# Development of a Novel Gear-like Disk Resonator Applied in Gyroscope
###### Abstract  
This paper proposes a novel gear-like disk resonator (GDR). The design, fabrication, and characterization of GDR are presented. In comparison with a ring-like disk resonator (RDR), a GDR replaces the circular rings with meander-shaped rings consisting of linear beams. The finite element method (FEM) is implemented, and the simulation results show that the GDR has a much lower frequency and effective stiffness, higher quality factor (\(Q\)), and better immunity to crystal orientation error. Affected by high \(Q\) and small frequency splits, the mechanical sensitivity (\(S_{mcd}\)) is shown to increase greatly. GDR and RDR with the same structure parameters are built side-by-side on the same wafer, and prototypes are fabricated through the SOI fabrication technique. The frequency response test and ring-down test are implemented using a readout circuit under a vacuum condition (5 Pa) at room temperature. The frequency split (9.1 Hz) of the GDR is about 2.8 times smaller than that (25.8 Hz) of the RDR without electrostatic tuning. Compared with the RDR, the \(Q\) (19.2 k) and decay time constant (0.59 s) of the GDR are improved by 145% and 236%, respectively. The experimental results show great promise for the GDR being used as a gear-like disk resonator gyroscope (GDRG).  
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The optimum spoke length distribution (SLD) is obtained by using the particle swarm optimization method, and the \(Q\) is greatly improved [8; 9]. Some studies use the stiffness-mass decoupling method to improve \(Q\) by adding masses on the DRG frame [10; 11]. This method can also be used to lower the resonant frequency for a given resonator size or reduce the resonator size for a given resonant frequency. Thermoelastic dissipation occurs in any material that experiences thermal expansion, and selecting materials with a low coefficient of thermal expansion can also improve \(Q_{TED}\)[12].  
Frequency split is the difference between driving frequency and sensing frequency, which is mainly caused by the non-uniformity of a material and asymmetry in the structure [11]. Structural design and optimization can influence the immunity of DRGs to crystal orientation error and fabrication error [13]. Honeycomb-like disk resonator gyroscopes (HDRGs) [14] and cobweb-like disk resonator gyroscopes (CDRGs) [15] have achieved good results in reducing frequency split. Electrostatic tuning is a widely used method to reduce frequency split in DRGs because of its simple implementation and wide applicability [16; 17; 18; 19]. However, this method requires a high tuning voltage when the frequency split is large, and it is difficult to control the magnitude of large voltages with high precision [20]. Structural trimming is a common practice for the frequency matching of the two operating modes. Laser trimming [21], directional lapping [22], and tailored etch profiles [23] are widely used in the precision mode matching of micro-gyroscopes. Improving processing technology can obtain a higher symmetry resonator; this method has the ability to keep the frequency split below 1 Hz [24].  
In this paper, we proposed a novel resonator for gyroscope application. The structure was made up of meander-shaped rings consisting of linear beams that were interconnected by spokes, which had the potential to provide higher \(Q_{TED}\). The expression of \(S_{mech}\) considering frequency split was derived, and it was proved via the FEM that the GDR had higher \(S_{mech}\) than the RDR because of its lower stiffness, higher \(Q\) value, and smaller frequency split. The prototypes of GDR were fabricated through the SOI fabrication technique, together with a conventional RDR on the same wafer. The basic characterizations of the GDR and RDR are presented for comparison.