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.
In this paper, analytical couple-stress solution for size-dependent large-amplitude vibrations of FG (functionally-graded) taperednanobeams is presented. Using the modified couple-stress theory, the small scale effects are accounted for. Employing the Homotopy-Pade Analysis Method, efficient and accurate analytical expressions for the deflection and non-linear frequencies of the both single and double tapered nanobeam are presented. Very good agreement is observed between the present work results and some available results reported in the literature. This study may be helpful to investigate the size-dependent mechanical properties of MEMS\NEMS. Therefore, the proposed analytical solution can be used as an efficient tool for the material or geometrical parametric studies of small scale devices consisting of beams for their design and optimization which involves a large number of simulations.
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