A modal superposition method for vibration responses of atomic force microscope cantilevers using the Timoshenko beam model

Responses of the flexural vibration of an atomic force microscope (AFM) cantilever are evaluated using the Timoshenko beam theory. An analytical expression for the response of an AFM subjected to a sampling force with an excitation force of an arbitrarily chosen frequency is obtained with the help of the modal superposition method. In this study, the governing equations of the Timoshenko beam model with coupled differential equations expressed in terms of the flexural displacement and the bending angle were uncoupled to produce the fourth order equation. The effects of shear deformation were adopted to solve the dynamic model. A validity comparison for AFM modeling between the Timoshenko beam model and the Bernoulli-Euler beam model was conducted by using the ratios of the Young's modulus to the shear modulus. Based on the results, the Bernoulli-Euler beam model for AFM applies to the small effects of transverse shear deformation, but not to ratios greater than 1000. Moreover, one can reduce the response at the end of AFM by decreasing the shear modulus when the frequencies of processing are far away from the modal frequencies, and by increasing the shear modulus when the frequencies of processing are close to the modal frequencies. Furthermore, an AFM with a large tip width and length is suitable for reducing the response at the end of the AFM.