1 Introduction

With the development of aviation engines towards high rotational speeds and high thrust-to-weight ratios, the failure of main spindle bearings is often caused by sliding-induced frictional wear in addition to normal fatigue damage [1, 2]. This can greatly affect the engine, impairing the normal operation of rolling bearings and seriously jeopardizing the safety of aviation engines. Therefore, it is paramount to investigate the evolution of the sliding wear process of bearing materials.

Generally, bearing steels have completely different wear processes under different lubrication conditions. He et al. investigated the tribological performance of GCr15 bearing steel under boundary lubrication and dry sliding conditions and found that from lean lubrication to dry friction, the tribological properties of GCr15 bearing steel change sharply and the wear mechanism changes from abrasive wear to mixed wear of adhesive wear and oxidation wear [3]. Su et al. studied the sliding friction and wear behavior of GCr15 steel under dry sliding and grease lubrication and found that the friction coefficient decreased with an increasing load while the wear rate decreased with time. The surface exhibited higher wear under dry sliding than under lubrication [4]. Sedlacek et al. investigated the correlation between surface roughness and friction coefficient in dry and oil-lubricated pin-on-disk tests. They observed that the friction coefficient was low when the roughness was high in the dry and lubricated tests. Similarly, the wear process of GCr15 bearing steel varies significantly with the sliding counterpart’s load, sliding speed, and surface roughness [5]. He et al. studied the influence of load on the sliding friction and wear performance of GCr15 steel and found that the friction coefficient decreased with an increasing load while the width of the wear track and wear rate increased. The wear mechanism was mainly a mixture of adhesive wear and oxidative wear, and the degree of adhesion and oxidation increased with increasing load [6]. Qiao et al. studied the influence of sliding speed and surface roughness on the sliding friction and wear performance of GCr15 bearing steel. The results showed that the wear mechanism of the friction pair of GCr15 bearing steel was mainly abrasive wear at high sliding speeds and a mixture of abrasive wear and adhesive wear at lower speeds. An optimal range of surface roughness exists, within which both the friction coefficient and wear rate are relatively low. When the surface roughness of the specimen is too high or too low, the wear rate is relatively large; When the roughness is large, the specimen damage is mainly adhesive damage [7, 8]. A. Albers et al. found that surface roughness is a decisive factor affecting friction behavior. Therefore, it is necessary to study the influence of surface roughness on the tribological behavior of GCr15 bearing steel [9, 10].

Proper running-in is beneficial for the contact surfaces in relative motion, improving their consistency, enhancing performance, and prolonging their lifespan [11]. Running-in behavior is closely related to load, temperature, and initial surface roughness, with roughness being an important parameter [12, 13]. Ma et al. studied that the surface morphology of the friction pair has an important influence on the running-in performance, and the results show that the appropriate initial roughness and surface texture can effectively improve the running-in quality. With the increase of the initial surface roughness, the effect of surface texture to improve the running-in performance begins to appear gradually [14]. In most studies, running-in quality is characterized by measuring surface morphology after the experiment, such as the surfaces of bearing rings and cylinder liners with piston rings [15]. However, studies have found that friction signals exhibit better stability after high-quality running-in and are more convenient to measure than surface morphology [16]. Therefore, the quality can be studied from the perspective of friction and wear by quantifying the stability of friction signals. Chaos theory provides a novel method for quantifying running-in quality. Zhu Hua et al. found that the friction system has chaotic characteristics, and the friction process is the process of mutual matching of friction pairs. The rise, fall and stability of the friction coefficient correspond to the process of divergence, convergence and dispersion of the attractor, and proposed the running-in attractor based on chaos theory [17, 18]. Ding et al. investigated the dynamic characteristics of surface roughness during sliding by means of the running-in attractor and revealed the evolution of the running-in attractor [19]. The running-in attractor has characteristic parameters, including convergence CON, correlation dimension D and enclosing radius R, which reflect information about run-in quality from the perspective of system stability [20]. Thus, the characterization parameters of the running-in attractor bridge the gap between the running-in quality and the working condition parameters.

Compared with all-steel bearings, hybrid ceramic bearings are more likely to meet the prerequisites of high speed, high efficiency, high reliability, high precision, and long life [21, 22, 23]. In this study, Si3N4 balls and GCr15 bearing steel were selected as the rotating friction pairs, and pin-on-disc wear tests were conducted under different loads and surface roughness. Four different surface roughness gradients were obtained by grinding GCr15 bearing steel using a grinder. The friction and wear characteristics of the bearing material were analyzed in terms of friction coefficient, wear width, and wear morphology. Finally, to reveal the high-dimensional dynamic features of the friction system composed of the bearing material and study the influence of surface roughness on the running-in quality, the dynamic characteristics of the friction coefficient signal were qualitatively characterized using phase trajectories, which were quantitatively characterized using correlation dimension and enclosed radius to study the quality of running-in.

2 Materials and tests

In this experiment, the material of the disc specimen was bearing steel (GCr15), and its chemical composition was shown in Table 1. The GCr15 bearing steel was subjected to quenching and tempering treatment, with the specific heat treatment process as follows: first, it was kept at 780 ~ 810℃ for 4 h, then cooled to 650℃, and then air-cooled to room temperature; then it was reheated to 840℃, kept for 30 min, and oil-quenched; finally, it was tempered at 160℃ for 1.5 h. As shown in Fig. 1, the microstructure of the bearing steel after heat treatment mainly consists of tempered martensite, uniformly dispersed carbides, and residual austenite. The hardness of heat treated bearing steel was 58-60HRC. The bearing steel disks were prepared by utilizing a grinding machine and sand disc for polishing, resulting in four different levels of surface roughness, measured as Rq values of 1.192, 0.584, 0.186, and 0.009 μm. Among them, Rq values of 1.192, 0.584, and 0.186 μm were directly obtained through grinding, while the Rq value of 0.009 μm was obtained through polishing. The Rq value was measured using laser confocal microscopy (OLYMPUS OLS4000) according to ISO: 4827. The surface profile of the GCr15 bearing steel are shown in Fig. 2. From the figure, it can be seen that the peaks and valleys on the surface of GCr15 are more obvious when Rq is 1.192 μm, and the maximum distance between the peaks and valleys is about 15 μm. As the roughness decreases, the maximum distance between peaks and valleys gradually decreases, and when Rq is 0.009 μm, the peaks and valleys are no longer visible. The pin specimen is made of 87% silicon nitride (Si3N4) balls (G5 grade) with a diameter of 7.5 mm, a Vickers hardness of 1700HV, and a surface roughness (Rq) of 0.012 μm.

Table 1 Chemical composition of GCr15 bearing steel(mass fraction,%)
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Fig. 1
figure 1

Microstructure of GCr15 bearing steel

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Fig. 2
figure 2

Surface profile of GCr15 bearing steel: a1-a4 are three-dimensional surface profiles, b1-b4 are two-dimensional surface profiles perpendicular to the grinding direction

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The experiment was conducted on a self-developed pin-on-disc friction and wear testing machine. As shown in Fig. 3(a), the Si3N4 ball is fixed on the ball fixture, the GCr15 bearing steel sample is fixed on the disc, and the application of the load is controlled by an electric system. During the test, the disc rotates at a certain speed, driving the GCr15 bearing steel sample to rub against the silicon nitride ball. Figure 3(b) shows the wear radius for this test was 10 mm. For each surface roughness level, loads of 20 N, 40 N, and 60 N were applied at a speed of 300 r/min. Before the test, 0.05 ml of SAE-30 lubrication oil was added to the friction pair surface. The wear test was conducted for 60 min. During the test, the friction force signal was collected by a force transducer with an accuracy of ± 0.2% FS and a sampling frequency of 100 Hz. The coefficient of friction signal was derived by dividing the friction force by the load.

Fig. 3
figure 3

Friction and wear test: a schematic diagram of the device, b wear scar

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3 Results and discussion

3.1 Friction coefficient

The friction coefficient signals for all the tests are shown in Fig. 4, it is obvious that the friction coefficient signals exhibit similar changes within a certain range regardless of the surface roughness or the load. With the continuation of the friction and wear process, the friction coefficient increases continuously at first, then decreases slightly, and finally tends to a dynamic balance. The friction coefficient change is generally divided into two stages: the running-in and stable stages, corresponding to Stages I and II in Fig. 4, respectively. The running-in stage includes the increasing and decreasing process of the friction coefficient. In the early stage, the friction pairs are mainly in point contact, with less frictional resistance and a smaller friction coefficient. With the time extension, the micro-protrusions on the friction pairs surface are continuously worn down or detached during the friction process. Hence, the point contact transforms into face contact, leading to an actual contact area increase and a rapid increase for the friction coefficient. As time progresses, the accumulation of a large number of wear debris plays a certain lubricating role, and the friction coefficient gradually drops. In the stabilization stage, wear debris’s continuous production and overflow reach an equilibrium state, and the frictional resistance and friction coefficient tends to stabilization. In addition, it can also be observed that under the same load, the greater the surface roughness, the longer the running-in time, as shown in Fig. 4(a1), (c1), and (d1). This is because the rougher the surface roughness, the greater the number of micro-protrusions and the longer the time required for them to be worn down or detached during friction. Interestingly, when the surface roughness is the same, if the load differs significantly, the running-in time becomes longer with an increase in load, as shown in (c1) and (c3) in Fig. 4. This is because although an increase in load can intensify the wear of the micro-protrusions on the surface, it also intensifies the wear of the Si3N4 ball, leading to an increase in the wear radius of the ball, and wear mainly occurs during the running-in period, resulting in an extended running-in time.

Fig. 4
figure 4

Friction coefficient signals: a1-d1 are Rq = 0.009 μm, 0.186 μm, 0.584 μm and 1.192 μm under 20N; a2-d2 are Rq = 0.009 μm, 0.186 μm, 0.584 μm and 1.192 μm under 40N; a3-d3 are Rq = 0.009 μm, 0.186 μm, 0.584 μm and 1.192 μm under 60N

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Figure 5 displays the average friction coefficients during the stabilization stage. It can be observed that the same roughness levels, the average friction coefficients decrease as the load increases. Part of the reason is that the friction pairs are not sufficiently compacted when the load is low, and the lubricating oil is stored between the micro-convex peaks. After the lubricating film formed during the friction process breaks, it is difficult to restore, and the lubricating oil cannot be replenished during the friction process, resulting in a higher friction coefficient. As the load increases, the convex peaks on the friction pairs are smoothed and compacted, and the lubricating oil stored within is easily released and replenished into the broken oil film. This dynamic equilibrium of oil film destruction and formation during the friction process decreases the friction coefficient. Another reason is that the increasing load intensifies the wear of Si3N4 balls, increasing the wear radius of the balls, increasing the contact area during the stabilization wear, and decreasing the friction coefficient. Additionally, it can be seen from the figure that, with an increase in the surface roughness Rq, the average friction coefficient during the stabilization stage increases. This is because an increase in surface roughness leads to a decrease in the thickness of the oil film, an increase in the number of contact points between the surfaces of friction pairs, and an increase in the likelihood of oil film rupture and local lubrication failure, increasing the friction coefficient.

Fig. 5
figure 5

Average friction coefficient in the stabilization stage

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3.2 Wear performance

Figure 6 shows the cross-sectional profiles of the wear scars on the GCr15 bearing steel after testing. It can be observed from the figure that both the width and depth of the wear scars increase with an increase in surface roughness and load. This is because when the load is constant, a rougher surface leads to a smaller effective contact area between the friction pairs, resulting in higher contact pressure and more severe wear. When the surface roughness is constant, a higher load leads to higher contact pressure and more severe wear. In addition, when the surface roughness Rq of the GCr15 bearing steel disk is 0.009 μm, only slight wear occurs on the disk surface after the friction and wear test, while more severe wear occurs at other surface roughness levels. This is mainly related to the lubrication status of the four different surface roughness levels during service under contact loads. Under oil lubrication, there are mainly three lubrication states between the friction pairs: boundary lubrication, elastohydrodynamic lubrication, and mixed lubrication. Boundary lubrication refers to the insufficient formation of a lubricating oil film between the mating surfaces, resulting in direct contact between the surfaces of the two friction pairs, either locally or over a large area. Elastohydrodynamic lubrication refers to forming a lubricating oil film with a certain thickness between the friction pairs, allowing the two friction pairs to separate without any local micro-contact. Mixed lubrication refers to the simultaneous existence of the above two states. The lubrication state mainly depends on the oil film parameter λ. When λ < 1, it belongs to the boundary lubrication state; when λ > 3, it belongs to the elastohydrodynamic lubrication state; and when 1 ≤ λ ≤ 3, it belongs to the mixed lubrication state [24].

Fig. 6
figure 6

Cross-sectional profile of wear scar: a1-d1 are the abrasion profiles at Rq = 0.009 μm, 0.186 μm, 0.584 μm and 1.192 μm under 20N, respectively; a2-d2 are the abrasion profiles at Rq = 0.009 μm, 0.186 μm, 0.584 μm and 1.192 μm under 40N, respectively; a3-d3 are the abrasion profiles at Rq = 0.009 μm, 0.186 μm, 0.584 μm and 1.192 μm under 60N, respectively

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The oil film parameter λ is calculated by the following formula, and then the lubrication state is classified.

$$\lambda =\frac{{h}_{min}}{\sqrt{{R}_{qb}^{2}+{R}_{qc}^{2}}}$$
(1)

where Rqb and Rqc are the root mean square values of the Si3N4 ball and GCr15 bearing steel disc surface, respectively; hmin is the minimum oil film thickness formed between the friction pairs under load, which is calculated by the following formula:

$${h}_{min}=3.63{U}^{0.68}{M}^{0.49}{W}^{-0.073}(1-{e}^{-0.68k})R$$
(2)

where U is the velocity parameter and is given by \(U={n}_{0}u/({E}_{eq}R)\); M is the material parameter and is given by \(M=\alpha {E}_{eq}\); W is the load parameter and is given by \(W=F/({E}_{eq}{R}^{2})\); n0 is the dynamic viscosity of the lubricating oil used in the experiment at atmospheric pressure (0.06081 Ns/m2); u is the rotational speed of the silicon nitride ball on the surface of the bearing steel disk (0.19 m/s); Eeq is the equivalent elastic modulus and is given by \({E}_{eq}=\frac{2{E}_{b}{E}_{c}}{(\left(1-{v}_{b}^{2}\right){E}_{c}+(1-{v}_{c}^{2}){E}_{b})}\). Eb (300 GPa) and Ec (200 GPa) are the elastic moduli of the Si3N4 ball and the bearing steel disk, respectively. vb (0.26) and νc (0.3) are their respective Poisson ratios. R is the radius of the Si3N4 ball (3.75 mm). α is the viscosity-pressure coefficient (0.0236 mm2/N). F is the applied load in the experiment. k is the ellipticity, which is equal to 1 in this study because the contact is of the point-to-plane type, and under the assumption of elastic deformation, the contact area is circular.

The lubrication state of the disc under different surface roughness under different loads can be calculated by Eqs. (1) and (2), and the calculation results are shown in Table 2. When the surface roughness Rq = 0.009 μm, 1 ≤ λ ≤ 3, which is a typical mixed lubrication state; when Rq = 0.186, 0.584 and 1.192, λ < 1, which is a typical boundary lubrication state. In addition, when Rq = 0.009 μm, the value of λ ranges from 2.3933–2.5932 under the load range of 20N-60N. Although it is in a mixed lubrication state, the value of λ is relatively high, indicating that the system is approaching the state of elastic fluid hydrodynamic lubrication, and the separation between the sliding surfaces is sufficient. During the entire friction and wear process, only slight wear occurs during the running-in period, and there is almost no wear during the subsequent stable wear process, as shown in Fig. 7(a). When Rq = 0.186, 0.584 and 1.192, the value of λ takes the range of 0.0334–0.2314 and is in the boundary lubrication state, a more obvious rough contact occurs between the Si3N4 ball and GCr15 disc surface, and the micro-convex bodies on the surface frustrate each other, forming local micro-fracture, generating a large number of abrasive particles, and generating a large number of plow grooves under the plowing action of abrasive particles. The wear mode is typical abrasive wear, as shown in Fig. 7 (a), (b), (c) and (d). The remaining test parameters also show the wear pattern of abrasive wear, and the SEM diagrams of the corresponding tests are omitted due to space limitations.

Table 2 λ values and lubrication status for different loads and surface roughness
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Fig. 7
figure 7

SEM image under 60N load: a Rq = 0.009 μm; b Rq = 0.186 μm; c Rq = 0.584 μm; d Rq = 1.192 μm

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3.3 Running-in quality

Although all 12 experimental trials are in the stable wear stage, there are differences in their running-in quality. Therefore, it is necessary to investigate the running-in quality. The extracted friction coefficients from the friction system contain important system behavior information. The friction coefficient signal Y = {x1, x2, x3, …, xn} is chosen as the research object for univariate time series analysis to reveal the nonlinear behavior during the running-in process. The phase space reconstruction is then conducted to generate phase trajectories [25]. The high-dimensional matrix Y reconstructed from this process is presented below:

$$Y=\left[\begin{array}{c}{Y}_{1}\\ {Y}_{2}\\ \vdots \\ {Y}_{N}\end{array}\right]=\left[\begin{array}{ccc}\begin{array}{cc}{x}_{1}& {x}_{1+\tau }\end{array}& \cdots & {x}_{1+(m-1)\tau }\\ \begin{array}{cc}{x}_{2}& {x}_{2+\tau }\end{array}& \dots & {x}_{2+(m-1)\tau }\\ \begin{array}{c}\begin{array}{cc}\vdots & \vdots \end{array}\\ \begin{array}{cc}{x}_{N}& {x}_{N+\tau }\end{array}\end{array}& \begin{array}{c}\ddots \\ \dots \end{array}& \begin{array}{c}\vdots \\ {x}_{N+(m-1)\tau }\end{array}\end{array}\right]$$
(3)

where N is the number of vectors Yi and is given by N = n − (m − 1)τ; and n is the number of data points xi. τ is the time delay calculated using the mutual information method, as expressed by Eq. (4), and m is the optimal correlation dimension calculated using the false nearest neighbors method as expressed by Eq. (5) [26]:

$$I\left(X,Y\right)=\sum\nolimits_{i}\sum\nolimits_{j}{P}_{xy}\left({x}_{i},{y}_{i}\right){\text{log}}_{2}\left[\frac{{P}_{xy}\left({x}_{i},{y}_{i}\right)}{{P}_{s}\left({x}_{i}\right){P}_{q}\left({y}_{i}\right)}\right]$$
(4)

where X is a discrete-time series (X = { x1, x2, x3, ⋯, xn}); Y is the time series of delaying τ of X (Y = {x1+τ, x2+τ, x3+τ,⋯, xn+τ}); and Pxy (xi, yj) is the joint distribution probability; At X = xi, Y = yj; PX(xi) and PY(yj) are the edge distribution probabilities.

$${\left[\frac{{R}_{m+1}^{2}\left(i,r\right)-{R}_{m}^{2}(i,r)}{{R}_{M}^{2}(i,r)}\right]}^\frac{1}{2}=\frac{\left|{X}_{i}-{X}_{i}^{r}\right|}{{R}_{m}(i,r)}>{R}_{tol},i=\text{1,2},\dots ,N$$
(5)

where Rm is the distance between any two points in m-dimensional phase space; Rm+1 is the distance between any two points in the (m + 1)-dimensional phase space; Yi is a point in the m-dimensional phase space; \({Y}_{i}^{r}\) is the nearest neighbor of Yi, r = 1, and Rtol is the threshold value, with Rtol ∈ [10, 50], [27]. In this paper, Rtol was set to 10 (Table 3).

Table 3 Presents the values of the delay time τ and the optimal correlation dimension m for all experiments conducted
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Three principal vectors were extracted from the reconstructed matrix using principal component analysis, and the phase trajectories were plotted in a 3D space. Figure 8 shows the phase trajectories under different surface roughness tests at 60N. In Fig. 8(a1)-(a4) are the 3D phase trajectories for Rq = 0.009, 0.186, 0.584, 1.192 μm, and their corresponding projections onto the x–y plane are shown in Fig. 8(b1)-(b4). During the running-in process, the phase trajectories first expand outward from the initial position indicated by the solid purple dot and then gradually converge to form a closed space, which is marked with a dashed circle as the running-in attractor. Although the space occupied by the running-in attractor is small, most points are concentrated in this stage. The blue line depicts the divergence process of the phase trajectories, corresponding to the rising stage of the friction coefficient one-dimensional time series, and the convergence process is shown by the red line, corresponding to the falling stage of the friction coefficient one-dimensional time series. These two parts constitute the running-in stage of the friction system, that is, stage I in Fig. 4. The phase trajectories converge to form a running-in attractor in a closed space, corresponding to stage II in Fig. 4. The macroscopic motion of the phase trajectories transforms into microscopic motion in a very small space. It is worth noting that for Rq = 0.009 μm, only the rapid convergence process can be observed in Fig. 8, and the divergence process is almost invisible due to the short running-in time. The formation time of the running-in attractor is equal to the running-in time, which is the time it takes for the phase trajectories to converge to the running-in attractor. In the following, a quantitative study will be conducted on the running-in attractor to investigate the effect of surface roughness and load on the running-in quality.

Fig. 8
figure 8

Phase trajectory under different roughness tests at 60N: a1-a4 correspond to Rq = 0.009, 0.186, 0.584, 1.192 μm, respectively, b1-b4 is the corresponding projection on the x0y plane

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The correlation dimension is the fractal dimension of the running-in attractor, which portrays the fractal structure and complexity of the running-in attractor [28]. The correlation dimension D is defined as follows [29, 30]:

$$C\left(r\right)=\frac{1}{N(N-1)}\sum\nolimits_{i=1}^{N}\sum\nolimits_{j=1,j\ne i}^{N}H\left(r-\Vert {Y}_{i}-{Y}_{j}\Vert \right)$$
(6)
$$D=\underset{r\to \infty }{\text{lim}}\frac{lnC(r)}{lnr}$$
(7)

where C(r) is the correlation function, and r is any positive number. Yi = {xi, xi+τ,\(\cdots ,\) xi+(m-1)τ}, i = 1, 2,\(\cdots\), N, N = n—( m—1)τ. H(x) is the Heaviside step function, when x > 0, H(x) = 1; when x < 0, H(x) = 0. D is the correlation dimension, which is obtained by selecting the linear portion of the ln(r)-ln(C(r)) curve with the best linearity and performing linear regression using the least squares method. The slope of the regression line represents the value of the correlation dimension D.

As the correlation dimension increases, the macroscopic motion of the phase points weakens while the microscopic motion strengthens [31]. This indicates an increase in the degree of intrinsic randomness, leading to fine structures on the trajectory and microscopic fluctuations in the friction signal. In this case, the phase points move randomly and uniformly in the phase space, while the data points of the friction signal are uniformly distributed around the mean in the time domain. Therefore, the larger the correlation dimension, the more stable the running-in attractor and the better the running-in quality.

Enclosing radius R is used to quantify the boundary range of the phase trajectory, where each phase point in the phase trajectory represents the state of the friction system at a certain moment and is defined as follows:

$$R=Max\left|{Y}_{i}-{Y}_{j}\right|$$
(8)

where Yi and Yj are the i-th and j-th phase points on the phase trajectory.

For a large enclosing radius, when the phase trajectory has a wide boundary, the phase points are more scattered, making it easier to find two states with significant differences, and the system state changes drastically. The phase trajectory at this time is unstable, as shown in the rising and converging stages in Fig. 8, and the corresponding running-in quality is poor. On the contrary, for a small enclosing radius, when the phase trajectory has a narrow boundary, the distribution of phase points is relatively concentrated, and it is not easy to find two states with significant differences, so the system state is relatively stable. At this time, the phase trajectory tends to be stable, as shown in the stable stage in Fig. 8, and the corresponding running-in quality is high. Therefore, high-quality phase trajectories are characterized by small enclosing radii.

Figure 9 shows the correlation dimension D and enclosing radius R of the friction coefficient in the stable stage of friction and wear tests under different surface roughnesses and loads. It can be seen from the figure that the correlation dimension D and enclosing radius R exhibit opposite corresponding trends. This is consistent with the conclusion above that the larger the correlation dimension, the smaller the enclosing radius, and the higher the running-in quality. When the surface roughness is 0.009 μm, the correlation dimension under different loads is between 6.0–6.5, and the enclosing radius is between 5.0–5.5. When the surface roughness is 0.186 μm, the correlation dimension under different loads is between 5.4–5.6, and the enclosing radius is between 5.5–6.0. When the surface roughness is 0.584 μm, the correlation dimension under different loads is 5.2–5.4, and the enclosing radius is 5.7–6.6. When the surface roughness is 1.192 μm, the correlation dimension under different loads is between 5.1–5.5, and the enclosing radius is between 5.6–6.65. The correlation dimension increases with an increase in surface roughness, while the enclosing radius decreases with an increase in surface roughness. The running-in quality is related to the initial surface roughness; the lower the initial surface roughness, the better the running-in quality.

Fig. 9
figure 9

Correlation dimension D and enclosing radius R

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When the initial surface roughness is 0.009 μm, 0.186 μm, 0.584 μm, and 1.192 μm, the maximum correlation dimension and minimum enclosing radius correspond to loads of 40N, 20N, 60N, and 60N, respectively. Under different surface roughness, it may be better to have a higher, lower, or moderate load, which is caused by the complexity of the friction system. In addition, when the initial surface roughness is 0.009 μm, the maximum correlation dimension and minimum enclosing radius correspond to a load of 40N, indicating the best running-in quality; when the initial surface roughness is 1.192 μm, the minimum correlation dimension and maximum enclosing radius correspond to a load of 20N, indicating the poorest running-in quality.

4 Conclusion

In order to study the effects of surface roughness and load on frictional wear behavior, a series of frictional wear tests were carried out on a pin-and-disc friction and wear testing machine. Then, the effects of surface roughness and load on the wear resistance of the material were analyzed. In addition, the friction coefficient signal is analyzed by using chaos theory to study the influence of surface roughness and load on the running-in quality. The main conclusions are as follows:

  1. (1)

    In the stable wear stage, the average friction coefficient decreases with increasing load and increases with increasing surface roughness. This is because an increase in load promotes the formation of the oil film, which reduces the friction coefficient. On the other hand, an increase in surface roughness results in a thinner oil film, which increases the likelihood of oil film rupture and local lubrication failure, leading to an increase in the friction coefficient.

  2. (2)

    With an increase in the initial surface roughness and load, the wear of the GCr15 bearing steel surface intensifies. This is because the rougher the surface at the same load, the smaller the effective contact area between the friction pair and the greater the pressure, leading to more severe wear. Similarly, a higher load results in greater pressure at the same surface roughness, leading to more severe wear. The wear mechanism is all abrasive wear caused by the grinding particles.

  3. (3)

    To study the effect of surface roughness and load on running-in quality, the friction coefficient of the one-dimensional time series was reconstructed into phase space, and a phase trajectory was generated. The phase trajectory’s divergence, convergence, and stability correspond to the friction coefficient’s rising, falling, and stable stages, respectively. The running-in quality was characterized quantitatively by the phase trajectory’s correlation dimension and enclosing radius. The running-in quality is related to the initial surface roughness, where a lower initial surface roughness leads to a higher running-in quality. The effect of load on running-in quality is complex due to the friction system’s complexity.