The power series method for studying the thermal and surface effects on the vibration of nanobeams based on the novel Timoshenko beam formulation

This paper presents the influence of the surface tension and the temperature on the vibration frequencies of a Timoshenko nanobeam which lays on the Winkler-Pasternak foundation. The local and Eringen nonlocal elasticity theories are considered for modelling nanobeam system. The Hamilton’s principle is firstly used to get the differential equations system as well as the boundary conditions. Based on the power series method (PSM), the recursive relation of the final single differential equation is obtained. Next, boundary condition equations enable to get and solve the eigenvalue problem for the fully supported (SS), clamped supported (CS), fully clamped (CC) and clamped free(CF) ends. After comparing our results with those existing, the PSM appears to excellently compute the vibration frequencies of a local and nonlocal nanobeam. Moreover, we observed a decreasing effects of the surface tension and temperature on a frequency of a local cantilever nanobeams and their increasing effects on the frequency for other boundary conditions. For nonlocal CF nanaobeam, the temperature and the surface tension decrease the frequency when the small scale parameter is close to zero. The contrary effect is observed when the scale parameter is bigger than 0.1. The previous parameters have an increasing role on the frequency for SS, CS, CC nonlocal nanobeams for both local and nonlocal approaches.