A fresh insight into the non-linear vibration of double-tapered atomic force microscope cantilevers by considering the Hertzian contact theory

Abstract

The non-linear flexural vibration for a double-tapered atomic force microscope cantilever has been investigated by using the Timoshenko beam theory. In this article, the normal and tangential tip—sample interaction forces are found from the Hertzian contact model, and the effects of the contact position, normal and lateral contact stiffness, height of the tip, thickness of the beam, angle between the cantilever and the sample surface, and breadth and height taper ratios on the non-linear frequency to linear frequency ratio have been studied. The differential quadrature method is employed to solve the non-linear differential equations of motion. The results show that the softening behaviour is seen for all cases. The non-linear frequency to linear frequency ratio increases by increasing the Timoshenko beam parameter and breadth and height taper ratios, but decreases by increasing the contact position for the first and second modes. For the first vibrational mode, the non-linear frequency to linear frequency ratio increases by increasing the height of the tip and the angle α between the cantilever and sample surface. By increasing the normal contact stiffness, the frequency ratio increases for the first mode.