JP2000104732A - Linear guide device - Google Patents

Abstract

PROBLEM TO BE SOLVED: To improve the operational accuracy of linear guide devices by lowering stresses acting on contact portions between their bearing races and their guide rail and slider. SOLUTION: A pair of stationary bearing races 5 is arranged in a stationary groove 3 on each side of a guide rail 1 so that their arcuately expanding face is placed in contact with an associated arcuately recessed face of the groove 3. A pair of movable bearing races 6 is arranged in a movable groove 4A in a slider 2 so that their arcuately expanding face is placed in contact with an associated arcuately recessed face of the groove 4A. Another pair of movable bearing races 6 is fitted in another movable groove 4B on the opposite side so that their flat face is placed in contact with an associated inner flat face of the groove 4B and their arcuately expanding face is with an arcuately expanding face of a pressure member 8 pressed on them by an adjusting screw 9. A plurality of balls 7 are interposed for rolling motion between each couple of the stationary and movable bearing races 5 and 6.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a linear guide device in which a moving part is linearly moved through a bearing mechanism in which rolling elements are rotatably arranged between a plurality of bearing races, and more particularly, to the linear guide device. The present invention relates to a technique for reducing a compressive stress generated at a contact portion between a race, a moving portion, and a fixed portion.

[0002]

2. Description of the Related Art Conventionally, there has been developed a linear guide device which secures the hardness of a rolling surface of a rolling element by heat-treating the entire moving part and fixed part made of high carbon chromium bearing steel or martensitic stainless steel. As is known, there is a problem that such a linear guide device is not only expensive but also has a large mass. For this reason, a linear guide device as shown in FIG. 8 is used in applications in which a load acting is relatively small, such as various optical measuring instruments, small electronic component assembling machines, and electronic computers.

A linear guide device of this type includes a slider 51 as a moving portion and a guide rail 52 as a fixed portion made of a lightweight aluminum alloy or the like.
1 and a pair of movable bearing races 55, 55 and fixed bearing races 56, 56 arranged in linear grooves 53, 54 extending in a straight line and provided on the guide rail 52, respectively. A required number of balls 57 as rolling elements are interposed between the bearing races 55, 55 and the fixed-side bearing races 56, 56 so as to roll freely, whereby the slider 51 moves on the guide rail 52 as shown in the figure. It is capable of linearly reciprocating in a direction orthogonal to the cross section. The moving bearing races 55 and 55 are adjusted by screwing them into the slider 51 so that play is not generated in the moving and fixed bearing races 55 and 56 and the ball 57 and the operation accuracy of the slider 51 is impaired. The screw 58 is pressed against the fixed bearing races 56, 56 via a pressing member 59 (see, for example, JP-A-49-80448).

[0004]

However, the following problems are pointed out in the linear guide device according to the prior art. That is, since the slider 51 and each moving-side bearing race 55 and the guide rail 52 and each fixed-side bearing race 56 are in contact with each other on a flat surface and a curved surface and are in a line contact state, a large compressive stress is generated at the contact portion. However, since the slider 51 and the guide rail 52 are made of a soft material such as an aluminum alloy for weight reduction, the slider 51 and the guide rail 52 have low mechanical strength and are easily deformed by the compressive stress. Therefore, the bearing races 55, 56, the ball 57, and the pressing member 59 made of a hard material such as high carbon chromium bearing steel or martensitic stainless steel have high rigidity as a linear guide device despite high mechanical strength. May be reduced, and the operation accuracy required in an optical measuring instrument, a small electronic component assembler, or the like may be impaired.

The present invention has been made in view of the above points, and its main technical problem is that a compressive stress generated at a contact portion between a bearing race and a slider and a contact portion between a bearing race and a guide rail are provided. It is an object of the present invention to provide a linear guide device capable of increasing the operation accuracy even if the slider and the guide rail are made of materials having low mechanical strength.

[0006]

Means for Solving the Problems The technical problems described above are:
This can be effectively solved by the present invention. That is, the linear guide device according to the present invention includes a pair of fixed-side bearing races arranged in the fixed-side groove provided in the fixed portion, and a movable-side groove provided in the moving portion corresponding to the fixed-side groove. A pair of moving-side bearing races disposed therein, and a required number of rolling elements rotatably interposed between the fixed-side bearing race and the moving-side bearing race. Both widthwise corners of at least one of the concave grooves form an arc-shaped concave surface having an arc-shaped cross-section, and a contact surface of the bearing race with the arc-shaped concave surface forms an arc-shaped convex surface having an arc-shaped cross-section, The radius of curvature of the arc-shaped concave surface is made equal to or slightly larger than the radius of curvature of the arc-shaped convex surface.
Here, the “transverse section” is a section cut on a plane orthogonal to the extension direction (moving direction of the moving unit) of the fixed-side and moving-side bearing races.

[0007] According to the linear guide device having the above-described configuration, the arc-shaped convex surface of the bearing race comes into contact with the corner portion forming the arc-shaped concave surface of the groove. At this time, if the radius of curvature of the arc-shaped concave surface of the groove is equal to the radius of curvature of the arc-shaped convex surface of the bearing race, the two are in surface contact with each other.
Even when the moving part is made of a soft material such as an aluminum alloy, the compressive stress generated on the concave groove side is reduced. Also, when the arcuate concave surface of the concave groove has a slightly larger radius of curvature than the arcuate convex surface of the bearing race, the two are in line contact, but the contact area increases sharply due to a slight contact pressure, and the surface contact state occurs. , The compressive stress is dispersed.

Further, according to the above configuration, since the arc-shaped concave surface which comes into contact with the arc-shaped convex surface of the bearing race is a portion corresponding to a corner of the concave groove, the load from the bearing race due to the contact with the rolling element is reduced by the concave shape. Almost 45 ° to groove
Acts at an angle of Therefore, the thickness of the groove on both sides in the width direction can be reduced, and the linear guide device can be downsized.

[0009]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Hereinafter, a linear guide device according to the present invention will be described in detail with reference to a preferred embodiment shown in FIGS.

First, as shown in the perspective view of FIG. 1 and the exploded perspective view of FIG.
It has a guide rail 1 as a fixed part and a slider 2 as a moving part. On the upper surface of the guide rail 1, a convex portion 1a extending over the entire length in the longitudinal direction is formed, and on both outer surfaces of the convex portion 1a, fixed side grooves 3 extending over the entire length in the longitudinal direction are formed. On the other hand, the slider 2 has an inverted concave shape which is disposed so as to straddle the convex portion 1a.
Are provided over the entire length in the moving direction of the slider 2.

The guide rail 1 and the slider 2 are each made of an extruded material such as an aluminum alloy which is lightweight and has excellent corrosion resistance and workability. In the process of manufacturing the extruded material, the convex cross section of the guide rail 1 and the concave cross section of the slider 2 are formed, so that the fixed side groove 3 and the movable side grooves 4A, 4B are finished after the extruded material is manufactured. After processing and cutting to a required length, mounting portions 14 and 15, opening 17, and piping ports 29 A and 29 to be described later are used.
B, a hole such as the female screw hole 2b or the like, or tapping may be performed. For this reason, the manufacturing cost can be significantly reduced.

As shown in the sectional view of FIG. 3 and FIGS.
The fixed-side concave grooves 3 on both side surfaces of “a” are in the width direction (vertical direction).
Are formed as arc-shaped concave surfaces 3a having an arc-shaped cross section. A pair of upper and lower fixed-side bearing races 5 and 5 are arranged in the fixed-side groove 3. Each of the fixed-side bearing races 5 has a rod shape having a substantially D-shaped cross section, that is, the surface thereof has an arc-shaped convex surface 5a having a circular cross section.
And the flat portion 5b, and the arc-shaped convex surface 5a of each fixed-side bearing race 5 is in contact with the upper and lower arc-shaped concave surfaces 3a of the fixed-side groove 3. The arcuate concave surface 3a
It is formed with the same radius of curvature as the arc-shaped convex surface 5a or a slightly larger radius of curvature.

One of the movable side grooves 4A (left side in FIG. 3) of the movable side grooves 4A and 4B on both inner side surfaces of the slider 2 is symmetrical to the fixed side groove 3 as shown in FIG. The shape, that is, both corners in the width direction (vertical direction) are arc-shaped concave surfaces 4a having an arc-shaped cross section, and a pair of upper and lower moving-side bearing races 6, 6 are arranged in this moving-side groove 4A. Have been. Each of the moving-side bearing races 6 has a rod shape having the same shape and the same size as the fixed-side bearing race 5, that is, the surface thereof has an arc-shaped convex surface 6a having an arc-shaped cross section.
And a flat portion 6b.
a is in contact with the upper and lower arc-shaped concave surfaces 4a of the moving-side groove 4A.

As shown in FIG. 6, the other one (the right side in FIG. 3) of the movable side grooves 4A and 4B of the movable side grooves 4A and 4B on the inner side surface of the slider 2 has upper and lower corners at right angles. It does not have an arcuate concave surface, and the upper and lower inner surfaces are formed as flat portions 4b, 4b, respectively. A pair of upper and lower moving side bearing races 6, 6 are provided in the moving side groove 4A.
Are arranged in a state where the plane portion 6b is in contact with the plane portion 4b.

Between the fixed bearing races 5, 5 and the movable bearing races 6, 6, a plurality of balls 7 as rolling elements are rotatably interposed in a state of maintaining a proper interval. Each ball 7 is in contact with the flat portion 5b of the fixed bearing race 5 and the flat portion 6b of the moving bearing race 6 between the fixed groove 3 on the left side and the moving groove 4A on the left side in FIG. Between the other fixed-side groove 3 and the moving-side groove 4B, the flat portion 5b of the fixed-side bearing race 5 and the arc-shaped convex surface 6 of the moving-side bearing race 6 are provided.
a.

Further, the right moving side groove 4B in FIG.
Is a pressing member accommodating portion in which the bottom central portion in the width direction is appropriately formed in a deep step shape, and a rod-shaped pressing member 8 is disposed in this accommodating portion. The pressing member 8 has the same rod shape as the fixed-side bearing race 5 and the moving-side bearing race 6, and its arc-shaped convex surface 8a is formed with the arc-shaped convex surfaces 6a, 6a of the moving-side bearing races 6, 6. In contact. A plurality of female screw holes 2b are formed at predetermined intervals in the side wall portion 2a of the slider 2 in which the moving-side concave grooves 4B are formed, and are screwed into these female screw holes 2b. The flat tip surface of the adjusted screw 9 is in contact with the flat portion 8 b of the pressing member 8. Then, depending on the screwing amount of the adjusting screw 9, both the fixed groove 3 and the movable groove 4A and the fixed groove 3 and the movable groove 4B on the fixed side are fixed. The contact state between the bearing races 5, 5 and the moving-side bearing races 6, 6 and the ball 7 can be appropriately adjusted.

The fixed-side bearing race 5, the moving-side bearing race 6, the ball 7, and the pressing member 8 are made of a hard material such as high-carbon chromium bearing steel or martensitic stainless steel. Precise finishing is performed by polishing, improving the accuracy and rigidity of the linear guide device. The fixed-side bearing race 5, the moving-side bearing race 6, and the pressing member 8 have the same cross-sectional shape, size, and length as described above, that is, they are the same members, and thus are shared with each other. Is what you can do.

At the longitudinal ends of the fixed-side bearing race 5 arranged at the bottom of the fixed-side groove 3, small pan screws 11 are mounted. Further, end plates 12 are fixed to both ends of the slider 2 with pan screws 13 to prevent the balls 7 from falling off.

A mounting portion 14 for connecting a tool or a jig is provided on the upper surface of the slider 2.
Is provided with a mounting portion 15 for fixing the linear guide device to equipment or the like.

The guide rail 1 is provided with a through hole 16 extending over the entire length thereof in the longitudinal direction, and has a projection 1a.
An opening 17 having an elongated shape in the longitudinal direction of the projection 1a is formed from the upper surface of the projection toward the through hole 16. In the through hole 16, an engaging member 21 integrally having a piston 23 at both ends in the axial direction is disposed so as to be movable in the axial direction,
As shown in the sectional view of FIG. 4, a piston seal 24 is provided on the outer peripheral surface of each of the pistons 23 so as to be in airtight sliding contact with the inner surface of the through hole 16. An engaging pin 18 is disposed vertically through the opening 17, and the lower end of the engaging pin 18 is loosely inserted into an engaging hole 22 formed substantially at the center of the engaging member 21 in the axial direction. The male screw portion 19 formed at the upper end is fitted and engaged with the slider 2 by screwing into a female screw hole 20 formed substantially at the center of the top plate portion 2c of the slider 2. . Openings at both ends of the through hole 16 are closed by closing members 25, and O-rings 26 are attached to the outer periphery of the closing members 25, respectively. Each of the closing members 25 is a retaining ring 2 for a hole.
7 fixed.

The inside of the through hole 16 is formed by a pair of cylinder chambers 28A, 28A by pistons 23 on both sides of the engaging member 21.
B. A pair of piping ports 29A and 29B are provided on the side surface of the guide rail 1, and the piping ports 29A and 29B are respectively connected to the orifices 30A and 30B.
B communicates with the cylinder chambers 28A and 28B. Each of the piping ports 29A and 29B is connected to a compressed air supply source via a piping and a switching valve (not shown).

That is, according to the linear guide device according to the above-described embodiment, the compressed air from the compressed air supply source is supplied to one of the piping ports 29A and 29B and the other is opened to the atmosphere.
In addition, the engaging member 21 integrated therewith moves axially in the through hole 16 of the guide rail 1 toward the low pressure side. For this reason, the slider 2 guided by the convex portion 1a of the guide rail 1 operates linearly via the engaging pin 18 engaged with the engaging member 21. The stroke distance at this time is defined by the engagement pin 18 abutting on the inner surface of one end 17a or the other end 17b in the longitudinal direction.

FIG. 4 shows that compressed air is supplied to the cylinder chamber 28A from the pipe port 29A via the orifice 30A, and the cylinder chamber 28B is evacuated to the atmosphere via the orifice 30B, the pipe port 29B and a switching valve (not shown). By releasing, the engagement pin 18 is moved to the opening 17.
Until the piston 23 and the engaging member 21 contact the inner surface of the other end 17b in the longitudinal direction of the cylinder chamber 28 in the through hole 16.
The state moved to B is shown. Then, the output direction of the switching valve is switched from this state,
When compressed air is supplied to the cylinder chamber 28B through the orifice 30B and the compressed air filled in the cylinder chamber 28A is released to the atmosphere through the orifice 30A, the piping port 29A and the switching valve, the piston 23
Then, the engaging member 21 moves inside the through hole 16 toward the cylinder chamber 28A, and the slider 2 operates via the engaging pin 18 engaged with the engaging member 21. Then, the engaging pin is moved to the opening 17.
The stroke end is reached at the time of contact with the inner surface of one end 17a in the longitudinal direction, and the movement stops.

In the operation described above, the fixed-side bearing races 5 and 5 arranged in the fixed-side concave grooves 3 of the guide rail 1 are used.
The ball 7 rolls between the moving bearing groove 5 and the moving bearing races 6 and 6 arranged in the moving groove 4A or the moving groove 4B on the inner surface of the slider 2, whereby the slider 7 is moved as shown in FIG. It can move smoothly in the direction orthogonal to the cross section.

Here, the stress generated at the contact portion between the upper and lower arc-shaped concave surfaces 3a of the fixed-side concave groove 3 and the arc-shaped convex surface 5a of the fixed-side bearing race 5 will be described. When two elastic bodies having a curved surface are pressed against each other, a large concentrated stress that cannot be obtained by a general elasticity calculation formula occurs at the contact portion. This stress is Hel
It is known to guide by the formula of tz. Helt
According to the formula of z, the stress σbg generated when the arc-shaped convex surface 5a and the arc-shaped concave surface 3a are pressed together is expressed by the following equation (1).

(Equation 1) Here, rg is the radius of curvature of the arc-shaped concave surface 3a, rb is the radius of curvature of the arc-shaped convex surface 5a, vg is the Poisson's ratio of the guide rail 1, νb is the Poisson's ratio of the fixed-side bearing race 5, and Eb is the The longitudinal elastic coefficient, Eb, is the longitudinal elastic coefficient of the fixed-side bearing race 5, and q is the pressing force per unit length.

Similarly, the upper and lower arc-shaped concave surfaces 4a of the moving side groove 4A shown in FIG.
According to Heltz's formula, the stress σbs generated when the contact portions with the arc-shaped convex surface 6a are pressed against each other is expressed by the following equation (2).

(Equation 2) Here, rs is the radius of curvature of the arc-shaped concave surface 4a, rb is the radius of curvature of the arc-shaped convex surface 6a, VS is the Poisson's ratio of the slider 2, νb is the Poisson's ratio of the moving-side bearing race 6, and Es is the longitudinal elasticity of the slider 2. The coefficient, Eb, is the longitudinal elastic coefficient of the moving-side bearing race 6, and q is the pressing force per unit length.

In this embodiment, the fixed side and moving side bearing races 5 and 6 are used as common parts, the guide rail 1 and the slider 2 are made of the same material, and the arcuate concave surface 3 is formed.
a, 4a, and the arcuate convex surfaces 5a, 6a have the same radius of curvature, so from the above equations (1) and (2), σbg = σbs
Becomes

Next, the contact portions between the upper and lower plane portions 4b of the moving side groove 4B shown in FIG. 6 and the plane portions 6b of the moving side bearing race 6 arranged in the moving side groove 4B are pressed against each other. Σbsh generated when applied
Can be obtained by an ordinary elasticity calculation formula, that is, it is expressed by the following formula (3).

(Equation 3) Here, d is the contact width between the plane portion 4b of the moving side groove 4B and the plane portion 6b of the moving side bearing race 6, and qh is the pressing force per unit length. Since σbsh is a contact between planes, σbsh is much smaller than σbg and σbs.

When this is compared with the prior art shown in FIG. 8, in the prior art, the flat portion of the fixed side bearing race 56 having a circular cross section and the fixed side groove 54 formed in the guide rail 52 are formed. The moving-side bearing race 55 having a circular cross section and the plane portion of the moving-side groove formed in the slider 51 are in contact with each other. For this reason, He
According to the ltz formula, the stress σbgo generated when the outer peripheral surface of the fixed-side bearing race 56 and the plane portion of the guide rail 52 are pressed against each other is expressed by the following equation (4).

(Equation 4) Similarly, the stress σbso generated when the outer peripheral surface of the moving-side bearing race 55 and the plane portion of the slider 51 are pressed against each other is expressed by the following equation (5).

(Equation 5) Here, rb is the radius of the outer peripheral surface of the bearing races 55 and 56, vg is the Poisson's ratio of the guide rail 52, ν
s is the Poisson's ratio of the slider 51, νb is the Poisson's ratio of the bearing races 55 and 56, and Eg is the guide rail 5.
Es is the longitudinal elastic coefficient of the slider 51,
b is the longitudinal elastic modulus of the bearing races 55 and 56, and q 'is the pressing force per unit length.

Assuming that the bearing races 55 and 56 are the same parts, the material of the guide rail 52 and the slider 51 are the same, and the radius of the bearing races 55 and 56 is the same, from the above equations (4) and (5), σbgo = σbso. Becomes

Since the bearing races 55 and 56 come into contact with two mutually orthogonal surfaces of the concave grooves 53 and 54, the pressing force q 'per unit length is expressed by the following equation (6).

(Equation 6) Therefore, the stress σbg generated at the contact portion between the arc-shaped convex surface 5a and the arc-shaped concave surface 3a in the present embodiment and the stress σbgo generated at the contact portion between the outer peripheral surface of the fixed-side bearing race 56 and the flat portion of the guide rail 52 in the related art. Is expressed by the following equation (7).

(Equation 7) In this equation (7), for example, rg = 2.05 mm, r
Assuming that b = 2 mm, σbg is slightly less than 19% of σbgo, and it can be seen that in the embodiment, the stress at the contact portion is significantly reduced as compared with the prior art. Equation (7)
As can be seen from FIG. 7, it goes without saying that the stress decreases as rg approaches rb. By the way rg = 2.01
In the case of mm, σbg is reduced to about 8% of σbgo.

That is, according to the configuration of the present embodiment, in the fixed-side groove 3 and the one moving-side groove 4A, the arc-shaped convex surfaces 5a, 6 of the fixed-side and moving-side bearing races 5, 6 are formed.
a is in contact with the arcuate concave surfaces 3a, 4a having the same or a slightly larger radius of curvature as in this case. Since they are in surface contact with the portion 4b, the compressive stress generated in these contact portions is significantly smaller than in the prior art. Therefore, even when the guide rail 1 and the slider 2 are made of a soft material such as an aluminum alloy, the movement accuracy can be improved.

As shown in FIG. 3, the arc-shaped concave surfaces 3a,
4a and the pressing force q of the arc-shaped convex surfaces 5a, 6a act at an angle of approximately 45 ° with respect to the width direction of the fixed-side groove 3 and the moving-side groove 4A. Therefore, to reduce the thickness t ₂ of the end of the slider 2 which constitutes the wall thickness t ₁ or the moving side groove 4A of the end portion of the convex portion 1a of the guide rail 1 which constitutes the fixed side groove 3, The size of the linear guide device can be reduced.

In the above-described embodiment, the pressing member 8 that is shared with the bearing races 5 and 6 is disposed in the moving-side concave groove 4B. Plate-shaped pressing member 1 having a rectangular cross section with strength
0 may be used. Also, by adjusting the clearance between the bearing races 5 and 6 by selecting the size of the ball 7,
By omitting the pressing member, the movable side groove 4B can be configured in the same manner as the movable side groove 4A, that is, both sides of the convex portion 1a can be configured as shown in FIG.

[0035]

As described above, according to the linear guide device of the present invention, the arc-shaped convex surface formed on the bearing race is brought into contact with the arc-shaped concave surface formed on the groove of the fixed portion or the moving portion. This reduces the stress generated at the contact portion. Therefore, the linearity of the operation can be significantly improved by increasing the rigidity of the linear guide device.

Further, the mutual pressing force at the contact portion between the arc-shaped convex surface of the bearing race and the arc-shaped concave surface of the concave groove acts in a direction inclined with respect to the width direction of the concave groove.
The thickness of the end of the fixed portion or the moving portion forming the concave groove can be reduced. Therefore, the size of the linear guide device can be reduced.

[Brief description of the drawings]

FIG. 1 is an external perspective view showing a preferred embodiment of a linear guide device according to the present invention.

FIG. 2 is an exploded perspective view of the embodiment.

FIG. 3 is a cross-sectional view illustrating the above-described embodiment cut along a plane orthogonal to a moving direction of a slider.

FIG. 4 is a cross-sectional view showing the slider according to the embodiment cut along a plane parallel to a bottom surface thereof.

FIG. 5 is an enlarged partial cross-sectional view showing a part of FIG. 3;

FIG. 6 is an enlarged partial cross-sectional view showing another part of FIG. 3;

FIG. 7 is a partially enlarged sectional view showing another embodiment of the linear guide device according to the present invention.

FIG. 8 is a partial cross-sectional view showing a linear guide device according to the related art.

[Explanation of symbols]

 1 Guide rail (fixed portion) 2 Slider (moving portion) 2b, 20 Female screw hole 3 Fixed concave groove 3a, 4a Arc-shaped concave surface 4A, 4B Moving concave groove 4b, 5b, 6b, 8b Flat portion 5 Fixed bearing Race 5a, 6a, 8a Arc-shaped convex surface 6 Moving-side bearing race 7 Ball (rolling body) 8,10 Pressing member 9 Adjusting screw 11,13 Pan head screw 12 End plate 14,15 Mounting part 16 Through hole 17 Opening 18 Mating pin 19 Male thread part 20 Female thread hole 21 Engagement member 22 Engagement hole 23 Piston 24 Piston seal 25 Closing member 26 O-ring 27 Retaining ring for hole 28A, 28B Cylinder chamber 29A, 29B Piping port 30A, 30B Orifice

Claims (1)

[Claims] 1. A fixed-side concave groove (3) provided in a fixed portion (1).
A pair of fixed-side bearing races (5, 5
5) and the moving part (2) corresponding to the fixed-side groove (3).
A pair of moving-side bearing races (6, 6) disposed in the moving-side concave grooves (4) provided in the fixed-side bearing races (5, 5) and the moving-side bearing races (6, 6).
And a required number of rolling elements (7) rotatably interposed therebetween, wherein both widthwise corners of at least one of the concave grooves (3, 4) of the fixed side and the moving side are circular in cross section. An arc-shaped concave surface (3a, 4a), and the arc-shaped concave surface (3) of the bearing race (5, 6).
a, 4a) has an arc-shaped convex surface (5
a, 6a), wherein the radius of curvature of the arcuate concave surface (3a, 4a) is the same as or slightly larger than the radius of curvature of the arcuate convex surface (5a, 6a).