A tangent stiffness–based approach to study free vibration of shear-deformable functionally graded material rotating beam through a geometrically non-linear analysis

Abstract

An improved mathematical model to study the free vibration behavior of rotating functionally graded material beam is presented, considering non-linearity up to second order for the normal and transverse shear strains. The study is carried out considering thermal loading due to uniform temperature rise and using temperature-dependent material properties. Power law variation is assumed for through-thickness symmetric functional gradation of ceramic–metal functionally graded beam. The effects of shear deformation and rotary inertia are considered in the frame-work of Timoshenko beam theory. First, the rotating beam configuration under time-invariant centrifugal loading and thermal loading is obtained through a geometrically non-linear analysis, employing minimum total potential energy principle. Then, the free vibration analysis of the deformed beam is performed using the tangent stiffness of the deformed beam configuration, and employing Hamilton’s principle. The Coriolis effect is considered in the free vibration problem, and the governing equations are transformed to the state-space to obtain the eigenvalue problem. The solution of the governing equations is obtained following Ritz method. The validation is performed with the available results, and also with finite element software ANSYS. The analysis is carried out for clamped-free beam and for clamped–clamped beam with immovably clamped ends. The results for the first two modes of chord-wise and flap-wise vibration in non-dimensional speed-frequency plane are presented for different functionally graded material compositions, material profile parameters, root offset parameters and operating temperatures.