Harmonic Solution of a Weakly Non-linear Second Order Differential Equation Governed the Motion of a TM-AFM Cantilever
A. M. Elnaggar
Department of Mathematics, Faculty of Science, Benha University, B.O.13518, Egypt.
K. M. Khalil
Department of Mathematics, Faculty of Science, Benha University, B.O.13518, Egypt.
A. S. Rahby *
Department of Mathematics, Faculty of Science, Benha University, B.O.13518, Egypt.
*Author to whom correspondence should be addressed.
Abstract
The harmonic solution of a weakly non-linear second order differential equation governed the dynamic behavior of a micro cantilever based on TM (Tapping mode) AFM (Atomic force microscope) is investigated analytically by applying the method of multiple scales (MMS). The modulation equations of the amplitude and the phase are obtained, steady state solutions, frequency response equation, the peak amplitude with its location and the approximate analytical expression are determined. The stability of the steady state solutions is calculated. Numerical solutions of the frequency response equation and its stability condition are carried out for different values of the parameters in the equation. Results are presented in a group of figures. Finally discussion and conclusion are given.
Keywords: Micro-electro-mechanical system (MEMS), atomic force microscopy (AFM), differential equation, harmonic solution, multiple scales method