A Closed Form Solution for Non-linear Deflection of Non-straight Ludwick Type Beams Using Lie Symmetry Groups

Abstract

Micro Electro Mechanical Systems (MEMS) have found a large range of applications over the recent years. One of the prodigious application of micro-cantilever beams that is in use is represented by AFM probes (Atomic Force Microscopy). The AFM principle is based on the real-time measurement of the deflection of a micro-beam while following a surface profile. Hence, the prior knowledge of the deflection of beams has been of great interest to designers. Although both analytical and numerical solutions have been found for specific type of loads, there is no general solution specifically formulated for micro-cantilever beams that are not geometrically perfectly straight. Hence, the problem has not been specifically considered so far. The current work presents an analytical method based on Lie symmetry groups. The presented method produces an exact analytical solution for the deflection of Ludwick type beams subjected to any point load for non-straight beams. The Lie symmetry method is used to reduce the order of the Ordinary Differential Equation (ODE) and formulate an analytical solution of the deflection function. The result is compared with an analytical solution for a particular case that is available in the open literature. It was found that the two results coincide.