Effect of Winkler Elastic Foundation on Free Vibration of Tapered Beam Based on Non-Local Elasticity Theory

In this paper, the influence of Winkler type elastic foundation on the free vibration behavior of non-local Euler-Bernoulli beam with varying cross-section is semi-analytically investigated. For this purpose, the governing equation of motion is derived based on nonlocal elasticity Eringen’s model and Hamilton’s principle. The power series approximation is applied to solve the fourth order differential equation, since in the presence of variable cross-section, stiffness quantities are not constant. Based on this semi-analytical methodology, displacement component and cross-section properties should be expanded in terms of power series of a known degree. The natural frequencies of non-local beam with variable cross-section are then derived by imposing the boundary conditions and solving the eigenvalue problem. The results of this research are compared with the obtained results by other researchers and there is a good agreement. At the end, the effects of different parameters such as: end conditions, nonlocal Eringen’s parameter, tapering ratio and Winkler spring constant on non-dimensional natural frequency of nano-beam are studied in detail. The outcomes of this study indicate that the increase of nonlocal parameter and non-uniformity ratio causes to decrease the dimensionless natural frequency. However, with increasing the Winkler elastic constant, the non-local frequencies increase. Furthermore, it is shown that the effect of Winkler elastic foundation is predominate than the tapering ratio and Eringen’s parameter on the natural frequency of nano-beams.