# A Hybrid Model for Freight Train Air Brake Condition Monitoring
## 2 Materials and Methods
### Main Components Implemented in the Model
#### 2.3.2 Distributor  
The distributor (labelled with (2) in Figure 6) pressure (p\({}_{\text{d}}\)) is modelled as the depression of the main brake pipe pressure (p\({}_{\text{mbp}}\)) from the control reservoir pressure (p\({}_{\text{cr}}\)), ranging between 0.1 bar and 1.5 bar. The first pressure parameter (identified as Act in Table 1) is related to the distributor activation: for a pressure drop lower than this value, the braking action is not activated, and the distributor valve remains closed with a zero-pressure reference. The activation threshold has been empirically tuned to model the time lag of the brake plant mechanical system, resulting in a braking activation time in the main brake pipe that is different from the activation time in the brake cylinder. With a pressure drop of 1.5 bar (identified as Dmax in Table 1), the brake cylinder develops the maximum braking force, resulting in the distributor pressure saturation. This parameter comes from the UIC 540-0 normative [33]. In Figure 7a, the simplified scheme of the distributor is reported, while in Figure 7b, the free body diagrams of the forces acting on the piston-valve assembly of the distributor are shown. Figure 6: Simplified scheme of the air brake system. The equation of motion of the piston-valve assembly of the distributor is reported in Equation (1). \[\left(\mathrm{M_{p}}+\mathrm{M_{v}}\right)\mathrm{\ddot{x}}=\left(\mathrm{p_{cr}}- \mathrm{P_{mbp}}\right)\mathrm{A_{1}}-\mathrm{P_{ar}}\mathrm{A_{3}}-\mathrm{p_{ d}}\mathrm{A_{2}}-\mathrm{k_{s}}\mathrm{\Delta x}-\mathrm{F_{pl}}-\mathrm{F_{f}} \tag{1}\]  
where \(\mathrm{M_{p}}\) and \(\mathrm{M_{v}}\) are the distributor piston mass and opening valve mass connecting the auxiliary reservoir to the distributor outlet pipe, \(\mathrm{p_{cr}}\) is the pressure at the control reservoir, \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Component** & **Parameters** & **Value** & **Description** & **Source** \\ \hline \multirow{3}{*}{Brake plant} & Vmin & 0.005 m\({}^{3}\) & Brake cylinder minimum volume & T3000e specs \\  & Vmax & 0.01 m\({}^{3}\) & Brake cylinder maximum volume & T3000e specs \\ \hline \multirow{3}{*}{Distributor} & Tchar & 24 s & Normative braking time based on G mode & UIC Normative \\  & Act & 0.1 bar & Distributor activation pressure & T3000e specs \\  & Dmax & 1.5 bar & Distributor saturation pressure for full-service & UIC Normative \\ \hline \multirow{3}{*}{Braking phase} & Fb & 0.4 bar & First phase braking ending pressure at the & T3000e specs \\  & EQP & 0.15 bar &
\begin{tabular}{c} Braking time based on G mode \\ activation \\ \end{tabular} & T3000e specs \\  & Pmax & 3.6 bar & Maximum braking pressure at the brake & UIC Normative \\ \hline \multirow{3}{*}{Kink valve} & K0 & 0.2 bar & Distributor pressure for kink valve S-mode & T3000e specs \\  & K1 & 0.7 bar & Distributor pressure for kink valve S-mode & T3000e specs \\  & LIM1 & 1.36 bar & Distributor pressure for kink valve load & T3000e specs \\  & LIM2 & 1.2 bar & Distributor pressure for kink valve load & T3000e specs \\  & & & & saturation in the releasing phase & T3000e specs \\ \hline \multirow{3}{*}{Releasing phase} & Sact & 0.68 bar & Distributor pressure for release activation & T3000e specs \\  & S1 & 0.175 bar & Brake cylinder pressure for ending efflux & T3000e specs \\  & Tin & 273.18 K & Ambient temperature & Physical parameter \\  & R & 287.1 1/kgK & Gas constant & Physical parameter \\  & patm & 1 bar & Atmospheric pressure & Physical parameter \\  & A & 0.0000125 m\({}^{2}\) & Orifice area for release in G mode & T3000e specs \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model brake system parameters. Figure 7: Schematic overview of (**a**) the distributor and (**b**) a free body diagram of the forces acting on the piston-valve assembly of the distributor.