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

Abstract

A rolling bearing 1 comprises raceway surfaces 11a, 12a, and spherical rolling elements 13 that roll on the raceway surfaces. In the rolling bearing 1, the raceway surfaces and each of the rolling elements are brought into point contact with each other, an oil film is formed at a point-contact portion, and a contact area where the raceway surfaces 11a, 12a come into contact with the rolling element 13 and a non-contact area where the raceway surfaces do not come into contact with the rolling element are provided. An estimated value Frs,w of rolling viscous resistance is calculated from a regression equation, where a calculation range of a contact ellipse in the major axis direction is set to a length selected within a range exceeding the major axis diameter of the contact ellipse and less than or equal to the equivalent diameter in a direction y orthogonal to a rolling direction x of the rolling element, and a negative calculation range of the contact ellipse in the minor axis direction is set to a length selected within a range exceeding the minor axis radius of the contact ellipse and less than or equal to the equivalent radius in the rolling direction x.

Description

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

[0001] The present invention relates to a method, a system, and a program for estimating rolling viscous resistance in a rolling bearing, particularly rolling viscous resistance under starved lubrication, and to a computer-readable recording medium storing the program.

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

[0003] Estimating the magnitude of rolling viscous resistance is an important issue in the design and performance evaluation of rolling bearings. Patent Document 1 below discloses a method for estimating rolling viscous resistance with high accuracy, in which the calculation is performed with an arbitrary axial calculation range that exceeds the major axis diameter of the contact ellipse and is equal to or less than the axial equivalent radius.

[0004] On the other hand, with the method of Patent Document 1, it is expected that during high-speed rotation or when the amount of lubricating oil is small, a large deviation will occur between the calculated value and the actually measured value of the rolling viscous resistance generated in the bearing. This is considered to be mainly caused by the decrease in rolling viscous resistance due to the reduction of the inlet meniscus distance. The inlet meniscus distance refers to the distance between the contact point and the liquid meniscus (liquid interface) formed when lubricating oil enters the contact area between the rolling element and the raceway surface. The state in which rolling viscous resistance decreases along with such a reduction in inlet meniscus distance is generally called a starved lubrication (starvation) state. It is known that rolling viscous resistance under starved lubrication can be expressed numerically mainly by changing the calculation range in the circumferential direction (Non-Patent Document 2).

[0005] Japanese Patent Publication No. 2025-17692

[0006] T. Nogi, H. Shiomi and N. Matsuoka, Starved Elastohydrodynamic Lubrication With Reflow in Elliptical Contacts, J. Tribol., 140,1(2018), 11501

[0007] However, the analysis method in Patent Document 1 calculates rolling viscous resistance under sufficient lubrication and does not address rolling viscous resistance under depleted lubrication. The calculation formula described in Non-Patent Document 2 does not specify the axial calculation domain, and is therefore thought to deviate from the actual rolling viscous resistance value of bearings. Thus, a practical method for calculating rolling viscous resistance under depleted lubrication has not yet been established. If a method for estimating rolling viscous resistance under depleted lubrication can be established, it will be possible to improve the accuracy of estimating the rolling viscous resistance of bearings by using it in conjunction with the method for estimating rolling viscous resistance under sufficient lubrication disclosed in Patent Document 1.

[0008] Therefore, the present invention aims to provide a method for estimating rolling viscous resistance, an estimation system, an estimation program, and a computer-readable recording medium on which the program is stored, which can estimate the rolling viscous resistance of a real bearing with high accuracy even under depleted lubrication conditions.

[0009] To solve the above problems, the present invention provides a method for estimating rolling viscous resistance in a rolling bearing comprising a raceway surface and spherical rolling elements that roll on the raceway surface, wherein the raceway surface and the rolling elements are in point contact, an oil film is formed at the point contact portion, and the raceway surface comprises a contact portion in contact with the rolling elements and a non-contact portion that does not contact the rolling elements. The method for estimating the rolling viscous resistance in such a rolling bearing is characterized in that the calculation range in the direction of the major axis of the contact ellipse is set to a length selected within a range that exceeds the major axis diameter of the contact ellipse and is less than or equal to the equivalent diameter of the direction y perpendicular to the direction x in which the rolling elements roll, and the negative calculation range in the direction of the minor axis of the contact ellipse is set to a length selected within a range that exceeds the minor axis radius of the contact ellipse and is less than or equal to the equivalent radius of the rolling direction x, and the estimated value of rolling viscous resistance Frs,w is calculated from a regression equation.

[0010] In this estimation method, the regression equations Fr,c and Fr,nc for rolling viscous resistance and the starvation coefficient equation φ are used for the contact portion and the non-contact portion, respectively. s,c , φ s,nc The regression equation Fr,c for the rolling viscous resistance of the contact portion is defined, and the starvation coefficient equation φ for the contact portion is defined. s,c The result obtained by multiplying by the regression equation Fr,nc for the rolling viscous resistance of the non-contact portion, and the starvation coefficient equation φ for the non-contact portion. s,nc The sum of the result of multiplying by and the result of can be given as the estimated rolling viscous resistance Frs,w.

[0011] In this estimation method, the estimated value Frs,w can be calculated from the following formula: however, Here, k is the contact elliptic ratio (k = a / b), η ０ ∫ is the viscosity of the lubricating oil at atmospheric pressure [Pa·s], u is the average velocity [m / s], E' is the equivalent Young's modulus [Pa], w is the load [N], and α is the viscosity-pressure coefficient [Pa] －１ ], R ｘ R is the equivalent radius in the x direction [m]. ｙ is the equivalent radius in the y direction [m], l is the distance in the y direction from the center of the contact ellipse to the edge of the analysis domain [m], a is the major axis radius of the contact ellipse [m], b is the minor axis radius of the contact ellipse [m], the subscript left represents the left side of the analysis domain, and the subscript right represents the right side of the analysis domain.

[0012] In this estimation method, the regression equation of the contact area can be multiplied by a thermal correction coefficient.

[0013] Further, a rolling viscous resistance estimation system according to the present invention is for estimating rolling viscous resistance of a rolling bearing comprising a raceway surface and a spherical rolling element that rolls on the raceway surface, wherein the raceway surface and the rolling element are in point contact, an oil film is formed at the point contact portion, and the raceway surface comprises a contact portion that contacts the rolling element and a non-contact portion that does not contact the rolling element, wherein a calculation range in a major axis direction of a contact ellipse is set to a length selected within a range exceeding a major axis diameter of the contact ellipse and not more than an equivalent diameter in a direction y orthogonal to a rolling direction x in which the rolling element rolls, a negative calculation range in a minor axis direction of the contact ellipse is set to a length selected within a range exceeding a minor axis radius of the contact ellipse and not more than an equivalent radius in the rolling direction x, and the system comprises: a calculation unit that calculates an estimated value Frs,w of the rolling viscous resistance from a regression equation; and an output unit that outputs the estimated value Frs,w.

[0014] In this estimation system, regression equations Fr,c, Fr,nc of rolling viscous resistance and starvation coefficient formulas φ s,c , φ s,nc are defined for each of the contact portion and the non-contact portion, and a sum of a product obtained by multiplying the regression equation Fr,c of rolling viscous resistance for the contact portion by the starvation coefficient formula φ s,c for the contact portion and a product obtained by multiplying the regression equation Fr,nc of rolling viscous resistance for the non-contact portion by the starvation coefficient formula φ s,nc for the non-contact portion can be used as the estimated value Frs,w of the rolling viscous resistance.

[0015] In this estimation system, the estimated value Frs,w can be calculated from the following formula: Provided that , k is a contact ellipse ratio (k = a / b), η ０ is atmospheric viscosity of lubricating oil [Pa·s], u is average speed [m / s], E' is equivalent Young's modulus [Pa], w is load [N], α is viscosity-pressure coefficient [Pa －１], R ｘ R is the equivalent radius in the x direction [m]. ｙ is the equivalent radius in the y direction [m], l is the distance in the y direction from the center of the contact ellipse to the edge of the analysis domain [m], a is the major axis radius of the contact ellipse [m], b is the minor axis radius of the contact ellipse [m], the subscript left represents the left side of the analysis domain, and the subscript right represents the right side of the analysis domain.

[0016] In this estimation system, the regression equation for the contact area can be multiplied by a thermal correction coefficient.

[0017] Furthermore, the rolling viscous resistance estimation program according to the present invention is a rolling bearing comprising a raceway surface and spherical rolling elements that roll on the raceway surface, wherein the raceway surface and the rolling elements are in point contact, an oil film is formed at the point contact portion, and the raceway surface comprises a contact portion in contact with the rolling elements and a non-contact portion that does not contact the rolling elements, and the program for estimating the rolling viscous resistance of the rolling bearing is characterized in that the computer is instructed to perform the following processes: calculate an estimated value Frs,w of rolling viscous resistance from a regression equation, and output the estimated value Frs,w, by setting the calculation range in the direction of the major axis of the contact ellipse to a length selected within a range that exceeds the major axis diameter of the contact ellipse and is less than or equal to the equivalent diameter of the direction y perpendicular to the direction x in which the rolling elements roll, and set the negative calculation range in the direction of the minor axis of the contact ellipse to a length selected within a range that exceeds the minor axis radius of the contact ellipse and is less than or equal to the equivalent radius of the rolling direction x.

[0018] In this estimation program, the regression equations Fr,c and Fr,nc for rolling viscous resistance and the starvation coefficient equation φ are used for the contact portion and the non-contact portion, respectively. s,c , φ s,nc The regression equation Fr,c for the rolling viscous resistance of the contact portion is defined, and the starvation coefficient equation φ for the contact portion is defined. s,c The result obtained by multiplying by the regression equation Fr,nc for the rolling viscous resistance of the non-contact portion, and the starvation coefficient equation φ for the non-contact portion. s,nc The sum of the result of multiplying by and the result of can be given as the estimated rolling viscous resistance Frs,w.

[0019] Furthermore, in this estimation program, the estimated values ​​Frs,w can be calculated from the following formula: however, Here, k is the contact elliptic ratio (k = a / b), η ０ ∫ is the viscosity of the lubricating oil at atmospheric pressure [Pa·s], u is the average velocity [m / s], E' is the equivalent Young's modulus [Pa], w is the load [N], and α is the viscosity-pressure coefficient [Pa] －１ ], R ｘ R is the equivalent radius in the x direction [m]. ｙ is the equivalent radius in the y direction [m], l is the distance in the y direction from the center of the contact ellipse to the edge of the analysis domain [m], a is the major axis radius of the contact ellipse [m], b is the minor axis radius of the contact ellipse [m], the subscript left represents the left side of the analysis domain, and the subscript right represents the right side of the analysis domain.

[0020] In this estimation program, the regression equation for the contact area can be multiplied by a thermal correction coefficient.

[0021] The estimation program described above may be recorded on a computer-readable recording medium.

[0022] According to the present invention, the rolling viscous resistance of a real bearing can be estimated with high accuracy even under depleted lubrication conditions.

[0023] This is a cross-sectional view of the ball bearing according to this embodiment. This is an unfolded view of the raceway surface showing the analysis range and integration range used in EHL analysis. This is a table showing the calculation control parameters. This is a table showing the calculation conditions (bearing model 6320). This is a table showing the calculation conditions (bearing model 6210). This is a table showing the calculation conditions (bearing model 6004). This is a table showing the physical properties of the materials used in the calculation. This is a table showing the physical properties of the lubricating oil used in the calculation. This is a graph showing the relationship between the analytical value and dimensionless number of the starvation coefficient at the contact area. This is a graph showing the relationship between the analytical value and dimensionless number of the starvation coefficient at the non-contact area. This is a graph comparing the calculated value and analytical value of the starvation coefficient at the contact area. This is a graph comparing the calculated value and analytical value of the starvation coefficient at the non-contact area.

[0024] Embodiments of the present invention will be described in detail below with reference to the drawings.

[0025] Figure 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 circumference, an outer ring 12 having an outer ring raceway surface 12a on its inner circumference, 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 that holds the rolling elements 13 at predetermined intervals in the bearing circumferential direction. An oil film is formed between the raceway surfaces 11a, 12a of the ball bearing and the rolling elements 13 by lubricating oil (including grease) supplied into the bearing.

[0026] In the ball bearing 1, the inner ring raceway surface 11a and the outer ring raceway surface 12a are machined so that their radii of curvature are slightly larger than the radius of curvature of the ball 13. Therefore, the inner ring raceway surface 11a and the ball 13, and the outer ring raceway surface 12a and the ball 13, all make point contact. The point contact area takes on an elliptical shape (called a contact ellipse) due to the elastic deformation of the raceway surface and the ball. The size of the contact ellipse can be determined by Hertz's elastic contact theory or by contact analysis using the semi-infinite elastic body approximation.

[0027] The present invention enables the highly accurate estimation of the rolling viscous resistance value within a bearing, particularly the rolling viscous resistance value under depleted lubrication conditions, from a regression equation. This value is crucial information for the design or performance evaluation of rolling bearings as described above. The derivation process of the regression equation is explained below.

[0028] The following are definitions of the symbols used in the following explanation: a: Major axis radius of a tangential ellipse [m] a １ ~a ５ : Constants in the dimensionless rolling viscous resistance equation b: Minor axis radius of the contact ellipse [m] b １ ~b ５ : Constants in the dimensionless rolling viscous resistance equation E': Equivalent Young's modulus [Pa] F ｒ　 : Dimensionless rolling viscous resistance (sufficient lubrication) F ｒｓ : Dimensionless rolling viscous resistance (depletion lubrication) f ｒ　 : Rolling viscous resistance (sufficient lubrication) [N] f ｒｓ: Rolling viscous resistance (depletion lubrication) [N] G: Dimensionless material parameter h ∞ : gap [m] h ｃ : Central 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 contact ellipse to the edge of the analysis region [m] p: Pressure [Pa] R ｘ : Equivalent radius in the x-direction [m] R ｙ : Equivalent radius in the y direction [m] U: Dimensionless velocity parameter U': Dimensionless velocity parameter of the non-contact part u: Average velocity [m / s] W: Dimensionless load parameter w: Load [N] x: Coordinate in the rolling direction [m] y: Coordinate in the direction perpendicular to the rolling direction [m] α: Viscosity-pressure coefficient [Pa －１ The specified φ ｔ : Thermal correction coefficient η ０ Viscosity of lubricating oil at atmospheric pressure [Pa·s]

[0029] [Subscripts] c: Contact area nc: Non-contact area left: Left side of the analysis area right: Right side of the analysis area nl: Left side of the non-contact area nr: Right side of the non-contact area w: Entire analysis area

[0030] Note that the equivalent radius R is calculated by taking the radius of the rolling element (ball) as R. １ Let R be the radius of the orbital plane. ２ Therefore, R = R １ ×R ２ / (R １ +R ２ The equivalent radius R in the x-direction is obtained from the above. ｘ When calculating R ２ The radius of curvature of the orbital surface in the x-direction is used, and the equivalent radius R in the y-direction is used. ｙ When calculating R ２ The radius of curvature of the track surface in the y-direction is used as the equivalent radius. The equivalent diameter is twice the value of the equivalent radius. The equivalent Young's modulus E' is calculated by taking the Young's modulus of the ball and multiplying it by E. １ , Poisson's ratio ν １ Let the elastic modulus of the raceway be E ２ , Poisson's ratio ν ２ Therefore, 2 / E' = (1 - ν １ ２) / E １ + (1-ν) ２ ２ ) / E ２ It is required from.

[0031] In the following, the oil film pressure and oil film thickness distribution within the analysis domain are determined by isothermal EHL (Elastohydrodynamic Lubrication) analysis. This analysis uses the Roelands equation as the viscosity-pressure relationship and the Wu-Klaus-Duda equation as the viscosity-pressure coefficient. A multilevel method is used as the numerical analysis algorithm.

[0032] Figure 2 shows the analysis range and integration range used in EHL analysis, and is an unfolded view of one of the bearing raceway surfaces (e.g., the inner raceway surface 11a) laid out on a plane. As shown in Figure 2, a contact ellipse is formed on the raceway surface, which is the contact area with the rolling elements. In Figure 2, the x-direction (the minor axis direction of the contact ellipse) is the direction in which the rolling elements roll, and the y-direction (the major axis direction of the contact ellipse), which is perpendicular to the x-direction, corresponds to the axial direction in radial rolling bearings and the radial direction in thrust rolling bearings. The method for calculating rolling viscous resistance will be explained below.

[0033] Rolling viscous resistance f r This is the shear resistance of the lubricating oil due to the Poiseuille flow in the rolling direction, It is represented as follows.

[0034] In order to calculate the rolling viscous resistance using equation (1), it is necessary to define the integration range. In this embodiment, the integration range in the x-direction is set to the range from -x' to 2b. The negative calculation range in the x-direction exceeds the minor axis radius b of the contact ellipse and is less than or equal to the equivalent radius of the rolling direction x. Regarding the y-direction, in the analysis of Non-Patent Literature 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 range that exceeds the major axis diameter of the contact ellipse and is less than or equal to the equivalent diameter of the direction y which is perpendicular to the rolling direction x, thus avoiding such a problem by making the integration range in the y-direction a constant region independent of the size of the contact ellipse.

[0035] The rolling viscous resistance is calculated by dividing it into contact and non-contact areas, as shown in equations (2) to (5) below, and constructing separate regression equations for the contact and non-contact areas. Within the analysis domain, the rectangular area between the two vertices on the major axis side of the contact ellipse (with the center of the contact ellipse as the origin, in the range of -x' to +2b and ±a) becomes the contact area, and the two rectangular areas flanking the contact area from both axial sides become the non-contact areas. Note that x' is greater than the minor axis radius b of the contact ellipse and can be set to any length within the range of the equivalent radius Rx in the x direction. Dividing the analysis range into contact and non-contact areas offers advantages such as being able to consider the effect of shear heat generation of the lubricating oil that occurs only around the entrance of the contact point only in the contact area, and enabling calculations even when there is no contact ellipse in the center of the raceway surface. In this embodiment, the sum of the y-direction width dimensions of the contact area and the two non-contact areas is equal to the width dimension of the raceway surface. In other words, the analysis range in the y direction coincides with the width dimension of the raceway surface.

[0036] The following describes the numerical analysis (EHL analysis) of rolling viscous resistance. In this analysis, a parameter study of rolling viscous resistance was first performed under the calculation control parameters shown in Figure 3. The calculation conditions are shown in Figures 4 to 6. Note that the ball radius and pitch circle radius in Figures 4 to 6 correspond to bearing models 6320, 6210, and 6004, respectively. The physical properties of the materials and lubricants used in the calculations are shown in Figures 7 and 8.

[0037] Figure 9 shows the relationship between the analytical value of the starvation coefficient and the dimensionless number at the contact area, and Figure 10 shows the relationship between the analytical value of the starvation coefficient and the dimensionless number at the non-contact area. Both relationships were obtained by EHL analysis. The horizontal axis of both figures represents the dimensionless number Φ for the contact area and the non-contact area. ｃ , Φ ｎｃ These are determined by the following equations (6) and (9).

[0038] However, H in equation (8) c,frepresents the average oil film thickness under sufficient lubrication. Furthermore, U is a dimensionless velocity parameter, W is a dimensionless load parameter, and G is a dimensionless material parameter, while U' in equation (9) is the dimensionless velocity parameter of the non-contact area. These dimensionless parameters are expressed by the following equations, respectively.

[0039] From the analysis results shown in Figures 9 and 10, the starvation coefficient formula φ in the contact and non-contact areas is obtained. s,c , Φ s,nc (Equations (15) and (16)) are obtained. This starvation coefficient equation φ s,c , φ s,nc The dimensionless number Φ used in both coefficient equations is c , Φ nc The following is also shown.

[0040] Figure 11 shows a comparison of the calculated (calc.) and analytical (Numerical) starvation coefficients for the contact area, and Figure 12 shows a comparison of the calculated (calc.) and analytical (Numerical) starvation coefficients for the non-contact area. In both figures, the calculated and analytical values ​​are in general agreement.

[0041] In the present invention, the starvation coefficient formula obtained by the above procedure is combined with the formula for estimating rolling viscous resistance under sufficient lubrication (Patent Document 1). The formula for rolling viscous resistance under sufficient lubrication is, at the contact point, It is represented as follows, and in the non-contact area, It is expressed as follows. However, the dimensionless shape parameter L is expressed by the following equation (19). The dimensionless shape parameter L is a parameter that represents the size of the gap in a specific region in the non-contact part. Since the parameters that affect the shape of the non-contact part are the length of the non-contact part l-a, the equivalent radius Rx in the rolling direction, and the equivalent radius Ry in the axial direction, the dimensionless shape parameter L was defined as shown in equation (19).

[0042] Thus, the regression equation F for rolling viscous resistance is obtained for both the contact area and the non-contact area. r,c F r,nc and starvation coefficient formula φ s,c , φ s,nc After determining this, the regression equation F for the rolling viscous resistance of the contact area is calculated. r,c Starvation coefficient formula φ of the contact area s,c The result obtained by multiplying by and the regression equation F of the rolling viscous resistance of the non-contact area r,nc The starvation coefficient formula for the non-contact portion is φ s,nc The sum of the product of and is denoted as the estimated rolling viscous resistance Frs,w under depleted lubrication. r,nc = F r,nl = F r,nr as, φ s,nc = φ s.nl＝ = φ s,nr Therefore, the estimated rolling viscous resistance Frs,w under depleted lubrication can be expressed by the following equation.

[0043] The dimensionless rolling viscous resistance Frs,w calculated from this formula is the estimated (dimensionless) rolling viscous resistance of the rolling bearing under depleted lubrication conditions. ｘ By multiplying by 2 / α, it becomes possible to estimate the dimensional rolling viscous resistance value (frs,c + frs,nl + frs,nr) (unit: N) of a real bearing under depleted lubrication conditions.

[0044] When calculating the total torque of ball bearing 1, the bearing torque can be estimated with high accuracy by calculating the sum of bearing torque factors other than rolling viscous resistance (such as traction and elastic hysteresis loss) that are calculated separately.

[0045] In Patent Document 1, which considers only sufficient lubrication, the calculation range in the direction of the minor axis of the contact ellipse is limited to +2b to -30b. This is because the rolling viscous resistance is generally saturated outside the range of +2b to -30b, so the boundary between sufficient lubrication and depleted lubrication is considered to be around -30b in the x direction. Therefore, if the estimation formula of Patent Document 1 is used under depleted lubrication conditions, it is expected that the calculated value of rolling viscous resistance will deviate from the actual bearing value. In contrast, in this embodiment, the negative calculation range in the x direction (the direction of the minor axis of the contact ellipse) is selected as a length that exceeds the radius of the minor axis of the contact ellipse and is 30b or less (less than or equal to the equivalent radius in the x direction), and the estimated value of rolling viscous resistance Frs,w is calculated from the regression equation. Therefore, even under depleted lubrication conditions, the estimated value of rolling viscous resistance can be brought closer to the actual value.

[0046] The negative integral range in the x-direction (-x') should preferably be determined to coincide with the inlet meniscus distance, which is derived by observation or estimation during bearing operation.

[0047] If the rolling viscous resistance formula is applied to the torque calculation of a ball bearing, it is ideal that the integration range in the y-direction (the major axis direction of the contact ellipse) be the width of the bearing raceway. In this embodiment, regression equations for the rolling viscous resistance under depleted lubrication are defined for both the contact and non-contact portions, and the sum of the values ​​calculated based on each regression equation is taken as the estimated rolling viscous resistance Frs,w of the rolling bearing. This makes it possible to match the integration range with the width of the bearing raceway. As a result, it becomes possible to construct a calculation formula for determining rolling viscous resistance with an arbitrary raceway width as the calculation domain, and rolling viscous resistance can be estimated with high accuracy.

[0048] Furthermore, when applying the present invention to ball bearings, especially ball bearings that rotate at high speed while using low-viscosity oil, the effect of heat generation due to shear of the lubricating oil present in and near the contact area becomes significant, and this effect of heat generation should be taken into consideration. Therefore, in such cases, the heat correction coefficient φ should be applied only to Fr and c, as shown in equation (21) below. ｔ It is preferable to multiply by . The reason for specifying "only for Fr,c" is that shear heat generation of the lubricating oil occurs only around the entrance of the contact area.

[0049] Temperature correction coefficient φ ｔ As the temperature correction coefficient φ, various known formulas can be used. For example, the temperature correction coefficient φ represented by the following formula ｔ (Mihir K. Ghosh, Raj K. Pandey, "Thermal Elastohydrodynamic Lubrication of Heavily Loaded Line Contacts - An Efficient Inlet Zone Analysis") can be used. In the above formula, W represents a dimensionless load parameter, Q represents a thermal load parameter, and S represents a sliding parameter.

[0050] The estimation of rolling viscous resistance under starved lubrication described above can be performed inside a system (or apparatus). As such a system, for example, it is assumed that the system comprises: an arithmetic unit that calculates an estimated value Frs,w of rolling viscous resistance from a regression equation, wherein the calculation range in the major axis direction of the contact ellipse is a length selected within a range that exceeds the major axis diameter of the contact ellipse and is equal to or less than the equivalent diameter in the direction y orthogonal to the rolling direction x of the rolling element, and the negative calculation range in the minor axis direction of the contact ellipse is a length selected within a range that exceeds the minor axis radius of the contact ellipse and is equal to or less than the equivalent radius in the rolling direction x of the rolling element; and an output unit that outputs the estimated value Frs,w. As the arithmetic unit, for example, a CPU built in a computer can be used. The output unit includes not only a display for displaying calculation results, but also an output interface for transmitting output signals to an external device such as a display.

[0051] In this estimation system, for the contact part and the non-contact part respectively, the regression equations Fr,c, Fr,nc for rolling viscous resistance and the starvation coefficient formulas φ s,c , φ s,nc are determined, and the sum of a product obtained by multiplying the regression equation Fr,c for rolling viscous resistance of the contact part by the starvation coefficient formula φ s,c of the contact part and a product obtained by multiplying the regression equation Fr,nc for rolling viscous resistance of the non-contact part by the starvation coefficient formula φ s,nc of the non-contact part is treated as the estimated value Frs,w of the rolling viscous resistance of the rolling bearing. This processing can be performed by the arithmetic unit.

[0052] Furthermore, the estimation of rolling viscous resistance described above can also be performed on a computer that has loaded the program. In this case, the program will instruct the computer to perform the following processes: calculate the estimated rolling viscous resistance Frs,w from a regression equation, and output the estimated Frs,w, by selecting a length in the calculation range along the major axis of the contact ellipse that exceeds the major axis diameter of the contact ellipse and is less than or equal to the equivalent diameter in the direction y perpendicular to the direction x in which the rolling element rolls, and selecting a length in the negative calculation range along the minor axis of the contact ellipse that exceeds the minor axis radius of the contact ellipse and is less than or equal to the equivalent radius in the direction x in which the rolling element rolls.

[0053] This program uses regression equations Fr,c and Fr,nc for rolling viscous resistance and a starvation coefficient equation φ for the contact and non-contact areas, respectively. s,c , φ s,nc The following equations are established: the regression equation Fr,c for the rolling viscous resistance of the contact area, and the starvation coefficient equation φ for the contact area. s,c The result obtained by multiplying by the regression equation Fr,nc for the rolling viscous resistance of the non-contact area, and the starvation coefficient equation φ for the non-contact area. s,nc The process can be performed to obtain the estimated rolling viscous resistance of the rolling bearing, Frs,w, by multiplying it by a certain factor and then summing the result.

[0054] In both this estimation system and estimation program, the estimated value Frs,w can be calculated from equation (23). The estimated value Frs,w can also be calculated from an equation obtained by multiplying the regression equation of the contact area by a thermal correction coefficient.

[0055] In the above explanation, a method for estimating rolling viscous resistance was described using a ball bearing as an example. However, this method is widely applicable to rolling bearings in which the raceway surface and rolling elements make point contact, and can be applied to other types of bearings, such as self-aligning roller bearings.

[0056] 1 Rolling bearing (ball bearing) 11 Inner ring 11a Inner ring raceway surface 12 Outer ring 12a Outer ring raceway surface 13 Rolling elements (balls) 14 Cage

Claims

1. A method for estimating the rolling viscous resistance of a rolling bearing comprising a raceway surface and spherical rolling elements that roll on the raceway surface, wherein the raceway surface and the rolling elements are in point contact, an oil film is formed at the point contact portion, and the raceway surface comprises a contact portion in contact with the rolling elements and a non-contact portion that does not contact the rolling elements, characterized in that the calculation range in the direction of the major axis of the contact ellipse is set to a length selected within a range that exceeds the major axis diameter of the contact ellipse and is less than or equal to the equivalent diameter of the direction y perpendicular to the direction x in which the rolling elements roll, and the negative calculation range in the direction of the minor axis of the contact ellipse is set to a length selected within a range that exceeds the minor axis radius of the contact ellipse and is less than or equal to the equivalent radius of the rolling direction x, and the estimated value of the rolling viscous resistance Frs,w is calculated from a regression equation.

2. Regression equations Fr,c and Fr,nc for rolling viscous resistance and starvation coefficient equation φ for the contact portion and the non-contact portion, respectively. s,c , φ s,nc The regression equation Fr,c for the rolling viscous resistance of the contact portion is defined, and the starvation coefficient equation φ for the contact portion is defined. s,c The result obtained by multiplying by the regression equation Fr,nc for the rolling viscous resistance of the non-contact portion, and the starvation coefficient equation φ for the non-contact portion. s,nc The method for estimating rolling viscous resistance according to claim 1, wherein the sum of the result of multiplying by and the estimated rolling viscous resistance Frs,w is used.

3. The method for estimating rolling viscous resistance according to claim 2, wherein the estimated value Frs,w is calculated from the following formula: however, Here, k is the contact elliptic ratio (k = a / b), η ０ ∫ is the viscosity of the lubricating oil at atmospheric pressure [Pa·s], u is the average velocity [m / s], E' is the equivalent Young's modulus [Pa], w is the load [N], and α is the viscosity-pressure coefficient [Pa] －１ ], R ｘ R is the equivalent radius in the x direction [m]. ｙ is the equivalent radius in the y direction [m], l is the distance in the y direction from the center of the contact ellipse to the edge of the analysis domain [m], a is the major axis radius of the contact ellipse [m], b is the minor axis radius of the contact ellipse [m], the subscript left represents the left side of the analysis domain, and the subscript right represents the right side of the analysis domain.

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

5. A rolling bearing comprising a raceway surface and spherical rolling elements that roll on the raceway surface, wherein the raceway surface and the rolling elements are in point contact, an oil film is formed at the point contact portion, and the raceway surface comprises a contact portion in contact with the rolling elements and a non-contact portion that does not contact the rolling elements, the rolling bearing comprising: a calculation unit that calculates an estimated value Frs,w of rolling viscous resistance from a regression equation, wherein the calculation range in the direction of the major axis of the contact ellipse is a length selected within a range that exceeds the major axis diameter of the contact ellipse and is less than or equal to the equivalent diameter of the direction y perpendicular to the direction x in which the rolling elements roll, and the negative calculation range in the direction of the minor axis of the contact ellipse is a length selected within a range that exceeds the minor axis radius of the contact ellipse and is less than or equal to the equivalent radius of the rolling direction x; and an output unit that outputs the estimated value Frs,w.

6. A regression expressions Fr,c and Fr,nc of rolling viscous resistance and a starvation coefficient expression φ for each of said contact portion and said non-contact portion s,c , φ s,nc are defined, and the sum of a product obtained by multiplying the regression expression Fr,c of rolling viscous resistance of said contact portion by the starvation coefficient expression φ of said contact portion s,c and a product obtained by multiplying the regression expression Fr,nc of rolling viscous resistance of said non-contact portion by the starvation coefficient expression φ of said non-contact portion s,nc is taken as the estimated rolling viscous resistance Frs,w, the rolling viscous resistance estimating system according to claim 5.

7. The rolling viscous resistance estimation system according to claim 6, wherein the estimated value Frs,w is calculated from the following formula: however, Here, k is the contact elliptic ratio (k = a / b), η ０ ∫ is the viscosity of the lubricating oil at atmospheric pressure [Pa·s], u is the average velocity [m / s], E' is the equivalent Young's modulus [Pa], w is the load [N], and α is the viscosity-pressure coefficient [Pa] －１ ], R ｘ R is the equivalent radius in the x direction [m]. ｙ is the equivalent radius in the y direction [m], l is the distance in the y direction from the center of the contact ellipse to the edge of the analysis domain [m], a is the major axis radius of the contact ellipse [m], b is the minor axis radius of the contact ellipse [m], the subscript left represents the left side of the analysis domain, and the subscript right represents the right side of the analysis domain.

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

9. A rolling bearing comprising a raceway surface and spherical rolling elements that roll on the raceway surface, wherein the raceway surface and the rolling elements are in point contact, an oil film is formed at the point contact portion, and the raceway surface comprises a contact portion in contact with the rolling elements and a non-contact portion that does not contact the rolling elements, wherein the program for estimating the rolling viscous resistance of the rolling bearing is characterized by causing the computer to perform the following processes: calculate an estimated value of rolling viscous resistance Frs,w from a regression equation, with the calculation range in the direction of the major axis of the contact ellipse being a length selected within a range that exceeds the major axis diameter of the contact ellipse and is less than or equal to the equivalent diameter of the direction y perpendicular to the direction x in which the rolling elements roll, and set the negative calculation range in the direction of the minor axis of the contact ellipse to be a length selected within a range that exceeds the minor axis radius of the contact ellipse and is less than or equal to the equivalent radius of the rolling direction x; and output the estimated value Frs,w.

10. Regression equations Fr,c and Fr,nc for rolling viscous resistance and starvation coefficient equation φ for the contact portion and the non-contact portion, respectively. s,c , φ s,nc The regression equation Fr,c for the rolling viscous resistance of the contact portion is defined, and the starvation coefficient equation φ for the contact portion is defined. s,c The result obtained by multiplying by the regression equation Fr,nc for the rolling viscous resistance of the non-contact portion, and the starvation coefficient equation φ for the non-contact portion. s,nc The rolling viscous resistance estimation program according to claim 9, wherein the sum of the result of multiplying by and the estimated rolling viscous resistance Frs,w is used.

11. The rolling viscous resistance estimation program according to claim 10, which calculates the estimated value Frs,w from the following formula: however, Here, k is the contact elliptic ratio (k = a / b), η ０ ∫ is the viscosity of the lubricating oil at atmospheric pressure [Pa·s], u is the average velocity [m / s], E' is the equivalent Young's modulus [Pa], w is the load [N], and α is the viscosity-pressure coefficient [Pa] －１ ], R ｘ R is the equivalent radius in the x direction [m]. ｙ is the equivalent radius in the y direction [m], l is the distance in the y direction from the center of the contact ellipse to the edge of the analysis domain [m], a is the major axis radius of the contact ellipse [m], b is the minor axis radius of the contact ellipse [m], the subscript left represents the left side of the analysis domain, and the subscript right represents the right side of the analysis domain.

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

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

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