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

Abstract

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.

Dynamic Modeling and Performance Trade-offs in Flexure-based Positioning and Alignment Systems

“…For flexural beam lengths smaller than the shear approximation length factor c ≈ 0.6 in, the Timoshenko model matches the trend from the FEA data better than the Euler-Bernoulli model. This confrms the prediction that shear effects dominate at small beam lengths and agrees with similar observations supporting the Timoshenko beam bending models for depicting the natural frequencies of short AFM cantilevers in [25]. Closed-form expressions for the natural frequency of the first three-dominant modes in the decoupled case, for large flexure lengths, are presented in Table 1.…”

Vibration Responses of Atomic Force Microscope Cantilevers

“…The governing equations of the Timoshenko beam model are two coupled differential equations expressed in terms of the flexural displacement and the angle of rotation due to bending. When the beam support is constrained to be fixed and all other external influences are set to zero, we obtain the classical coupled Timoshenko-beam partial differential equations (Hsu, et al, 2007):…”

“…Neglecting the effects of transverse shear deformation and rotary inertia in the vibration analysis may result in less accurate results. Hsu et al (Hsu, et al, 2007) studied the modal frequencies of flexural vibration for an AFM cantilever using the Timoshenko beam theory and obtained a closed-form expression for the frequencies of vibration modes. However, the solution of the vibration response obtained using the modal superposition method for AFM cantilever modeled as a Timoshenko beam, and the response of flexural vibration of a rectangular AFM cantilever which has large shear deformation effects, are absent from the literature.…”

Lateral-Mode Vibration of Microcantilever-Based Sensors in Viscous Fluids Using Timoshenko Beam Theory

“…While there exist a few attempts to utilize Timoshenko beam models in MEMS/NEMS applications, their focus has tended to be on the transverse flexural mode and atomic force microscopy (AFM) applications [32][33][34][35] or on the influence of surface effects [36][37]. Moreover, with the exception of one reference [34], none of these models incorporates the effects of a surrounding fluid.…”