Mode coupling in nanomechanical string resonators

Towards Fermi-Pasta-Ulam-Tsingou mechanics

Master thesis (2021)

Authors

T. Jansen Mechanical Engineering

Contributors

A. Keşkekler Dynamics of Micro and Nano Systems - Mechanical, Maritime and Materials Engineering (supervisor 1)
F. Alijani Dynamics of Micro and Nano Systems - Mechanical, Maritime and Materials Engineering (supervisor 1)
R.A. Norte QN/Groeblacher Lab - AS (supervisor 1)
Faculty
Mechanical Engineering
Copyright: © 2021 Tim Jansen
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Published Date

13-07-2021

Language

English

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Faculty

Mechanical Engineering

Abstract

In the early years of numerical simulation methods, Fermi, Pasta, Ulam and Tsingou (FPUT) discovered that an undamped, weakly nonlinear equation describing the motion of a chain of masses and springs could show complex dynamics. Integration of these equations froma n initial displacement in the formof the fundamental mode resulted in significant mode coupling: energy was transferred from the fundamental mode to several other modes, before the energy would return to the initial condition. To date, very little observations of such behavior in mechanical vibrations have been reported. Recent developments in fabrication of high stress Silicon-Nitride (Si3N4) string resonators have shown that it is possible to generate resonators with extremely high Q-factors, proving a potential testbed for these mechanics. This research shows, through modal conversionof the FPUT potential, that one may observe significant FPUT behavior in systems with non-integer frequency ratios and certain coupling coefficients. In addition, it is shown that for the default FPUT beta-model, the effect of damping is negligible for fundamental mode Q-factors higher than 10,000. Simulations of the experimental
frequency response of a high-Q Si3N4 string resonator show that the nonlinear dynamics of these resonators may be approximated by an analytical model that does not possess the required frequency ratios and coupling coefficients for FPUT behavior. Another string model, for which no mechanical equivalent
has (yet) been found, may potentially show FPUT behavior. Several string-like resonator designs are tested using a numerical tool which can extract the modal coefficients. These resonators are modelled using simplified
deformation models, which account only for axial deformation of the structure. The results for various string-like designs show that the eigenfrequencies and nonlinearity may be engineered easily, but these do
not generate the required coupling coefficient for FPUT behavior.