Flexural vibration frequency of atomic force microscope cantilevers using the Timoshenko beam model

The modal frequencies of flexural vibration for an atomic force microscope (AFM) cantilever have been evaluated using the Timoshenko beam theory, and a closed-form expression for the frequencies of vibration modes has been obtained. In the analysis, the effect of the ratios of different cantilever dimensions on frequency for the cantilever were studied. The results show that increasing the ratio of cantilever thickness and length decreases the vibration frequency. Increasing the ratio of the tip length and cantilever length also increases the frequency. In addition, results using the Bernoulli-Euler beam theory and the Timoshenko beam theory are compared. It can be found that the Timoshenko beam theory is able to predict the frequencies of flexural vibration of the higher modes with higher contact stiffness for the AFM cantilever.