JP5473814B2 - Dynamic stress analysis method and analysis system for bearing cage - Google Patents

Description

  The present invention relates to a stress analysis method and a stress analysis system for a cage in a rolling bearing such as a tapered roller bearing, and relates to a technique for numerically calculating a stress generated in the cage of the rolling bearing during operation by dynamic analysis.

  For the design of rolling bearing cages, it is important to understand the dynamic behavior and stress history of the cages, and kinetic analysis is effective in elucidating them. Conventionally, there is a dynamic analysis in which the cage remains a rigid body, and a technique for obtaining a force acting on the cage by numerical calculation has been put to practical use (for example, see Non-Patent Document 1). It is conceivable to calculate the stress applied to the cage by introducing the interference force to the cage obtained by the above numerical calculation into FEM (finite element method) analysis. It is also conceivable to introduce all the bearing parts on the FEM software and perform dynamics calculation therein (for example, see Non-Patent Document 2).

In the technique according to Non-Patent Document 1, since the dynamic analysis is performed with a rigid body, the stress of the cage cannot be calculated. In addition, in the technology that calculates the stress applied to the cage based on the interference force to the cage obtained by the numerical calculation, it is difficult to set the boundary condition in the static case, and the introduced interference force in the dynamic case. Since the effect of dynamic deformation is not included in itself, a cage having low deformation rigidity increases from the value at the moment of contact, and the calculated stress value is excessive. Therefore, it is difficult to calculate with high analysis accuracy.
The technology related to Non-Patent Document 2 introduces all bearing parts on the FEM software and calculates the dynamics in it, but the degree of freedom of calculation increases in proportion to the number of elements, so the calculation time is reduced. As a result, it is difficult to calculate the cage stress with high accuracy.

  In order to solve these problems, a dynamic analysis method that takes into account the elastic deformation of the cage has been proposed (Patent Document 1). The stress calculation of the cage uses the mode synthesis method.

JP 2008-1106040 A

Japan Tribology Society Tribology Conference Proceedings, 2004-11, 3D Dynamic Analysis of Tapered Roller Bearing Cage 2nd Report Japan Tribology Society Tribology Conference Proceedings, 2001-11, Dynamic Analysis of Ball Bearing by FEM Stress Analysis of Cage (1st Report)

In any of the above conventional techniques, it has been difficult to accurately and efficiently predict the stress of the cage of the rolling bearing during operation.
The method disclosed in Patent Document 1 uses a mode synthesis method for the stress calculation of the cage, and although the calculation speed is excellent, the prediction accuracy is not so high. In particular, when contact loads from the rollers are applied to various points on the surface of the pocket that contacts the rollers, the calculation accuracy of the generated stress may be reduced.

In the paragraph [0031] of Patent Document 1, it is described that the accuracy of calculating the stress of the cage is improved by installing at least one degree of freedom of the mode synthesis method on the pillar of the cage. In this technology, the case where the actual column deformation can be reproduced only by the primary deformation of the column is considered. In this case, a degree of freedom is provided near the middle in the longitudinal direction where the column deformation is greatest. Propose that.
The calculation amount of the dynamic analysis greatly depends on the number of degrees of freedom, and setting of unnecessary degrees of freedom should be avoided. However, it is not possible to obtain a highly accurate cage stress with only one degree of freedom. .

  In the same paragraph, the provision of a degree of freedom in at least one of the end portions in the longitudinal direction of the column is mentioned for the time being, but the effect is not described. Moreover, although there exists description with the edge part of the longitudinal direction of a pillar, it does not specify clearly which site | part shows with the edge part. In addition, when setting the degree of freedom of the mode composition method to the pillar of the cage, the rolling element to the cage will be determined according to which part of the pillar, such as the outer diameter surface, inner diameter surface, and thickness center, of the cage. Although it depends on the mode of the action of the load on the load, the setting point of the appropriate degree of freedom for the load action mode has not been considered.

  An object of the present invention is to achieve a dynamic stress analysis method and analysis system for a bearing cage that can achieve both high accuracy and efficiency in stress calculation and enable highly accurate and efficient calculation of cage stress during operation. Is to provide.

The dynamic stress analysis method for a bearing cage of the present invention is based on a dynamic analysis model of a rolling bearing in which a bearing component is regarded as a rigid body as a process of analyzing the stress of the cage of a rolling bearing composed of roller bearings. Of the dynamic elastic deformation mode and natural deformation mode of the cage obtained by the method based on the mode synthesis method, the degree of freedom of elastic deformation introduced in the process, and the motion of the specified bearing components The cage stress is calculated by calculating the deformation history of the cage including the dynamic characteristics of deformation by numerically integrating the degrees of freedom at the same time and converting the deformation history calculated in this process into a stress distribution. In the dynamic stress analysis method for a rolling bearing cage including a process of performing the following steps, the installation position of the degree of freedom to be introduced into the cage by the mode synthesis method is the outer diameter surface of the cage between the cage pockets and the cage. Inner surface In other words, it is a point on the surface where the displacement due to the translation and torsion of the column becomes larger due to the load from the roller, and multiple points on the central cross section that is the center in the width direction of the column between the pockets of the cage. ,
Two points of these multiple places of freedom of installation are the two points on the central cross section of the column that are both ends in the axial direction of the roller within a range in which a load acts from the roller. Features.

  According to this configuration, similar to the method described in Patent Document 1, the dynamic analysis characteristics of the cage (that is, the natural deformation mode and its frequency obtained by the super element method) are combined into the dynamic analysis model. Introduced based on the law, the degree of freedom of elastic deformation and the degree of freedom of movement of predetermined bearing components are numerically integrated simultaneously. Thereby, the deformation history of the cage including the dynamic characteristics of the deformation is obtained. Cage stress is obtained by converting this deformation history into a stress distribution. In the dynamic analysis model, the other components of the cage are regarded as rigid bodies. Therefore, it is possible to calculate the cage stress during operation with high accuracy and efficiency as compared with other conventional techniques.

  In introducing the eigendeformation mode, it is important to reproduce the deformation of the cage column, but it is necessary to introduce eigendeformation modes up to a very high order for this consideration. It takes a long time. According to the configuration of this invention, as in the invention of Patent Document 1, by setting the degree of freedom of the super element method for each column of the cage, the column deformation mode is always output to the constraint deformation mode. By reducing the number of remaining modes of the other natural deformation modes, an efficient and appropriate cage stress value can be obtained. In other words, the time required for the dynamic analysis can be shortened as compared with the method of introducing the inherent deformation mode up to the very high order described above. Alternatively, the accuracy of the solution can be maintained at a high level by reliably introducing the column deformation mode.

In particular, in the configuration of the present invention, unlike the invention of Patent Document 1, when the dynamic elastic deformation characteristics of the cage are introduced into the above dynamic analysis model based on the mode synthesis method, A plurality of degrees of freedom are set on the axial central section. By doing so, even if the load from the roller approaches the corner where the stress is concentrated, the stress at the corner can be predicted with relatively high accuracy.
Two points of the multiple degree of freedom installation points are the two points that are both ends of the column in the axial direction of the roller within the range where the load is applied from the roller. As a result of various studies, it has been found that when the above two points are calculated by the above mode synthesis method, the stress acting on the cage appears most accurately. Therefore, the cage stress can be calculated with higher accuracy by including the above two points in the installation places with a plurality of degrees of freedom.
In addition, place the degree of freedom on the outer surface of the cage between the pockets of the cage and the inner surface of the cage on the side where displacement due to translation and torsion of the column becomes larger due to the load from the rollers. Therefore, the cage stress can be calculated with higher accuracy.

In this invention, it is preferable that only two points at both ends of the pillar are installed at a plurality of places on the pillar of the cage.
In order to examine the analysis results of the cage stress, calculation was performed by changing the number of places and number of degrees of freedom, and the results were compared with the results calculated by the finite element method. In this case, a value close to the same level as the calculation result by the finite element method was obtained even when compared to the case where a large number of installation places with a degree of freedom were installed at about 10 places, for example. The finite element method is a highly accurate analysis method that can be estimated to be an actual stress value, although the calculation amount is large and the calculation time is long. In addition, the smaller the number of installation locations with the degree of freedom, the smaller the calculation amount and the more efficient the calculation. Therefore, using the above two points where the results close to the results of analysis by the finite element method are set as the installation locations with the degree of freedom is the most excellent in terms of both high accuracy and efficiency in cage stress calculation. .
It should be noted that the above-mentioned two points for setting the degree of freedom are more excellent in terms of both higher accuracy and higher efficiency in calculating the cage stress when the correction described later is used together. Become.

In this invention, it is good also considering the installation location of the freedom degree installed in the multiple places of the pillar of the said cage | basket as 3 points | pieces, 2 points | pieces of the said pillar both ends, and the center part between these 2 points | pieces.
When the number of degrees of freedom is set to the above three points, the analysis result by the finite element method is slightly smaller than the above two points, but a close result is obtained. Even if three points are used, the calculation efficiency is in a satisfactory range. Therefore, even if the above three points are used, the calculation of the cage stress is excellent in terms of both high accuracy and high efficiency. In the case where correction described later is not performed, the above three points are preferable in terms of increasing the accuracy of cage stress calculation.

  In the present invention, the dynamic analysis model may be a three-dimensional model. Compared to the case where the analysis target is two-dimensional, if the three-dimensional model is used, the axial displacement and inclination of the rollers and cages can be taken into consideration, so the cage stress during operation that assumes such behavior is highly accurate. Can be calculated. For example, stress due to axial contact between the roller and the cage can be considered.

In the method of the present invention, a method may be adopted in which a preliminary investigation is performed to obtain and correct a correction coefficient. The following two types can be adopted for the correction by the preliminary survey.
One of the corrections by the preliminary survey is a method using a correction coefficient related to the finite element length. As a preliminary survey, a finite number used in the dynamic analysis model for each point that is a predetermined portion, that is, the installation position of the degree of freedom. Using the element length and a finite element length shorter than this finite element length, obtain the stress under the same load condition by the finite element method, find the ratio of both stresses, and correct this ratio by the finite element length. This is a method of correcting the cage stress obtained in the process of calculating the coefficient by converting the deformation history into a stress distribution by multiplying the correction coefficient related to the finite element length.
In this case, with respect to the finite element length of the predetermined portion used for the dynamic analysis, the stress of the predetermined portion when the length is shortened to a necessary and sufficient length is obtained. By obtaining the ratio between the obtained stress and the stress at the finite element length used in the dynamic analysis as a correction coefficient, the stress estimation accuracy is improved. As described above, once the correction coefficient is obtained as a preliminary investigation, the correction coefficient is simply applied to the cage stress output in the cage stress output process without shortening the finite element length of the predetermined portion each time. Stress multiplication accuracy can be improved only by multiplication. As a result, the processing load on the computing means can be reduced, and the cage life can be determined more accurately and easily.

Another one of the corrections by the preliminary investigation is a method using a correction coefficient according to the mode synthesis method. As preliminary investigations, the process of introducing the elastic deformation characteristics, the process of calculating the deformation history, and the stress of the cage The cage stress is obtained by dynamic calculation using the calculation process, the stress of the cage is obtained by the finite element method, and the stress obtained by the finite element method and the stress obtained by the dynamic calculation are calculated. Cage obtained in the process of calculating the ratio by converting the deformation history into stress distribution in the stress analysis process of the cage to be analyzed, using this ratio as the correction coefficient for the mode synthesis method. The stress is corrected by multiplying the correction coefficient according to the mode synthesis method.
In the above dynamics calculation, the cage deformation is reproduced by the mode synthesis method, but the stress value is smaller than the analysis by the finite element method, that is, the FEM analysis. found. Ideally, by increasing the number of residual modes in the eigendeformation mode and the number of degrees of freedom of the constrained deformation mode in the super element method, the stress value at the target position is the same in the FEM analysis and the mode synthesis method. However, when the cage shape changes, the optimum analysis conditions of the super element method change, so it is necessary to consider each time. Therefore, the stress estimation accuracy is improved by obtaining, as a correction coefficient, the ratio of the stress obtained by the finite element method and the stress of the problematic part obtained by the dynamic calculation. As described above, once the correction coefficient is obtained as a preliminary investigation, the stress estimation accuracy can be improved only by multiplying the stress value obtained by the mode synthesis method by the correction coefficient. As a result, the processing load on the computing means can be reduced, and the cage life can be determined more accurately and easily.

  The correction coefficient related to the finite element length and the correction coefficient related to the mode composition method can be used in combination. For example, a correction coefficient obtained by multiplying the correction coefficient related to the finite element length and the correction coefficient related to the mode composition method may be used.

  In the present invention, when none of the above correction coefficients is used, it is preferable to set the following three points for the degree of freedom. That is, in this invention, the installation places with a degree of freedom to be installed at a plurality of locations on the central cross section of the retainer column are three points, two points on both ends of the column and a central part between these two points. The cage stress obtained in the process of calculating by converting the deformation history into the stress distribution is set as the cage stress output as the analysis result. When the correction coefficient is not used, it is easier to obtain a highly accurate stress calculation result when the degree of freedom is set to the above three points than when the above two points are set.

As described above, the above-described three correction factors are not used and the degree of freedom is set at the above three points because the degree of freedom of movement of the rigid body mode of the roller, the race, and the cage, and the elasticity of the cage. This is preferably applied when the degree of freedom of deformation is limited to two dimensions. When the calculation is performed in two dimensions, the accuracy and the type of stress required are inferior to those in the calculation in three dimensions, but the calculation amount is small and the calculation can be performed efficiently. In this case, by setting the number of freedom installation locations as the above three points, the cage stress can be obtained with as high accuracy as possible while the number of freedom installation locations is small and the calculation is small.
In this case, the degree of freedom of movement of the rigid body mode of the roller, the race, and the cage, and the degree of freedom of elastic deformation of the cage are limited to two dimensions,
And the installation position of the degree of freedom to be introduced into the cage by the mode synthesis method is based on the translation and torsion of the column due to the load from the roller among the cage outer diameter surface and the cage inner diameter surface between the pockets of the cage. It is a point on the surface on the side where displacement becomes larger, and a plurality of locations on the central cross section that is the center in the width direction of the column between the pockets of the cage,
One point of the plurality of degrees of freedom of installation places is a point that becomes an axial end portion of the roller within a range in which a load is applied from the roller on the central cross section of the column. good.
It is good also considering only the two points of the one point of the said column end, and the center point of the longitudinal direction of a column as the installation location of the freedom degree installed in the multiple places of the column of the said holder | retainer.

In this invention, the rolling bearing is a tapered roller bearing, and the cage is a conical shape including a pair of annular portions at both ends in the axial direction and a plurality of columns provided between the annular portions. It is a window shape and may be a flange shape in which the annular portion on the large diameter side protrudes toward the outer diameter side.
In the case of such a cage shape and bearing type, the accuracy of the stress calculation is improved and the efficiency is improved by setting the above-described two or three degrees of freedom as the installation location, and the cage stress during operation It is possible to obtain an advantageous effect of enabling highly accurate and efficient calculation.

The dynamic stress analysis system for a bearing cage according to the present invention is a dynamic stress analysis device for a bearing cage that analyzes the stress of the cage of a rolling bearing comprising roller bearings, and is operated with the input means 2 (FIG. 3). Means 3 and output means 4 are provided.
The calculation means 3 regards the bearing components excluding the cage as a rigid body, and an analysis model setting unit 3a that defines a dynamic analysis model of the rolling bearing capable of giving the cage freedom of elastic deformation; ,
A process for introducing a dynamic elastic deformation mode and a natural deformation mode of the cage obtained by the super element method into the dynamic analysis model of the rolling bearing based on the mode synthesis method, and the degree of freedom of the introduced elastic deformation, A process to calculate the deformation history of the cage including the dynamic characteristics of deformation, and the deformation history calculated by this process to stress distribution by numerically integrating the specified degree of freedom of movement of the bearing components at the same time A stress calculator 3b that performs a process of calculating the cage stress, an output processor 3c that outputs the cage stress obtained by the stress calculator to the output unit,
Have
The stress calculation unit 3b determines the installation position of the degree of freedom to be introduced into the cage by the mode synthesis method by the load from the roller among the cage outside diameter surface and the inside diameter surface of the cage between the pockets of the cage. Multiple points on the center cross-section that are points on the side where the displacement due to translation and torsion of the column becomes larger, and that are the center in the width direction of the column between the pockets of the cage. Two points of the installation locations of the degree are two points which are both ends in the axial direction of the roller within a range where the load is applied from the roller on the central cross section of the column.

  According to the system of this configuration, as described in the dynamic stress analysis method of the present invention, high accuracy and efficient calculation of cage stress during operation can be achieved by achieving both high accuracy and efficiency of stress calculation. Can be obtained.

  The dynamic stress analysis method for a bearing cage of the present invention is based on a dynamic analysis model of a rolling bearing in which a bearing component is regarded as a rigid body as a process of analyzing the stress of the cage of a rolling bearing composed of roller bearings. Of the dynamic elastic deformation mode and natural deformation mode of the cage obtained by the method based on the mode synthesis method, the degree of freedom of elastic deformation introduced in the process, and the motion of the specified bearing components The cage stress is calculated by calculating the deformation history of the cage including the dynamic characteristics of deformation by numerically integrating the degrees of freedom at the same time and converting the deformation history calculated in this process into a stress distribution. In the dynamic stress analysis method of the rolling bearing cage including the process of performing the following steps: the degree of freedom to be introduced into the cage by the mode synthesis method, the outer diameter surface of the cage between the pockets of the cage and the cage Inner surface The point on the surface where the displacement due to the translation and torsion of the column becomes larger due to the load from the roller, and a plurality of locations on the central cross section that is the center in the width direction of the column between the pockets of the cage, In order to make two points of the installation places of these multiple degrees of freedom to be two points that are both ends in the axial direction of the roller within the range where the load acts on the center cross section of the column, It is possible to achieve both high accuracy and efficient stress calculation, and highly accurate and efficient calculation of cage stress during operation.

  A dynamic stress analysis system for a bearing cage according to the present invention is a dynamic stress analysis device for a bearing cage that analyzes the stress of a cage of a rolling bearing composed of a roller bearing, and includes an input means, an arithmetic means, an output means, The calculation means regards the bearing component excluding the cage as a rigid body, and an analysis model setting unit that defines a dynamic analysis model of the rolling bearing that allows the cage to be given a degree of freedom of elastic deformation. And a process for introducing the dynamic elastic deformation mode and the natural deformation mode of the cage obtained by the super element method into the dynamic analysis model of the rolling bearing based on the mode synthesis method, the degree of freedom of the elastic deformation introduced, and Calculating the deformation history of the cage including the dynamic characteristics of the deformation by numerical integration of the determined degrees of freedom of movement of the bearing components at the same time, and the deformation history calculated by this processing into the stress distribution In other words, a stress calculation unit that performs processing for calculating cage stress, and an output processing unit that outputs the cage stress obtained by the stress calculation unit to the output unit, The position of the degree of freedom to be introduced into the cage by the mode synthesis method is determined by the translation and torsion of the column due to the load from the roller among the cage outer diameter surface and the cage inner diameter surface between the pockets of the cage. It is a point on the surface on the side where displacement becomes larger, and a plurality of locations on the central cross section that is the center in the width direction of the column between the pockets of the cage, and among the installation locations of these multiple degrees of freedom These two points are the two axial ends of the roller within the range where the load is applied from the roller on the central cross section of the column, so that the stress calculation is highly accurate and efficient. Achieving both high accuracy and efficient calculation of cage stress during operation That.

(A) is a perspective view showing a dynamic analysis model of a bearing used in the dynamic stress analysis method according to the first embodiment of the present invention. (B) is a dynamic analysis model of a cage of the rolling bearing. It is the perspective view seen in the reverse direction to a figure (A). It is a flowchart showing the stress analysis method concerning the embodiment. (A) is a block diagram showing an electrical configuration of the stress analysis system according to the embodiment, and (B) is a block diagram showing a conceptual configuration of the stress analysis system. It is a partial expansion perspective view which shows the example of the location which sets a freedom degree to each pillar of the dynamics analysis model of the cage. It is a figure showing the calculation result example of the interference force to the holder | retainer of the bearing in driving | operation, and a holder | retainer stress. It is a figure showing the relationship between the measured value of a cage life, and the calculated value of the maximum Mises stress of a cage. (A) is explanatory drawing of the holder explaining the constraint condition limited to a two-dimensional degree of freedom and the setting method of the degree of freedom in the stress analysis system according to the second embodiment of the present invention, (b) is the holder FIG. It is the perspective view cut | disconnected by the radial plane of a bearing center about the dynamic analysis model of the rolling bearing in 2nd Embodiment. It is explanatory drawing showing the calculation result example of the interference force to the holder | retainer of the bearing in operation | movement which concerns on 2nd Embodiment, and a holder | retainer stress. It is a figure showing the relationship between the finite element length of a pocket corner part, and its stress. It is a figure showing the relationship between the cage | basket stress calculated | required by the finite element analysis, and the cage | basket stress calculated | required by the dynamics analysis by a mode synthesis method. It is a figure showing the relationship between the measured value of a cage life, and the calculated value of the maximum Mises stress of a cage which considered the correction coefficient. It is a perspective view showing the stress generation | occurrence | production location in the pillar center part of a holder | retainer. It is a graph which shows ratio of the maximum stress by the finite element method with respect to the mode composition method in the pocket large-diameter side corner part of a holder | retainer when a freedom degree installation place is an outer diameter side and a load action point is an outer diameter side. It is a graph which shows ratio of the maximum stress by the finite element method with respect to the mode composition method in the pocket small diameter side corner part of a holder | retainer when a freedom degree installation place is an outer diameter side and a load action point is an outer diameter side. It is a graph which shows the ratio of the maximum stress by the finite element method with respect to the mode composition method in the pocket large-diameter side corner of the cage when the degree of freedom installation place is the outer diameter side and the load application point is the inner diameter side. It is a graph which shows ratio of the maximum stress by the finite element method with respect to the mode composition method in the pocket small diameter side corner part of a cage | basket | case, when a freedom degree installation place is an outer diameter side and a load action point is an inner diameter meaning. It is a graph which shows the change rate of the influence for every load action position which affects the ratio of the maximum stress by the finite element method with respect to the mode synthetic | combination method in a pocket corner. It is a graph which shows ratio of the maximum stress by the finite element method with respect to the mode composition method in the pocket large diameter side corner of a cage when the degree of freedom installation place is the inner diameter side and the load application point is the outer diameter side. It is a graph which shows the ratio of the maximum stress by the finite element method with respect to the mode composition method in the pocket small diameter side corner part of a holder | retainer when a freedom degree installation place is an inner diameter side and a load action point is an outer diameter side. It is a graph which shows the ratio of the maximum stress by the finite element method with respect to the mode composition method in the pocket large-diameter side corner of the cage when the degree of freedom installation place is the inner diameter side and the load application point is the inner diameter side. It is a graph which shows the ratio of the maximum stress by the finite element method with respect to the mode synthetic method in the pocket small diameter side corner part of a holder | retainer when a freedom degree installation place is an inner diameter side and a load action point is an inner diameter meaning. It is a perspective view which shows the state before and behind a deformation | transformation of the pillar of a holder | retainer. It is sectional drawing which shows the state before and behind a deformation | transformation of the pillar of a holder | retainer. It is a partially broken side view of an example of the tapered roller bearing to be analyzed. It is a partially broken front view of an example of the tapered roller bearing used as an analysis object.

  Hereinafter, a plurality of embodiments for carrying out the present invention will be described with reference to the drawings. In the following description, the same reference numerals are given to portions corresponding to the matters described in the preceding forms in each embodiment, and overlapping description may be omitted. When only a part of the configuration is described, the other parts of the configuration are the same as those described in the preceding section. Not only the combination of the parts specifically described in each embodiment, but also the embodiments can be partially combined as long as the combination does not hinder.

  The stress analysis system according to the first embodiment of the present invention is applied to a tapered roller bearing retainer, for example. However, the present invention is not necessarily limited to the cage for tapered roller bearings, and is applied to stress analysis of various rolling bearing cages. The following description also includes a description of a cage stress analysis method in the dynamic analysis of a tapered roller bearing in three dimensions.

Tapered roller bearings expressed by the dynamic analysis model of FIG. 1 are shown in FIGS. In this tapered roller bearing, a plurality of rollers 14 which are rolling elements held by a cage 13 are interposed between raceways facing the inner ring 11 and the outer ring 12. The inner race 11 and the outer race 12 have conical surfaces facing each other, and the rollers 14 are tapered rollers. The inner ring 11 has both hooks having a large end hook 11a and a small end hook 11b on both sides of the raceway surface, and the outer ring 12 has no hooks.
The cage 13 has a conical window shape composed of a pair of annular portions 13a and 13b at both ends in the axial direction and a plurality of pillars 13c provided between the annular portions 13a and 13b, and is adjacent to each other. A space between the columns 13c is a pocket for holding the rollers 14. The large-diameter side annular portion 13a has a flange shape protruding toward the outer diameter side, and the small-diameter side annular portion 13b has a flange shape protruding toward the inner diameter side. The cage 13 is, for example, a pressed product of a steel plate.

  A dynamic analysis model of a rolling bearing will be described. FIG. 1 is a diagram showing a dynamic analysis model and a coordinate system of a rolling bearing. FIG. 1A shows an entire model of the rolling bearing, and FIG. 1B shows a cage model of the bearing. In addition, the same figure (A) and (B) is shown in reverse with respect to each other. That is, the drawing (A) is an overhead view of the bearing from the large diameter side of the roller, the direction from the small diameter side of the roller toward the large diameter side is z +, and (B) is the small diameter of the cage. It is the figure which looked down at the retainer from the side, Comprising: The direction which goes to the large diameter side from the small diameter side of a retainer is z +. The coordinate system is expressed by arrows x, y, and z in three axis directions orthogonal to each other. The x and y directions represent radial directions orthogonal to each other, and the z direction represents an axial direction. As shown in FIG. 5B, the cage was fixed in the space by restraining displacement at three points (P1, P2, P3) away from the pocket where the load is applied. Specifically, the three translational displacements at the two end portions (P1, P2) of the annular portion intersecting the YZ plane and the Z displacement at the end point (P3) in the −X direction were constrained.

  The tapered roller bearings to be analyzed (FIGS. 25 and 26) have main dimensions (inner diameter × outer diameter × width) of, for example, φ75 mm × φ160 mm × 40 mm, the number of rollers is 14, and the cage 13 is a rolling element guide type. Regarding the cage, two types of bearings in which the plate thicknesses of the columns are different from B1 and B2 (B1 = 3.5 mm, B2 = 5.0 mm) were examined. Grease was applied as the lubricant. The rotational speed of the inner ring was 1000 rpm, the axial displacement between the inner and outer rings was fixed, and the representative temperature was set to 60 ° C. when a radial load of 1000 N was applied. However, it is not necessarily limited to these conditions.

The assumed conditions for the three-dimensional dynamic analysis were defined as follows. As a premise, bearing components excluding the cage are regarded as rigid bodies.
(I) Give 6 degrees of freedom to the rollers and cage.
(Ii) The outer ring is fixed in the space.
(Iii) A constant axial displacement and a constant rotational angular displacement are forcibly given to the inner ring to give a degree of freedom of two translational displacements in the radial direction (degree of freedom is 2).
(Iv) Includes all apparent forces such as centrifugal force.
(V) Use a bearing orientation around the horizontal axis and consider gravity.
(Vi) Interference force distribution on the roller rolling surface is evaluated by the slice method.

(Vii) The traction coefficient μhd under fluid lubrication is a simple theoretical formula of Muraki et al. Masayoshi Muraki, Koji Kimura; Study on Traction Characteristics of Lubricating Oil (2nd Report), Lubrication, 28, 10 (1983) 753-760. }, But the conditions are isothermal.
(Viii) The friction coefficient μbd under boundary lubrication is given by Equation (1) obtained by correcting the Kragelskii model {Kragelskii, I, V., Friction and Wear, Butterworths, London (1965) 178-184}.
(Ix) The tangential force between the roller and the raceway surface is given by equation (2) in consideration of the change in the lubrication region. In addition, rolling viscous resistance {RSZhou, MRHoeprich; Trans. ASME, J. Trb, 113, 7 (1991) 590.} is considered under the elastohydrodynamic lubrication (abbreviated as EHL) state.

ここで、「ｓ」は、すべり率である。 (X) It is assumed that the interference force between the roller large end face and the inner ring large brim surface acts on the maximum proximity point. The tangential force coefficient is given by equation (2).
(Xi) The tangential force coefficient between the roller and cage is assumed only under boundary lubrication.
(Xii) In contact between the small brim surface and the pocket surface with respect to the roller end surface, all contact forces and tangential forces act on the maximum penetration point.

Here, “s” is a slip rate.

ここで、「μｒ」は、接線力係数であり、「Λ」は、油膜パラメータである。

Here, “μr” is a tangential force coefficient, and “Λ” is an oil film parameter.

  FIG. 2 is a flowchart showing the stress analysis method according to the first embodiment step by step. First, in step s1, a finite element model simulating an actual cage is created. The dynamic characteristic information of the cage is obtained by executing analysis by the super element method described later using software (for example, I-DEAS, Nastran, etc.) for executing FEM analysis. The result of the dynamic characteristic information is introduced into the above-described dynamic analysis model based on a so-called mode synthesis method in which a plurality of mode shapes are overlapped to produce an appropriate elastic deformation.

  Next, the process proceeds to step s2, where the degree of freedom of movement of the bearing components and the degree of freedom of elastic deformation of the cage introduced in step s1 are simultaneously numerically integrated, so that A deformation history is calculated. In this step s2, the interference force is calculated based on a calculation routine for the interference force between the cage and the "roller" which is a bearing component, which is a calculation routine incorporated in the means for performing the processing of step s2. To do. Here, the nodes (the vertices of the finite elements) of the introduced cage pocket surface cause the interference force to act when geometrical interference with the rollers occurs.

  Specifically, a reference point on a node position on the pocket surface and an Euler angle indicating its inclination are acquired in an inertial coordinate system, and after converting the node to a coordinate system fixed to the roller, the contour information of the roller From this, the amount of interference with the node was determined. When interference occurs, the frictional force is obtained from the contact force and the sliding speed at the interference point, and the resultant force may be handled as the interference force. In this method, it is possible to calculate the interference force with the roller in a state where all the behavior and elastic deformation of the cage are taken into consideration. In the above numerical calculation, the history of moments during numerical integration of the equation of motion is recorded to obtain the behavior of the cage and the deformation history of each deformation mode. Thereafter, the process proceeds to step s3, where the calculated deformation history is converted into strain and stress distribution and output. Thereafter, the processing shown in the flowchart of FIG.

FIG. 3A is a block diagram showing an electrical configuration of the stress analysis system according to this embodiment, and FIG. 3B is a block diagram of a conceptual configuration of the stress analysis system. The stress analysis system 1 mainly includes an input unit 2, a calculation unit 3, and an output unit 4. The computing means 3 is composed of a computer such as a personal computer and an analysis program (not shown) executed on the computer. The input means 2 is realized by, for example, a keyboard, a pointing device, means for reading a recording medium, means for connecting to a communication line, or the like.
In FIG. 3B, the calculation means 3 includes an analysis model setting unit 3a, a stress calculation unit 3b, and an output processing unit 3c. The analysis model setting unit 3a sets the above-described dynamic analysis model, and inputs the dynamic elastic deformation characteristics (natural deformation mode and its frequency) of the cage based on the mode synthesis method to the dynamic analysis model. It is possible.
The analysis model setting unit 3a is a unit that performs the process of step s1 in the flowchart of FIG. 2, and the stress calculation unit 3b converts the deformation history of the process of s2 and the process of step s3 into cage stress. It is a means for performing processing.

  The stress calculation unit 3b simultaneously numerically integrates the degree of freedom of elastic deformation introduced by the analysis model setting unit 3a and the degree of freedom of movement of a predetermined bearing component (for example, a roller), thereby performing deformation motion. A deformation history calculation unit 3ba that calculates the deformation history of the cage including the characteristics, and a stress distribution conversion unit 3bb that converts the calculated deformation history into a stress distribution. The output processing unit 3c outputs the stress distribution converted by the stress distribution conversion unit 3bb to the output unit 4. In addition, you may provide the correction calculating part 3bc demonstrated later in the stress calculating part 3b.

  The arithmetic means 3 includes, for example, a central processing unit 5 (CPU), memories 6 and 7 (read only memory 6 (ROM), random access memory 7 (RAM)), bus 8, input / output interface 9, and output means 4. Drive circuit 10 for driving the.

  A central processing unit 5 and memories 6 and 7 are connected to the input / output interface 9 via a bus 8. The input means 2 is electrically connected to the input / output interface 9, and the output means 4 is electrically connected via the drive circuit 10. The output unit 4 is realized by, for example, a display or a printer capable of display output. For example, the memory 6 or the memory 7 stores an analysis program for calculating the deformation history of the cage including the above-described deformation dynamic characteristics and converting the calculated deformation history into a stress distribution. The memory 7 temporarily stores input values, calculated values, and the like.

  FIG. 4 is a perspective view showing a place where a degree of freedom is set for each column of the retainer of the tapered roller bearing. The cage 13 includes a plurality of columns 13c at regular intervals in the circumferential direction, and rollers as rolling elements are disposed between adjacent columns 13c. Each column 13c is formed in a rectangular parallelepiped shape extending substantially parallel to the axial direction of the roller to be disposed. In the dynamic analysis model of the cage, the degree of freedom of the super element method (abbreviated as DOF: Degree Of Freedom, here, 3 translation) is set for each column 13c of the cage 13.

  The DOF is installed at a point on the surface on the side where the displacement due to translation and torsion of the column 13c becomes larger due to the load from the roller, of the outer diameter surface of the cage and the inner diameter surface of the cage. Thus, a plurality of points on the central cross section that is the center in the width direction of the column 13c are shown (in FIG. 4, each point (-2, -1, 0, +1, +2) is shown. Two points of the installation places with the degree of freedom are two points (−2, +2) that are on both ends of the column 13c in the axial direction of the roller and within the central cross section of the column within the range where the load is applied from the roller. The above-mentioned “surface on the side of the outer diameter surface of the cage 13c and the inner diameter surface of the cage where the displacement due to translation and torsion of the column 13c due to the load from the roller is larger” refers to various types of rolling bearings. Although it depends on conditions, in the illustrated example, the outer diameter surface of the cage.

It is preferable that the degree of freedom DOF is installed at only two points (−2, +2) that are both ends in the axial direction of the roller, but the two points (−2, +2) at both ends and these two points. It may be three points with the central portion (0) between (−2, +2), or more points. For example, a point (−1, +1) serving as the center between the two points (−2, +2) serving as both end portions and the center portion (0) may be added.
When the above-mentioned “surface on which the displacement due to translation and twist of the column 13c becomes larger” is the cage inner diameter surface, the installation position of the DOF degree of freedom is on the inner cross section on the center cross section, And it is good to set it as two points (points on the surface of the column on the inner diameter side of the cage and corresponding to -2 and +2) that are both ends in the axial direction of the roller within the range in which the load acts from the roller. Alternatively, a central portion on the inner diameter surface of the cage (a point corresponding to 0 on the surface of the column on the inner diameter side of the cage) may be added, and two points (both on the inner diameter side of the cage). A point (center of the cage inner diameter side) between the center of the column (point corresponding to -2, +2) and the center (point of the column on the inner diameter side of the cage and corresponding to 0). A point on the column surface and corresponding to -1, +1) may be added.
The reason why it is preferable to install the DOF with the degree of freedom as described above will be described later by comparing with the result of the FEM analysis.

  Here, the super element method (also referred to as super element method) will be described. This method is a method in which the characteristic matrix is formulated by the finite element method (FEM), and Guyan's static reduction is performed for analysis. A mass matrix of a partial structure is represented by [M], a stiffness matrix is represented by [K], a degree of freedom to be deleted in the reduction process of the super element method is represented by a subscript a, and a remaining degree of freedom is represented by a subscript b. If an external force does not act on the degree of freedom a, the equation of motion of undamped vibration is as follows in the frequency domain. The degree of freedom DOF of the super element method is the degree of freedom {xb}.

  In Expression (3), {x} represents a displacement amplitude vector, and {fb} represents an external force vector. In equation (3), the inertia term is omitted, and the displacement {xa} of the degree of freedom eliminated from the upper half of the relational expression is expressed as equation (4) by the displacement {xb} of the remaining degree of freedom.

  The original mass matrix and stiffness matrix are reduced to a matrix equal to {xb} degrees of freedom using the relationship of equation (4). The mass matrix after the reduction is expressed as Equation (5).

The stiffness matrix is expressed as Equation (6).

  A segregated model in which the matrices of these equations (5) and (6) are expressed is called a super element. Superimposes the characteristic matrix of the superordinate system as a coupled state of the structure, creates a reduced equation of motion for the entire system, and solves it.

  By the way, in introducing the deformation mode, it is important to reproduce the deformation of the cage column. To take this into account, it is necessary to introduce a natural deformation mode up to a very high order. Was getting longer. In this embodiment, by setting the degree of freedom of the super element method described above for each pillar of the cage, the deformation mode of the pillar is always included in the dynamic characteristics, and the remaining of the other inherent deformation modes remains. By reducing the number of modes, an efficient and reasonable cage stress value can be finally obtained. When the degree of freedom of the super element method is set for each column of the cage, the time required for the dynamic analysis can be shortened as compared with the method of introducing a higher-order natural deformation mode.

  Specifically, the mode composition method described in the present invention means that the elastic deformation of the cage is simulated by a linear sum of the constraint deformation mode by the super element method and the eigen deformation mode by the eigenvalue analysis. The introduction to the dynamic analysis is realized by using the Lagrangian equation of motion considering the mode deformation shown below.

  Next, an aspect in which the degree of freedom of movement of the bearing component and the degree of freedom of dynamic elastic deformation of the cage are numerically integrated simultaneously will be described based on the equation of motion. In the following Lagrangian equation of motion, the degrees of freedom of elastic deformation are introduced into the generalized coordinates and Lagrangian L and solved.

ここでｘ、ｙ、ｚは並進変位、ψ、θ、φは角変位、ｑi は、各変形モードのモーダル座標である。
以下、ラグランジアンＬの定義について説明する。
Ｌ＝Ｔ−Ｖ
ここでＴは運動エネルギである。弾性体の運動エネルギは微小要素ｄＶの運動エネルギの積分により次式のように表される。

Here, x, y, z are translational displacements, ψ, θ, φ are angular displacements, and qi are modal coordinates of each deformation mode.
Hereinafter, the definition of Lagrangian L will be described.
L = TV
Here, T is kinetic energy. The kinetic energy of the elastic body is expressed by the following equation by integrating the kinetic energy of the minute element dV.

ここで、式（１０）のｖは速度、ρは密度、ｍｐは有限要素の微小要素ノードｐの質量、ｖｐはノードｐの速度、ＧωＢｐはその角速度、Ｉｐはその慣性モーメントのテンソルである。速度ｖｐの算出は下式に基づく。

ここで、ＧＡＢは物体に固定した座標系から、慣性座標系への変換行列である。ｓｐはノードｐの弾性変形が無い場合の位置ベクトルで、ｕｐはその弾性変形ベクトルである。
変換行列の変化量については、下式の変換が可能である。

ここで、記号の上に付くチルダ（〜）はスキュー演算子であり、行列の積で外積演算を表すことができる。
また、一般化座標上の角変位の時間微分を慣性座標系に変換すれば角速度がえられるため、

の関係も成立する。ここで、Ｂは慣性から物体の基準点への座標変換行列である。
よって、速度は下式となる。

Here, v in equation (10) is the velocity, ρ is the density, mp is the mass of the microelement node p of the finite element, vp is the velocity of the node p, GωBp is the angular velocity, and Ip is the tensor of the moment of inertia. The calculation of the speed vp is based on the following equation.

Here, GAB is a transformation matrix from a coordinate system fixed to an object to an inertial coordinate system. sp is a position vector when there is no elastic deformation of the node p, and up is its elastic deformation vector.
The amount of change in the conversion matrix can be converted using the following equation.

Here, a tilde (˜) attached to the symbol is a skew operator, and can express a cross product operation by a matrix product.
Also, since the angular velocity can be obtained by converting the time derivative of angular displacement on generalized coordinates to the inertial coordinate system,

The relationship is also established. Here, B is a coordinate transformation matrix from inertia to the reference point of the object.
Therefore, the speed is as follows.

If the equation (10) is simplified by replacing v and ω, the following equation is derived on the generalized mass matrix and generalized coordinates.

If the contents of the mass matrix in the above equation are clarified, they are classified as shown in Equation (12). The subscripts t, r, and m represent translation, rotation, and mode displacement, respectively.

The nine M matrices in Expression (12) are expressed by the following expression.

  V is potential energy and is expressed by the following equation.

The stiffness matrix K in Equation (20) is Equation (21).

Here, Kmm is a generalized stiffness matrix in the mode coordinate system, where the mode matrix of the object is Φ and the stiffness matrix is K,
Kmm = Φ ^T KΦ
It becomes.

ここで、ｇは重力加速度ベクトルである。ラグランジェの運動方程式に入る一般化された重力による力は下式となる。

Vg is the potential energy due to gravity.

Here, g is a gravitational acceleration vector. The generalized gravity force that falls into Lagrange's equation of motion is

  F is dissipative energy, and is expressed by the following equation from Rayleigh's dissipative function. Here, D is a modal attenuation matrix.

  When the above equation is introduced into the Lagrangian equation of motion, the equation of motion in the generalized coordinate system of the elastically deformed portion is finally given by the following equation.

  FIG. 5 is a diagram illustrating a calculation result example of the interference force of the bearing during operation to the cage and the cage stress. In this figure, the color contour of the cage is Mises stress, and the inner ring, outer ring, and rollers are not displayed. As shown in FIG. 5, the interference force vector 12 from the roller acts radially outward from the retainer column and in a substantially tangential direction. In the stress analysis according to the present embodiment, the interference force greater than the columns existing in the other circumferential portions other than the top vicinity of the plurality of columns near the top of the cage disposed around the horizontal axis. Has been shown to be working. In addition, it is clear that the interference force vector 12 from the rollers acts on the plurality of columns near the top in the radially outward direction and in a direction different from the substantially tangential direction. . In the upper half of the cage, stress is applied along the inner periphery of the cage, and stress is applied from the inner periphery to the column and the outer periphery.

  According to the cage stress analysis method and stress analysis system described above, the dynamic elastic deformation characteristics of the cage (inherent deformation) are added to the three-dimensional dynamic analysis model of a rolling bearing in which the bearing components are regarded as rigid bodies. Mode and its frequency) are introduced based on the mode synthesis method, and the degree of freedom of elastic deformation and the degree of freedom of movement of the roller which is a bearing component are numerically integrated simultaneously. Thereby, the deformation history of the cage including the dynamic characteristics of the deformation is obtained. The cage stress can be obtained based on such deformation history. In the above dynamic analysis model, the other components of the cage are regarded as rigid bodies.

  Therefore, it is possible to calculate the cage stress during operation with high accuracy and efficiency as compared with the prior arts such as Non-Patent Documents 1 and 2. Therefore, for example, design changes that increase the load capacity by increasing the number of rolling elements without increasing the bearing size by reducing the cross-sectional area of the pillar of the cage or changing the pitch circle diameter at the center of the pillar. Can be calculated with high accuracy and efficiency. With this, for example, a rolling bearing retainer that reduces the contact surface pressure at high load, improves the life under severe lubrication conditions and foreign matter contamination lubrication conditions, and can achieve high rigidity at the same time, It can be obtained in a short time. Therefore, it is possible to reduce the cost of manufacturing the bearing. With this stress analysis method and stress analysis system, it is possible to establish an experimental alternative technique.

  In addition, since the degree of freedom of the super element method is set for each column 11a of the cage 11, the deformation mode of the column 11a is always included in the dynamic characteristics, and the remaining mode number of other natural deformation modes is set. By reducing it, an efficient and reasonable cage stress value can be obtained. In other words, the time required for the dynamic analysis can be shortened as compared with the conventional method of introducing a deformation mode up to a very high order.

In particular, in this embodiment, when the dynamic elastic deformation characteristics of the cage are introduced into the dynamic analysis model based on the mode synthesis method, a plurality of axial cross sections of the cage pillars are arranged. Set the degree of freedom. By doing so, even if the load from the roller approaches the corner where the stress is concentrated, the stress at the corner can be predicted with relatively high accuracy.
Two points of the installation locations with multiple degrees of freedom are the two points that are both ends of the roller in the axial direction within the range where the load is applied from the roller, or only these two points. The cage stress can be calculated with high accuracy while reducing the amount of calculation by reducing the number of installation points as much as possible. In addition, place the degree of freedom on the outer surface of the cage between the pockets of the cage and the inner surface of the cage on the side where displacement due to translation and torsion of the column becomes larger due to the load from the rollers. Therefore, the cage stress can be calculated with higher accuracy.

Next, methods (1) and (2) for improving the calculation accuracy of the cage stress by correction in the above embodiment will be described.
(1) Stress Correction by Finite Element Length FIG. 10 is a diagram showing the relationship between the finite element length of the pocket corner of the cage and the stress. This figure shows the case where the length of the pocket corner as a predetermined part used in the dynamic analysis is sufficiently short with respect to a finite element length of 0.7 mm, for example, 0.35 mm or less. The stress at the corner of the pocket is calculated by the finite element method.
That is, as a preliminary investigation, with respect to the pocket corner that becomes a stress concentration portion, a finite element length (see the dotted line in FIG. 10) used in the dynamic analysis model and a finite element length shorter than this finite element length (the same figure, The stress is obtained by the finite element method using each of the arrows.

  The ratio of these two stresses is obtained in advance. In this embodiment, the stress at the corner of the finite element used in the dynamic analysis, for example 0.7 mm, is 6 × 10 −3 KPa, and when this finite element length is shortened sufficiently and sufficiently, for example, 0.07 mm The stress at the corner of the pocket is 1 × 10 −2 KPa. The ratio of both stresses obtained by dividing 1 × 10 −2 KPa by 6 × 10 −3 KPa is “1.66”. Using this ratio as a correction coefficient α, the cage stress output in the cage stress calculation process is corrected by multiplication of the correction coefficient α. The correction coefficient α is stored in a rewritable manner in the RAM 7 or the like in the calculation means 3 shown in FIG. The cage stress correction is performed by the stress calculation unit 3b in FIG. For example, a correction calculation unit 3bc is provided in the stress calculation unit 3b, the cage stress output from the stress distribution conversion unit 3bb is corrected by the correction calculation unit 3bc, and the correction result is output from the output processing unit 3c. The correction coefficient α is not limited to “1.66”.

  As described above, once the correction coefficient α is obtained as a preliminary survey, the cage stress output in the cage stress output process is simply reduced without shortening the finite element length of the pocket corner each time. The stress estimation accuracy can be improved only by multiplying the correction coefficient α. As a result, the processing load on the computing means 3 can be reduced, and the cage life can be determined more accurately and easily. In this example, the example of the pocket corner portion has been described as the stress concentration portion, but the stress concentration portion is not limited to the pocket corner portion. For example, when there is a shape in which stress is likely to concentrate at the central part of the column (FIG. 13), the influence of the finite element length on the stress at that part may be analyzed as described above. In FIG. 13, a portion Hc surrounded by a round dotted line is a stress generation portion at the center of the column.

(2) Stress Correction by Mode Synthesis Method FIG. 11 is a diagram showing the relationship between the cage stress obtained by finite element analysis and the cage stress obtained by dynamic analysis by the mode synthesis method. In the dynamic analysis, the dynamic deformation of the cage is introduced into the dynamic analysis model based on the mode synthesis method to reproduce the cage deformation. It was found by carrying out the calculation that the stress value at the corner of the pocket in the dynamic analysis by the mode synthesis method is smaller than the analysis by the finite element method, that is, the FEM analysis. Ideally, by increasing the number of residual modes in the eigendeformation mode and the number of degrees of freedom of the constrained deformation mode in the super element method, the stress value at the target position is the same in the FEM analysis and the mode synthesis method. However, when the cage shape changes, the optimum analysis conditions of the super element method change, so it is necessary to consider each time. Therefore, the stress estimation accuracy can be improved by obtaining the ratio of the stress at the problematic part such as the pocket corner obtained by the finite element method and the stress at the same part obtained by the dynamic calculation as the correction coefficient β. Improve.

  In the present embodiment, two types of bearings B1 and B2 in which the cage thicknesses are mainly different were studied. In FIG. 11, the stress calculation result of the bearing B1 is represented by a one-dot chain line, and the stress calculation result of the bearing B2 is represented by a long dotted line. The stress value obtained by the FEM analysis was 1.6 times that of the bearing B1 and 1.2 times that of the bearing B2 with respect to the stress value obtained by the dynamic calculation. That is, as a preliminary survey, the correction coefficient β “1.6” in the bearing B1 and the correction coefficient β “1.2” in the bearing B2 can be obtained. This correction coefficient β is stored in a rewritable manner in the RAM 7 or the like in the calculation means 3 shown in FIG. 3, and is used for calculation. The correction of the cage stress is performed by the correction calculation unit 3bc. The correction coefficient β is not limited to “1.6” and “1.2”.

Therefore, once the correction coefficient β is obtained as a preliminary investigation, the stress estimation accuracy can be improved only by multiplying the stress value obtained by the mode synthesis method by the correction coefficient β. As a result, the processing load on the computing means 3 can be reduced, and the cage life can be determined more accurately and easily.
Note that the correction calculation unit 3bc in FIG. 3 may perform the stress correction by the finite element length or the stress correction by the mode synthesis method, and further perform both stress corrections. It may be. When the correction calculation unit 3bc performs both of the above corrections, the cage stress output in the cage stress calculation process is corrected by, for example, multiplying both the correction coefficient α and the correction coefficient β. To do. It is preferable to perform both corrections in terms of further improving the stress estimation accuracy. In this case, the life of the cage can be obtained more accurately.

  Next, a comparison example between a calculation example by actual correction and a cage life test is shown below. In the cage life test, radial vibration was forcibly applied to a pair of bearings during operation, and the cage breakage time was measured. The cage breakage time is synonymous with the cage life. The tapered roller bearing to be tested has main dimensions (inner diameter × outer diameter × width) of, for example, φ75 mm × φ160 mm × 40 mm, the number of rollers is 14, and the plate thickness (B1, B2) of the cage is different (B1 = 3). .5 mm, B2 = 5.0 mm) Two types of bearings were carried out. The bearing axial gap was 0.2 mm, and grease was applied as the lubricant. The rotation speed of the inner ring was 1000 rpm, and excitation was performed at a frequency of 100 Hz and 196 m / s2. A radial load of 1000 N per bearing was applied in the horizontal direction.

FIG. 12 is a diagram showing the relationship between the measured value of the cage life and the calculated value of the maximum Mises stress of the cage that allows for the correction coefficient. In these two types of bearings B1 and B2, the calculated values of the maximum Mises stress obtained by multiplying the two types of correction coefficient α and correction coefficient β are shown. The cage stress SB1 of the bearing in which the cage is broken exceeds the material fatigue limit FL indicated by the dotted line. Conversely, the cage stress SB2 of the bearing in which the cage is not damaged does not exceed the material fatigue limit FL. Therefore, the validity of the stress analysis of the cage according to the present embodiment can be confirmed by this cage life test. The cage breakage occurs from the corner on the small diameter side of the pocket, but high stress is generated at the same position in the stress analysis. High stress occurs when the rollers accelerate the cage upward.
In this embodiment, since the cage stress is corrected by multiplying by the correction coefficients α and β, it is possible to more accurately determine whether the cage stress exceeds or does not exceed the material fatigue limit FL. In this result, the consistency with the experimental results could be found for the first time in consideration of the correction coefficient β, but it is possible to estimate the cage stress level without the correction coefficient, which is useful information for strength design. It becomes. If the shape of the cage is unchanged, the correction coefficients α and β are not necessary when examining the influence of operating conditions on the cage stress.

Next, a description will be given of the results of a comparative examination of the calculation accuracy of the cage stress due to the difference in the position and number of the installation positions of the degree of freedom.
With respect to the cage of FIG. 1 (B), the degree of freedom is set in each of the five points (−2, −1, 0. + 1, +2) in FIG. In each case, a load simulating a roller load is sequentially applied to 10 points on the pocket surface, and the maximum principal stress at the corner of the pocket is determined by the finite element method and the mode synthesis method of the embodiment (however, the installation place of the degree of freedom is (Including the case not included in the embodiment).

The ratio of the stress of the finite element method to the mode synthesis method is shown in FIGS. The horizontal axis represents a load point, and each one is loaded at only one point. However, ALL is a case where a load is applied to all 10 points.
As can be seen from these figures, if the load application point is 0 U or 0 L which is the center of the column, the stress ratio is relatively small even in the pattern 1 provided with one degree of freedom. However, the stress ratio increases when it is biased to the end in the longitudinal direction, such as -2 or +2. In the case of the patterns 2B and 2C having two degrees of freedom, the increase width of the stress ratio due to the bias of the load points is reduced as compared with the pattern 1. As can be seen from FIG. 14 and FIG. 15, in the pattern 2C with the degrees of freedom at both ends, the stress ratio when the load points are at both ends is smaller than the pattern 2B, but when the load acts between −1 and +1. The pattern 2B having a shorter degree of freedom has a smaller stress ratio. With 3 degrees of freedom, the pattern 3E has a relatively small stress ratio regardless of the position at which the load is applied.
In pattern 5 in which the degree of freedom was set to all five points, the stress ratio was the lowest in the entire region. Therefore, in order to calculate a stress value close to the finite element method at any contact position on the contact surface of the pocket, the pattern 5 having a greater degree of freedom is preferable. However, the calculation cost in the dynamic analysis is the largest, and the difference from the pattern 3E is small. Therefore, in consideration of calculation efficiency and accuracy, the pattern 3E having a total of three points including two points at both ends and one point at the center is preferable.

  When estimating the stress at the pocket corner using the stress correction coefficient, it is desirable that the influence of the load application position on the stress ratio is small. FIG. 18 shows the ratio of the maximum value to the minimum value of the stress ratio when the load application position is changed. If this ratio is small, the finite element method, that is, the actual stress value can be estimated with relatively high accuracy by multiplying the stress value obtained by the mode synthesis method by the correction coefficient even if the load application position changes. From FIG. 18, pattern 5 is the best. The next smallest (good) is the pattern 2C, and the difference from the pattern 5 is very small. The calculation cost of the pattern 2C can be calculated efficiently because the degree of freedom is reduced by 60% compared to the pattern 5.

  In the above, the degree of freedom of the mode synthesis method was set on the outer diameter surface side of the cage pillar. However, it is also possible to actually set the degree of freedom on the inner diameter surface side of the column. Therefore, when the stress correction ratio when the installation location of the degree of freedom is set to the positions of −2 and +2 on the inner diameter side is obtained, FIGS. 19 to 22 are obtained. In the figure, the results of patterns 1 and 2C are shown for comparison. The one installed on the inner side is described as 2CI. The stress ratio is larger than that of the pattern 2C. It is appropriate that the degree of freedom is set at a location where displacement due to deformation is large. As shown in FIGS. 23 and 24, the deformation of the cage column this time is composed of a translational component in the diagonally upper left direction and a counterclockwise torsional component, and the outer diameter is larger than the inner diameter side (point I). The side (point O) has a larger displacement. Therefore, the selection of the inner diameter side or the outer diameter side in the installation of the degree of freedom should be the side on which the direction of translational deformation by the load from the roller coincides with the direction of deformation by torsion and the displacement increases. .

  Based on the above, the most preferable calculation method is to use a roller within the range where the load acts from the roller on the side where the displacement due to translation and torsion of the column becomes larger due to the load from the roller and on the central cross section between the pockets. Install two total degrees of freedom of the mode synthesis method at both ends of the shaft direction, perform dynamic analysis of the rolling bearing to calculate the stress of the cage, and add two types of stress correction factors determined in advance. It is to multiply.

  If the stress correction factor is not used, one more degree of freedom is added to the central part of the roller in the axial direction within the range where the column load is applied, and the cage stress is calculated by dynamic analysis of the rolling bearing. It is preferable to calculate in order to achieve both calculation accuracy and calculation speed.

  Next, a stress analysis system for a cage that numerically calculates stress generated in the cage of the rolling bearing during operation by two-dimensional dynamic analysis that requires a short calculation time will be described. The cage stress analysis system according to this embodiment is applied to a cage of a needle bearing or a cylindrical roller bearing, for example. However, the present invention is not necessarily limited only to the cage for these bearings, and is applied to stress analysis of a cage of a rolling bearing which is various roller bearings. The following description includes a description of a cage stress analysis method in the dynamic analysis of a two-dimensional needle bearing or cylindrical roller bearing.

  The above-described three-dimensional rolling bearing cage stress dynamics analysis system can be applied to various types of rolling bearings. However, in the three-dimensional dynamic analysis, since all the six degrees of freedom of each part and the three-dimensional degree of freedom of elastic deformation of the cage are simultaneously numerically integrated, the calculation cost is high. By the way, in a needle bearing, a cylindrical roller bearing, etc., there are cases where it is desired to handle only the behavior of an object on a radial plane. In this case, the dynamic analysis considering all the three-dimensional degrees of freedom is not efficient. In addition, although the degree of freedom of movement of the rigid parts can be easily constrained, it is difficult to constrain the axial displacement of the cage handled as an elastic body and the angular displacement around two axes excluding rotation.

Therefore, in the dynamic analysis system for the cage stress of the rolling bearing according to the second embodiment of the present invention, the degree of freedom of elastic deformation of the cage is limited to two dimensions, and the motion of the rolling elements and the bearing rings is limited. Limiting the degree of freedom to two dimensions also reduces the amount of processing required for numerical integration. Thereby, calculation can be completed in a shorter time than the case where the cage stress of a rolling bearing is performed by three-dimensional analysis. Similar to the first embodiment, the cage stress analysis system of the second embodiment includes an input unit 2, a calculation unit 3, and an output unit 4, as shown in FIG.
The calculation means 3 defines a dynamic analysis model of a rolling bearing that makes it possible to give the cage a degree of freedom of elastic deformation, assuming that the bearing components excluding the cage are rigid bodies. The analysis model setting unit 3a, and a process for introducing the dynamic elastic deformation mode and the natural deformation mode of the cage obtained by the super element method into the dynamic analysis model of the rolling bearing based on the mode synthesis method, and this introduction A process for calculating the deformation history of the cage including the dynamic characteristics of deformation by simultaneously numerically integrating the degree of freedom of elastic deformation and the degree of freedom of movement of the determined bearing components, and is calculated by this process By converting the deformation history into a stress distribution, a stress calculation unit 3b that performs processing for calculating the cage stress, and an output processing unit 3c that outputs the cage stress obtained by the stress calculation unit to the output unit. Have
However, in this embodiment, the dynamic analysis model set in the analysis model setting unit 3a and the stress calculation unit 3b limit the degree of freedom of elastic deformation of the cage to two dimensions, and the rolling elements and the raceway The degree of freedom of movement is also limited to two dimensions. This limitation to two dimensions is a plane including the axis, for example, the yz plane in FIG.

  When the degree of freedom of movement of the dynamic analysis is limited to two dimensions, the installation position of the deformation degree of freedom DOF of the cage will be described by taking the three-dimensional model in FIG. 4 as an example. Assuming the case where only the right side of FIG. 4 is left after cutting, there are a plurality of locations on the central cross section which is the center in the width direction of the column 13c, specifically, within the range in which the load acts from the rollers. The two points, one point (+2) which is both ends in the axial direction of the roller and the center part (0U). Moreover, it is set as the point on the surface of the side where the displacement by translation and torsion of column 13c becomes larger by the load from a roller among the cage outer diameter surface and cage inner diameter surface of column 13c. Each point (0, +2) for setting the above-mentioned degree of freedom is a point set when the outer diameter surface of the cage becomes a surface on the side where displacement due to translation and torsion of the column 13c becomes larger due to the load from the roller. If the surface on which the displacement due to translation and torsion of the column 13c becomes larger is the cage inner diameter surface, the one end on the inner diameter surface and the two points (0, +2) at the center thereof are set. .

  According to this configuration, dynamic elastic deformation characteristics of the cage are introduced into the dynamic analysis model based on the mode synthesis method, and the degree of freedom of elastic deformation and the degree of freedom of motion of the bearing components are numerically integrated simultaneously. Thus, the deformation history of the cage including the dynamic characteristics of the deformation can be obtained. Cage stress can be obtained based on the deformation history. In particular, the structure that restricts the elastic deformation of the cage and the motion of the rolling element, the raceway, and the rigid body mode of the cage to two dimensions in a shorter time than the case of the three-dimensional analysis. Calculation results can be obtained.

  As shown in FIG. 7, at the time of calculating the dynamic elastic deformation characteristics (natural deformation mode and its frequency) of the cage 11A to be introduced into the dynamic analysis model, the cage 11A is connected to each column 11b. Cut along the radial direction, that is, the yz plane, in the vicinity of the middle in the longitudinal direction (so-called cage center), and only one of the cuts is the analysis target. As shown in FIGS. 7 and 8, in the x, y, and z coordinate systems orthogonal to each other, the x direction represents the axial direction based on the rolling bearing, and the y and z directions are radial directions, respectively. Represents. The radial plane is synonymous with the yz plane.

  By giving a two-dimensional constraint (two translations in the radial direction and a degree of freedom of rotation about an axis perpendicular to the radial direction) to the cut surface sm of one of the cages 11A, and performing analysis by the super element method, the cage 11A Get dynamic characteristic information. The result of this dynamic characteristic information is introduced into the dynamic analysis model based on the mode synthesis method. Next, the deformation history of the cage 11A including the dynamic characteristics of the deformation is calculated by simultaneously numerically integrating the degree of freedom of movement of the bearing component and the degree of freedom of elastic deformation of the cage 11A.

  In the dynamic analysis, each part of the roller, the inner ring, and the outer ring is cut along a radial plane in the same manner as the cage 11A, and only one of the cut pieces is set as an analysis target. Then, for these parts and the cage analyzed by the super element method, a constraint condition limited to a two-dimensional degree of freedom is given to one representative point on each cross section in the dynamic analysis. The cut surface has already been two-dimensionally restricted on the super element method, and all the cut surfaces sm of the pillars 11b are constrained on this cut plane. If a two-dimensional constraint is applied to an arbitrary point, the degree of freedom of the rigid movement of the cage can be limited to only three degrees of freedom on the radial plane. Similarly to the first embodiment described above, if two translational degrees of freedom are provided for each degree of freedom installation location of each column 11b of the retainer 11A, the deformation mode of each column 11b of the retainer 11A is necessarily set. Even if the number of remaining normal eigenmodes is reduced, the deformation of each column can be reproduced well by the dynamic analysis, and the calculation time of the numerical integration can be shortened compared to the case of performing the three-dimensional analysis. Therefore, the time required for the cage design can be shortened, and the number of man-hours can be reduced.

  In the dynamic analysis according to the present embodiment, only one of the cage 11A cut along the yz plane in the vicinity of the middle in the longitudinal direction of each column 11b is an analysis target. For this reason, since the mass of the cage is half of the actual mass of the cage, each part of the roller, inner ring, and outer ring is cut along the radial plane in the same manner as the cage, and the mass of each part is actually Is half the mass of the parts. Therefore, the load and moment acting on the bearing are also half of the actual load and moment.

  In particular, in a needle bearing, the cage is often guided by a raceway, and it is necessary to evaluate the interference force between the cage and the raceway. In the needle bearing, when the cage is guided by the raceway, half of the interference force that should be originally introduced may be introduced by dynamic analysis. By doing this, in the equation of motion, the mass, contact spring (ratio of the magnitude of the interference force to the interference amount of the two objects), and force components are all half of the original, and the resulting behavior and elastic deformation are as much as possible. Correct value.

  FIG. 9 is a diagram illustrating a calculation result example of the interference force of the bearing during operation to the cage and the cage stress according to the second embodiment. In the second embodiment, an example is shown in which the cage stress of a rolling bearing under planetary motion by a planetary gear mechanism is calculated. In the planetary gear mechanism, a plurality of (for example, 3 to 5) planetary gears mesh with the externally toothed sun gear and the internal gear, and each planetary gear is rotatably supported on the outer periphery of the shaft integral with the carrier via a rolling bearing. It is a thing.

  In the rolling bearing, the outer diameter surface of the shaft and the inner diameter surface of the planetary gear serve as raceway surfaces on which the rolling elements roll. This rolling bearing includes a cage 14 in which pockets 13 are formed at a plurality of locations in the circumferential direction, and holds rolling elements 15 in each pocket 13. The rolling element 15 includes rollers such as needle rollers. For the degree of freedom of each component of the planetary gear mechanism, a model is used only for the degree of freedom in the radial plane. Moreover, in this model, the load peculiar to the planetary gear mechanism which acts on a rolling bearing shall be expressed by the following assumptions.

・ Consider the inertial force and centrifugal force of parts subject to dynamic analysis.
-Consider the contact force and tangential force of the rolling elements, cage, shaft, and planetary gear.
The rotation angular velocity of the carrier and the planetary gear is given as a constant or a known condition such as a known function. Therefore, the revolution angular velocity of the planetary gear is defined as the degree of freedom in dynamic analysis.
-The center positions of the external sun gear, the internal gear, and the carrier are fixed and coincide with each other. -Since the meshing clearance of the tooth surface contact portion between the planetary gear, the external sun gear, and the internal gear is larger than the radial internal clearance of the rolling bearing that supports the planetary gear, two translations of the planetary gear for this radial internal clearance Give a degree of freedom.
The interference force from the internal gear and the external sun gear to the planetary gear can be reduced to the translational force and rotation moment of the planetary gear in the revolution direction. Further, under the steady operation state, this translational force is determined by the transmission torque.

  In this figure, the collar contour of the cage is the maximum principal stress, and the interference force vector 16 from the outer ring to the roller 15 acts from the outer side in the radial direction toward the inner side. Thus, the interference force vector 17 from the inner ring to the roller 15 acts. The above is for planetary motion, but can be applied to all rolling bearings to be analyzed on the assumption of two-dimensional motion. In addition to planetary motion, the operating conditions can also be applied to connecting rod parts that perform crank motion and idler bearings frequently used in automobile manual transmissions.

  According to the cage dynamic analysis system and the dynamic analysis method according to the second embodiment described above, each of the rollers, the inner ring, the outer ring, and the cage analyzed by the super element method, In the dynamic analysis, a constraint condition limited to the two-dimensional degree of freedom is given to the representative points which are three points or two points. If two translational degrees of freedom are given to the two points at both ends of each pole of the cage, or the three points at both ends and the middle part in the longitudinal direction, the deformation mode of each pole of the cage is always output, and the normal eigenmode Even if the number of residuals is reduced, the deformation of each column can be reproduced well by the dynamic analysis, and the calculation time of the numerical integration can be shortened compared with the case of performing the three-dimensional analysis. Therefore, the time required for the cage design can be shortened, and the number of man-hours can be reduced. In particular, when only the behavior of an object on a radial plane is to be handled with a needle bearing or a cylindrical roller bearing, it is not necessary to undesirably consider all three-dimensional degrees of freedom. In the super element method, the radial plane (cut plane) of the cage is two-dimensionally restricted, and all the cut faces of the pillars are restricted on the cut plane. A two-dimensional constraint can be easily realized by cutting the cage along the radial plane in the vicinity of the middle in the longitudinal direction of each column, and using only the cut one as an analysis target.

1: dynamic stress analysis system 2: input means 3: calculation means 3a: analysis model setting unit 3ba: deformation history calculation unit 3bb: stress distribution conversion unit 3bc: correction calculation unit 4: output means 11: inner ring 12: outer ring 13: Cage 13a, 13b: Ring portion 13c: Pillar 14: Roller

Claims (11)

As a process of analyzing the stress of a roller bearing cage consisting of roller bearings, the dynamic analysis mode of the cage obtained by the super element method is added to the dynamic analysis model of the rolling bearing in which the bearing components are regarded as rigid bodies. The process of introducing the natural deformation mode based on the mode synthesis method, the degree of freedom of elastic deformation introduced in the above process, and the degree of freedom of motion of the specified bearing components are numerically integrated simultaneously, so that Dynamic stress analysis of rolling bearing cages including the process of calculating the deformation history of the cage including characteristics and the process of calculating the cage stress by converting the deformation history calculated in this process into a stress distribution In the method
The position of the degree of freedom to be introduced into the cage by the mode synthesis method is the displacement of the column between the cage pockets by the translation and torsion of the column due to the load from the roller, between the outer diameter surface of the cage and the inner diameter surface of the cage. Are points on the surface on the larger side, and a plurality of locations on the central cross section that is the center in the width direction of the column between the pockets of the cage,
Two points of these multiple places of freedom of installation are the two points on the central cross section of the column that are both ends in the axial direction of the roller within a range in which a load acts from the roller. A dynamic stress analysis method for a bearing cage.   The dynamic stress analysis method for a bearing cage according to claim 1, wherein installation positions with a degree of freedom to be installed at a plurality of locations of the pillar of the cage are only two points at both ends of the pillar.   2. The operation of the bearing cage according to claim 1, wherein the installation locations of the degree of freedom to be installed at a plurality of locations of the pillar of the cage are three points, two points at both ends of the pillar and a central portion between the two points. Stress analysis method.   4. The dynamic stress analysis method for a bearing cage according to claim 2, wherein the dynamic analysis model is a three-dimensional model.   5. The finite element length used in the dynamic analysis model and a shorter length than the finite element length for each point that is the installation location of the degree of freedom as a preliminary survey according to claim 2. Obtain stress under the same load condition by the finite element method using each finite element length, find the ratio of both stresses, use this ratio as a correction factor by the finite element length, and convert the deformation history into stress distribution A dynamic stress analysis method for a bearing cage, wherein the cage stress obtained in the calculation process is corrected by multiplication of a correction coefficient related to the finite element length.   6. Dynamic calculation using the process of introducing the elastic deformation characteristics, the process of calculating the deformation history, and the process of calculating the cage stress as preliminary investigations according to any one of claims 2 to 5. To obtain the cage stress by the finite element method, obtain the ratio of the stress obtained by the finite element method and the stress obtained by the dynamics calculation, , The correction coefficient according to the mode composition method, and the cage stress obtained in the process of calculating the deformation history into the stress distribution in the stress analysis process of the cage to be analyzed is related to the mode composition method. A method for stress analysis of a cage, wherein correction is performed by multiplication of a correction coefficient.   In Claim 1, the installation location of the degree of freedom to be installed at a plurality of locations on the central cross section of the pillar of the cage is three points, two points on both ends of the pillar, and a central portion between these two points, A dynamic stress analysis method for a bearing cage, wherein the cage stress obtained in the process of calculating by converting the deformation history into a stress distribution is a cage stress output as an analysis result. In Claim 7, the freedom degree of movement of the rigid body mode of the roller, the race, and the cage, and the degree of freedom of elastic deformation of the cage are limited to two dimensions,
And the installation position of the degree of freedom to be introduced into the cage by the mode synthesis method is based on the translation and torsion of the column due to the load from the roller among the cage outer diameter surface and the cage inner diameter surface between the pockets of the cage. It is a point on the surface on the side where displacement becomes larger, and a plurality of locations on the central cross section that is the center in the width direction of the column between the pockets of the cage,
One point of the plurality of degrees of freedom of installation points is one point that becomes an axial end portion of the roller within a range in which a load is applied from the roller on the central cross section of the column. A dynamic stress analysis method for a bearing cage.   9. The dynamic stress of the bearing cage according to claim 8, wherein installation positions with a degree of freedom to be installed at a plurality of locations of the pillar of the cage are only two points, that is, one point of the column end and a center point in the longitudinal direction of the column. analysis method.   The rolling bearing according to any one of claims 1 to 5, wherein the rolling bearing is a tapered roller bearing, and the cage is provided between a pair of annular portions at both ends in the axial direction and the annular portions. A dynamic stress analysis method for a bearing retainer having a conical window shape including a plurality of pillars and a flange shape in which a ring portion on the large diameter side protrudes toward the outer diameter side. A dynamic stress analysis device for a bearing cage that analyzes the stress of a cage of a rolling bearing composed of a roller bearing, comprising an input means, a calculation means, and an output means,
The calculation means regards the bearing component excluding the cage as a rigid body, and an analysis model setting unit that defines a dynamic analysis model of the rolling bearing capable of giving the cage flexibility of elastic deformation;
A process for introducing the dynamic elastic deformation mode and natural deformation mode of the cage obtained by the super element method into the dynamic analysis model of the rolling bearing based on the mode synthesis method, and the degree of freedom of the elastic deformation introduced. Processing to calculate the deformation history of the cage including the dynamic characteristics of deformation, and to convert the deformation history calculated by this processing into a stress distribution By doing so, a stress calculation unit that performs processing to calculate the cage stress,
An output processing unit that outputs the cage stress obtained by the stress calculation unit to the output unit;
Have
The stress calculation unit is arranged in accordance with the load from the roller on the outer diameter surface of the cage between the pockets of the cage and the inner diameter surface of the cage. The points on the surface on the side where the displacement due to translation and torsion becomes larger, and a plurality of locations on the central cross section that is the center in the width direction of the column between the pockets of the cage,
Two points of these multiple places of freedom of installation are the two points on the central cross section of the column that are both ends in the axial direction of the roller within a range in which a load acts from the roller. A dynamic stress analysis system for bearing cages.

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