Comparative study on free vibration analysis of rotating bi-directional functionally graded beams using multiple beam theories with uncertainty considerations

The present study investigates the free vibration behavior of rotating beams made of functionally graded materials (FGMs) with a tapered geometry. The material properties of the beams are characterized by an exponential distribution model. The stiffness and mass matrices of the beams are derived using the principle of virtual energy. These matrices are then evaluated using three different beam theories: Bernoulli-Euler (BE) or Classical Beam Theory (CBT), Timoshenko (T) or First-order Shear Deformation Theory (FSDT), and Reddy (R) or Third-order Shear Deformation Theory (TSDT). Additionally, the study incorporates uncertainties in the model parameters, including rotational velocity, beam material properties, and material distribution. The mean-centered second-order perturbation method is employed to account for the randomness of these properties. To ensure the robustness and accuracy of the probabilistic framework, numerical examples are presented, and the results are compared with those obtained through the Monte Carlo simulation technique. The investigation explores the impact of critical parameters, including material distribution, taper ratios, aspect ratio, hub radius, and rotational speed, on the natural frequencies of the beams is explored within the scope of this investigation. The outcomes are compared not only with previously published research findings but also with the results of 3-Dimensional Finite Element (3D-FE) simulations conducted using ANSYS to validate the model's effectiveness. The comparisons demonstrate a strong agreement across all evaluations. Specifically, it is observed that for thick beams, the results obtained from FSDT and TSDT exhibit a greater agreement with the 3D-FE simulations compared to CBT. It is shown that the coefficient of variation (C.O.V.) of first mode eigenvalue of TSDT, FSDT and CBT are approximately identical for random rotational velocity and discernible deviations are noted in CBT compared to FSDT and TSDT in the case of random material properties. The findings suggest that TSDT outperforms FSDT by eliminating the need for a shear correction coefficient, thereby establishing its superiority in accurately predicting the natural frequencies of rotating, tapered beams composed of FGMs.