# Research on Magnetic Coupling Flywheel Energy Storage Device for Vehicles
## 4 Principles and Strategies of Energy Recovery  
### The Principle of Energy Recovery  
The vehicle dynamics model with the magnetic coupling flywheel energy storage device is simplified to obtain an equivalent dynamic model, as shown in Figure 9. The main reducer and the gear train are simplified to the driven half shaft in the magnetic coupling mechanism, and a simplified model with the equivalent moment of inertia \(J_{b}\) is obtained. According to the states of the magnetic coupling mechanism of the car under different driving conditions, the relevant mechanical equations are derived from the three stages of starting, braking and acceleration for the equivalent model, and the mechanical analysis is carried out to obtain the energy recovery of the driving conditions. The establishment of the equivalent mechanical model involves the following formulas [25]:  
\[M=\sum_{i=1}^{k}F_{i}v_{i}\frac{\cos\theta_{i}}{\omega}+\sum_{i=1}^{k}+M_{i} \frac{\omega_{i}}{\omega} \tag{3}\]  
\[J=\Sigma_{i=1}^{k}m_{i}(\frac{\gamma_{si}}{\omega})^{2}+\Sigma_{i=1}^{k}J_{si} (\frac{\omega_{i}}{\omega})^{2} \tag{4}\]  
The state of the magnetic coupling mechanism of the car based on different driving conditions is shown in Table 1:  
Figure 9: Simplified model diagram. (1) Initial stage  
When the car starts, the magnetic coupling flywheel energy storage device does not work, and the magnetic ring is at the leftmost end of the driving shaft. At this time, the two half shafts are in a separated state, and there is no torque transmission. The angular velocity of the energy storage flywheel is 0. Since the driven half shaft is connected to the vehicle through the gear train, the device and the vehicle transmission system, the following can be obtained:  
\[\left\{ \begin{array}{l}\omega_{a}=0\\ \omega_{b}=\frac{v_{g}i\omega_{l}}{3 br}\\ J_{b}=\frac{d\omega_{T}}{l\omega_{T}}\\ \delta=1+\frac{\sum I_{b}I_{b}}{mr^{2}}+\frac{1f_{b}i^{2}i^{2}i^{2}\eta_{T}}{mr^ {2}}\end{array} \right. \tag{5}\]  
where \(J_{b}\) is the equivalent moment of inertia of the right drive axle; \(\omega_{a}\) is the flywheel angular velocity; \(\omega_{b}\) is the left drive axle angular velocity; \(v_{a}\) is the vehicle speed; \(i_{0}\) is the main reducer gear ratio; \(i_{g}\) is the transmission gear ratio; \(\eta_{T}\) is the automobile transmission efficiency; \(i_{m}\) is the energy storage flywheel gear ratio (gear train transmission ratio); \(I_{W}\) is the moment of inertia of the wheel; \(m\) is the mass of the vehicle; \(r\) is the radius of the wheel; \(\delta\) is the conversion factor of the rotating mass of the vehicle. (2) Braking phase  
Figure 10 is a schematic diagram of the mechanical model under the braking state of the vehicle:  
Among them, \(T_{m}\) is the friction torque of the rolling bearing; \(T_{b}\) is the vehicle braking torque equivalent to the right drive axle; \(T_{f}\) is the friction torque; \(T_{r}\) is the vehicle resistance torque equivalent to the right drive axle; \(T_{n}\) is the energy storage flywheel air friction torque. The formula for calculating the equivalent moment in the above figure is as follows:  
\[\left\{ \begin{array}{l}T_{b}=\frac{T_{b1}+T_{b2}}{\frac{1}{l\omega_{m}}}\\ T_{r}=\frac{r_{T}^{m}}{l\omega_{m}}+\frac{M_{m}}{l\omega_{m}}\end{array} \right. \tag{6}\]  
where \(T_{b1}\) and \(T_{b2}\) is the braking torque of the two driving wheels; \(M_{m}\) is the motor anti-drag torque; \(F_{m}\) is the driving resistance of the vehicle. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Vehicle Status** & **Start** & **Brake Deceleration** & **Accelerate after Deceleration** \\ \hline Permanent magnet & & & \\ coupling transmission & Disconnection & & \\ mechanism status & & & \\ Energy storage & There is damping for & & \\ Flywheel status & free rotation & & \\ Drive mode & Motor & No & \\ Time division & (0,\(t1\)) & & (\(t1\),\(t2\)) and (\(t2\),\(t3\)) & (\(t3\),\(t4\)) and (\(t4\),\(t5\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: State diagram.