Solution a Problem of Nonlinear Elasticity Using Power Series in Complex Time - Shakir, Akyl (Nazarbayev University School of Science and Technology, 2018)
We apply singularity analysis in complex time to investigate the solutions of a dynamical system of one degree of freedom related to the oscillations of a Micro-Electro-Mechanical System (MEMS) of nonlinear elasticity. This problem is expressed mathematically by a second order differential equation for the position variable x(t ) of the oscillator. Using the fact that this equation is connected with an energy integral (kinetic plus potential energy), we first study its solutions by plotting the graph of its potential function for different values of an important parameter of the problem K>0. Then, we analyze these solutions by expanding x(t using power series in complex time. Our solutions are expanded about a singularity of the equation of motion at x = 1, which constitutes a very important point in the analysis, and complements solutions found by other researchers using real time. As a first application, our complex time expansions can be used to estimate locations in x space where the solutions are bounded and periodic and regions where the solutions are unbounded (and hence non-physical). We also examine our approach as an extension of the well–known Frobenius theory for linear second order ODEs. Finally, we investigate the locations of the singularities near the periodic solutions using a Newton-Raphson method near the singularity of the equation at x =1.
Keywords: differential equation; Micro-Electro-Mechanical System (MEMS); Nonlinear Ordinary Differential Equations (NODEs)