Free vibrations of graphene platelet reinforced composite skew plates resting on point supports

摘要

In the present investigation, vibration behaviour of skew plates which are located on point supports is investigated. The point supports may be on the edges on even in the plate domain. A composite laminated plate is reinforced with graphene platelets. Weight fraction of graphene platelets may be different in the layers which results in a piecewise functionally graded media. The elasticity modulus of the composite is estimated by means of the Halpin-Tsai rule which also captures the dimensions of the nano-reinforcement. Mass density and Poisson's ratio, on the other hand are obtained via the simple rule of mixtures approach. Total strain and kinetic energies of the skew plate are established using the first-order shear deformation theory of plates. After that, by means of the general idea of Ritz method where the shape functions are constructed with the aid of Chebychev polynomials, the elements of mass and stiffness matrices are obtained. To impose the effects of point supports with arbitrary number and arbitrary locations, the method of Lagrangian multipliers is adopted. According to this technique, the number of degrees of freedom in the system increases with the introduction of point supports. Using the developed formulation and solution method, the in-plane and out-of-plane frequencies of the plate as well as the mode shapes will be obtained with high accuracy. Numerical results are devoted to show the effects of number of layers of composite plate, weight fraction of graphene platelets, distribution pattern of graphene platelets, position and number of point supports, geometries of the plate and also the boundary conditions on the frequencies of the skew plate located on point supports.