Mathematical modeling and numerical computation for the vibration of a magnetostrictiveactuator

Abstract

Mathematical modeling to analyze the vibration problem of a Terfenol-D actuator ispresented, using Hamilton’s variational principle. The total kinetic, strain and magneticenergy stored in the rod is expressed as a function of the displacement. The materialconstitutive relation of the magnetostrictive rod is assumed to be cubic nonlinear. Boththe governing equation and the boundary condition derived are of time variablecoefficient. A numerical approach which combines the finite difference method with thetransfer matrix method is proposed for solving the excited vibration problem of amagnetostrictive rod. By discretizing the displacement in space and making thefinite difference formulation, the nodal displacement is obtained in terms of asystem of linear second-order ordinary differential equations (ODEs), which can besubsequently 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 solutionbeing expressed by the transfer matrix method. As a numerical example, thevibration of a Terfenol-D rod excited by a harmonic current is analyzed. Thenumerical results show that the induced displacement in the rod is periodic, withits frequency being roughly twice that of the exciting current. Such a doublefrequency effect has been observed in experiments. The stress behavior and the peakdisplacement within the rod are also numerically analyzed, with an emphasison the effect of the involved magnetostrictive and magnetoelastic parameters.The good agreement between the numerical results and the experimental dataavailable in the literature verifies the validity of the present modeling and method.