Analysis of free vibration characteristics of non-uniform graphene-reinforced axially functionally graded nanocomposite beam

This paper presents the changing effect on the free vibration behaviour of graphene-reinforced axially functionally graded nanocomposite (Gr-AFG) beam with non-uniform geometry. A generalized finite element method (GFEM) and Euler Bernoulli’s theory have been used for parametric analysis and the subsequent formulation of governing differential equations. The proposed approach (for uniform geometry) has been compared with the existing gradation schemes, e.g. ‘X’, ‘O’, ‘UD’, ‘A’, and ‘V’, and found to be well in agreement. Furthermore, a new harmonic ‘M’ function has been introduced in the current analysis, and the non-uniformity in the geometry has been implemented by varying the cross-section along the axial direction of the beam. The ‘M’ gradation in the axial direction results in an increase in the natural frequency with an increase in the slenderness ratio for the first four modes. Compared to uniform geometry, the non-uniform AFG-M beam with exponential geometry variation shows a lower fundamental frequency for all values of the slenderness ratio of the beam with an increase in the value of the exponent under C–S and C–C end support conditions.