Figure 4 Definition of topological modification and example.
- Click image to enlarge
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Figure 5 Measurement grid convention of Klingelnberg machine format.
- Click image to enlarge
The example data map defines
the progressive tip relief on Side I and
no modification on Side II. Note that
the modification values in the data map
are normalized and the actual local
modifications are calculated with Ca_
local = fij * Ca, where fij is the modification
factor at (i, j) node and Ca is the
amount of modification. The intermediate
values in between the data can be
interpolated by linear, quadratic, or spline
approximation along the tooth width and
height, respectively.
The adjustment of the bevel gear models
to any predetermined measurement
grid points should now be possible by
applying the topological modification.
That is, the modification can be calculated
as the deviation between the surface
of the 3-D model and the measurement
grid points of the target model. The measurement
grid points report contains the
Cartesian coordinates and the normal
vectors of the grid points with the format
of (XP YP ZP XN YN ZN). The reference
coordinate system of the data is
different according to the measurement
machines. For example, the reference
coordinate system of Klingelnberg format
is using the convention shown in Figure
5. The order of the indexes for the points
and the sections are defined according
to ISO/TR 10064-6 (Ref. 5) as well as the
convention from the manufacturers such
as Klingelnberg (Ref. 6). Here the index
of the lines starts from the root to the tip,
and the index of the columns from Side II
(heel) to Side I (toe).
In applying the modification, however,
various problems have arisen. The definition
of topological modification surface
in helical gears is located between the
tip and the root form diameters, but the
diameters over the tooth width for bevel
gears are changing.
On the other hand, the effort to transform
the measured grid points to the
format of the topological modifications
is greatly increased. While the measurement
direction of the distance between
the two corresponding grid points for
adjustment calculation is different from
the normal of the tooth form (that is, the
path of contact) along which the modification
is applied. Moreover, even the
deviation values are given correctly, we
cannot easily reach to the exact surface
points because the target modification
can have highly nonlinear pattern.
Thus, the procedure to get the topological
modification, so that the final
model becomes equivalent with the target
model, cannot be finished in just a single
step but requires rather several iterations
(Fig. 6).
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Figure 6 Procedure to get topological modification for target model.
- Click image to enlarge
_400.jpg)
Figure 7 Modifications for right flank at Step 1 (left) and Step 11 (right).
- Click image to enlarge
In each step, the distance
between the corresponding measurement
points are calculated and converted into
the dimension in the virtual cylindrical
gear. Then the topological modification
is calculated based on these values
and applied to generate a new measurement
grid. The procedure iterates until
the given acceptance criteria are met. The
acceptance criteria are given as the maximum
distance between the surface of the
3-D model and the corresponding measurement
points is smaller than the userdefined
tolerance.
Application and Result
We used 11×7 points for the measurement
and topology template definition;
that is, 11 points starting from Side I
(toe) to Side II (heel), and 7 points from
the root form diameter to the tip diameter without margins. The position of
each measurement point is defined as the
length factor of the path of contact from
the root form diameter to the tip diameter
(column values in yellow in Table 1)
and the face width factor from Side I to
Side II (row values in yellow in Table 1).
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- Click image to enlarge
Topological modification for the right
flank. Table 1 shows the topological deviation
and modification template values
for the right flank according to the calculation
steps. In the calculation, we set the
acceptable maximum deviation to 5 μm.
Step1. In the first step we measure
the deviation by the normal distance
between the measurement points of the
Gleason model with the flank surface of
3-D model (see Deviation 1 in Table 1).
Then, we use the Deviation 1 as the initial
topological template, Modification
1. The green-colored fields in the table
indicate the border of the tooth flank. In
our modeling strategy we use a slightly
bigger surface area to cover the real gear
surface and it’s not possible to measure
correct distances at the borders. Thus we
ignore the border values in the acceptance
checking in the calculation procedure
and use the extrapolated values for
the values. The maximum distance of
the initial step gives 575 μm at the position
(0.965, 0.696). The deviation shows
relatively big values because we intentionally
increased the tooth thickness of the
KISSsoft model to completely cover the
surface of the target model and to give
positive distances. Thus, the final model
is compensating not only the topological
deviation of the surface but also the tooth
thickness deviation of the model.
Step 2. After applying the topological
modification of the first step, the maximum
distance at the position (0.965,
0.696) reduced to 65 μm and the new
maximum distance is 135 μm at the position
(0, 0.879) (see Deviation 2 in Table
1). From Deviation 2 you will see the
three points at (0, 0.089), (0.522, 0.089)
and (0.965, 0.193) have deviations less
than the acceptance criteria of 5 μm (values
in blue). In this case we use the same
topological modification values of the last
step at those positions. For the remaining
positions we build a new topological
modification by linear summation of the
deviation of each point and the last topological
modification, which is:
Modification 2 = Modification1 + Deviation 2
Step 3. Now Deviation 3 after applying
Modification 2 shows smaller distances
than Deviation 2, and more positions
fitting into the acceptance deviation.
The new maximum distance is
70 μm at the position (0.965, 0.879) (see
Deviation 2 in Table 1). However, the
deviation in several positions — such as
the positions at (0.956, 0.089) and (0.956,
0.193) — increased because the surface is
generated by spline approximation from
the topological modification template
(values in red). In this case we build a
new topological modification from the
last topological modification, that is:
Modification3 = Modification2 – SIGN
(Modification2 – Modification1) *
(Deviation2) + (SIGN(Deviation2) + SIGN
(Deviation3))/2*(Deviation2–Deviation3).
Step 11 (final step).We then needed
to iterate 11 steps until all deviations
fit into the acceptance criteria. You can
find the final topological modification as
Modification 11, and the final deviation
as Deviation 12, in Table 1. Now all the
deviation values are less than the maximum
deviation of 5 μm — except the values
at the border.
The graphical comparison of the modification
surfaces of Step 1 and the Step 11
(final step) are shown in Figure 1. As you
can expect, the final modification surface
doesn’t not show a regular pattern, and
it’s impossible to achieve the modification
by simple combination of the conventional
modification types such as crowning
and barrelling.
Topological Modification for the Left
Flank
After finishing the calculation for the
right flank, we applied the same procedure
for the left flank. Table 2 shows the
topological deviation and modification
template values according to the calculation
steps for the left flank.
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- Click image to enlarge
Step 1. In the first step the maximum
distance of the left flank shows 570 μm at
the position (0.965, 0.789).
Step 14 (final step).We could reach
the final topological modification after
14 steps for the left flank. You can find
the final modification as Modification
14 and the final deviation as Deviation
15. You can see all the deviation values are less than the maximum deviation of
5 μm, except the values at the border. The
graphical comparison of the modification
surfaces of Step 1 and the Step 14 (final
step) are shown in Figure 8.
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Figure 8 Modifications for left flank at Step 1 (left) and Step 14 (right).
- Click image to enlarge
Conclusions
The developed method makes it possible
to incorporate any desired flank form
of a bevel gear given by grid points, and
provides the model for the CAM machining
in a very short time from the simplest
way. That is, the macrogeometry is
generally assumed by existing standards
or data sheets, and the microgeometry is
created by a difference of unmodified real
flank-to-the-flank created by topological
modifications with the help of KISSsoft.
The results showed that the final flank
with the topological modification gives
the deviation of less than 5 μm, which can
be ignored, considering the manufacturing
tolerance in practical situations.
The method presented here has considerably
high potential for practical usage
because it allows not only the modeling
of all existing flank forms into 3-D models,
but also can be applied for various
other purposes, such as to compensate
hardening distortions and cutting deviations
of 5-axis milling models. These are
very important features in practice, and
were unresolved issues in the 5-axis
milling process.
References
- KISSsoft AG. Calculation Program for Machine
Design. http://www.kisssoft.ch.
- ISO 23509 (2006). Bevel and Hypoid Gear
Geometry. http://www.iso.org.
- Bae, I. and J. Langhart. “Können 5-Achsgefräste
3D-Kegelräder mit Konventionell Hergestellten
Kegelrädern Gepaart Werden?” 2013, Dresdner
Maschinenelemente Kolloquium 2013, 135-152
- ISO 21771 (2007). Gears: Cylindrical Involute
Gears and Gear Pairs — Concepts and Geometry.
- ISO/TR 10064-6 (2009). Code of Inspection
Practice — Part 6: Bevel Gear Measurement
Methods.
- Klingelnberg, J. Kegelräder, 2008, Springer.
- Breton S.p.A., http://www.breton.it
About Author
Inho Bae, Ph.D, received his doctorate in 2002 from
Hanyang University in Korea by the research on the
design of multi-stage gearboxes. After working as a postdoctoral
research fellow at Kyoto University, he moved
in 2008 to KISSsoft AG in Switzerland as a development
engineer. Dr. Bae is the head of technical support and
also working on the development of the KISSsoft and
KISSsys software suites.
Virgilio Schirru was trained as a mechanical engineer
at the University of Cagliari in Italy and at the Glasgow
University of Scotland. After working for M.g. Mini Gears
S.p.a. in their sintering and cut steel department for
cylindrical and bevel gears, Schirru joined KISSsoft AG
as a support/development engineer
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