Analysis of Dimensional Tolerances on Hydraulic and Acoustic Properties of a New Type of Prototypal Gear Pumps

: This study focuses on the construction of a prototype series of pumps. The technological capabilities of the entire series of gear pumps with a three-poly-involute outline were determined. We developed neural networks to analyze the dimensional tolerance and composition of the pump components and impact on the distribution for the constructed units. The most crucial dimensions to control were then determined—namely, dimensional and form tolerance were necessary—with a reduction in accuracy classification where it is less important. Measurements of acoustic quantities and of vibrations were also carried out. In conclusion, after positive verification, printed polyethylene wheels can be manufactured in greater, mass-produced quantities. Optimization techniques can then be applied, leading to reduced manufacturing costs and increased efficiency.


Introduction
The development of modern pumping units is currently concerned with two trends: the overall efficiency, associated with the minimization of mass, durability, reliability, vibrations, and pump performance, as well as decreasing the noise emission. Working pressures in the hydrostatic system determine the pumping efficiency of the entire installation. In modern machines and devices with a hydrostatic drive, one can observe a tendency to increase the pressure of displacement at the expense of a reduced flow rate for the working medium. The flow rate significantly affects the hydraulic losses occurring in local ducts and at points of resistance.
Modern manufacturing technique gear pump designs have attained an efficiency of around 80-90% within a range of pressures up to 28 MPa. An important reason for such a wide range is the tolerance achieved by different production techniques [1][2][3][4][5]. Another important aspect is the pressure of pulsation and variable dynamic load, which are a main reason for the creation of sound-associated vibrations. The noise in hydrostatic drives is one of the most important issues determining the area of their application and the possibility for further improvement in regard to the reduction of consumable lubricants, etc. via the possibility of using new and innovative low-friction materials in hydrostatic drives. Where striving to minimize their use is aimed at improving the power-to-weight ratio. As a result, minimization and, then, through a process of optimization, small structures are achieved while maintaining optimal hydraulic parameters [6][7][8].

Development of Gear Unit Design
The development of a gear unit design tends to desire a reduction of performance pulsation, which may be the reason for: an improper operation of control elements, vibrations, and increased noise of machines and devices with hydrostatic drive. Ear units with an external toothing are characterized by a coefficient of unevenness in efficiency at an average level of approx. 18%. Compared to other positive displacement units, gear pumps have the highest pulsation rates. The most effective method of pulsation reduction is obtained as a result of using the active method, i.e., by mitigating the course of the instantaneous performance function and by reducing the amplitude of the pulsation of efficiency transmitted to the hydraulic system. The second important direction of gear pumps development is the minimization of energy losses and increase of transmitted power; thus, the tendency of any changes is aimed at increasing the energy efficiency of the generator even more.
The analysis of numerous patents, literary sources, and other currently manufactured gear units could indicate that technical methods have already been exhausted to ensure the optimal internal tightness with maximum operating pressures, minimum efficiency pulsation, and noise emissions [7][8][9][10][11][12][13][14]. The conducted tests and considerations prevent achieving peak operational parameters by comparisons to modern gear units.
Therefore, new tasks and goals have been set, one of which is presented and discussed in this article. Two basic goals set in this prototype gear pump research are: to present theoretical possibilities, as well as methods, to reduce performance pulsation and, then, to propose new design solutions to increase working pressures while ensuring high internal tightness. These tasks required solving several technological, design, and manufacturing problems. As a result, innovative design solutions were proposed to reduce the performance pulsation, increase the working pressure, and improve the volumetric and total efficiency, as well as reducing the noise emission to the environment. Eight patents, including copyrights, were filed during this research, with much time having also been devoted to the cognitive aspects. New mathematical models describing, among others, the bearing and gear wheel loads for new methods of compensation of clearance, the determination of presented stresses for compensating elements, relief of sealed space, shaping tooth contours, or performance pulsation modeling. The theoretical considerations carried out were verified by numerous acoustic and hydraulic tests.
There are many research efforts and production methods concerned with and regarding the design of pump profiles only to be used in gear units. It is instigated mainly by striving for better hydraulic and acoustic properties [15][16][17][18][19]. This is the next part of the article related to the identification of sensitive dimensions of the importance of measuring points. The tests were carried out for four pump prototype units: 1PWR, 2PWR, 3PWR, and 2PW-SEW. The results were compared with a conventional pump to determine the impact of the proposed innovative design solutions.
For example, in the work described in [19], the identification of sensitive control dimensions (value/tolerance) of examined pumps (3PWR-SE) was determined by means of the multi-valued logic trees.
In the present work, a neural network is used to analyze the dimensional tolerance and the impact of the performance of prototype 2PW-SE [15,20] gear pump components on the overall performance. Optimization then minimized the acoustic noise and vibration. In addition, the theoretical considerations and experimental results presented here open new possibilities for gear pumps.

Basics of Gear Pump Design
Considering the recommended sizes when designing a gear pump-specific performance, determined by Renard [17], the technology in manufacturing, the availability of manufacturing machinery, and the resulting restrictions are related to the standardization of machining tools, e.g., with a specific module, buttress angle, tooth height coefficient, and others. Tool cutting teeth are among the most complex and expensive. An important aspect is the correct selection of the cuttingedge contour, including the subsequent regeneration of this edge in the sharpening process. The accuracy of machined wheels depends on this, which is the basic condition for the mass production of gears. At the design stage, in addition to the production technology, the design should assume the correct operating conditions. Newly designed units should be adapted to the working fluids used in typical hydraulic systems.
The original pump tested was a conventional unit, manufactured by the Wytwórnię Pomp Hydraulicznych Sp. z o. o. located in Wrocław, Poland. The experimental pump was designed for the technological capabilities of WPH S.A., where this specific pump has been studied many times, and the results are published within the scientific literature [1,18,20].
The newly designed and realized prototype pump is a three-plate structure shown schematically in Figure 1. The front plate (1) is used for mounting the pump on the drive unit. The middle plate (2) contains gear wheels, slide bearing housings, and suction and forcing holes for connecting to a hydraulic system. The whole construction is closed with a rear plate (3). The main meshing parameters for the pump with a unit delivery q = 40 cm 3 are listed in Table 1. The original pump had gears for both active and passive rollers with an involute (or cycloid) outline, and wheels with or without side clearance were considered.

Optimization of the Technology of Poly-Involutedly Shaped Pump Construction
To obtain longevity in the accuracy of manufactured tolerances at the micrometer level, a whole series of errors should be continuously checked and compensated for, the kinematic and geometric errors caused by cutting forces of the accepted machining parameters. These errors can be significantly reduced, but they cannot be entirely eliminated. Currently, increasing the accuracy of machine tools is carried out by implementing new design and software solutions, applying error correction and compensation.
In the process of innovation, the involute profile was modified at its bottom by the so-called fillet. The modification was done by means of a cutting tool with protuberance or by the appropriate correction of teeth. This type of analysis was presented in a number of co-authored works, e.g., [1,18,20]. The improved pump model included two modifications: The outline of the tooth was optimized using multi-valued logic trees. The optimization process was carried out considering five basic criteria: technology of the tool, minimum compression ratio, small changes in dynamic forces on the cavity, minimum efficiency of the pulsation coefficient, and high energy efficiency. From the viewpoint of the accepted criteria, a type X1 profile was selected among the several alternative combinations of the three-involute outlines. The profile selected through the optimization process was characterized by the occurrence of two ordinary involutes and one elongated involute. Such a procedure was described in the work described in [1,18,20]. The optimization of the contour of the polyevolvent tooth was made taking into account multi-valued logical trees. The main focus was on the values of the pressure angle between the pressure section and the tangent line to the rolling wheels at the rolling point as the most important evolvent parameter.
The literary and patent analysis of existing solutions offered by gear pump manufacturers showed the absence of a design with three-involute, oblique outline cuts and made with a backlashfree technology. Therefore, the developed design was submitted to the Patent Office of the Republic of Poland to ensure the protection of the intellectual property.
Subsequently, the technology for making new outline gear teeth was developed.Before making the wheels, the kinematics of the optimized three-involute mesh was studied for gear wheels fabricated using 3D technology. The model gear wheels shown in Figure 4a were the equivalent of the group II pump with a unit output of q = 8cm 3 /rpm. After positive verification of polyethyleneprinted wheels, the fabrication process commenced in commercial industrial conditions, where the surface of the three-involute outline was manufactured by ground and chip technologies (Figure 4b).
The second modification was optimization of the machining technology for components affecting the overall efficiency of the newly designed unit (a topic discussed in this article). The dimensional analysis and shape tolerances of the fully realized gear wheels enabled the selection of the following groups of the control dimensions: critical, important, and non-important. This resulted in a rational narrowing for the tolerances of the manufactured shapes and dimensions where necessary by lowering of the required class of accuracy in spaces of less importance. Optimization of manufacturing technology helped to cut production costs and to increase productivity.

The Research Object
The research concerns the development of a prototype gear pump from the 2PW-SEW series belonging to group II. The pump was designed by a HYDROTOR S.A. Pumps prototype entirely executed by the company HYDROTOR S.A., Tuchola, Poland. All 10 tested pumps have the model number 2PW-SEW-08-28-2-776 and a serial number (from 1 to 10). The efficiency of each model pump is 8 (cm 3 /rev). Figure 5 exploded the views of the 2PW-SEW prototype gear pump. The following hydraulic measurements were determined: volumetric efficiency characteristic ηvol, total efficiency ηc, hydraulic-mechanical efficiency ηhm, and delivered power N. Hydraulic tests were carried out for all ten pump units, while two 2PW-SEW-08-28-2-776 gear units (serial numbers: No. 2 and No. 4) were selected for comprehensive acoustic tests.

Measuring Rig
Static characteristics were determined on the test stand shown in Figure 6. In this system, the tested pump 1 is driven by a 100-kW DC motor 2 cooperating with the territorial control system. The DC motor Pxob-94a and the territorial control system type DSI-0360/MN-503 enable smooth changes of the pump speed in the range from 0 to 2000 rpm. Figure 7 shows a block diagram of the measuring path.   Tables 2 and 3 present the values of total efficiency for model pumps 2PW-SEW-08-28-2-776 at the discharge pressure pt(MPa) for the example rotational speed n = 500 (rev/min) and n= 2000 (rev/min) [20]. Table 2. Tested gear pumps for n= 500 (rev/min) [20]. Pt: discharge pressure.  Figure 8 shows a comparison of the total efficiency of the ten gear pumps from Table 2 for a rotational speed of n = 500 (rev/min).  Table 2 for a rotational speed of n = 500 (rev/min) [20]. Pt: discharge pressure

Application of Neural Network in Extracting Measurement Points
The two main software applications used for the analysis were: 1. AitechSPHINX artificial intelligence software (program with a license purchased for the Faculty of Production Engineering and Logistics, Opole University of Technology, Opole, Poland). The Neuronix module was used for creation of the neural network.
The following types of files were used: training file.lrn, test file.tst, weight file.wgt, knowledgebased file.bw, data file.vts, representation file.pre, multi-task learning system.aut, and chart.vtc.
The training file was created for the results of the control measurements (values/tolerances) for the prototype series gear pumps. The measurements included the following elements: body, active gear, passive gear, and plate and set of bearings (Tables 4-8).
The data elements were assumed to be vectors containing any finite number of coordinates. In contrast, the data on which the learning accuracy of the network was examined was the test set. All measurement points were included [24][25][26][27][28].
The set contained 1080 data points. The analysis was done for different training/test ratios: 50/50, 60/40, 70/30, 80/20, and 90/10 for three and four layers with one and two hidden layers, respectfully. To calculate the accuracy, in addition, 5-and 10-part cross-validations were performed. The last step is the selection of the best performing model. That model is chosen based on the accuracy of the analysis for all cases, counting the overall average and selecting the model closest to that average.
The final size of the training set was 90%, whereas the test set size was 10%. The other parameters of the neural network are: -learning process parameters: learning rate: 0.9, torque factor 0.7, and learning cycle size: 1.
-conditions for completing the learning process: testing tolerance: 0.25 and Epsilon: 0.01.
The mean square error was used as a measure of error on the output neurons. The loops start from the initial value of 0 (not from 1), so the index denoting the output layer of lw is (lw-1). This procedure is repeated until the error generated by the network is smaller than the asserted value. Then, another input vector is fed to the network input, and the process is repeated. After the entire learning sequence is processed (this is called an epoch), an error for the epoch is calculated.
The entire cycle was repeated until the error falls below the acceptable margin. The regression graphs in Figures 9-11 represent the network output data with respect to the learning goal, validation, and test sets. To get a perfect match, the data should lie along a line with a 45-degree positive-angle slope, i.e., where the network outputs are equal to the targets (patterns).   In our analysis, the qualitative approximation agrees very reasonably with the fitted data for all sets, with values of R in each case being 0.93 or higher. The neural network classified the following parameters as the most important: parameters= (19,20,21,22,30,  The identification of the impact of the manufacturing technology for the model units showed that the important dimensions affecting the efficiency of the pumps are generally repeated in all details, regardless of the analyzed group. Table 9 shows the results of the hydraulic measurements for the original pump and Table 10 for the improved unit. Table 9. Results of the hydraulic measurements for the original pump [15], where: n-rotation speed, p(t)-discharge pressure, Qrz-flow rate, M-torque, Nh-hydraulic power, Nm-mechanical power, ηv-volumetric efficiency, ηhm-hydraulic and mechanical efficiency, ηc-total efficiency.  Other selected hydraulic (and acoustic) results for selected pumps were presented, e.g., in works [15,17]. The comparison of the hydraulic performance for the pumps before modification show that the total efficiency of the currently produced gear pumps ranges from 74% to 88%. These values are lower than the efficiency of the prototype units by 4% to 18%. The increase in energy efficiency of the prototype units is mainly due to the relatively high internal tightness. The slightly higher efficiency of a unit with a fitting angle of c = 130 is due to the low hydraulic-mechanical torque loss. Even greater differences in the overall performance favor prototype pumps that are observed at low rotational speeds. This is due to a high drop in the volumetric efficiency of conventional pumps at speeds below n = 800 rpm. The volumetric efficiency of the conventional units did not exceed 60% for the nominal pressures, and rotational speed n = 500 rpm. The relatively high internal tightness of the prototype pumps ensures that the volumetric efficiency is maintained at a minimum of 90% throughout the pressure range, regardless of the rotational speed.

Acoustic Measurements of Prototype Pumps
Acoustic measurements were carried out in a reverberation chamber at the Department of Hydraulic Drives and Automation, Wroclaw University of Science and Technology. In rooms of this type, there is a perfectly dispersed field characterized by the fact that all acoustic energy reflected from the walls returns towards the source, such that the sound intensity at each point in this field is equal [27]. This property is achieved by lining the ceiling and floor walls with hard and smooth elements that perfectly reflect sound waves. To prevent standing waves, opposite walls are made at an angle to each other. The reverberation chamber shown in Figure 13 has a volume of V = 102 m 3 . The room meets the requirements of ANSI S1.  and PN-85/N-01334 and provides the possibility of testing machines and devices for vibration and noise. The chamber's sound insulation in relation to external interference is within the 20-20,000 Hz frequency range of 50 dB. Such insulation ensures the elimination of interference from the propulsion system and the hydraulic system supplying the tested unit. Figure 13. Block diagram of gear pump noise measurement: KA-calibrator, MC-eight free sound field microphones, MU-multiplexer, WP-instrumentation amplifier, AF-two-channel frequency analyzer, PC-computer, PZ-gear pump, and KO-chamber [15,17].
The chamber was made of two similar irregular polyhedrons located "one in the other". The uniformity of the sound field distribution in the chamber is within acceptable limits, starting from the center frequency for an octave of 125 Hz. Eight fixed measuring points were determined within the chamber based on the sound field distribution tests [17]. The microphones were set in accordance with the recommendations of the abovementioned standards at a height of 1.3 m from the floor, where this height corresponded to the position of the drive shaft axis.
The measuring microphones were spaced at eight points, from which the sound pressure level was read and then averaged. Measuring microphones were selected when reading data using a multiplexer and the level together with the spectrum was stored in the memory of a two-channel analyzer. The data was edited on a PC in the B&K program type 5306.
Acoustic measurements of an experimental version of the pump, respectively, for the value of the discharge pressure pt: 0, 2, 4, ..., 30 MPa and the frequency f: 25÷20k Hz were obtained in the analysis. Table 11 shows the exemplary acoustic measurements of a gear pump after tooth root undercutting for pt = 12 MPa. Table 11. Acoustic measurements of a gear pump after tooth root undercutting for p(t) = 12 MPa [15]., where Lmj-sound pressure level, Smj-average sound pressure level, LAj-corrected sound pressure level.

f (Hz)
Microphone As part of the research, the following measurements were performed: the sound pressure level Lp, the A-weighted sound pressure level LA, the sound power level LW, and the A-weighted sound power level LWA. The average value of the sound pressure level Lp was calculated according to the following formula:      The measurement of the aRMS vibration acceleration was located on the rear cover of the gear pump in the axis of the driving wheel shaft. The choice of the measurement point resulted from the previous tests with the use of the acoustic probe. The distribution of sound intensity on the surface of the pump body, determined by the energy method, showed a local increase in sound vibrations. This fact proves the transmission of sound-generating vibrations mainly from the pump drive.
Comparing the acoustic characteristics of the prototype pumps with the basic units, it turned out that the solution with a polyevolvent outline is characterized by 3 to 5-dB lower noise emission to the environment. The beneficial effect of noise reduction was observed in the range from 0 to 20 MPa. In the extreme case, for 8 MPa, the sound level A was reduced to 5 dB. Acoustic tests of model pumps with dimensional optimization and comparable units without modification showed similar noise emission parameters at low operating pressures. In the case of an increase in working pressure, units with the optimization of production technology are more favorable (Table 12). * values estimated based on the tooth geometry measurement, ** the parameter was determined for the pump operating with the highest available power, and *** the parameter was determined for the pump working with the highest available power in group II with a cover and plate made of PA9.
After optimizing the technology of the prototype pumps, a smaller dispersion is observed in the results of the efficiency course. The characteristics are more grouped. Comparing the acoustic characteristics of the prototype pumps ( Figure 17), it turned out that the solution with ground wheels has a 3 to 5-dB reduction in noise emitted to the environment. The advantage of shaved pumps is a higher overall efficiency due to low hydraulic-mechanical losses. A higher relative hydraulic and mechanical efficiency is mainly associated with surface roughness. The use of carburizing and hardening of the teeth reduces the total height of the roughness profile (St). After stripping, the parameter St is at the level of 14 pom, decreasing after a treatment of carbonizing and hardening to the value of about 8 pom. The phenomenon of lowering the roughness can be explained by the high carburizing and hardening temperatures conducive with oxidation of a sharp surface roughness. Low values of the tooth profile roughness parameters help to improve the lubrication conditions of mating teeth.

Conclusions
The objective of this research was the construction of a model and prototype pumps. The final analysis of the dimensional and geometric tolerances allowed for the selection of dimensions critical, important, and not important to the control. This resulted in a rational narrowing of dimensional and geometric tolerances where necessary and a reduction in the accuracy classification in areas of little importance. Optimizing the production technology contributed to lowering the production cost and increasing its efficiency. The results obtained for both modeled and prototype pumps were referenced against other constructions manufactured by the implemented entity or by other leading gear pump manufacturers. Due to the large number of units tested, this monograph presents only selected aspects in a fragmentary way. In addition, during testing, the impact of the environment on the measuring system is negligible. Uniform measurement conditions were ensured, i.e., tests were carried out with the same apparatus while maintaining a constant temperature and humidity. This approach significantly simplifies the interpretation of results that are used for comparative purposes, i.e., they allow direct determination of the differences in measured physical quantities between the tested objects. The patent and analysis of the structures offered by gear pump manufacturers show a lack of solutions with a poly-involute outline.
At the design stage, in addition to selecting the tooth geometry, the designer should assume the correct operating conditions and consider the technological capabilities of the developed structure. In connection with the above, the research presents technological and operational conditions that should be met by newly designed constructions. The adopted construction, technological parameters, and operating conditions decide, amongst other factors, the efficiency, durability, reliability, and noise emission to the environment. There is a possibility of further generalizations and modifications with a detailed presentation of the technological method of making tooth shapes, referring, for example, to publications [29,30].
Author Contributions: A.D. was involved in the conception and design of the study, drafted the manuscript, with support from P.O., compiled all the results. P.O. prepared data from hydraulic and acoustic tests and supervised the tests with a cooperating company. M.A.P. prepared data from hydraulic and acoustic tests, and supervised the tests with a cooperating company. The corresponding author attests that all listed authors meet authorship criteria and that no others meeting the criteria have been omitted. All authors have read and agreed to the published version of the manuscript