A Variational Principle Approach for Vibration of Non-Uniform Nanocantilever Using Nonlocal Elasticity Theory

In this article free vibration of a nanocantilever with nonuniform cross section is studied using nonlocal elasticity within the scope of continuum mechanics. Based on an exact variational principle approach, an asymptotic partial differential equation of infinite order is derived along with the corresponding boundary conditions. These equations involve essential higher-order differential terms which, if neglected, have previously led to some rather intriguing observations and conclusions. A reduced sixth-order differential equation is, then, solved for a nanocantilever by applying finite element method using quintic spline interpolation functions. The finite element model developed will be of practical use and reference to physicists and engineers, alike, in the analysis and design of more complicated nanostructures.The paper also resolves some of the strange observations from similar studies like: (i) the non-existence of real eigenvalues (for the first and second modes) for nonlocal parameter (e0a/L)>0.62; and (ii) the existence of a height ratio where frequency becomes independent of size effects, this ratio was defined as the ‘critical height ratio’. It is clear from this study that these observations were a result of using a governing equation and boundary conditions which were not exact thus leading to erroneous observations and conclusions.