Transverse Vibration of Functionally Graded Tapered Double Nanobeams Resting on Elastic Foundation

Abstract

The natural vibration behavior of axially functionally graded (AFG) double nanobeams is studied based on the Euler−Bernoulli beam and Eringen’s non-local elasticity theory. The double nanobeams are continuously connected by a layer of linear springs. The oscillatory differential equation of motion is established using the Hamilton’s principle and the constitutive relations. The Chebyshev spectral collocation method (CSCM) is used to transform the coupled governing differential equations of motion into algebraic equations. The discretized boundary conditions are used to modify the Chebyshev differentiation matrices, and the system of equations is put in the matrix-vector form. Then, the dimensionless transverse frequencies and the mode shapes are obtained by solving the standard eigenvalue problem. The effects of the coupling springs, Winkler stiffness, the shear stiffness parameter, the breadth and taper ratios, the small-scale parameter, and the boundary conditions on the natural transverse frequencies are carried out. Several numerical examples were conducted, and the authors believe that the results may be interesting in designing and analyzing double and multiple one-dimensional nano structures.

Keywords

• free vibration
• axially functionally graded beams
• eringen’s nonlocal elasticity theory
• chebyshev collocation method
• eigenvalue problem