Thermoelastic Damping in Vibrating Beam Accelerometer: a New Thermoelastic Finite Element Approach

In order to respond to the demand of accurate miniature inertial navigation systems, ONERA has been working on the design of a vibrating beam accelerometer called the Vibrating Inertial Accelerometer (VIA). The accuracy of the VIA is directly related to the thermoelastic quality factor of its sensitive element, which is a beam made of quartz. In this work, thermopiezoelectric finite element analyses of the beam are carried out in order to determine the thermoelastic quality factor. These finite element results are compared to the analytical and experimental quality factors. Due to their inherent restrictive assumptions, analytical models overestimate the quality factor while the finite element results are in good agreement with the experimental values. As the finite element model allows to take into account the real geometry of the beam and the piezoelectricity of the material, it allows to quantify more precisely the thermoelastic quality factor. ∗Address all correspondence to this author. INTRODUCTION Thermoelastic damping has been identified as an important loss mechanism in numerous high-Q micro-resonators, see for example Refs. [1–5]. The ability to accurately model and predict energy loss due to the thermoelastic effects is therefore a key requirement in order to improve the performance of high-Q resonators. Although most studies of thermoelastic quality factor till date have been based on analytical models, which are subject to very restrictive assumptions so that they are not sufficiently accurate to predict the behavior of complex 3-D structures. In this paper, a finite element formulation has been developed in order to analyze the behavior of systems that are not analytically tractable. The resonator devices used in this study are accelerometers fabricated at ONERA. In order to respond to the demand of accurate miniature inertial navigation systems, ONERA has been working on the design of a vibrating beam accelerometer called the Vibrating Inertial Accelerometer (VIA) [6]. The present applications of this device are the guidance and the attitude control of tactical missiles as well as aircraft inertial navigation. The ac1 Copyright c © 2006 by ASME Figure 1. VIA DESIGN WITH THE DECOUPLING FRAME curacy of the VIA is directly related to the quality factor of its sensitive element, which is a beam made of quartz. The aim of this paper is to study the influence of the thermoelastic effects on the behavior of the VIA. Firstly, the VIA is introduced and the importance of the thermoelastic effects on its performances is highlighted. Then, thermoelastic damping in beam resonators is briefly reviewed and the thermo-piezoelectric finite element formulation is derived. Finally, finite element analyses are carried out and the results are compared to the analytical and experimental quality factors. THE VIBRATING INERTIAL ACCELEROMETER The Vibrating Inertial Accelerometer (VIA) [6–8] is a Vibrating Beam Accelerometer (VBA) made of monocrystalline quartz. Its concept is based on the resonance frequency shift of a beam when submitted to axial stresses induced by acceleration. More precisely, in the VIA design, a micrometric beam (cross section 30 μm x 60 μm, length 2.26 mm) is clamped at one of its ends and is connected to proof mass at the other (see Fig. 1). When an acceleration is applied along the sensitive axis of the sensor (perpendicular to the transducer plane), the proof mass generates an axial stress into the beam, which modifies its bending resonance frequency. As quartz is a piezoelectric material, it is possible to actuate and detect the oscillations of the beam by metallic electrodes which are deposited on it. An electronic oscillator, with gain and phase control, is used to excite the beam at its resonance. The output of VIA is thus the frequency of the oscillator signal, and its variations represent the applied acceleration. Bias stability, i.e. beam frequency without acceleration, requires a resonator with high quality factor, in order to reduce the sensitivity of electronic phase drift. The quality factor (Q) is defined by the ratio of the stored energy in the resonator (W ) and the total dissipated energy per cycle of vibration (∆W ):