Natural frequencies and mode shapes of slacked carbon nanotube NEMS resonators

Estimating accurately the natural frequencies of electrically actuated carbon nanotubes (CNTs) has been the center of research attention over the past few years. Despite this important topic, a robust knowledge is still missing to understand the role of various physical parameters that affect the natural frequencies, such as the stretching of doubly-clamped CNTs, the DC electrostatic force, and the curvature of CNTs with slacks. In this investigation, we use a 2D nonlinear curved beam model (arch) to simulate the coupled in-plane and the out-of-plane motions of a CNT with curvature (slack). We calculate the variation of its natural frequencies and mode-shapes with the level of slack and the DC electrostatic load. Toward this, we derive a reduced-order model using a multi-mode Galerkin procedure based on the mode shapes of the straight unactuated CNT. We calculate the natural frequencies of the slacked CNT for a given voltage by substituting the static solution into the Jacobian of the reduced-order-model and then finding the corresponding eigenvalues. We show various scenarios of mode crossing and mode veering as the levels of slack and DC load are varied.