Nonlocal vibration of fractional order viscoelastic nanobeam with attached nanoparticle

Abstract

We propose a mathematical framework to examine the free damped transverse vibration of nanobeams with attached nanoparticles by using the nonlocal theory of Eringen and fractional calculus. Governing equation of nanobeam with arbitrary attached nanoparticle is derived by considering the viscoelastic constitutive equation involving fractional order derivative and nonlocal Euler-Bernoulli beam theory. The solution is proposed by using the method of separation of variables. Eigenvalues and mode shapes are determined for two typical boundary conditions. The solution of fractional order differential equation in terms of a time function is found via Laplace transform method. The time dependent behavior is examined by observing the time function for different values of fractional order parameter and different ratios of other parameters in the model.