Non-linear vibration of dagger-shaped atomic force microscope cantilevers by considering the Hertzian contact theory

The non-linear flexural vibration for a dagger-shaped atomic force microscope cantilever has been investigated using the Timoshenko beam theory. In this article, the normal and tangential tip–sample interaction forces are found from Hertzian contact model and the effects of the geometry, normal and lateral contact stiffness, height of the tip, thickness of the beam, the 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 (DQM) is employed to solve the non-linear differential equations of motion. The results show that the softening behaviour is seen for the most cases. The non-linear frequency to linear frequency ratio increases by increasing the lateral contact stiffness, the central (double-tapered) cantilever length, and breadth and height taper ratios, but decreases by increasing the right side (double-tapered) cantilever length for the first and second modes. The influence of Timoshenko beam parameter on the frequency ratio is dependent on the amount of height taper ratio for the first and second modes. For the first mode, by increasing the breadth and height taper ratios, the softening behaviour changes to the hardening behaviour and by increasing the normal contact stiffness, the frequency ratio increases.