Dynamic analysis of tapping atomic force microscopy considering various boundary value problems

An accurate understanding of the microcantilever motion and tip-sample force is needed to generate accurate images in Atomic Force Microscopy (AFM). In this paper, different methods to apply the tip-sample force to the dynamic equations of motion and boundary conditions are derived and compared to determine the superior method for dynamic analysis of these systems. Hamilton's principle and the Galerkin method are employed to investigate the vibration of the microcantilever probe used in tapping mode AFM. Three different methods of including contact and excitation force in the equations of motion and boundary conditions are analyzed then compared. The first case considers the contact force at the tip and the inertial force due to tip mass to be a part of the boundary conditions of the microbeam. The second case assumes that the force is a concentrated force that is applied in the equations of motion, and the boundary conditions are the same as for the free end of a microcantilever beam. The third case is a combination where the contact force is included in the equation of motion, but the inertial force due to the tip mass is included in the boundary conditions. For the three cases, the equations of motion, the modal shape functions including the natural frequencies, and the time and frequency response functions are obtained. The numerical results are compared to experimental results obtained from the Bruker Innova AFM. Results show that the first and third methods produce results that accurately match the experimental outcomes. However, since including the forces in the boundary conditions is considerably more complex mathematically, this research indicates including the forces in the equations of motion is preferable unless tip mass is relatively large.