A Study of Tribological Performance Prediction Based on Surface Texture Parameters

Abstract

Surface texture parameters are a quantitative way of characterising surface topographical features and are closely related to tribological properties. In this paper, the correlation between surface topographic features and friction coefficient is investigated on the basis of the proposed improved correlation analysis model for high-speed milling surface topography of hardened steel. It was found that the friction coefficient could not be accurately reflected by a single parameter, so a prediction model for the friction coefficient based on Sxp, Sq, Sp, Sz, Sku and Sdq was developed. In this paper, the parameter screening was completed based on the changing characteristics of the data, and a multi-parameter prediction model of the friction coefficient in the stable wear stage was established, which provides a new idea to investigate the influence of the characteristics of surface topography on tribological performance.

1. Introduction

Surface topography is a fingerprint of the workpiece manufacturing process, which has an important impact on the service life and service performance of the workpiece, and it is an important factor in evaluating the functional properties of the surface and revealing friction and wear mechanisms [1,2,3,4]. The surface texture parameter is the most common and important digital characterization of surface topography, which is important for assessing and controlling the functional properties of surfaces [5].

Along with the development of surface characterisation techniques, scholars have carried out a series of studies on the correlation between 3D surface texture parameters and tribological properties. Dzierwa combined ball-on-disc wear experiments and found that surface topography has a significant effect on tribological properties [6]. Pawlus investigated the characterisation analysis of surface topography and concluded that Sp/Sz and Sq/Sa are better than Ssk and Sku in characterising the vertical coordinate distribution of surface textures [7]. Venkata analysed the effect of three-dimensional surface texture parameters on the friction coefficient and wear depth with pin disk wear tests at different weights and sliding speeds [8]. Yang et al. combined finite element analysis with experimental studies to investigate the correlation between surface roughness parameters and wear properties and proposed a wear performance determination method [9]. In recent years, scholars based their studies on the ISO25178-2 standard [10], in the surface topography characterisation analysis [11], functional evaluation [12] and other aspects of research. Although scholars have investigated related issues, the influence mechanism of surface topography on tribological properties is still unclear.

Surface topography is an important influence on tribological properties, but there is a wide variety of surface topography characterisation parameters, so it is particularly important to select the appropriate parameter variables. The correlation analysis between parameters is the basis of parameter selection; therefore, some scholars have studied the correlation between parameters.

Mieczyslaw discusses the correlation between surface roughness parameters using linear correlation analysis [13]. Pawlus has established a parameter set for surface topography based on the screening of surface texture parameters [5]. Basil found a significant negative correlation between Smrl and contact angle in his study on the effect of surface texture parameters on contact angle [14]. Yang has researched the correlation of surface texture parameters in grinding and established a parameter set for surface characterisation in grinding [15]. The current study widely used Pearson’s linear correlation coefficient to complete the correlation analysis, but it was found in the study that the parameters were not all linearly correlated with each other, and therefore the selection of parameters based on the correlation analysis was often inaccurate.

Ball-end milling is an important method in machining and is widely used in aerospace manufacturing, mould machining, automotive manufacturing and many other fields [16]. In ball-end milling, the machined surface exhibits pit-like topographical features under the influence of machining parameters such as cutting width (a_e), tool cut-in angle and feed per tooth (f_z) [17]. In this paper, taking the ball-end milling machined surface as the research object, the correlation between 3D surface texture parameters and the friction coefficient is studied based on the established new correlation analysis model, and the prediction model of the friction coefficient based on surface texture parameters is established on this basis. The research in this paper provides new ideas for further revealing the correlation between surface topographic features and tribological performance.

2. Experimental Process

2.1. Ball-End Milling Experiment

In this paper, the hardened mould steel, Cr12MoV, with a hardness of 60HRC was used as an object to carry out a group of high-speed ball milling experiments, and the experimental equipment is a five-axis machining centre(Bielefeld Germany), as shown in Figure 1.

The workpiece surface is a flat surface, the machining process adopts dry cutting without cutting oil, the angle between the tool and the workpiece is 30 degrees, and the surface is shaped by a single milling process. The experimental parameters for ball milling are shown in

Table 1. Ball-end milling experiment plan.

	A	B	C	D	E	F	G	H	I
fz (mm/tooth)	0.4	0.4	0.4	0.4	0.4	0.4	0.4	0.4	0.4
ae (mm)	0.4	0.5	0.6	0.4	0.5	0.6	0.4	0.5	0.6
ap (mm)	0.3	0.3	0.3	0.4	0.4	0.4	0.5	0.5	0.5

, where spindle speed S is 1000 r/min and a_p is the depth of cut.

Then, the CCI Map Taylor Hobson (Leicester England) non-contact white light interferometer was used to extract the topographical characteristic parameters of different machined surfaces and measurement equipment as shown in Figure 2. Its main parameters are as follows: vertical resolution of 0.01nm, maximum data points over 10⁶, noise floor less than 0.08 nm, and filtering technology using ISO/TS 16610 standard [17].

2.2. Sliding Friction Test

In order to study the tribological properties of different surfaces, reciprocating sliding friction tests were carried out on nine sets of surfaces obtained by machining according to Table 1. The workpiece after the ball-end milling process was cut into a rectangular piece with a length of 30 mm, width of 16 mm and height of 6 mm. The surface with the machined topography is the lower block, and the upper block is made of YS8 tungsten steel with a hardness of 80 HRC. The normal load was 80 N, the frequency was 2 Hz, the sliding wear time was 30 min and the single stroke was 20 mm. The reciprocating sliding friction test was completed using the MFT-5000 multifunctional friction and wear tester (San Jose, CA, USA), and this experiment was dry friction test at room temperature, so it did not use lubricant. The experimental equipment is shown in Figure 3.

3. Correlation Analysis and Prediction

Grey correlation analysis is an essential branch of grey system theory, which can effectively determine the degree of correlation between small samples of data series [18,19]. Grey correlation analysis is a method of achieving correlation evaluation based on the degree of similarity between data series, but it tends to ignore the effect of data magnitude. In this study, it was found that the changing characteristics of the data were important for the analysis of the data, which was not adequately taken into account by the grey correlation analysis method. For this reason, this paper establishes an improved grey correlation analysis model, which is based firstly on the idea that the data series are non-dimension and all of them are placed in the same [0, 1] interval, and then based on the analysis of the characteristics of the data changes in the correlation.

3.1. Non-Dimension Processing

Different data series tend to have different dimensions, and their value ranges are bound to be different, so direct correlation analysis of data series with different dimensions is not accurate. Therefore, the data series need to calculated non-dimensionally before correlation analysis can be carried out. Let X_i = (x_i(1), x_i(2), …, x_i(n)) be the behavioural sequence of factor X_i. The non-dimensional calculation is shown in the following equation:

$${\overline{x}}_i={\displaystyle \frac{x_i\left(k\right)-\underset{k}{\min } x_i\left(k\right)}{\underset{k}{\max } x_i\left(k\right)-\underset{k}{\min } x_i\left(k\right)}} k=1,2,\cdots ,n$$

(1)

After non-dimensional processing of the data sequence, it becomes ${\overline{X}}_i=\left({\overline{x}}_i\left(1\right),{\overline{x}}_i\left(2\right),\cdots ,{\overline{x}}_i\left(n\right)\right)$. The non-dimensional calculation allows the data series to all have the same dimensions and the data to all be placed in the same [0, 1] interval, laying the foundation for subsequent correlation analysis.

3.2. The Rate of Change of Data

When studying the problem of correlation analysis of data series, we tend to focus more on the changing characteristics of the data series. Therefore, on the basis of placing all of the data series in the same interval, calculating the rate of change of the data series can better analyse the change characteristics of the data series and more accurately calculate the degree of correlation between different data series. The rate of change of data is calculated as shown in Equation (2).

$$\mathsf{\Delta }{\overline{x}}_i={\displaystyle \frac{d{\overline{x}}_i\left(k\right)}{\mathsf{\Delta }k}} k=1,2,\cdots ,n$$

(2)

The computed data sequence is $\mathsf{\Delta }{\overline{X}}_i=\left(\mathsf{\Delta }{\overline{x}}_i\left(1\right),\mathsf{\Delta }{\overline{x}}_i\left(2\right),\cdots ,\mathsf{\Delta }{\overline{x}}_i\left(n\right)\right)$.

3.3. Grey Relational Grade

Grey correlation analysis is an important branch of grey system theory; along with the development of grey system theory, numerous grey correlation analysis models have appeared, such as absolute relational grade, relative relational grade, synthetic relational grade and so forth [20]. Based on the basic idea of absolute relational grade, this paper analyses the correlation characteristics of different data series by using the sequence of data change rates. Taking the correlation calculation of two data series as an example, the calculation steps are as follows:

3.3.1. Zeroing of the Starting Point

Different data series $\mathsf{\Delta }{\overline{X}}_i$ often have different starting points, and in order to calculate their degree of correlation more accurately, the data series should all have the same starting point. After zeroing the starting point, the starting data sequence can be expressed as $\mathsf{\Delta }{\overline{X}}_1^0$ and $\mathsf{\Delta }{\overline{X}}_2^0$.

$$\mathsf{\Delta }{\overline{X}}_1^0=\left(\mathsf{\Delta }{\overline{x}}_1^0\left(1\right),\mathsf{\Delta }{\overline{x}}_1^0\left(2\right),\cdots ,\mathsf{\Delta }{\overline{x}}_1^0\left(n\right)\right)$$

(3)

$$\mathsf{\Delta }{\overline{X}}_2^0=\left(\mathsf{\Delta }{\overline{x}}_2^0\left(1\right),\mathsf{\Delta }{\overline{x}}_2^0\left(2\right),\cdots ,\mathsf{\Delta }{\overline{x}}_2^0\left(n\right)\right)$$

(4)

3.3.2. Calculation of Characteristic Parameters

Based on the basic idea of grey correlation analysis, the characteristic parameters of the correlation are defined as

$$s_1={\displaystyle {\int }_1^n\left|\mathsf{\Delta }{\overline{X}}_1^0\right|}dt$$

(5)

$$s_2={\displaystyle {\int }_1^n\left|\mathsf{\Delta }{\overline{X}}_2^0\right|}dt$$

(6)

$$s_2-s_1={\displaystyle {\int }_1^n\left|\mathsf{\Delta }{\overline{X}}_1^0-\mathsf{\Delta }{\overline{X}}_2^0\right|}dt$$

(7)

3.3.3. Calculating the Degree of Correlation for Improved Correlation Analysis Models

$${\epsilon }_{12}={\displaystyle \frac{\left|s_1\right|+\left|s_2\right|}{\left|s_1\right|+\left|s_2\right|+2\times \left|s_1-s_2\right|}}$$

(8)

The correlation calculation method is based on the idea of grey absolute relational grade; paying more attention to the characteristics of the change between the data, the calculated grey relational grade can better accurately the change law between the data series, and more accurately analyse the degree of correlation between the data series. The grey relational grades all lie between (0, 1], and the closer the value of grey relational grade is to 1, the stronger the correlation.

3.4. Multi-Parameter Prediction Model

Grey systems theory enables accurate prediction of future changes in information systems by mining information. For the change characteristics of the system under the influence of multi-parameters, the zeroth order multi-parameter GM(0,N) is usually used for prediction analysis. The GM(0,N) models tend to focus on the data themselves, ignoring the effect of dimension and range of values on the data. For this reason, this paper improves GM(0,N) based on the basic theory of grey system theory.

Let the system data sequence be $Y_0=\left(y_0\left(1\right),y_0\left(2\right),\cdots ,y_0\left(m\right)\right)$; the relevant factors of the data sequence are $Y_i=\left(y_i\left(1\right),y_i\left(2\right),\cdots ,y_i\left(m\right)\right)i=1,2,\cdots ,m$.

3.4.1. Dimensionless Processing of the Data

Since the data series often have different dimensions from each other, all of the data are firstly non-dimensional and computed using initialling operator, i.e.,

$${\overline{y}}_i\left(k\right)={\displaystyle \frac{y_i\left(k\right)}{y_i\left(1\right)}}, y_i\left(1\right)\ne 0, k=1,2,\cdots ,m$$

(9)

3.4.2. Generate the Data Sequence into an Accumulation Data Sequence

$${\overline{y}}_i^{\left(1\right)}\left(k\right)={\displaystyle \sum _1^k{\overline{y}}_i\left(k\right)}, k=2,3,\cdots ,m$$

(10)

Then, the data sequence is transformed into

$${\overline{Y}}_0^{\left(1\right)}\left(k\right)=\left(1,{\overline{y}}_0^{\left(1\right)}\left(2\right),{\overline{y}}_0^{\left(1\right)}\left(3\right),\cdots ,{\overline{y}}_0^{\left(1\right)}\left(m\right)\right)$$

$${\overline{Y}}_i^{\left(1\right)}\left(k\right)=\left(1,{\overline{y}}_i^{\left(1\right)}\left(2\right),{\overline{y}}_i^{\left(1\right)}\left(3\right),\cdots ,{\overline{y}}_i^{\left(1\right)}\left(m\right)\right),i=1,2,\cdots ,m$$

3.4.3. Calculate the Cumulative Data Sequence Matrix

Establishing a matrix of correlating factors

$$B=\left[\begin{array}{cccc}{\overline{y}}_1^{\left(1\right)}\left(2\right)& {\overline{y}}_2^{\left(1\right)}\left(2\right)& \cdots & {\overline{y}}_M^{\left(1\right)}\left(2\right)\\ {\overline{y}}_1^{\left(1\right)}\left(3\right)& {\overline{y}}_2^{\left(1\right)}\left(3\right)& \cdots & {\overline{y}}_M^{\left(1\right)}\left(3\right)\\ ⋮& ⋮& \ddots & ⋮\\ {\overline{y}}_1^{\left(1\right)}\left(m\right)& {\overline{y}}_2^{\left(1\right)}\left(m\right)& \cdots & {\overline{y}}_M^{\left(1\right)}\left(m\right)\end{array}\right]$$

(11)

The data matrix of the system is $Y={\left[\begin{array}{cccc}{\overline{y}}_0^{\left(1\right)}\left(2\right)& {\overline{y}}_0^{\left(1\right)}\left(3\right)& \cdots & {\overline{y}}_0^{\left(1\right)}\left(m\right)\end{array}\right]}^T$.

3.4.4. Calculation of Model Coefficients

$$\widehat{b}=\left[b_1,b_2,\cdots ,b_m,a\right]={\left(B^{T}B\right)}^{-1}{B}^T{Y}_0$$

(12)

3.4.5. Building the GM(0,N) Prediction Model

$${\overline{y}}_0^{\left(1\right)}\left(k\right)=b_1{\overline{y}}_1^{\left(1\right)}\left(k\right)+b_2{\overline{y}}_2^{\left(1\right)}\left(k\right)+\cdots +b_m{\overline{y}}_m^{\left(1\right)}\left(k\right)+a$$

(13)

Thus, the GM(0,N) prediction model can be obtained in relation to the accumulated system data, and then the model data can be restored and finally multiplied with the initial value to obtain the predicted value of the system information data sequence.

The GM(0,N) model in grey system theory can complete the prediction of system characteristics under multiple influencing factors, but it does not take into account the effects of different meanings and different dimension between parameters on the prediction model. The improved GM(0,N) model fully takes into account the effect of dimension, which makes its physical meaning clearer and the prediction results more accurate.

4. Results and Discussion

4.1. Surface Topography Characteristics

The topographic characteristics of the nine groups’ surfaces after ball-end milling are shown in Figure 4.

From Figure 4, it can be seen that under different cutting parameters and cutting conditions, the surface of the ball-end milling process presents pit-like topographical characteristics. The surface topography after ball-end milling will be affected by multiple factors such as cutting parameters, initial phase difference, tool vibration, etc. [12]. These influencing factors are coupled with each other to form the final surface topography, so it is difficult to analyse the topography characteristics of the machined surface based on single-factor experiments.

Three-dimensional surface texture parameters are currently the most widely used method for characterising surface topography, which can more accurately and clearly reflect the topographic features of the surface compared to two-dimensional surface textures. Therefore, in this paper, the nine groups’ machined surfaces in Section 2 were measured by a white light interferometer according to ISO251 78-2 standard [10], and the values are shown in

Table 2. Surface texture parameters.

	A	B	C	D	E	F	G	H	I
Sq (μm)	2.9887	3.7560	4.3216	3.6161	3.9576	4.0488	3.3873	3.6020	3.6396
Sa (μm)	2.4933	3.1024	3.5646	3.0286	3.2978	3.3672	2.8589	3.0506	3.0045
Sp (μm)	10.0332	12.2234	14.0057	12.1746	13.5993	12.9835	11.0383	11.6043	11.7628
Sv (μm)	6.3562	8.9953	10.1273	7.7764	8.6241	8.1852	7.7245	7.6129	7.0672
Sz (μm)	16.3894	21.2187	24.1330	19.9510	22.2234	21.1687	18.7628	19.2173	18.8300
Ssk	0.5191	0.3095	0.2796	0.4616	0.3707	0.5351	0.3340	0.3171	0.5218
Sku	2.3829	2.4397	2.4634	2.4205	2.4430	2.4144	2.2115	2.1535	2.4708
Sxp (μm)	6.8284	8.0198	9.2207	8.0060	8.6450	9.3353	7.0135	7.3515	8.3320
Sdq	0.3732	0.4093	0.4319	0.4298	0.4333	0.4452	0.4265	0.4363	0.3603
Sdr (%)	5.8182	6.9580	7.7202	7.4161	7.8620	8.1512	7.5074	7.9503	5.4522

.

From Figure 4 and Table 2, it can be seen that the surface texture parameters are not the same for different surface topographies. Surface texture parameters are overall characterisation parameters of the surface topography, which vary with the surface topography. As the surface texture parameters are derived from surface topographic features, the study of correlations between surface texture parameters is fundamental to surface characterisation and performance prediction.

4.2. Degree of Relevance Analysis

Three-dimensional surface texture features can provide more comprehensive topographic information, so they are the hotspot of current research. Based on the ISO25178-2 standard, there are many kinds of 3D surface texture parameters with different meanings, but all of the parameters are derived from the real surface topography, and the correlation between different parameters has not been effectively studied, so it is of great significance to explore the correlation between 3D surface texture parameters to accurately characterise and functionally evaluate the surface topography. Therefore, the correlation between different 3D surface texture parameters is important for the accurate characterisation and functional assessment of surface topography. To this end, the correlation between different 3D surface texture parameters is calculated in this paper according to the basic principle of grey correlation analysis using Equations (1)–(8), as shown in

Table 3. Correlation between 3D surface texture parameters.

	Sq	Sa	Sp	Sv	Sz	Ssk	Sku	Sxp	Sdq	Sdr
Sq	1	0.9592	0.8823	0.7265	0.8408	0.5411	0.6493	0.7005	0.6299	0.6339
Sa		1	0.8817	0.7269	0.8421	0.5407	0.6698	0.6976	0.6435	0.6508
Sp			1	0.7148	0.8258	0.5334	0.6528	0.6997	0.5976	0.6157
Sv				1	0.8400	0.6845	0.6110	0.5620	0.6148	0.6059
Sz					1	0.6031	0.6324	0.6217	0.6200	0.6201
Ssk						1	0.4670	0.4450	0.5440	0.5390
Sku							1	0.5697	0.7246	0.7180
Sxp								1	0.5352	0.5456
Sdq									1	0.9086
Sdr										1

.

Let the critical value of correlation δ₁ = 0.9; i.e., let the degree of correlation be greater than δ₁. This means that the parameters have a strong degree of correlation with each other. From the data in Table 3, it can be seen that the surface texture parameters with strong correlation are as follows:

4.2.1. Sq and Sa

The parameters Sq and Sa are all height parameters, and their variation characteristics are shown in Figure 5.

The height parameters Sa and Sq are defined as the arithmetical mean height and the root mean square height of the surface profile, respectively, and are the most widely used three-dimensional parameters in the evaluation of surface topography analysis.

It was found that for completely different surface topographies, the values of Sa may be exactly the same, and the Sa reflection is not sensitive when there are small changes in the surface topography. The height parameter Sq reflects the standard deviation of the surface profile height, and it is more sensitive to deviations in surface topography than Sa. As can be seen from Figure 5, Sq and Sa are strongly correlated with each other and the trends are similar, so that Sq can be used instead of Sa to characterise the surface topographic features.

4.2.2. Sdq and Sdr

The 3D surface texture parameters Sdq and Sdr both belong to the composite parameter, which is the characterisation parameter of the surface topography integrity, and their variation characteristics are shown in Figure 6.

The parameter Sdq is defined as the root mean square gradient of the surface topography, which characterises the slope properties of the points in the surface topography. Sdr is the developed interfacial area ratio of the surface topography and can be used to characterise changes in the surface area of the surface topography. Both the composite parameters Sdq and Sdr can be used to characterise the overall tilt variation of the surface topography, and there is a strong correlation and similar trend between Sdq and Sdr. It was found that the variation of Sdq is more sensitive and meaningful to study; therefore, Sdq can be used instead of Sdr for analytical studies when characterising the surface topography. From the above analysis, it is clear that there is a strong correlation between the parameters Sa and Sq and Sdr and Sdq in the ball-end milling surface.

4.3. Correlation between Friction Coefficients and Surface Texture Parameters

For the study of the effect of surface topography on tribological properties, the friction coefficients (μ) were extracted for the stabilisation phase of the sliding friction process, as shown in Figure 7.

As can be seen in Figure 5, the friction coefficient increases rapidly during the abrasive wear stage, while the change of the friction coefficient tends to flatten out during the stable wear stage. The surface topography after ball-end milling is not flat and shows the polygonal crater-like topography; i.e., the surface topography is uneven, so the friction coefficient measured during the experiment is not a definite value, but a continuous oscillating change. The average friction coefficient for 28–29 min of the steady wear phase was extracted and is shown in

Table 4. Sliding friction coefficient of each group surface.

	A	B	C	D	E	F	G	H	I
μ	0.8578	0.8120	0.8059	0.8493	0.8913	0.8222	1.0375	0.8780	0.8320

.

Combining the friction coefficient in Table 4 and the 3D surface texture parameters in Table 2, based on the basic idea of grey correlation analysis, the correlation between the friction coefficient and the 3D surface texture parameters was calculated using Equations (1)–(8), as shown in

Table 5. The degree of correlation between the friction coefficient and surface texture parameters.

	Sq	Sa	Sp	Sv	Sz	Ssk	Sku	Sxp	Sdq	Sdr
ε	0.6367	0.6242	0.6003	0.5556	0.5886	0.4367	0.5843	0.6579	0.5771	0.5433

.

As shown in Table 5, there is no strong correlation feature between the friction coefficient and a single parameter, which indicates that the tribological performance of the surface is an overall evaluation parameter of the surface topography features, and that it does not depend entirely on a single parameter. From Table 5, it is observed that the correlation between the friction coefficients and the surface texture parameters are, in descending order, Sxp, Sq, Sa, Sp, Sz, Sku, Sdq, Sv, Sdr, Ssk.

The parameters Sxp, Sp, Sz are all peak characterisation parameters of the surface topography; Sku can be used to evaluate the convex peak characteristics, Sdq is the root-mean-square gradient, which can reflect the degree of inclination of the peaks of the surface topography, and Sq is the overall characterisation parameter of the topographic high and low characteristics of the surface. It can be seen that the friction coefficient depends mainly on the peak properties of the surface topography. Zhu [21] found that the friction coefficient under dry friction conditions is positively related to the bearing area ratio, and the peak characteristics of the surface profile have an important influence on the bearing area ratio, so the friction coefficient of the surface mainly depends on the peak characteristics of the surface topography. Relevant content has been revised with additions to the manuscript.

4.4. Prediction of Friction Coefficient

In order to establish the friction coefficient prediction model based on surface texture parameters, the critical value of correlation δ₂ = 0.5 is selected; i.e., the surface texture parameters whose correlation with the friction coefficient is greater than δ₂ are selected for constructing the prediction model of friction coefficient.

From the previous analyses, it can be seen that Sq and Sa and Sdq and Sdr are extremely correlated with each other and are substitutes for each other in building the prediction model, whereas Sv = Sz − Sp; Sv can be represented by Sq and Sz. Based on Table 5, the selected 3D surface texture parameters are Sxp, Sq, Sp, Sz, Sku, Sdq. A GM(0,6) prediction model for the friction coefficient was established according to Equations (9)–(13).

Let the GM(0,6) model be

$$\overline{\mu }=b_1{S}_{\mathrm{x}\mathrm{p}}^{\left(1\right)}+b_2{S}_{\mathrm{q}}^{\left(1\right)}+b_3{S}_{\mathrm{p}}^{\left(1\right)}+b_4{S}_{\mathrm{z}}^{\left(1\right)}+b_5{S}_{\mathrm{k}\mathrm{u}}^{\left(1\right)}+b_6{S}_{\mathrm{d}\mathrm{q}}^{\left(1\right)}+a;$$

(14)

then,

$$B=\left[\begin{array}{ccccccc}2.1745& 2.2567& 2.2183& 2.2947& 2.0238& 2.0967& 1\\ 3.5248& 3.7027& 3.6142& 3.7671& 3.0576& 3.2540& 1\\ 4.6973& 4.9126& 4.8277& 4.9844& 4.0734& 4.4057& 1\\ 5.9633& 6.2368& 6.1831& 6.3404& 5.0986& 5.5667& 1\\ 7.3304& 7.5915& 7.4771& 7.6320& 6.1118& 6.7596& 1\\ 8.3576& 8.7249& 8.5773& 8.7768& 7.0399& 7.9025& 1\\ 9.4342& 9.9301& 9.7339& 9.9494& 7.9436& 9.0715& 1\\ 10.6544& 11.1479& 10.9063& 11.0983& 8.9805& 10.0370& 1\end{array}\right]$$

(15)

$$\overline{\mu }={\left[\begin{array}{cccccccc}1.9466& 2.8861& 3.8762& 4.9152& 5.8737& 7.0832& 8.1068& 9.0767\end{array}\right]}^T$$

(16)

The estimated values of the predictive model coefficients are then

$$\widehat{b}=\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\\ b_4\\ b_5\\ b_6\\ a\end{array}\right]={\left(B^{T}B\right)}^{-1}{B}^T\overline{\mu }=\left[\begin{array}{c}-0.7261\\ 0.6967\\ -0.8414\\ -0.3005\\ 1.2455\\ 1.0566\\ -0.2385\end{array}\right]$$

(17)

Then, the dimensionless coefficient of the friction coefficient μ is predicted as

$$\overline{\mu }=-0.7261S\mathrm{x}{\mathrm{p}}^{\left(1\right)}+0.6967S{\mathrm{q}}^{\left(1\right)}-0.8414S{\mathrm{p}}^{\left(1\right)}-0.3005S{\mathrm{z}}^{\left(1\right)}+1.2455S\mathrm{k}{\mathrm{u}}^{\left(1\right)}+1.0566S\mathrm{d}{\mathrm{q}}^{\left(1\right)}-0.2385$$

From this, the predicted values of the friction coefficient can be obtained, as shown in

Table 6. Error table for the prediction model.

Group	Actual Value	Simulation Value	Relative Error (%)
B	0.8120	0.8020	1.2315
C	0.8059	0.7794	3.2882
D	0.8493	0.9058	6.6525
E	0.8913	0.8792	1.3576
F	0.8222	0.8428	2.5055
G	1.0375	0.9964	3.9614
H	0.8780	0.8967	2.1298
I	0.8320	0.8270	0.6010

.

From Table 6, it can be seen that the average error of the friction coefficient prediction model is 2.72%, which illustrates that the prediction model is highly accurate and can accurately represent the data characteristics.

It should be noted that the friction coefficient prediction model in this paper was developed under specific conditions. When the external conditions change, such as normal load, lubrication conditions, friction time and so forth, the friction coefficient will be changed, and the conclusions obtained by applying the model may be very different, but this does not affect the application of the model established in this paper to complete the comparative evaluation of the tribological performance of different surfaces. The prediction model of friction coefficient enables tribological performance analysis based on surface topography features and lays the foundation for functional manufacturing of machined surfaces.

5. Conclusions

This paper takes the surface topography of a ball-end milling machined surface as the research object and analyses the influence of the surface texture parameter on tribological performance, and the specific conclusions are shown as follows.

(1)

Based on the improved correlation analysis model, the strong correlation between the Sa and Sq and Sdq and Sdr of ball-end milling machining is demonstrated.

(2)

The friction coefficient is a holistic response of the surface topography characteristics and cannot be accurately characterised by a single parameter. The correlations between friction coefficients and surface texture parameters are, in order from strongest to weakest, Sxp, Sq, Sa, Sp, Sz, Sku, Sdq, Sv, Sdr, Ssk.

(3)

The GM(0,6) prediction model for the friction coefficient was developed based on 3D surface texture parameters, which had an average prediction accuracy of 2.72%.