# Development of a Novel Gear-like Disk Resonator Applied in Gyroscope
## 2 Design and Analysis  
### Structure Design  
A GDR is topologically reformed from the traditional nested-ring disk resonator. It consists of a central anchor, concentric meander-shaped rings, spokes, and some electrodes around the resonator for driving and detection, as shown in Figure 1a. In contrast to an RDR, a GDR consists of two kinds of straight beams with different lengths instead of a concentric ring frame, and the lengths of all short beams are the same. Compared with the arc-shaped structure, the micromachining technique has higher adaptability to the linear structure [25]. Therefore, a GDR is more immune to fabrication error. The straight beams between the same layer are connected end to end, and different layers are interconnected by shared spokes. The meander-shaped rings are parallel to each other, and the spokes are perpendicular to the short beams at the junction with the rings. The side of the electrodes facing the resonator is paralleled to the outermost ring; it is more similar to a parallel plate capacitor. A GDR operates at the second elliptic degenerate mode, which is composed of a 0-degree mode and a 45-degree mode. Ideally, the two degenerate modes will have the same resonant frequency and similar modal shape, as shown in Figure 1b. When the gyroscope is operating, an electrostatic force is applied to the driving electrodes, and the 0-degree mode is stimulated to vibrate with a constant amplitude. Meanwhile, the vibration of the 45-degree mode is excited by the Coriolis force induced by the rotational angular velocity. The amplitude of the 45-degree mode is proportional to the input angular velocity, so the angular velocity can be measured by detecting the amplitude.  
The basic structure parameters of the GDR studied in this paper included the radius of the anchor (\(r\)), radius of the outermost ring (\(R\)), long beam length (\(L_{ib}\)), short beam length (\(L_{sh}\)), bending angle (\(\theta\)), ring width (\(H\)), structure thickness (\(T\)), spoke length (\(L\)), and the number of rings (\(N\)), as shown in Figure 1a. \(L_{lo}\) was determined by \(\theta\) and \(L_{sh}\). The resonant frequency of the GDR could be changed by adjusting \(\theta\) and \(L_{sh}\), while other structure parameters were determined. A low-resonant frequency would reduce the shock resistance of the device, and a higher resonant frequency would reduce the quality factor. In this paper, a compromise design scheme was chosen: \(\theta\) was set as 130\({}^{\circ}\) and \(L_{sh}\) was set as 80 \(\upmu\)m. The final design scheme is shown in Table 1.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Structure Parameters** & **Value** \\ \hline Anchor radius (\(r\)) & 1300 \(\upmu\)m \\ Outer radius (\(R\)) & 3615 \(\upmu\)m \\ Short beam length (\(L_{sh}\)) & 80 \(\upmu\)m \\ Bending angle (\(\theta\)) & 130\({}^{\circ}\) \\ Ring width (\(H\)) & 20 \(\upmu\)m \\ Structure thickness (\(T\)) & 60 \(\upmu\)m \\ Spoke length (\(L\)) & 190 \(\upmu\)m \\ Ring number (\(N\)) & 10 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Structure parameters of GDR.  
Figure 1: Design strategy: (**a**) the structure of GDR; (**b**) the working mode of GDR.