used the trigonometric shear flexible beam theory and direct iterative procedure to explore the nonlinear flexural free vibration behavior of size-dependent curved nano/microbeams with reinforcement of graphene platelets. Tabatabaei-Nejhada and colleagues investigated the out-of-plane vibration characteristics of laminated functionally graded graphene platelets-reinforced composite curved beams bonded by piezoelectric layers using the first-order shear deformation theory. employed a higher-order laminated beam model to find an analytical solution for the nonlinear vibration behavior of thick sandwich nanocomposite beams reinforced by functionally graded graphene nanoplatelet sheets. Liu introduced an exact solution of the vibrational characteristics of multilayer magnetic nanocomposite beams reinforced by graphene nanoplatelets. also used Timoshenko beam theory to conduct a vibration analysis of single-/three-layered micro sandwich beams with porous core and graphene platelet-reinforced composite face sheets under magnetic field and elastic foundation. Mojiri and Salami combined both Timoshenko beam theory and generalized differential quadrature method to examine the free vibration and dynamic transient response of a multilayer polymer nanocomposite beam resting on an elastic foundation reinforced by graphene platelets nonuniformly distributed through the thickness direction in a thermal environment.
Wang and his co-workers used a new Ritz-solution shape function and an improved third-order shear deformation theory to capture the solution for free and forced vibrations of a functionally graded polymer nanocomposite beam reinforced with a low content of graphene oxide and excited by a moving load with a constant velocity.
Barati and Shahverdi studied the forced vibration problem of a nanocomposite beam reinforced with different distributions of graphene platelets in thermal environments using the development of a higher-order refined beam element. investigated the nonlinear free vibration of edge-cracked graphene nanoplatelet- (GPL-) reinforced composite laminated beams resting on a two-parameter elastic foundation in thermal environments. Also, based on the first-order shear deformation beam theory combined with the differential quadrature method, Song et al. Yas and Rahimi carried out the thermal vibration analysis of functionally graded porous nanocomposite beams reinforced by graphene platelets using the Timoshenko beam theory and the generalized differential quadrature method. As a result, researchers worldwide are interested in the mechanical behavior of these structures. In addition to the low production cost, the material reinforced by GPL is a relatively good choice to fabricate details and structures in a high load-bearing environment. Graphene-reinforced materials have remarkable characteristics such as a very high Young’s modulus, great strength, and excellent thermal conductivity. Recently, due to the development of science and technology, new materials have been invented and widely applied in engineering practice, in which, materials reinforced by graphene platelets (GPLs) are one of the next-generation structural forms. The numerical results show numerous new points that have not been published before, especially the influence of the rotational speed, temperature, and material distribution on the free vibration response of the structure.
The beam material is reinforced with graphene platelets (GPLs) with three types of GPL distribution ratios. Equations for determining the fundamental frequencies as well as the vibration mode shapes of the beam are established, as mentioned, by the finite element method. This work is the first exploration using the new shear deformation theory-type hyperbolic sine functions to carry out the free vibration analysis of the rotating functionally graded graphene beam resting on the elastic foundation taking into account the effects of both temperature and the initial geometrical imperfection. These structures commonly operate in hot weather as a result, the arising temperature significantly changes their mechanical response, so studying the mechanical behavior of these structures in a temperature environment has great implications for design and use in practice. In such cases, it can be seen as a moving beam that rotates around a fixed axis. Rotating structures can be easily encountered in engineering practice such as turbines, helicopter propellers, railroad tracks in turning positions, and so on.