Straight bevel gears are among the most common transmission components used in differential drives, reducers, machine tools and agricultural equipment. In many shops, the bore of a straight bevel gear is finished by broaching on an arbor press or by internal grinding. The greatest difficulty is not the bore itself, but the requirement that the pitch cone of the gear must run true with the bore axis. If the bore is ground from an arbitrary datum, the pitch cone may be out of limits and the gear will be rejected. I have used the fixture described in this article for a long time, and more than one thousand straight bevel gears have been ground successfully with it. The fixture is simple, inexpensive and reliable. It uses the tooth spaces of the straight bevel gear itself as the locating datum, which is the only natural datum that preserves the relation between the pitch cone and the finished bore.
1. Why a Special Fixture Is Needed for Straight Bevel Gears
In ordinary cylindrical gears, the bore can often be ground by mounting the gear on an expansion arbor or on a precision mandrel. The outside diameter, the face, or the root circle can be used as a reference; then the bore is ground and the tooth profile is subsequently generated relative to the bore. In contrast, a straight bevel gear has a conical pitch surface. The axis of the pitch cone is the same as the rotational axis of the gear, but the outer cone surface and the large end face cannot always be used as reliable locating datums after heat treatment. The tooth flanks of a straight bevel gear are cut on a generating machine, so the tooth spaces contain a very accurate record of the gear axis. The fixture that I developed makes full use of that fact.
Before the fixture was introduced, the common production method was to push a cylindrical broach through the bore in an arbor press. This is productive, but it cannot correct the existing eccentricity of the pitch cone with respect to the bore. The broach follows the pre-machined bore; it cannot centre the workpiece from the teeth. Grinding is necessary for higher accuracy, but the operator has to spend time indicating the tooth cone or the pitch cone on the grinding machine. This is difficult when the bore is the only surface already machined. The fixture described here removes that problem by providing a positive, self-centring location between the tooth spaces of the straight bevel gear and the grinding machine spindle.
2. Locating Principle of the Fixture
The fixture uses three precision cylindrical pins, or rollers, inserted into three tooth spaces that are as evenly spaced as possible around the straight bevel gear. Each cylinder contacts the two flanks of one tooth space. Because the cylinder touches both flanks, it is located in the angular position of the tooth space and at a certain radial height with respect to the cone surface. The three cylinders together form an external three-point centring system. The centre of the circle passing through the cylinder centres is the axis of the pitch cone of the straight bevel gear. When the gear and the cage are clamped in the fixture body, the bore can be ground concentric with that axis.
For most straight bevel gears, the three tooth spaces are selected from the complete set of tooth spaces. They should be separated by an angle close to \(120^\circ\). If the gear has \(z\) teeth, the ideal tooth-space index for the \(j\)-th cylinder is
\[
n_j = \mathrm{round}\left(\frac{j z}{3}\right), \quad j = 0, 1, 2
\]
where \(\mathrm{round}\) gives the nearest integer. The corresponding angular position of the cylinder is
\[
\phi_j = \frac{2\pi n_j}{z}
\]
In this way, even when \(z\) is not divisible by three, the three cylinders still give a nearly equilateral triangle of locating points. This principle is especially useful for small and medium sized straight bevel gears with an odd number of teeth.
3. Nomenclature and Symbols
In order to present the design rules clearly, I use the following notation throughout the article.
| Symbol | Description | Typical unit |
|---|---|---|
| \(z\) | Number of teeth of the straight bevel gear | – |
| \(m\) | Large-end module | mm |
| \(\alpha\) | Nominal pressure angle | deg |
| \(b\) | Face width of the gear | mm |
| \(R_e\) | Outer cone distance | mm |
| \(R_m\) | Mid-face cone distance | mm |
| \(d\) | Diameter of the locating cylinder | mm |
| \(l\) | Length of the locating cylinder | mm |
| \(\lambda\) | Cylinder diameter coefficient, \(\lambda = d/m\) | – |
| \(\delta\) | Pitch cone angle of the gear | deg |
| \(\delta_a\) | Face cone angle of the gear | deg |
| \(\theta_j\) | Measured angle of the exposed cylinder generator for cylinder \(j\) | deg |
| \(\theta_m\) | Mean measured angle | deg |
| \(\beta_f\) | Taper half-angle of the fixture body bore | deg |
| \(n_j\) | Tooth-space index for the \(j\)-th cylinder | – |
| \(T\) | Required pitch-cone runout relative to bore axis | mm |
| \(t_f\) | Perpendicularity tolerance of fixture datum face | mm |
| \(A_c\) | Contact area between taper and cage | mm² |
| \(A_t\) | Total taper contact area | mm² |
4. Determination of the Cylinder Diameter
The most important design parameter is the diameter \(d\) of the locating cylinder. The cylinder must be large enough to protrude above the tooth space, but small enough to enter the tooth space at both the large end and the small end of the straight bevel gear. If the cylinder is too large, it will contact the flanks only at the large end and will not seat stably. If it is too small, it will sink too deeply and lose the accurate relation to the cone surface.
At the middle of the face width, the cone distance is
\[
R_m = R_e – \frac{b}{2}
\]
For a standard straight bevel gear with a module \(m\), the circular tooth-space width at the outer end can be approximated by
\[
s \approx \frac{\pi m}{2}
\]
At the middle of the face width this width is reduced in proportion to the cone distance:
\[
s_m = \frac{\pi m}{2} \cdot \frac{R_m}{R_e}
\]
A useful theoretical estimate for the cylinder diameter is therefore
\[
d_{\mathrm{th}} = s_m \cos\alpha
\]
In practice I use a safety factor \(K\) to avoid excessive contact at the outer edge of the tooth:
\[
d = \frac{s_m \cos\alpha}{K}
\]
For most straight bevel gears, \(K\) lies in the range \(1.2\) to \(1.6\). This gives the useful workshop formula
\[
d = \lambda m
\]
with
\[
0.65 \le \lambda \le 0.90
\]
For a first trial, I take \(\lambda = 0.80\). The cylinder should protrude above the root and above the tooth faces far enough to allow the retainer to hold it firmly. The protrusion height \(h_p\) should satisfy
\[
h_p \ge 0.1 d
\]
In my experience, the following starting values are reliable for straight bevel gears with a pressure angle of \(20^\circ\).
| Module \(m\) (mm) | Minimum \(d\) (mm) | Maximum \(d\) (mm) | Recommended starting \(d\) (mm) |
|---|---|---|---|
| 1.0 | 0.65 | 0.90 | 0.80 |
| 1.25 | 0.81 | 1.12 | 1.00 |
| 1.5 | 0.98 | 1.35 | 1.20 |
| 2.0 | 1.30 | 1.80 | 1.60 |
| 2.5 | 1.63 | 2.25 | 2.00 |
| 3.0 | 1.95 | 2.70 | 2.40 |
| 4.0 | 2.60 | 3.60 | 3.20 |
| 5.0 | 3.25 | 4.50 | 4.00 |
| 6.0 | 3.90 | 5.40 | 4.80 |
These values are starting points. The final cylinder diameter must be checked on the actual straight bevel gear. I make a set of trial cylinders with a diameter range of \(0.1\) mm steps, place them in the tooth spaces at the large end and the small end, and observe the contact marks. The best diameter gives a small, symmetrical contact mark on both flanks near the middle of the tooth space. The cylinder must not touch the root, and it must not rock when pressed lightly by hand.
5. Cylinder Material, Length and Heat Treatment
The locating cylinders are precision parts. They must be hard, wear resistant, and ground to a very tight diameter tolerance. I use high-carbon chromium bearing steel such as AISI 5210 or 100Cr6. The manufacturing sequence is:
- Turn the blank to a diameter slightly larger than the final diameter.
- Drill a small centre hole at one end. The centre hole is useful for holding the cylinder during grinding and for attaching a wire during the final assembly.
- Harden and temper the cylinders to a hardness of \(58\) to \(62\) HRC.
- Finish grind the outside diameter with a centreless grinder or between centres.
- Lap the outside diameter if the shape error is too large.
The length \(l\) of the cylinder should be related to the face width \(b\) of the straight bevel gear. From my experience,
\[
l = (0.35 \sim 0.55) b
\]
A length of about half the face width is usually satisfactory. If the cylinder is too long, it may contact the root or the face cone at more than the intended points. If it is too short, it may tilt in the slot and reduce the accuracy of the locating system.
The final grinding tolerances for the cylinders are critical. I use the following limits:
| Characteristic | Recommended tolerance |
|---|---|
| Diameter tolerance | \(\pm 0.002\) mm |
| Out-of-roundness | \(0.001\) mm |
| Taper of cylinder surface | \(0.001\) mm over length |
| Difference between the three cylinders | \(0.002\) mm maximum |
| Surface roughness | \(R_a \le 0.2\) μm |
| Straightness of generator line | \(0.001\) mm |
The three cylinders used in one fixture must be finished in the same grinding setup, or at least selected so that their diameters do not differ by more than \(0.002\) mm. If one cylinder is larger than the other two, the straight bevel gear will be pushed off-centre in the fixture and the bore will be ground with an unacceptable runout.
6. Manufacture of the Cylinder Retainer Cage
The retainer cage is similar to a bearing cage. It keeps the three cylinders in their correct positions while allowing them to seat freely in the tooth spaces of the straight bevel gear. The cage is a conical shroud whose inner cone matches the face cone of the gear being machined. I make the cage from a low-carbon steel shell or an aluminium alloy shell. The cone angle of the cage shell must be correct, because the cage rests against the face cone of the straight bevel gear.
The face cone angle \(\delta_a\) of the gear is
\[
\delta_a = \delta + \theta_a
\]
where \(\theta_a\) is the addendum angle. For many straight bevel gears, \(\theta_a\) is small, but it must not be neglected. If the drawing of the straight bevel gear is not available, the cage can be copied from the face cone of the blank. The important point is that the cage should not obstruct the protrusion of the cylinders from the tooth spaces.
After the conical shell has been made, I mill three slots in the conical surface. The slots are positioned over the selected tooth spaces. The angular position of each slot is
\[
\phi_j = \frac{2\pi n_j}{z}
\]
The width of the slot is made slightly larger than the cylinder diameter:
\[
W = d + 0.1 \ \mathrm{mm}
\]
The length of the slot is made slightly larger than the cylinder length:
\[
L_s = l + (2 \sim 4) \ \mathrm{mm}
\]
This clearance allows the cylinder to move radially a little when it enters the tooth space, so that it can seat against both flanks. The cylinders are then retained by thin spring steel wire. I drill small holes in the cage around each slot and pass a \(0.3\) to \(0.5\) mm music wire over the cylinders. The wire must not press the cylinders so hard that they cannot seat freely.
For a given straight bevel gear, the selected tooth space indices for the three cylinders can be calculated from the formula given earlier. The table below gives practical values for common tooth numbers.
| Number of teeth \(z\) | \(n_0\) | \(n_1\) | \(n_2\) | Approximate spacing |
|---|---|---|---|---|
| 17 | 0 | 6 | 11 | 127° / 106° / 127° |
| 18 | 0 | 6 | 12 | 120° / 120° / 120° |
| 20 | 0 | 7 | 13 | 126° / 108° / 126° |
| 21 | 0 | 7 | 14 | 120° / 120° / 120° |
| 24 | 0 | 8 | 16 | 120° / 120° / 120° |
| 26 | 0 | 9 | 17 | 125° / 111° / 124° |
| 30 | 0 | 10 | 20 | 120° / 120° / 120° |
| 32 | 0 | 11 | 21 | 124° / 112° / 124° |
| 36 | 0 | 12 | 24 | 120° / 120° / 120° |
| 40 | 0 | 13 | 27 | 117° / 126° / 117° |
7. Machining the Fixture Body Taper Bore
The fixture body has a conical bore that receives the cage together with the straight bevel gear. The taper of this bore must be cut to match the angle made by the exposed generator lines of the three cylinders when the cage is seated on the gear. This angle is not exactly equal to the pitch cone angle or the face cone angle, because the cylinder protrudes from the tooth space and is inclined by the local tooth-space geometry. Therefore I measure it directly.
I place the assembled cage and the straight bevel gear on a surface plate. Then I use a universal bevel protractor to measure the angle between the exposed cylinder generator and the gear axis. This is done for each of the three cylinders. The mean value is
\[
\theta_m = \frac{1}{3}\sum_{j=1}^3 \theta_j
\]
If the protractor reads the acute angle with the axis, the included taper angle of the fixture body is
\[
2\beta_f = 2\theta_m
\]
In other words, the taper half-angle of the fixture body is set equal to \(\theta_m\):
\[
\beta_f = \theta_m
\]
I then machine the conical bore in the fixture body to this taper. The final taper is checked with a taper gauge or by the blueing method. The contact area ratio is
\[
C_c = \frac{A_c}{A_t} \times 100\%
\]
and I require
\[
C_c \ge 80\%
\]
This ensures that the cage is not distorted when the cover is tightened. If the contact is only at one side, the straight bevel gear will be tilted in the fixture and the ground bore will be inclined relative to the pitch cone.
One of the most important requirements for the fixture body is the perpendicularity between the axis of the taper bore and the clamping face of the fixture body. In order to control the final runout of the straight bevel gear, I use
\[
t_f \le \frac{T}{3}
\]
where \(T\) is the allowable pitch-cone runout. For example, if the drawing of the straight bevel gear requires \(T = 0.03\) mm, the perpendicularity of the fixture face must be no larger than \(0.01\) mm. The taper bore and the datum face should be ground in the same mounting of the fixture body on the grinding machine.
8. Pressure Cover and Clamping
The pressure cover is the upper part of the fixture. It presses the straight bevel gear and the cage into the taper bore of the fixture body. I drill several small holes in the cover so that a pin spanner can be used to rotate the cover and apply the clamping force. The cover should also have a small clearance bore to admit the grinding wheel through the already machined bore of the gear, if the bore is to be ground from the opposite side.
The clamping force must be axial and should be applied uniformly around the circumference. An excessive force may deform the cone or damage the tooth flanks. A useful estimate of the required axial force is
\[
F_a = \frac{M_c}{\mu r_m}
\]
where \(M_c\) is the grinding torque, \(\mu\) is the friction coefficient between the cage and the fixture body, and \(r_m\) is the mean radius of the conical seating surface. In normal grinding of small to medium straight bevel gears, the axial clamping force is modest. I usually tighten just enough to prevent the gear from rotating during the cut.
9. Grinding Procedure
The procedure that I use in the workshop is simple:
- Clean the taper bore of the fixture body and the cone of the cage carefully. Any chip or dirt will change the location of the straight bevel gear.
- Mount the fixture body on the internal grinding machine spindle or on the magnetic table.
- True the taper bore of the fixture body with a diamond dressing wheel or a grinding wheel. The axial runout of the datum face should be less than \(0.005\) mm.
- Select the three locating cylinders and check their diameters with a micrometer. The difference between them must be within \(0.002\) mm.
- Place the straight bevel gear into the fixture body.
- Put the cage over the gear so that the slots in the cage are aligned with the selected tooth spaces.
- Insert the three cylinders through the slots into the tooth spaces. They should fall into contact with both flanks without force.
- Fasten the retaining wire so that the cylinders cannot fall out during grinding.
- Put the pressure cover on top and tighten it with a pin spanner.
- Grind the bore in light passes. Allow two or three spark-out passes at the end.
- Release the cover, remove the gear, and check the pitch-cone runout on the inspection fixture.
The grinding conditions naturally depend on the bore size and material. For hardened straight bevel gears with a bore diameter between \(20\) and \(80\) mm, the following parameters are a good starting point.
| Parameter | Recommended range |
|---|---|
| Wheel speed | 25 – 35 m/s |
| Work speed | 10 – 20 m/min |
| Infeed per pass | 0.005 – 0.015 mm |
| Spark-out passes | 2 – 4 |
| Coolant | Water-soluble grinding fluid |
| Bore tolerance | H7 or tighter if required |
10. Error Analysis and Tolerances
The accuracy of the fixture depends mainly on three factors: the errors of the three cylinder diameters, the errors of the tooth spaces of the straight bevel gear, and the errors of the cage slots. The effect of cylinder diameter errors can be estimated with a simple three-point model.
Assume that the three cylinders are placed at angles \(0^\circ\), \(120^\circ\) and \(240^\circ\). Let \(\delta r_1\), \(\delta r_2\) and \(\delta r_3\) be the radial errors of the three cylinder positions. The displacement of the centre of the straight bevel gear in the \(x\) and \(y\) directions is then
\[
\delta x = \frac{2\delta r_1 – \delta r_2 – \delta r_3}{3}
\]
\[
\delta y = \frac{\delta r_2 – \delta r_3}{\sqrt{3}}
\]
The resultant eccentricity is
\[
\delta e = \sqrt{\delta x^2 + \delta y^2}
\]
This formula is useful for estimating the influence of unequal cylinder diameters. For example, if the diameters of the three cylinders differ by \(0.002\) mm, the resulting eccentricity is only about \(0.0013\) mm. That is negligible for most commercial straight bevel gears. A larger error comes from the tooth-spacing error of the gear itself, because the cylinder will sit in a slightly different angular position if the tooth space is wide or narrow.
The error caused by an angular error \(\Delta \theta\) in the seating cone can be estimated as
\[
\delta_s = R_e \Delta \theta
\]
where \(\Delta \theta\) is in radians. Therefore the taper angle of the fixture body must be measured carefully. A small measurement error of \(0.1^\circ\), which is \(0.00175\) radians, will produce a displacement of about \(0.05\) mm if the outer cone distance is \(30\) mm. That is too large for high-accuracy straight bevel gears. In practice I measure the angle at three positions around the gear and take the mean, and I also check the taper with blueing before grinding the first piece.
The following error budget is typical for the fixture when it is used for a module 3 straight bevel gear with an outer cone distance of \(40\) mm.
| Error source | Typical value | Resulting eccentricity |
|---|---|---|
| Cylinder diameter difference | \(0.002\) mm | \(0.0013\) mm |
| Cylinder roundness error | \(0.001\) mm | \(0.0007\) mm |
| Slot clearance in cage | \(0.1\) mm | \(0.01\) mm if not seated |
| Tooth-space composite error | \(0.01\) mm | \(0.006\) mm |
| Taper measurement error | \(\pm 0.1^\circ\) | \(0.02\) mm at \(R_e=12\) mm |
| Dirt or burr on cone | \(0.005\) mm | \(0.005\) mm |
In practice, I have found that the fixture can easily hold a pitch-cone runout of \(0.02\) to \(0.04\) mm. For straight bevel gears with a required runout of \(0.05\) mm, the fixture is completely safe. For the highest precision gears, the locating cylinders and the taper bore must be lapped and the cage must be cleaned before every workpiece.
11. Practical Results with Straight Bevel Gears
I have applied this fixture to several batches of straight bevel gears. The table below summarises one of the production runs that I carried out in my shop.
| Item | Value |
|---|---|
| Type of workpiece | Differential straight bevel gear |
| Module | 2.5 mm |
| Number of teeth | 20 |
| Pressure angle | 20° |
| Face width | 22 mm |
| Outer cone distance | 28.6 mm |
| Bore diameter before grinding | 24.5 mm |
| Bore diameter after grinding | 25H7 |
| Required pitch-cone runout | 0.03 mm |
| Observed runout range | 0.008 – 0.025 mm |
| Number of pieces ground | 1000 |
| Rejected pieces | 0 |
The fixture was also used for straight bevel gears with modules from 2 to 6 mm and with bore diameters from 18 to 60 mm. In every case, the main advantage was the reduction of setting time. The operator no longer needed to indicate the pitch cone point by point. He simply placed the gear in the fixture, tightened the cover, and started the grinding cycle. The accuracy of the bore was stable throughout the batch.
12. Troubleshooting and Adjustments
No fixture is completely trouble-free. The following table gives the most common problems that I have observed with this type of fixture for straight bevel gears, together with their causes and remedies.
| Symptom | Probable cause | Remedy |
|---|---|---|
| Pitch cone runout is too large | The three cylinder diameters are not equal | Select cylinders with a maximum difference of \(0.002\) mm |
| Pitch cone runout is too large | Chips or burrs under the cage | Clean the taper bore and the cage before each workpiece |
| The gear rocks in the fixture | The cage cone angle does not match the face cone | Recut the cage cone or the fixture taper bore |
| Cylinders fall out during grinding | Retaining wire is too loose or broken | Use a thicker spring wire or replace the wire |
| Cylinders do not seat in the tooth spaces | Cylinder diameter is too large | Reduce the cylinder diameter by \(0.05\) mm steps |
| Contact marks are only on one flank | Slot positions are not aligned with the tooth spaces | Remill the slots according to the calculated \(n_j\) values |
| Taper contact is less than 80% | Taper angle of fixture body is incorrect | Regrind the taper and check with blueing |
| Bore is tapered | Grinding wheel or quill is deflecting | Reduce the grinding allowance and use a sharper wheel |
| The workpiece is difficult to remove | The taper is too steep or the cover is overtightened | Check the taper angle; reduce the clamping force |
13. Further Design Considerations for Straight Bevel Gears
When I first made this fixture, I believed that the cylinder diameter was the only important parameter. Later I found that the length of the cylinders and the shape of the cage slots also influence the result. If the cylinder is too long, it may contact the high parts of the face cone or the root at the small end. This can lift the gear out of the taper bore and produce a large runout. Therefore I now always machine a small chamfer at both ends of each cylinder. The chamfer should be about \(0.2\) mm at \(45^\circ\). It allows the cylinder to slide into the tooth space without scraping the flanks.
The slots in the cage should have rounded corners. A sharp corner can wear the cylinder surface and create tiny metal particles. The particles may be pressed into the flanks of the straight bevel gear and cause measuring errors. I use a small radius of \(0.5\) mm at the corners of the slots. The retaining wire should also be slightly elastic, so that it can hold the cylinder without preventing it from finding its natural seating position.
For straight bevel gears with a very small number of teeth, such as \(z = 12\) or \(14\), the tooth space angle is large and the tooth space is relatively open. In that case, a smaller cylinder diameter must be used. For gears with many teeth, for example \(z = 40\), the tooth spaces are narrow and the cylinder diameter is closer to the upper limit. The formula \(d = \lambda m\) with \(\lambda = 0.80\) is only a starting point. The final decision is always made by trial on the gear itself.
14. Rigidity and Repeatability of the Fixture
Rigidity is important because internal grinding forces are not large, but the fixture must not allow the straight bevel gear to move during the grinding spark-out. I make the fixture body from steel that is hardened and tempered to about \(45\) HRC. The taper bore is ground after hardening. The cage is made from a softer material so that it does not damage the face cone of the straight bevel gear. The pressure cover is hardened and ground on both faces.
The repeatability of the fixture can be checked in a simple way. Grind a test ring, remove it from the fixture, then put it back and measure the runout again. If the runout remains within \(0.005\) mm, the fixture is considered stable. In my experience, this fixture can repeat within \(0.003\) to \(0.008\) mm when the parts are clean and the cylinders are carefully selected.
15. Economic Advantages of the Fixture
The fixture is cheap to make. The main cost is the time needed to make the cage and the fixture body. The locating cylinders are simple parts and can be used for a long time if they are hardened and kept clean. Compared with buying or maintaining an expensive precision mandrel, this fixture has almost no consumable cost. It also reduces inspection time, because the operator can see immediately when the gear is seated properly from the position of the cylinders.
Another advantage is that the same fixture can be used for straight bevel gears that have slightly different face angles, as long as the taper bore of the fixture body is adjusted. If the fixture body is made with a replaceable taper seat, it can be adapted to several gear sizes. This is useful in a job shop where many small batches of straight bevel gears are processed.
16. Relation Between the Fixture and Conventional Methods
The conventional method of push broaching the bore of a straight bevel gear is fast but cannot guarantee the pitch-cone runout. The conventional method of grinding on a mandrel is accurate but slow, because the operator must indicate the gear teeth or the cone surface. The fixture that I have described is a compromise. It keeps the productivity of the grinder and at the same time makes use of the tooth flanks as the locating datum. For straight bevel gears that have already been hardened, this is the most reliable solution that I know.
It should be noted that the fixture does not require a special grinding machine. It can be used on an ordinary internal grinder, a jig grinder, or a universal cylindrical grinder with an internal grinding attachment. The only requirement is that the spindle nose or the work table on which the fixture is mounted must have a low axial runout. In my shop, I use it on a small internal grinder with a magnetic chuck. The fixture body is mounted on the chuck, and the taper bore is trued in place before every setup.
17. Calculation of the Taper Angle with a Universal Protractor
One point that deserves special attention is the angle measurement. The exposed generator line of the locating cylinder is not necessarily parallel to the pitch cone generator of the straight bevel gear. It is the line that is tangent to the cylinder surface and visible outside the tooth space. The cylinder touches the tooth flanks at two points; because of the pressure angle and the local involute shape, the cylinder centre lies along a line that is slightly offset from the theoretical cone generator. Therefore the angle \(\theta_j\) is not exactly the face angle \(\delta_a\). It must be measured, not assumed.
I measure the angle with the cage assembled and lightly held by a rubber band. The universal bevel protractor is placed against the side of the cylinder and the axis of the gear. The reading is recorded. After rotating the gear to the second and third cylinders, the mean angle is calculated:
\[
\theta_m = \frac{\theta_1 + \theta_2 + \theta_3}{3}
\]
The cone half-angle of the fixture body is then machined to
\[
\beta_f = \theta_m
\]
If the cage is not clamped in the fixture but is merely placed on the gear, a small axial clearance should be removed by pressing the cage lightly toward the gear. The contact area between the cage and the face cone should be uniform. Otherwise the measuring angle will be too large or too small.
18. Blueing Test for the Fixture Body Taper
The blueing test is quick and reliable. I coat the taper bore of the fixture body with a thin layer of Prussian blue, insert the cage, rotate it about one quarter of a turn, and then remove it. The blueing marks on the cage show the contact pattern. The contact area ratio is calculated as
\[
C_c = \frac{A_c}{A_t} \times 100\%
\]
If the contact area is not at least 80%, I regrind the taper bore. A common fault is that the contact is heavy at the large end or the small end. This means that the taper angle is too large or too small. The angle is corrected by moving the compound rest of the lathe or by adjusting the taper attachment. In a production situation, the fixture body should be checked with a taper gauge before it is used. I also recheck the taper after the fixture has been used for several hundred pieces, because the seating surfaces may wear.
19. Influence of Straight Bevel Gear Tooth-Surface Accuracy
Because the fixture locates on the tooth flanks of straight bevel gears, its accuracy is limited by the accuracy of the tooth flanks themselves. If the gear has large tooth-spacing errors, the three cylinders will not produce a perfect centre. However, the averaging effect of three tooth spaces is better than the effect of one tooth space. The formula for the selected tooth-space indices \(n_j\) ensures that the sample of tooth spaces is spread around the whole gear. This reduces the influence of cyclic tooth-spacing errors.
In one test, I measured the runout of the same straight bevel gear using different combinations of tooth spaces. I found that the best combination was the one closest to \(120^\circ\) spacing. The worst combination was obtained when the three tooth spaces were adjacent or nearly adjacent. Therefore I always calculate the slot positions before making the cage. For straight bevel gears that are cut on a good generating machine, the tooth-space composite error is very small and the fixture is extremely accurate.
20. Conclusions
I have described a practical grinding fixture for the bores of straight bevel gears. The fixture consists of three precision cylinders, a retaining cage, a conical fixture body and a pressure cover. The cylinders are inserted into three tooth spaces of the straight bevel gear and therefore reproduce the pitch-cone axis from the tooth flanks. The angle of the exposed cylinder generator is measured with a universal bevel protractor, and the taper bore of the fixture body is machined to that angle. In use, the fixture is trued on the grinding machine, the gear is clamped with the cage and cover, and the bore is ground in the usual way.
The fixture has been used successfully for more than one thousand straight bevel gears. All of the ground pieces were within the required pitch-cone runout. The design is simple, inexpensive and easy to adapt to different gear sizes. It eliminates the tedious setting operation that is normally needed when the bore of a straight bevel gear is ground on a mandrel. I am certain that this method will be useful in any workshop that manufactures or repairs straight bevel gears in small or medium batches.