Modeling and analysis of nonlinear Oscillations of atomic force microscopy in liquid environment

Atomic force microscopy is a powerful tool in imaging and detection of nanoscale materials. The performance of this device in liquid environment has a great advantage such as ability of imaging biological samples, reducing the van der Waals forces and elimination of capillary forces. Understanding of dynamic behavior of the device in liquid environment under various conditions is essential. Therefore in this paper the atomic force microscopy is modeled as a cantilever beam and its dynamic behavior is investigated using the analytical method of multiple scales. For this purpose the equation of motion of microcantilever is obtained using the continuous beam model. Subsequently the hydrodynamic force due of fluid incorporates into the equation using additional mass and damping. Then using the method of separation of variables and use of the Galerkin approximation method, partial differential equation of motion of the beam is turned to an ordinary differential equation. Finally, using the method of multiple scales, the frequency response of the micro-beam near its resonance frequencies is obtained. Moreover the effect of the fluid viscosity and beam dimensions on beam frequency response is studied. In addition, since in nonlinear systems, addition to the primary resonance, it is possible to there is another resonant frequencies, so in this paper the resonant frequencies other than the natural frequency of the linear system is studied and the existence of two resonant frequencies lower than the natural frequency is shown.