Influence of non-circular cross section shapes on torsional vibration of a micro-rod based on modified couple stress theory

Abstract Torsional micro-rods are key components in many micro-sensors and micro-resonators in aviation industry but in the literature proposed models have usually circular cross section which can be a limitation. In the torsional vibrations of micro-structures with non-circular cross section shape, the displacement fields and governing equations are completely dependent on the cross section shape of the structure which makes the analysis procedure more complicated. The aim of this study is to demonstrate the impact of geometrical parameters of the micro-rod especially its cross section shape, the type of medium and the presence of the external fields such as magnetic field in the torsional vibration behavior of the micro-rod. For this purpose, using the Hamilton's principle, the governing equation of the torsional vibration of the micro-rod with non-circular cross section embedded on elastic medium in the presence of a magnetic field is derived. To show the effects of size dependency of the micro-rod, modified couple stress theory is used. For instance, elliptical and rectangular cross section shapes with the same area are chosen for a micro-rod. After calculating the natural frequency of the micro-rod by the Galerkin method, the effect of cross section shape on the non-dimensional natural frequency of the micro-rod is investigated for changes in the aspect ratio of the cross section. In addition, the influence of length scale parameter, stiffness of torsional medium and magnetic field permeability on the natural frequency of the micro-rod are evaluated. The obtained results demonstrate that the natural frequency of the micro-rod is completely affected by the shape and aspect ratio of the cross section. These results can be useful in the micro-structure design stage. Also, a comparison is made between the obtained analytical results and those of semi-analytical differential transform method (DTM).