Sealing Performance Analysis of Lip Seal Ring for High-Speed Micro Bearing

Abstract

This article focuses on the problem of sealing failure in high-speed micro bearings. Based on a thermal-stress coupled finite element model, the distribution of equivalent stress and contact pressure of the sealing ring and the influence of various factors on the sealing performance are analyzed. Based on this, the Latin Hypercube sampling method, Kriging surrogate model and genetic algorithm are used to find the optimal combination of sealing performance. Finally, the accuracy of the model and method is verified through orthogonal experiments. Research has found that the maximum equivalent stress of the seal ring is 0.59234 MPa, and it increases first and then decreases with the increase in lip inclination angle, friction coefficient and radial interference amount, increases with the increase in lubricant temperature, and decreases with the increase in bearing rotation speed. The maximum contact pressure is 0.20433 MPa, and it decreases with the increase in the lip inclination angle, increases first and then decreases with the increase in the friction coefficient, and decreases first and then increases with the increase in the lubricant temperature, bearing rotation speed and radial interference amount. The most significant factor affecting the equivalent stress of the seal ring is the lubricant temperature, and the most significant factor affecting the contact stress is the interference fit amount. When the seal lip inclination angle is 43.99°, the friction coefficient is 0.01 mm, the lubricant temperature is 111.5 °C, the bearing rotation speed is 28,853 rpm and the radial interference amount is 0.04 mm, the sealing performance of the sealing ring is optimal.

1. Introduction

Vacuum cleaners, with their excellent cleaning ability, wide applicability and efficient work efficiency, have become an indispensable cleaning tool in modern households, and their market prospects are very broad [1]. When the vacuum cleaner works, it is in an environment with a lot of dust and impurities. Once these small particles enter the rolling bearing chamber, they intensify bearing wear, reduce service life, and even cause the entire mechanical system to malfunction. Therefore, it is particularly important to analyze the sealing performance of rolling bearing seals, and optimize their design to achieve the best sealing effect, effectively preventing the invasion of external impurities and the leakage of internal lubricating grease.

Lip seal originated in the 1930s and is made of synthetic rubber with good oil resistance and wear resistance. To date, lip seal has been applied in a variety of industrial devices and has become one of the most extensive sealing methods [2,3]. Many scholars have conducted in-depth research on its sealing performance. Jagger [4] first proposed that the microscopic fluid lubrication film between the sealing lip and the rotating shaft is the main reason for the sealing lip to avoid friction. Poll [5,6] obtained the oil film thickness of the seal gap by means of fluorescence imaging technology and optical open-loop focusing error signal detection technology, which proved that there was fluid lubrication between the seal lip and the rotating shaft, but failed to explain the mechanism of the formation of fluid lubrication. Hamilton [7], Johnston [8] and Horve [9] show that the surface roughness of the sealing lip is the main reason for the formation and retention of an oil film between contact surfaces. Salant [10,11] studied the influence of surface micro-fringe and micro-convex shape on the oil film pressure and pumping rate through numerical calculation, based on the consideration of the influence of lip deformation on the oil film. Then, Salant [12] established the elastohydrodynamic lubrication model of lip oil seal based on Reynolds equation and elastic deformation equation, and studied the influence of high-speed micro-convex on the shaft surface on lubrication performance. He [13] analyzed the effects of different texture shapes and their parameters on the sealing performance and found that a rectangular texture with certain parameters enhances the buoyancy of the air film by 81.2%.

On the macro side, Fern A.G [14] used a two-dimensional axisymmetric model of the sealing ring and a rubber Mooney Rivlin model strain energy density function to simulate the mechanical properties of the sealing structure. Shuang [15] analyzed the influence of sealing parameters on sealing performance based on the two-dimensional axisymmetric finite element model of the X–O composite ring seal and found that the height of the X ring seal is the root cause of the pressure, stress and deformation of the seal ring. Yan [16] calculated the variation laws of static contact pressure, radial force, reverse pumping rate and friction torque with lubricant temperature, and verified the simulation results through bench experiments. Zhang [17] analyzed the influence of sealing ring material and structural parameters on Von Mises and contact stress based on the finite element model of rigid flexible coupling sealing groove, but did not consider the interaction of various factors. Soldatenkov [18] describe a stochastic model of the wear process of a lip seal that takes into account random changes in the temperature and external load, providing numerical analysis results of the seal wear process. Liang [19] analyzed the influence of rubber material parameters and working conditions on the performance of seals. Huang [20] found that high shear stress caused by an excessive compression ratio of the sealing ring is the main reason for bearing sealing ring damage. The structural parameters of the sealing ring were optimized through finite element analysis, and the optimized compression rate of the sealing ring was reduced to 14% while the wear rate was reduced to about 28% of the original. However, the interaction between various factors was not considered during the optimization. Engelfried [21], focusing on the phenomenon of seal contact leakage or insufficient lubrication caused by “lead structures” on the sealing surface, modeled the interaction of various geometric features of the lead structures and their combined effect on the pumping capacity and obtained the relationship between shaft lead and its pumping. Lee [22] studied the distribution of lip contact stress and contact width under different interference amounts and conducted experimental verification. Wang [23] used finite element software to establish a two-phase flow model of lubricating grease and air in the bearing fluid domain, and determined the amount of lubricating grease filling and the interference fit of the sealing ring. However, the influence of temperature on the sealing performance was not considered in the stress analysis of the sealing ring. Zhang [24] used the PDS module of ANSYS software to analyze the influence of structural parameters of bearing sealing rings on sealing performance, and optimized them to obtain an optimization sequence. However, the optimized structure was not experimentally verified. Shabbir [25] tested the sealing performance of four types of sleeve coatings, namely tungsten carbide, chromium oxide, hard chromium and stainless steel, as well as the radial lip seal performance pressurized by preloaded retaining springs. The optimal sleeve coating sealing combination was found and experimentally verified using an experimental testing platform. Romanik [26] measured the friction coefficient of flat rubber samples with different surface layer textures and measured the contact width and the clamping force of the seal lip on the rotary shaft before and after the test of the rotating shaft lip seals in standard and prototype versions, determined the pressure and width of the seal lip through finite element analysis, and finally combined the experimental and simulation results to calculate the friction torque. This research method has achieved satisfactory results in industrial application. Zhang [27] used a thermal-stress–wear coupled finite element model based on oil seals to simulate and analyze the sealing ring of rolling bearings using mesh reconstruction method, and obtained the wear law of the sealing ring. Liu [28] proposed a multi-scale wear simulation method for the rotary shaft lip ring seal considering micro-interface interaction under mixed lubrication conditions, and found that increasing the circumferential roughness density on the basis of reducing the axial roughness can reduce the wear of the oil seal. In a specific range, the axial roughness has a great influence on the sealing performance. Zhou [29] designed a bionic non-smooth surface sealing ring and found through finite element analysis that the stress and contact pressure distribution of the sealing ring was uniform and reasonable.

In summary, most of the existing studies on bearing seals focus on the influence of a single structural field or a single factor on the sealing performance, and few consider the influence of multi-factor interaction on the sealing performance of the bearing seal ring based on the coupling of temperature field and structural field. In this paper, aiming at the problem of seal failure caused by a large amount of heat generated by high working speed of micro bearings in vacuum cleaners, a thermal-stress coupled finite element model was established, and the response surface optimization design method was adopted to optimize the parameters of the sealing ring. The response surface optimization results were compared with the orthogonal experiment results, and the values of each parameter when the sealing performance of the seal ring was optimal were obtained to improve the sealing performance of the sealing ring. It also provides guidance and reference for the optimization design of high-speed micro bearing seal ring in the future.

2. Thermal-Stress Coupled Finite Element Analysis

In order to facilitate the solution, the following assumptions should be made for this paper:

(1)

The effect of sealing ring wear on the maximum contact stress and equivalent stress is not considered.

(2)

Seal lip Young’s modulus, Poisson’s ratio and other material parameters do not change with temperature and deformation.

(3)

Because the temperature of the internal lubricant during bearing operation is difficult to measure, the temperature of the internal lubricant during bearing operation is approximately equal to the average temperature of the inner and outer ring during bearing operation during the experiment.

2.1. Geometric Model and Material Parameters

2.1.1. Geometric Model

Figure 1a shows the geometric model of high-speed micro bearing. In order to improve the calculation efficiency, only the 15° model was used in the simulation analysis, as shown in Figure 1b. Figure 2 is the bearing seal structure and local enlarged picture.

Table 1. Bearing structure parameters.

Parameter	Value (mm)	Parameter	Value (mm)
Bearing outside diameter	13.00	Bearing bore diameter	5.00
Bearing width	4.00	Roller diameter	2.00
Bearing outer ring flange diameter	10.29	Bearing inner ring flange diameter	7.81

shows the geometric structure parameters of rolling bearings.

2.1.2. Material Parameters

Rubber material has very good elasticity, chemical resistance and wear resistance [30], so it is widely used in the sealing unit of rolling bearings [31].

Table 2. Material parameters.

Material	Poisson’s Ratio	Density (g/cm3)	Young’s Modulus (MPa)	Isotropic Thermal Conductivity (W/(m·k))	Coefficient of Thermal Expansion (/°C)
GCr15	0.29	7.81	219	35	1.25 × 10−6
SPCC	0.28	7.85	2 × 105	55	1.12 × 10−6
NBR	0.49	0.98	7.8	0.2	180 × 10−6

shows the material parameters of rolling bearings. The outer ring and inner ring materials of rolling bearings are GCr15, the sealing ring material is NBR, and the metallic framework materials is SPCC. The Mooney–Rivlin model can effectively characterize the mechanical behavior of nonlinear superelastic material NBR [20], and its elastic strain energy function is [32]:

$$W=C_{10}(I_2-3)+C_{01}(I_2-3)+(1/d){(J-1)}^2$$

(1)

$$C_{10}=E/7.5, C_{01}=0.25C_{10}$$

(2)

where W is strain energy, C₁₀ and C₀₁ represent the material constant, I₁ and I₂ are two principal invariants of the strain tensor, d is the incompressibility factor, J is the determinant of the elastic deformation gradient, and E is the elastic modulus of rubber.

The elastic modulus E of rubber is [33]:

$$E=(15.75+2.15H_A)/(100-H_A)$$

(3)

$$d=(1-2\mu )/(C_{10}+C_{01})$$

(4)

where $H_A$ is Shore hardness of rubber materials and $\mu$ is Poisson’s ratio, 0.29.

Table 2 shows the specific values of material parameters of the sealing structure.

2.2. Mesh Partitioning and Contact Pair Creation

Figure 3 shows the mesh division of the finite element model. The shape of the inner ring, outer ring and metallic framework of the model is regular, so a hexahedron is used to divide the grid elements, in which the bearing outer ring mesh size is set to 0.1 mm, the sealing ring and metallic framework is set to 0.03 mm, and the bearing inner ring is set to 0.07 mm. In order to accurately capture the lip stress distribution, it is necessary to refine the lip mesh of the seal ring and ensure that the mesh size between the contact surfaces is consistent. In this paper, the mesh between the contact surfaces is divided into 0.02 mm.

To avoid the number of grids affecting the result, it is necessary to verify the grid independence of the model.

Figure 4 shows the relationship between the number of grids in the 15° model and the equivalent stress and contact pressure. As can be seen from Figure 4, when the number of grids is greater than 54,626, the contact stress and equivalent stress values tend to be stable. In order to improve the calculation efficiency, the number of grids is selected as 54,626 for calculation.

When establishing contact pairs, the sealing ring acts as the contact surface, with the inner ring of the bearing designated as the target surface, and the interaction between them is defined as a frictional contact. The augmented Lagrange formula was used to reduce the generation of ill-conditioned matrix. The interference fit was treated with a ramped effect. The bonded contact was set between the seal ring and the metallic framework.

2.3. Setting of Boundary Conditions

2.3.1. Setting of Boundary Conditions of the Temperature Field

The performance parameters of rubber materials will change with the ambient temperature, thus changing the structural stress distribution of the sealing ring, thus affecting the sealing performance of the sealing ring [34]. The transmission of temperature is usually heat conduction, heat convection and heat radiation. In addition, the influence of the friction power consumption inside the bearing on the temperature is also considered.

(1)

Heat conduction

Heat conduction is a process in which an object only relies on the thermal movement of microscopic particles to generate heat transfer [35]. It follows Fourier’s law:

$$q^{″}=-k{\displaystyle \frac{dT}{dx}}$$

(5)

where q″ is heat flux, (W/m²); and k is thermal conductivity, (W/(m·°C)).

(2)

Thermal convection

Thermal convection refers to the process of heat transfer caused by the relative displacement of cold and hot fluids caused by the macroscopic movement of fluids [35]. Thermal convection satisfies Newton’s cooling equation:

$$q″=h(T_{\mathrm{s}}-T_{\mathrm{b}})$$

(6)

where h is convective heat transfer coefficient, T_s is solid surface temperature; and T_b is the ambient fluid temperature.

(3)

Thermal radiation

Thermal radiation refers to the energy exchange process in which an object emits electromagnetic energy and is absorbed as heat by other objects [35]. It satisfies the Stefan–Boltzmann equation:

$$q=\epsilon \sigma A_1{F}_{12}(T_1^4-T_2^4)$$

(7)

where q is the heat flow rate, $\epsilon$ is emissivity, $\sigma$ is the blackbody radiation constant, A₁ is the radiation area, F₁₂ is the form factor from radiation 1 to radiation surface 2, T₁ is the absolute temperature of radiant surface 1, and T₂ is the absolute temperature of radiant surface 2.

(4)

Bearing friction power consumption

In addition to heat conduction, convection and radiation, the influence of internal friction heat generation in bearings on the results is also considered in the thermal boundary conditions. The formula for calculating the frictional power consumption of bearings is [36]:

$$Q=1.05\times 10^{-4}\mathrm{nM}$$

(8)

where Q is the bearing friction power consumption (W), n is the bearing speed (r/min), and M is friction torque (N·mm).

$$M=M_0+M_1$$

(9)

where M₀ is friction torque (N·mm) related to the bearing type, speed and lubricant properties; and M₁ is friction torque (N·mm) in relation to the load on the bearing.

When $vn\ge 2000$,

$$M_0={10}^{-7}{f}_0(vn{)}^{2/3}{d}_m^3$$

(10)

When $vn<2000$,

$$M_0=160\times {10}^{-7}{f}_0{d}_m^3$$

(11)

where f₀ is the coefficient related to the bearing type and lubrication method, v is the kinematic viscosity of the lubricant (mm²·s⁻¹), n is the bearing speed (r/min), and d_m is the bearing pitch circle diameter.

$$M_1=f_1{P}_1{d}_m$$

(12)

where f₁ is the coefficient related to the bearing type and load, and P₁ is the determined calculated load of the bearing friction torque (N).

The calculated friction power consumption is added to the steady-state thermal module of ANSYS in the form of the heat generation rate, and then the stress of the sealing ring is analyzed using the thermal-stress coupling method.

2.3.2. Setting of Boundary Conditions of Structural Field

In the course of setting the boundary conditions of the structural field, the outer ring of the bearing is set as a fixed support to ensure it remains stationary. The inner ring of the bearing is subjected to a rotational speed of 30,000 rpm. The inner ring’s freedom is restricted by remote displacement, allowing rotation only about the X-axis. Due to the elastomer nature of the seal, large deflection needs to be opened in the analytical setting. Figure 5a shows the rotational speed setting for the inner ring of the bearing. Figure 5b shows the boundary condition settings for the bearing structure field.

3. Analysis of Thermal-Stress Coupling Results

3.1. Distribution of Maximum Equivalent Stress and Contact Pressure

The temperature, stress and deformation of the sealing element are coupled with each other during the operation of the bearing [31], and the thermal stress complete coupling calculation should be carried out. However, the calculation of the complete coupling is huge, which can easily cause non-convergence. Considering that the stress field has an influence on the temperature field but the influence is not large, thermal-stress sequential coupling is adopted in this paper. Figure 6 and Figure 7 show the equivalent stress distribution diagram and contact pressure distribution diagram of the sealing ring when friction power consumption is considered.

It can be seen from Figure 6 that the maximum value of equivalent stress occurs at the junction of the seal wall from thick to thin, because there is a right-angle transition and stress concentration is prone to occur there. Changing this transition area to an arc transition would reduce the stress value here. In addition, the equivalent stress value in the contact area between the sealing ring and the bearing outer ring, metallic framework and inner ring is also large, because the hardness and elastic modulus of the bearing outer ring material GCr15 are greater than that of the oil seal material NBR, and the rigid support of the bearing outer ring limits the deformation direction of the seal ring, thus increasing the equivalent stress of the rubber in the contact area. The contact area between the seal lip and the inner ring is elastic deformation due to the presence of interference at the seal lip. NBR is a kind of highly elastic material, and the rubber stiffness increases under large deformation, resulting in an increase in equivalent stress.

As shown in Figure 7, the maximum contact pressure between the lip and the bearing inner ring contact area appears at the lip tip, and the contact pressure decreases from the lip tip to both sides of the sealing ring. This is because there is a certain amount of interference in the contact zone of NBR, resulting in a certain deformation, and the recovery force generated by the deformation causes greater pressure on the inner surface of the lip, ultimately leading to an increase in the contact pressure. The deformation from the lip tip of the sealing ring to both sides gradually decreases, so the contact pressure also decreases in turn.

In this section, the establishment method of the thermal-stress coupled finite element model and the distribution of equivalent stress and contact pressure are described, the phenomenon is analyzed, and the reasons for this phenomenon are explained.

3.2. Influence of Various Factors on Sealing Performance

Contact pressure is generated between the contact surface of the sealing lip and the bearing inner ring due to the existence of interference, and the size and distribution of this contact pressure has a crucial impact on the sealing performance of the oil seal [37]. If the contact pressure is too small, the thickness of the oil film between the contact surfaces increases, which easily causes grease leakage. If the contact pressure is too large, the thickness of the oil film between the contact surfaces will be reduced, and the wear between the two will be aggravated, which will reduce the service life of the oil seal [38]. Equivalent stress is an index to measure the degree of stress under complex stress conditions. Excessive equivalent stress will accelerate the fatigue damage of the sealing ring and reduce its fatigue life. Therefore, the maximum equivalent stress of the seal ring and the maximum contact pressure of the lip are used as indicators to measure the sealing performance of the oil seal.

Figure 8 is the variation diagram of the maximum equivalent stress of the seal ring and the maximum contact pressure of the different seal lip inclination angle. It can be seen from Figure 8 that with the increase in lip inclination angle, the equivalent stress and contact pressure both increase first and then decrease, and the maximum equivalent stress is 0.62 MPa and the maximum contact stress is 0.28 MPa. This is because when the seal lip inclination angle begins to increase, the deformation of the sealing ring is mainly concentrated in the area near the lip, and when it increases to a certain extent, the deformation begins to transition to the overall uneven, so it shows a trend of first rising and then falling. As for the contact pressure, with the increase in the seal lip inclination angle, the normal force of the contact surface increases, as does the contact pressure. When the seal lip inclination angle increases to a certain extent, friction and wear will be aggravated due to the excessive radial pressure in the contact area, thus reducing the contact pressure.

Figure 9 shows the variation in the maximum equivalent stress of the seal ring and the maximum contact pressure of the seal lip under different friction coefficients. As can be seen from Figure 9, with the increase in the friction coefficient, the equivalent stress first increases and then decreases, reaching its peak at the friction coefficient of 0.06, with a peak value of 0.55 MPa. The contact pressure decreases with the increase in the friction coefficient. This is because, as the friction coefficient increases, the oil seal needs more force to overcome the friction, so the equivalent stress shows a rising trend, and when the friction coefficient increases to a certain extent, the lip wear intensifies, and the equivalent stress decreases. As for the contact pressure, with the increase in the friction coefficient, the seal lip produces a greater tangential force in the contact area to resist the rotation of the shaft, and the degree of wear is intensified, resulting in a decrease in the contact pressure.

Figure 10 shows the variation in the maximum equivalent stress of the seal ring and the maximum contact pressure of the seal lip under different lubricant temperatures. As can be seen from Figure 10, with the increase in lubricant temperature in the bearing chamber, the equivalent stress and contact pressure both increase, but the equivalent stress increases more evenly, and the contact pressure increases slowly at the beginning and rapidly after 100 °C. This is because as the temperature of the lubricant increases, the viscosity of the oil decreases, and the friction in the contact area increases. At the same time, the temperature of the lubricant increases the deformation of the seal ring, so the equivalent stress and contact pressure will increase.

Figure 11 shows the variation in the maximum equivalent stress of the seal ring and the maximum contact pressure of the seal lip at different bearing speeds. As can be seen from Figure 11, the equivalent stress decreases with the increase in inner ring speed, and the contact pressure increases with the increase in inner ring speed. This is because the increase in speed will generate more heat, and the heat dissipation will be accelerated, and the generation and dissipation of this heat reduces the stiffness of the material, thus reducing the equivalent stress. For the contact pressure, as the speed increases, the pumping effect of the sealing ring is enhanced, and more fluid is pumped into the sealing zone, thereby increasing the contact pressure in the contact zone.

Figure 12 shows the variation in the maximum equivalent stress of the seal lip and the maximum contact pressure of the seal lip under different interference quantities. As can be seen from Figure 12, the equivalent stress gradually decreases with the increase in interference amount, the seal lip contact pressure decreases first and then increases with the increase in radial interference amount, and the contact pressure is the minimum 0.16 MPa. This is because with the increase in interference, the deformation is dispersed from the initial local area to the whole, so that the equivalent stress shows a trend of decreasing first and then increasing. As far as the contact pressure is concerned, the deformation generated by the seal lip at the beginning is not enough to make the contact surface very tight. With the increase in the interference amount and the adjustment of the sealing ring structure, the contact stress decreases briefly. With the further increase in the interference amount, the contact state becomes tighter beyond the critical value, and the contact pressure increases accordingly.

This section is based on a thermal-stress coupled finite element model for analyzing the influence of a single factor on equivalent stress and contact pressure, explaining the reasons for this change, and determining a reasonable optimization range for response surface optimization in the following text.

4. Response Surface Optimization Design

4.1. The Value Range of Optimized Variables

The response surface method is a statistical method that analyzes the relationship between independent factors that affect a specific response and selects the best factor from a series of potential solutions through an empirical model [39,40], calculates the response value through an approximate function, avoids repeated modeling and simulation, and improves the efficiency of optimization [41]. The range of optimization variables is crucial to the calculation efficiency and response surface accuracy. In this section, based on the influence law of the above single factor on the sealing performance, the contact pressure and equivalent stress are taken as responses, and the optimal value range of each factor is selected, as shown in

Table 3. The value range of the designed variable.

Factor	Level
	Initial Value	Lower Bound	Upper Bound
P2: Seal lip inclination angle (°)	44	42	44
P3: Friction coefficient	0.03	0.01	0.03
P5: Lubricant temperature (°C)	110	110	130
P8: Bearing speed (rpm)	28,000	22,000	30,000
P9: Radial interference (mm)	0.01	0.01	0.04

.

4.2. Response Surface Analysis

Commonly used agent models include Kriging, Neural Network, etc. [42]. As a spatial interpolation method, Kriging predicts data of unknown points according to data of known points, which has outstanding fitting ability and can obtain satisfactory fitting effect for high-dimensional nonlinear problems with high precision, so a Kriging agent model is adopted for response surface fitting [43,44,45,46]. In this section, contact pressure and equivalent stress are taken as optimization objectives. The result of the equivalent stress should be less than 10.5 MPa of the allowable stress of NBR under this working condition, and the contact pressure should be as large as possible. Figure 13 presents the equivalent stress response surface diagram, and Figure 14 presents the contact pressure response surface diagram.

It is not difficult to see from Figure 13 and Figure 14 that under the interaction of various factors, the equivalent stress first increases and then decreases with the increase in seal lip inclination angle P2, friction coefficient P3 and interference quantity P9, increases with the increase in lubricant temperature P5, and decreases with the increase in speed P8. The contact pressure decreases with the increase in lip angle P2, first increases and then decreases with the increase in friction coefficient P3, and first slightly decreases and then increases with the increase in lubricant temperature P5, speed P8 and interference amount P9.

In Figure 13, the response surfaces (b), (f) and (j) are curved and steep, and the contours are nearly elliptical, indicating that the interaction between P3 and P9, P3 and P8, and P3 and P2 is significant and the influence of the equivalent effect force is large. The response surfaces in Figure 14b,f,g,i,j are curved and steep, indicating that P9 and P3, P8 and P5, P5 and P3, and P3 and P2 have a great influence on the contact pressure, and the contours of (g), (h) and (j) are nearly elliptical. The interaction between P8 and P5, P2 and P5, and P3 and P2 is significant.

4.3. Sensitivity Analysis

Sensitivity analysis is a method used to evaluate the degree of influence of input variables on output results in a certain system. Its connotation is to study the influence of coefficient changes on the optimal solution [47]. This paper uses the response surface optimization module in the software Ansys Workbench to analyze the sensitivity of each factor.

Figure 15 shows the sensitivity analysis diagram of optimization variables to the objective function. As can be seen from Figure 15, for equivalent stress, P5 has the greatest influence, followed by P9 and P3, and P2 and P8 have the least influence; for contact pressure, P9 has the greatest influence, followed by P3, and P2, P5 and P8 have the least influence.

4.4. Candidate Points from Response Surface Optimization

Based on the Latin hypercube sampling method and Kriging proxy model, a genetic algorithm was used to find the optimal solution satisfying the constraints in the design space. The three candidate points were obtained as shown in

Table 4. Candidate points from response surface optimization.

Name	Seal Lip Inclination Angle (°)	Friction Coefficient	Lubricant Temperature (°C)	Bearing Speed (rpm)	Radial Interference (mm)	Von Mises Maximum		Pressure Maximum
						Parameter Value	Variation Form Reference	Parameter Value	Variation Form Reference
Candidate point 1	44.00°	0.010005	112.02	27,930	0.04	0.73020	−0.21%	1.4518	−7.67%
Candidate point 2	43.99°	0.010023	111.47	28,853	0.04	0.73183	0.00%	1.5724	0.00%
Candidate point 3	43.99°	0.010002	111.48	27,590	0.04	0.73144	−0.05%	1.4698	−6.52%

. The equivalent stress values of the three groups of data were similar, and all of them are smaller than 10.5 MPa of allowable stress of NBR under this working condition, but the contact pressure of candidate point 2 was the largest, and the contact pressure of candidate point 1 was 7.67% lower than that of candidate point 2. Candidate 3 is 6.52% smaller than 2, so candidate 2 is chosen.

In this section, the optimal range of response surface is determined based on the influence rule of single factor on sealing performance. The response surface is sampled by the Latin Hypercube sampling method, and the response surface is fitted using the Kriging proxy model. The influence rule of interaction of various factors on sealing performance is analyzed, and the sensitivity analysis is carried out to find the best combination of sealing performance.

5. Experimental Verification

The lubricant leakage rate of grease-lubricated bearings is an important index for measuring the sealing performance [48]. Therefore, in this paper, the seal lip inclination angle, friction coefficient, lubricant temperature, speed and radial interference amount are taken as experimental variables, and the grease leakage rate is taken as the sealing performance index to conduct the experiment. The specific experimental method is employed to carry out orthogonal tests on the above variables and compare the optimal solution obtained from the orthogonal test with the optimal solution obtained from the response surface. If the two results are consistent, the thermal-stress coupled finite element model and analysis method are considered correct; if the results are inconsistent, the model and method are improved until the two results are consistent.

5.1. Experimental Equipment

Figure 16 shows the high-speed bearing lubrication performance testing machine, which is composed of a drive system, loading system, lubrication system, measuring system and control system. When the testing machine is working, the motor provides high-speed rotating power, the loading system simulates the bearing’s stress in the actual working condition, the lubrication system applies the lubricating medium in the designated area according to the requirements, the measuring system detects various parameters of the bearing during operation through various sensors, and the control system controls and data acquisition and processing the entire experiment process. The experimental stand is temporarily built, can measure the bearing diameter range of Φ5–30 mm, can provide a maximum speed of about 5000 rpm, and can test 1–4 bearings at the same time.

The orthogonal experiment is a design method for arranging and analyzing a multi-factor experiment by using an orthogonal table that has balanced dispersion and neat comparability. Figure 17 presents a flow chart of the experiment. In this experiment, the optimal combination of sealing performance will be obtained by orthogonal experiment, and it will be compared with the optimal combination of response surface optimization. Since the orthogonal experiment can only select the optimal solution from the existing sample level, and the response surface rule is to fit the entire spatial range through the regression equation, when the experimental average value of the candidate points on the response surface is not greater than the average value of the orthogonal experiment, it can be shown that the experimental results are consistent with the optimization results.

5.2. Experimental Results

The experiment was carried out using three-factor and three-level orthogonal experiments, with the friction coefficient, lubricant temperature and speed as experimental variables and the lubricant leakage rate of the test bearing as an index of sealing performance. The experimental bearing grease filling capacity is 30%, radial load is 55 N, and axial load is 85 N. The experiment adopts the weighing method, the lubricant leaked on the bearing surface after the experiment is wiped off and weighed, and the bearing weight is compared with that before the experiment. The difference is the weight of the leaked lubricant, and the lubricant leakage rate of the bearing is calculated accordingly.

Table 5. Orthogonal experimental factor level table.

Procedure	Friction Coefficient	Lubricant Temperature (°C)	Bearing Speed (rpm)	Grease Leakage Rate (%)
1	0.01	110	22,000	2.67%
2	0.01	120	26,000	4.10%
3	0.01	130	30,000	3.21%
4	0.02	110	26,000	2.35%
5	0.02	120	30,000	2.43%
6	0.02	130	22,000	2.38%
7	0.03	110	30,000	2.98%
8	0.03	120	22,000	3.57%
9	0.03	130	26,000	3.85%

is the orthogonal experimental factor level table. According to the experimental results in Table 5, the values of $\overline{K_1}$, $\overline{K_2}$, $\overline{K_3}$ and $\mathrm{R}$ in

Table 6. Optimal point of orthogonal experiment.

Mean Value	Friction Coefficient	Lubricant Temperature (°C)	Bearing Speed (rpm)
$\overline{K_1}$	3.33%	2.67%	2.87%
$\overline{K_2}$	2.39%	3.37%	3.43%
$\overline{K_3}$	3.47%	3.15%	2.87%
R	1.08	0.7	0.56
Optimal scheme	0.02	110	22,000 or 30,000

are calculated, where $\overline{K_1}$, $\overline{K_2}$ and $\overline{K_3}$ are the average values of the test results with the same level of each bit, and $\mathrm{R}$ is the range values of $\overline{K_1}$, $\overline{K_2}$ and $\overline{K_3}$ corresponding to each factor. As can be seen from Table 6, the $\mathrm{R}$ value of the friction coefficient is the largest, followed by the temperature of the lubricant and the bearing speed. According to this, the optimal point of the orthogonal experiment is a friction coefficient of 0.02, lubricant temperature of 110 °C, and bearing speed of 22,000 or 30,000 rpm.

Comparison experiments were conducted between the optimal values obtained by the orthogonal experiments and the optimal values obtained from the response surface.

Table 7. Comparison experiment table.

Procedure	Friction Coefficient	Lubricant Temperature (°C)	Bearing Speed (rpm)	Grease Leakage Rate (%)
1	0.02	110	22,000	2.34
2	0.02	110	22,000	2.23
3	0.02	110	22,000	2.47
4	0.02	110	30,000	2.32
5	0.02	110	30,000	2.18
6	0.02	110	30,000	2.13
7	0.01	111.5	28,900	2.11
8	0.01	111.5	28,900	2.35
9	0.01	111.5	28,900	1.94

is the comparison experiment table of the optimal values of the two experiments. Figure 18 shows the comparison of the experimental results between the two optimal values of the orthogonal experiment and the optimal value of the response surface. In order to reduce the error caused by uncertain factors in the experiment, five experiments were conducted on each group of data, and the average value was taken as the experimental result.

As can be seen from Figure 18, experiments in groups 1–5 are optimal value 1 of the orthogonal experiment, experiments in groups 6–10 are optimal value 2 of the orthogonal experiment, and 11–15 are candidate points 2 among the optimal points on the response surface. The average lubricant leakage rate of experiments in groups 11–15 is 2.12%, which is lower than the average lubricant leakage rate of experiments in groups 1–5 and 6–10. Since the orthogonal experiment can only select the optimal solution from the existing sample level, and the response surface method fits the entire spatial range through regression equation, when the experimental average value of the candidate points on the response surface is not greater than the average value of the orthogonal experiment, it can be shown that the experimental results are consistent with the optimization results, indicating that the parameter values determined by candidate point 2 correspond to the minimum lubricant leakage rate. In other words, the parameter values determined by candidate point 2 are the optimal solution of sealing performance, and the correctness of the thermal-stress coupled finite element model and analysis method is also demonstrated.

6. Conclusions

This article focuses on the problem of sealing failure in high-speed micro bearings; based on the thermal-stress coupled finite element model, the distribution of the equivalent stress and contact pressure of the sealing ring and the influence of each factor on the sealing performance were analyzed. Based on this, the Latin Hypercube sampling method, Kriging proxy model and genetic algorithm were used to find the best combination of sealing performance. Finally, the optimal solution of response surface optimization and the optimal solution of orthogonal experiment are compared to verify the accuracy of the model and method, and the conclusions are as follows:

• Based on the thermal-stress coupled finite element model, it is found that the maximum equivalent stress of the sealing ring appears at the junction of the thin region and the thick region, followed by the sealing ring lip region, and the contact pressure is the largest at the sealing ring lip and decreases from the lip to both sides in turn.
• Through response surface optimization analysis, it is found that under the interaction of various factors, the equivalent stress of the seal ring increases first and then decreases with the increase in seal lip inclination angle, friction coefficient and radial interference amount, increases with the increase in lubricant temperature, and decreases with the increase in speed. The contact pressure decreases with the increase in seal lip inclination angle, increases first and then decreases with the increase in friction coefficient, and decreases first and then increases with the increase in lubricants temperature, speed and radial interference.
• The sensitivity analysis of each factor shows that the most important factor of the equivalent stress is the temperature of the lubricants in the bearing chamber, followed by the radial interference amount and friction coefficient, and the least important factor is the seal lip inclination angle and rotation speed. The radial interference amount has the most significant influence on contact pressure, followed by the friction coefficient, and the least significant factor is the seal lip inclination angle, lubricant temperature and rotation speed.
• The experimental results and simulation results agree that when the seal lip inclination angle is 43.99°, the friction coefficient is 0.01 mm, the lubricant temperature is 111.5 °C, the rotation speed is 28,853 rpm and the radial interference volume is 0.04 mm, the sealing performance of the sealing ring is the best, with the orthogonal experiment used to verify it.