JP2009104492A - Method and apparatus for calculating contact surface pressure and subsurface stress under conformal contact - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a method capable of calculating contact surface pressure and subsurface stress under conformal contact in a relatively short period of time and with accuracy without using finite element method or boundary element method. <P>SOLUTION: A reference curved surface S is set between two contacting objects M1, M2 and is divided into a mesh where cells are arranged vertically and horizontally. Calculation points for discretely calculating the distance between the contacting objects M1, M2 are defined on the cells. The amount of influence that a point corresponding to each calculation point on the surfaces of the contacting objects M1, M2 has on displacement when a distributed load acts on each cell is calculated. The surface pressure of each cell is calculated using balancing conditional expressions that require that the total of the products of the surface pressures and the amounts of influences of the cells balance a difference between the amount that the contacting objects approach each other and an initial gap between them, and that the total of the surface pressures of the cells balance an external force. In a process for calculating the influence of the displacement and a process for calculating contact surface pressure, the distance over the reference curved surface is used in the calculation of the distance between measuring points, and surface pressure is calculated by making a correction to a normal direction on the basis of a difference in position on the reference curved surface. <P>COPYRIGHT: (C)2009,JPO&INPIT

Description

  The present invention relates to a calculation method and apparatus for calculating contact surface pressure and subsurface stress between contact objects whose contact area cannot be regarded as flat without using a finite element method or a boundary element method.

The contact pressure calculation problem can be divided into the following two fields.
One is a non-conformal contact problem (Counter-formal (or Non-Conforming) Problem), which is a problem when the contact area is smaller than the object radius of the contact portion.
The other is a conformal contact problem (Conformal (or Conforming) Problem), which is a problem when the contact area is larger than the object radius of the contact portion.

In the non-conformal contact problem (Counter-formal Problem), the contact area can be regarded as being almost on a flat surface, so that it is relatively easy to solve.
In particular, when the contact surface is a curved surface represented by a quadratic function, the Hertz contact theory is widely used, and calculation can be performed at the calculator level using a numerical table. For curved surfaces that cannot be expressed by quadratic functions, a method is known in which the surface pressure distribution is calculated numerically by superimposing analytical solutions when a concentrated load acts on a semi-infinite body (Boussinesq). In addition, various documents (for example, Non-Patent Document 1 and Non-Patent Document 2) have been issued with solutions similar to these, and surface pressure calculation software based on this method is also commercially available.
JP 59-69519 A Harnett, MJ, “The Analysis of Contact Stresses in Rolling Element Bearings”, ASMEJ. Of Lubr., 101, Jan. p105, (1979), Author Sakurai, “Boundary element analysis of three-dimensional elastic contact problems”, Tribologist, 36, 6, (1991), 429-433

However, for the conformal problem, there is no calculation method other than solving by using the finite element method or the boundary element method because the above analysis method cannot be applied.
When performing calculations using the finite element method or the boundary element method, there is a problem that the calculation time becomes long.

As a method and apparatus for solving such a problem, the present applicant has previously proposed the following method and apparatus (Japanese Patent Application No. 2007-026577).
This proposed example is a method for calculating surface pressure and subsurface stress by applying analytical solutions of displacement and subsurface stress when concentrated load is applied to a semi-infinite body to a conformal contact problem. When acting, superimpose using the Boussinesq solution. At this time, a reference curved surface is set in the vicinity of the curved surface of the two contact objects, the reference curved surface is divided into meshes, and a point for measuring the distance between the two objects is set on the mesh. The influence of the displacement amount on the measurement point between the two objects when acting is calculated. A contact area is determined by contact determination at that position, and an amount of approach between the two objects is determined so that the total surface pressure of each cell is balanced with the external force, and the contact surface pressure is calculated.

According to the proposed example, the contact surface pressure in conformal contact and subsurface stress can be calculated relatively easily and in a short time and with high accuracy compared to methods using the finite element method or boundary element method. Can do.
However, in the above proposed example, since the displacement in three directions is taken into consideration, the calculation method is still complicated, and it takes time to perform numerical integration. Therefore, it is desired to enable faster calculation.

  The object of the present invention is to extend the numerical calculation method performed in the non-conformal contact problem without using the finite element method or the boundary element method, and to apply it to the conformal contact problem, and the above proposal A method that improves the example and allows calculation in consideration of displacement in one direction, and enables calculation of contact surface pressure in conformal contact and calculation of subsurface stress in a short time and with high accuracy. And an object to provide an apparatus.

  The calculation method of contact surface pressure and subsurface stress under conformal contact of the present invention applies the analytical solution of displacement and subsurface stress when a concentrated load is applied to a semi-infinite body to the conformal contact problem. And a subsurface stress calculation method. As an analytical solution, a Boussinesq solution is used when a normal load is applied.

Specifically, the contact surface pressure calculation method under conformal contact according to the present invention includes two contacts that perform conformal contact with each other when the external force is applied (contact that the contact region cannot be regarded as substantially on a plane). A method for calculating the surface pressure at the contact surfaces of the objects M1 and M2 using a computer,
An input process (S1) in which a shape near the contact surface of the contact object, an elastic coefficient, a Poisson's ratio, and an acting external force are input and stored in a storage means, and an analysis for calculating the surface pressure using the input value It consists of a process (S2) and an output process (S3) for outputting the calculated result.
The analysis process (S2) includes the following processes (T1, T2, T4, T5).

That is, a reference curved surface S having a shape close to the contact surface shape is set in the vicinity of the surfaces of the two contact objects M1 and M2 that are in contact with each other, and the reference curved surface S is divided into meshes in which cells C are arranged vertically and horizontally. , M2 is a mesh division process (T1) for defining a calculation point A on the cell, which is a point for calculating the distance between M2,
A displacement influence calculation step (T2) for calculating an influence amount on a displacement of a point corresponding to each calculation point A on the surface of the contact object M1, M2 when a unit distribution load is applied to each cell C;
The sum of the product of the surface pressure P and the influence amount due to the unit distributed load for each cell C balances the difference between the approach amount between the contact objects and the initial gap in the contact area, and the sum of the surface pressure P of each cell C is the external force. It includes a contact surface pressure calculation process (T3) in which the surface pressure P of each cell is calculated from a balance conditional expression of “F”.

  In the displacement influence calculation process (T2) and the contact surface pressure calculation process (T3), the distance on the reference curved surface is used to calculate the distance between measurement points. Also, the normal pressure is corrected by the difference in position on the reference curved surface, and the surface pressure is calculated. That is, the surface pressure is calculated by correcting the normal direction of the reference curved surface in calculating the influence coefficient.

The reference curved surface S is preferably a shape close to the contact surface shape of the surfaces of the two contact objects M1 and M2 that are in contact with each other. In the displacement influence calculation process (T2), the unit distribution load acting on each cell C is, for example, a uniform surface pressure, but a surface pressure that varies according to a linear function or a quadratic function may be assumed. The distance between the contact objects M1 and M2 is evaluated at discrete measurement points A. The measurement point A is, for example, at the center of the mesh.
In the displacement influence calculation process (T2), a displacement analysis solution when a unit concentrated load is applied (for example, a Boussinesq solution for a load in the normal direction is integrated over the entire area of one element to obtain one element surface. A displacement amount of a point on the surface of the two contact objects M1 and M2 (corresponding to the measurement point A) generated by the pressure distribution is obtained, and the “element” is a cell.

The method of the present invention can be applied when the two objects M1 and M2 are an inner ring and a ball of a rolling bearing. In that case, in the mesh division process (T1), the reference curved surface S is a cylindrical surface.
In the displacement influence amount calculation step (T2), the coordinate position in the direction along the circular arc of the cross section cut by the plane perpendicular to the cylindrical central axis on the reference curved surface S composed of the cylindrical surface is x, and the direction along the cylindrical central axis. Where y is the coordinate position, z is the coordinate position away from the normal direction on the reference curved surface S, and the displacements in these three axes are u, v, and w, respectively.
When the external force is a force in the normal direction, the influence amount is calculated by superimposing displacement analysis solutions w at the time of unit concentrated load action expressed by the following equation (1) for all cells.

That is, in the method of the present invention, in the displacement influence amount calculation step (T2), the displacement analysis solution (Boussinesq's solution) at the time of unit concentrated load action is integrated over the entire area of one element, and is generated by the surface pressure distribution of one element. Displacement amounts of points on the surfaces of the two contact objects M1 and M2 (corresponding to the measurement point A) are obtained.
The displacement at the time of unit surface pressure load exerted on the position (θ ′, y ′) by the element e at the position (θ, y) is given by the equation (1). At this time, since the normal direction of the reference curved surface S changes depending on the position, COS (θ−θ ′) is applied.
What is important here is that (1) the distance (R0θ−R0θ ′) on the reference curved surface is used as the θ direction distance between elements, and (2) the correction of the normal direction of the reference curved surface S is performed by the COS (θ−θ ') (That is, the surface pressure is calculated by correcting the normal direction of the reference curved surface S in calculating the influence coefficient).
The above equation (1) can be integrated analytically, and numerical integration is not necessary.

The contact surface pressure calculation process (T3) is a process of calculating the surface pressure P of each cell from the balance condition equation, and is calculated as follows, for example.
That is, the surface pressure distribution of each cell is determined so that the following equation (2) is balanced.

  Further, the approach amount α is obtained so that the following equation (3) is balanced.

  The formula (2) is a balance condition formula that the sum of the product of the surface pressure and the influence amount due to the unit distribution load for each cell balances the difference between the approach amount between the contact objects and the initial gap inside the contact region, Further, the displacement w calculated by the equation (1) is overlaid for all elements. The expression (3) is a balance condition expression that the total surface pressure of each cell is balanced with the external force.

According to the method of the present invention, an analytical solution of displacement when a concentrated load is applied to a semi-infinite body is applied to a conformal contact problem, and the distance between measurement points at the time of contact surface pressure calculation is calculated on the reference curved surface. Since the surface pressure is calculated by using the distance and correcting the normal direction of the reference curved surface to calculate the influence coefficient, the contact surface pressure in conformal contact can be calculated relatively easily and in a short time. It can be carried out.
In addition, it was confirmed by comparing the calculation results between the method of the present invention and the finite element method that the calculation results were as accurate as when the contact surface pressure was calculated using the finite element method. .
In this way, the present invention applies a solution of the Buenesque semi-infinite approximation to the conformal contact problem, which was previously considered to be inapplicable to the conformal contact, and provides an analysis of good surface pressure. The result can be obtained.
Although the method of the present invention can be applied to non-conformal contact, the surface pressure can be obtained by simpler calculation in the case of non-conformal contact. It becomes.

In the method of the present invention, when the subsurface stress is calculated, the subsurface stress is calculated after calculating the surface pressure P of each cell C by any of the above methods.
In this subsurface stress calculation process,
When the load on the cell is a force in the normal direction, it is integrated over the entire area of each cell C using the subsurface stress at the time of unit concentrated load action given by the following equations (4) to (9). The subsurface stress is obtained by taking the sum of the effects of all the cells C.

When the load on the cell is a normal force, it is calculated as follows. When the two contact objects M1 and M2 are a ball and an inner ring of a rolling bearing, the reference curved surface S is a cylindrical surface as described above in the mesh division process (T1). In the displacement influence amount calculation step (T2), the coordinate position in the direction along the circular arc of the cross section cut by the plane perpendicular to the cylindrical central axis on the reference curved surface S made of the cylindrical surface is θ, and the direction along the cylindrical central axis Where y is the coordinate position of z and z is the coordinate position away from the normal direction on the reference curved surface.
The shear stress τzy at the time of unit surface pressure load exerted on the position (θ ′, y ′, z) by the element e at the position (θ, y) is obtained by the following equation (10).

  As described above, the subsurface stress can be calculated relatively easily and in a short time, and the subsurface stress can be calculated with high accuracy.

The method of the present invention may include a mesh correction process (T6) in which the size of the mesh is corrected during the analysis process, and each process after the mesh division is performed again according to the correction result.
Depending on the shape of the contact surface of the contact object, the error in discretization may increase due to the edges of the contact area becoming jagged, but the error in discretization is reduced by correcting the mesh during the analysis. Can be made.

  The method of the present invention can be effectively applied when the two contact objects M1 and M2 are a race ring which is an inner ring or an outer ring in a rolling bearing and a rolling element such as a ball. Even in the contact between the bearing ring of the rolling bearing and the rolling element such as a ball, the contact area is not considered to be substantially flat, and it is necessary to obtain the surface pressure as a problem of conformal contact. The method can be applied effectively.

The contact surface pressure calculation device under conformal contact according to the present invention is a device for calculating the surface pressure at the contact surface of two contact objects that make conformal contact with each other when an external force is applied.
Input processing means 4 for inputting the shape of the contact surfaces of the contact objects M1, M2 in the vicinity of the contact surface, the elastic coefficient, the Poisson's ratio, and the acting external force F and storing them in the storage means 6, and the surface pressure using the input values The analysis means 7 for calculating the output and the output processing means 5 for outputting the calculated result,
As the analysis means 7, the following means 8, 9, 11, 12 are provided.

That is, a reference curved surface S is set in the vicinity of the surfaces of the two contact objects M1 and M2 in contact with each other, the reference curved surface S is divided into meshes in which cells C are arranged vertically and horizontally, and the distance between the contact objects M1 and M2 is determined. Mesh dividing means 8 for defining a calculation point A, which is a point to be calculated, on a cell;
A displacement influence calculating means 9 for calculating an influence amount on a displacement of a point corresponding to each calculation point A on the surface of the contact object M1, M2 when a uniform unit distribution load is applied to each cell C;
The sum of the product of the contact pressure P and the influence amount due to the unit distributed load for each cell C is balanced with the difference between the approach amount between the contact objects and the initial gap in the contact area, and the sum of the contact pressure P of each cell C is And a contact surface pressure calculating means 10 for calculating the surface pressure P of each cell from a balance conditional expression of balancing with an external force.
The displacement influence calculation means 9 and the contact surface pressure calculation means 10 use the distance on the reference curved surface S for calculating the distance between the measurement points, and correct the normal direction by the difference in position on the reference curved surface S. Go to calculate the surface pressure. That is, the surface pressure is calculated by correcting the normal direction of the reference curved surface S in calculating the influence coefficient.

In the subsurface stress calculation apparatus according to the present invention, when the two contact objects M1 and M2 are an inner ring and a ball of a rolling bearing, the following configuration may be adopted.
The mesh dividing means 8 uses the reference curved surface S as a cylindrical surface.
The displacement influence amount calculation means 9 has a coordinate position in the direction along the circular arc of the cross section cut by a plane perpendicular to the cylindrical central axis on the reference curved surface S made of the cylindrical surface, and x in the direction along the cylindrical central axis. The position is y, the coordinate position away from the normal direction on the reference curved surface is z, and the displacements in these three axes are u, v, and w, respectively.
When the external force is a force in the normal direction, the influence amount is calculated by superimposing displacement analysis solutions w at the time of unit concentrated load action expressed by the following equation (1) for all cells.

  In this case, the contact surface pressure calculation means 10 calculates the following equation (2), which is an equation for superimposing the displacement w in the z-axis direction calculated by the equation (1) for all the cells, To obtain the surface pressure distribution of each cell.

  Moreover, the said approach amount (alpha) is calculated | required using following Formula (3) as another one of the said balance conditional expressions.

  According to the surface pressure calculation apparatus having this configuration, the surface pressure at the contact surface that performs conformal contact can be calculated relatively easily in a short time and with high accuracy by implementing the method of the present invention.

In the surface pressure calculation device according to the present invention, when a subsurface stress calculation function is further added, the subsurface stress calculation device is configured by providing the following subsurface stress calculation means 13.
When the load on the cell is a normal force, the subsurface stress calculation means 13 uses the subsurface stress at the time of unit concentrated load action given by the following equations (4) to (9) to individually The subsurface stress is obtained by integrating over the whole area of the cell and summing the influence of all the cells.

  In the case of this configuration, the subsurface stress calculation method of the present invention can be performed, and the subsurface stress on the contact surface that performs conformal contact can be calculated relatively easily in a short time and with high accuracy.

According to the method and apparatus for calculating the contact surface pressure under conformal contact according to the present invention, the analytical solution of displacement when a concentrated load is applied to a semi-infinite body is applied to the conformal contact problem, and Since the distance on the reference curved surface is used to calculate the distance between measurement points, and the normal pressure direction of the reference curved surface is corrected to calculate the influence coefficient, the surface pressure is calculated relatively easily, and therefore short. It is possible to calculate the contact surface pressure in conformal contact with time, and to obtain an accurate calculation result. In particular, when an analysis solution is made to superimpose a business solution in the case of a normal direction load action, it is possible to obtain a calculation result as accurate as that calculated using the finite element method.
According to the method and apparatus for calculating the contact surface pressure and the subsurface stress under conformal contact of the present invention, in addition to the contact surface pressure, the subsurface stress is also relatively simple and accurate in a short time. Calculations of subsurface stress in conformal contact can be made.

An embodiment of the present invention will be described with reference to FIGS. FIG. 1 is a flowchart of each process of the method of the present invention, FIG. 2 is a block diagram of a conceptual configuration of an apparatus for carrying out the method of the present invention, FIG. 3 is a specific example of a contact surface pressure calculation process, and FIG. FIG. 5 shows examples of reference curved surfaces.

This contact surface pressure and subsurface stress calculation method under conformal contact is based on the calculation of the surface pressure at the contact surfaces of the two contact objects M1 and M2 (FIG. 4) that perform conformal contact with each other when an external force is applied. It is a method of calculating using. The two contact objects M1 and M2 are, for example, a ball and an inner ring in a rolling bearing.
This calculation method is an input process in which the shape near the contact surface of the contact objects M1 and M2, the elastic modulus (here, Young's modulus E and Poisson's ratio ν), and the acting external force F are input and stored in the storage means. (S1), an analysis process (S2) for calculating the surface pressure using the input value, and an output process (S3) for outputting the calculated result. The analysis process (S2) includes the following mesh division process (T1), displacement influence calculation process (T2), contact surface pressure calculation process (T3), and subsurface stress calculation process (T4), which are sequentially performed.

A computer 1 (FIG. 2) serving as a computing device has a central processing unit (CPU) and storage means (not shown) such as a memory, and the hardware (including an operation system) of the computer 1 and this Each means shown in a conceptual configuration in FIG. 2 is configured by a contact surface pressure / subsurface stress calculation program (not shown) executed by the computer 1.
The input processing means 4 is a means for inputting the shape near the contact surface of the contact objects M1 and M2, the elastic coefficient, the Poisson's ratio, and the acting external force F from the input device 2 and storing them in the input information storage means 6. The “shape” here is a concept including size. For example, the shape data in the vicinity of the contact surfaces of the contact objects M1 and M2 include the radius R1 of the contact object M1 made of a ball, the groove radius R2 of the raceway surface of the contact object M2 made of an inner ring, and the PCD and contact angle of the bearing. Enter. In addition to the information on the shape and the like, the input processing unit 4 inputs, from the input device 2, the size of each mesh in the mesh direction divided by the analysis unit 7 described later, the number of cells (NA, NB), and the like. It is good also as a thing provided with the function memorize | stored in the input information storage means 6. Further, the input processing means 4 may have a function of performing preprocessing for calculation by the analysis means 7 on the information input from the input device 2. The input device 2 may be a keyboard or the like, or may be a communication unit or a storage element. The input information storage unit 6 is a predetermined storage area in the storage unit of the computer 1. The output processing means 5 is a means for outputting the result calculated by the analyzing means 7 to the output device 3 composed of a liquid crystal display, a printer, a communication device or the like.

The analysis unit 7 is a unit that calculates the surface pressure and the subsurface stress using the data input by the input processing unit 4 and stored in the input information storage unit 6, and includes a mesh dividing unit 8 and a displacement influence calculation unit 9. A contact surface pressure calculating means 10 and a subsurface stress calculating means 11. The means 8 to 12 provided in the analyzing means 7 are respectively a mesh division process (T1), a displacement influence calculation process (T2), a contact surface pressure calculation process (T3), and a surface in the flowchart of FIG. It is means for executing the lower stress calculation process (T4), and stores calculation formulas, set values, and the like necessary for the execution. The specific contents of the means 8 to 11 are as described in the steps (T1) to (T4) described below.
In addition to the above, the analysis means 7 has a mesh correction means 12 described later.

The analysis process (S2) of FIG. 1 (A) will be described with reference to FIG. 1 (B) and FIGS.
This analysis assumption (S2) is a method for solving the conformal contact problem by extending the Hartnett method when normal direction weighting is applied.

First, as a mesh division process (T1), a reference curved surface S is set between the contact objects M1 and M2 as shown in FIG. The reference curved surface S is divided into meshes in which cells C are arranged vertically and horizontally as shown in FIGS. 5 and 6, and a calculation point A, which is a point for calculating the distance between the contact objects M1 and M2, is determined on the cell C.
The reference curved surface S is not necessarily between the two contact objects M1 and M2, but is set in the vicinity of the surfaces of the two contact objects M1 and M2 that are in contact with each other. The reference curved surface S is a curved surface having a shape close to the contact surface shape of the two contact objects M1 and M2.
The contact objects M1 and M2 are, for example, a ball 23 and an inner ring 21 in the rolling bearing shown in FIG. The rolling bearing shown in the figure is an angular ball bearing, a deep groove ball bearing, or the like in which a plurality of balls 23 are interposed between the raceways having a circular arc cross section of the inner ring 21 and the outer ring 22 and a cage 24 for holding the balls 23 is provided. Ball bearing.
When the contact objects M1 and M2 are the ball and the inner ring in the rolling bearing as described above, the reference curved surface S is regarded as a cylindrical surface. In the figure, the reference curved surface S is a cylindrical surface interposed between the track surface of the contact object M2 that is an inner ring and the outer spherical surface of the contact object M1 that is a ball, and the cylindrical central axis is a contact object that is an inner ring. It is parallel to the axis of M2.

  In addition, regarding the reference curved surface S, the exact shape of the contact surface shape is not originally known, but it can be considered that the contact surface shape is close to a part of the cylindrical surface in the contact between the ball and the inner ring (this reference surface is also used for analysis). ). Further, when the convex spherical surface and the concave spherical surface are in contact, the contact surface shape can be regarded as a substantially spherical surface. The conventional “contact surface can be regarded as flat” is a flat surface. There are cases where complicated curved surfaces cannot be expressed with simple curved surfaces as described above. How the reference surface shape is made affects the analysis accuracy, so it is better to be as close as possible to the contact surface shape, but there is a demerit that calculation becomes difficult if it is complicated. Accordingly, the “reference curved surface S” is appropriately determined by the person who analyzes it on a case-by-case basis.

The calculation point A is a point that represents the position of the cell C in order to evaluate the distance between the contact objects M1 and M2 as discrete points. The calculation point A may be set at an appropriate position in the cell C. Here, the calculation point A is set at the center of the cell C.
The size of the mesh to be divided into meshes, that is, the size of the cell C may be set as appropriate, but may be freely set by input from the input processing means 4 or the like. Further, the width of the reference curved surface S is appropriately set wider than the contact area so as to include all the contact areas of the contact objects M1 and M2. Here, it is assumed that the reference curved surface S includes NA cells C in the X-axis direction and NB cells in the Y-axis direction.

  In the displacement influence calculation process (T2), after the mesh division process (T1), the points corresponding to the respective calculation points A on the surfaces of the contact objects M1 and M2 when the uniform distributed load P is applied to each cell C. The amount of influence on the displacement is calculated. In this case, it is assumed here that the uniform surface pressure P acts on the individual cells C as a whole, but the surface pressure that varies depending on the position in the cell C according to a predetermined function such as a linear function or a quadratic function. You may assume that it works.

At this time, the coordinate positions in the three axial directions orthogonal to each other are calculated as x, y, z, and the displacements in these three axial directions are calculated as u, v, and w, respectively. Note that the displacements u and v in the x and y axis directions are described later in order to explain the displacement analysis solution (Boussinesq's solution) when the unit concentrated load is applied. The calculation is performed using only the displacement w in the z-axis direction without using the displacements u and v.
When the two contact objects M1 and M2 are rolling bearing balls and inner rings as described above and the reference surface S is a cylindrical surface, the reference surface S is a cylindrical surface perpendicular to the central axis of the cylinder. The coordinate position in the direction along the circular arc of the cross section cut by the plane is x, the coordinate position in the direction along the cylindrical central axis is y, the coordinate position away from the normal direction on the reference curved surface is z, and the displacement in these three axis directions is Let u, v, and w respectively. The x coordinate position on the reference plane S is represented by polar coordinates Ro θ expressed by an angle θ.

In the displacement influence calculation process (T2), the surface of the two-contact objects M1 and M2 generated by the surface pressure distribution of one element is obtained by integrating the displacement analysis solution (Boussinesq's solution) at the time of unit concentrated load action over the entire area of one element. A displacement amount of the upper point (corresponding to the measurement point A) is obtained.
That is, the displacement at the time of unit surface pressure load exerted on the position (θ ′, y ′) by the element e at the position (θ, y) is given by the following equation (1). At this time, since the normal direction of the reference curved surface changes depending on the position, COS (θ−θ ′) is applied. Here, one cell C is calculated as one element.
What is important here is that (1) the distance on the reference plane (R0θ−R0θ ′) is used as the θ direction distance between elements, and (2) the normal direction correction of the reference plane is corrected by COS (θ−θ ′ ) (That is, the surface pressure is calculated by correcting the normal direction of the reference curved surface S in calculating the influence coefficient).

  In addition, said Formula (1) is guide | induced as mentioned later from the displacement analysis solution (solution of Boussinesq) at the time of a unit concentrated load effect | action.

  In the contact surface pressure calculation process (T3), the sum of the product of the surface pressure and the influence amount due to the unit distributed load for each cell C is balanced with the difference between the approach amount between the contact objects M1 and M2 and the initial gap in the contact region. Further, the surface pressure of each cell is calculated from a balance conditional expression that the total surface pressure of each cell C is balanced with the external force.

In the contact surface pressure calculation process (T3), specifically, the surface pressure distribution is determined so that the following equation (2) is balanced, and the approach amount α is determined so that the equation (3) is balanced. Equation (2) is a condition that the sum of the product of the surface pressure Pkl and the influence amount fijkl due to the unit distribution load for each cell Cij is balanced with the difference Dij between the approach amount between the contact objects M1 and M2 and the initial gap. This is an equation that superimposes the displacement w obtained by equation (1) for all elements. Expression (3) is a conditional expression that the sum of the surface pressure Pij of each cell Cij is balanced with the external force F.
Note that Cij indicates the cell C that is i-th in the x-axis direction and j-th in the y-axis direction. Qkl indicates the k-th point in the x-axis direction and the l-th point A in the y-axis direction (where i: 1 to NA, j: 1 to NB, k: 1 to NA, l: 1 to NB).

If the surface pressure value of each cell C is determined so that the above equations (2) and (3) are balanced, the discrete surface pressure distribution can be obtained.
In the process of solving the equation (2) as described above, for the cell Cij in which the difference Dij between the approach amount between the two contact objects M1 and M2 and the initial gap is a positive value, the surface pressure of the cell Cij is The contact area is the arrangement range of the cells Cij in which the difference Dij between the approach amount and the initial gap is zero or a negative value.
Equation (2) can be integrated analytically, and numerical integration is not necessary.

FIG. 3 shows an example of an algorithm for solving the equations (2) and (3) by the contact surface pressure calculation means 10 in the computer.
First, the initial value of the α value (: the approaching amount of the two contact objects M1 and M2) is set (step H1).
The contact surface pressure of each cell Cij is set according to the α value (step H2). In this case, the contact pressure Pij of all the cells Cij is set in the range of i = 1 to NA and j = 1 to NB. It is determined whether or not the equation (2) is established based on the set contact surface pressure Pij of each cell Cij, and if not, the process returns to the contact surface pressure setting process (step H2) of the cell Cij (H3). In step H2, the contact surface pressure Pij of each cell Cij is set again, and it is determined whether or not the formula (2) is satisfied by the set contact surface pressure Pij of the cell Cij. If not, the contact of the cell Cij is determined. The surface pressure Pij is returned to the setting process (H2). Such processing is repeated until equation (2) is satisfied, and when equation (2) is satisfied, the process proceeds to step H4.
In Step H4, it is determined whether or not the expression (3) is satisfied. If not, the process returns to Step H1 to reset the α value again.
Thereafter, the processes in steps H2 to H4 and H1 are repeated in the same manner as described above, and the process ends when the expression (3) is established.
The contact surface pressure Pij of each cell Cij set in step H2 when the expressions (2) and (3) are obtained in this way is the value of the contact surface pressure that is the output of the contact surface pressure calculation means 10. It is. Note that the initial value of the contact surface pressure of the cell Cij may be any value, but the convergence is quicker when a value as close to the actual surface pressure as possible is set.

According to this contact surface pressure calculation method, the analytical solution of displacement when a concentrated load is applied to a semi-infinite body is applied to the conformal contact problem, and the distance between measurement points when calculating the contact surface pressure is calculated. Since the distance on the reference curved surface C is used for the calculation and the normal pressure direction of the reference curved surface C is corrected for the calculation of the influence coefficient, the surface pressure is calculated relatively easily. It is possible to calculate the contact pressure in the shape contact.
In addition, it is confirmed by comparing the calculation result between the method of this embodiment and the finite element method that the calculation result is as accurate as when the contact surface pressure is calculated using the finite element method. did it.
As described above, this embodiment applies the solution of the Buenesque's semi-infinite approximation to the conformal contact problem, which is conventionally considered not to be applicable to the conformal contact. Analysis results can be obtained.
Although the method of the present invention can be applied to non-conformal contact, the surface pressure can be obtained by simpler calculation in the case of non-conformal contact. It becomes.

  In the subsurface stress calculation process (T4), the surface pressure distribution obtained as described above is used to calculate the individual stress by using the subsurface stress at the time of unit concentrated load action given by the following equations (4) to (9). Subsurface stress is obtained by integrating over the entire area of cell C and summing the effects of all cells C.

When the load on the cell C is a force in the normal direction, the calculation is performed as follows. When the two contact objects M1 and M2 are a ball and an inner ring of a rolling bearing, the reference curved surface S is a cylindrical surface as described above in the mesh division process (T1). In the displacement influence amount calculation step (T2), the coordinate position in the direction along the circular arc of the cross section cut by the plane perpendicular to the cylindrical central axis on the reference curved surface S made of the cylindrical surface is θ, and the direction along the cylindrical central axis Where y is the coordinate position of z and z is the coordinate position away from the normal direction on the reference curved surface.
The shear stress τzy at the time of unit surface pressure load exerted on the position (θ ′, y ′, z) by the element e at the position (θ, y) is obtained by the following equation (10).

  Also in this case, since the integral is obtained analytically, numerical integration is not necessary.

  As described above, the subsurface stress can be calculated relatively easily and in a short time, and the subsurface stress can be calculated with high accuracy.

In the above embodiment, depending on the shapes of the contact objects M1 and M2, the edge of the contact area may be jagged. In such a case, during the analysis, for example, in the contact surface pressure calculation process (T3) Thereafter, the mesh division may be corrected, that is, the size of each cell in the x and y directions may be corrected, and a mesh correction process (T5) may be performed in which each process after mesh division is performed again according to the correction result. .
The mesh correction means 13 in FIG. 2 is a means for executing such a mesh correction process (T5).

  FIG. 9 shows the calculation results of this embodiment, the above-described proposal example (Japanese Patent Application No. 2007-026577), and the finite element method (FEM) when the contact objects M1 and M2 are a ball and an inner ring in a rolling bearing. Show. All three are in good agreement and all are at a usable level. On the other hand, the conventional method (analysis on the premise that the contact surface is flat) is as shown in FIG.

In the method of the above-mentioned proposal example, since the displacement in three directions was taken into consideration, it took a long calculation time (faster than FEM). However, in this embodiment, this can be one direction. In addition, the proposed examples in all orders were performed by numerical integration. However, in this embodiment, it can be replaced by analytical integration, and the calculation can be simplified and speeded up.
An example of the time required for calculation is 197.08 seconds in the method of the above-mentioned proposed example, whereas it is 112.45 seconds in the method of the embodiment, which is 57% of the conventional method.

  Next, the relationship between the above equation (1) and the solution of Boussinesq will be described.

The solution of Bousnesinesq, which is a displacement analysis solution when a unit concentrated load is applied when the applied load is in the normal direction, is expressed by the following equations (11) to (13).
FIG. 7 shows the coordinate system of the Binesque solution. When a unit load is applied to the origin O of the semi-infinite plane as shown in the figure, the displacements in the x, y, and z directions of an arbitrary point Q are u, v, and w, respectively.

From this equation, the above equation (1) is derived as follows.

Equation (1) is obtained by integrating this using the relative distance as the displacement of element e on point Q. In this method, since the distance on the reference plane is used for x in equation (1), R ₀ θ is used instead of x. For integration, the following variable conversion x = R ₀ θ ⇒ dx = R ₀ dθ
(1) is obtained using.

(A) is a flowchart which shows the contact surface pressure under a conformal contact and subsurface stress calculation method based on one Embodiment of this invention, (B) is a flowchart of the analysis process. It is a block diagram which shows the conceptual structure of the contact surface pressure under conformal contact which concerns on one Embodiment of this invention, and a subsurface stress calculation apparatus. It is a flowchart of the specific example of a contact surface pressure calculation process. It is explanatory drawing of an example of a contact model. It is explanatory drawing which shows the example of a form of a reference | standard curved surface and mesh division | segmentation. It is an expanded view of a reference | standard curved surface. It is explanatory drawing of the coordinate system of a solution of a business. It is explanatory drawing of the initial stage clearance gap between contact objects. It is a graph which shows the result of having calculated | required the surface pressure distribution at the time of heavy load of a contact object by the method of this embodiment, the method of a proposal example, and FEM analysis. It is explanatory drawing of the result of having calculated the surface pressure distribution at the time of the heavy load of the contact object with the software for the light load on the market. It is a fragmentary sectional view of the rolling bearing used as the object which calculates | requires the same surface pressure.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 ... Computer 4 ... Input processing means 5 ... Output processing means 6 ... Input information storage means 7 ... Analysis means 8 ... Mesh division means 9 ... Displacement influence calculation means 10 ... Contact surface pressure calculation means 11 ... Subsurface stress calculation means M1, M2 ... contact object S ... reference curved surface C ... cell

Claims (10)

A method of calculating a surface pressure at a contact surface of two contact objects that conformally contact each other when an external force is applied using a computer,
An input process for storing the shape near the contact surface of the contact object, an elastic coefficient, a Poisson's ratio, and an external force to be applied and storing it in the storage means, and an analysis process for calculating the surface pressure using the input value; It is an output process that outputs the calculated result.
As the analysis process,
A reference curved surface is set in the vicinity of the surface where the two contact objects contact each other, the reference curved surface is divided into meshes in which cells are arranged vertically and horizontally, and a calculation point which is a point for calculating the distance between the contact objects is placed on the cell. A predetermined mesh division process;
A displacement influence calculation process for calculating an influence amount on a displacement of each of the calculation points on the surfaces of the two contact objects when a unit distribution load is applied to each cell;
The balance that the sum of the product of the surface pressure and the influence amount due to the unit distributed load for each cell balances the difference between the approach amount between the contact objects and the initial gap inside the contact area, and the sum of the surface pressure of each cell balances with the external force. From the conditional expression, including the contact surface pressure calculation process of calculating the surface pressure of each cell,
In the displacement influence calculation process and the contact surface pressure calculation process, the distance between the measurement points is calculated using the distance on the reference curved surface, and the normal direction is corrected by the difference in the position on the reference curved surface to obtain the surface pressure. To calculate,
The contact surface pressure calculation method under conformal contact characterized by the above-mentioned. In Claim 1, the two contact objects are an inner ring and a ball of a rolling bearing,
In the mesh division process, the reference curved surface is a cylindrical surface,
In the displacement influence amount calculation process, the coordinate position in the direction along the circular arc of the cross section cut by the plane perpendicular to the cylinder central axis on the reference curved surface made of the cylindrical surface is x, and the coordinate position in the direction along the cylinder central axis is y, z is the coordinate position away from the normal direction on the reference curved surface, and w is the displacement in the z-axis direction.
When the external force is a force in the normal direction, the influence amount is calculated by a displacement analysis solution at the time of unit concentrated load action represented by the following equation (1).

Contact surface pressure calculation method under conformal contact. 3. In the contact surface pressure calculation process according to claim 2, the following equation (2), which is an equation for superimposing w in the z-axis direction calculated by the equation (1) for all cells, Use as one to find the surface pressure distribution of each cell,

And, as another one of the balance condition expressions, the approach amount α is obtained using the following expression (3).

Contact surface pressure calculation method under conformal contact.   The mesh correction process according to any one of claims 1 to 3, wherein, in the middle of the analysis process, a size to be divided into the meshes is corrected, and each process after mesh division is performed again according to the correction result. Including contact surface pressure calculation method under conformal contact. A method of calculating, using a computer, the surface pressure and the subsurface stress at the contact surface of two contact objects that make conformal contact with each other when an external force is applied,
The method for calculating the contact surface pressure according to any one of claims 1 to 4, and the step of calculating the subsurface stress after calculating the surface pressure of each cell by this calculation method,
As this subsurface stress calculation process,
When the external force is a force in the normal direction, it integrates over the entire area of each cell using the subsurface stress at the time of unit concentrated load action given by the following equations (4) to (9), and Calculate the subsurface stress by taking the sum of the effects of all cells.

Calculation method of contact surface pressure and subsurface stress under conformal contact. In Claim 5, the two contact objects are an inner ring and a ball of a rolling bearing,
In the mesh division process, the reference curved surface is a cylindrical surface,
In the displacement influence amount calculation process, the coordinate position in the direction along the circular arc of the cross section cut by the plane perpendicular to the cylinder central axis on the reference curved surface made of the cylindrical surface is θ, and the coordinate position in the direction along the cylinder central axis is y, z is the coordinate position away from the normal direction on the reference curved surface,
The shear stress τzy at the time of unit surface pressure load that the element e at the position (θ, y) exerts on the position (θ ′, y ′, z) is obtained by the following equation (10).

Calculation method of contact surface pressure and subsurface stress under conformal contact. A device for calculating the surface pressure at the contact surface of two contact objects that make conformal contact with each other when an external force is applied,
An input processing means for inputting a shape in the vicinity of the contact surface of the contact object, an elastic coefficient, a Poisson's ratio, and an acting external force and storing them in a storage means; and an analysis means for calculating the surface pressure using the input value; And output processing means to output the calculated results,
As the analysis means,
A reference curved surface is set in the vicinity of the surface where the two contact objects contact each other, the reference curved surface is divided into meshes in which cells are arranged vertically and horizontally, and a calculation point which is a point for calculating the distance between the contact objects is placed on the cell. Mesh dividing means to be determined;
A displacement influence calculating means for calculating an influence amount on a displacement of each of the calculation points on the surfaces of the two contact objects when a unit distribution load is applied to each cell;
The balance that the sum of the product of the surface pressure and the influence amount due to the unit distributed load for each cell balances the difference between the approach amount between the contact objects and the initial gap inside the contact area, and the sum of the surface pressure of each cell balances with the external force. Contact surface pressure calculating means for calculating the surface pressure of each cell from the conditional expression,
The displacement influence calculating means and the contact surface pressure calculating means use the distance on the reference curved surface for calculating the distance between the measurement points, and correct the normal direction due to the difference in position on the reference curved surface to obtain the surface pressure. To calculate,
An apparatus for calculating contact pressure under conformal contact. In Claim 7, the two contact objects are an inner ring and a ball of a rolling bearing,
The mesh dividing means has a cylindrical surface as the reference curved surface,
The displacement influence amount calculation means x is a coordinate position in a direction along a circular arc of a cross section cut on a plane perpendicular to the cylinder central axis on a reference curved surface made of the cylindrical surface, and a coordinate position in a direction along the cylinder central axis. y, z is the coordinate position away from the normal direction on the reference curved surface, and w is the displacement in the z direction.
When the external force is a force in the normal direction, the influence amount is calculated by a displacement analysis solution at the time of unit concentrated load action represented by the following equation (1).

Contact surface pressure calculation device under conformal contact. 9. The contact surface pressure calculation means according to claim 8, wherein the following equation (2), which is an equation for superimposing the displacement w in the z-axis direction calculated by the equation (1) for all cells, To obtain the surface pressure distribution of each cell,

And, as another one of the balance condition expressions, the approach amount α is obtained using the following expression (3).

Contact surface pressure calculation device under conformal contact. An apparatus for calculating the surface pressure and the subsurface stress at the contact surface of two contact objects that make conformal contact with each other when an external force is applied,
A contact surface pressure calculation device according to any one of claims 7 to 9, and a subsurface stress calculation means,
This subsurface stress calculation means
When the external force is a force in the normal direction, it integrates over the entire area of each cell using the subsurface stress at the time of unit concentrated load action given by the following equations (4) to (9), and Calculate the subsurface stress by taking the sum of the effects of all cells.

Contact surface pressure and subsurface stress calculation device under conformal contact.

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