# Finite Element Analysis of the Distribution Parameters of a Metal Dot Array in a SAW Gyroscope
## 2 Theoretical Analysis  
Previous research showed that distributing a metal dot array on the propagation path of the SAW is an effective method to improve the detection sensitivity performance of a SAW gyroscope [24]. The typical structure of a SAW gyroscope, as shown in Figure 2, includes interdigital transducers (IDTs), a metal dot array and a piezoelectric substrate. The structure consists of two parallel and reverse SAW delay lines to form a differential sensing mode [7]. When the SAW gyroscope rotates, the vibrating particles, in dual delay lines arranged in opposite directions, are subjected to a Coriolis force in the opposite direction [25]. Taking the frequency difference of dual delay lines as the output signal can not only realize temperature compensation, but also obtain stronger detection sensitivity performance [26]. Figure 1: Schematic of the gyroscopic effect on the Rayleigh wave. Compared with a metal dot array, the thickness of the piezoelectric substrate is much larger; therefore, we can regard the piezoelectric substrate in SAW gyroscope devices as a semi-infinite space [27]. Taking a single metal dot element on a semi-infinite piezoelectric substrate as the research object (as shown in Figure 3), we studied the relationship between the intensity of the SAW gyroscopic effect and the distribution parameters of the metal dot array. The value \(h\) in Figure 3 represents the thickness of the metal dot, and \(z<0\) represents the piezoelectric substrate in the semi-infinite space. SAW propagates along the \(x\)-axis direction with speed \(V_{0}\), and the whole model rotates around the \(y\)-axis with angular velocity \(\Omega\)[28; 29]. When the substrate model rotates around the \(y\)-axis at a constant angular velocity \(\Omega\), the vibrating particles are subjected to the action of the Coriolis force and centrifugal force, which can be, respectively, expressed as [30]:  
\[F_{cor}=2 mV\times\Omega \tag{1}\]  
\[F_{cent}=m\Omega\times(\Omega\times u) \tag{2}\]  
In which, \(F_{cor}\) represents the Coriolis inertia force, \(F_{cent}\) represents the centrifugal inertia force, \(m\) represents the mass, \(V\) represents the vibration velocity, \(u\) represents the elastic displacement and \(\Omega\) represents the angular velocity. Since the angular frequency of the SAW is much larger than the angular velocity, the influence of the centrifugal force on the particle is much smaller than that of the Coriolis force which can be ignored. Equation (1) shows that the strength of the Coriolis force depends on three factors: the mass \(m\) of the vibrating particle, the vibration velocity \(V\) and the angular velocity \(\Omega\). Therefore, adding a metal dot array on the path of the acoustic wave propagation can increase the mass of the particles; in particular, the metal dots with an appropriate size can make the Coriolis force superimpose in the same direction to enhance the sensitivity. On the contrary, an improper metal dot array will cause Coriolis force cancelling in the opposite direction and weaken the sensitivity. In the SAW gyroscope devices distributed with a metal dot array, the periodic metal dot array plays a role similar to the interdigital electrode and causes the characteristic frequency to split into two, namely the symmetric frequency (\(f_{sc+}\)) and the anti-symmetric  
Figure 3: Solve object. Figure 2: SAW sensor structure. frequency (\(f_{sc-}\)). When the substrate model rotates around the \(y\)-axis, the symmetric and anti-symmetric frequencies will change accordingly. However, it should be noted that with the change of the normalized angular velocity \(\Omega/\omega\), the variation of symmetric frequency and anti-symmetric frequency is not the same, which corresponds to the change in the characteristic frequencies of the SAW propagating along the \(+\)x and \(-\)x axes of the dual delay lines with \(\Omega/\omega\), respectively, in Figure 2[11; 31].