Free vibration of non-uniform nanobeams using Rayleigh–Ritz method

• We study vibration of Euler–Bernoulli nanobeams with non-uniform cross section using Rayleigh–Ritz method.
• We examine the effect of nonlocal parameter, boundary condition, non-uniform parameter and length-to-diameter ratio on the frequency parameters.
• Frequency parameters decrease with increase in scaling effect parameter and increase with length-to-diameter ratio.
• We plot mode shapes for two types of boundary conditions.

In this article, boundary characteristic orthogonal polynomials have been implemented in the Rayleigh–Ritz method to investigate free vibration of non-uniform Euler–Bernoulli nanobeams based on nonlocal elasticity theory. Non-uniform cross section of nanobeams has been considered by taking linear as well as quadratic variations of Young's modulus and density along the space coordinate. Detailed analysis has been reported for all the possible cases of such variations. The objective of the present study is to analyze the effects of nonlocal parameter, boundary condition, length-to-diameter ratio and non-uniform parameter on the frequency parameters. It is found that clamped nanobeams are having highest frequency parameters than other types of boundary conditions for a particular set of parameters. It is also observed that frequency parameters decrease with increase in scaling effect parameter. First four deflection shapes of non-uniform nanobeams have also been incorporated. In this analysis, some of the new results in terms of boundary conditions have also been included.