# Development of a Novel Gear-like Disk Resonator Applied in Gyroscope
## 2 Design and Analysis
### Theory  
Considering the asymmetry between the drive mode and sense mode, which is caused by the unequal stiffness and damping of the drive axis and the sense axis, the dynamic equation of the GDR can be expressed as follows [26]:  
\[\begin{array}{l}m_{eff}\ddot{q}_{A}(t)+\gamma_{A}\dot{q}_{A}(t)+k_{A}q_{A}(t)=F _{drive}\\ m_{eff}\ddot{q}_{B}(t)+\gamma_{B}\dot{q}_{B}(t)+k_{B}q_{B}(t)=F_{Coriolis}\end{array} \tag{1}\]  
where \(\gamma_{A}\), \(k_{A}\), and \(q_{A}(t)\) are the damping coefficient, effective stiffness, and vibration displacement of the drive mode, respectively; \(\gamma_{B}\), \(k_{B}\), and \(q_{B}(t)\) are the damping coefficient, effective stiffness and vibration displacement of the sense mode, respectively; \(m_{eff}\), \(F_{drive}\), and \(F_{Coriolis}\) are the effective mass, driving force generated from applied voltage, and Coriolis force caused by input angular velocity, respectively.  
When the DC part and double-frequency part in \(F_{drive}\) is ignored, \(F_{drive}\) can be simplified as [27]:  
\[F_{drive}=F_{0}\sin(w_{A}t) \tag{2}\]  
where \(F_{0}\) is the amplitude of drive force; \(w_{A}=\sqrt{k_{A}/m_{eff}}\) is the resonant frequency of drive mode.  
When angular velocity (\(\Omega\)) is inputted, Coriolis force is generated, which is given as [28]:  
\[F_{Coriolis}=2 nA_{g}m_{eff}\Omega\dot{q}_{A}(t) \tag{3}\]  
where \(n\) is the circumferential wave number and \(A_{g}\) is the angular gain. For a DRG working at \(n=2\) wineglass mode, Equation (1) can be simplified according to Equations (2) and (3) as below:  
\[\begin{array}{l}m_{eff}\ddot{q}_{A}(t)+\gamma_{A}\dot{q}_{A}(t)+w_{A}^{2}m_ {eff}q_{A}(t)=F_{0}\sin(w_{A}t)\\ m_{eff}\ddot{q}_{B}(t)+\gamma_{B}\dot{q}_{B}(t)+w_{B}^{2}m_{eff}q_{B}(t)=4A_{g} m_{eff}\Omega\dot{q}_{A}(t)\end{array} \tag{4}\]  
where the \(w_{B}=\sqrt{k_{B}/m_{eff}}\) is the resonant frequency of sense mode. The quality factor of the drive mode and sense mode can be given as:  
\[\begin{array}{l}Q_{A}=(w_{A}m_{eff})/\gamma_{A}\\ Q_{B}=(w_{B}m_{eff})/\gamma_{B}\end{array} \tag{5}\]  
The driving amplitude (\(q_{A0}\)) and sensing amplitude (\(q_{B0}\)) can be derived from Equations (4) and (5):  
\[\begin{array}{l}q_{A0}=\frac{Q_{A}F_{0}}{K_{A}}\\ q_{B0}=\frac{4A_{g}m_{eff}\Omega w_{A}/k_{B}}{\sqrt{\left[1-\left(\frac{w_{A}} {B}\right)^{2}\right]^{2}+\left(\frac{w_{A}}{B}\frac{w_{A}}{B}\right)^{2}}} \end{array} \tag{6}\]  
Then, the mechanical sensitivity \(S_{mech}\) can be derived [29]:  
\[S_{mech}=\frac{4A_{g}Q_{A}F_{0}w_{A}}{k_{A}\sqrt{\left(w_{A}^{2}-w_{B}^{2} \right)^{2}+\left(\frac{w_{A}w_{B}}{Q_{B}}\right)^{2}}} \tag{7}\]  
The mechanical sensitivity can be maximized when there is no frequency split (\(w_{A}=w_{B}\)):  
\[S_{mech}=\frac{4Q^{2}A_{g}F_{0}}{k_{A}w_{0}} \tag{8}\]  
where \(w_{0}\) is the resonant frequency when there is no frequency split (\(w_{0}=w_{A}=w_{B}\)); \(Q\) is the quality factor when there is no frequency split (\(Q=Q_{A}=Q_{B}\)). It follows that the mechanical sensitivity should be improved by increasing the quality factor or reducing the frequency split.