## 2 Modeling and Mechanical Analysis of EMS Inertial Anti-Rolling Mass Block
### Mechanical Analysis of Single Mass Block of EMS Inertia Stabilizer  
The inertial anti-rolling mass block is suspended between the upper and lower electromagnets without an external disturbance force. The air gap between the mass block and the upper and lower electromagnets is basically the same and is set to \(g_{0}\). Assuming that the ship is subjected to external interference forces such as sea waves with counterclockwise \(F_{\text{in}}\omega t\), the ship would roll counterclockwise, as shown in Figure 2, while the mass block would remain unchanged due to inertia. Because of the symmetry of the structure, the forces on the left and right inertial anti-rolling mass blocks are also symmetrical, so we only analyzed the forces on the right mass block. The gap between the right mass block and the upper right electromagnet becomes larger. Assuming that the gap becomes \(g_{0}+x\), the gap between the right mass block and the lower right electromagnet becomes smaller. Assuming that the gap becomes \(g_{0}-x\), the right mass blockwould be subjected to three forces, namely gravity \(F_{\text{G}}\), electromagnetic force \(F_{\text{M}}\) and Coriolis force \(F_{\text{C}}\). Assuming that  
Figure 1: Structural diagram of EMS inertial stabilizer. the static equilibrium position of the inertial anti-rolling mass block as the coordinate origin, the dynamic model of the right inertial anti-rolling mass block and the ship can be described as follows:  
\[\left\{ \begin{array}{c}m\ddot{x}_{1}(t)=F_{\rm MR}+F_{\rm C}-F_{\rm G}+(k_{1}+k_ {2})[x_{2}(t)-x_{1}(t)]\\ M\ddot{x}_{2}(t)=Fsin\omega t-F_{\rm MR}-F_{\rm C}-(k_{1}+k_{2})[x_{2}(t)-x_{1 }(t)]\end{array} \right. \tag{1}\]  
In Equation(1), \(k_{1}\) and \(k_{2}\) are the support stiffness coefficients between the mass block and the upper right and lower right electromagnet, \(k_{1}=\frac{N^{2}A_{\rm H}i_{0}i_{0}}{\dot{s}_{0}^{2}}\frac{di}{dx}+\frac{N^{2} A_{\rm H}i_{0}^{2}i_{0}^{2}}{\dot{s}_{0}^{2}}\)[19]; \(k_{2}=\frac{N^{2}A_{\rm H}i_{0}i_{0}}{\dot{s}_{0}^{2}}\frac{di}{dx}+\frac{N^{2 }A_{\rm H}i_{0}^{2}i_{0}}{\dot{s}_{0}^{2}}\)[19]. Gravity: \(F_{\rm G}=\rm mg\); electromagnetic force: \(F_{\rm M}=\frac{N^{2}i^{2}A_{\rm H}i_{0}}{4g_{0}^{2}}\), Coriolis force: \(F_{\rm C}=-2m\omega\times v\). \(N\) is the number of turns of the electromagnet coil, \(A\) is the cross-sectional area of the electromagnet core, \(\mu_{0}\) is the relative permeability of air, \(i_{0}\)U and \(i_{0}\)D are the coil currents of upper right electromagnet and lower right electromagnet, respectively, \(m\) is the mass of the right mass block, \(g\) is the acceleration of gravity, \(\omega\) is the angular speed of the ship rotating counterclockwise, and \(v\) is the speed of the mass block along the Z coordinate direction \(dx/dt\). The Coriolis force exists because of the relative motion of particles and the rotation of the reference frame, and its direction is perpendicular to the direction of angular speed and relative motion speed. Along the horizontal direction of the ship, that is, the Y coordinate direction in Figure 2, because the angular speed of the ship is constantly changing with external disturbances such as sea waves, the direction of the Coriolis force \(F_{\rm C}\) is also constantly changing. In this direction, the force on the mass block is:  
\[F_{\rm Y}=F_{\rm C}-F_{\rm G}sin\omega t=2m\omega\times v-mgsin\omega t \tag{2}\]  
According to Equation (2), the force of the right mass block in the Y coordinate direction mainly depends on the angular speed of ship rotation and the mass of the mass block. In order to prevent the mass block from moving along the Y direction due to the force in the Y direction, a positioning axis is set in the middle of the mass block, which passes through the geometric center of the upper right and lower right electromagnets and is fixed with the mass block as a whole.