The present Part I of the comprehensive study is dedicated to establishing the fundamental mathematical and experimental apparatus required for the multi-criteria optimization of the geometric parameters of an innovative three-rotor hydraulic pump with bilateral lantern meshing, subjected to the actual volumetric flow rate $Q$ and the volumetric efficiency ${\eta }_v$. A complex approach integrating similarity theory, dimensional analysis, and mathematical modeling is employed to define and refine the two objective functions subject to optimization. Based on the area of geometric existence of the gearing and additionally imposed geometric and operational constraints, the exact domain for seeking the optima of $Q$ and ${\eta }_v$ is defined. Based on the statistical processing of experimental data, empirical dependencies of the objective functions are derived, accounting for the influence of the pump’s geometric parameters, the operational conditions and the physical properties of the fluid. The criterion equation of the volumetric efficiency, approximated using all experimental data, was obtained with a very high coefficient of determination $R^2=0.9898$. The rest of the study, related to parameter optimization, is contained in Part II. In it, through numerical investigation and analytical proof, the universal optimal parametric values of the dimensionless geometric coefficients (the relative lantern radius $r_{c,opt}^{*}$ and the shortening coefficient ${\lambda }_{opt}$) are identified to achieve maximum flow rate and volumetric efficiency. Furthermore, in Part II, a multi-criteria Pareto optimization (MINLP) is conducted to resolve the engineering conflict regarding the number of teeth $z$, and a direct simple algebraic dependency $z=f\left(p,n\right)$. The generalized results from both parts provide a methodological toolkit and recommendations for the optimal selection of the geometric parameters in the design of pumps of this type with respect to the actual flow rate and volumetric efficiency, in accordance with the operating conditions and regimes.

Part II of this study builds upon the mathematical framework developed and validated in Part I for describing the geometry and volumetric performance indicators of an innovative three-rotor hydraulic pump with bilateral lantern meshing. This part focuses on the numerical investigation and multi-objective optimization of these indicators through the proper selection of geometric parameters. The aim of the study is to establish the isolated and combined influence of the dimensionless geometric parameters—number of teeth $z$, relative lantern radius $r_c^{\ast }$, and cycloid shortening coefficient $\mathsf{\lambda }$—on the actual flow rate $Q$ and the volumetric efficiency ${\mathsf{\eta }}_v$ under various operating conditions, while maintaining the overall dimensions of the pump element in the radial and axial directions. Through detailed numerical analysis and subsequent rigorous analytical proof, it has been established that the optimal values of the geometric coefficients $r_{c,opt}^{\ast }$ and ${\mathsf{\lambda }}_{opt}$ are strictly determined and provide a simultaneous global maximization of both indicators ($Q,{\eta }_v$), regardless of the operating pressure $p$, rotational speed $n$, or the viscosity of the working fluid. However, an analytically irresolvable conflict regarding the number of teeth $z$ has been identified: a small number maximizes the flow rate, whereas a large number increases the volumetric efficiency. To overcome this contradiction, the problem is formulated within the class of mixed-integer nonlinear programming (MINLP), and multicriteria Pareto optimization is applied, combined with the PSIMS method for the selection of optimal compromise solutions. An empirical relationship (with a coefficient of determination of $R^2=0.9603$) has been derived, which defines the optimal number of teeth $z_{opt}$ as a function of the operating pressure and rotational speed. The proposed methodology provides a reliable and applicable tool for designing highly efficient three-rotor pumps tailored to specific operational requirements.

The article presents an accurate mathematical model of an electro-pneumatic drive system for positioning a pneumatic cylinder with a double-acting rod, controlled by PWM and four high speed On/Off pneumatic valves. The main elements of the system are included in the model: screw compressor, high-speed pneumatic valves, pneumatic cylinder with double-out rod, PID controller. A simulation of the pneumatic positioning system was carried out using the Simscape language in MATLAB. The system is built using the Simscape Foundation gas domain, implementing simulations of piston rod displacement. An analysis of the results obtained and the simulation model is performed.

This paper presents a demonstration experimental test bench for investigating the dynamic characteristics of an electropneumatic positioning system with High Speed Pneumatic Valves (HSPV) and Pulse Width Modulation (PWM) control. A specialized virtual instrument has been developed for real-time control, data acquisition, and processing. The transient response of the electropneumatic positioning system is examined under various single-step input signals with frequencies ranging from 30 Hz to 70 Hz and varying duty cycles from 30% to 90%. The results are presented graphically, demonstrating that the system exhibits better linear characteristics compared to conventional pneumatic actuation systems.