Development of a mathematical model and analytical solution of a coupled two-beam array with nonlinear tip forces for application to AFM

Utilising arrays of cantilevers has been identified as a method of increasing AFM sensitivity and throughput beyond the limitations of current non-contact AFM. In order to develop the technology, we investigate the change in eigenstates of arrays due to interactions that occur between cantilever tip and sample using a model of a base-coupled array of cantilevers with individual mass and stiffness properties influenced by a quadratic nonlinear force to represent tip–sample interactions. An analytical expression is developed for a coupled two-beam array utilising Multiple Scale Lindstedt–Poincare perturbation theory to assess how a selected parameter space alters the array dynamics. Experiments are carried out using a macroscale set-up to validate the developed model. We show that the model captures the eigenstates of the system with good qualitative accuracy and that the perturbation expansion performs well in both the weakly (far from surface) and strongly (near the pull-in point) nonlinear regimes. The results demonstrate that utilising changes in eigenstates due to force interactions could have significant benefits to non-contact AFM with regard to measurement sensitivity and speed.