Thermoelastic Damping in Flexural-mode Ring Gyroscopes

This paper provides a comprehensive derivation for thermoelastic damping (TED) in flexural-mode ring gyroscopes, in light of recent efforts to design high rateresolution gyroscopes. Imposing an upper limit on the attainable mechanical noise floor of a vibratory gyroscope, thermoelastic damping in a ring gyroscope is extracted from the equations of linear thermoelasticity. By assuming that it is small and therefore has negligible effect on the flexural-mode vibrations in a ring, thermoelastic damping manifests itself through temporal attenuation, where a complex frequency is used to quantitatively evaluate this damping. The exact solution to thermoelastic damping is derived and verified with experimental data in the literature. This work not only provides significant insight to the geometrical design in high-Q ring gyroscopes, but also defines their performance limit. INTRODUCTION Micromechanical flexural-mode ring vibratory gyroscopes are of great interest for sensing rotation rate, due to their inherent symmetric structures and better temperature sensitivity [1-4]. One key determinant of performance for a vibratory gyroscope is its mechanical quality factor (Q). Since a higher Q in a gyroscope translates to higher rate-resolution, better bias stability, and lower power consumption, the design of a ring gyroscope with high Q or little energy loss is consistently pursued. Identified as a fundamental loss mechanism, thermoelastic damping (TED) imposes an upper limit on the attainable Q and further determines mechanical noise floor (Ωmechanical) of a ring gyroscope. Thus, it is of significant importance to understand thermoelastic damping in ring gyroscopes, not only for improving their performance, but also for establishing their performance limit. Although the physical mechanism and theory of thermoelasticity has been well established [5], analytical studies on thermoelastic damping in flexural-mode vibrations of different finite-geometries are few. In the 1930’s, by introducing the resonant mode shapes into the equation of heat conduction and obtaining the corresponding transverse thermal modes, Zener derived an approximate expression for thermoelastic damping in rectangular beams undergoing flexural-mode vibrations [6,7]. Through keeping the first transverse thermal mode and neglecting the rest thermal modes, Zener’s theory showed that thermoelastic damping in a flexural beam exhibits a Lorentzian behavior with a single thermal relaxation time. This relaxation time is related to the width b of the beam and the thermal diffusivity χ of the material used. Without neglecting any transverse thermal modes, Lifshitz and Rouke’s recent work [8] provided an exact solution to the linear thermoelastic equations in flexural-mode beam resonators, predicting a modified Lorentzian behavior of the thermoelastic damping. Both the above-mentioned works have a fundamental assumption that thermoelastic coupling is very weak and thus has negligible influence on the uncoupled elastic resonant modes of a beam, so the elastic and thermal problems are essentially decoupled. This assumption is necessary, since severe thermoelastic coupling would cause noticeable influence on the uncoupled elastic behavior and lead to third-order phenomenon, where quality factor does not exist. Following this assumption, the uncoupled resonant mode shapes are further assumed, in order to find the temperature variation in the beam. Section II will give an overview of thermoelastic damping in a flexural-mode rectangular beam resonator. In light of recent efforts to design high Q gyroscopes, one particularly interesting aspect of the physical behavior of a ring gyroscope is its thermoelastic damping at different frequencies