Microresonator designs based on nonlinear 1:2 internal resonance between flexural -flexural and torsional -flexural structural modes

Ashwin Vyas, Purdue University

Abstract

A great majority of the micromechanical resonators in the literature are based on linear response under external resonance conditions of micro-structures vibrating in some flexural or torsional modes. In this dissertation, a novel nonlinear microresonator concept is proposed which utilizes “internal resonance” phenomena and non-linear interactions between structural modes for its essential operation. The focus in this work is on systems with 1:2 internal resonance between two structural natural frequencies and modal interactions in the presence of quadratic nonlinearities. Two such electrostatically actuated microresonator designs are considered: a T-shaped resonator with three beam elements in T configuration, and a pedal resonator with plate type pedal structure supported by two beams. The T-resonator is designed to have in-plane motions with two flexural modal natural frequencies in 1:2 ratio, while the pedal microresonator has an out-of-plane torsional mode which has a natural frequency smaller than the natural frequency of an in-plane flexural mode. The two natural frequencies are again in a 1:2 ratio. In both the designs, the higher frequency mode autoparametrically excites the lower frequency mode through inertial quadratic nonlinearities. Analytical models are developed for the resonators using Lagrangian formulations. The T-resonator model includes: inertial quadratic nonlinearities, cubic nonlinearities due to mid-plane stretching and curvature of the beam, electrostatic potential, and effects of thermal pre-stress. The pedal resonator model in addition to flexural deflections includes torsional rotations of the beam which contribute to kinetic and elastic energy. Linear equations of motion are used to define resonator designs so as to achieve 1:2 internal resonance between the desired modes. A nonlinear two-mode reduced-order model for each design is derived using linear structural modes with desired internal resonance. These nonlinear models are used to describe the static pull-in voltages that cannot be exceeded in the resonator’s operations. Asymptotic methods are then used to study the responses for the resonators and nonlinear frequency responses are developed in each case for the resonant actuation of higher frequency mode. It is shown that the lower frequency mode is excited for actuation levels only above a certain threshold, and generates a response component at half the frequency of resonant actuation. The effects of damping, thermal pre-stress, and mass and geometric perturbations from nominal design are thoroughly discussed. The nonlinear resonators show a high sensitivity to mass perturbations and, thus, hold great potential as RF filter-mixers and mass sensors. The application of the pedal resonator as a mass sensor is explored in some detail. Experiments are performed on a macro sized T-structure with the flexural modes tuned to 1:2 resonance. Response of this T-structure is measured using Laser displacement sensors and piezoelectric patches are used for actuation. Experimental results show a large response component at half the excitation frequency when the higher frequency flexural mode is resonantly excited. The tuned T-structure response is also observed to be very sensitive to any mass perturbations. The macro T-structure behavior verifies the basic concept of T-shaped microresonator design based on 1:2 internal resonances.

Degree

Ph.D.

Advisors

Bajaj, Purdue University.

Subject Area

Mechanical engineering

Off-Campus Purdue Users:
To access this dissertation, please log in to our
proxy server
.

Share

COinS