Criteria for Evaluating the Tribological Effectiveness of 3D Roughness on Friction Surfaces

: A new technique for finishing the surfaces of friction pairs has been proposed, which, in combination with the original test method, has shown a significant influence of the initial roughness configuration (surface texture) on friction and wear. Two types of finishing processing of the shaft friction surfaces were compared, and it was found that the friction and wear coefficients differ by more than 2–5 and 2–4 times, respectively. Based on a new methodology for analyzing standard roughness parameters, the tribological efficiency criteria (in the sense of reducing friction and wear) are proposed for the initial state of the friction surface of a radial plane sliding bearing shaft relative to the friction direction, which is consistent with its frictional characteristics. Comparison of the laboratory test results with the surface tribological efficiency criteria showed that these criteria are very promising for controlling existing technologies and optimizing new technologies for friction surface finishing in various friction systems.


Introduction
From a tribology perspective, mechanical devices consist of various friction units with distinct contact geometry and friction kinematics.The overall efficiency and operability of the product depend on the efficiency, reliability, and durability of each of these friction units, especially in case of critical wear [1].A special place among them is occupied by aviation fuel control equipment, on which flight safety directly depends.These units use fuel as a lubricating medium, making it impossible to lubricate tribological contacts with modern oils that have highly effective additive packages.Therefore, traditional approaches are used to increase the reliability and service life of such units.These include selecting a material combination for the parts of the friction pairs, using various chemical and thermal technologies for processing friction surfaces, and applying various effective coatings through various physico-chemical methods [2,3].
The technology of finishing friction surfaces is the most common method to improve tribological properties in the manufacture of friction parts, especially fuel control equipment on surfaces, which can retain boundary-lubricating layers.This texturing technique is widely used in various sliding bearings, as evidenced by numerous publications [4][5][6][7][8][9] and their references.However, there are significantly fewer publications dedicated to shaft texturing [10][11][12][13].
The microgeometry of the rough surfaces of contacting parts changes rapidly at the start of the tribological interaction process [14].At the same time, the intensive processes that occur during the initial stage of run-in, such as microplastic deformation, local thermal flares (triboplasma), and desorption of boundary layers, significantly affect the behavior of the tribological contact, including wear, friction force, and durability [15].Therefore, the Lubricants 2024, 12, 209 2 of 18 reliability and service life of a product are not only determined by the accuracy of its size or shape but also by the state of the micro-or nano-geometry of the surface layer.
The adhesion-deformation or molecular-mechanical theory of friction and wear has been widely used to describe the mechanisms and causes of wear of rubbing surfaces [16].According to this theory, the boundary layers of lubricant are almost solid bodies that are destroyed by tangential mechanical stresses and heat flows.However, this contradicts the fundamental theories of boundary layer adsorption.Based on the identified fundamental phenomena of extrusion and rarefaction in boundary layers of lubricant, we propose an adhesion-deformation-hydrodynamic (ADH) model of friction and wear [17].The ADH model explains that the primary cause of adhesion interaction and destruction of friction surfaces is the desorption of boundary layers in diffuser elastically deformed discrete micro-and macro-curved contacts.Rarefaction occurs there under the influence of negative gradients of contact stresses.Therefore, the diffuser elastically deformed regions are the most dangerous, as rarefaction always occurs there during friction.Intense desorption of the boundary layers causes quasi-dry friction conditions, leading to primary adhesion and subsequent tearing of the material from the bearing surface.
To prevent the destructive effects of friction in diffuser regions, the ADH model of friction and wear proposes creating a structure of protrusions and indentations on the surface of the shaft.This structure could increase the pressure in the diffuser zones of microcontacts.This approach was tested in [18].The authors suggest that this is due to the lateral flow of the boundary lubricant from areas of high pressure, such as the confusor zones, to areas of lower pressure, such as the diffuser microcontact zones, through bypass channels on the bearing.This paper presents an approach based on the ADH model to create such closed microcrater zones on the friction surface of the shaft, in the depressions of which, with natural pre-compression in the confusor and transition zones of the elastically deformed contact, it is possible to increase the pressure of the boundary lubricant.It is assumed that in the rarefied zones of tribological contacts, pressure increases due to the flow of previously compressed lubricant fragments from closed cavities through the contact contours.This results in a decrease in the intensity of their desorption in accordance with the Langmuir and BET (Brunauer-Emmett-Teller) adsorption isotherms.
To implement the proposed approach, two objectives must be met: to develop a method for creating micro-craters on the surface of the shaft and to determine methods for evaluating their 3D configuration relative to the direction of friction.
The technological process for creating the friction surfaces of sliding bearing shafts involves sequentially grinding, fine-tuning, and polishing at high rotational speeds using special lapping and abrasive media of various grain sizes.The resulting mirrored surfaces are believed to have better tribological properties when friction occurs in the bearing.However, it has been observed that such tribological sliding systems experience significant adhesive-deformation wear during operation.To improve wear resistance and decrease friction force, we propose creating micro-crater formations on the mirrored surfaces, as suggested by the ADH model of friction and wear.The primary objective in creating surfaces with uniformly distributed micro-crater formations on the shafts is to utilize a specific abrasive medium and finishing modes.
The task also involves determining the tribological properties of surfaces after finishing and establishing their relationship with the initial roughness parameters.The tribological properties of the surface are determined through laboratory tests using specialized equipment, for example, as in [17].These tests ensure that identical samples experience the same type of tribological contact under the same initial conditions throughout the friction process of the model shaft along the model bearing, including during a single revolution.
To determine the surface roughness parameters, a well-known control and analysis methods according to ISO 4287 [19], ISO 13565 [20], and ISO 25178 [21] are utilized.Modern measuring equipment enables the acquisition of information regarding the volumetric configuration of nano-roughness and its corresponding parameters in 3D space.As demon-strated in [22], this information provides a more comprehensive understanding of the statistical properties of roughness than existing standard roughness parameters.
Preliminary research suggests that these systems can be used to measure local surface defects, including assessing volumetric wear of machine parts.This is particularly important in laboratory tribological tests [18].The 3D parameters or characteristics obtained from this information should allow for both qualitative and quantitative evaluation of the variation in roughness properties in different directions for shaft surfaces, whether obtained by traditional or proposed new finishing techniques.
The purpose of this paper is therefore (1) to experimentally confirm the possibility of reducing the frictional force and increasing the wear resistance of frictional surfaces by applying a surface-finishing technology that provides the necessary configuration of the nano-geometry of the roughness of the shaft, and (2) to search for parameters of the volumetric roughness configuration that allow for the evaluation of the tribological efficiency of the surface in terms of reducing friction and wear, taking into account the direction of friction.

Objects, Materials, and Research Method
The Timken linear contact scheme was chosen for laboratory modeling of friction pairs because it corresponds to the operation of the sliding friction unit, which is widely used in mechanical engineering, including plunger pumps, multiple spool pairs, and other friction units of fuel control equipment of various types of engines.This scheme is also the most problematic in terms of wear resistance.To ensure good repeatability of the results and avoid misinterpretation of data after testing, it is extremely important to ensure several important requirements regarding the preparation of test samples and the test methodology itself.We used a technique for processing the surfaces of shafts assembled in one cassette that allows us to create a certain texture on their rubbing surfaces with a high degree of uniformity.Also, when physically modeling tribocontacts, our test methodology assumes the need to ensure constant contact and constant contact stresses at any time of friction.
The BIT-TRIBO-01 laboratory system, shown in Figure 1, was chosen from the available Timken friction-testing machines [23].The BIT-TRIBO-01 friction laboratory setup is designed to determine the wear and friction resistance of lubricants and their additives, structural materials, and coatings.It also evaluates the effect of friction surface texture on their friction performance by testing a model friction system with a single linear contact (see Figure 2).Its notable feature is that this single contact is complete and constant within a circle of the axis, which closely corresponds to its theoretical ideas.On this device, the counter-sample 1 (model shaft) slides over the surface of the planar sample 2 while maintaining a constant contact length (indicated by the white contact line) (see Figure 2).At the same time, displacement of the contact line relative to the sensitive element of the friction force measurement sensor is almost completely eliminated.The design of the device is such that all axes along which the contact may have changed (OX, OY, and OZ) intersect at the centroid of the counter-sample 1-at point O (Figure 2). Figure 2 shows the axis of action of load N at the center of linear contact (OY axis).Figure 2 also shows the OX 1 and OZ 1 axes associated with the orientation of the linear contact.At the same time, all axes are coordinated to the centroid O of the model shaft.Therefore, at the slightest displacement of the contact, all axes simultaneously deviate relative to the centroid of the counter-sample O, maintaining perpendicularity to each other and providing constant and complete single linear contact.The BIT-TRIBO-01 device achieved high reproducibility of friction and wear laboratory test results due to the high stability of the initial contact conditions.Additionally, this device allows for observation of a uniform secondary structure on the friction surfaces of counter-samples that corresponds to the properties of the friction pair material, lubricant, and surface texture under specified friction conditions.This is a difficult observation to make for other devices with Timken schemes.Flat stationary specimens (model flat bearings) were made of steel G52986 with a hardness of HRC 59-62 and the nano-sized roughness of a model flat bearing.The surface roughness, determined by the Ra parameter, was less than 0.01 µm and achieved through sequential multistage polishing on cast iron lapping plates with different diamond pastes ranging from 40 µm to 0.5 µm in grain size.
Two types of friction pairs were used for tribological tests within the framework of this paper, differing only in the friction surfaces of the model shafts.The first type had surfaces obtained through the traditional method of grinding and polishing with diamond  The BIT-TRIBO-01 device achieved high reproducibility of friction and wear laboratory test results due to the high stability of the initial contact conditions.Additionally, this device allows for observation of a uniform secondary structure on the friction surfaces of counter-samples that corresponds to the properties of the friction pair material, lubricant, and surface texture under specified friction conditions.This is a difficult observation to make for other devices with Timken schemes.Flat stationary specimens (model flat bearings) were made of steel G52986 with a hardness of HRC 59-62 and the nano-sized roughness of a model flat bearing.The surface roughness, determined by the Ra parameter, was less than 0.01 µm and achieved through sequential multistage polishing on cast iron lapping plates with different diamond pastes ranging from 40 µm to 0.5 µm in grain size.
Two types of friction pairs were used for tribological tests within the framework of this paper, differing only in the friction surfaces of the model shafts.The first type had surfaces obtained through the traditional method of grinding and polishing with diamond The BIT-TRIBO-01 device achieved high reproducibility of friction and wear laboratory test results due to the high stability of the initial contact conditions.Additionally, this device allows for observation of a uniform secondary structure on the friction surfaces of counter-samples that corresponds to the properties of the friction pair material, lubricant, and surface texture under specified friction conditions.This is a difficult observation to make for other devices with Timken schemes.
Flat stationary specimens (model flat bearings) were made of steel G52986 with a hardness of HRC 59-62 and the nano-sized roughness of a model flat bearing.The surface roughness, determined by the Ra parameter, was less than 0.01 µm and achieved through sequential multistage polishing on cast iron lapping plates with different diamond pastes ranging from 40 µm to 0.5 µm in grain size.
Two types of friction pairs were used for tribological tests within the framework of this paper, differing only in the friction surfaces of the model shafts.The first type had surfaces obtained through the traditional method of grinding and polishing with diamond paste AFM.The second type had surfaces obtained through additional finishing of previously polished shafts using silicon carbide and the same laps.
Counter-samples that simulate the shaft were prepared from steel G52986 with HRC 59-62 hardness.The surfaces of the samples were prepared using glass laps with increased flatness (no more than 0.5 µm on an area of 120 × 40 mm 2 ).The laps were made using the well-known three-plate method and manually in a suspension of silicon carbide (5 µm) in mineral oil with a ratio of 1:3.More than a dozen control samples were collected in cassettes, fixed, and ground in processing centers (see Figure 3a).The cassettes were manually polished using various diamond paste AFMs and rotated at a frequency of 1000 rpm.This followed the traditional step-by-step diamond-polishing technology with a grain size ranging from 40 µm to 0.5 µm.As a result, we obtained several highly polished model shafts.
Lubricants 2024, 12, x FOR PEER REVIEW 5 paste AFM.The second type had surfaces obtained through additional finishing of p ously polished shafts using silicon carbide and the same laps.
Counter-samples that simulate the shaft were prepared from steel G52986 with 59-62 hardness.The surfaces of the samples were prepared using glass laps with incre flatness (no more than 0.5 µm on an area of 120 × 40 mm 2 ).The laps were made usin well-known three-plate method and manually in a suspension of silicon carbide (5 µ mineral oil with a ratio of 1:3.More than a dozen control samples were collected in settes, fixed, and ground in processing centers (see Figure 3a).The cassettes were m ally polished using various diamond paste AFMs and rotated at a frequency of 1000 This followed the traditional step-by-step diamond-polishing technology with a grain ranging from 40 µm to 0.5 µm.As a result, we obtained several highly polished m shafts.
(a) (b) To create micro-crater formations on the surface of the model shafts, we utilize additional finishing technology called Si-technology.This involved finishing with a suspension of silicon carbide.
It has been experimentally established that when silicon carbide with a maxi grain size of 5 µm is in a free state in an oil emulsion in a ratio of 1:3 and is located bet a flat glass lap and the processed steel surface of the shaft along the line, a micro-c texture forms on the steel surface.Given the plate-like structure of silicon carbide cr lites and their relatively high hardness (ranked fourth after diamond), it is reasonab assume that large silicon carbide crystals may crumble into smaller fractions durin ishing, resulting in craters on the surface of the shaft.
After polishing the diamonds, the cassettes were washed thoroughly with petro ether, a mixture of low-molecular-weight hydrocarbons (C5H12 and C6H14), and were wiped with moistened filter paper before being dried.The same procedure was rep for the cassettes after additional finishing with Si-technology.
Five samples were taken from the central part of the cassettes, which were proc differently and tested to avoid the influence of edge effects after finishing.The m shafts were prepared and installed on the friction machine according to the desc methods, with radial deviations from the axis of rotation not exceeding 3 µm.Upon tact with the flat sample, it was fully immersed in aviation kerosene.
The hardness of the samples was determined using a standardized DECHUAN ness tester (Laizhou, China).Radial runout of the model shafts after installation o spindle was eliminated using a special device, which measured micro-displacem when the spindle shaft was slowly rotated using a MITUTOYO (Kanagawa, Japan To create micro-crater formations on the surface of the model shafts, we utilized an additional finishing technology called Si-technology.This involved finishing with an oil suspension of silicon carbide. It has been experimentally established that when silicon carbide with a maximum grain size of 5 µm is in a free state in an oil emulsion in a ratio of 1:3 and is located between a flat glass lap and the processed steel surface of the shaft along the line, a micro-crater texture forms on the steel surface.Given the plate-like structure of silicon carbide crystallites and their relatively high hardness (ranked fourth after diamond), it is reasonable to assume that large silicon carbide crystals may crumble into smaller fractions during finishing, resulting in craters on the surface of the shaft.
After polishing the diamonds, the cassettes were washed thoroughly with petroleum ether, a mixture of low-molecular-weight hydrocarbons (C 5 H 12 and C 6 H 14 ), and were then wiped with moistened filter paper before being dried.The same procedure was repeated for the cassettes after additional finishing with Si-technology.
Five samples were taken from the central part of the cassettes, which were processed differently and tested to avoid the influence of edge effects after finishing.The model shafts were prepared and installed on the friction machine according to the described methods, with radial deviations from the axis of rotation not exceeding 3 µm.Upon contact with the flat sample, it was fully immersed in aviation kerosene.
The hardness of the samples was determined using a standardized DECHUAN hardness tester (Laizhou, China).Radial runout of the model shafts after installation on the spindle was eliminated using a special device, which measured micro-displacements when the spindle shaft was slowly rotated using a MITUTOYO (Kanagawa, Japan) dial indicator with a division value of 1 µm.Measurements of deviations in the shape of the surfaces of the samples were carried out on a "KEYENCE" (Osaka, Japan) coordinate-measuring machine.
Test conditions: The linear sliding speed was 0.3 m/s and the axial load was 110 N. Based on G. Hertz's formulas [24] for this linear contact of surfaces made of G52986 steel, the maximum initial design contact stresses were 70 MPa, which is realistic for fuel equipment.However, it should be noted that the theory of G. Hertz assumed perfectly smooth surfaces without boundary layers.The calculated initial contact width was approximately 0.16 mm, which is over 50 times greater than the radial runout of the shaft, which was 0.003 mm.
Friction and wear test method: Tribological comparison tests were performed using Timken test methods based on ASTM D2509 [25] for greases and ASTM D2782 [26] for oils.However, in our case, we compared not lubricating media but the anti-wear and anti-friction properties of surfaces having different three-dimensional states in the same environmentaviation kerosene at the same initial contact loads, at the same sliding speed, for the same four-stage tests, with the same time, and in the same external environmental conditions.
After applying a load of 110 N and filling the chamber with aviation kerosene TS-1, the shaft rotated at a tangential speed of 0.3 m/s.The first stage of friction lasted for 15 min.Subsequently, two more such tests (2 nd and 3 rd stages) were carried out using the same model shaft, but on new sections of the surface of the flat model bearing, each lasting 15 min.Finally, the fourth stage of friction was carried out for 3 h.It was possible to construct accurate wear rate dependencies by assuming complete running-in occurred in the first three stages.The similar wear values after friction in the second and third stages indicate that the additional work in kerosene was completed.The tests monitored the volumetric temperature of the kerosene and friction force, and wear tracks were measured and investigated afterward.
If the surface roughness is described in accordance with ISO 4287 [19], the data obtained on the average value and distribution of the values of the roughness parameters along a plane do not provide information on the angular characteristics of these parameters.However, raster surface scanning and additional data processing made it possible to obtain the angular distribution of the surface roughness parameters, taking into account the direction of friction.This paper introduces some angular characteristics that allow us to describe the standard roughness parameters.
For example, Figure 4 shows the results of the calculation for the parameter Ra of the standard surface roughness.The calculation is presented in the form of a standard regular profile with Rz = 0.1 µm.The relief of the profile is shown in Figure 4a, taking into account the angular distribution of the average value of this parameter, as shown in Figure 4c.Additionally, a histogram of the distribution of the Ra parameter along raster lines is presented in Figure 4b, and its dependence on the direction is shown in Figure 4d. Figure 4b shows a histogram of the distribution of the Ra parameter value for angle φ = 0 • .The profile is measured in the direction perpendicular to the grooves on the surface of the test sample, as shown in Figure 4a.The blue curves in Figure 4c show the standard deviation of the Ra parameter from its average value.Figure 4d shows the quantitative distribution of surface profiles based on the Ra parameter value and the direction of the profile study.The red color Indicates the maximum number of profiles (N).The contours of the color image characterizing the distribution of the Ra parameter in Figure 4d

Analysis of the Results
During multiple tribological laboratory tests of friction pairs with various finishing work surfaces, it was discovered that these pairs have fundamentally different tribological properties, all other factors being equal (in the same environment, at the same initial contact loads, at the same sliding speed, during the same duration of the friction process, and under the same external normal environmental conditions).Friction pairs in which the working surfaces of the model shafts were adjusted using Si-technology exhibited abnormally low values for the friction coefficient and wear rate.At the same time, these friction pairs exhibited a slight variation in tribological characteristics during the burnishing stage and maximum resistance to setting, with the critical specific load being at least 200 MPa.
Figure 5 show some results of wear tests on friction pair samples (5 control samples from different batches) in four stages (I, II, III, and IV).The model shaft surfaces were finished using various technologies.The temporal dependencies of wear of a flat bearing are presented in Figure 5a, while the histograms of wear of a flat bearing depending on the batch number are shown in Figure 5b.The wear is presented as the depth I of the trace formed on the surface of the flat bearing as result of the material removing during the friction process.

Analysis of the Results
During multiple tribological laboratory tests of friction pairs with various finishing work surfaces, it was discovered that these pairs have fundamentally different tribological properties, all other factors being equal (in the same environment, at the same initial contact loads, at the same sliding speed, during the same duration of the friction process, and under the same external normal environmental conditions).Friction pairs in which the working surfaces of the model shafts were adjusted using Si-technology exhibited abnormally low values for the friction coefficient and wear rate.At the same time, these friction pairs exhibited a slight variation in tribological characteristics during the burnishing stage and maximum resistance to setting, with the critical specific load being at least 200 MPa.
Figure 5 show some results of wear tests on friction pair samples (5 control samples from different batches) in four stages (I, II, III, and IV).The model shaft surfaces were finished using various technologies.The temporal dependencies of wear of a flat bearing are presented in Figure 5a, while the histograms of wear of a flat bearing depending on the batch number are shown in Figure 5b.The wear is presented as the depth I of the trace formed on the surface of the flat bearing as result of the material removing during the friction process.Significant differences were observed in the behavior of friction coefficient oscillograms during the friction of shafts prepared using traditional processing methods and during the friction of the same shafts processed using Si-technology in the most informative fourth stage of the tests (see Figure 6).The results show that there were both quantitative and qualitative differences between the two technologies.The friction coefficient Cf decreased by 2-5 times in the case of the Si-technology compared to the traditional technology both in terms of amplitude and frequency of oscillations.Additionally, there were differences in the amplitude and frequency of oscillations, indicating a decrease in the intensity of the adhesive forces in the interaction of the friction surfaces in contact (submicron adhesions) in the case of Si-technology. Figure 6    Significant differences were observed in the behavior of friction coefficient oscillograms during the friction of shafts prepared using traditional processing methods and during the friction of the same shafts processed using Si-technology in the most informative fourth stage of the tests (see Figure 6).The results show that there were both quantitative and qualitative differences between the two technologies.The friction coefficient C f decreased by 2-5 times in the case of the Si-technology compared to the traditional technology both in terms of amplitude and frequency of oscillations.Additionally, there were differences in the amplitude and frequency of oscillations, indicating a decrease in the intensity of the adhesive forces in the interaction of the friction surfaces in contact (submicron adhesions) in the case of Si-technology. Figure 6  Significant differences were observed in the behavior of friction coefficient oscillograms during the friction of shafts prepared using traditional processing methods and during the friction of the same shafts processed using Si-technology in the most informative fourth stage of the tests (see Figure 6).The results show that there were both quantitative and qualitative differences between the two technologies.The friction coefficient Cf decreased by 2-5 times in the case of the Si-technology compared to the traditional technology both in terms of amplitude and frequency of oscillations.Additionally, there were differences in the amplitude and frequency of oscillations, indicating a decrease in the intensity of the adhesive forces in the interaction of the friction surfaces in contact (submicron adhesions) in the case of Si-technology. Figure 6    The graphs in Figure 6 show that samples treated with Si-technology have a friction coefficient reduced by up to five times as well wear resistance increased by two to three times (Figure 5b).The linear roughness parameters (ISO 4287) obtained from the surfaces under study using one profilogram for samples with Si-technology surface treatment differed from those with traditional surface treatment, showing a decrease.The surfaces of model shafts manufactured using Si-technology had, on average, 30-50% lower Ra and Rz parameter values compared to those manufactured using traditional technology.However, this alone does not account for the significant differences in wear rates and friction coefficients.Therefore, a search was carried out for more informative parameters using a profilograph-profilometer.
The study investigated cylindrical surface sections of a series of model shafts with surface finishing using both traditional technology and Si-technology. Figure 7 shows typical surface reliefs of the indicated areas with a size of 300 × 300 µm 2 , which were obtained using a profilograph-profilometer. The obtained images (Figure 7b,c) clearly show that the three-dimensional configuration of roughness on the cylindrical surfaces of the investigated shafts has different characteristics.On the surface of the shaft processed using traditional technology, grooves in the form of a structure are clearly visible.These grooves run perpendicular to the axis of the model shaft and coincide with the direction of the X axis in Figure 7. Similarly, grooves are visible on the surface of the shaft processed using Si-technology, but they are less noticeable and lack a clearly defined direction.In addition, it is worth mentioning that the surface irregularities on the shaft, which were processed using Si-technology, are distributed more uniformly.
Lubricants 2024, 12, x FOR PEER REVIEW 9 of 18 The graphs in Figure 6 show that samples treated with Si-technology have a friction coefficient reduced by up to five times as well wear resistance increased by two to three times (Figure 5b).The linear roughness parameters (ISO 4287) obtained from the surfaces under study using one profilogram for samples with Si-technology surface treatment differed from those with traditional surface treatment, showing a decrease.The surfaces of model shafts manufactured using Si-technology had, on average, 30-50% lower Ra and Rz parameter values compared to those manufactured using traditional technology.However, this alone does not account for the significant differences in wear rates and friction coefficients.Therefore, a search was carried out for more informative parameters using a profilograph-profilometer.
The study investigated cylindrical surface sections of a series of model shafts with surface finishing using both traditional technology and Si-technology. Figure 7 shows typical surface reliefs of the indicated areas with a size of 300 × 300 µm 2 , which were obtained using a profilograph-profilometer. The obtained images (Figure 7b,c) clearly show that the three-dimensional configuration of roughness on the cylindrical surfaces of the investigated shafts has different characteristics.On the surface of the shaft processed using traditional technology, grooves in the form of a structure are clearly visible.These grooves run perpendicular to the axis of the model shaft and coincide with the direction of the X axis in Figure 7. Similarly, grooves are visible on the surface of the shaft processed using Si-technology, but they are less noticeable and lack a clearly defined direction.In addition, it is worth mentioning that the surface irregularities on the shaft, which were processed using Si-technology, are distributed more uniformly.The angular spatial spectrum of the test surfaces analyzed due to their pronounced dependence on the direction was considered first.It was found that the angular spatial spectrum has almost identical characteristics in different areas of the cylindrical surface of samples processed using the same technology.However, the angular spectra of the surfaces of the samples processed by different technologies differed significantly.Figure 8 shows typical graphs of angular spectra for the test surfaces of model shafts.The direction with an angle of 0 • in the graphs of the angular spectra corresponds to the X-axis in Figure 7.
Lubricants 2024, 12, x FOR PEER REVIEW 10 of 18 The angular spatial spectrum of the test surfaces analyzed due to their pronounced dependence on the direction was considered first.It was found that the angular spatial spectrum has almost identical characteristics in different areas of the cylindrical surface of samples processed using the same technology.However, the angular of the surfaces of the samples processed by different technologies differed significantly.Figure 8 shows typical graphs of angular spectra for the test surfaces of model shafts.The direction with an angle of 0 in the graphs of the angular spectra corresponds to the X-axis in Figure 7.The ISO 25178 [21] provides a quantitative characterization of the angular spatial spectrum of the test surface using the parameters Std and Stdi.The texture direction, Std, is defined as the angle of the dominating texture of the rough surface.The texture direction index, Stdi, is a measure of how dominant the dominating direction is.The Stdi value is always between 0 and 1. Surfaces with very dominant directions will have Stdi values close to zero, and if all directions are similar, Stdi will be close to 1. Figure 8 shows the values of Std and Stdi parameters for different technologies used to finish the surfaces of model shafts.The Std parameter confirms the main direction of the surface texture, which is consistent with the conclusions drawn from Figure 7.With traditional surface-processing technology (Figure 8a), directional roughness is clearly expressed.The grooves on the surface are parallel to the Y axis in Figure 7a, the maximum values of which are achieved perpendicular to the grooves on the surface in the direction of 0 (along the X axis in Figure 7a).Similar reasoning can be repeated regarding Si-technology. Figure 8b shows that the angular spectrum has the largest values in several selected directions, namely, 0, 45, and 90.In this case, the Std parameter shows the greatest dominance in the direction of 90.Additionally, the roughness direction with Si-technology is significantly less dominant compared to with traditional technology, as evidenced by the values of the Stdi parameter.With traditional technology, the Stdi is much closer to zero (Stdi = 0.178) than with Si-technology, where it is closer to unity (Stdi = 0.409).This is supported by the shape of the angular spectrum graph.With traditional technology, the angular spectrum is narrow, with a clear maximum in the direction of roughness dominance.However, with Si-technology, the angular spectrum is significantly wider.
Figure 9 shows typical angular dependences of the histogram of the distribution of parameters Ra, Rz, and Rk over the surface for model shaft samples processed using traditional technology (Figure 9a,c,e) and Si-technology (Figure 9b,d,f).The orientation of the shaft samples with different surface-polishing technologies and the test areas of their surfaces coincide with the designations shown in Figure 7a.This case does not consider the fact that the average values of the parameters are slightly different for both technologies, as previously noted.The obtained images show that for both surface pre-treatment The ISO 25178 [21] provides a quantitative characterization of the angular spatial spectrum of the test surface using the parameters Std and Stdi.The texture direction, Std, is defined as the angle of the dominating texture of the rough surface.The texture direction index, Stdi, is a measure of how dominant the dominating direction is.The Stdi value is always between 0 and 1. Surfaces with very dominant directions will have Stdi values close to zero, and if all directions are similar, Stdi will be close to 1. Figure 8 shows the values of Std and Stdi parameters for different technologies used to finish the surfaces of model shafts.The Std parameter confirms the main direction of the surface texture, which is consistent with the conclusions drawn from Figure 7.With traditional surface-processing technology (Figure 8a), directional roughness is clearly expressed.The grooves on the surface are parallel to the Y axis in Figure 7a, the maximum values of which are achieved perpendicular to the grooves on the surface in the direction of 0 • (along the X axis in Figure 7a).Similar reasoning can be repeated regarding Si-technology. Figure 8b shows that the angular spectrum has the largest values in several selected directions, namely, 0 • , 45 • , and 90 • .In this case, the Std parameter shows the greatest dominance in the direction of 90 • .Additionally, the roughness direction with Si-technology is significantly less dominant compared to with traditional technology, as evidenced by the values of the Stdi parameter.With traditional technology, the Stdi is much closer to zero (Stdi = 0.178) than with Si-technology, where it is closer to unity (Stdi = 0.409).This is supported by the shape of the angular spectrum graph.With traditional technology, the angular spectrum is narrow, with a clear maximum in the direction of roughness dominance.However, with Si-technology, the angular spectrum is significantly wider.
Figure 9 shows typical angular dependences of the histogram of the distribution of parameters Ra, Rz, and Rk over the surface for model shaft samples processed using traditional technology (Figure 9a,c,e) and Si-technology (Figure 9b,d,f).The orientation of the shaft samples with different surface-polishing technologies and the test areas of their surfaces coincide with the designations shown in Figure 7a.This case does not consider the fact that the average values of the parameters are slightly different for both technologies, as previously noted.The obtained images show that for both surface pretreatment technologies the parameters Ra, Rz, and Rk have similar angular dependencies within the same technology.However, these dependencies differ between different surface treatment technologies.This applies to both the average values of the parameters and to the distributions of parameters across surface profiles.With traditional technology, there is a wide and non-uniform scatter of all parameters (40-60% of the average value), which depends on the angle and on individual surface profiles in a given direction.For Si-technology, the dispersion of parameter in terms of angle and individual profiles in a given direction is significantly smaller (25-35% of the average value) and more consistent (the peaks of the predominant parameter values are less fragmented in terms of angle compared to the case of standard technology).It is important to note that the level of roughness with Si-technology is approximately the same over the entire surface of the sample and is not significantly by the direction, as the direction of roughness is weakly expressed.technologies the parameters Ra, Rz, and Rk have similar angular dependencies within the same technology.However, these dependencies differ between different surface treatment technologies.This applies to both the average values of the parameters and to the distributions of parameters across surface profiles.With traditional technology, there is a wide and non-uniform scatter of all parameters (40-60% of the average value), which depends on the angle and on individual surface profiles in a given direction.For Si-technology, the dispersion of parameter in terms of angle and individual profiles in a given direction is significantly smaller (25-35% of the average value) and more consistent (the peaks of the predominant parameter values are less fragmented in terms of angle compared to the case of standard technology).It is important to note that the level of roughness with Si-technology is approximately the same over the entire surface of the sample and is not significantly affected by the direction, as the direction of roughness is weakly expressed.The ISO 13565 [20] identifies the bearing area curve (BAC) as a significant characteristic of a rough surface.The BAC, also known as the Abbott-Firestone curve, was proposed as a general measure of surface quality for machined parts.It is a simple and The ISO 13565 [20] identifies the bearing area curve (BAC) as a significant characteristic of a rough surface.The BAC, also known as the Abbott-Firestone curve, was proposed as a general measure of surface quality for machined parts.It is a simple and practical tool for assessing surface quality.The BAC is an integral characteristic of contacting rough surfaces and provides a visual representation of the primary operational characteristics of the contact, including rigidity, wear resistance, and tightness.The BAC determines several crucial parameters, such as Rk, Rpk, and Rvk, which establish the limits of the profile core, the average height of the protrusions, and the average depth of the depressions, respectively.Additionally, the parameters Mr1 and Mr2 determine the proportion of surface protrusions and depressions, respectively.
Figure 10 shows typical images illustrating the change in BAC based on the direction.The color-coded amplitude values of the points on the BAC are shown for both surfaceprocessing technologies of model shafts.The orientation of shaft samples with different surface-polishing technologies and the areas of surface examination coincides with the designations shown in the Figure 7a.The images obtained also illustrate the presence of a dominant roughness direction on surfaces manufactured using traditional technology (Figure 10a).The average depth of the depressions and the height the protrusions increase in these surfaces.This is indicated by the increase in BAC values in Figure 10a, as well as in the Rk values in Figure 10c, in the direction of the dominant roughness.In contrast, surfaces manufactured using Si-technology (Figure 10b) do not exhibit a pronounced dominant direction, which is consistent with the previously obtained characteristics.Figure 10c,d present typical graphs of the parameters Mr1 and Mr2 depending on the direction for two finishing technologies.The graphs show that the proportion of protrusions and depressions on the surface of shafts processed using traditional technology can vary significantly depending on the direction.In most cases, depressions predominate in the direction of maximum roughness.On shaft surfaces processed using Si-technology, depressions are more prevalent in the direction of greatest roughness, while the proportion of protrusions and depressions is roughly equal in other directions.It is worth noting that the parameters Rpk and Rvk exhibit similar behavior (Figure 10c,d).For shaft surfaces processed using traditional technology, in the direction of greatest roughness, the ratio Rvk/Rpk is usually greater than one in the direction of greatest roughness.This ratio varies significantly across different samples and areas of the surface.For shaft surfaces processed using Si-technology, the ratio Rvk/Rpk remains relatively constant.Specifically, this ratio is greater than 1 and close to 1.3 in angle ranges from 0 to 50 • and from 150 • up to 180 • .In the range of angles from 50 • up to 150 • , the ratio is close to 1.
Lubricants 2024, 12, x FOR PEER REVIEW 12 of 18 practical tool for assessing surface quality.The BAC is an integral characteristic of contacting rough surfaces and provides a visual representation of the primary operational characteristics of the contact, including rigidity, wear resistance, and tightness.The BAC determines several crucial parameters, such as Rk, Rpk, and Rvk, which establish the limits of the profile core, the average height of the protrusions, and the average depth of the depressions, respectively.Additionally, the parameters Mr1 and Mr2 determine the proportion of surface protrusions and depressions, respectively.Figure 10 shows typical images illustrating the change in BAC based on the direction.The color-coded amplitude values of the points on the BAC are shown for both surfaceprocessing technologies of model shafts.The orientation of shaft samples with different surface-polishing technologies and the areas of surface examination coincides with the designations shown in the Figure 7a.The images obtained also illustrate the presence of a dominant roughness direction on surfaces manufactured using traditional technology (Figure 10a).The average depth of the depressions and the height of the protrusions increase in these surfaces.This is indicated by the increase in BAC values in Figure 10a, as well as in the Rk values in Figure 10c, in the direction of the dominant roughness.In contrast, surfaces manufactured using Si-technology (Figure 10b) do not exhibit a pronounced dominant direction, which is consistent with the previously obtained characteristics.Figure 10c,d present typical graphs of the parameters Mr1 and Mr2 depending on the direction for two finishing technologies.The graphs show that the proportion of protrusions and depressions on the surface of shafts processed using traditional technology can vary significantly depending on the direction.In most cases, depressions predominate in the direction of maximum roughness.On shaft surfaces processed using Si-technology, depressions are more prevalent in the direction of greatest roughness, while the proportion of protrusions and depressions is roughly equal in other directions.It is worth noting that the parameters Rpk and Rvk exhibit similar behavior (Figure 10c,d).For shaft surfaces processed using traditional technology, in the direction of greatest roughness, the ratio Rvk/Rpk is usually greater than one in the direction of greatest roughness.This ratio varies significantly across different samples and areas of the surface.For shaft surfaces processed using Si-technology, the ratio Rvk/Rpk remains relatively constant.Specifically, this ratio is greater than 1 and close to 1.3 in angle ranges from 0 to 50 and from 150 up to 180.In the range of angles from 50 up to 150, the ratio is close to 1.The angular dependencies of standard surface roughness parameters can identify certain directions on the surface where these parameters exhibit features that make it possible to describe the fundamental differences between the surface-finishing technologies of the model shafts considered here.

Criteria for Evaluating the Tribological Efficiency of Roughness
The analysis of the above angular dependencies of standard roughness parameters enables the formulation of criteria for evaluating the tribological efficiency of roughness in the sense of reducing friction and wear.
When comparing the images in Figure 9, it can be seen that the distributions of average values of standard roughness parameters over an angle (as shown in Figure 4c) and over individual surface profiles (similarly, as shown in Figure 4b) for Si-technology surface finishing appear more symmetrical relative to angles 0 and 90 than for traditional technology.It is assumed that this feature contributes to Si-technology's superior tribological friction efficiency.Therefore, when evaluating the tribological efficiency of a surface, the symmetry of the surface texture relative to the direction of friction can be considered a characteristic of the surface.This characteristic depends on the direction of friction and is determined based on a standard roughness parameter (such as Ra or Rz).To obtain this characteristic objectively, we will consider the coefficient of symmetry of the surface texture in the direction of friction.The calculation is proposed to be performed as follows.
Using the parameter Ra as an example, it is evident from Figure 9a that traditional technology exhibits a significant scatter of values for individual profiles taken in one direction relative to its average value.Additionally, the maximum of the distribution of this parameter also changes significantly depending on the direction in which it is determined.For Si-technology (as shown in Figure 9b), the Ra parameter values are less spread out over the surface in different directions of its determination.Furthermore, the maximum distribution of this parameter is characterized by a lower dependence on the observation angle.Thus, it can be concluded that for Si-technology, there exists a dominant value of the parameter Ra over the entire test surface, as determined along various profiles and directions.This dominant value can be used to calculate the dominance coefficient, which characterizes the degree of dominance of this parameter over the surface in a given direction.We propose calculating the dominance coefficient using the following formula: The angular dependencies of standard surface roughness parameters can identify certain directions on the surface where these parameters exhibit features that make it possible to describe the fundamental differences between the surface-finishing technologies of the model shafts considered here.

Criteria for Evaluating the Tribological Efficiency of Roughness
The analysis of the above angular dependencies of standard roughness parameters enables the formulation of criteria for evaluating the tribological efficiency of roughness in the sense of reducing friction and wear.
When comparing the images in Figure 9, it can be seen that the distributions of average values of standard roughness parameters over an angle (as shown in Figure 4c) and over individual surface profiles (similarly, as shown in Figure 4b) for Si-technology surface finishing appear more symmetrical relative to angles 0 • and 90 • than for traditional technology.It is assumed that this feature contributes to Si-technology's superior tribological friction efficiency.Therefore, when evaluating the tribological efficiency of a surface, the symmetry of the surface texture relative to the direction of friction can be considered a characteristic of the surface.This characteristic depends on the direction of friction and is determined based on a standard roughness parameter (such as Ra or Rz).To obtain this characteristic objectively, we will consider the coefficient of symmetry of the surface texture in the direction of friction.The calculation is proposed to be performed as follows.
Using the parameter Ra as an example, it is evident from Figure 9a that traditional technology exhibits a significant scatter of values for individual profiles taken in one direction relative to its average value.Additionally, the maximum of the distribution of this parameter also changes significantly depending on the direction in which it is determined.For Si-technology (as shown in Figure 9b), the Ra parameter values are less spread out over the surface in different directions of its determination.Furthermore, the maximum distribution of this parameter is characterized by a lower dependence on the observation angle.Thus, it can be concluded that for Si-technology, there exists a dominant value of the parameter Ra over the entire test surface, as determined along various profiles and directions.This dominant value can be used to calculate the dominance coefficient, which characterizes the degree of dominance of this parameter over the surface in a given direction.We propose calculating the dominance coefficient using the following formula: where for the selected direction of friction the following α values are determined: C p (α) = N p (α)/N-ratio of the maximum number of profiles N p (α) in the direction α with the same value Ra (determined from the parameter distribution histogram in Figure 9) to the total number of profiles N on a surface, ∆Ra(α)-the standard deviation of the Ra parameter over all surface profiles in the direction α, Ra (α)-the average value of the parameter over all surface profiles in the direction α.The coefficient C p (α) in Formula (1) describes the maximum amplitude distribution of parameter values Ra over the surface in a given direction α.The second factor in Formula (1) describes the influence of the width of the distribution function of the parameter Ra over the surface in the direction α.By utilizing the data shown above to characterize the dominance of the roughness parameter Ra on the surface, it is possible to calculate the symmetry coefficient of the distribution of this roughness parameter in relation to the chosen direction of friction.To do this, you need to determine the degree of symmetry (similarity) of the functional dependence Ω Ra (α) on both sides (within the range of ±90 • ) relative to the selected direction α using a normalized autocorrelation function.We propose determining the symmetry coefficient with respect to the selected direction of friction as follows: As a result, we obtain the following graphs (Figure 11) for the coefficient of symmetry of the surface texture depending on the direction of friction within the range of angles α = (−90 • . . .+90 • ).By utilizing the data shown above to characterize the dominance of the roughness parameter Ra on the surface, it is possible to calculate the symmetry coefficient of the distribution of this roughness parameter in relation to the chosen direction of friction.To do this, you need to determine the degree of symmetry (similarity) of the functional dependence () Ra   on both sides (within the range of 90) relative to the selected direction  using a normalized autocorrelation function.We propose determining the symmetry coefficient with respect to the selected direction of friction as follows: As a result, we obtain the following graphs (Figure 11) for the coefficient of symmetry of the surface texture depending on the direction of friction within the range of angles  = (−90 … +90).The graphs indicate that traditional technology exhibits maximum symmetry in two directions, i.e., 0 • and 90 • .This is due to the dominant direction of the roughness on the surface, which takes the form of regular grooves that run perpendicular to the axis of the model shaft.Therefore, the roughness in traditional technology is most symmetrical in the aforementioned directions.For Si-technology, symmetry is uniform and maximal in all directions.This characteristic can serve as a criterion for determining the optimal surface texture for friction in a specific direction.
The second criterion is related to the analysis of the BAC and its directional dependence.As shown in Figure 10, Rk roughness parameters determined from the BAC exhibit several features that are characteristic of the surface-finishing technologies used for model shafts.This analysis utilized all Rk parameters and their angular dependencies.To evaluate the tribological efficiency of the friction surface in a specific direction in terms of reducing friction and wear using Rk parameters, the dependencies shown in Figure 10 were used as a basis.
The dependencies presented in Figure 10 clearly show that for traditional technology, the parameter Rk has a minimum value only in the direction perpendicular to the shaft axis, which coincides with the direction of friction.The parameter itself varies within a wide range, from 0.1 µm to 0.2 µm.In addition to the observations made along the shaft axis, there is also a noticeable divergence in the values of Rpk and Rvk when measured perpendicular to the shaft axis (90 • ).The fractional components of the protrusions and depressions corresponding to the parameters Mr1 and Mr2 also show a similar divergence.It is worth noting that for Si-technology, the parameter Rk exhibits two insignificant minima at 0 • and 90 • , and its value is less than that of traditional technology, varying within a significantly smaller range (0.085 to 0.1).In the direction perpendicular to shaft axis, similar values of the parameters Rpk and Rvk, as well as Mr1 and Mr2, respectively, are observed.
It is known that the samples prepared using the Si-technology have significantly higher friction efficiency (in the sense of reducing friction and wear) in the direction perpendicular to the axis of the model shaft compared to traditional technology.To account for the behavior of the parameters Rk, Rpk, Rvk, Mr 1, and Mr 2 described above, a friction roughness efficiency coefficient is introduced.This coefficient varies in the range of 0 to 1.In this case, a value of 0 corresponds to the maximum ineffective roughness (in the sense of increasing friction), while a value of 1 corresponds to the maximum effective roughness (in the sense of reducing friction).When selecting a suitable functional dependence for calculating this coefficient, it is important to consider that in order to achieve maximum efficiency of the tribological interaction between the surfaces of the friction pair, the coefficient should quickly approach unity.The coefficient of roughness efficiency with respect to friction should tend to zero, except when the parameters Rpk and Rvk, as well as the parameters Mr1 and Mr2, respectively, have close values.In the direction of maximum friction efficiency, the parameters Rk, Rpk, and Rvk should take a minimum value.Based on the given conditions, the coefficient of roughness efficiency with respect to friction should tend to zero.Therefore, we propose representing the function describing the angular dependence of the roughness efficiency coefficient with respect to friction as follows: where the parameters Rk, Rpk, Rvk, Mr1, and Mr2 are determined for a given direction at the angle α, and the parameter [Rk • Rpk • Rvk] min represents the minimum value of the product of the parameters Rk, Rpk, and Rvk in the angle α range from 0 to 180 • .If we take α as an angle relative to the axis of the model shaft, the dependencies χ(α) for both the traditional technology and the additional Si-technology for finishing the surfaces of model shafts will have the form shown in Figure 12.
the product of the parameters Rk, Rpk, and Rvk in the angle  range from 0 to 180.If we take  as an angle relative to the axis of the model shaft, the dependencies  () for both the traditional technology and the additional Si-technology for finishing the surfaces of model shafts will have the form shown in Figure 12.Using the obtained dependencies, a criterion for the efficiency of roughness in terms of reducing friction and wear can be established.It is observed that in the direction perpendicular to the axis of the model shaft ( 90  = ), both surface-finishing technologies exhibit a pronounced maximum.This indicates that surface wear and friction force will be minimized when rubbing in this direction.This statement is consistent with tribological test data and the general understanding that the surface's tribological efficiency will be better in the direction where the surface roughness parameters have the lowest value.However, it is important to note that the maximum roughness efficiency coefficient with respect to friction  () for Si-technology (Figure 12, curve 2) is much closer to 1 than the maximum of the similar coefficient for traditional technology (Figure 12, curve 1).Based on the tribological test data, it can be assumed that the surfaces of friction pairs exhibit the best tribological efficiency when rubbing in the direction with a higher roughness efficiency coefficient  ().A coefficient closer to 1 indicates greater tribological efficiency of the friction pair.Using the obtained dependencies, a criterion for the efficiency of roughness in terms of reducing friction and wear can be established.It is observed that in the direction perpendicular to the axis of the model shaft (α = 90 • ), both surface-finishing technologies exhibit a pronounced maximum.This indicates that surface wear and friction force will be minimized when rubbing in this direction.This statement is consistent with tribological test data and the general understanding that the surface's tribological efficiency will be better in the direction where the surface roughness parameters have the lowest value.However, it is important to note that the maximum roughness efficiency coefficient with respect to friction χ(α) for Si-technology (Figure 12, curve 2) is much closer to 1 than the maximum of the similar coefficient for traditional technology (Figure 12, curve 1).Based on the tribological test data, it can be assumed that the surfaces of friction pairs exhibit the best tribological efficiency when rubbing in the direction with a higher roughness efficiency coefficient χ(α).A coefficient closer to 1 indicates greater tribological efficiency of the friction pair.

Conclusions
The original technique for performing tribological tests of friction pairs in combination with a new method of additional finishing to the surfaces of friction pairs has shown a significant influence of the configuration of the initial relief of rough surfaces on the parameters of their tribological interaction (friction and wear).The surface of the shaft was prepared using both traditional diamond paste polishing techniques and additional finishing with silicon carbide.Tribological testing was performed on a single permanent linear contact with other factors being equal.In a comparative experiment on sliding friction of linear contact friction pairs composed of model shafts with different states of friction surfaces on a fixed plane, significant differences in friction performance were found between shafts with different surface textures.Additional finishing of the shaft surface with silicon carbide resulted in a 2-5 times reduction in friction the coefficient and a 2-4 times reduction in wear.In particular, the friction coefficient decreased from a value of 0.1 to a value of less than 0.02.Along with this, the standard parameters of the initial roughness of the surfaces of friction pairs processed using different technologies did not show significant differences.
The abnormally low values of friction coefficient and wear strength observed during the additional finishing process on the shaft surface are consistent with the adhesive deformation fluid dynamics model of friction and wear.This is due to the random micro cracks on the polished nano surface.The new technology applied forms a uniform surface relief.The reduction in the degree of rarefaction and micro-adhesive interaction of local areas of surfaces in the elastic deformation contact area of the diffuser can explain this phenomenon.
A new technique for analyzing rough friction surfaces in polar coordinates has been developed.This technique enables the acquisition of roughness angular characteristics and the evaluation of their impact on the frictional characteristics of the surface relative to the friction direction using standard roughness parameters.The technique was developed considering the frictional test results of the model friction pair and the study of the influence of various machining techniques on the three-dimensional-state characteristics of roughness generated on the shaft surface.Criteria for evaluating the tribological efficiency of rough surfaces (in the sense of reducing friction and wear) in a given friction direction are proposed based on roughness angular characteristics.These criteria are consistent with the results of frictional tests and enable the control of existing technologies for machining friction pair working surfaces.They also facilitate the development of new technologies to increase the wear resistance of high load friction sliding systems operating under boundary fuel lubrication conditions.This method shows promise and could be instrumental in optimizing various friction system friction surface-finishing technologies.

Figure 3 .
Figure 3. Appearance of a cassette tape with 22 counter-samples.After the usual diamond finishing is completed, the counter-samples in the cassette tape have a mirrorlike surface sta One counter-sample extracted from the cassette tape after additional Si-finishing has a matte d ing surface (b).

Figure 3 .
Figure 3. Appearance of a cassette tape with 22 counter-samples.After the usual diamond slurry finishing is completed, the counter-samples in the cassette tape have a mirrorlike surface state (a).One counter-sample extracted from the cassette tape after additional Si-finishing has a matte diffusing surface (b).
correlate with the curves shown in Figure 4c.Following this example, one can analyze the angular dependences of the different parameters of the test sample surfaces such as Ra, Rz, and Rk.

Figure 4 .
Figure 4. Angular dependence (by angle) distribution of the average value (red line) and the boundaries of the range of change (blue lines) of the parameter Ra (c) for the standard surface roughness with a regular profile (a), including considering histograms of the distribution of this parameter along raster lines (b) and along the angle directions of profile research (d).

Figure 4 .
Figure 4. Angular dependence (by angleφ) distribution of the average value (red line) and the boundaries of the range of change (blue lines) of the parameter Ra (c) for the standard surface roughness with a regular profile (a), including considering histograms of the distribution of this parameter along raster lines (b) and along the angle directions of profile research (d).

Figure 5 .
Figure 5.The time dependences of wear of a flat bearing (a); wear histograms of a flat bearing depending on the batch number (b) for two finishing technologies.
shows dotted lines that limit the range of trends in oscillograms of friction coefficients when testing five batches of model shafts under other equal friction conditions with different processing technologies on a stationary sample.The minor discrepancies between the friction force values during the five identical tests for each case of the surface condition of the model shafts (no more than 10%) indicate a high level of reproducibility of the experiments performed.

Figure 6 .
Figure 6.Change in the coefficient of friction Cf over time during sliding friction of model shafts at the fourth stage of testing using various technologies for finishing shaft surfaces: traditional processing method (1); processing using Si-technology (2).The coefficients of friction trends during the testing of five batches of model shafts are shown with dashes.

Figure 5 .
Figure 5.The time dependences of wear of a flat bearing (a); wear histograms of a flat bearing depending on the batch number (b) for two finishing technologies.

Figure 5 .
Figure 5.The time dependences of wear of a flat bearing (a); wear histograms of a flat bearing depending on the batch number (b) for two finishing technologies.
shows dotted lines that limit the range of trends in oscillograms of friction coefficients when testing five batches of model shafts under other equal friction conditions with different processing technologies on a stationary sample.The minor discrepancies between the friction force values during the five identical tests for each case of the surface condition of the model shafts (no more than 10%) indicate a high level of reproducibility of the experiments performed.

Figure 6 .
Figure 6.Change in the coefficient of friction Cf over time during sliding friction of model shafts at the fourth stage of testing using various technologies for finishing shaft surfaces: traditional processing method (1); processing using Si-technology (2).The coefficients of friction trends during the testing of five batches of model shafts are shown with dashes.

Figure 6 .
Figure 6.Change in the coefficient of friction C f over time during sliding friction of model shafts at the fourth stage of testing using various technologies for finishing shaft surfaces: traditional processing method (1); processing using Si-technology (2).The coefficients of friction trends during the testing of five batches of model shafts are shown with dashes.

Figure 7 .
Figure 7.The reliefs of cylindrical surface section (a) of model shaft samples processed using traditional technology (b) and Si-technology (c) were scanned by a profilograph-profilometer.Figure 7. The reliefs of cylindrical surface section (a) of model shaft samples processed using traditional technology (b) and Si-technology (c) were scanned by a profilograph-profilometer.

Figure 7 .
Figure 7.The reliefs of cylindrical surface section (a) of model shaft samples processed using traditional technology (b) and Si-technology (c) were scanned by a profilograph-profilometer.Figure 7. The reliefs of cylindrical surface section (a) of model shaft samples processed using traditional technology (b) and Si-technology (c) were scanned by a profilograph-profilometer.

Figure 8 .
Figure 8. Angular spectra of the surfaces of model shaft cylinders processed using traditional technology (a) and Si-technology (b).

Figure 8 .
Figure 8. Angular spectra of the surfaces of model shaft cylinders processed using traditional technology (a) and Si-technology (b).

Figure 9 .
Figure 9. Angular dependences of the histogram of the distribution of the parameters Ra (a,b), Rz (c,d), and Rk (e,f) over the surface for samples of model shafts, the surfaces of which are processed using traditional technology (a,c,e) and Si-technology (b,d,f).

Figure 9 .
Figure 9. Angular dependences of the histogram of the distribution of the parameters Ra (a,b), Rz (c,d), and Rk (e,f) over the surface for samples of model shafts, the surfaces of which are processed using traditional technology (a,c,e) and Si-technology (b,d,f).

Figure 10 .
Figure 10.The 3D images of the BAC dependencies were averaged over the surface and determined based on the direction of study of the surface profile for samples of model shafts manufactured using traditional technology (a) and using Si-technology (b).The parameters Rk, Rpk, Rvk, Mr1, and Mr2 were determined based on the direction of study of the surface profile for samples of model shafts manufactured using traditional technology (c) and using Si-technology (d).

N
selected direction of friction the following  values are determined:  in the di- rection  with the same value Ra (determined from the parameter distribution histo- gram in Figure9) to the total number of profiles N on a surface, the Ra parameter over all surface profiles in the direction  , Ra (  )-the average value of the parameter over all surface profiles in the direction  .The coefficient () p C  in Formula (1) describes the maximum amplitude distribution of parameter values Ra over the surface in a given direction  .The second factor in For- mula (1) describes the influence of the width of the distribution function of the parameter Ra over the surface in the direction  .

Figure 11 .
Figure 11.Angular dependencies of the symmetry coefficient SRa of surface texture on the direction of friction  , obtained for traditional technology (curve 1) and Si-technology (curve 2) of model shaft surface finishing.

Figure 11 .
Figure 11.Angular dependencies of the symmetry coefficient SR a of surface texture on the direction of friction α, obtained for traditional technology (curve 1) and Si-technology (curve 2) of model shaft surface finishing.

Figure 12 .
Figure 12.Angular dependencies of the roughness efficiency coefficient with respect to friction  on the direction of friction  obtained for traditional technology (curve 1) and Si-technology (curve 2) for finishing the surfaces of model shafts.

Figure 12 .
Figure 12.Angular dependencies of the roughness efficiency coefficient with respect to friction χ on the direction of friction α obtained for traditional technology (curve 1) and Si-technology (curve 2) for finishing the surfaces of model shafts.