Nonlinear vibrations in atomic force microscopy

Recent atomic force microscopy research has focused on dynamical methods in which AFM probes are vibrated while in contact with a specimen during scanning. The nonlinear tip-sample interactions can induce nonlinear features into the dynamic response. Nonlinear responses observed experimentally include the DC shift (or lift-off) and primary response softening as well as the development of subharmonics and superharmonics. Here, this problem is formulated in terms of a nonlinear boundary value problem which is solved using the method of multiple scales. The main result of this analysis is the amplitude-frequency relation for all vibration modes. The nonlinear normal modes are comprised of terms representing the softening effect of the resonance, the static offset, and harmonics. The softening effect on the primary response is shown to be a function of the particular vibration mode as expected. The contact mechanics model used here is restricted to Hertzian contact, but can be generalized to more complex models. Results of the primary response for various excitations are presented. The amplitude-frequency behavior is dependent on the linear contact stiffness, the forcing amplitude, and contact damping. It is also shown that the modes have a differing sensitivity to the nonlinearities present in the contact.