magnetic energy stored in the rod is expressed as a function of the displacement. The material constitutive relation of the magnetostrictive rod is assumed to be cubic nonlinear. Both the governing equation and the boundary condition derived are of time variable coefficient. A numerical approach which combines the finite difference method with the transfer matrix method is proposed for solving the excited vibration problem of a magnetostrictive rod. By discretizing the displacement in space and making the finite difference formulation, the nodal displacement is obtained in terms of a system of linear second-order ordinary differential equations (ODEs), which can be subsequently transformed into a system of linear first-order ODEs in the time domain. The time domain within a period is then discretized with the numerical solution being expressed by the transfer matrix method. As a numerical example, the vibration of a Terfenol-D rod excited by a harmonic current is analyzed. The numerical results show that the induced displacement in the rod is periodic, with its frequency being roughly twice that of the exciting current. Such a double frequency effect has been observed in experiments. The stress behavior and the peak displacement within the rod are also numerically analyzed, with an emphasis on the effect of the involved magnetostrictive and magnetoelastic parameters. The good agreement between the numerical results and the experimental data available in the literature verifies the validity of the present modeling and method.