JP4980031B2 - Rolling bearing crowning design method - Google Patents

Description

  The present invention relates to a crowning design method for forming at least one of an inner ring raceway surface, an outer ring raceway surface and a roller rolling surface of a roller bearing.

  Conventionally, in roller bearings, the outer ring raceway surface, the inner ring raceway surface or the roller rolling surface is formed with a crowning to prevent edge load from occurring at the end of the contact portion between the raceway surface and the rolling surface. The fatigue life is extended.

  The shape of the crowning formed on the roller bearing may be a curve expressed by a logarithmic function, and the one proposed by Lundberg is widely known as the crowning curve expressed by this logarithmic function ( Non-patent document 1: Lundberg, G., Elastic Contact Between Two Semi-Infinite Bodies, Forschung auf den Gebiete des Ingenieurwesen, 5 (1939), pp. 201-211. As a practical improvement of this crowning curve, the Johns-Gohar equation (Non-Patent Document 2: Johns, PM and Gohar, R., Roller bearings under radial and eccentric loads, Tribology International, 14 (1981), pp.131-136) is known.

  However, the crowning curve according to the Johns-Gohar equation has a tendency that the contact pressure between the raceway surface and the rolling surface at the end of the crowning formation portion is somewhat high, and edge load prevention is insufficient.

Therefore, in the past, the present inventor has proposed a crowning curve in which a new design parameter is introduced into the Johns-Gohar equation in order to make the contact pressure between the raceway surface and the rolling surface uniform (patent). Reference 1: Japanese Patent Laid-Open No. 2006-52790). In designing a roller bearing to which this crowning curve is applied, an initial value search range and a division number of the design parameter are determined, and an objective function is obtained for a combination of design parameters obtained by the initial value search range and the division number. A combination of design parameters that optimizes the objective function is adopted as an initial value, and further optimized by a mathematical optimization method to identify a crowning curve and design a crowning of a roller bearing.
Lundberg, G., Elastic Contact Between Two Semi-Infinite Bodies, Forschung auf den Gebiete des Ingenieurwesen, 5 (1939), pp. 201-211. Johns, PM and Gohar, R., Roller bearings under radial and eccentric loads, Tribology International, 14 (1981), pp.131-136. JP 2006-52790 A

  In the conventional method for designing the crowning of a roller bearing, optimization calculation for determining initial values of design parameters is performed by a computer. However, this optimization calculation requires a large amount of calculation for the conditions included in the initial value search range of the design parameter. Therefore, the conventional design method has a problem of requiring a lot of time and labor.

  Therefore, an object of the present invention is to provide a method for designing a crowning of a roller bearing that can greatly reduce the time and effort required for the design.

  In order to solve the above-mentioned problem, a method for designing a crowning of a roller bearing according to claim 1 includes the above-described inner ring raceway, outer ring raceway, A crowning design method for forming on at least one of the roller rolling surfaces, wherein the amount of crowning drop at the end of the effective length of the roller is determined based on the nominal size and design load of the roller, Apply the drop amount at the end of the effective length of the roller to a table in which multiple drop amounts corresponding to multiple positions in the bus direction are expressed as dimensionless amounts, and specify multiple drop amounts at multiple positions in the bus direction And defining a contour line of a crowning formed on at least one of the outer ring raceway surface, the inner ring raceway surface and the roller rolling surface from the plurality of identified drop amounts. It is.

  The present inventor performed optimization calculation of an objective function using a predetermined design parameter for a conventional logarithmic function crowning curve. As a result, the nominal size and design load of the roller and the crowning at the end of the effective length of the roller were calculated. It was found that there is a correlation between the amount of drops. Based on this discovery, the present invention has been made.

  According to the first aspect of the present invention, the amount of the crowning drop at the end of the effective length of the roller is determined based on the nominal size of the roller and the design load. By applying the drop amount at the end of the effective length of the roller to the table, a plurality of drop amounts at a plurality of positions in the bus direction can be specified. From this identified drop amount, it is possible to determine the contour line of the crowning formed on at least one of the outer ring raceway surface, the inner ring raceway surface and the roller rolling surface. Thus, according to the present invention, it is not necessary to perform optimization calculation for a large number of parameter values. Therefore, the time and labor required for designing the crowning of the roller bearing can be greatly reduced as compared with the conventional case.

  The amount of crowning drop refers to the distance in the direction perpendicular to the bus line from the generatrix of the raceway surface or rolling surface on which the crowning is performed to the crowning surface.

According to a second aspect of the present invention, in the crowning design method for the roller bearing according to the first aspect, the amount of the crowning drop at the end of the effective length of the roller is obtained using the following equation (2). It is said.
Where z _m (μm) is the amount of crowning drop at the end of the effective length of the roller, x (%) is the ratio of the design load to the basic dynamic load rating, d (mm) is the roller diameter, and L (mm) is The effective length of the roller.

According to the second aspect of the present invention, the roller diameter can be easily obtained simply by substituting the sum of the diameter d and the effective length L as the nominal dimension of the roller and the ratio x of the design load to the basic dynamic load rating as the design load. The crowning drop amount z _m at the end of the effective length is obtained.

According to a third aspect of the present invention, in the roller bearing crowning designing method according to the first aspect, the table includes values described in Table 2 below.

  According to the invention of claim 3, the drop amount at each position in the busbar direction can be specified only by multiplying the drop amount at the end of the effective length of the roller by the dimensionless value of the drop amount in Table 2 above.

In Table 2 above, the value in the busbar direction position is a dimensionless value with half the effective length of the roller (L / 2), and the value of the drop amount is the end of the effective length of the roller. Is a dimensionless value with the drop amount (z _m ).

  According to a fourth aspect of the present invention, in the roller bearing crowning design method according to the second aspect, the value of x in the formula (2) is 25 or more and 35 or less, and the crowning drop at the end of the effective length of the roller It is characterized by determining the quantity.

  According to invention of Claim 4, the crowning curve corresponding to the actual use condition of a roller bearing can be designed. Here, if the value of x is smaller than 25, the prevention of edge load due to the crowning curve is insufficient. On the other hand, when the value of x is larger than 35, the amount of processing at the time of producing the crowning increases, resulting in an increase in manufacturing cost.

  According to a fifth aspect of the present invention, in the roller bearing crowning design method according to the first aspect, the plurality of identified drop amounts may be calculated based on the drop amount of the inner ring raceway surface or the outer ring raceway surface and the roller rolling surface. A contour line of a crowning formed on the inner ring raceway surface or the outer ring raceway surface and the roller rolling surface is defined by being distributed to the drop amount.

  According to the invention of claim 5, the plurality of specified drop amounts are distributed to the drop amount of the inner ring raceway surface or the outer ring raceway surface and the drop amount of the roller rolling surface at each position in the generatrix direction. From this distributed drop amount, it is possible to determine the crowning contour line formed on the inner ring raceway surface or the outer ring raceway surface and the crowning contour line formed on the roller rolling surface. Thereby, the crowning formed on both the inner ring raceway surface or the outer ring raceway surface and the roller rolling surface can be designed.

  According to the present invention, the crowning contour line is obtained by obtaining the crowning drop amount at the end of the effective length of the roller based on the nominal size of the roller and the design load, and applying this drop amount to a predetermined dimensionless table. Therefore, it is not necessary to perform an optimization calculation for a large number of parameter values. Therefore, the time and labor required for designing the crowning of the roller bearing can be greatly reduced.

  DESCRIPTION OF EMBODIMENTS Hereinafter, embodiments of a crowning design method for a roller bearing according to the present invention will be described in detail with reference to the drawings.

  FIG. 1 is a cross-sectional view showing a cylindrical roller bearing to which crowning is performed by the design method of the embodiment of the present invention. As shown in FIG. 1, the cylindrical roller bearing includes an inner ring 11, an outer ring 12, and a plurality of cylindrical rollers 13, 13,... That are rotatably interposed between the inner ring raceway surface 11a and the outer ring raceway surface 12a. And a retainer 14 for holding the cylindrical rollers 13, 13,... At a predetermined interval in the bearing circumferential direction. In this embodiment, cut crowns 13b, 13c are provided on the rolling surfaces 13a, 13a, ... of the cylindrical rollers 13, 13, ..., and the raceway surface 11a of the inner ring 11 and the raceway surface 12a of the outer ring 12 are respectively cylindrical. It is formed.

  FIG. 2 is a diagram showing the crowning of the cylindrical roller 13 on a yz coordinate system in which the extending direction of the generatrix of the cylindrical roller 13 is the y-axis and the direction orthogonal to the generatrix (the radial direction of the roller) is the z-axis. is there. This yz coordinate system is on the generatrix of the cylindrical roller 13 and the origin O is the center of the effective contact portion between the inner ring 11 or the outer ring 12 and the cylindrical roller 13. The effective contact portion is a contact portion between the inner ring 11 or the outer ring 12 and the cylindrical roller 13 when it is assumed that the cut crowns 13 b and 13 c are not formed on the cylindrical roller 13. Moreover, since each crowning 13b, 13c of the cylindrical rollers 13, 13,... Is normally formed symmetrically with respect to the z axis passing through the center of the effective contact portion, only one crowning 13b is shown in FIG. .

The crowning 13b can be represented by the following equation (3) using a logarithmic function.

K ₁ is a parameter representing the degree of curvature of crowning. A is represented by 2Q / πLE ′, Q is the load, L is the length of the effective contact portion in the generatrix direction, and E ′ is the equivalent elastic modulus. Z _m is a crowning drop amount at the end of the effective length of the roller, and means a maximum drop amount of the crowning 13b. The point P _{1 in} FIG. 2 is a position indicating the maximum drop amount z _m of the crowning 13b. a is the length from the origin O to the end of the effective contact portion. K ₂ is a parameter that represents the ratio of the crowning length to a. In the crowning 13b of FIG. 2, since the origin O is the center of the effective contact portion, a = L / 2. Further, since the coordinates of the starting point O ₁ of the crowning 13b are (a−K ₂ a, 0), the range of y in the equation (3) is y> (a−K ₂ a).

In equation (3), z (y) is the drop amount of the crowning 13b at the position y in the generatrix direction of the cylindrical roller 13. In equation (3), the values of Q, L, E ′ and a are given as design conditions. Since the region from the origin O to the starting point O _{1 of the} crowning 13b is a straight portion formed in a cylindrical surface shape, z (y) = 0 when 0 ≦ y ≦ (a−K ₂ a). Become. When K ₂ = 1, since the starting point O ₁ coincides with the origin O, Equation (3) represents full crowning without a straight portion.

In the conventional crowning design method, the crowning curve is specified by giving design conditions such as the load Q and appropriate design parameters K ₁ , K ₂ , and z _m to Equation (3). In order to specify the design parameters K ₁ , K ₂ , and z _m , optimization calculation of an objective function is performed within a range that each parameter can take. Therefore, it is necessary to perform a large amount of optimization calculation for each cylindrical roller bearing to be designed, and much labor and time are required.

In contrast, in the present embodiment, by using a function to nominal size and the design load variables of time, determine the drop amount z _m of the crowning at the end of the effective length of the roller, the drop amount z _m The shape of the entire crowning is specified by applying to a predetermined table.

  The above function is obtained as follows.

First, the above parameters are optimized by a mathematical optimization method. In the optimization of this parameter, when the roller is tilted, K ₂ = 1 under most conditions, and the surface pressure or Mises equivalent stress becomes smaller as it approaches the full crowning. On the other hand, it is desirable that the cylindrical roller has a straight portion of at least about 50% of the entire length for manufacturing reasons. Therefore, in this embodiment, K ₂ is not optimized, and K ₂ is fixed so that the straight portion is 50% of the total length.

  Subsequently, optimization of the crowning curve is performed in the same manner as in the past for the rollers having dimensions of φ5 × 5 to φ24 × 38. As the optimization objective function, the maximum value of Mises equivalent stress in the vicinity of the contact portion was adopted. As for the design condition, the roller tilt is 1/1000 (the value corresponding to 2/1000 in the misalignment of the inner ring and the outer ring). The design load is set for the contact load between one roller and the inner ring. This contact load is set to the maximum rolling element load when the load acting on the bearing is 20% to 50% of the basic dynamic load rating Cr. Hereinafter, a design condition in which the maximum rolling element load when a load of x% of the basic dynamic load rating Cr is applied to the bearing is a design load is referred to as x% Cr design. The objective functions to be optimized are the maximum contact pressure applied to the inner ring raceway surface, outer ring raceway surface or roller rolling surface, maximum value of Mises equivalent stress, maximum value of equivalent stress of Tresca, rolling fatigue life At least one of them can be used. When the maximum contact pressure, the maximum value of Mises' equivalent stress, or the maximum value of Tresca's equivalent stress is used as an objective function, design parameters are determined so that these values are minimized. When the rolling fatigue life is an objective function, design parameters are determined so that the rolling fatigue life is maximized.

FIG. 3 is a diagram showing the calculation result of the drop amount z _m at the end of the effective length from the result of optimization using the maximum value of Mises' equivalent stress for each design condition as an objective function. In FIG. 3, the horizontal axis is the sum of the roller diameter d (mm) and the effective length L (mm), and the vertical axis is the drop amount z _m (μm) at the end of the effective length. Fig. 3 shows the optimization results of the roller of 64 types of φ24 or less for 25% Cr design and 35% Cr design. About 30% Cr design, 40% Cr design and 50% Cr design Fig. 6 shows the optimization results of 20 types of rollers out of 64 types of dimensions of φ24 or less. As can be seen from FIG. 3, the correlation coefficient is 0 between the sum of the roller diameter d and the effective length L and the drop amount z _m at the end of the effective length of the roller under any load condition. There is a linear relationship of .997 or more.

From the calculation results shown in FIG. 3, the relationship between d + L (mm) and z _m (μm) under each load condition is expressed as the following equations (4) to (8).
Here, Equation (4) is a 25% Cr design, Equation (5) is a 30% Cr design, Equation (6) is a 35% Cr design, and Equation (7) is a 40% Cr design. (8) is a 50% Cr design.

Further, when the above formulas (4) to (8) are generalized to the form z _m = a (d + L) + b, the constant parts a and b are considered to have a linear relationship with the load. And b can be approximated by the following equations (9) and (10).

FIG. 4 is a diagram in which the above equations (9) and (10) are superimposed on the coordinates. In FIG. 4, the horizontal axis represents the ratio x (%) of the design load to the basic dynamic load rating, and the vertical axis represents a and b. When the equations (4) to (8) are generalized using the equations (9) and (10), the following equation (11) is obtained.

From the above equation (11), the optimum value of z _m (μm) can be obtained from d + L (mm) as the nominal dimension of the roller and the ratio x (%) as the design load.

  Next, a table for specifying the overall shape of the crowning is created.

First, the result of optimization calculation of the crowning curve is made non-dimensional by dividing the value in the busbar direction position by half of the effective length L, and the value of the drop amount in each busbar direction position is set to the end of the effective length. Dividing by the drop amount z _m in FIG. FIGS. 5A to 5E are diagrams showing the crowning curves after dimensionless for each load condition. 5A to 5E, the horizontal axis represents the dimensionless bus direction position, and the vertical axis represents the dimensionless drop amount. The position 0 in the dimensionless bus direction indicates the center of the roller.

  FIGS. 5A to 5E are diagrams showing optimization results related to rollers of 64 kinds of design dimensions having a diameter of φ24 or less, and the design conditions other than the design load are the same. The design loads are 25% Cr in FIG. 5A, 30% Cr in FIG. 5B, 35% Cr in FIG. 5C, 40% Cr in FIG. 5D, and 50% Cr in FIG. 5E.

  As can be seen from FIGS. A to 5E, the crowning curves under any load condition are represented in substantially the same shape when shown in a dimensionless manner. FIG. 6 is a diagram in which a curve indicating the maximum value and a curve indicating the minimum value are extracted from all the crowning curves in FIGS. 5A to 5E. As can be seen from FIG. 6, the crowning can be designed by specifying a dimensionless curve from a relatively narrow region regardless of the roller dimensions and load conditions.

FIG. 7 is a diagram showing the distribution of the maximum value of Mises equivalent stress, which is the objective function of the optimization calculation, using K ₁ and z _m as parameters. As can be seen from FIG. 7, it is preferable that the value of the parameter K ₁ is larger than the optimum line Lb because the maximum value of Mises equivalent stress tends to be reduced. Here, as the parameter K ₁ increases, the curvature of the crowning curve decreases. Therefore, if a crowning curve that meets a wide range of conditions is selected from the area between the maximum value and the minimum value of the curve shown in FIG. 6, a curve with the maximum drop amount is selected at all busbar direction positions. It is preferable to do this. Such a point on the curve is expressed in Table 3 below when expressed in a dimensionless amount between the position in the busbar direction and the drop amount.
Table 3 Points on the curve where the drop amount is the maximum at all bus direction positions

The shape of the entire crowning can be specified by applying the drop amount z _m obtained by the above equation (11) to Table 3 above. In other words, the dimensionless amount in the column of bus-line direction position, the multiplied value of L / 2 is half the effective length of the roller, the dimensionless amount of drop amount column is multiplied by the drop amount z _m. Thereby, a point on the yz coordinate system formed by the generatrix direction (y-axis) of the roller and the generatrix orthogonal direction (z-axis) can be specified. By setting a curve passing through this point, the crowning curve can be specified.

  The crowning design method of this embodiment does not require optimization calculation of the crowning curve every time the design is performed. Therefore, since it is not necessary to perform a large amount of calculations for the conditions that can be taken by the design parameters as in the conventional case, it is possible to effectively reduce the labor and time for designing the crowning.

In the above embodiment, in the case of high load, it is better z _m is large in terms of the effect of preventing edge load, the greater the amount of processing time crowning fabrication when unnecessarily large drop amount z _m, uneconomical It is. Moreover, under most practical use conditions, the load is 25% or less of the basic dynamic load rating, and exceeding 35% is extremely rare. Therefore, in the formula (11), x is preferably 25 or more and 35 or less.

  The crowning may be provided on either the rolling surface of the roller and the raceway surface of the outer ring or the inner ring. When the crowning is formed on both the rolling surface of the roller and the raceway surface of the outer ring or inner ring, the drop amount at each position in the generatrix direction is distributed to the rolling surface side and the raceway surface side and formed on each surface. What is necessary is just to specify the crowning curve to be performed.

  In Table 3 showing the dimensionless amount of the busbar direction position and the drop amount, the busbar direction position is shown every 0.05, but the specific interval of the busbar direction position is a value other than 0.05. There may be.

It is sectional drawing which shows the cylindrical roller bearing by which the crowning by the design method of embodiment of this invention is given. It is the figure which showed the outline of the crowning of the cylindrical roller on the yz coordinate system. From the results of the optimization calculation is a diagram showing by extracting the drop amount z _m at the end of the effective length. Generalized formula constants a drop amount z _m at the end of the effective length, and b, is a diagram showing the relationship between the ratio x to the basic dynamic load rating of the design load. It is a figure which shows the optimization result of a crowning shape in case a design load is 25% of a basic dynamic load rating. It is a figure which shows the optimization result of crowning shape in case a design load is 30% of a basic dynamic load rating. It is a figure which shows the optimization result of a crowning shape in case a design load is 35% of a basic dynamic load rating. It is a figure which shows the optimization result of crowning shape in case a design load is 40% of a basic dynamic load rating. It is a figure which shows the optimization result of crowning shape in case a design load is 50% of a basic dynamic load rating. It is the figure which extracted and showed the curve which shows the maximum value, and the curve which shows the minimum value among all the crowning curves of FIG. 5A thru | or 5E. The K ₁ and z _m as parameters, a diagram showing the distribution of the maximum value of Mises equivalent stress.

Explanation of symbols

DESCRIPTION OF SYMBOLS 11 Inner ring 11a Inner ring raceway surface 12 Outer ring 12a Outer ring raceway surface 13 Cylindrical roller 13a Cylindrical roller rolling surface 13b, 13c Cut crowning

Claims (5)

A method for designing a crowning formed on at least one of the inner ring raceway surface, the outer ring raceway surface and the roller rolling surface for a roller bearing in which a plurality of rollers are interposed between an inner ring raceway surface and an outer ring raceway surface. ,
Based on the nominal size and design load of the roller, obtain the amount of crowning drop at the end of the effective length of the roller,
Applying the drop amount at the end of the effective length of the roller to a table in which a plurality of drop amounts corresponding to a plurality of positions in the busbar direction are expressed as dimensionless amounts, a plurality of drop amounts at a plurality of positions in the busbar direction are obtained. Identify,
A crowning design method for a roller bearing, wherein a contour line of a crowning formed on at least one of an outer ring raceway surface, an inner ring raceway surface and a roller rolling surface is determined from the plurality of specified drop amounts. In the crowning design method of the roller bearing according to claim 1,
A crowning design method for a roller bearing, wherein a drop amount of the crowning at the end of the effective length of the roller is obtained using the following formula (1).
Where z _m (μm) is the amount of crowning drop at the end of the effective length of the roller, x (%) is the ratio of the design load to the basic dynamic load rating, d (mm) is the roller diameter, and L (mm) is The effective length of the roller. In the crowning design method of the roller bearing according to claim 1,
The said table contains the value described in following Table 1, The design method of the crowning of a roller bearing characterized by the above-mentioned.
In the design method of the crowning of the roller bearing according to claim 2,
A method for designing a crowning of a roller bearing, wherein the value of x in the above formula (1) is 25 or more and 35 or less, and the amount of crowning drop at the end of the effective length of the roller is obtained. In the crowning design method of the roller bearing according to claim 1,
The plurality of identified drop amounts are distributed to the drop amount of the inner ring raceway surface or the outer ring raceway surface and the drop amount of the roller rolling surface, and the inner ring raceway surface or outer ring raceway surface, and the roller rolling A crowning design method for a roller bearing, characterized by defining a contour line of a crowning formed on a surface.