Shock and Vibration

Volume 2018, Article ID 6901268, 10 pages

https://doi.org/10.1155/2018/6901268

## Energy Dissipation Contribution Modeling of Vibratory Behavior for Silicon Micromachined Gyroscope

^{1}Research & Development Institute of Northwestern Polytechnical University in Shenzhen, China^{2}Northwestern Polytechnical University, Xi’an, China^{3}The MOE Key Laboratory of Micro and Nano Systems for Aerospace, Northwestern Polytechnical University, Xi’an 710072, China^{4}Shanghai Institute of Radio Equipment, Shanghai 200090, China

Correspondence should be addressed to Q. Shen; moc.621@7191gnaiqnehs

Received 25 January 2018; Accepted 6 March 2018; Published 14 June 2018

Academic Editor: Md Abdul Halim Miah

Copyright © 2018 J. Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Energy dissipation contribution of micro-machined Coriolis vibratory gyroscope (MCVG) is modeled, numerically simulated, and experimentally verified in this paper. First, the amount of independent damping dissipation consisting of thermoelastic loss, anchor loss, surface loss, Akhiezer loss, and air damping loss during vibration is obtained by simulation model, PML-based method, and numerical calculation, respectively. Then, temperature and pressure dependence characteristic of the corresponding quality factor (Q) for the MCVG are obtained. Meanwhile, dominant sources of damping dissipation are determined, which paves the way to improve Q. Finally, the temperature-dependent and pressure-dependent characteristics of the total Q are measured with errors of less than 10% and 18% compared with the simulated total Q, respectively, in which accuracy is acceptable for predicting the damping dissipation behavior of MCVG in design stage before high-cost fabrication.

#### 1. Introduction

Vibration behavior is an important feature of MEMS two-order mass-damping-stiffness system. Energy conversion is a fundamental characteristic of these vibrational system. For some type MEMS devices, vibration regarded as energy is collected as power of electrical components, such as vibration energy harvest [1–3]. For another type devices, this vibration behavior can be localized to improve signal noise ratio (SNR) greatly [4]. Nevertheless, more common but inevitable phenomenon for all of MEMS devices during vibration occurrence is vibration energy dissipation behavior, typically such as micromachined Coriolis vibratory gyroscope (MCVG) [5–7]. The performance of this type of vibration system is significantly limited by the rate at which the vibration energy is dissipated, which is called time constant. is generally used as a measurement of this time constant of the vibration energy and is defined as ratio of the total strain energy to the dissipated energy per vibration cycle. Higher means larger time constant and lower vibration energy dissipation, for the MCVG, which means achievement of excellent bias stability [8–12].

In order to achieve high performance of the MCVG, analysis of energy dissipation mechanisms of the Q value of the MCVG during vibration becomes an important issue [13–16]. For the single-mode mass-damping-stiffness resonator, the value has been widely regarded as a combination of five energy dissipation mechanisms and inverse proportion to the total energy dissipation [17–19]. Energy dissipation usually consists of two categories: intrinsic dissipation such as thermoelastic damping loss* 1/Q*_{the}, anchor damping loss* 1/Q*_{anc}, and Akhiezer damping loss* 1/Q*_{akh} and extrinsic dissipation such as surface damping loss* 1/Q*_{sur} and air damping loss* 1/Q*_{air}.

These damping loss mechanisms of the single-mode resonators have been extensively investigated for achieving the expected* Q*. In summary, influencing factors on damping losses can contribute to three aspects: vacuum packaging technology, materials properties, and mechanical structure topology. First, vacuum packaging scheme as the most common one is used to greatly decrease air damping, such as* epi-seal* encapsulation process [20–25], hermetic lid sealing process [13, 26–28], Through-Silicon Via (TSV) process [29, 30], and SPIL MEMS WLP process [31]. Second, some different kinds of materials are tried to yield lower the thermoelastic, surface deflect, and Akhiezer damping. For example, crystalline and doped silicon are chosen frequently to make high-Q resonators because of their excellent semiconductor properties [32–37]. Researchers also take advantage of single crystal (SC) [38], microcrystalline (MC) [39–42], nanocrystalline (NC) [43, 44], and ultra NC [45] diamonds to achieve high-Q resonators by trading off contributions of different damping loss items, respectively. As for the structure topologies which mainly affect anchor damping, anchor loss models of simple cantilever beam have been established [46–48]. However, for some complex fully 3D resonators, contributions of another geometrical resonators including DETF [21], tether geometry [49–51], microsphere [52], cupped [52, 53], wineglass [54], hemispherical [39], and disk shapes [55–57] to anchor loss are calculated hard by the analytical solution if no simplifying assumption is present. Damping loss model on accurately estimating the experiment results for these complex structure topology is very limited. Main reason is that it is difficult to obtain precise a priori knowledge of the stress profile at the anchor area of these design. Especially for the MCVG with both the drive and sense modes, the elastic beams connecting with anchors exist mechanical vibration coupling between the drive and sense modes.

Therefore, we will try to model and simulate the energy dissipation mechanisms for MCVG by numerical analysis technologies in this paper. Furthermore, the contribution of each damping dissipation to the total quality factor of the MCVG is predicted. The experiments are also implemented to verify the theoretical analyzes.

This paper is organized as follows: in Section 2, modeling and simulation of the independent damping dissipation for the MCVG are presented. In Section 3, temperature and pressure dependence characteristic measurements of the quality factor are implemented. Finally, the conclusions are provided in Section 4.

#### 2. Numerical Simulation Analysis of Damping Dissipation

Here, we consider our single-mass z-axis MCVG as the example to [8, 58–60] to analyze the contribution of the damping loss, as shown in Figure 1. The symmetrical SOI-based MCVG structure consists of a set of the electrodes, anchors, elastic beams, and three masses including the drive-mode mass, the sense-mode mass, and the proof mass. The structure is fixed on the substrate through four anchors, which connect with eight elastic beams. The vibration between both modes is decoupled through eight other elastic beams located at the proof mass. In order to enhance the performance,* Q* value of the MCVG should be as high as possible.