Flexural resonant frequencies of an AFM cantilever in viscoelastic surface contact mode using modified nonlocal elasticity theory

This paper studies the dynamics of an atomic force microscopy (AFM) cantilever that is considered to be operating under continuous viscoelastic surface contact with material profiles based on the modified nonlocal theory of elasticity. The contact between the cantilever’s tip and the sample surface is modeled using a linear stiffness–damper pair and a lumped mass at the beam’s free end. The higher-order partial differential equation (PDE) governing the AFM nanocantilever transverse motion and its associated higher-order boundary conditions (BCs) are derived employing extended Hamilton’s principle based on the nonlinear nonlocal higher-order constitutive relation in Euler–Bernoulli beam model. The Galerkin’s decomposition method is applied to discretize the higher-order PDE and BCs of motion into a set of ordinary differential equations (ODEs) via the mode summation technique using eigenfunctions (mode shapes) of a classic cantilever thin beam. Then, using state-space form of ODEs of motion the frequency analysis is performed based on the eigenvalues of vibration motion. The obtained results are validated with the literature works. The impact of various parameters including nonlocal nanoscale elasticity parameter, added point mass, contact stiffness and viscous damping factors and the specific position where the concentrated mass and the contact stiffness–damper pair are attached to the beam on the resonant frequencies of AFM cantilever is comprehensively investigated. Numerical simulations showed that the resonance frequencies of the AFM cantilever increase by increasing the value of nonlocal nanoscale parameter. Also, it was concluded that an increase in the nonlocal parameter and surface contact stiffness leads the AFM cantilever to be more stiffened. Moreover, it was seen that by increasing the position distance of lumped mass on the beam and contact spring–damper pair from the beam’s fixed end, the resonant frequency reduction in the larger values of the surface contact stiffness is more noticeable.