## 2 Modelling the Brake Pressure Response
### Modelling the Brake Chamber  
Figure 4a shows the structure of the brake chamber, which consists of a diaphragm, a push plate, a push rod, and a spring, thereby forming two chambers. When braking is applied, the rod-less chamber is charged, the air expands to push the piston and push rod, and then the rod chamber is compressed. However, there are four holes in the rod chamber with diameters of 6-8 mm so that the air can be discharged to the atmosphere as quickly as possible, and this means that there is no back pressure in the rod chamber. Thus, we need only consider the rod-less chamber when modelling the brake chamber.  
The main parameters are shown in Figure 4b. By differentiating the gas state equation \(pV=mR\theta\) with respect to time, the pressure response in the rod-less chamber is obtained as  
\[\frac{dp_{c}}{dt}=\frac{1}{V_{c}}\bigg{[}G_{c}R\theta_{c}-p_{c}\frac{dV_{c}}{ dt}+\frac{p_{c}V_{c}}{\theta_{c}}\frac{d\theta_{c}}{dt}\bigg{]} \tag{8}\]  
To improve the accuracy of the model, we consider here the heat exchange in the brake chamber. Thus, according to the first law of thermodynamics, the energy flow in or out of the chamber always satisfies  
\[dU=dH+dW+Q \tag{9}\]  
where \(U\) is the internal energy, \(H\) is the enthalpy that flows in the chamber, \(W\) is the work done by the rod, and \(Q\) is the heat exchanged between the inner chamber and the surroundings through the wall, which can be expressed as follows according to [PERSON]'s law of cooling:  
\[Q=hS_{h}(\theta_{a}-\theta_{c}) \tag{10}\]  
The heat-exchange coefficient \(h\) can be measured using the stop method [32]. However, it changes during the charging process, so we adopt an average value here of \(h=20\). In addition, the model is adiabatic for \(h=0\), tending to isothermal has \(h\rightarrow\infty\).  
By differentiating Equation (9) with respect to time and combining it with the equations \(U=c_{p}\theta\), \(H=c_{p}\theta\) and the Mayer equation \(c_{p}-c_{p}=R\), we obtain the temperature response as  
\[\frac{d\theta_{c}}{dt}=\frac{1}{m_{c}c_{p}}\bigg{[}c_{p}G_{c}(\theta_{a}- \theta_{c})+R\theta_{a}G_{c}-p_{c}\frac{dV_{c}}{dt}+hS_{hc}(\theta_{a}-\theta _{c})\bigg{]} \tag{11}\]  
Figure 4: Structural diagram of brake chamber.  
The volume \(V_{c}\) of the diaphragm changes as that of a cone rather than a cylinder, and is given approximately by  
\[V_{c}=A\Big{(}x_{0}+\frac{\pi}{3}x\Big{)} \tag{12}\]  
The force on the piston is shown diagrammatically in Figure 4b, from which the kinematic equation of the push rod can be obtained as follows according to [PERSON]'s second law of motion:  
\[\left\{\begin{array}{l}a=(p_{c}-p_{a})A_{p}-k(x_{0k}+x)-F_{d}\\ \frac{dt}{dt}=\frac{1}{M}\left[(p_{c}-p_{a})A_{p}-k(x_{0k}+x)-F_{d}\right]\\ (a>0\cap x=s)\cup(0<x<s)\\ \frac{dt}{dt}=0\qquad(a<0\cap x=0)\cup x=s\end{array}\right. \tag{13}\]  
where \(A_{p}\) is the effective loaded area, which changes during the stroke of the piston [33] and is expressed as  
\[A_{p}=\left\{\begin{array}{ll}A\qquad 0\leq x\leq\frac{2}{3}s\\ \frac{7}{5}A-\frac{3A}{56}x\quad\frac{2}{3}s<x\leq s\end{array}\right. \tag{14}\]  
In Equation (13), \(F_{d}\) is the deforming force on the diaphragm, which also changes during the stroke and is expressed as  
\[F_{d}=\left\{\begin{array}{ll}F_{d0}+\frac{4F_{0k}}{s}x\quad 0\leq x\leq \frac{1}{3}s\\ 2F_{d0}\qquad\frac{1}{3}s\leq x\leq\frac{3}{3}s\\ 0\qquad\frac{1}{3}s\leq x\leq s\end{array}\right. \tag{15}\]  
Through the three models established above, we obtain an analytical model for the brake pressure response. The method for obtaining solutions to this model is discussed in detail in Section 3.