Effective dampings and frequency shifts of several modes of an inclined cantilever vibrating in viscous fluid

In this study, one investigates the dynamic behavior of an inclined non-uniform cantilever vibrating in fluid and close to a sample's surface. The closed-form solution of this beam model is presented. In fact, there must exist the hydrodynamic loading to the cantilever. Its effects include the viscous shear damping, the squeeze film damping and the added liquid mass attached to the cantilever. These depend on the material and geometrical properties and the operational conditions, e.g. the inclined angle of a cantilever to a sample's surface. For simplicity, the effective damping and the added mass are usually expressed as some formula. It is found here that these conventional formula are inaccurate for the case of the cantilever close to a sample's surface. For understanding the detailed mechanism of motion, Basak and Raman (2006) [1] analyzed the 3D fluid-structure interaction of a cantilever vibrating in liquid and close to a solid surface. The Q-factors and the resonant frequencies of different modes were presented. But the effective damping and the added mass attached to the cantilever were not presented. Via the present solution method the effective damping and the added mass are easily determined. It is very helpful for constructing the mathematical model and understanding the AFM behavior clearly.