Determination of Pressure Jump Dependence and Time Constants of Hydraulic Pumps with Constant Pressure and Variable Flow

Abstract

Simulations of the pump response time refer to the determination of the time constant of the transient process when the flow and pressure change. The changes mentioned in the standards are precisely defined and prescribed by the “MIL-P-19692E” norms. The simulation showed that the time constants are within the permissible limits prescribed by these norms. The diagram shows the responses for the considered pump and for the case when the flow changes from Q_n to Q_min and changes from Q_min to Q_n. The time constants t₁ and t₂ are defined on the diagrams, with the parameters that the pump has. In addition, the standards define the pressure jump that occurs during the transient process as well as time constants. From the flow change diagram, it can be seen that the flow change is also very fast and takes place in time intervals shorter than 0.1 s. By evaluating the size of the time constants, t₁ and t₂, it can be concluded that they have a value below 0.05 s, which meets the regulations in that area. Also, the size of the jump pressure meets the regulations because it is only 1.2 M Pa above the nominal pressure.

1. Introduction

Hydraulic systems play a vital role in diverse industrial and mobile applications, offering precise control and high-power density. Central to these systems are hydraulic pumps, which determine system performance by converting mechanical energy into hydraulic energy [1]. Among various types, pumps with constant pressure and variable flow have garnered significant attention due to their ability to maintain consistent output pressure while dynamically adjusting flow rates to meet varying operational demands.

A critical aspect of these pumps is their dynamic response to changes in operating conditions, which is often characterized by two key parameters: pressure jump and time constants. The pressure jump represents transient pressure spikes that occur during sudden variations in flow or load, while time constants define the system’s responsiveness and stabilization period after such disturbances [2]. These parameters are integral to ensuring stable and efficient operation, particularly in applications requiring rapid response and minimal energy loss.

Despite their importance, the interplay between pressure jump and time constants remains a complex and underexplored area [3]. Factors such as pump design, control mechanisms, fluid properties, and external system dynamics can significantly influence this relationship.

Understanding these dependencies is crucial for improving pump performance, enhancing system stability, and preventing issues such as cavitation, noise, and excessive wear [4].

The MIL-P-19692E standard specification and detailed pump specification establish requirements for variable flow hydraulic pumps for use in aircraft hydraulic systems that conform to and are defined in MIL-H-5440 and MIL-H-8891, as applicable. General requirements for Type I and Type II hydraulic system pumps are specified in MIL-H-8775, and for Type III system pumps in MIL-H-8890.

Testing and simulation software were used to model different operating scenarios and optimize pump parameters [5]. It helps to identify potential problems and improve pump performance under different conditions. Aircraft hydraulic system design is considered so that the system effectively minimizes pressure losses. The work is devoted to the issues of analysis and minimization of pressure losses in aircraft hydraulic systems. Aircraft hydraulic power losses and operational delay associated with piping and fitting connections. It was pointed out that pressure losses can be analyzed using CFD methods, and that hydraulic pipes and corner joints are an important source of pressure losses. The equivalent length and equivalent length of the same shape methods were used in the calculations. We would like to emphasize that an important source of losses in hydraulic systems is the hydraulic lines, and the reduction in pressure losses can be performed at the system design stage [6]. Safety mechanisms are built into the hydraulic system to ensure reliable operation, even in the event of pump failure.

The appearance of vibrations and noise during pump operation the reliability of the aircraft’s hydraulic system. In addition to vibrations, the following parameters also have a significant effect: flow (K), pressure (p), number of revolutions of the pump drive shaft (n), type of hydraulic oil (viscosity), and oil temperature (T). Operating parameters should be adjusted to maximize pump efficiency and reduce mechanical and volumetric losses.

Testing of constant-pressure, variable-flow hydraulic pumps typically involves specialized equipment and test setups to accurately assess performance, efficiency, and reliability.

The pressure control valve (relief valve) maintains a constant pressure in the system, regardless of the flow. Adjustable to set desired test pressure level, ensuring stable conditions. Load simulation includes devices such as orifice plates, restrictors, or dynamic actuators that simulate different loads on the pump while keeping the pressure constant. It helps in evaluating the performance of the pump under real operating conditions. Pressure transducers and gauges include high-precision sensors that measure pressure within the system. Real-time data are key to ensuring pump operation under constant pressure.

The temperature control unit ensures that the oil temperature remains stable, as temperature fluctuations can affect both pump efficiency and test results. The data acquisition system collects data from pressure transducers, flow meters, and other sensors to monitor pump performance in real time. It enables in-depth analysis of flow, pressure, efficiency, and other key performance indicators.

A dynamometer measures the power required by the pump and can also simulate load conditions [7]. A torque sensor measures the torque produced by the pump to assess mechanical efficiency, especially when changing the flow rate.

Monitoring vibration and noise is an important assessment of mechanical integrity and noise performance, especially in applications where pumps are sensitive to vibration. Computational fluid dynamics (CFD) plays a key role in various engineering applications in multiple fields. CFD is used to analyze airflow over wings and fuselage to optimize aerodynamic performance and reduce drag. CFD enables engineers to visualize complex fluid behaviors, optimize design and make informed decisions, significantly reducing development time and cost while improving performance and safety [8].

This study aims to analyze the relationship between pressure jump and time constants in hydraulic pumps with constant pressure and variable flow. By combining experimental data with theoretical modeling, the research seeks to identify key influencing factors, provide insights into dynamic behavior, and establish guidelines for optimizing pump design and operation. This work is expected to contribute to the development of more reliable and efficient hydraulic systems across various industries.

2. Mathematical Model

Section 2 presents process models in a piston axial pump. The simultaneous integration of nonlinear differential equations of boundary conditions and partial differential equations of flow in the pressure pipeline required the application of a computer and the development of a suitable computer program.

The program that connects and simultaneously solves all the mentioned differential equations, equations of changes in characteristic flow sections and changes in the physical characteristics of the fluid, required an appropriate structure and organization. The original AKSIP software system was specially developed for mathematical modeling of current and hydrodynamic processes during the entire working cycle of a piston axial pump with combined separation of the working fluid. The AKSIP program is modularly designed and consists of the main AKSIP program and its modules. The program is written in the programming language Digital Visual Fortran 5.0. The principles of structural and modular programming are used [9]. The program consists of a main program and modules.

More important programs are written as enclosed modules that connect to each other or to the main program but can also be used independently.

The parameters of the hydrodynamic process of the piston axial pump (flow of pressure, suction and compression) cannot be accurately determined purely experimentally, nor purely mathematically. Sufficiently accurate parameters can be obtained by the combined application of pressure flow measurement in the cylinder, mathematical modeling of the actual hydrodynamic process, whereby systematic measurement errors and unknown parameters can be simultaneously determined.

Equations describing hydrodynamic processes are the basis for mathematical modeling of axial piston pumps with constant pressure and variable flow [10]. System assumptions and definitions have been adopted:

• pump type (axial piston pump with fixed displacement along the stroke of the piston);

• working conditions (constant pressure but variable flow);

• components (pistons, cylinder block, valve plate angle, valve plate and hydraulic fluid properties);

• physical limitations (mechanical limitations of pistons, fluid compressibility, leakage and friction).

Kinematic modeling includes the angle of the flap that regulates the variable flow. The movement along the stroke of the piston is proportional to the angle of the folding plate, and thus the length of the stroke of the piston is determined [11]. The motion of each piston is typically modeled using the angular velocity of the cylinder block and the angle of the swash plate. The volume of liquid displaced by each piston as a function of time also depends on the angular velocity of the rotating block of cylinders.

Fluid dynamics and total flow depend on the number of pistons, the displacement per piston, and the rotational speed of the cylinder block [12]. Variable flow control is performed by the angle of the swash plate which can be adjusted based on the requirement to maintain constant pressure while changing the flow. This is usually achieved by a feedback control mechanism.

Variable displacement pumps take into account the compressibility of the hydraulic fluid and the flow in the system when modeled. The fluid modulus, effective volume, and leakage must be considered.

Leakage flow is typically modeled as a function of differential pressure and can be included to reflect efficiency loss [13].

The constant pressure control strategy is based on the use of a pressure control valve or an electronic feedback mechanism. The model includes a feedback loop to compare the measured pressure with a reference value and adjust the swash plate angle accordingly.

Flow is modulated based on system demand, and the control system adjusts the angle of the baffle plate to achieve the required flow while maintaining the desired constant pressure [14].

2.1. Description of the Hydrodynamic Processes of the Pump

Differential equations describe the time-varying behavior of the pump in different operating conditions.

Solving these equations requires numerical methods (e.g., Runge–Kutta) to simulate the dynamic response of the pump in different scenarios [15].

• mass flow of fluid through suction opening 1, at the entrance to the suction chamber of the pump, Figure 1:

(1)${\displaystyle \frac{d{m}_1}{dt}}={\mathsf{\sigma }}_1\cdot {\mathsf{\mu }}_1\cdot A_1\cdot \sqrt{2\cdot {\rho }_s\left|p_u-p_s\right|}$

where

• σ₁ = 1 for p_u ≥ p_s, σ₁ = −1 for p_u < p_s

• $A_1$—geometrical flow section of the intake pipe

• ${\mu }_1$—flow coefficient of the intake pipe

Figure 1

Pump components [16]. (1) suction line connection, (2) pump suction space, (3) cylinder block, (4) pump delivery chamber, (5) delivery line connection, (6) piston, (7) panel with socket, (8) inclined panel, (9) drive pump shaft, (10) drive pump shaft base.

[Figure omitted. See PDF]

• mass flow of fluid through the split pump organ during filling one of the pump cylinders:

(2)${\displaystyle \frac{d{m}_u}{dt}}={\mathsf{\sigma }}_u\cdot {\mathsf{\mu }}_u\cdot A_u\sqrt{2\cdot {\rho }_s\cdot \left|p_s-p_c\right|}$

where

• σ_u = 1 for p_s ≥ p_c, σ_u = −1 for p_s < p_c

• A_u—geometrical flow section of the intake split organ

• ${\mu }_u$—flow coefficient of the intake split organ

• mass balance of the intake space is:

(3)${\displaystyle \frac{d{m}_s}{dt}}={\displaystyle \frac{d{m}_1}{dt}}-{\displaystyle \sum _{j=1}^z\frac{d{m}_{u,j}}{dt}}$

tags are:

• j = 1, 2, …, z_c, z_c—piston numbers.

The pressure in the suction chamber of the pump is described by a differential equation

(4)${\displaystyle \frac{d{p}_s}{d\mathsf{\phi }}}={\displaystyle \frac{E}{V_s\cdot {\mathsf{\rho }}_s}}\cdot \left({\displaystyle \frac{d{m}_1}{d\mathsf{\phi }}}-{\displaystyle \sum _{j=1}^{z_c}\frac{d{m}_{u,j}}{d\phi }}\right)$

The compressibility of the hydraulic fluid is derived from the principles of fluid mechanics and thermodynamics. A common approach is to use the fluid volume modulus E (4).

• $E$—modulus of elasticity;

• $\phi$—angle of the pump drive shaft.

The pressure in the pump cylinder is described by the equation:

(5)${\displaystyle \frac{d{p}_c}{d\mathsf{\phi }}}={\displaystyle \frac{E}{V_c}}\cdot \left[{\displaystyle \frac{A_c\cdot v_k}{\mathsf{\omega }}}+{\displaystyle \frac{1}{{\mathsf{\rho }}_c}}\cdot \left({\displaystyle \frac{d{m}_u}{d\mathsf{\phi }}}-{\displaystyle \frac{d{m}_i}{d\mathsf{\phi }}}\right)\right]$

where

• $V_c=V_{c\min }+V_{cx};V_{cx}\hspace{0.17em}=A_c\cdot x_k$—current volume of the cylinder; the change in the pump cylinder volume caused by piston moving: ${\displaystyle \frac{d{V}_c}{dt}}=-A_c\cdot v_k$, $x_k$—current displacement of the piston.

• mass balance of the delivery chamber is:

(6)${\displaystyle \frac{d{m}_v}{dt}}={\displaystyle \sum _{j=1}^{z_c}\frac{d{m}_{i,j}}{dt}}-{\displaystyle \frac{d{m}_2}{dt}}$

where j = 1, 2, …, z_c—the numbers of cylinders

• mass flow out of the delivery chamber into the delivery pipe is:

(7)${\displaystyle \frac{d{m}_2}{dt}}={\mathsf{\sigma }}_2\cdot {\mathsf{\mu }}_2\cdot A_2\cdot \sqrt{2\cdot {\mathsf{\rho }}_t\cdot \left|p_v-p_n\right|}$

where σ₂ = 1 for p_v ≥ p_n, σ₂ = −1 for p_v < p_n

• A₂—geometrical flow section of the delivery pipeline

Differential equation for pressure in the delivery pump chamber:

(8)${\displaystyle \frac{d{p}_v}{d\mathsf{\phi }}}={\displaystyle \frac{E}{V_v\cdot {\mathsf{\rho }}_v}}\cdot \left({\displaystyle \sum _{j=1}^{z_c}\frac{d{m}_{i,j}}{d\mathsf{\phi }}}-{\displaystyle \frac{d{m}_2}{d\mathsf{\phi }}}\right)$

• mass flow through a concentric clearance between the cylinder and the piston:

(9)${\displaystyle \frac{d{m}_z}{dt}}={\displaystyle \frac{\mathsf{\pi }\cdot D_c\cdot \Delta r^3}{12\cdot \mathsf{\eta }\cdot x_k\left(\mathsf{\phi }\right)}}\cdot \left(p_c-p_s\right)\cdot {\mathsf{\rho }}_c$

where $D_c$—diameter of cylinder, Δr—radial clearance between the piston and the cylinder, η—dynamic viscosity, $x_k\left(\mathsf{\phi }\right)$—current displacement of the piston, p_c—the pressure in the cylinder, p_s—pressure in the suction area of the pump, ρ_c—fluid density in the pump cylinder.

2.2. Flow in the Suction and Delivery Pipeline of the Pump

Modeling efficiency and losses (mechanical and hydraulic) includes friction between moving parts, which can be modeled as a damping force or torque. Losses due to leakage, flow restriction and fluid compressibility can be modeled as pressure-dependent flow loss [17].

The model is validated by comparing simulation results with experimental data, and parameters are adjusted (such as leakage coefficients or friction terms) to improve accuracy. Analysis of the sensitivity of pump performance to various parameters (e.g., swash plate angle, fluid properties, leakage) is a summary of the key equations. Displacement per piston, flow rate, pressure dynamics, and leakage flow are working parameters that significantly affect the efficiency of the pump [18].

This general approach can be applied in various simulation software to study the performance of axial piston pumps under different operating conditions, and further refined to take into account effects from real operating conditions.

Continuity Equations

The equation of continuity with the functions p, v, ρ during an isentropic change in state is:

(10)${\displaystyle \frac{\partial p}{\partial t}}+v\cdot {\displaystyle \frac{\partial p}{\partial x}}+a^2\cdot \rho \cdot {\displaystyle \frac{\partial v}{\partial x}}=0$

where $\rho$ and v—density and fluid velocity per cross section.

$a=\sqrt{{\left({\displaystyle \frac{\partial p}{\partial \rho }}\right)}_s},\mathrm{speed\; of\; sound\; in\; fluid},$

where $p=p(t,x)$ and $\rho =\rho (t,x)$, the t and x coordinates are functions of time.

Momentum conservation equation

(11)$v\cdot {\displaystyle \frac{\partial \rho }{\partial t}}+\rho \cdot {\displaystyle \frac{\partial v}{\partial t}}+v{\displaystyle \frac{\partial }{\partial x}}\left(\rho \cdot v\right)+\rho \cdot v\cdot {\displaystyle \frac{\partial v}{\partial x}}+{\displaystyle \frac{\partial p}{\partial x}}=-f_r\cdot \rho$

where $f_r$—unit force of friction.

Flows are unsteady in the flow fiber.

3. Results of Measuring of Parameters

Experimental testing of a constant-pressure, variable-displacement hydraulic pump focuses on performance such as efficiency, flow rate, pressure stability, noise levels, and temperature effects.

It was observed how the flow rate responds to changes in system requirements while maintaining a constant pressure. The flow rate decreases as system demand increases, confirming the pump’s ability to adjust volume to meet desired conditions.

Maintaining a constant output pressure even with different flow rates.

It was observed that the pressure in the system remains relatively stable, which indicates the correct functioning of the compensation mechanism in the pump.

Pump efficiencies were measured in different operating conditions. A drop in efficiency occurs at higher flow rates or pressure adjustments due to increased friction and fluid leakage. The efficiency curves show the optimum operating point at medium flow rates.

When oil viscosity decreases, internal leakage increases and the volumetric efficiency of the pump decreases. In addition, lubrication conditions deteriorate through a decrease in oil film thickness.

The experimental tests were carried out on a real object that was designed according to the model of a hydraulic system in which the hydropump works in the system under real conditions (Figure 2). The experiments mainly included tests of dynamic characteristics such as dynamic processes of transient phenomena and changes during flow regulation. The mentioned characteristics are extremely important for application on hydro systems of aircraft, considering that these systems have extremely fast processes [19].

The hydraulic system has a reservoir with a volume of about 7 × 10⁻³ m³ of hydraulic fluid. The fluid participates in the transformation and transmission of approximately 7.4 kW power. The hydraulic fluid is mineral-based, AMG10, with a kinematic viscosity of ν = 15 × 10⁻⁶ m²/s, at a temperature of 400 °C. The characteristics of AMG10 correspond to the hydraulic fluid used in aviation. The pump is designed to work with the liquid “Aero Shell 40, which is used for the temperature range from −550 °C to 1350 °C. At times when the system does not need hydrostatic energy, the pump reduces the flow to approx. 2 × 10⁻⁵ m³/s. Then, it absorbs far less power compared to the case when the excess liquid is returned to the tank after the pressure is reduced. In the hydraulic system (Figure 3) there is an electromagnetic distributor, (pos. 6), with which the flow of the pump can be very quickly transferred from the maximum to the minimum value. There is a safety valve in the pressure line, (pos. 3), which, if necessary, relieves the system to the set level. The indirect pressure regulator, (pos. 7), enables precise pressure regulation in the pressure line, while the pressure variations are small, they are regulated by variable resistance dampers, (pos. 8).

The following four parameters were tested:

• pressure in the discharge line;

• pump flow;

• temperature of the working fluid in the tank;

• temperature at the outlet of the choke with variable resistance.

3.1. Pressure Pulsations

Axial piston pumps are prone to pressure pulsation, which occurs as a result of pump cycling and inherent design features. These pulsations can be categorized into high-pressure pulsations and low-pressure pulsations, both of which have different causes and effects. The subject of this article is the examination of the influence of mechanical vibrations on changes in the pressure pulsation spectrum of hydraulic systems. The paper shows that components and equipment equipped with hydraulic systems are a source of vibrations of a wide frequency spectrum. In addition, hydraulic valves are also subject to vibrations. Vibrations of the base on which the hydraulic valve is mounted force the control element of the hydraulic valve to vibrate. The vibration of the control element produced in this way causes changes in the pressure pulsation spectrum of the hydraulic system. Understanding these pulsations is critical to improving pump performance (Figure 4).

High-pressure pulsations generally occur in the discharge line of the pump, where the liquid exits under high pressure.

Low-pressure pulsations occur on the suction side of the pump, where fluid is drawn into the pump chamber [21]. Cavitation occurs when the pressure in the suction line drops below the vapor pressure of the hydraulic fluid, causing vapor bubbles to form. As these bubbles collapse, they create pressure spikes and pulsations on the suction side, which can be transmitted to the discharge side.

The pump is tested in a hydraulic system that simulates the actual system in which the pump will be installed, as defined in the detailed pump specification. The volume of the system is simulated using a pipe of the diameter of the discharge line. The length of the pipeline is chosen so that the natural frequency is resonant with the pulsation frequency.

3.2. Response Time

Measuring the response time of a constant-pressure, variable-displacement hydraulic pump involves estimating how quickly the pump adjusts its flow to maintain a desired pressure in response to changing system demands (Figure 5). Pressure and flow sensors are installed at the pump’s inlet and outlet to capture real-time data.

A high-speed data acquisition system was used to record pressure and flow rate with high temporal resolution. The pump operates in a steady state (constant pressure and steady flow) until a sudden change (flow demand by opening a valve or activating a downstream actuator followed by closing a valve or reducing the downstream load).

By comparing the response times under different conditions, the dynamic performance of the pump can be evaluated and its suitability for applications that require rapid flow adjustments.

For the tested pump, it took 200 ms to adjust from the initial flow rate of 50 L/min to the new flow rate of 30 L/min. So, its response time is 200 ms.

When the pump initially drops to 25 L/min before settling to 30 L/min, an overshoot of 5 L/min is noted. Measurement tools and software provide high accuracy and sampling rates, to capture real-time data with millisecond resolution [16]. The pressure transducer is located in the discharge line of the pump towards the outlet connection.

Indirect pressure regulator, pos. 7 (Figure 3), the nominal and maximum pump pressure values p_n = 20 MPa and p_max = 21 MPa are set. With the electromagnetic distributor, pos. 6, the pump receives a command for two operating modes: with nominal flow and pressure (Q_n = 3.7 × 10⁻⁴ m³/s, p_n = 20 MPa), and then with minimum flow and maximum pressure (Q_min = 2 × 10^{- 5} m³/s, p_max = 21 MPa). The diagram in Figure 7 shows flow of parameters.

When we talk about the diagram of the time constant of pressure change in hydraulic systems, especially in relation to hydraulic pumps, we can look at how changes in flow rates—especially from nominal to minimum flow and vice versa—affect the dynamics of pressure (Figure 6).

The time constant in hydraulic systems is a measure of how quickly the system responds to changes in flow. It is defined by the product of the system resistance and capacitance. In hydraulic terms, resistance can be related to flow limitations in a system, and capacitance refers to the ability of a fluid system to store energy (as in batteries).

An initial sudden drop or spike in pressure, followed by a gradual approach to a steady value, illustrating the time constants associated with each transition.

Understanding the behavior of the time constant when transitioning between nominal and minimum flow is critical to optimizing hydraulic system performance, ensuring efficient operation, and preventing damage from rapid pressure changes. Proper analysis of these dynamics can help design more resilient hydraulic systems that respond effectively to changing demands.

Figure 7 shows a diagram of the entire flow regulation process with changed pressure jump values. From that diagram, a significantly enlarged part of the diagram that defines more precisely the size of the jump is separated and it amounts to p = 2 MPa above the nominal pressure. Figure 7

Flow rate when the pressure changes from p_r = 3 MPa to p_max = 21 MPa [20].

[Figure omitted. See PDF]

The diagrams in Figure 8a,b are significantly enlarged details from the diagram in Figure 7. Magnification was carried out in order to be able to accurately read the considered magnitudes of pressure jump and time constants. By evaluating the size of the time constants, t₁ and t₂, it can be concluded that they have a value below 0.05 s, which meets the regulations in that area as well [20]. Also, the size of the jump pressure meets the regulations because it is only 1.2 MPa above the nominal pressure.

In order for the pump to have the smallest possible time constant t₂ and return to full flow, the operating parameter that affects the efficient supply of liquid to the hydraulic system was changed.

The parameter was changed, with the aim of correcting the time constant t₂.

The operating parameter of the reactive piston, which defines the stiffness of the spring, significantly affects the process of switching the pump from the regime of minimum flow and maximum pressure to the regime of nominal flow and nominal pressure. By reducing the stiffness of the spring, C₂ shows a longer transient process during the experiment and thus an increase in the time constant.

The change diagram is shown in Figure 9a,b, the right side of the diagram is shown greatly enlarged in order to read the time constant, t₂, which is t₂ = 0.2 s.

Tests of dynamic characteristics such as dynamic processes of transient phenomena and changes during pump flow regulation are extremely important for application on aircraft hydraulic systems, considering that these systems have extremely fast processes.

4. Conclusions

This study provides a comprehensive analysis of the dynamic behavior of hydraulic pumps with constant pressure and variable flow, specifically focusing on the determination of pressure jump dependence and time constants. The findings contribute to a deeper understanding of the pump’s response characteristics under varying load conditions, which has direct implications for optimizing hydraulic system performance in real-world applications. A comprehensive study of constant pressure and variable flow hydraulic pumps covers their application in various industries. Emphasizing the importance of maintaining constant pressure and variable flow in aerospace hydraulics, this study specifically addresses the determination of pump response time.

Key insights gained from this research include:

1. Pressure Jump Dependence: The results highlight that pressure jumps in constant pressure pumps are primarily influenced by the rate of flow change and the system’s control dynamics. A significant relationship was observed between pressure spikes and flow transitions, underscoring the importance of precise control mechanisms to minimize pressure oscillations, which can otherwise lead to system instability or component fatigue.

2. Time Constants: The identification of time constants in the pump’s response behavior offers valuable data for tuning control systems. The ability of the pump to rapidly adjust its flow rate while maintaining constant pressure demonstrates its suitability for high-demand applications where swift response times are crucial. However, the time constants measured in this study suggest that system designers must account for slight delays to optimize performance, especially in applications with strict timing requirements.

3. Implications: For recording the time constant, the pump is prepared for operation with nominal pressure p_n = 20 MPa and maximum pressure p_max = 21 MPa. By means of an indirect pressure regulator, the nominal pressure value is set. The electromagnetic distributor pump receives a command to operate in two modes: with nominal flow and pressure (K_n = 3.7 × 10⁻⁴ m³/s, p_n = 20 MPa), then with minimum flow and maximum pressure (K_min = 2 × 10⁻⁵ m³/s, p_max = 21 MPa) and again with nominal flow and nominal pressure. The described changes are shown diagrammatically.

Significantly enlarged details of the transition process in the first and second cases are presented. The time constants t₁ and t₂ are within the permitted limits, below 50 ms. The time constant t₁, is slightly smaller than the time constant t², but both are within the required limits.

• 4.. Recommendations for Future Work: To expand upon this research, future studies could explore the effect of fluid properties, varying environmental conditions (temperature, viscosity), and different control strategies on pressure jump behaviors and time constants. Additionally, the integration of advanced sensors and real-time monitoring systems would allow for more precise measurements and control in high-speed applications.

During the development of application software for mathematical modeling of piston axial pumps, special attention was paid to the real needs of the practice of engineers. For this purpose, the original graphic 2D and 3D application software was used in real-time with simultaneous display and processing in 24 high-resolution windows. It should be noted here that during the optimization and identification of the parameters of the reciprocating axial pump, hundreds of complex 2D diagrams are automatically formed and displayed, which enables the investigation of hydrodynamic processes at any time, if necessary, to intervene by changing the input data, thereby changing the subsequent flow of identification and optimization.

In conclusion, this study’s findings contribute to the applied science of fluid power systems by offering quantifiable data on the dynamic performance of hydraulic pumps with constant pressure and variable flow. This knowledge can serve as a foundation for improving the reliability, efficiency, and longevity of hydraulic systems across a wide range of industrial applications.

Author Contributions

Conceptualization, R.P.; Methodology, M.A. and R.P.; Software, M.A., R.P., I.I. and V.P.; Validation, M.A. and R.P.; Formal analysis, R.P.; Investigation, R.P. and J.G.; Resources, R.P.; Data curation, R.P.; Writing—original draft, R.P.; Writing—review & editing, R.P.; Visualization, R.P.; Supervision, R.P.; Project administration, R.P.; Funding acquisition, R.P. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures

Figure 2. Hydraulic system for testing the pump [20]. 1—electric motor, 2—pump, 3—safety valve, 4—manometer, 5—pressure regulator, 6—electromagnetic distributor, 7—indirect action regulator, 8—variable resistance damper, 9—elastic coupling, 10—suction filter, 11—pressure regulator, 12—intake filter, 13—flow converter, 14—reservoir, 15—air source, 16—temperature converter.

Figure 3. Original hydraulic pump test system [20]. 1—electric motor, 2—hydraulic pump, 3—safety valve, 4—manometer, 5—pressure transducer, 6—electromagnetic distributor, 7— pressure regulator, 8—muffler, 9—flow transducer, 10—reservoir.

Figure 4. Pressure in the cylinder (pc) and pressure chamber of the pump (pH) for one working cycle obtained by mathematical modeling.

Figure 5. Response times as a function of system impedance.

Figure 6. Pressure changes when the flow rate changes from Q to Qmin and vice versa.

Figure 8. (a) Pressure increase in phase flow reductions. (b) Reduction in pressure in the phase flow increases.

Figure 9. (a) Pressure increase in phase flow reduction. (b) Reduction in pressure in the phase flow increase.

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