Nonlinear dynamics of a noncontacting atomic force microscope cantilever actuated by a piezoelectric layer

The nonlinear equations of motion for a silicon cantilever beam, covered by a piezoelectric lead–zirconate–titanate layer, subjected to a Lennard-Jones type boundary condition, are derived for voltage excitation. The Lagrangian of the system is obtained from the electric enthalpy density, including the virtual work of the Lennard-Jones potential, assuming the beam undergoes only small displacements. By application of Hamilton’s principle, the nonlinear equations of motion are consistently derived and truncated to third order for perturbation analysis. The evolution equations are obtained by the multiple scales method and periodic solutions to the equations of motion are determined and discussed with respect to different tip to sample distances. An analytically obtained frequency response function enables determination of the frequency shift of individually piezoactuated microbeams, which are proposed as fundamental elements of parallel atomic force microscopy, undergoing forced vibration in a dissipative environment.