Analysis of nonlinear free vibration of a beam with magnetic shape memory alloy elements

Abstract

In this study, nonlinear vibration of a clamped–clamped beam containing magnetic shape memory alloy is investigated through combining the constitutive relations of magnetic shape memory alloy and the large deformation response of Euler–Bernoulli beam. Due to the presence of moderately large strains, the effect of mid-plane stretching is taken into consideration. The magnetic shape memory alloy elements are bounded on the top surfaces in the clamped roots of the beam. Since magnetic shape memory alloy units should be always under compressive stress during the vibration, an appropriate compressive pre-strain is applied in magnetic shape memory alloy elements. In order to derive the governing equation of motion, Hamilton’s principle is utilized, and Galerkin’s method is applied in order to numerically solve the set of nonlinear equations. Variation of the strain in magnetic shape memory alloy elements during the beam vibrations makes it necessary to consider the influence of magnetic shape memory alloy on the beam response as well as the damping effects. Vibration-induced deformation in the beam is transferred to the magnetic shape memory alloys causing reorientation in them. It is shown that as a result of hysteresis behavior of magnetic shape memory alloys, the total energy of the beam dissipates until the amplitude of deformation is low enough that reorientation may no longer be induced in the magnetic shape memory alloys. A detailed study is carried out to investigate the effects of magnetic shape memory alloy elements on the damping as well as the natural frequencies of the beam. The influence of the initial conditions, magnetic field, number of magnetic shape memory alloy elements, and also the pre-strain in magnetic shape memory alloy is numerically investigated.