# Suboptimal Analysis of the Differential System of the Conceptual Trailer Air Brake Valve
## 3 Results  
### Determination of Characteristic Operating Times and Maximum Displacement of the Valve Piston  
Implemented in the Matlab-Simulink environment, the calculation procedures based on Equation (10) and Table 1 enabled the determination of pressure waveforms in the valve chambers, chamber volume changes, and piston displacement. Example waveforms are shown in Figure 4. The correlation of the pressure waveforms (Figure 4a) with the change in chamber volume (Figure 4b) and the displacement of the valve piston (Figure 4c) can be seen, which confirms the correctness of the adopted mathematical description and calculation methodology. At the point where the piston reaches its maximum displacement (\(l_{\textit{pmax}}\)), chamber A increases its volume to its maximum value (\(V_{A\max}\)), the opposite of chamber B (\(V_{B\min}\)). At the moment of actuation of the differential part of the valve \(p_{\textit{in}}\), due to the proximity of chamber A and the relatively high value of the inlet capacity (\(\mu A\))in, the course \(p_{\textit{in}}\) differs little from the course \(p_{\textit{in}}\) (Figure 4a). The initial increase in pressure in chamber B until the piston reaches maximum displacement \(h_{\textit{pmax}}\) is due to air compression, since this chamber is perfectly sealed and has no outlet. The reason why the pressure \(p_{A}\) is not able to equalize with \(p_{B}\) at the point where the piston reaches displacement \(h_{\textit{pmax}}\) is due to the presence of a spring in chamber B. After the piston reaches \(h_{\textit{pmax}}\), the pressure in chamber B increases with a smaller gradient than until it reaches \(h_{\textit{pmax}}\). The reason for this fact is the throughput of the opening in the piston (\(\mu A\))\({}_{\textit{noz}}\) and the associated possibility of air flow between chambers A and B. From the waveforms shown in Figure 4, one can see the dominant effect of spring stiffness and piston bore throughput on the performance of the differential part of the brake valve. Increasing the spring stiffness will delay the opening of the valve \(t_{o}\) but shorten its closing \(t_{c}\) (Figure 4c). In this case, \(t_{o}\) and \(t_{c}\) were determined when the piston exceeded \(0.25\) mm in the opening and closing processes. It may also prevent the maximum piston displacement \(h_{\textit{pmax}}\) from being achieved, according to Table 1. The differential valve should accelerate the braking process at the initial stage but also  
Figure 3: Forcing and pressure waveforms at selected points of the pneumatic braking system: (**a**) forcing and pressure waveforms at selected points in the pneumatic system; and (**b**) input pressure waveforms before and after correction (description in text). to tracking action. The accelerating action will be realized only if the pressure in chamber A builds up with a greater gradient than in chamber B and is also able to overcome the force coming from the spring. The result will be a displacement of the piston. With a smaller input pressure gradient \(p_{in}\), the throughput (\(\mu A\))\({}_{max}\), combined with the stiffness of the spring \(k\), should only provide tracking action for the valve. This is confirmed by the results presented in [45]. This confirmed the validity of the objective adopted in this study. The evaluation of the influence of the spring stiffness and the throughput of the piston bore opening of the valve's differential part on its functional characteristics is important and can determine the direction of design development for this type of device. Guaranteeing a short response time (\(t_{o}\) = 0.24 s) of the valve in the differential member to the forcing was possible with a spring with the smallest stiffness of 0.5 \(\times\)\(10^{5}\) N/m and the smallest throughput value of 0.05 \(\times\)\(10^{-6}\) m\({}^{2}\) among those analyzed (Figure 5).