The mode-dependent dynamics of nonlinear nanomechanical resonators

With the extreme miniaturization of NEMS, the role of nonlinear dynamics has become increasingly important -- even when the dynamics are driven by the Brownian force. This nonlinearity has imposed a mechanical noise floor on the linear frequency measurements made in sensing applications. Given that NEMS also become more sensitive as they become smaller, this floor has resulted in a complex interplay between the nonlinear regime and the linear sensitivity required to make continued advancements in exercising ultra-sensitive measurements. Recently, this has led to efforts to more accurately characterize the edge of the linear regime. Inside of the nonlinear regime, there are also ongoing fundamental studies in theory and experiment to partially characterize the nonlinear behavior of NEMS. Theoretically, these systems are frequently studied by decomposing the nonlinear continuous system into one or more nonlinear oscillators. However, in many of these works, the nonlinear spring constants are estimated at the lowest order. As such, there is a clear need to more accurately characterize and scale the nonlinear coefficients for NEMS.

This work considers a long and slender NEMS resonator in the form of a doubly-clamped beam in tension. Using nonlinear Euler-Bernoulli beam theory, the geometric nonlinearity due to the stretching of the neutral axis is considered. We extensively explore simplifications in using a Galerkin discretization of the continuous system, where a single mode's dynamics are described as a damped, Duffing oscillator. We examine limitations of current approaches and find that using a tensioned and doubly-clamped mode shape for the trial function more accurately predicts experiment. Additionally, we find that doubly-clamped beams of finite tension may have their boundary conditions modified to that of a hinged-hinged beam in tension with little to no loss of generality. This modification allows for closed-form scaling of the critical amplitude and dynamic range at arbitrary mode number and tension. We extend this approach to scale the relative influence of bending, tension, and nonlinearity with increasing mode number n, finding that bending and nonlinear influences quickly outgrow the contributions of the intrinsic tension. Where applicable, these results are compared with experiment, and we obtain good agreement. To validate the approximate Galerkin formulations, we develop a finite element method to calculate the nonlinear coefficients of symmetric NEMS resonators. Unlike previous works, the present formulation may include all nonlinearity due to geometry, and the nonlinear amplitude-frequency backbone described by the Duffing oscillator is found as an excellent approximation for large amplitude beams. For beams of zero intrinsic tension, the finite element method obtains excellent agreement with the literature. For beams of finite tension and varying mode number, we find the error from the Galerkin discretization is small (≈5%).

In addition, we theoretically explore the stochastic dynamics of a Duffing oscillator driven nonlinearly by the Brownian force. To access this regime experimentally with current nanomechanical systems, we motivate an experimental "synthetic noise" to approximate the Brownian force in the proximity of a single mode. As a measure of drive magnitude, we vary an effective temperature to explore the linear and nonlinear stochastic dynamics of a doubly-clamped nanoresonator. Using similitude and the Fluctuation-Dissipation theorem, we show that varying the effective temperature of the synthetic noise offers a window into the fundamental limits of thermally-driven nonlinearities. We compare theory, numerics, and experiment where applicable, obtaining good agreement for both limits of frequency shifts in the weakly nonlinear case.

This research was supported by the National Science Foundation, grant number CMMI-2001559, and portions of the computations were conducted using the resources of Virginia Tech's Advanced Research Computing (ARC) center.