technical

The Importance of Profile Shift, Root

Angle Correction and Cutter Head Tilt
Hermann J. Stadtfeld

Bevel Gear Technology

Chapter 2, Continued
In the previous sections, development of conjugate, face milled — as well as face hobbed — bevel gearsets — including the applica-
tion of profile and length crowning — was demonstrated. It was mentioned during that demonstration that in order to optimize
the common surface area, where pinion and gear flanks have meshing contact (common flank working area), a profile shift must
be introduced. This concluding section of chapter 2 explains the principle of profile shift; i.e. — how it is applied to bevel and hyp-
oid gears and then expands on profile side shift, and the frequently used root angle correction which — from its gear theoretical
understanding — is a variable profile shift that changes the shift factor along the face width. The end of this section elaborates on
five different possibilities to tilt the face cutter head relative to the generating gear, in order to achieve interesting effects on the
bevel gear flank form. This installment concludes chapter 2 of the Bevel Gear Technology book that lays the foundation of the fol-
lowing chapters, some of which also will be covered in this series.
— Hermann J. Stadtfeld

Introduction offset and unique facts regarding gen- Principle of Applying Profile Shift
The goal of the following sections is to eral spatial transmissions are also dis- In the design of bevel gearsets the profile
develop a deeper understanding for the cussed in this chapter. In the next sec- shift of pinion and gear is always applied
function, limits and, perhaps, not fully tion (November/December 2015 Gear as a so called “V0 shift.” If not otherwise
utilized possibilities of bevel and hypoid Technology), length and profile crowning specified, a positive profile shift value of
gears. The gear mathematics developed are added to the Formate bevel gearset to x increases the pinion addendum and
by the author is based on a triangular vec- deliver a practically usable angular trans- reduces the pinion dedendum. The fol-
tor model that presents a comprehensive mission as it is used in industrial gear lowing formal definition will achieve a
tool for simple observations in the gener- boxes. The reader will be able to apply the gear addendum reduction and a gear
ating gear up to complex, three-dimen- derivations to any other bevel and hyp- dedendum increase with the same abso-
sional developments. All different kinds oid gearset. With results for each calcu- lute amounts:
(1)
of bevel and hypoid gears can be observed lation step, basic settings are computed, x = x1 = –x2
and manipulated with this model, without as they are commonly used by modern
alteration of the notation. However, in the CNC bevel gear generators in order to cut Where:
x Nominal profile shift factor is based
most complex level, the lengths and direc- or grind real bevel gearsets. on a normal section at mid-face
tions of the vectors change according to To complete the explanations and x1 Pinion profile shift factor, equal to the
higher-order functions and depending on examples discussed so far, it seems appro- nominal profile shift factor
the rotational position of the generating priate to elaborate with some graphics on x2 Ring gear profile shift factor, equal
gear (Refs. 1–2). the three most commonly used geometri- to the negative nominal profile shift
The first chapter of this book, cal features in bevel gear optimization. factor
“Nomenclature and Definition of
Symbols,” should help to avoid or mini-
mize the interruption of the flow in the
gear theoretical developments with defi-
nitions of formula symbols.
In previous sections, the develop-
ment of a face milled, conjugate spi-
ral bevel gearset is conducted (August
2015 Gear Technology). In the second
step, an analogue face hobbed bevel gear-
set is derived (September/October 2015
Gear Technology) that is converted to a
non-generated (Formate) version in the
third step. In step 4 an offset is added
to the Formate spiral bevel gearset that
results in a hypoid gearset. Consequences
regarding the introduction of the hypoid
Figure 1 Impact of positive profile shift on pinion blank.

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 As visible in Figures 1and 2, there is
no influence from the profile shift on
the position of the pitch line, the pitch
angle and the mean cone distance RM.
The pitch line is defined by the gearing
law and represents the surface generator
of a cone that rolls with the pitch cone of
the mating gear without slippage, satisfy-
ing the ratio given by the number of teeth
of the two mating members.
In the case of non V0 cylindrical gear
pairs (V+ or V–), the center distance will
change, which will establish new, effective
pitch cylinders in the interplay between
the two profile-shifted cylindrical gears,
according to the gearing law. The two
effective pitch cylinders roll upon each
other without slippage, with the ratio
defined by the number of teeth of the two
mating cylindrical gears.
For bevel gears, the analogy to the cen-
ter distance change is a change to the
shaft angle with unchanged mean cone
Figure 2 Impact of positive profile shift onto ring gear blank.
distance RM. Because a shaft angle change
in the course of a gear optimization is not
acceptable, it seems a valid conclusion
that for bevel gear systems only a V0 pro-
file shift is physically acceptable.
The aim of a profile shift is to increase
or reduce the profile depth portions above
and below the pitch line in order to use
other parts of the involute, i.e. — octoide.
For bevel gears this means that all
machine settings remain the same, while
the blanks and the axial blade profile loca-
tions are changed. The blade reference
point location S890 changes as follows:
(2)
S890 = hF * (fDepth + fSPFK + x) * mn Figure 3 Profile shapes in case of positive and negative profile shift.

The three graphics in Figure 3 show employed, where: radial positions of the cutting edges have
(3)
the initial tooth without profile shift to to be corrected accordingly. Attention
xS = xS1 = –xS2
the left. In the middle of Figure 3 a tooth (4) has to be paid, to that the blade top width
with positive profile shift is shown, which dZS = π * mn + 2 * xS * mn does not become too small. Referring
(5)
has a larger tooth root thickness, but dZS = dZ + 2 * xS * mn to Figure 4, the difference RCOW 2 –
exhibits an almost pointed topland. The RCOW 1 for a module m n above 4 mm
Where:
tooth with negative profile shift is to the dZ nominal tooth thickness should not be smaller than 0.8 mm (face
right in Figure 3 has a weakened tooth dZS corrected tooth thickness by profile milling). The new blade point radii are
thickness in the root but shows a larger side shift calculated with:
(6)
topland. These effects would be more RCOW1S = RCOW1 + xS * mn
significant if the original tooth thickness The value of xS is applied with oppo- (7)
dz wasn’t defined at the actual reference site signs to pinion and ring gear in order RCOW2S = RCOW2 – xS * mn
(8)
circle for each profile shifted example. to maintain functionality and the origi- RCOW1S = RCOW3 + xS * mn
This convention is meaningful in order nal backlash of the gearset. The resulting (9)
to maintain balanced tooth thickness- tooth thickness change xS is based on the RCOW1S = RCOW4 – xS * mn
es between pinion and ring gear in the normal module and is applied in the nor- The definition of the profile side shift is
course of profile shift optimizations. mal tooth section at mid-face, if not spec- based on the reference profile of the gen-
In the case of a desired tooth thick- ified differently. erating gear as shown in Figure 4.
ness change, profile side shifts x S are In the course of a profile side shift the

January/February 2016 | GEAR TECHNOLOGY 49

 technical

The Root Angle Correction

Bevel pinions with a bearing hub on the
small diameter often pose a problem
in that the hub is “sliced” by the bevel
gear cutter in the course of the cutting
process. The first suspicion of a possi-
ble interference problem occurs when
the extension of the root tooth line is
viewed as cutting through a part of the
hub. The calculated and graphically rep-
resented cutter path shows the relation-
ship between the roll position and the
closest distance to the work gear axis.
Figure 4 Pinion reference profile with positive profile side shift.
Each calculated point is rotated into the
drawing plane, where the sum of points is
drawn as a curve.
In order to eliminate a cutter/hub inter-
ference, the idea of a root angle correc-
tion was developed. Figure 5 shows in
the top section a cross sectional view of
a bevel pinion with a pitch angle that is
calculated from the relationship between
the pinion and gear number of teeth (see
“Basics of Gear Theory, Part II,” July 2015
Gear Technology, Equations 10-12). The
lower part of Figure 5shows the alteration
of the original pinion by the root angle
correction δK. The auxiliary cone with the
angle GATK was rotated around the ref-
erence point in the middle of the tooth
about – δK. Face and root angle follow this
rotation as well. Although the pitch cone
is not influenced by this rotation, the gen-
erating roll motion in the manufacture of
root angle-corrected bevel gears happens
around the auxiliary cone. The side effects
that occur due to rolling on an incorrect
cone can be partially eliminated by tilting Figure 5 Principle of root angle correction.
the cutter head, as explained in (see “Basics
of Gear Theory, Part II,” July 2015 Gear the explanation about profile shift. As a that generally only cutter head tilt is men-
Technology, Figure 23). matter of fact, an analogy is justified to tioned where in reality the cutter head tilt
A large part of the influence on the view a root angle correction as a linear has a certain orientation in space relative
flank form due to the root angle cor- variable profile shift which is zero at mid to the generating gear. The following list
rection cancels out between pinion and face and which reaches on the toe a maxi- summarizes the five known kinds of cut-
gear. The remaining part consists of flank mum and on the heel a minimum. ter head tilt, i.e. — the effects achieved by
twisting, which with same limitations Root angle corrections in the vicin- implementing certain tilt directions.
can be used for Ease-Off optimizations. ity of 2° can be reliably used for opti- Effects due to cutter head tilt:
A further interesting aspect of the root mizations without any negative effect to • Generation of length crowning
angle correction can be observed on pin- the rolling behavior of the bevel gearset. • Correction of pressure angles
ions with undercut. A reduction of the However, it is recommended not to apply • Root angle tilt for pinions mated with
root angle enlarges the root diameter at root angle corrections above 4°. Formate gears
• Root angle tilt to achieve flank twisting
the toe (with remaining mean pinion
• Improved generating gear orientation
diameter), which reduces or even elimi- The Cutter Head Tilt in case of tapered depth teeth
nates existing under-cut. Together with Cutter head tilt is applied to achieve vari-
profile shift the introduction of the root ous effects in the mathematical generat- Length crowning and pressure angle
angle correction increases the risk of ing model for bevel gears as well as in corrections are realized with a rotation
pointed toplands at the pinion toe. older mechanical bevel gear generators. of the cutter head around the tangent to
The last paragraph reminds much of It has to be mentioned in this context, the cutter track at the mean face position.

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The principle is shown in Figure 6. The
left photograph shows the un-tilted ref-
erence position where in the right photo,
the cutter tilt can be recognized.
Pinions which roll with Formate ring
gears require large tilt angles around the
vertical axis of the generating gear model.
Figure 7 shows in the left photo the start-
ing point of an un-tilted cutter head. At
the right side, a cutter head tilt with the
amount of the pitch angle of the generat-
ed pinion is symbolically represented (see
also “Basics of Gear Theory, Part 2,” July
2015 Gear Technology).
A root angle change for the generation
of a determined amount of flank twist
can be achieved with the tilt principle in Figure 6 Cutter head tilt for length crowning and blade angle correction: No tilt in left graphic;
Figure 8. A cutter head tilt with the same right graphic with cutter tilt.
amount of the root angle change (Fig.
8, left to right) will lead to the desired
flank twist. It is required to recalculate
the ratio of roll, since the pinion rolls
now on a generating gear with a changed
cone angle. The new ratio of roll is calcu-
lated using equations 10-13 from “The
Basics of Gear Theory, Part 2.” (July 2015
Gear Technology). The cutter head tilt for
establishing an improved generating gear
orientation consists also of the same prin-
ciple shown in Figure 8.

Dr. Hermann J.
Stadtfeld received
in 1978 his B.S. and in
1982 his M.S. degrees in
mechanical engineering Figure 7 Cutter head tilt of a Formate pinion member.
at the Technical University
in Aachen, Germany; upon
receiving his Doctorate,
he remained as a research scientist at the
University’s Machine Tool Laboratory. In
1987, he accepted the position of head of
engineering and R&D of the Bevel Gear
Machine Tool Division of Oerlikon Buehrle AG
in Zurich and, in 1992, returned to academia
as visiting professor at the Rochester Institute
of Technology. Dr. Stadtfeld returned to the
commercial workplace in 1994 — joining The
Gleason Works — also in Rochester — first as
director of R&D, and, in 1996, as vice president
R&D. During a three-year hiatus (2002–2005)
from Gleason, he established a gear research
company in Germany while simultaneously
accepting a professorship to teach gear
technology courses at the University of
Ilmenau. Stadtfeld subsequently returned to
the Gleason Corporation in 2005, where he
currently holds the position of vice president,
Figure 8 Cutter head tilt for the creation of generating crowning.
bevel gear technology and R&D. A prolific
author (and frequent contributor to Gear
Technology), Dr. Stadtfeld has published more For Related Articles Search
than 200 technical papers and 10 books on
bevel gear technology; he also controls more bevel gears
than 50 international patents on gear design,
gear process, tools and machinery. at www.geartechnology.com

January/February 2016 | GEAR TECHNOLOGY 51