Dynamical Analysis Of The Atomic Force Microscope

In 1986 Binnig, C.F.Quate who comes from Stanford University and Christoph Gerber form IBM Su Lishi laboratory cooperated to develop Atomic Force Microscope (AFM) successfully. It is a kind of scanner probe microscope that does not need the electric test specimen. This microscope can be used to inspect the surface of object on the extremely near distance through its probe as large as an atomic. And it can distinguish the surface features and details in the minimum criterion, while other microscopes can't. So this microscope can be used to detect the shape of the atom and the molecular, identify electrical, magnetic and mechanical features by the high resolving power, even determine the temperature changes. The test specimen doesn't need to be changed and conducted the high-energy destructive radiation by this microscope. Due to the extremely widespread application of Atomic Force Microscope, it is significative to research and analyze its dynamics behavior.For the dynamics analysis model of Atomic Force Microscope, there had been three types: single degree of freedom spring-quality system, the vibration of the line elastic cantilever beam under outside load function, the nonlinear vibration of the linear beam with the nonlinear boundary condition. The non-linearity is caused by the intermolecular force. In this paper, we analyze the stress of beam and obtain nonlinear oscillation partial differential equation using the Newton second law. Based on the non-linear model of cantilever beam, we can obtain the single degree of freedom and two degree of freedom model by Galerkin interruption method. In the former studies of boom, the researchers just focused on the single degree of freedom model. But in third kind of model, we manage to solve the problem separately regarding the exciting force as the restraint at the end of pole or the boundary condition.With the single degree of freedom model, we solve the equation by the method of multiple scales, obtain the approximate solution and the frequency response curve, then compare with the original numerical solution, and analyze its dynamics characteristic with the Runge-Kutta numerical method. By the Poincarémap, the dynamics behaviors are identified based on the numerical solutions of the ordinary differential equations. The parameter bifurcation is presented. The Lyapunov exponent is calculated to identify chaos. With two degree of freedom model, the dynamics behaviors also are analyzed by the Runge-Kutta numerical method.By taking excitation frequency as base frequency, we can obtain the bifurcation diagrams of vibration parameter and compare with results of the single degree of freedom model. The comparison shows that there is similar furcation situation based on these two models.For the vibration of line elastic beam with non-linear boundary condition, the perturbation methods are the effective ways. Because the continuous medium is an infinite dimension system, the separation must create error. The traditional perturbation methods to the discretization equation have certain limitation. Therefore the perturbation method used to be applied in beam's control partial differential equation,and then given the solution following the soluble condition. In this paper, we analyze the beams which have Derjaguin-Muller-Toporov and the Hertzian boundary condition by the method of multiple scales, and obtain the response under different mode. The results show that the frequency response is decided the number of mode and the line contact rigidity. And we have studied furcation behavior and stable based on first frequency resonating. The conclusion is tally with the experimental result.It is the first time to analyze the beam with Hertzian boundary condition by the asymptotic perturbation method; we can obtained frequency responds in the first frequency and the half frequency. And we analyze the frequency response changes with various parameters. With the stable analysis, we also analyze the reason of the jump phenomenon and the various parameters impact to stability.