Closed-form solutions for the frequency shift of V-shaped probes scanning an inclined surface

The analytical method to determine the frequency shift of an AFM V-shaped probe scanning the relative inclined surface in non-contact mode is proposed. If the tip is not perpendicular to the surface plane, the lateral force to the tip will occurs. Consequently, there exists a moment to the end of probe. The closed-form solution of the partial differential equation with a nonlinear boundary condition is derived. The error of transforming the distributed-masses system into lumped-masses one in the force gradient method or the perturbation method is eliminated. The dimensionless parameters are introduced for reducing the numerical transaction error. The limiting case such as a uniform or tapered beam can be obtained easily from the general system. The assessments of the force gradient method, the perturbation method and the propose method determining the frequency shift of a V-shaped probe are made. It is discovered that increasing the absolute inclined angle θ decreases the frequency shift especially for a small tip-surface distance