Method for estimating viscous rolling resistance, system for estimating viscous rolling resistance, program for estimating viscous rolling resistance, and computer-readable recording medium having the program recorded thereon

Abstract

To precisely estimate the rolling viscosity resistance of an actual bearing.SOLUTION: A rolling bearing 1 includes trace surfaces 11a and 12a and a rolling body 13 for rolling on the trace surfaces. The rolling bearing 1 includes: a contact part in which the trace surfaces and the rolling body are in point-contact with each other, an oil film is formed in the point-contact part, and the trace surfaces 11a and 12a are in contact with the rolling body 13; and a contactless part as a part which is not in contact with the rolling body. The estimated value Fr,w of the rolling viscosity resistance is calculated from a regression equation on the condition that the calculation range of a contact ellipse in the direction of a longer axis is a length selected in the range from larger than a longer axis radius of a contact ellipse and to the equivalent radius in the axial direction.SELECTED DRAWING: Figure 2

Description

[Technical field]

[0001] The present invention relates to a method for estimating viscous rolling resistance in a rolling bearing, a system for estimating viscous rolling resistance, a program for estimating viscous rolling resistance, and a computer-readable recording medium having the program recorded thereon. [Background technology]

[0002] The main purpose of lubrication in rolling bearings is to reduce friction and prevent damage caused by the formation of an oil film between elements. In properly lubricated rolling bearings, an EHL (Elasto-hydrodynamic Lubrication) oil film is formed between the raceway surface and the rolling elements, separating the two surfaces and preventing early surface damage. From the perspective of preventing surface damage, a thicker oil film is better, but this is not necessarily the case from the perspective of low friction, which is one of the important performance indicators of rolling bearings. Under conditions where the oil film is thick, resistance due to the viscosity of the oil increases when the fluid flows into the contact surface. This resistance is called rolling viscous resistance.

[0003] Estimating the magnitude of viscous rolling resistance is an important issue in the design and performance evaluation of rolling bearings. In order to estimate the viscous rolling resistance with high accuracy, past research has constructed a regression equation for the viscous rolling resistance by calculating the vicinity of the contact point between the ball and the raceway using EHL analysis. When constructing the regression equation, it is known that the calculation range for the viscous rolling resistance is the range of the contact ellipse or the range of the semi-major axis on both sides of the axial direction centered on the contact ellipse (Non-Patent Document 1). [Prior art documents] [Non-patent literature]

[0004] [Non-Patent Document 1] N. Biboulet and L. Houpert, Hydrodynamic force and moment in pure rolling lubricated contacts. Part 1: point contacts, J. Eng. Tribol., 224,8(2010), 777-788 Summary of the Invention [Problem to be solved by the invention]

[0005] However, in the analysis of Non-Patent Document 1, the calculation range of viscous rolling resistance is narrow compared to that of actual bearings, so the viscous rolling resistance of actual bearings is underestimated. This is because in a bearing under sufficient lubrication, the space between the balls and the raceway is filled with oil, so viscous rolling resistance occurs not only in the contact area where the raceway surface and the rolling elements come into contact (the area where the contact ellipse is formed) but also in the non-contact area, which is the area axially outside the contact ellipse within the width of the raceway surface, whereas in Non-Patent Document 1, the calculation range in the axial direction is limited to the range of the contact ellipse, etc. Therefore, the calculated values ​​in Non-Patent Document 1 tend to deviate from the viscous rolling resistance values ​​of actual bearings.

[0006] Therefore, an object of the present invention is to provide a method, an estimation system, and an estimation program for estimating viscous rolling resistance, which are capable of estimating the viscous rolling resistance of an actual bearing with high accuracy, and a computer-readable recording medium having the program recorded thereon. [Means for solving the problem]

[0007] In order to solve the above-mentioned problems, a method for estimating viscous rolling resistance according to the present invention is a method for estimating viscous rolling resistance of a rolling bearing having a raceway surface and rolling elements that roll on the raceway surface, wherein the raceway surface and the rolling elements are in point contact with each other, an oil film is formed at the point contact portion, and the raceway surface has a contact portion in which the rolling elements are in contact with each other and a non-contact portion in which the rolling elements are not in contact with each other, the method being characterized in that a calculation range in the major axis direction of the contact ellipse is set to a length selected within a range that exceeds the major axis radius of the contact ellipse and is equal to or smaller than the axial equivalent radius, and an estimated value Fr,w of the viscous rolling resistance is calculated from a regression equation.

[0008] In this estimation method, a regression equation for the viscous rolling resistance is determined for each of the contact portion and the non-contact portion, and the sum of the values ​​calculated based on each regression equation can be used as the estimated value Fr,w of the viscous rolling resistance of the rolling bearing.

[0009] In this estimation method, the estimated value Fr,w can be calculated from the following formula:

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[0010] In this estimation method, the regression equation for the contact portion can be multiplied by a thermal correction factor.

[0011] Furthermore, the system for estimating viscous rolling resistance according to the present invention is a rolling bearing having a raceway surface and rolling elements that roll on the raceway surface, wherein the raceway surface and the rolling elements are in point contact with each other, an oil film is formed at the point contact portion, and the raceway surface has a contact portion where the raceway surface is in contact with the rolling elements and a non-contact portion where the raceway surface is not in contact with the rolling elements, the system is characterized in that it comprises a calculation unit that calculates an estimated value Fr,w of the viscous rolling resistance from a regression equation, with a calculation range in the major axis direction of the contact ellipse being a length selected within a range that exceeds the major axis radius of the contact ellipse and is equal to or smaller than the axial equivalent radius, and an output unit that outputs the estimated value Fr,w.

[0012] In this estimation system, a regression equation for viscous rolling resistance is determined for each of the contact portion and the non-contact portion, and the sum of the values ​​calculated based on each regression equation can be used as the estimated value Fr,w of the viscous rolling resistance of the rolling bearing.

[0013] In this estimation system, the estimate Fr,w can be calculated from the following formula:

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[0014] In this estimation system, the regression equation for the contact portion can be multiplied by a thermal correction factor.

[0015] Furthermore, the present invention provides a program for estimating viscous rolling resistance for a rolling bearing having a raceway surface and rolling elements that roll on the raceway surface, wherein the raceway surface and the rolling elements are in point contact with each other, an oil film is formed at the point contact portion, and the raceway surface has a contact portion in which the rolling elements are in contact with each other and a non-contact portion in which the rolling elements are not in contact with each other, the program causing a computer to perform the following steps: calculate an estimated value Fr,w of the viscous rolling resistance from a regression equation, with a calculation range in the major axis direction of the contact ellipse being a length selected within a range that exceeds the major axis radius of the contact ellipse and is equal to or smaller than the axial equivalent radius; and output the estimated value Fr,w.

[0016] In this estimation program, a regression equation for the viscous rolling resistance is determined for each of the contact portion and the non-contact portion, and the sum of the values ​​calculated based on each regression equation can be used as the estimated value Fr,w of the viscous rolling resistance of the rolling bearing.

[0017] In addition, in this estimation program, the estimated value Fr,w can be calculated from the following formula:

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[0018] In this estimation program, the regression equation for the contact portion can be multiplied by a thermal correction coefficient.

[0019] The above-described estimation program may be recorded on a computer-readable recording medium. Effect of the Invention

[0020] According to the present invention, the rolling viscous resistance of an actual bearing can be estimated with high accuracy. [Brief description of the drawings]

[0021] [Figure 1] 1 is a cross-sectional view of a ball bearing according to an embodiment of the present invention. [Diagram 2] FIG. 1 is a development of the orbital plane showing the analysis range and integration range used in the EHL analysis. [Diagram 3] 1 is a table showing calculation control parameters. [Figure 4] 13 is a table showing calculation conditions (bearing model number 6320). [Diagram 5] 13 is a table showing calculation conditions (bearing model number 6210). [Figure 6]1 is a table showing calculation conditions (bearing model number 6004). [Figure 7] 1 is a table showing physical property values ​​of materials used in calculations. [Figure 8] 1 is a table showing physical property values ​​of lubricating oils used in calculations. [Figure 9] The figure shows the oil film pressure and oil film thickness distribution at y = 0 mm. Figure (b) shows an enlarged view of the area x = ±0.4 mm in Figure (a). [Figure 10] FIG. 13 is a graph showing the oil film pressure and oil film thickness distribution at y=3 mm. [Figure 11] This is a graph showing the rolling viscous resistance and oil film thickness distribution per unit length at x = 0 mm. [Figure 12] This figure shows the oil film pressure distribution and the change in the pressure generation position at y = 0 mm due to changes in average velocity. Figure (b) shows an enlarged view of the area x = ±0.4 mm in Figure (a). [Figure 13] The figure shows the oil film pressure distribution at y = 0 mm and the change in the pressure generation position due to load change. Figure (b) shows an enlarged view of the area x = ±0.4 mm in Figure (a). [Figure 14] This figure shows the oil film pressure distribution and the change in the pressure generation position at y = 0 mm due to changes in the axial equivalent radius. Figure (b) shows an enlarged view of the area x = ±0.4 mm in Figure (a). [Figure 15] This is a diagram showing the oil film pressure distribution and the change in pressure generation position at y = 3 mm due to changes in average velocity. [Figure 16] FIG. 13 is a diagram showing the oil film pressure distribution and pressure generation position change at y=3 mm due to load change. [Figure 17] FIG. 13 is a diagram showing the change in oil film pressure distribution and pressure generation position at y = 3 mm due to the change in axial equivalent radius. [Figure 18] Figures showing the influence of non-dimensional parameters on the non-dimensional viscous rolling resistance Fr,c of the contact area: (a) shows the non-dimensional velocity parameter U, (b) shows the non-dimensional load parameter W, (c) shows the non-dimensional material parameter G, and (d) shows the contact ellipticity ratio. [Figure 19] 1 is a table showing initial values ​​used in searching for optimal values ​​of constants. [Figure 20]11 is a table showing the results of searching for optimal values ​​of constants. [Figure 21] FIG. 13 is a diagram showing the effect of the dimensionless speed parameter U′ on the dimensionless rolling viscous resistance Fr,nc of the non-contact portion. [Figure 22] FIG. 13 is a diagram showing the effect of the dimensionless shape parameter L on the dimensionless rolling viscous resistance Fr,nc of the non-contact portion. [Figure 23] 1 is a table showing initial values ​​used in searching for optimal values ​​of constants. [Figure 24] 11 is a table showing the results of searching for optimal values ​​of constants. [Diagram 25] 1 is a table showing the range of use of dimensionless parameters. [Figure 26] FIG. 13 is a diagram showing the relative error distribution between analytical and calculated values ​​of Fr,c. [Figure 27] FIG. 13 is a diagram showing the relative error distribution between analytical and calculated values ​​of Fr,w. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0022] Hereinafter, an embodiment of the present invention will be described in detail with reference to the drawings.

[0023] 1 is a cross-sectional view showing a ball bearing 1, which is a type of rolling bearing. This ball bearing 1 comprises an inner ring 11 having an inner ring raceway surface 11a on its outer peripheral surface, an outer ring 12 having an outer ring raceway surface 12a on its inner peripheral surface, a plurality of rolling elements (balls) 13 interposed between the inner ring raceway surface 11a and the outer ring raceway surface 12a so as to be able to roll, and a cage 14 which holds the rolling elements 13 at predetermined intervals in the circumferential direction of the bearing. An oil film is formed between the raceway surfaces 11a, 12a of the ball bearing and the rolling elements 13 by lubricating oil supplied inside the bearing.

[0024] In ball bearing 1, inner ring raceway surface 11a and outer ring raceway surface 12a are machined so that their radii of curvature are very slightly larger than the radius of curvature of balls 13. As a result, inner ring raceway surface 11a and balls 13, and outer ring raceway surface 12a and balls 13 are in point contact. The point contact area assumes an elliptical shape (called a contact ellipse) due to elastic deformation of the raceway surfaces and balls. The size of the contact ellipse can be determined by Hertz's elastic contact theory or contact analysis using the semi-infinite elastic body approximation.

[0025] The present invention makes it possible to estimate with high accuracy from a regression equation the value of the rolling viscous resistance inside a bearing, which is important information when designing or evaluating the performance of a rolling bearing as described above. The process of deriving the regression equation will be described below.

[0026] The definitions of symbols used in the following description are listed below. a: semimajor axis radius of osculating ellipse [m] a1~a5: Constants of the non-dimensional rolling viscous resistance formula b: minor axis radius of osculating ellipse [m] b1~b5: Constants of the non-dimensional rolling viscous resistance formula E': Equivalent Young's modulus [Pa] F r : Dimensionless rolling viscous resistance f r : Rolling viscous resistance [N] f t : Viscous rolling resistance taking into account thermal correction coefficient [N] G: Dimensionless material parameter h ∞ : Gap [m] h c : Center oil film thickness [m] k : Contact ellipse ratio=a / b L: Dimensionless shape parameter l: Distance in the y direction from the center of the osculating ellipse to the edge of the analysis domain [m] p: Pressure Pa R x : equivalent radius in x direction [m] R y : equivalent radius in y direction [m] U: Dimensionless velocity parameter U': Dimensionless velocity parameter of the non-contact part u: average speed [m / s] W: Dimensionless load parameter w: Load [N] x: Coordinate in the rolling direction [m] y: Coordinate perpendicular to the rolling direction [m] α: Viscosity-pressure coefficient [Pa -1 ] φ t : Thermal correction factor η0: Normal pressure viscosity of lubricating oil [Pa s]

[0027] [Subscript] c: Contact area nc: non-contact part left: to the left of the center of the analysis region right: Right side of the center of the analysis area nl: Left side of non-contact area nr: Right side of non-contact area w: Entire analysis area

[0028] The equivalent radius R can be calculated by R=R1×R2 / (R1+R2), where R1 is the radius of the rolling element (ball) and R2 is the radius of the raceway. The equivalent Young's modulus E' is calculated by 2 / E'=(1-ν1), where E1 is the Young's modulus of the ball, ν1 is the Poisson's ratio, E2 is the elastic coefficient of the raceway, and ν2 is the Poisson's ratio. 2 ) / E1+(1-ν2 2 ) / E2.

[0029] In the following, the oil film pressure and oil film thickness distribution in the analysis region are obtained by isothermal EHL analysis. In this analysis, the Roelands equation is used as the relationship between viscosity and pressure, and the Wu-Klaus-Duda equation is used as the viscosity-pressure coefficient. The multilevel method is used as the numerical analysis algorithm.

[0030] Figure 2 shows the analysis range and integration range used in EHL analysis, and is a development diagram of one of the raceway surfaces of a bearing (for example, the inner raceway surface 11a) developed onto a plane. As shown in Figure 2, a contact ellipse is formed on the raceway surface by contact with the rolling elements. The x direction in Figure 2 is the direction in which the rolling elements roll, and the y direction (the major axis direction of the contact ellipse) perpendicular to the x direction corresponds to the axial direction for radial rolling bearings and the radial direction for thrust rolling bearings. A method for calculating viscous rolling resistance will be explained below.

[0031] Rolling viscous resistance f r is the shear resistance of the lubricant due to the Poiseuille flow in the rolling direction,

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[0032] In order to calculate the rolling viscous resistance using formula (1), it is necessary to define the integration range. In this embodiment, the integration range in the x direction is set to -30b to 2b. This is because the rolling viscous resistance is almost saturated outside the range of -30b to 2b. Regarding the y direction, in the analysis of Non-Patent Document 1, the integration range in the y direction depends on the major axis radius a of the contact ellipse, so there is a problem that the integration range changes depending on the load and the contact ellipse ratio. However, in this embodiment, the integration range in the y direction is set to the axial width of the raceway surface, which is a constant region that is unrelated to the size of the contact ellipse.

[0033] The rolling viscous resistance is calculated by dividing the calculation into the contact part and the non-contact part as shown in the following formulas (2) to (5), and constructing separate regression equations for the contact part and the non-contact part. In the analysis region, the rectangular region (range of -30b to 2b and ±a) between the two vertices on the major axis side of the contact ellipse is the contact part, and the two rectangular regions sandwiching the contact part from both axial sides are the non-contact parts. By dividing the analysis region into the contact part and the non-contact part in this way, advantages are obtained such as the effect of shear heat of the lubricant oil generated only around the inlet of the contact point can be considered only in the contact part, and calculation is possible even when the contact ellipse does not exist in the center of the raceway surface. In this embodiment, the sum of the y-direction width dimensions of the contact part and the two non-contact parts is equal to the width dimension of the raceway surface. In other words, the analysis region in the y direction coincides with the width dimension of the raceway surface.

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[0034] When applying the present invention to ball bearings, especially ball bearings that use low-viscosity oil and rotate at high speeds, the effect of heat generation due to shearing of the lubricating oil at and near the contact points becomes significant, so the effect of this heat generation must be taken into consideration. Therefore, in such cases, the thermal correction coefficient φ should be applied only to fr,c, as shown in the following formula (6). t It is preferable to multiply by "fr,c only" because the shear heat of the lubricant occurs only around the inlet of the contact area. Thermal correction coefficient φ t Details will be given later.

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[0035] The numerical analysis (EHL analysis) of rolling viscous resistance is explained below. In this analysis, first, a parameter study of rolling viscous resistance was performed under the calculation control parameters shown in Fig. 3. The calculation conditions are shown in Figs. 4 to 6. Note that the ball radii and pitch circle radii in Figs. 4 to 6 correspond to bearing model numbers 6320, 6210, and 6004, respectively. The physical properties of the materials and lubricant used in the calculation are shown in Figs. 7 and 8.

[0036] The oil film pressure, rolling viscous resistance per unit length, and oil film thickness distribution obtained by EHL analysis are shown in Figures 9(a)(b), 10, and 11. The calculation conditions are the "reference conditions" in Figure 5. Figures 9(a)(b) show the oil film pressure and oil film thickness distribution at y=0mm, and Figure 10 shows the oil film pressure and oil film thickness at y=3mm. y=0mm is the contact area, and y=3mm is the non-contact area. At y=0mm, the presence of a pressure spike, which is characteristic of EHL, and the presence of a minimum oil film thickness near the outlet can be confirmed. In addition, the pressure generation position where the pressure on the inlet side is 0 is x=-3.89mm. Since the pressure generation position exists within the analysis region, this calculation condition can be said to be an analysis under sufficient lubrication, that is, an analysis at a position where the calculated value of the rolling viscous resistance does not change by setting the length of the inlet side of the analysis region in the rolling direction to a certain length or more. At y=3mm, no pressure spike occurs, and the position of the maximum pressure has moved significantly toward the inlet side compared to the contact area. Therefore, at y = 3 mm, the pressure distribution is what is known as fluid lubrication. Also, the pressure generation position does not change significantly from the contact area, but the maximum value of the oil film pressure is smaller than that of the contact area.

[0037] Figure 11 shows the rolling viscous resistance per unit length and the oil film thickness distribution at x = 0 mm. The rolling viscous resistance per unit length is the value obtained by integrating the shear stress caused by the Poiseuille flow generated in each grid only in the rolling direction, and by integrating this in the axial direction, the rolling viscous resistance in any range can be calculated. The rolling viscous resistance distribution fluctuates slightly in the axial direction due to calculation errors occurring near the outlet due to the number of grid divisions, but this is not considered a problem because the amplitude is small and the rolling viscous resistance of the entire calculation area is integrated in the axial direction, and the study was continued in this state. The rolling viscous resistance per unit length is significantly different between the contact area and the non-contact area, but the rolling viscous resistance of the entire contact area and non-contact area is fr,c = 0.5N and fr,nc = 0.082 × 2 = 0.164N, respectively, and it became clear that the rolling viscous resistance of the non-contact area cannot be ignored. Even in deep groove ball bearings that are generally used in motors, etc., the minimum clearance of the non-contact parts is often similar to the conditions for this calculation, so when calculating torque, the rolling viscous resistance of the non-contact parts must be taken into account.

[0038] Changes in the oil film pressure distribution and pressure generation position at the contact area due to changes in various parameters are examined. The calculation conditions were changed from the "reference conditions" in Figure 5 for average speed, load, and axial equivalent radius. Figures 12(a)(b) to 14(a)(b) show the changes in the oil film pressure distribution and pressure generation position at y = 0 mm due to changes in average speed, load, and axial equivalent radius. At the contact area, the pressure generation position moves forward due to increases in the average speed and the minor axis radius of the contact ellipse. In addition, regarding Figures 14(a)(b), if the minor axis radius of the contact ellipse is the same, it was expected that the pressure generation position would move away because the lateral leakage of pressure is suppressed by increasing the contact ellipse ratio. However, under the above calculation conditions, the surface pressure and the minor axis radius of the contact ellipse become smaller due to an increase in the axial equivalent radius, so it is considered that the pressure generation position moves closer as the axial equivalent radius increases.

[0039] Next, we will consider the change in the oil film pressure distribution and pressure generation position in the non-contact area at y = 3 mm due to changes in various parameters. The calculation conditions are the same as those for the change in various parameters in the contact area already described. Figures 15 to 17 show the change in the oil film pressure distribution and pressure generation position due to changes in average speed, load, and axial equivalent radius. In the non-contact area, the oil film pressure in the non-contact area increases with an increase in average speed, and the pressure generation position moves away. Also, the oil film pressure increases as the minimum gap becomes smaller due to an increase in load and axial equivalent radius. Also, the pressure generation position moves away due to an increase in load, and moves closer due to an increase in axial equivalent radius. This is thought to be due to various factors, such as the increase in the rolling direction equivalent radius at y = 3 mm due to an increase in the axial equivalent radius, the effect of the pressure generation distance in the contact area, and the relationship between the analysis area and the size of the contact ellipse.

[0040] Based on the above analysis results, the regression equation for the rolling viscous resistance Fr,c of the contact area is derived. The following equations (7) to (9) are used as dimensionless numbers in the regression equation for the rolling viscous resistance Fr,c of the contact area.

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[0041] Moreover, the dimensionless rolling viscous resistance Fr,c of the contact area is expressed by equation (10).

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[0042] Figures 18(a) to (d) show the influence of non-dimensional parameters on the non-dimensional viscous rolling resistance Fr,c of the contact area. The plots are analytical values ​​and the dotted lines are approximation lines. Fr,c is proportional to the 0.68 to 0.72 power of U, the 0.43 to 0.45 power of W, the 1.07 to 1.1 power of G, and the 0.57 to 0.6 power of κ. From the above results, the regression equation for the non-dimensional viscous rolling resistance Fr,c of the contact area is given by the following equation (11), where a1 to a5 are constants.

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[0043] We will consider how to determine the optimal values ​​of constants a1 to a5 and their values. When searching for the optimal values ​​of constants a1 to a5, it is necessary to give appropriate initial values ​​in advance. Figure 19 shows the initial values ​​used in searching for the optimal constant values. The initial values ​​were determined using the approximation function of Microsoft Excel.

[0044] The algorithm for finding the optimal values ​​of the constants a1 to a5 is shown below. (i) Fix a2 to a5 and search for a1 that minimizes the sum of squares of the difference between the analytical solution and the regression solution. (ii) Using a1 and a3 to a5 in (i), search for a2 that minimizes the sum of squares of the difference between the analytical solution and the regression solution. (iii) In the same manner, search for a3 to a5. (iv) If the rate of change of the sum of squares of the difference between the analytical solution and the regression solution has converged, then the process ends. If not, return to (i).

[0045] As a result of searching for the optimal value of each constant in this way, the constants shown in Figure 20 were obtained. From the above, the regression equation for the rolling viscous resistance Fr,c of the contact area is expressed as equation (12).

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[0046] Next, the regression equation for the rolling viscous resistance Fr,nc of the non-contact part is derived. The dimensionless numbers used in the regression equation for the rolling viscous resistance Fr,nc of the non-contact part are shown in the following equations (13) and (14).

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[0047] For the non-dimensional velocity parameter U', it was determined that the effect of the Young's modulus of the material in the non-contact area is extremely small, and so it was changed from the equivalent Young's modulus to the viscosity-pressure coefficient. The non-dimensional shape parameter L is a parameter that represents the size of the gap in a specific area in the non-contact area. The parameters that affect the shape of the non-contact area are the length la of the non-contact area and the equivalent radius R in the rolling direction. x , equivalent axial radius R y Since the three parameters are as follows, the dimensionless shape parameter L is defined as shown in equation (14).

[0048] In addition, the dimensionless rolling viscous resistance Fr,nc of the non-contact part is defined as shown in equation (15).

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[0049] Figures 21 and 22 show the results of analyzing the effect of non-dimensional parameters on the non-dimensional rolling viscous resistance Fr,nc of the non-contact part. Fr,nc is roughly proportional to the 0.89 to 0.94 power of U' and the 0.78 to 0.83 power of L, respectively. Even if there are changes in the contact ellipse ratio, load, and raceway width under the calculation conditions shown in Figures 4 to 6, the plots are roughly on the straight line of each U'. Therefore, it can be seen that Fr,nc can be approximated only by U' and L, and the effects of load and surface pressure can be ignored. However, since the power approximation of L can have a large error between the analytical value and the approximation line, an approximation formula with a smaller error was considered. When expressed using this approximation formula, the regression formula for the non-dimensional rolling viscous resistance Fr,nc is given as formula (16). However, b1 to b5 are constants. The dotted line in Figure 22 indicates that U'=7.8×10-7 This is an approximation using equation (16) in the above. Although the error appears to be somewhat large when L is small, the absolute value is very small, so it is unlikely to result in a large error in the torque calculation.

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[0050] We will consider how to determine the optimum values ​​of constants b1 to b5 and their values. When searching for the optimum values ​​of constants b1 to b5, it is necessary to give appropriate initial values ​​in advance. Figure 23 shows the initial values ​​used in searching for the optimum values ​​of the constants. The initial values ​​were determined using the approximation function of Microsoft Excel.

[0051] The optimum values ​​for constants b1 to b5 are also searched for in the same way as for the contact area. Note that in order to reduce constant error and improve convergence, the order in which the constants are searched is different from that of the contact area, and they are searched in the order b4 → b5 → b1 → b2 → b3. As a result of searching for the optimum values ​​of the constants using the above method, the constants shown in Figure 24 are obtained. From the above, the regression equation for the rolling viscous resistance Fr,nc of the non-contact area is expressed as equation (17).

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[0052] Based on the above explanation, the regression equation of rolling viscous resistance and the dimensionless numbers used are shown below. Also, Fig. 25 shows the range of use of the dimensionless parameters of the regression equation constructed in this embodiment. · Dimensionless number used for Fr,c

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[0053] By unifying the method of non-dimensionalizing the non-dimensional rolling viscous resistance into equation (10) and combining equations (12) and (17) with reference to equation (15), the regression equation for the non-dimensional rolling viscous resistance Fr,w at any raceway width is expressed by the following equation (18). In equation (18), the sum of the values ​​calculated based on the regression equations for the contact area and the non-contact area is taken as the estimated value of the non-dimensional rolling viscous resistance Fr,w. Note that the above "unifying the method of non-dimensionalizing into equation (10)" involves multiplying the dimensional rolling viscous resistance by α / R x 2 This means that the method of non-dimensionalization will be unified so that non-dimensionalization will be achieved by multiplying by .

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[0054] Figure 26 shows the relative error distribution between the analytical and calculated values ​​of Fr,c, and Figure 27 shows the relative error distribution between the analytical and calculated values ​​of Fr,w. The relative error is calculated as (calculated value - analytical value) / analytical value. The vertical axis shows the proportion of relative error present under all analytical conditions. For both Fr,c and Fr,w, 90% or more of all analytical conditions were within a relative error of 10%, which shows that the formulas are sufficiently accurate.

[0055] The value of rolling viscous resistance Fr,w calculated from equation (18) is the estimated value (dimensionless) of the rolling viscous resistance of the rolling bearing. x 2 By multiplying by / α, the dimensional rolling viscous resistance value of the actual bearing (f r,c +f r,nl +f r,nr) (unit N) can be estimated (f r,c +f r,nl +f r,nr =Fr,w×R x 2 / α).

[0056] When calculating the total torque of the ball bearing 1, the bearing torque can be estimated with high accuracy by calculating the sum of bearing torque factors other than the rolling viscous resistance (traction, elastic hysteresis loss, etc.) that are calculated separately.

[0057] In existing regression equations of rolling viscous resistance, including the equation in Non-Patent Document 1, the integral range is limited to the range of the major axis radius centered on the contact ellipse. As already mentioned, since a considerable amount of rolling viscous resistance occurs not only in the contact portion but also in the non-contact portion, when using existing regression equations, the calculated value of rolling viscous resistance tends to deviate from the value of the actual bearing. In contrast, in this embodiment, the calculation range in the major axis direction of the contact ellipse is set to a length selected within the range exceeding the major axis radius of the contact ellipse and not exceeding the axial equivalent radius, and the estimated value Fr,w of rolling viscous resistance is calculated from the regression equation. Therefore, the rolling viscous resistance in the non-contact portion can be included in the calculated value, and the estimated value of rolling viscous resistance can be made closer to the actual value.

[0058] If the rolling viscous resistance equation is applied to the torque calculation of a ball bearing, the integral range in the y direction (the major axis direction of the contact ellipse) is ideally set to the raceway width of the bearing. In this embodiment, a regression equation for rolling viscous resistance is determined for each of the contact portion and the non-contact portion, and the sum of the values ​​calculated based on each regression equation is set to the estimated value Fr,w of the rolling viscous resistance of the rolling bearing, so that it is possible to match the integral range with the raceway width of the bearing. This makes it possible to construct a calculation equation for calculating rolling viscous resistance with an arbitrary raceway width as the calculation region, and to estimate the rolling viscous resistance with high accuracy.

[0059] As mentioned above, when the temperature correction factor is taken into consideration, the temperature correction factor φ is added to the regression equation for the contact part in equation (18) as shown below. t Use the formula multiplied by

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[0060] Temperature correction coefficient φ t For example, the following temperature correction coefficient φ t (Mihir K. Ghosh, Raj K. Pandey, "Thermal Elastohydrodynamic Lubrication of Heavily Loaded Line Contacts - An Efficient Inlet Zone Analysis") can be used.

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[0061] The estimation of the rolling viscous resistance described above can be performed inside a system (or device). As such a system, for example, a system is assumed that includes a calculation unit that calculates an estimated value Fr,w of the rolling viscous resistance from a regression equation, with the calculation range in the major axis direction of the contact ellipse being a length selected within a range that exceeds the major axis radius of the contact ellipse and is equal to or smaller than the axial equivalent radius, and an output unit that outputs the estimated value Fr,w. As the calculation unit, for example, a CPU or the like built into a computer can be used. The output unit includes a display that displays the calculation results, as well as an output interface for transmitting an output signal to an external device such as a display.

[0062] In this estimation system, regression equations for the rolling viscous resistance are determined for each of the contact and non-contact parts, and the sum of the values ​​calculated based on each regression equation is treated as the estimated value Fr,w of the rolling viscous resistance of the rolling bearing. This processing can be performed by the calculation unit.

[0063] The estimation of the rolling viscous resistance described above can also be performed by a computer loaded with a program. In this case, the program causes the computer to perform a process of calculating an estimated value Fr,w of the rolling viscous resistance from the regression equation, with the calculation range in the major axis direction of the contact ellipse being a length selected within a range exceeding the major axis radius of the contact ellipse and not exceeding the axial equivalent radius, and a process of outputting the estimated value Fr,w.

[0064] This program determines a regression equation for the viscous rolling resistance for each of the contact and non-contact parts, and the sum of the values ​​calculated based on each regression equation can be processed to obtain the estimated value Fr,w of the viscous rolling resistance of the rolling bearing.

[0065] In either of the estimation systems and programs, the estimated value Fr,w can be calculated from equation (18). The estimated value Fr,w can also be calculated from an equation that multiplies the regression equation for the contact area by a thermal correction coefficient.

[0066] In the above explanation, a method for estimating viscous rolling resistance has been explained using a ball bearing as an example, but this method is widely applicable to rolling bearings in which the raceway surface and rolling elements are in point contact, and can also be applied to other bearings besides ball bearings, for example, self-aligning roller bearings. [Explanation of symbols]

[0067] 1. Rolling bearings (ball bearings) 11. Inner Circle 11a Inner ring raceway surface 12 Outer ring 12a Outer ring raceway 13 Rolling elements (balls) 14 Cage

Claims

1. A method for estimating rolling viscous resistance of a rolling bearing including a raceway surface and rolling elements that roll on the raceway surface, the raceway surface and the rolling elements being in point contact with each other, an oil film being formed at the point contact portion, and the raceway surface including a contact portion that contacts the rolling elements and a non-contact portion that does not contact the rolling elements, comprising: A calculation range in the major axis direction of the contact ellipse is set to a length selected within a range exceeding the major axis radius of the contact ellipse and not exceeding an equivalent radius in a direction y perpendicular to the direction x in which the rolling element rolls, and an estimated value Fr,w of the rolling viscous resistance is calculated from the regression equation; A method for estimating rolling viscous resistance, wherein the calculation range is an integral range in deriving a regression equation by EHL analysis.

2. 2. The method for estimating viscous rolling resistance according to claim 1, further comprising determining a regression equation for viscous rolling resistance for each of the contact portion and the non-contact portion, and setting the sum of values ​​calculated based on each regression equation as the estimated value Fr,w of the viscous rolling resistance of the rolling bearing.

3. The method for estimating rolling viscous resistance according to claim 2, wherein the estimated value Fr,w is calculated from the following formula: [0026] however, [0070] [0080] [0090] ##EQU00013## ##EQU14## where k is the contact ellipse ratio (k=a / b), η ０ is the normal pressure viscosity of the lubricant [Pa s], u is the average speed [m / s], E' is the equivalent Young's modulus [Pa], w is the load [N], α is the viscosity-pressure coefficient [Pa －１ ], R ｘ is the equivalent radius in the x direction [m], R ｙ represents the equivalent radius in the y direction [m], l represents the distance in the y direction from the center of the osculating ellipse to the edge of the analysis region [m], a represents the major axis radius of the osculating ellipse [m], b represents the minor axis radius of the osculating ellipse [m], the subscript "left" represents the left side of the analysis region, and the subscript "right" represents the right side of the analysis region.

4. 4. The method for estimating the rolling viscous resistance of a rolling bearing according to claim 3, further comprising multiplying the regression equation of the contact portion by a thermal correction coefficient.

5. A system for estimating rolling viscous resistance of a rolling bearing having a raceway surface and rolling elements that roll on the raceway surface, the raceway surface and the rolling elements being in point contact with each other, an oil film being formed at the point contact portion, and the raceway surface having a contact portion that contacts the rolling elements and a non-contact portion that does not contact the rolling elements, comprising: a calculation unit which calculates an estimated value Fr,w of rolling viscous resistance from a regression equation by setting a calculation range in the major axis direction of the contact ellipse to a length selected within a range exceeding the major axis radius of the contact ellipse and not exceeding an equivalent radius in a direction y perpendicular to the rolling direction x of the rolling element, and sets the calculation range as an integral range in deriving the regression equation by EHL analysis; and an output unit that outputs the estimated value Fr,w.

6. 6. The system for estimating rolling viscous resistance according to claim 5, wherein a regression equation for rolling viscous resistance is determined for each of the contact portion and the non-contact portion, and the sum of values ​​calculated based on each regression equation is set as the estimated value Fr,w of the rolling viscous resistance of the rolling bearing.

7. The rolling viscous resistance estimation system according to claim 6, wherein the estimated value Fr,w is calculated from the following formula: [0026] however, [0070] [0080] [0090] ##EQU00013## ##EQU14## where k is the contact ellipse ratio (k=a / b), η ０ is the normal pressure viscosity of the lubricant [Pa s], u is the average speed [m / s], E' is the equivalent Young's modulus [Pa], w is the load [N], α is the viscosity-pressure coefficient [Pa －１ ], R ｘ is the equivalent radius in the x direction [m], R ｙ represents the equivalent radius in the y direction [m], l represents the distance in the y direction from the center of the osculating ellipse to the edge of the analysis region [m], a represents the major axis radius of the osculating ellipse [m], b represents the minor axis radius of the osculating ellipse [m], the subscript "left" represents the left side of the analysis region, and the subscript "right" represents the right side of the analysis region.

8. The system for estimating rolling viscous resistance according to claim 7 , wherein the regression equation for the contact portion is multiplied by a thermal correction coefficient.

9. A program for estimating rolling viscous resistance of a rolling bearing having a raceway surface and rolling elements that roll on the raceway surface, the raceway surface and the rolling elements being in point contact with each other, an oil film being formed at the point contact portion, and the raceway surface having a contact portion that contacts the rolling elements and a non-contact portion that does not contact the rolling elements, comprising: On the computer, a process of calculating an estimated value Fr,w of rolling viscous resistance from a regression equation by setting a calculation range in the major axis direction of the contact ellipse to a length selected within a range exceeding the major axis radius of the contact ellipse and not exceeding an equivalent radius in a direction y perpendicular to the rolling direction x of the rolling element, and setting the calculation range as an integral range in deriving the regression equation by EHL analysis; and outputting the estimated value Fr,w.

10. 10. The program for estimating rolling viscous resistance according to claim 9, wherein a regression equation for rolling viscous resistance is determined for each of the contact portion and the non-contact portion, and the sum of values ​​calculated based on each regression equation is set as the estimated value Fr,w of the rolling viscous resistance of the rolling bearing.

11. The rolling viscous resistance estimation program according to claim 10, wherein the estimated value Fr,w is calculated from the following formula: [0026] however, [0070] [0080] [0090] ##EQU00013## ##EQU14## where k is the contact ellipse ratio (k=a / b), η ０ is the normal pressure viscosity of the lubricant [Pa s], u is the average speed [m / s], E' is the equivalent Young's modulus [Pa], w is the load [N], α is the viscosity-pressure coefficient [Pa －１ ], R ｘ is the equivalent radius in the x direction [m], R ｙ represents the equivalent radius in the y direction [m], l represents the distance in the y direction from the center of the osculating ellipse to the edge of the analysis region [m], a represents the major axis radius of the osculating ellipse [m], b represents the minor axis radius of the osculating ellipse [m], the subscript "left" represents the left side of the analysis region, and the subscript "right" represents the right side of the analysis region.

12. The rolling viscous resistance estimation program according to claim 11, further comprising multiplying the regression equation of the contact portion by a thermal correction coefficient.

13. A computer-readable recording medium having the estimation program according to any one of claims 9 to 12 recorded thereon.

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