Free vibration analysis of bidirectional-functionally graded and double-tapered rotating micro-beam in thermal environment using modified couple stress theory

Free vibration behavior of bidirectional-functionally graded, double-tapered rotating micro-beam is investigated. An improved mathematical model based on Timoshenko beam theory and modified couple stress theory is developed that includes the effects of geometric non-linearity, spin-softening, Coriolis acceleration and high operating temperature. The problem is formulated in two steps. In the first step, the problem involving time-invariant inertia force due to rotation of the beam with constant angular speed is formulated using minimum potential energy principle and the governing non-linear equations are solved employing an iterative algorithm. In the next step, the free vibration problem is formulated employing Hamilton's principle and using the tangent stiffness of the deformed configuration induced due to time-invariant inertial loading. The governing equations for free vibration are transformed to state-space to formulate an eigenvalue problem. The governing equations are solved by approximating the displacement fields following Ritz method. The model is successfully validated with the available results. Extensive sets of results are presented for the first two chord-wise and flapwise modes of vibration in non-dimensional speed versus frequency plane. The effects of different parameters such as size-dependent thickness, axial and thickness gradation indices, taperness parameters, hub parameter, length-thickness ratio, operating temperature and FGM composition are reported.