# Suboptimal Analysis of the Differential System of the Conceptual Trailer Air Brake Valve
## 5 Conclusions  
Suboptimal analysis of the differential system of the pneumatic valve made it possible to evaluate its performance. Spring stiffness and piston bore throughput were used as input variables. The mathematical model, adopted on the basis of the necessary simplifications and lumped method, allowed a number of groups of results (maps) to be obtained numerically. These results served as input data for further processing aimed at selecting the most optimal solution. On this basis, it should be concluded that the work has been achieved. The main conclusions of the analysis are:  
* The smallest spring stiffness analyzed of 0.5 \(\times\) 10\({}^{5}\) N/m showed an opening time of 0.24 (0.33) s and a valve closing time of 7.08 (7.5) s depending on the throughput of the piston bore;
* The highest stiffness (1.45 \(\times\) 10\({}^{5}\) N/m) resulted in a slight increase in opening time to 0.26 (0.37) s and a marked decrease in closing time to 1.32 (7.08) s, depending on the throughput of the piston bore;
* The smallest piston bore throughput of 0.05 \(\times\) 10\({}^{-6}\) m\({}^{2}\) resulted in an opening time of 0.24 (0.36) s and a closing time of 7.08 (7.5) s depending on the spring stiffness;
* The highest piston bore throughput of 0.525 \(\times\) 10\({}^{-6}\) m\({}^{2}\) showed a similar opening time of 0.24 (0.33) s with a significant reduction in closing time to 1.32 (1.59) s;
* Not all analyzed configurations were able to achieve a maximum piston displacement (valve opening) of 7.3 mm, which may affect the effectiveness of the valve acceleration mode;
* Different times of accelerating action calculated as the difference of closing and opening times did not guarantee constant efficiency due to different displacements of the piston;
* The optimization carried out using the genetic algorithm yielded results corresponding to its stated purpose but required complementary analysis to select the optimal variant.  
The results of calculations, analysis, and optimization show that the differential brake valve, although developed in the second half of the 20 th century, is still a functional challenge. However, the mathematical model of the differential part of the valve presented in the paper and the proposal to use genetic optimization show the way to increase the efficiency of this type of design. The purpose of the differential valve is to accelerate the system's operation during the first phase of sudden braking and to be able to switch to tracking operation. The suboptimal analysis included in this study indicated that this is possible.  
The widespread use of electronics in actuators and their appropriate control, especially in the throughput area of the inter-chamber opening, may result in the emergence of a design of this type of valve with adaptive capabilities to trailer loading conditions.  
Conceptualization, M.K. and D.S.; literature review, M.K., methodology, D.S. and M.K., software, D.S. and M.K.; visualization, M.K.; resources and supervision, D.S.; original draft preparation, M.K.; writing--review and editing, D.S. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The datasets generated and/or analyzed during the current study are available from the corresponding author upon reasonable request.  
These analyses were financed through a subsidy from the Ministry of Science and Higher Education of Poland for the discipline of mechanical engineering in the Faculty of Mechanical Engineering at Bialystok University of Technology (WZ/WM-IIM/5/2023).