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Boundary Conditions in Sliding Bearing Lubrication Analysis

Sliding bearings have the advantages of strong bearing capacity, stable and reliable operation, low noise and long life. They are widely used in the field of machinery, and their working conditions have a very important impact on the economy, reliability and durability of mechanical devices. In the design of plain bearings, the analysis of lubrication performance plays a very important role. In 1883, Tower tested the sliding bearing for the axle of the train, and found the phenomenon of fluid pressure in the bearing oil film for the first time. In response to Tower’s discovery, in 1886, Reynolds applied the theory of fluid mechanics to deduce the Reynolds equation to explain the formation mechanism of fluid dynamic pressure, which laid the foundation for the study of fluid lubrication theory. According to the fluid lubrication theory, the lubrication analysis of sliding bearings is generally completed by solving the Reynolds equation. In the specific sliding bearing lubrication analysis, it is necessary to adopt appropriate boundary conditions such as pressure in combination with the actual problem of the analysis. The research shows that the rationality of the boundary conditions used in the calculation of the oil film pressure of the sliding bearing is an important factor affecting the error of the results. Therefore, the key to the lubrication analysis of the sliding bearing is how to determine the boundary conditions for solving the actual problem. Whether the boundary conditions for solving the Reynolds equation can be reasonably determined will directly affect the calculation and prediction accuracy of the lubrication performance of the sliding bearing.

1. Boundary conditions adopted earlier

1.1 Sommerfeld boundary condition [1]

The Sommerfeld boundary conditions are shown in Figure 1. The set solution interval is: the lower limit φ1 = 0, the upper limit φ2 = 2π; the oil film pressure p = 0 when the angular position coordinate φ = φ1 is satisfied, and the oil film pressure p = 0 when φ = φ2. The oil film pressure in the oil film (gap) convergence area is positive pressure, and the oil film pressure in the oil film divergent area is negative pressure, and the pressure distribution is antisymmetric. The Sommerfeld boundary condition assumes that the entire oil film cavity of the bearing is filled with lubricating oil and the oil film is continuous, which has a large deviation from the actual situation. For the actual bearing, negative pressure will be generated in the divergent area of ​​the oil film, which will inevitably lead to the mixing of air, so the oil film is not continuous.

The Sommerfeld boundary condition is more convenient to apply in the solution of bearing lubrication analysis, but because the actual oil film cannot withstand large and long-term negative pressure, the Sommerfeld boundary condition is not physically satisfied, so it can only be used for bearing lubrication problems. qualitative analysis.

1. 2 Half-Sommerfeld boundary condition[1]

The half-Sommerfeld boundary condition is shown in Figure 2, which considers that the general oil film can only bear positive pressure, but cannot bear negative pressure, and the pressure distribution in the oil film (gap) convergence area is the same as the Sommerfeld boundary condition, but The oil film pressure in the oil film divergence area is zero. The semi-Sommerfeld boundary condition only calculates the bearing capacity corresponding to the positive pressure in the oil film convergence area, and does not consider the influence of the negative pressure. The set solution interval is: lower limit φ1 = 0, upper limit φ2 = π; oil film pressure p = 0 when φ = φ1 is satisfied, oil film pressure p = 0 when φ = φ2, dpdφ = 0 . The semi-Sommerfeld boundary condition considers that the oil film is discontinuous and the pressure of the oil film is also discontinuous. From the analysis of the mechanism and physical process of the dynamic pressure oil film formation, this does not conform to the actual situation of the sliding bearing.

It is more convenient to use the semi-Sommerfeld boundary condition to solve the bearing lubrication analysis. The calculated bearing pressure distribution is close to the actual situation and is relatively safe, and can generally be used in the calculation of bearing lubrication performance in engineering. The problems of the semi-Sommerfeld boundary condition are: (1) The influence of the unsteady whirl velocity on the boundary position of the oil film is ignored, which is only approximately true when the journal is purely rotating, and is generally not applicable to bearings with squeezing effects; (2) In fact, there is still pressure in the oil film after the minimum value of the oil film thickness, so the boundary conditions do not meet the continuous conditions of pressure and flow, and the analysis is not accurate enough.

1. 3 Hahn boundary conditions [2]

The Hahn boundary condition considers that the oil film rupture occurs in the negative pressure region, and the oil film pressure after rupture is zero, that is, p ( φ1 ) = p ( φ2 ) = 0. The Hahn boundary condition does not directly determine the position of the oil film rupture and its influence on the starting and ending angles of the oil film region, but according to the geometric dimensions of the bearing, the changes in the motion parameters ε and εt (t is time), it can be determined that the oil film ruptures after the oil film ruptures. The influence of the starting and ending angles and the oil film pressure distribution, and it is considered that the oil film pressure after rupture is equal to zero, φ1 and φ2 vary with the geometric dimensions of the bearing, and the motion parameters ε and εt.

The Hahn boundary condition is more practical, for example, it is more reasonable to solve the bearing lubrication performance, but the boundary condition does not provide a mathematical model for calculation that is suitable for practical use, nor does it consider the impact of the journal whirling motion on the boundary conditions. When it is used for dynamic load bearing lubrication analysis, the calculated eccentricity of the bearing axis track is generally too large.

2 The currently used boundary conditions

2. 1 Reynolds boundary condition [1-3]

Reynolds boundary condition considers that the oil film is discontinuous, the starting point of forming the oil film is at the position of the maximum clearance of the bearing, and the end point of the oil film is not determined artificially, but is determined by the natural rupture condition of the oil film. The Reynolds boundary condition can not only overcome the negative pressure problem of the oil film in the divergent zone, but also satisfy the flow continuity condition. However, the Reynolds boundary condition only satisfies the mass conservation condition of the flow rate at the oil film rupture boundary, and cannot satisfy the flow rate at the initial boundary of the oil film formation. the mass conservation condition. The Reynolds boundary condition is shown in Fig. 3, and the starting point of the treatment is to satisfy the flow continuity of the lubricating oil in the region where the oil film exists. In this way, in the oil film area, there is only shear flow and no pressure flow at the following two sections. One section is the section where the maximum oil film pressure pmax occurs, and the other section is in the oil film divergence area, that is, not at φ = π (minimum at the gap hmin). Assuming the position angle of this section φ = π + α (Fig. 3), since the oil film pressure p at this section drops to zero.

Compared with the previous boundary conditions, the Reynolds boundary condition is more reasonable to apply to the oil film rupture, and it conforms to the pressure and flow continuity conditions. The bearing lubrication performance analysis using Reynolds boundary conditions is close to the measured results. The problems of the Reynolds boundary conditions are: (1) The position of the oil film termination point must be determined according to the calculation, which is inconvenient to use; (2) The situation that the oil film is formed after the oil film rupture cannot be correctly explained, and it is only applicable to the movement speed of the oil film rupture boundary is less than the journal line speed half of the cases; (3) Assuming that there is no flow at all where the cavity occurs, and there is no lubricating oil even on a solid surface, this is not entirely realistic; (4) When the effect of temperature is introduced, or the bearing oil supply is insufficient, or This boundary condition will not be suitable when there is negative pressure in the oil film.

2. 2 Double Reynolds boundary conditions [4]

The double Reynolds boundary conditions are set to satisfy both the pressure and the derivative of zero at the starting and ending positions of the oil film pressure.

It is proved that the double Reynolds boundary condition and the double variational form of the Reynolds boundary condition are correspondingly equivalent, and the problem of solving the Reynolds equation and determining the boundary can be transformed into a variational problem. Reference [6] applied the double Reynolds boundary conditions to calculate the lubrication performance of sliding bearings with different eccentricities. The results show that the double Reynolds boundary conditions are better than the half Sommerfeld boundary conditions due to the consideration of the influence of oil film rupture on the starting and ending positions of oil film pressure formation. and Reynolds boundary conditions are more suitable for bearing lubrication analysis under unsteady conditions. The advantages and disadvantages of the double Reynolds boundary conditions for bearing performance analysis are basically the same as those of the Reynolds boundary conditions. However, compared with the Reynolds boundary conditions, the double Reynolds boundary conditions also consider the influence of oil film rupture on the upstream and downstream boundaries, and are more suitable for unstable conditions. Bearing lubrication analysis under dynamic conditions.

2. 3 Floberg boundary conditions [2]

The Floberg boundary condition determines the boundary conditions from the viewpoint of lubricating oil flow continuity. It is considered that the oil film satisfies the Reynolds boundary condition before rupture, and the oil film pressure in the region between the beginning and the end of rupture satisfies the oil film pressure not negative pressure and flow continuity conditions.

The Floberg boundary condition is not suitable for the actual working conditions of high-speed and large-disturbance bearings, and it is only suitable for the calculation of bearing lubrication performance with low-speed, small disturbance and low Reynolds number oil film.

2. 4 Mass Conservation Boundary Conditions [7-22]

The mass conservation boundary condition was proposed by Jakobsson, Floberg and Olsson, also known as the JFO boundary theory. This boundary condition considers that the oil film is in mass conservation at the boundary position of rupture and reformation, and the lubricating region is set to be divided into a complete oil film region and a cavity region. The Reynolds boundary condition is still used in the complete oil film region, and the fluid flows in the form of strips in the cavity region, which is not separated from the bearing and rotor surfaces, and the pressure in the cavity region remains unchanged. The mass conservation boundary condition overcomes the shortcomings of the Reynolds boundary condition, and provides not only the oil film rupture condition, but also the oil film reformation condition.

The numerical realization of the mass conservation boundary condition is relatively difficult, and many researchers have discussed its numerical processing. (1) Literature [9] proposes an algorithm, and literature [10-12] improves it. By introducing a new variable θ and switching function g, the original equations describing the complete oil film region and the cavity region respectively are unified into a general equation, and the dynamic boundary is automatically determined in the numerical calculation.

2) Literature [13-15] proposed a finite element cavitation algorithm based on the mass conservation boundary condition, which considered lubricating oil to be incompressible. The cavity zone is a two-phase mixture of oil and air/steam, and the viscosity and density of the mixture vary.

The complete oil film area 2 can be regarded as the transition area between the complete oil film area 1 and the cavity area, in which although the oil film is complete,

But the oil density change rate has changed from zero to less than zero. (3) Reference [16] proposes a combination of variational and finite element

Hole calculation method.

(4) Literature [7, 17] found that the pressure of the oil film of the sliding bearing usually cannot reach the critical pressure value to compress the lubricating oil, and the assumption of the incompressibility of the lubricating oil in the lubrication analysis of the sliding bearing is generally in line with the actual situation. Based on this, the literature [10] is improved, and the unified expression of the lubrication equations in the complete oil film region and the cavity region is deduced, so that the mass conservation boundary condition can be more easily and accurately applied to the bearing lubrication analysis. This method is called incompressible fluid. Hole algorithm.

5) Reference [18] applied the mass conservation boundary condition to analyze the lubrication performance of the sliding bearing under dynamic load. Compared with the calculation results under the Reynolds boundary condition, it was found that the maximum oil film pressure and the minimum oil film thickness of the bearing obtained under the two boundary conditions were very close. , but the cavity region of the calculation results under the mass conservation boundary condition is much larger, and the flow rate is quite different from the results under the Reynolds boundary condition.

The mass conservation boundary condition is one of the most realistic boundary conditions for bearing lubrication analysis, which not only provides the condition of oil film rupture, but also provides the condition of oil film reformation. The bearing capacity, lubricating oil flow rate, flow rate and power consumption can be predicted more accurately by using the mass conservation boundary condition. The calculated numerical results are in good agreement with the measured results.

3 Conclusion

At present, the boundary conditions used in the lubrication analysis of sliding bearings are mainly Reynolds boundary conditions and mass conservation boundary conditions. Compared with these two boundary conditions, the Reynolds boundary condition adopts the method of setting the negative pressure to zero to gradually approach the boundary of oil film rupture, which is more convenient for solving the lubrication analysis, but its biggest problem is that it cannot correctly explain the formation of oil film after rupture. Therefore, it is not suitable for the overall calculation accuracy that requires high bearing performance analysis. The mass conservation boundary condition is a lubrication analysis boundary condition that can basically reflect the oil film conditions in all actual bearing operations. Although its application in the specific solution is more complicated than the Reynolds boundary condition, the very high overall calculation accuracy makes it suitable for sliding bearing lubrication. more and more applications in analysis. The mass conservation boundary conditions also need to be continuously improved. For example, the shape of the oil film in the cavity region is determined, that is, whether the oil film is strip-shaped, bubble-shaped or fern-shaped; the oil film pressure in the cavity region is determined.

It is generally determined based on experimental calculations, and how to determine it through theoretical analysis and calculation methods remains to be resolved.

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