## 2 Modeling and Simulation Method
### Modeling of the Modified MHD ARS Combing with Coriolis Effect (Abbreviated as C-MHD ARS)  
#### 2.1.1 The Working Principle of C-MHD ARS  
The MHD ARS, which measures angular velocity, can also provide angular position and acceleration by integrating or differentiating the output with respect to time. Then, the linear accelerometers can be derived, which is important for navigation systems [25; 26]. The basic MHD ARS is mainly based on the magnetohydrodynamic effect as shown in Figure 1a. The magnetic field \(B_{z}\)* is fixed to a cylinder case and the whole attached to a body whose inertial angular motions are measured. As the case rotating with angular velocity \(\Omega\), the inertial-fixed fluid moves through the case-fixed magnetic field \(B_{z}\)* with relative velocity \(u_{\theta}\). The top and bottom of the fluid annulus are insulated, while the inner and outer perimeters of the fluid annulus in contact with electrodes to measure the voltage \(\varphi\). The electromagnetic force \(f_{e}\) and viscous force \(f_{v}\) are always opposite to the direction of the relative velocity. Assuming there is no secondary flow and induced magnetic field, the basic MHD ARS transfer function can be approximately simplified as [12; 27]:  
\[\left|\frac{\varphi(s)}{\Omega(s)}\right|=\frac{B_{z}WRs}{s+\sigma{B_{0}}^{2}/ \rho+\
u/h^{2}}=\frac{B_{z}WRs}{s+(\
u/h^{2})(1+\text{Ha}^{2})} \tag{1}\]  
where, the conducting fluid is assumed to be viscous with the kinematic viscosity coefficient \(\
u\), incompressible, and homogeneous with a constant electrical conductivity \(\sigma\) and mass density \(\rho\). The top and bottom sides are parallel insulating plates with height \(h\). Inside and outside walls are concentric conducting cylinders with radii \(r_{i}\) and \(r_{o}\). Here, the equivalent radius \(R=(r_{i}+r_{o})/2\) is defined and \(W=r_{o}-r_{i}\) is the width of the annular channel, and Ha is the Hartmann number (\(\text{Ha}=\sqrt{\sigma{B_{0}}^{2}h^{2}/\
u\cdot\rho}\)), which gives the ratio electromagnetic force \(f_{e}\) to viscous force \(f_{v}\). When rotating with high-frequency vibrations, the two forces \(f_{e}\) and \(f_{v}\) are small enough to be ignored and the voltage \(\varphi\) across the inertial-fixed fluid is linearly related to the inertial angular velocity \(\Omega\). While the angular velocity frequency is near or below the cutoff frequency, the electromagnetic induction is weakened. Compared with the basic one in Figure 1a, the C-MHD ARS introduces a radial velocity \(u_{r}\) to accelerate the relative circumferential velocity \(u_{\theta}\) due to the Coriolis effect at low frequency, as shown in Figure 1b. The Coriolis force \(f_{c}\) is induced by the radial flow as the fluid rotating with the walls of the sensor case, whereas the MHD effect dominates at high frequency for the inertial-fixed fluid. The measurement through the whole bandwidth is achieved by combing the Coriolis effect at low frequency with MHD effect at high frequency. Figure 1: Description of the annular channel in the basic magnetohydrodynamic angular rate sensor MHD ARS and C-MHD ARS. (**a**) The basic one (**b**) the C-MHD ARS (with Coriolis Effect). The simplified formula can be approximately written as (2) under three assumption. (1) The radial velocity in the annual channel is constant and uniform; (2) The axis and radial velocity can be negligible compared with the circumferential velocity; (3) Only taking the current induced by relative circumferential velocity into consideration and ignoring the induced magnetic fields under the assumption of a small Reynolds number. The specific deduction can be seen in our earlier paper [24].