Issue 
Mechanics & Industry
Volume 23, 2022



Article Number  8  
Number of page(s)  13  
DOI  https://doi.org/10.1051/meca/2022007  
Published online  08 June 2022 
Regular Article
Simulation on the effect of cylinder liner and piston ring surface roughness on friction performance
^{1}
College of Mechanical and Electrical Engineering, Shandong Jianzhu University, Jinan Shandong 250101, PR China
^{2}
Xintai Science and Technology Bureau, Taian Shandong 271200, PR China
^{*} email: liuna_sd@sdjzu.edu.cn
Received:
5
November
2021
Accepted:
4
March
2022
Considering the surface roughness of piston ring and cylinder liner and deformation of cylinder liner, the friction dynamics calculation model of cylinder linerpiston ring friction pair was established. The effects of top ring, other piston rings and cylinder liner surface roughness on friction performance were analyzed by grouping comparison and BoxBehnken response surface methodology. The calculation results show that when the surface roughness of the piston ring increases, the friction of the asperity increases slightly, the friction of the fluid decreases, and the total friction loss decreases. When the surface roughness of the cylinder liner increases, the friction of the asperity decreases slightly, the friction of the fluid increases, and the total friction loss increases.
Key words: Piston ring / cylinder liner / surface roughness / response surface analysis / friction performance
© N. Liu et al., Published by EDP Sciences 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
The cylinder linerpiston ring friction pair mainly plays a sealing role in the internal work of the engine, and the friction loss generated in the work accounts for approximately 26% of the total loss of the engine [1]. It can be seen from previous studies that the operating conditions of the engine [2], the structural parameters of the piston ring and the cylinder liner [3], and the elastic force of the piston ring [4] have different effects on the friction performance of the cylinder liner piston ring friction pair. Since the piston ring and cylinder liner cannot obtain a completely smooth surface in the processing and production process, the microconvex on the surface also has a certain impact on the friction performance of the friction pair with the cylinder liner piston ring. In addition, adding surface texture on the surface of the piston ring or cylinder liner is also one of the important means to improve friction performance. In recent years, researchers have also conducted indepth research in the fields of surface roughness and surface texture [5]. For example, Li [6] used AVL PR to establish the dynamic simulation model of the piston ring group and analyzed the influence of different surface roughness texture directions on the friction performance under different working conditions. The results showed that different surface roughness texture directions had significant indigenous effects on the sealing performance and friction performance of the diesel engine. When the piston ring surface was horizontally textured and the cylinder liner was vertically textured, the sealing performance and friction performance of the engine could be improved. Wu [7] studied the influence of the inner surface mesh of cylinder liner on the friction performance by solving the average Reynolds equation, and found that the a larger roughness or crosstype inner mesh was beneficial for improving the friction performance and reducing the friction loss. Rao [8] analyzed the friction performance of three cylinder liners with different surface textures through wear experiments. The results show that the surface texture shape has an impact on the friction performance, and the surface texture area also has a certain impact on the friction performance. In addition, Liu [9] found that each piston ring in the piston ring group would have mixed lubrication in the work, that is, all piston rings would produce microconvex friction in the movement, indicating that it is also necessary to study the influence of the structure of different piston rings on the friction performance.
In previous studies, the influence of a single piston ring on the friction performance of cylinder liner piston ring friction pair was mostly studied, but the overall study of the piston ring group was relatively rare. In this paper, taking a singlecylinder diesel engine as the research object, considering the deformation of the engine cylinder liner, surface roughness, and mixed lubrication, a simulation model of the cylinder linerpiston ring group is established. In addition to analyzing the influence of surface roughness of top ring and cylinder liner on friction force and friction loss of cylinder liner piston ring friction pair, considering the coupling effect of multiple piston rings, the influence of surface roughness of four piston rings in piston ring group on friction loss was analyzed by response surface methodology. It provides a reference for the optimization design of surface processing of piston ring and cylinder liner of diesel engine.
2 Theoretical basis
2.1 Mean reynolds equation
Considering the effect of roughness on friction performance, the pressure flow factor and shear flow factor are introduced, and the average Reynolds equation proposed by PATIR and CHENG [10] is used to calculate the oil film pressure distribution between the cylinder liner and piston ring. (1)
Therein: ϕ_{x}, ϕ_{y} are the axial and circumferential pressure flow factors respectively; ϕ_{s} is the shear flow factor; ρ is the lubricant density; p is the oil film pressure; U is the movement speed of the piston ring; α is the comprehensive roughness of the two surfaces of the piston ring and the cylinder liner, ; σ_{1} is the surface roughness of the piston ring; α_{2} is the surface roughness of the cylinder liner; h is the nominal oil film thickness, is the expected value of the actual oil film thickness; μ is the lubricant dynamic viscosity.
Regarding the solution method of the Reynolds equation [11], the finite difference method proposed by SOUTHWELL is one of the earliest classes of solution methods, and it is currently widely used in related research.
The Reynolds equation is solved by determining the boundary conditions, which are the axial boundary conditions of the upper and lower edges of the piston ring and the piston ring opening gap boundary conditions.
Axial boundary conditions: (2) (3)
Therein: p_{1} is the gas pressure at the upper edge of the piston ring; p_{2} is the gas pressure at the lower edge of the piston ring; b is the axial height of the piston ring.
Opening gap boundary conditions: (4)
Therein: L is the length of the piston ring in the circumferential direction.
2.2 Rough contact model
Due to the roughness between the two surfaces of the piston ring and cylinder liner, when the film thickness ratio H < 3 (H = h/α), the friction pair is in a mixed lubrication state, and the two surfaces will have microconvex contact during the relative motion, generating microconvex contact force. This paper adopts the rough surface contact model proposed by Greenwood and Tripp [12] to calculate the microconvex contact force between cylinder liner and piston ring [13]. (5) (6) (7)
Therein: W_{A} is the microconvex contact force; η is the microconvex density; β is the microconvex radius of curvature; H is the film thickness ratio; A is the nominal contact surface contact of the microconvex; A_{c} is the actual contact area of the microconvex; E is the integrated elastic modulus of the two surfaces of the piston ring and the cylinder liner; E_{1}, E_{2} are the modulus of elasticity of cylinder liner and piston ring; v_{1}, v_{2} are the Poisson's ratio of cylinder liner and piston ring.
The calculation formulas of F_{2.5}(H) and F_{2}(H) are as follows: (8) (9)
2.3 Piston ring force model
In addition to gravity, the piston ring is mainly subjected to the force of the combustion chamber gas, the piston ring groove, as well as friction and its own elastic force. Figure 1 shows the force diagram when the piston moves upward.
Fig. 1 Piston ring force diagram. 
2.3.1 Radial force balance of piston ring
The radial force equilibrium condition of the piston ring in operation is: (10)
Therein: W_{A} is the microconvex contact force; F_{L} is the lubricant fluid force between the cylinder liner piston ring; F_{g} is the piston ring backlash gas force; F_{e} is the piston ring's own elastic force.
The lubricant fluid force between the piston ring and the cylinder liner can be obtained by integrating the oil film pressure on the piston ring. The specific calculation formula is: (11)
The calculation formula of piston ring backlash gas force is: (12)
Therein: p_{g} is the gas pressure of piston ring backlash; D is the diameter of the piston ring; T is the radial thickness of the piston ring; b is the axial height of the piston ring.
The calculation formula of the elastic force of the piston ring is: (13)
Therein: E_{2} it is the elastic modulus of the piston ring; S is the opening gap of the piston ring.
2.3.2 Piston ring axial force balance
The axial force equilibrium condition of the piston ring in operation is: (14)
Therein: Mg is the piston ring gravity; F_{2} is the gas pressure on the upper surface of the piston ring; F_{H} is the fluid friction; F_{A} is the asperity friction; R_{x} is the support force of the piston ring groove on the lower surface of the piston ring; F_{1} is the gas pressure on the lower surface of the piston ring.
The calculation formula of asperity friction and fluid friction is as follows: (15) (16)
Therein: τ_{0} is the shear stress constant, the value is 2 X 10^{6}; α is the boundary friction coefficient; x_{1},x_{2} are the starting point and ending point of the oil film; τ_{1}, τ_{2} is the shear factor.
2.3.3 Total piston ring friction
The total friction force on the piston ring is mainly composed of lubricant fluid friction and asperity friction between the contact surface of the piston ring and the cylinder liner. (17)
Therein: F_{f} is the total friction force; F_{H} is the fluid friction force; and F_{A} is the asperity friction force.
3 Model building
3.1 Engine dynamics model
Taking a singlecylinder diesel engine as the research object, according to the main parameters of the engine in Table 1, the dynamic model of the piston ring friction pair of the cylinder liner is established in AVL EXCITE P&R. The dynamic model mainly consists of piston, cylinder liner, piston ring set, piston pin, connecting rod and other major components, as shown in Figure 2. Figure 3 gives the relative position of the piston and piston ring of the engine. Table 2 presents the mass and basic dimensions of each piston ring.
Fig. 2 Dynamics model. 
Fig. 3 Relative position of piston ring and piston. 
Main engine parameters.
Piston ring mass and dimensional parameters.
3.2 Engine thermodynamic model
In addition to the main structural parameters, it is necessary to obtain important parameters such as combustion chamber pressure, temperature, heat transfer coefficient and cylinder liner surface profile during engine operation when building the model.
GTPower is used to establish the thermodynamic model of the engine as shown in Figure 4. Calculation of the thermodynamic parameters of the engine under rated conditions (8.8 kw, 2000 r/min). The change curves of the combustion chamber pressure, temperature and heat transfer coefficient with the crankshaft rotation angle of the engine are calculated in Figures 5, 6 and 7. Based on the data in the figure, the average combustion temperature and the average heat transfer coefficient were obtained as 598 °C and 375 W/m^{2} °C, respectively.
The calculated data were compared with the experimental data in previous studies. In [14], combustion pressure reaches a maximum value of approximately 7.6 MPa at 7 °C after the upper dead center of the work stroke, while the cylinder temperature reaches a maximum value of approximately 1980 at 40 °C after the upper dead center. The maximum combustion pressure calculated by this model was 7.5 MPa at 14 °C after the upper dead center, and the maximum combustion temperature was 1939 °C at 34 °C after the upper dead center, indicating that the calculation results of this model were accurate.
Fig. 4 Engine thermodynamic model. 
Fig. 5 Combustion chamber pressure variation curve with crankshaft rotation angle. 
Fig. 6 Combustion chamber temperature variation curve with crankshaft rotation angle. 
Fig. 7 Heat transfer coefficient variation curve with crankshaft rotation angle. 
3.3 Cylinder liner deformation
During the engine operation, the cylinder liner is slightly deformed by the combustion chamber temperature and other factors for a long time, and its surface profile changes have some influence on the friction performance of the cylinder liner piston ring friction pair [15].
The deformation of the cylinder liner during operation is analyzed by finite element software considering the thermal load of the cylinder liner and its assembly preload. According to the formula given in [16] and the average combustion temperature and heat transfer coefficient of the combustion chamber obtained in Section 3.2, the distribution characteristics of the surface combustion temperature and heat transfer coefficient of the cylinder liner along the axial direction are calculated as: (18) (19)
Therein: T(s), α(s) are the cylinder liner surface combustion temperature and heat transfer coefficient along with the axial height change; T_{m}, α_{m} are the average combustion temperature in the combustion chamber and the heat transfer coefficient; k_{1}, k_{2} are constants, k_{1} = 0.537(l/d)^{0.24}, k_{2} = 1.45k_{1}; l is the piston stroke; d is the cylinder liner inner diameter; γ = s/l (0 ≤γ ≤ 1), s is the piston axial displacement, s = 0 when the piston is located in the upper stop.
The surface combustion temperature and heat transfer coefficient distribution are calculated, which are used as boundary conditions. The temperature field distribution of the cylinder liner is simulated by finite element software, and the experimental results are compared. Multiple temperature characteristic points are taken in the upper, middle and lower parts of the cylinder liner. The specific temperature measurement points are shown in Figure 8, and the average temperature measurement points in the same plane are calculated. The calculation results are shown in Table 3 with the experimental results [17]. The comparison shows that the temperature numerical error of the corresponding characteristic points is within 3%, indicating that the calculation results of the model are reliable.
After completing the calculation of the deformation of the cylinder liner, 36 paths are taken on the inner surface of the cylinder liner, the radial deformation information of all the nodes on each path is extracted, and the final radial deformation profile of the cylinder liner is shown in Figure 9.
Fig. 8 Position of cylinder liner temperature measuring point. 
Comparison of experimental values.
Fig. 9 Extraction of cylinder sleeve deformation profile. 
4 Results and discussion
4.1 Top ring surface roughness on friction performance
Taking the top ring of the piston ring as an example, the influence of piston ring surface roughness on the friction performance of the cylinder linerpiston ring friction pair is analyzed, and three different roughness of 0.8 μm, 1.2 μm and 1.6 μm are set and simulated to obtain the variation curves of asperity friction, fluid friction and total friction with engine crankshaft angle as shown in Figures 10 and 11.
As can be seen from the curves in Figure 10, the asperity friction mainly occurs near the upper dead center of the engine's work stroke. At this time, the piston ring and the cylinder liner are in the mixed lubrication state, and the friction force is generated by the microconvex contact between the surfaces. At other times, the piston ring and cylinder liner are in a fluid lubricated state, and the asperity friction is basically zero. The change in piston ring surface roughness has a slight effect on the asperity friction, and the magnitude of its friction force increases slightly with the increase in roughness. The curves in Figure 12 show that the fluid friction force acts throughout the cycle, and the change in piston ring surface roughness has a significant effect on the fluid friction force. With the increase in piston ring surface roughness, the fluid friction decreases. This is because the increase in piston ring surface roughness facilitates the storage of lubricant on the piston ring surface and improves its frictional properties.
Figure 13 shows that the radial hydrodynamic load carrying capacity of the piston ring changes with the crank angle. It can be seen that the hydrodynamic load carrying capacity of the piston ring with a roughness of 0.8 μm is significantly less than that of 1.2 μm and 1.6 μm piston rings before the stop point of the compression stroke.
The variation in the radial asperity load carrying capacity of the piston ring with the crankshaft angle is shown in Figure 14. Among them, when the roughness is 0.8 μm, the asperity load carrying capacity is slightly larger than the other two kinds of roughness, but the asperity load carrying capacity of the 1.2 μm and 1.6 μm piston rings is not much different.
Figure 15 shows the change curve of the friction loss of the top ring with the engine crankshaft angle. The friction loss of the friction pair is largest near the upper dead center of the engine work stroke, and its value increases with the increase in the roughness of the piston ring surface. At this time, the asperity friction between the piston ring and the cylinder liner plays a dominant role. The increase in roughness increases the surface area of the microconvex body peak element contact increase, which leads to an increase in friction loss. At the other times, fluid friction plays a dominant role, the roughness of the piston ring surface is larger, and the fluid friction is smaller, so the friction loss decreases with increasing roughness.
Fig. 10 Asperity friction. 
Fig. 11 Total friction. 
Fig. 12 Fluid friction. 
Fig. 13 Hydrodynamic load carrying capacity. 
Fig. 14 Asperity load carrying capacity. 
Fig. 15 Top ring roughness friction loss relationship curve. 
4.2 Effect of surface roughness of piston ring set on friction performance
In addition to the top ring, the engine has two other air rings and one oil ring, and the three piston rings will also generate friction losses in operation. Figure 16 shows the four piston rings friction loss curve with the crankshaft angle change.
As seen in Figure 16, the friction loss of the top ring accounts for a higher percentage of the total friction loss compared to the other piston rings, while the oil ring friction loss accounts for the lowest percentage, and the asperity friction loss of the top ring is significantly higher than that of the other piston rings. The friction loss generated by the first three rings during the work stroke of the piston has a significant increase, and the friction loss of the oil ring does not change much in each stroke of the piston.
To study the effect of surface roughness on friction performance of different piston rings, this paper adopts the BoxBehnken experimental design method [18], in which the surface roughness of each piston ring is used as the design factor and the average friction loss is used as the response. The experimental design factors and their levels (level 0, 1, 1 represent three different roughness values) are shown in Table 4 [19], specific experimental arrangements are shown in Table 5, and the final combination produced 29 sets of experimental scenarios, which are shown in Figure 17.
In response surface methodology, polynomial fitting is generally used to characterize the relationship between the test factors and the response value. In this paper, based on the test grouping and its calculation results, the relationship between each factor and the response is fitted with a secondorder polynomial to obtain the following regression model: (20)
Therein: W is the average friction loss; A is the top ring roughness μm; B is the second ring roughness μm; C is the third ring roughness μm; D is the oil ring roughness μm.
Generally, to ensure the accuracy of the model, the model needs to be tested for significance. Several parameters such as the Pvalue of the model, the coefficient of determination R^{2} and the adjusted coefficient of determination R^{2}_{adj} are generally used to evaluate the degree of approximation of the regression model [19]. If the Pvalue of the model is less than 0.05, it indicates that the model is significant. R^{2} reflects the explanatory power of the regression model to the response value, the value of R^{2} is generally required to be greater than 0.9. R^{2}_{adj} represents the correlation between all factors and the response value, the closer the value is to 1, the higher the accuracy of the model. In this paper, the Pvalue of the model is less than 0.0001, that is, the model is significant. The values of R^{2} and R^{2}_{adj} of this model are all above 0.99, which indicates that the model has high fitting accuracy and can predict the effect of piston ring surface roughness on friction performance.
Figure 18 shows the predicted values of the regression model results compared with the original values. The information in the figure further confirms the accuracy of the model.
As seen from Figure 19, the average friction loss between the piston ring and the cylinder liner decreases with the increase in the surface roughness of each piston ring. The influence of piston ring roughness on average friction loss in the order from small to large is: oil ring, second ring, third ring and top ring. The influence of the top ring roughness is the largest, and the influence of the oil ring roughness is the smallest.
Fig. 16 Friction loss of piston ring set. 
BoxBehnken design factors and levels.
Experimental design of BoxBehnken design.
Fig. 17 Experimental scheme and calculation results. 
Fig. 18 Comparison of predicted and original calculated value. 
Fig. 19 Effect of surface roughness of individual piston rings on the average friction loss. 
4.3 The effect of cylinder liner surface roughness on friction performance
The cylinder liner is another important part of the cylinder linerpiston ring friction pair in addition to the piston ring set, it is equally necessary to study the influence of its surface parameters on the friction performance. Three different surface roughness of the cylinder liner are selected for calculation and comparison, the surface roughness of the cylinder liner is 0.8 μm, 1.2 μm and 1.6 μm, and the surface roughness of each piston ring is 0.8 μm. When the surface roughness is 0.8 μm, the total friction force and friction loss are minimal. When the surface roughness of the cylinder liner increases, the friction force of the microconvex body near the upper dead center of the work stroke is slightly reduced, but the fluid friction force and friction loss increase with the increase in the surface roughness of the cylinder liner. The total friction force and friction loss of the cylinder linerpiston ring friction pair with the crankshaft angle for three different roughness are shown in Figure 20 and 21.
Fig. 20 Cylinder liner roughness friction relationship curve. 
Fig. 21 Cylinder liner roughness friction loss relationship curve. 
5 Conclusion
This paper briefly describes the theoretical model of the piston ring cylinder liner friction pair, and establishes the engine thermodynamic model and the piston ring group dynamics model to calculate the friction performance between the piston ring and the cylinder liner. The influence of surface roughness of piston ring and cylinder liner on friction performance was analyzed. Taking into account the coupling effect of multiple piston rings, the influence of roughness variation of four piston rings on friction loss in the piston ring group is calculated by response surface methodology, and the following conclusions are obtained:
The top ring roughness has a significant effect on the friction loss between the piston ring and the cylinder liner. With the increase in roughness, the friction loss generated by asperity friction increases, while the friction loss generated by fluid friction decreases.
Except for the top ring, the surface roughness of other piston rings also affects the friction loss. The average friction loss between the piston ring and cylinder liner decreases with the increase in surface roughness of each piston ring. The influence of piston ring roughness changes on average friction loss in the order from small to large is: oil ring, second ring, third ring and top ring.
The surface roughness of the cylinder liner has an equally important effect on the friction loss: as the surface roughness of the cylinder liner increases, the total friction force increases and the friction loss also increases.
Conflict of interest
The authors certify that there is no conflict of interest.
Variable  Definition  Unit 

ϕ_{x}  axial pressure flow factors  dimensionless quantities 
ϕ_{y}  circumferential pressure flow factors  dimensionless quantities 
ϕ_{s}  hear flow factor  dimensionless quantities 
ρ  lubricant density  Kg/m^{3} 
P  oil film pressure  MPa 
U  velocity of the piston ring movement  m/s 
α  integrated roughness of the two surfaces of the piston ring and cylinder line  μ m 
σ_{1}  surface roughness of the piston ring  μ m 
σ_{2}  surface roughness of the cylinder liner  μ m 
H  nominal oil film thickness  μ m 
expected value of the actual oil film thickness  μ m  
μ  lubricant dynamic viscosity  N·s/m^{2} 
P_{1}  the gas pressure at the upper edge of the piston ring  MPa 
P_{2}  the gas pressure at the lower edge of the piston ring  MPa 
B  the axial height of the piston ring  mm 
L  the length of the piston ring in the circumferential direction  mm 
W_{A}  the microconvex body contact force  N 
β  radius of curvature of the microconvex body  μ mm 
H  film thickness ratio  dimensionless quantities 
E  integrated modulus of elasticity of the two surfaces of the piston ring and the cylinder liner  MPa 
E_{1}  the elastic modulus of cylinder liner  MPa 
E_{2}  the elastic modulus of piston ring  MPa 
v_{1}  the Poisson's ratio of cylinder liner  dimensionless quantities 
v_{2}  the Poisson's ratio of piston ring  dimensionless quantities 
A  nominal contact area of the microconvex body  mm^{2} 
A_{c}  actual contact area of the microconvex body  mm^{2} 
F_{L}  lubricant fluid force between the cylinder liner and piston ring  N 
F_{g}  piston ring backlash gas force  N 
F_{e}  piston ring own elastic force  N 
P_{g}  the gas pressure of piston ring backlash  MPa 
D  the diameter of the piston ring  mm 
T  the radial thickness of the piston ring  mm 
S  the opening gap of the piston ring  mm 
Mg  the gravitational force of the piston ring  N 
F_{1}  gas pressure on the lower surface of the piston ring  N 
F_{2}  gas pressure on the upper surface of the piston ring  N 
F_{A}  asperity friction  N 
F_{H}  fluid friction  N 
R_{x}  support force of the piston ring groove on the lower end surface of the piston ring  N 
τ_{0}  the shear stress constant, the value is 2 × 10^{6}  constant 
α  the boundary friction coefficient  dimensionless quantities 
X_{1}  the starting point of the oil film  mm 
X_{2}  the ending point of the oil film  mm 
τ_{1}, τ_{2}  the shear factor  dimensionless quantities 
F_{f}  total frictional force  N 
T(s)  the cylinder liner surface temperature along with the axial height change  °C 
α(s)  the heat transfer coefficient along with the axial height change  W/m^{2} °C 
T_{m}  the average temperature  °C 
α_{m}  the average heat transfer coefficient  W/m^{2} °C 
k_{1}, k_{2}  k_{1}=0.537(l/d)^{0.24}, k_{2}=1.45k_{1}  dimensionless quantities 
1  the piston stroke  mm 
d  the cylinder liner inner diameter  mm 
γ  γ = s/l (0⩽γ⩽1)  dimensionless quantities 
s  the piston axial displacement  mm 
Acknowledgments
This work was supported by A Project of Shandong Province Higher Educational Science and Technology Program (No. J18KA006): Study on lubrication performance of piston ring friction pair in internal combustion engine under multifield coupling.
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Cite this article as: N. Liu, C. Wang, Q. Xia, Y. Gao, P. Liu, Simulation on the effect of cylinder liner and piston ring surface roughness on friction performance, Mechanics & Industry 23, 8 (2022)
All Tables
All Figures
Fig. 1 Piston ring force diagram. 

In the text 
Fig. 2 Dynamics model. 

In the text 
Fig. 3 Relative position of piston ring and piston. 

In the text 
Fig. 4 Engine thermodynamic model. 

In the text 
Fig. 5 Combustion chamber pressure variation curve with crankshaft rotation angle. 

In the text 
Fig. 6 Combustion chamber temperature variation curve with crankshaft rotation angle. 

In the text 
Fig. 7 Heat transfer coefficient variation curve with crankshaft rotation angle. 

In the text 
Fig. 8 Position of cylinder liner temperature measuring point. 

In the text 
Fig. 9 Extraction of cylinder sleeve deformation profile. 

In the text 
Fig. 10 Asperity friction. 

In the text 
Fig. 11 Total friction. 

In the text 
Fig. 12 Fluid friction. 

In the text 
Fig. 13 Hydrodynamic load carrying capacity. 

In the text 
Fig. 14 Asperity load carrying capacity. 

In the text 
Fig. 15 Top ring roughness friction loss relationship curve. 

In the text 
Fig. 16 Friction loss of piston ring set. 

In the text 
Fig. 17 Experimental scheme and calculation results. 

In the text 
Fig. 18 Comparison of predicted and original calculated value. 

In the text 
Fig. 19 Effect of surface roughness of individual piston rings on the average friction loss. 

In the text 
Fig. 20 Cylinder liner roughness friction relationship curve. 

In the text 
Fig. 21 Cylinder liner roughness friction loss relationship curve. 

In the text 
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