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

Abstract

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.

1. 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].

1.1. 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.

1.2. 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 cutting-edge 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 cm³ are listed in

Table 1. Meshing parameters.

Parameter	Symbol	Unit	Value
Number of teeth	z	-	9
Modulus	m0	(mm)	4.5
Pressure angle	α0	(°)	20
Gear wheel width	b	(mm)	32.2

.

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.

2. 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.

Taking into account the technological performance, it was assumed that the contour of the polyevolvent tooth will be made of three basic evolvents. With two ordinary evolvents and one elongated evolvent or three ordinary evolvents. The optimization process was performed in two stages. In the first step, multiple sets of profiles were analyzed, and then, five types of profiles were selected for the final step:

X1, X2, X3, X4, and X5, where for:

X1:

Ordinary evolvent: α₀¹ > α₀²|Ordinaryevolvent: α₀² = 20°|Elongated evolvent

X2:

Ordinary evolvent: α₀¹ > α₀²|Ordinaryevolvent: α₀² = 20°|Elongated evolvent

X3:

Ordinary evolvent: α₀¹ > α₀²|Ordinaryevolvent: α₀² = 20°|Ordinary evolvent: α₀¹ >α₀²

X4:

Ordinary evolvent: α₀¹ > α₀²|Ordinaryevolvent: α₀² = 20°|Ordinary evolvent: α₀¹ >α₀²

X5:

Ordinaryevolvent: α₀¹ > α₀²|Ordinary evolvent: α₀² = 20°|Ordinary evolvent: α₀¹ >α₀²

The determined effect can be obtained by rounding or chamfering the upper part of the outline (Figure 2). Such a solution is protected by a co-authoring patent. The upper ordinary involute in the polyevolvent contour is a modification of the vertex contour with respect to conventional constructions. The advantage of this solution is the maintenance of the same diameters inside the pump body. Figure 3 shows a polyevolvent outline cut with a trapezoidal rack.

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 backlash-free 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 = 8cm³/rpm. After positive verification of polyethylene-printed 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 (cm³/rev).

Figure 5 exploded the views of the 2PW-SEW prototype gear pump.

Ten pumps were studied (with serial numbers No.1–No.10). The hydraulic tests included:

-

determination of the actual efficiency of the gear pump Q_rz = f(pt) and

-

determination of the torque M = f(pt).

The following hydraulic measurements were determined: volumetric efficiency characteristic η_vol, total efficiency η_c, hydraulic-mechanical efficiency η_hm, 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.

3. 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.

Table 2. Tested gear pumps for n= 500 (rev/min) [20]. Pt: discharge pressure.

No.	pt (MPa)	n (min−1)	2PW No. 1	2PW No. 2	2PW No. 3	2PW No. 4	2PW No. 5	2PW No. 6	2PW No. 7	2PW No. 8	2PW No. 9	2PW No. 10
			%	%	%	%	%	%	%	%	%	%
1	0	500	0	0	0	0	0	0	0	0	0	0
2	2		92.0	96.3	99.9	95.8	98.9	96.3	99.4	97.8	99.4	98.9
3	4		97.8	96.9	98.1	96.2	96.8	97.6	96.4	95.0	95.4	96.6
4	6		94.8	96.5	96.1	94.3	92.3	95.0	93.6	92.2	93.1	93.0
5	8		91.9	93.3	92.8	89.2	90.8	90.3	92.6	89.0	84.8	89.3
6	10		91.4	89.3	90.2	84.0	85.7	87.5	89.1	85.7	82.0	85.2
7	12		88.6	88.7	89.7	79.1	83.8	84.1	86.0	82.8	80.8	83.6
8	14		89.2	88.9	87.0	75.0	83.2	84.3	84.7	80.7	77.9	80.4
9	16		88.2	86.6	84.4	72.3	80.3	81.1	83.8	79.1	76.7	77.0
10	18		85.8	86.3	82.8	68.8	77.8	78.7	81.2	76.6	75.2	73.8
11	20		83.7	84.3	80.2	66.4	74.3	75.3	77.9	73.8	73.9	69.4
12	22		81.0	82.0	78.0	63.1	71.7	72.2	75.1	70.5	71.9	64.2
13	24		78.5	78.8	74.3	59.3	67.3	68.4	72.3	66.8	69.5	58.3
14	26		71.8	73.8	67.3	55.1	59.9	59.8	65.1	61.0	64.0	49.2
15	28		69.1	74.3	64.8	54.2	54.1	54.0	63.4	58.1	62.6	40.0
16	30		61.5	68.7	57.2	49.4	43.9	41.5	54.5	52.2	57.0	27.8
17	32		47.1	60.6	49.1	45.6	29.0	24.2	41.9	41.9	50.2	0

and

Table 3. Tested gear pumps for n= 2000 (rev/min) [20].

No.	pt (MPa)	n (min−1)	2PW No. 1	2PW No. 2	2PW No. 3	2PW No. 4	2PW No. 5	2PW No. 6	2PW No. 7	2PW No. 8	2PW No. 9	2PW No. 10
			%	%	%	%	%	%	%	%	%	%
1	0	2000	0	0	0	0	0	0	0	0	0	0
2	2		62.1	64.1	62.1	61.0	75.3	63.7	62.5	62.1	59.1	63.7
3	4		77.6	77.6	77.6	78.1	86.4	80.1	76.9	77.1	77.6	80.9
4	6		81.0	81.9	82.8	82.8	89.0	83.2	80.1	81.7	80.5	84.1
5	8		85.1	84.2	83.7	83.8	83.7	84.6	82.1	83.3	84.4	85.3
6	10		85.1	85.7	85.1	83.4	82.9	85.7	84.5	84.0	85.1	85.2
7	12		85.6	85.1	84.6	83.5	87.5	84.5	85.1	84.4	84.0	84.9
8	14		86.3	86.0	85.8	84.7	87.5	85.6	85.4	85.0	84.3	88.1
9	16		86.8	85.7	86.6	84.4	87.2	85.1	85.9	84.9	85.3	85.3
10	18		87.2	85.9	86.1	84.3	87.0	85.2	85.4	85.3	84.2	85.1
11	20		86.4	85.1	85.6	84.1	85.4	85.6	84.4	84.7	84.4	84.8
12	22		86.2	85.3	85.0	83.9	85.3	84.4	84.2	84.2	83.7	83.8
13	24		84.1	84.2	85.0	83.2	82.4	83.0	83.5	84.1	83.6	82.6
14	26		80.1	80.8	80.8	79.3	77.9	79.2	79.7	80.3	79.4	78.3
15	28		82.4	83.2	82.5	81.6	79.1	80.2	81.8	82.5	81.4	79.5
16	30		81.1	81.1	81.7	79.8	74.8	77.7	79.4	80.6	79.2	76.5
17	32		78.4	79.7	78.5	80.4	70.2	74.2	75.9	78.4	77.5	72.7

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].

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).

4. Determination of the Most Important Dimensional Tolerances for 2PW-SEW Pumps Using a Convolution Neural Network

Sensitivity analysis of gear pump control points was carried out as part of the Innovative Operational Program, Priority 1. Research and development of modern technologies, Measure 1.4 Support for targeted projects, project no. POIG.01.04.00-04-345/13. The difference in performance values of the tested pumps is related to the degree of sensitivity regarding the control dimensions.

Four components of the prototype pump were tested: body, active/passive gears, and the bearing set. For each part, an appropriate number of control points are generated corresponding to the control dimension (value/tolerance):

• for gears (active and passive): 17 points,
• for the body (housing): 22 points,
• for a bearing set: 39 points, and
• for the plate: 7 points.

Table 4. Results of measurements of dimensional tolerance for the body—Kr [20].

No.	Symbol/ Tolerance	Measurement Results—Kr
		Dimension	1	2	3	4	5	6	7	8	9	10
1	KR1	55.7 (−0.03)	55.7009	55.6728	55.6787	55.6774	55.6534	55.6638	55.6680	55.6745	55.6804	55.6902
2	KR2	55.7 (−0.03)	55.6985	55.6708	55.6793	55.6753	55.6544	55.6575	55.6671	55.6691	55.6785	55.6878
3	KR3	55.7 (−0.03)	55.7024	55.6747	55.6837	55.6788	55.6565	55.6636	55.6688	55.6738	55.6754	55.6862
4	KR4	55.7 (−0.03)	55.7006	55.6770	55.6830	55.6792	55.6566	55.6623	55.6614	55.6759	55.6742	55.6830
5	KR5	55.7 (−0.03)	55.7042	55.6740	55.6833	55.6816	55.6561	55.6625	55.6625	55.6745	55.6753	55.6829
6	KR6	55.7 (−0.03)	55.7033	55.6758	55.6827	55.6824	55.6545	55.6679	55.6631	55.6760	55.6746	55.6823
7	KR7	55.7 (−0.03)	55.7050	55.6743	55.6840	55.6850	55.6559	55.6665	55.6654	55.6764	55.6755	55.6834
8	KR8	55.7 (−0.03)	55.7052	55.6727	55.6832	55.6815	55.6569	55.6641	55.6708	55.6741	55.6808	55.6911
9	KR9	55.7 (−0.03)	55.6998	55.6707	55.6750	55.6702	55.6500	55.6589	55.6652	55.6705	55.6761	55.6849
10	KR10	55.7 (−0.03)	55.7007	55.6692	55.6760	55.6742	55.6508	55.6546	55.6650	55.6674	55.6762	55.6856
19	KR19	Pr. (0.02)	0.0091	0.0095	0.0115	0.0142	0.0084	0.0108	0.0014	0.0138	0.0000	0.0020
20	KR20	Pr. (0.02)	0.0070	0.0023	0.0010	0.0012	0.0047	0.0021	0.0022	0.0005	0.0034	0.0046
21	KR21	Pr. (0.02)	0.0066	0.0015	0.0019	0.0015	0.0019	0.0036	0.0006	0.0016	0.0007	0.0046
22	KR22	Pr. (0.02)	0.0110	0.0025	0.0052	0.0039	0.0028	0.0037	0.0073	0.0018	0.0100	0.0097

Pr.—perpendicularity.

,

Table 5. Measurement results of dimensional tolerance for the active gear—KzP [20].

No.	Symbol/ Tolerance	Measurement Results—KzP
		Dimension	1	2	3	4	5	6	7	8	9	10
1	Kzp1	Ø18 (−0.025/−0.035)	17.9650	17.9660	17.9650	17.9650	17.9660	17.9655	17.9665	17.9655	17.9655	17.9650
2	Kzp2	Ø18 (−0.025/−0.035)	17.9640	17.9630	17.9650	17.9660	17.9660	17.9640	17.9640	17.9640	17.9660	17.9660
3	Kzp3	Ø24.8 (+/−0.2)	24.8800	24.8400	24.8300	24.8300	24.8500	24.8200	24.8400	24.8300	24.8300	24.8400
4	Kzp4	Ø39 (−0.08/−0.096)	38.9380	38.9320	38.9300	38.9320	38.9300	38.9320	38.9320	38.9300	38.9320	38.9320
5	Kzp5	11.7 (−0.01)	11.7010	11.6980	11.6960	11.7010	11.7020	11.7010	11.7010	11.7000	11.6980	11.6960
6	Kzp6	11.7 (−0.01)	11.6980	11.6970	11.6950	11.7000	11.7000	11.7000	11.7000	11.6990	11.6970	11.6950
7	Kzp7	C (0.003)	0.0032	0.0040	0.0037	0.0026	0.0034	0.0025	0.0024	0.0034	0.0039	0.0034
8	Kzp8	R (0.005)	0.0050	0.0100	0.0025	0.0025	0.0050	0.0050	0.0050	0.0075	0.0100	0.0050
9	Kzp9	C (0.003)	0.0041	0.0047	0.0083	0.0058	0.0041	0.0040	0.0038	0.0050	0.0047	0.0034
16	Kzp16	14.602	14.5490	14.5540	14.5380	14.5500	14.5490	14.5560	14.5500	14.5480	14.5490	14.5600
17	Kzp17	R	0.0170	0.0100	0.0060	0.0120	0.0060	0.0060	0.0080	0.0180	0.0110	0.0080

W—cylindricity and R—run out.

,

Table 6. Measurement results of the dimensional tolerance for a passive gear—Kzpn [20].

No.	Symbol/ Tolerance	Measurement Results—Kzpn
		Dimension	1	2	3	4	5	6	7	8	9	10
1	Kzpn1	Ø18 (−0.025/−0.035)	17.968	17.960	17.964	17.967	17.964	17.964	17.956	17.966	17.965	17.968
2	Kzpn2	Ø18 (−0.025/−0.035)	17.968	17.964	17.968	17.968	17.970	17.962	17.964	17.968	17.968	17.967
3	Kzpn3	Ø24.8 (+/−0.2)	24.800	24.970	24.970	24.930	24.740	24.930	24.980	24.980	24.900	24.780
4	Kzpn4	Ø39 (−0.08/−0.096)	38.920	38.918	38.920	38.918	38.924	38.918	38.916	38.920	38.916	38.922
5	Kzpn5	11.7 (−0.01)	11.697	11.694	11.692	11.699	11.705	11.702	11.698	11.698	11.694	11.692
6	Kzpn6	11.7 (−0.01)	11.696	11.693	11.690	11.698	11.700	11.700	11.698	11.697	11.694	11.690
7	Kzpn7	W (0.003)	0.006	0.007	0.003	0.003	0.007	0.006	0.005	0.003	0.003	0.005
8	Kzpn8	R (0.005)	0.005	0.003	0.005	0.005	0.005	0.003	0.005	0.000	0.005	0.000
9	Kzpn9	R (0.003)	0.006	0.005	0.004	0.005	0.005	0.007	0.007	0.007	0.003	0.007
16	Kzpn16	14.602	14.536	14.576	14.540	14.498	14.498	14.521	14.540	14.500	14.536	14.488
17	Kzpn17	R	0.004	0.006	0.011	0.012	0.007	0.006	0.011	0.006	0.006	0.009

Pr.—perpendicularity.

,

Table 7. Measurement results of the dimensional tolerance for the plate—Płt [20].

No.	Symbol/ Tolerance	Measurement Results—Płt
		Dimension	1	2	3	4	5	6	7	8	9	10
1	Płt1	F (0.02)	0.0354	0.0435	0.0391	0.0239	0.0220	0.0372	0.0437	0.0407	0.0351	0.0291
2	Płt2	P (0.03)	0.0296	0.0199	0.0299	0.0267	0.0386	0.0213	0.0187	0.0199	0.0255	0.0217
3	Płt3	19 (+/−0.2)	18.9989	19.0165	18.9837	18.9177	18.9311	18.9898	18.9686	19.0073	19.0161	19.0066
4	Płt4	19 (+/−0.2)	19.0049	19.0258	18.9908	18.9265	18.9183	18.9954	18.9830	19.0205	19.0152	19.0159
5	Płt5	19 (+/−0.2)	18.9791	19.0059	18.9772	18.9036	18.9124	18.9742	18.9656	19.0045	18.9962	18.9985
6	Płt6	19 (+/−0.2)	19.0087	19.0234	19.0071	18.9303	18.9511	18.9948	18.9844	19.0244	19.0217	19.0202
7	Płt7	19 (+/−0.2)	18.9901	19.0250	18.9951	18.9170	18.9188	18.9878	18.9752	19.0107	19.0183	19.0059

F—flatness and P—parallelism.

and

Table 8. Measurement results of the dimensional tolerance for a set of bearings—Kł [20].

No.	Symbol/ Tolerance	Measurement Results—Kł
		Dimension	1	2	3	4	5	6	7	8	9	10
1	Kł1	22 (−0.08/−0.09)	21.9077	21.9078	21.9103	21.9057	21.9030	21.9047	21.9041	21.9021	21.9031	21.9059
2	Kł2	22 (−0.08/−0.09)	21.9078	21.9113	21.9073	21.9059	21.9034	21.9081	21.9083	21.9118	21.9123	21.9141
3	Kł3	22 (−0.08/−0.09)	21.9075	21.9082	21.9057	21.9076	21.8999	21.9072	21.9071	21.9078	21.9116	21.9190
4	Kł4	22 (−0.08/−0.09)	21.9177	21.9096	21.9097	21.9076	21.9034	21.9083	21.9105	21.9108	21.9100	21.9091
5	Kł5	22 (−0.08/−0.09)	21.9063	21.9096	21.9092	21.9072	21.9053	21.9087	21.9186	21.9021	21.9025	21.9031
6	Kł6	22 (−0.08/−0.09)	21.9031	21.9058	21.9013	21.9050	21.9011	21.9042	21.9120	21.9060	21.9044	21.9058
7	Kł7	22 (−0.08/−0.09)	21.9039	21.9048	21.9026	21.9053	21.8990	21.9032	21.9093	21.9068	21.9064	21.9095
8	Kł8	P (0.004)	0.0146	0.0065	0.0090	0.0026	0.0062	0.0055	0.0145	0.0010	0.0098	0.0159
9	Kł9	F (0.003)	0.0083	0.0128	0.0087	0.0115	0.0086	0.0108	0.0104	0.0138	0.0164	0.0180
10	Kł10	71.5 (−0.015)	71.4897	71.4891	71.4620	71.4753	71.4905	71.4919	71.5106	71.5005	71.5027	71.4826
38	Kł38	Pr. (0.01)	0.0002	0.0008	0.0022	0.0028	0.0107	0.0037	0.0024	0.0029	0.0037	0.0001
39	Kł39	Pr. (0.01)	0.0001	0.0064	0.0040	0.0040	0.0007	0.0019	0.0105	0.0132	0.0171	0.0035

F—flatness, P—parallelism, and Pr.—perpendicularity.

show the selected measurement values for the test elements [20].

Application of Neural Network in Extracting Measurement Points

The two main software applications used for the analysis were:

• 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 second, commercially available software used was MATLAB, Opole, Poland [21,22,23].

The following types of files were used: training file.lrn, test file.tst, weight file.wgt (Listing 1), knowledge-based 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 (Table 4, Table 5, Table 6, Table 7, Table 8).

The data elements were assumed to be vectors containing any finite number of coordinates (Listing 2). In contrast, the data on which the learning accuracy of the network was examined was the test set (Listing 3). 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 (Listing 4). 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 (Listing 5).

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 (Listing 6).

$${\epsilon }_{ms}=\frac{{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{n_{NL}}{(E_{V_j}-O_{ij})}^2}}}{i\cdot n_{NL}}$$

(1)

where

• ${\epsilon }_{ms}$—mean square error,
• i—the number of all learner vectors (in the training sequence),
• $E_{V_j}$—the expected value at the output of the j neuron for the learning vector $V_j$,
• $O_{ij}$—output of the j neuron in layer i, and
• $n_{NL}$—number of neural network layers.

Fragment of the source code:

Listing 1. generating the weight values.

• / * * * Generation of the weight values ***/
• for(i=1;i<LW;i++)
• for(j=0;j<n(i);j++)
• for(k=0;k<=n(i-1);k++)
• {W(i)(j)(k) = (((rand() % 1000000L) / 1700.0) - 9.8)*0.0015;
• if(W(i)(j)(k) == 0.0) W(i)(j)(k) = 0.01492;}
• /**********************************/

Listing 2. network input signal processing.

• / * * * Single processing ***/
• for(i=1;i<LW;i++)
• for(j=0;j<n(i);j++)
• {I(i)(j) = 0.0;
• for(k=0;k<=n(i-1);k++)
• I(i)(j) += O(i-1)(k) * W(i)(j)(k);
• O(i)(j) = 1.0 / (1.0 + exp(beta*(-I(i)(j))));}
• /**********************************/
• for(j=0;j<n(LW-1);j++)
• B(LW-1)(j)
  = Wy(wu)(j) - O(LW-1)(j);
• /***********************************/

Listing 3. errors on network outputs.

• for(j=0;j<n(LW-1);j++)
• B(LW-1)(j)
  = Wy(wu)(j) - O(LW-1)(j);
• /***********************************/

Listing 4. calculating errors on every neuron in the network (back-propagation).

• for(i=1;i<LW;i++)
• for(j=0;j<n(i);j++)
• for(k=0;k<=n(i-1);k++)
• {
• W2(i)(j)(k)
  = W(i)(j)(k),
• (i)(j)(k) += eta * E(i)(j) * O(i-1)(k) + alfa*(W(i)(j)(k)-W1(i)(j)(k)),
• W1(i)(j)(k)
  = W2(i)(j)(k);
• }
• /**********************************/

Listing 5. adjustment of weights.

• /********** Adjustment of weights *********/
• for(i=1;i<LW;i++)
• for(j=0;j<n(i);j++)
• for(k=0;k<=n(i-1);k++)
• {
• W2(i)(j)(k)
  = W(i)(j)(k),
• (i)(j)(k) += eta * E(i)(j) * O(i-1)(k) + alfa*(W(i)(j)(k)-W1(i)(j)(k)),
• W1(i)(j)(k)
  = W2(i)(j)(k);
• }
• /**********************************/

Listing 6. calculating a network ERMS (Error Root Mean Square) error for a single epoch.

• / ***** Calculating network error *****/
• for(j=0;j<n(LW-1);j++)
• RMS += (Wy(wu)(j) - O(LW-1)(j))*(Wy(wu)(j) - O(LW-1)(j));
• ERMS = sqrt(RMS/(double)(ile_wek*n(LW-1)));
• /*********************************/

Designations adopted in listings—meaning of individual variables:

• lw—the number of layers in the network
• n(i)—the number of neurons in layer i
• i—layer number
• j—single neuron number
• k—weight number
• (i)(j)(k)—current weight k at j neuron in layer i
• w1(i)(j)(k)—previous weight k at neuron j in layer i
• i(i)(j)—input of neuron j in layer i
• o(i)(j)—output of neuron j in layer i
• e(i)(j)—delta (sigma) of neuron j in layer i
• b(lw-1)(j)—error on jth neuron output
• wy(wu)(j)—the expected value on jth neuron output for the learner vector wu
• ile_wek—the number of all learner vectors (in the training sequence)
• alfa—momentum factor
• beta—activation curve factor
• eta—learning rate
• erms—network error

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 Figure 9, Figure 10 and Figure 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, 31, 38, 39, 40, 41, 42, 44, 45, 46, 48, 57, 58, 59, 60, 61, 62, 65, 73, 74, 101, 102, 104, 111, 112, 113, 140, 141, 142, and 143)

which correspond to the measured points: KR19, KR20, KR21, KR22, Pkr1, Pkr2, Kzp7. Kzp8, Kzp9, Kzp10, Kzp11, Kzp13, Kzp14, Kzp15, Kzp17, Kzpn9, Kzpn10, Kzpn11, Kzpn14, Kzpn17, Kła8, Kła9, Kła37, Kła36, Kła37, Kła39, Kł7, Kł8, Kł9, Kł36, Kł37, Kł38, and Kł39.

Figure 12 presents the dimensions and critical deviations: (1) the beating of plugs and the lateral surface of the toothed rim and (2) the perpendicularity of the side surface of the tooth to the pivots (wheel rotation axis) for the details of the gears speeding.

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. Results of the hydraulic measurements for the original pump [15], where: n—rotation speed, p(t)—discharge pressure, Q_rz—flow rate, M—torque, N_h—hydraulic power, N_m—mechanical power, η_v—volumetric efficiency, η_hm—hydraulic and mechanical efficiency, η_c—total efficiency.

n (rpm)	pt (MPa)	Qrz (/min)	M (Nm)	Nh (kW)	Nm (kW)	nv (%)	nhm (%)	nc (%)
500	0	15.1	2.0	0.00	0.10	94.4	0.0	0.0
	5	13.1	25.5	1.08	1.34	81.7	99.5	81.2
	10	11.4	52.5	1.89	2.75	71.2	96.8	68.9
	15	10.0	81.0	2.50	4.24	62.5	94.2	58.9
	20	9.5	110.0	3.17	5.76	59.6	92.5	55.1
	24	9.4	139.0	3.89	7.28	58.4	91.5	53.5
	30	8.6	172.0	4.30	9.01	53.8	88.8	47.7
1000	0	31.7	2.2	0.00	0.23	99.2	0.0	0.0
	5	30.4	25.5	2.53	2.67	95.1	99.5	94.6
	10	28.8	53.0	4.79	5.55	89.9	95.9	86.2
	15	27.6	81.5	6.88	8.53	86.1	93.6	80.6
	20	26.5	108.0	8.84	11.31	82.9	94.2	78.1
	24	26.3	140.0	10.93	14.66	82.1	90.9	74.6
	30	25.4	172.0	12.70	18.01	79.5	88.8	70.5
1500	0	48.0	2.8	0.00	0.44	100.0	0.0	0.0
	5	47.5	26.0	3.95	4.08	99.0	97.5	96.6
	10	45.7	56.5	7.60	8.87	95.2	90.0	85.6
	15	44.5	85.0	11.10	13.35	92.7	89.8	83.2
	20	43.5	113.0	14.47	17.75	90.5	90.1	81.5
	24	42.8	144.0	17.82	22.62	89.2	88.3	78.8
	30	41.9	176.0	20.92	27.65	87.2	86.8	75.7
2000	0	64.0	3.2	0.00	0.67	100.0	0.0	0.0
	5	63.5	27.5	5.27	5.76	99.3	92.2	91.5
	10	61.9	56.6	10.30	11.83	96.8	90.0	87.1
	15	60.8	85.0	15.19	17.80	95.0	89.8	85.3
	20	59.8	113.0	19.92	23.76	93.5	90.1	84.2
	24	59.1	143.0	24.59	29.95	92.3	89.0	82.1
	30	58.0	177.0	29.00	37.07	90.7	86.3	78.2

shows the results of the hydraulic measurements for the original pump and

Table 10. Results of the hydraulic measurements for improved pump [15].

n (rpm)	pt (MPa)	Qrz (/min)	M (Nm)	Nh (kW)	Nm (kW)	nv (%)	nhm (%)	nc (%)
500	0	15.6	0.5	0.00	0.03	100.0	0.0	0.0
	5	14.6	28.0	1.21	1.47	93.6	88.1	82.5
	10	14.1	56.0	2.34	2.93	90.4	88.4	79.9
	15	14.1	83.0	3.52	4.35	90.4	89.6	80.9
	20	14.4	110.0	4.79	5.76	92.3	90.1	83.2
	24	14.7	140.0	6.12	7.33	94.2	88.6	83.5
	30	14.4	170.0	7.19	8.90	92.3	87.5	80.8
1000	0	32.3	1.0	0.00	0.10	99.1	0.0	0.0
	5	32.0	32.0	2.65	3.35	98.2	80.6	79.1
	10	31.8	60.0	5.28	6.28	97.5	86.2	84.1
	15	32.0	85.0	7.98	8.90	98.2	91.4	89.7
	20	32.6	116.0	10.85	12.15	99.9	89.3	89.3
	24	32.6	141.0	13.57	14.77	99.9	91.9	91.9
	30	32.5	171.0	16.23	17.91	99.7	90.0	90.7
1500	0	49.9	2.0	0.00	0.31	99.8	0.0	0.0
	5	49.9	31.0	4.13	4.87	99.8	85.1	84.9
	10	49.2	60.0	8.18	9.42	98.4	88.2	86.7
	15	49.6	89.0	12.38	13.98	99.2	89.2	88.5
	20	50.0	118.0	16.64	18.54	99.9	89.8	89.8
	24	50.0	144.0	20.81	22.62	99.9	92.0	92.0
	30	49.6	175.0	24.78	27.49	99.2	90.9	90.1
2000	0	66.9	4.0	0.00	0.84	99.1	0.0	0.0
	5	67.3	33.0	5.57	6.91	99.7	80.9	80.7
	10	67.2	62.0	11.17	12.99	99.6	86.4	86.0
	15	67.2	91.0	16.77	19.06	99.6	88.4	88.0
	20	67.5	120.0	22.47	25.13	99.9	89.4	89.4
	24	67.1	147.0	27.92	30.79	99.4	91.2	90.7
	30	66.8	177.0	33.37	37.07	99.0	91.0	90.0

for the improved unit.

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.

5. 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 m³. The room meets the requirements of ANSI S1.21-1972 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.

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 ÷ 20 k Hz were obtained in the analysis.

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 Numbers								Thirds				Octaves
	1	2	3	4	5	6	7	8	Lmj	Smj	KAj	LAj	Lmj	LAj
	−1.48	−0.15	−0.20	0.33	0.16	−0.10	0.26	0.42
25	84.1	81.8	62.5	46.8	65.4	83.8	82.4	78.1	80.1	15.4	−44.7	35.4
31.5	62.3	60.0	54.9	40.0	56.8	63.7	60.3	62.0	60.1	8.0	−39.3	20.8	80.2	40.9
40	58.9	49.4	58.5	55.7	52.9	51.8	46.4	53.7	54.7	4.3	−34.6	20.1
50	70.8	67.8	74.2	74.9	73.0	66.0	65.3	62.2	71.1	5.0	−30.2	40.9
63	71.7	76.4	76.1	75.2	71.9	66.0	78.4	59.6	74.4	6.8	−26.2	48.2	77.1	50.9
80	75.4	73.0	67.1	69.8	64.9	63.8	68.7	69.0	70.1	3.8	−22.5	47.6
100	69.9	64.8	60.9	61.9	65.4	65.1	60.8	62.8	64.6	2.7	−19.1	45.5
125	74.6	73.3	75.1	66.6	71.0	66.2	68.5	70.0	71.5	3.3	−16.1	55.4	75.0	58.9
160	69.8	69.1	70.5	69.5	68.6	74.1	71.9	75.1	71.8	2.8	−13.4	58.4
200	81.8	71.1	76.8	74.0	69.4	78.2	76.5	74.0	76.3	3.8	−10.9	65.4
250	83.0	72.6	79.8	76.3	73.0	80.2	79.5	75.9	78.4	3.5	−8.6	69.8	81.2	72.6
315	77.1	68.8	73.4	70.6	75.0	72.6	74.0	65.7	73.0	3.5	−6.6	66.4
400	83.4	80.6	80.2	69.9	69.7	79.6	73.2	74.2	78.2	5.2	−4.8	73.4
500	84.2	81.3	80.5	71.4	70.8	80.6	74.4	75.1	79.0	4.9	−3.2	75.8	81.9	78.7
630	65.4	67.1	69.4	71.3	73.8	68.2	70.4	71.8	70.5	3.4	−1.9	68.6
800	60.7	60.9	63.9	62.5	64.1	63.1	69.5	63.8	64.6	3.3	−0.8	63.8
1 k	61.8	63.4	65.9	63.5	63.1	63.4	71.7	65.7	66.2	3.8	0	66.2	71.4	71.4
1.25 k	68.8	68.8	71.2	69.7	67.8	66.5	65.8	66.6	68.4	1.8	0.6	69.0
1.6 k	72.5	69.3	68.9	68.1	70.8	70.8	67.1	65.4	69.3	1.9	1	70.3
2 k	72.3	72.0	72.2	70.4	69.6	72.9	71.1	71.3	71.5	0.9	1.2	72.7	74.8	76.0
2.5 k	70.2	69.2	70.3	69.2	67.1	68.3	69.3	67.3	68.9	1.0	1.3	70.2
3.15 k	66.2	68.5	67.4	65.9	65.9	66.9	65.0	66.0	66.5	1.1	1.2	67.7
4 k	69.4	69.8	69.7	68.2	66.0	68.9	67.7	67.4	68.4	1.1	1	69.4	71.6	72.6
5 k	67.5	65.1	65.2	65.4	62.5	63.5	64.3	63.0	64.6	1.2	0.5	65.1
6.3 k	67.7	64.0	63.1	64.9	63.5	62.5	66.1	63.8	64.6	1.5	−0.1	64.5
8 k	68.6	65.5	65.2	64.3	63.2	65.2	64.2	64.4	65.1	1.1	−1.1	64.0	69.3	68.2
10 k	65.5	65.6	63.5	63.1	64.3	63.8	62.9	61.9	63.8	0.9	−2.5	61.3
12.5 k	66.1	63.0	63.2	62.4	61.7	61.9	60.8	62.7	62.8	1.1	−4.3	58.5
16 k	59.6	57.7	56.3	58.3	56.3	54.9	55.0	55.2	56.8	1.4	−6.6	50.2	63.9	57.3
20 k	53.2	51.4	49.8	49.3	49.0	47.3	47.7	47.6	49.6	1.6	−9.3	40.3

shows the exemplary acoustic measurements of a gear pump after tooth root undercutting for pt = 12 MPa.

As part of the research, the following measurements were performed: the sound pressure level L_p, the A-weighted sound pressure level L_A, the sound power level L_W, and the A-weighted sound power level L_WA. The average value of the sound pressure level L_p was calculated according to the following formula:

$$L_p=10\lg \left(\frac{1}{n}{{\displaystyle \sum _{i=1}^{n}10}}^{0.1L_{pi}}\right)$$

(2)

• $L_{pi}$—sound pressure level at the ith measuring point and
• n—total number of measuring points.

The mean value of the A-weighted sound level was determined in the same manner:

$$L_A=10\lg \left(\frac{1}{n}{{\displaystyle \sum _{i=1}^{n}10}}^{0.1L_{Ai}}\right)$$

(3)

• $L_{Ai}$—sound pressure level at the i-th measuring point and
• n—total number of measuring points.

The value of the A-weighted sound level was determined on the basis of the measured sound pressure level $L_{Ai}$ in the j-th band and after the correction $K_A^j$ resulting from the characteristics of the weighted curve using the following formula:

$$L_A^j=L_p^j+K_A^j$$

(4)

• $L_p^j$—sound pressure level in the j-th frequency band,
• $L_A^j$—A-weighted sound level in the j-th frequency band, and
• $K_A^j$—correction according to the A characteristic for the j-th frequency band.

Figure 14 presents the value of the acoustic pressure level L_m, corrected acoustic pressure level L_A, acoustic power level L_p, and corrected acoustic power level L_pA in the function of the discharge pressure p_t at a constant rotational speed of a pump shaft n = 1500 rpm.

Figure 15 and Figure 16 present a tertian and an octave spectrum of the gear pump after tooth root undercutting for the nominal discharge pressure and nominal rotational speed.

The measurement of the a_RMS 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. Presentation of the operating parameters of pumps from the 2nd group of commercial model (prototype) for the nominal operating parameters.

Parameter	Commercial Pump	Model Pump
Sound level A(dB)	83–84	79
Total efficiency (%)	73–76	87.0
Coefficient of performance unevenness (%)	19.0–21.0 *	19.9
Sound vibration level (effective acceleration value) (m/s2)	0.058–0.074	0.049
Power to weight parameter (kW/kg)	3.2–3.5 **	4.5 ***
Energy efficiency q/V	0.27–0.29 *	0.31
Sealed space compression ratio Vmax/Vmin	1.1–1.2 *	1.0

* 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.

6. 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].