Application of the generalized differential quadrature method to study vibration and dynamic stability of tri-directional functionally graded beam under magneto-electro-elastic fields

Given some engineering structures such as high-speed rail and aerospace structures are often affected by multidirectional thermal loads, we propose tri-directional functionally graded materials (3D FGMs) to fabricate a magneto-electro-elastic (MEE) Timoshenko beam placed on the Pasternak foundation to investigate the free vibration and dynamic stability. Unlike traditional bi-directional (2D) and unidirectional (1D) FGMs, we suppose the physical properties continuously and smoothly vary along the longitudinal, thickness, and width orientations, which satisfy the requirements of tri-directional function graded design. Considering the coupling between the MEE fields and 3D FGMs, as well as the Timoshenko beam theory, one leads to the equation of motion with variable coefficients of the structures by virtue of Hamilton's principle. Given the closed-form solution of free vibration of the antisymmetric composites such as tri-directional FGMs structures is difficult to find, a powerful methodology named the generalized differential quadrature method (GDQM) is applied to numerically solve the governing equation for predicting the dynamic behaviors. Numerical simulations are carried out to study the influences of 3D FMGs indexes, magneto-electro-mechanical loading, and foundation parameters on the natural frequency and stable boundary in detail. It is found that the interaction between hinged-hinged and 3D FMGs abnormally affects the fundamental frequency and dynamic stability, and the beams with higher magneto-electro-elastic parameters or larger foundation parameters or a lower length-thickness ratio are more suscepti-ble to destabilizing for all end supports. The results emanate that the 3D FGMs indexes can provide much wider space in tuning/tailing mechanical behaviors, which is different from the dynamic and stability of 1D and 2D FGMs structures.