Free Vibration Analysis of Twisted Composite Beam Reinforced with Graphene Platelets

In this work, free vibrations of a composite beam is investigated. Considered composite beam is assumed with the presence of pre-twist. One of the beam is without twist while the other end of the beam has the maximum twist angle. Twist rate in the beam is assumed to be non-constant where a higher order polynomial variation of twist angle is assumed. Composite beam of this study is composed of a number of layers where each layer of the beam is reinforced with graphene platelets. Layers may have different amount of graphene which leads to the functionally graded composite laminated beam with pre-twist. Elasticity modulus of the beam is estimated by means of the Halpin-tsai rule while the mass density and Poisson's ratio are assumed using the simple rule of mixtures approach. Timoshenko beam theory is adopted to estimate the displacement field in the beam. Due to the present of twist, five degrees of freedom containing three displacements and two cross-section rotations are considered. Using the Hamilton principle and with the aid of the Ritz method, the equations of motion are discretized. Shape functions of the Ritz method are constructed by means of the Chebyshev polynomials. This set of functions are orthogonal and has high rate of convergency. Matrix representation of the governing equations is obtained using the Chebyshev-Ritz method. The governing equations of the free vibration motion are established and solved as an eigenvalue problem. Results of this study cover the case of free vibration of a graphene platelet reinforced composite laminated pre-twisted beam. At first results of this study are validated with the available data in the open literature and also the required number of shape functions in the Ritz method are estimated through the convergence studies. After that using parametric studies, the influences of different parameters such as number of layers of composite, boundary conditions, twist angle, twist rate, graphene weight fraction and their pattern are explored. Numerical results of this study show that by increasing the weight fraction of graphene platelets, natural frequencies of the beam are enhanced. Also through adoption of a proper pattern, the natural frequencies in the beam may be controlled. The effect of twist angle on the frequencies of the beam is shown. It is depicted that increasing the twist angle may increase/decrease the frequencies depending on the mode number.