Free vibration of nonhomogeneous Timoshenko nanobeams

Free vibration of nonhomogeneous nanobeams based on nonlocal Timoshenko beam theory has been studied using boundary characteristic orthogonal polynomial functions in the Rayleigh-Ritz method. Orthogonal polynomial functions satisfying essential boundary conditions have been generated with the help of Gram-Schmidt Process. Nonhomogeneity of nanobeams is assumed to arise due to linear and quadratic variations in Young's modulus and density of the nanobeams with space coordinate. The lowest three frequency parameters of nanobeams subjected to different boundary conditions have been computed for various values of nonhomogeneous parameters to demonstrate the effect of each parameters on the frequency parameters. A detailed investigation has been reported for all the possible cases of variations in Young's modulus and density to analyze the numerical results for different scaling effect parameters and four types of boundary conditions. Present results are compared with the results in special cases and are found to be in good agreement.