# 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
The air supply to chamber A encountered the smallest force from the spring among the analyzed cases, resulting in the fastest movement of the piston in the initial phase until \(h_{\text{pmax}}\) was reached. The smallest value of throughput of the piston opening delayed the flow of air between chambers. This case had the opposite effect on the values of closing time, obtaining the highest value of \(t_{c}\) = 7.50 s (Figure 6). The spring was responsible for this state of affairs, which was unable to move the piston back to its initial state, with the additional flow resistance resulting from the small throughput value of the piston bore. The extreme opposite case to that described above was found with a stiffness of 1.45 \(\times\)\(10^{5}\) N/m and a throughput of 0.525 \(\times\)\(10^{-6}\) m\({}^{2}\). The opening time increased by more than 54% to 0.37 s (Figure 5), while the closing time decreased by more than 82% to 1.32 s (Figure 6). Based on this, it was concluded that the choice of spring stiffness and throughput of the opening in the piston of the differential part of the valve must be a compromise. The opening times were arranged almost symmetrically on the map (Figure 5), indicating a relationship close to proportionality. The situation was different for closing times (Figure 6), where only throughputs of less than 0.15 \(\times\)\(10^{-6}\) m\({}^{2}\) resulted in a significant increase in this time. The differences in opening times \(t_{o}\) in the cases studied were just over 54%, while closing times \(t_{c}\) were over 82%. A lack of symmetry is evident, the reason for which, in addition to the parameters in question, could also be the difference in inlet throughput (\(\mu A\))\({}_{in}\) and (\(\mu A\))\({}_{max}\). This difference affected the intensification of the air filling and compression processes in chambers A and B.  
Figure 4: Examples of the resulting waveforms from the calculations: (**a**) pressure waveforms in the inlet and valve chambers; (**b**) changes in chamber volume; and (**c**) piston displacement (description in text). The piston did not achieve its maximum displacement in all analyzed cases (Figure 7). The part of the map with the lowest values of spring stiffness and piston bore throughput showed the achievement of the possible maximum piston displacement of 7.3 mm (Table 1), while the opposite part of the map showed only 0.97 mm. There is some correlation between the map of piston lift \(l_{p\text{max}}\) and the map of opening times \(t_{o}\) (Figure 5), indicating that a shorter opening time \(t_{o}\) favors the piston to achieve greater displacement, approaching \(h_{p\text{max}}\). However, this is not binding, as flow is possible between chambers A and B as the piston moves, especially at higher values of piston bore throughput. Therefore, the piston may not have reached its maximum displacement. This is confirmed by a map showing the times to reach maximum piston displacement (Figure 8), where several results showed \(t_{hp\text{max}}\) = 1.17 s. Juxtaposing this with Figure 7, one can see the different values of maximum displacements that accompany this. Figure 5: Determined map of opening times \(t_{o}\) with varying values of spring stiffness \(k\) and throughput of the piston bore \((\mu A)_{nnz}\) according to Table 1.  
An important functional parameter of the differential part of the valve was the duration of operation in acceleration mode. This mode meant assisting the braking system with an additional reservoir located close to the trailer brake valve. The acceleration action time was determined from the relationship \(t_{w}=t_{c}-t_{o}\) and is shown in Figure 9. The map \(t_{w}\) was similar in distribution to the \(h_{p\mathrm{max}}\) map (Figure 7). As can be seen in Figure 9, the acceleration action in the longest time dimension was possible at low throughputs, especially at 0.05 \(\times\) 10\({}^{-6}\) m\({}^{2}\).