## 3 Influence of Nonlinear Factors on Performance of EMS Inertial Stabilizer  
### Theoretical Analysis and Simulation of Nonlinear Factors  
From theoretical calculation and analysis, it is known that the relationship between electromagnetic force, displacement and current is nonlinear, and the farther the mass block deviates from the equilibrium position, the higher the nonlinearity. In order to accurately control the inertial mass block, it is necessary to improve the accuracy of the electromagnet to control the acceleration of the mass block. Near the balance position of the mass block, Taylor expansion is used to analyze the magnetic force. Because the influence above the third order is small, it is only reserved to the second order. We obtain:  
\[F_{\rm M}=F(g_{0},\ I_{0})+\frac{\mu_{0}N^{2}AI_{0}}{2g_{0}^{2}}(i-I_{0})-\frac {\mu_{0}N^{2}AI_{0}^{2}}{2g_{0}^{3}}(g-g_{0})+\frac{\mu_{0}N^{2}AI_{0}^{2}}{4g _{0}^{2}}(i-I_{0})^{2}+\frac{\mu_{0}N^{2}AI_{0}^{2}}{2g_{0}^{4}}(g-g_{0})^{2} \tag{15}\]  
when the currentin the upper and lower electromagnetic coils is basically stable, and the force on the mass block in the \(Z\) coordinate direction is:  
\[ \begin{array}{ll}F_{\rm Z}\!=\!\Sigma F_{M}&-F_{G}cos\omega t\\ &=F(g_{0},\ I_{0})+\frac{\mu_{0}N^{2}AI_{0}}{2g_{0}^{2}}(i-I_{0})-\frac{\mu_{0 }N^{2}AI_{0}^{2}}{2g_{0}^{3}}(g-g_{0})+\frac{\mu_{0}N^{2}AI_{0}^{2}}{4g_{0}^{ 2}}(i-I_{0})^{2}\\ &+\frac{\mu_{0}N^{2}AI_{0}^{2}}{2g_{0}^{4}}(g-g_{0})^{2}\!-mgcos\omega t\\ \end{array}  \tag{16}\]  
when the mass block is in equilibrium. \(F(g_{0},\ I_{0})\) is equal to the mass block gravity \(mg\), and Equation (15) is brought into Equation (16) to obtain:  
\[ \begin{array}{ll}F_{\rm Z}=mg&+\frac{\mu_{0}N^{2}AI_{0}}{2g_{0}^{4}}(i-I_{0 })-\frac{\mu_{0}N^{2}AI_{0}^{2}}{2g_{0}^{4}}(g-g_{0})+\frac{\mu_{0}N^{2}AI_{0} ^{2}}{4g_{0}^{2}}(i-I_{0})^{2}\\ &+\frac{\mu_{0}N^{2}AI_{0}^{2}}{2g_{0}^{4}}(g-g_{0})^{2}\!-mgcos\omega t\\ \end{array}  \tag{17}\]  
In the vicinity of the equilibrium position, the angle \(\omega t\) is small, \(mg\approx mgcos\omega t\). In order to control the inertial mass block accurately, it is necessary to improve the response speed of the electromagnet to the control force of the mass block, and the electromagnet is an inductive load, so it is very necessary to analyze the response speed characteristics of the control force of the electromagnet. When the voltage across the electromagnet coil is equal to the driving voltage, the current in the coil changes the fastest:  
\[\frac{di}{dt}=\frac{2g_{0}(U_{d}-iR_{0})}{N^{2}\mu_{0}A} \tag{18}\]  
From Equations (17) and (18), the response speed of the control force in the \(Z\) coordinate direction is:  
\[\frac{\partial F_{Z}}{\partial t}=\frac{\partial F_{Z}}{\partial i}\frac{di}{dt }=\frac{N^{2}iA\mu_{0}}{2g_{0}^{2}}\frac{2g_{0}(U_{d}-iR_{0})}{N^{2}\mu_{0}A}= \frac{i(U_{d}-iR_{0})}{g_{0}} \tag{19}\]  
From Equation (19), along the vertical direction of the ship, that is, along the \(Z\) coordinate direction in Figure 2, the control force response speed is directly related to the solenoid current, the voltage across the solenoid, the coil resistance and the air gap. The electromagnetic force response speed can be controlled by controlling the coil current, adjusting the voltage at both ends of the coil, series-parallel resistance and adjusting the air gap. Because the electromagnetic coil is an inductive load, when the electromagnetic coil current is controlled, the frequency of the current change should not be too large, as this would cause the coil current to lag behind the voltage change. The simulation was carried out after the device was machined. The resistance of the electromagnetic coil is 0.6 \(\Omega\)and the inductance is 2.49 mH. The number of turns of the electromagnetic coil is 138.