Theoretical modeling of the dynamic range of an elastic nanobeam under tension with a geometric nonlinearity

TL;DR

This paper develops a theoretical model for the nonlinear dynamics and dynamic range of tensioned nanoscale beams, comparing models and validating predictions with experiments to inform nanoscale device design.

Contribution

It introduces a comprehensive theoretical framework for analyzing the nonlinear mode-dependent dynamics of tensioned nanobeams, including analytical expressions and model validation.

Findings

Hinged beam model accurately describes a wide range of conditions.

Bending effects are significant for higher modes depending on tension.

Theoretical predictions align well with experimental data for multiple modes.

Abstract

A theoretical description of the weakly nonlinear and mode-dependent dynamics of a nanoscale beam that is under intrinsic tension is developed. A full analysis of the dynamic range of the beam over a wide range of conditions is presented. The dynamic range is bounded from below by the amplitude of vibration due to thermal motion and it is bounded from above by large amplitude oscillations where the geometric nonlinearity plays a significant role due to stretching induced tension. The dynamics are analyzed using a beam with clamped boundaries, a string model, and a beam with hinged boundaries. The range of validity for the different models is quantified in detail. A hinged beam model is found to provide an accurate description, with insightful closed-form analytical expressions, over a wide range of conditions. The relative importance of bending and tension in the mode-dependent dynamics of the beam is determined. Bending is shown to be important for the higher modes of oscillation with the onset of its importance dependent upon the amount of intrinsic tension that is present. The theoretical predictions are directly compared with experimental measurements for the first ten modes of two nanoscale beams. We discuss the accuracy of these approaches and their use for the development of emerging micro and nanoscale technologies that exploit the multimodal dynamics of small elastic beams operating in the linear regime.