# Research on Magnetic Coupling Flywheel Energy Storage Device for Vehicles
## 4 Principles and Strategies of Energy Recovery
### The Principle of Energy Recovery
Figure 10: Schematic diagram of mechanical model during braking. According to reference [26], the calculation formula is as follows:  
\[Tm=\frac{1}{2}\mu dP \tag{7}\]  
where \(P\) is the bearing load; \(\mu\) is the friction coefficient; \(d\) is the bearing inner diameter. According to reference [27], the air friction torque of the energy storage flywheel is calculated by the following formula:  
\[\left\{ \begin{array}{l}Tn=\frac{1}{2}\tau\pi D^{2}l\\ \tau=\frac{1}{2}C_{f}\rho v\\ C_{f}=\frac{\rho(6\omega_{b})}{(Re)^{0.5}}\\ \mathrm{Re}=\frac{\rho ND^{2}}{\mu}\end{array} \right. \tag{8}\]  
where \(\tau\) is the surface shear stress of the energy storage flywheel; \(D\) is the diameter of the energy storage flywheel; \(l\) is the thickness of the energy storage flywheel; \(C_{f}\) is the local friction coefficient; \(v\) is the linear velocity of the cylinder surface; \(N\) is the rotating speed; \(\mu\) is the fluid dynamic viscosity; \(\rho\) is the fluid density; \(D\) is the cylinder diameter. When braking, the car speed is \(v_{0}\), the driven half-shaft angular velocity is \(\omega_{b}\), and the driving half-shaft angular velocity is \(\omega_{a}\). According to the position change of the magnetic ring in the magnetic coupling mechanism, the vehicle braking deceleration is divided into two stages:  
The first stage: combining stage (\(t1\)-\(t2\))  
When the car starts to brake, the magnetic ring immediately moves to the right from the left end of the driving shaft, that is, it is immediately combined, the driven shaft decelerates under the action of \(M_{m}\), \(F_{m}\) and \(T_{r}\), and the energy-storage flywheel accelerates to \(t2\) under the action of \(T_{r}\). The angular velocities of the driving shaft and the driven shaft are equal at this time. Its mechanical equation is  
\[\left\{ \begin{array}{l}\int_{0}^{\omega_{a}^{t2}}J_{d}d\omega=\int_{t1}^{t2} \Big{(}T_{f}-T_{m}-T_{n}\Big{)}d_{t}\\ \int_{\omega_{b}^{t1}}^{\omega_{b}^{t2}}J_{b}d\omega=\int_{t1}^{t2}\Big{(}-T_{ f}-T_{r}-T_{b}\Big{)}d_{t}\\ \omega_{a}^{t2}=\omega_{b}^{t2}\\ w_{b}^{t1}=\frac{\tau_{t1}\omega_{in}}{3\delta\tau}\end{array} \right. \tag{9}\]  
Assuming that the road surface is flat, the tires are in normal condition, and the deceleration of the car is \(a_{1}\), there is no lock-up phenomenon during the braking process. The research object is the whole vehicle, and it can be known from [PERSON]'s second law:  
\[\frac{\Big{(}-T_{r}-T_{b}-T_{f}\eta_{0}\Big{)}i_{0}i_{m}}{r}=\delta ma \tag{10}\]  
where \(\eta_{0}\) is the mechanical transmission efficiency from the right transmission half shaft to the wheel. Thus,  
\[\left\{ \begin{array}{l}\omega_{a}^{t2}=\omega_{b}^{t2}=\frac{\big{(}T_{f}-T _{m}-T_{n}\big{)}\Delta t1}{I_{a}}\\ \Delta t1=t2-t1\end{array} \right. \tag{11}\]  
During the process from asynchronous to synchronous, the main and driven shafts will be accompanied by asynchronous friction, which will generate sliding and grinding work. The calculation formula of sliding and grinding work is as follows:  
\[W=\int_{t1}^{t2}T_{f}(\omega_{a}-\omega_{b})dt \tag{12}\]  
The second stage: separation stage (\(t2\)-\(t3\))When the angular velocity of the driving shaft and the driven shaft is equal, the energy storage ends, and the magnetic ring moves to the left immediately. The driven shaft is connected to the whole vehicle and will continue until the deceleration is completed under the action of braking torque and vehicle resistance torque. At the same time, the energy storage flywheel rotates freely due to the friction force and air resistance of the rolling bearing. In this process, the mechanics equation is  
\[ \begin{cases}\int_{\omega_{b}^{2}}^{\omega_{b}^{3}}I_{a}dt=\int_{t2}^{t3}(-T_{ m}-T_{n})dt\\ \int_{\omega_{b}^{2}}^{\omega_{b}^{3}}I_{b}dt=\int_{t2}^{t3}(-T_{b}-T_{r})dt \end{cases}