## 6 Conclusions  
The research results of this study can be applied to optimize the design of axial piston pumps based on specific requirements (e.g., balancing compactness, cost, and performance in hydraulic systems where noise reduction is critical). The importance of these results is relevant for industries where noise and vibration control is key and is particularly prominent in automotive, aerospace, and industrial machinery.  
To minimize structural loading on the pump casing, the relationship between pressure pulsations and transmitted vibrations was investigated, aiming to achieve a more uniform distribution of forces. The study demonstrated that increasing the number of pistons generally reduces pressure pulsations. Understanding the interrelationship between the number of pistons, pressure pulsations, and casing vibrations is crucial for designing more efficient and reliable axial piston pumps. Specifically, reducing pressure pulsations can enhance the performance and lifespan of both the pump and the connected hydraulic system.  
All the assumptions that could affect the generalization of the results were made during the simulation and modeling process and refer to idealized conditions (e.g., perfect fluid dynamics and specific operating range) and do not fully reflect variations in real conditions. Limited experimental data are related to experimental settings (e.g., using limited pump sizes, material limitations, and lack of long-term performance testing).  
The study is focused on specific operating conditions (e.g., specific flow rates or pressures), and it is important to note that the results may vary outside of these ranges. An extension of the study is planned to take into account the effects of additional parameters such as fluid temperature, fluid viscosity, and pump casing material properties on vibration and pulsation characteristics. The influence of parameters such as piston geometry and cylinder block size will be the subject of future research on pressure pulsations and vibrations. Long-term experimental studies are planned to evaluate the effects of different numbers of pistons on pump durability and performance over extended operating periods, including the effect of wear. It is planned to use more sophisticated Computational Fluid Dynamics (CFD) or Finite Element Analysis (FEA) methods to simulate complex fluid-structure interactions in more detail.  
The results of the study highlight the significant role that the number of pistons plays in the pressure pulsation and vibration characteristics of axial piston pumps, which have a direct impact on the performance and longevity of the hydraulic system. Optimizing the number of pistons is key to the development of axial piston pumps, specifically hydraulic systems.  
Conceptualization, M.R., L.S., S.R.M., N.G.D., R.P. and J.L.; methodology, M.R., L.S., S.R.M., N.G.D., R.P. and J.L.; software, M.R., L.S., S.R.M., N.G.D., R.P. and J.L.; validation, M.R., L.S., S.R.M., N.G.D., R.P. and J.L.; formal analysis, M.R., L.S., S.R.M., N.G.D., R.P. and J.L.; investigation, M.R., L.S., S.R.M., N.G.D., R.P. and J.L.; resources, M.R., L.S., S.R.M., N.G.D., R.P. and J.L.; data curation, M.R., L.S., S.R.M., N.G.D., R.P. and J.L.; writing--original draft preparation, M.R., L.S., S.R.M., N.G.D., R.P. and J.L.; writing--review and editing, M.R., L.S., S.R.M., N.G.D., R.P. and J.L.; visualization, M.R., L.S., S.R.M., N.G.D., R.P. and J.L.; supervision, M.R., L.S., S.R.M., N.G.D., R.P. and J.L.; project administration, M.R., L.S., S.R.M., N.G.D., R.P. and J.L.; funding acquisition, R.P. All authors have read and agreed to the published version of the manuscript.  
This paper is funded by the University \"Union-NikolaTesla\", Faculty of Information Technology and Engineering.  
Not applicable.  
Data available statement: Data are contained within the article.  
The authors would like to thank the Editor-in-Chief, Editor, and anonymous reviewers for their valuable reviews.  
The authors declare no conflicts of interest.