www.geartechnology.com September/October 2010 GEARTECHNOLOGY 47
Tribology Aspects
in Angular
Transmission Systems
Part II
Straight Bevel Gears
Dr. Hermann Stadtfeld
(This is the second of an eight-part series on the tribology aspects of angular gear drives. Each article will be presented
first and exclusively by Gear Technology; the entire series will be included in Dr. Stadtfeld’s upcoming book on the subject,
which is scheduled for release in 2011.)
Dr. Hermann Stadtfeld received a bachelor’s degree in 1978 and in 1982 a master’s degree
in mechanical engineering at the Technical University in Aachen, Germany. He then worked
as a scientist at the Machine Tool Laboratory of the Technical University of Aachen. In 1987,
he received his Ph.D. and accepted the position as head of engineering and R&D of the Bevel
Gear Machine Tool Division of Oerlikon Buehrle AG in Zurich, Switzerland. In 1992, Dr.
Stadtfeld accepted a position as visiting professor at the Rochester Institute of Technology.
From 1994 until 2002, he worked for The Gleason Works in Rochester, New York—first as
director of R&D and then as vice president of R&D. After an absence from Gleason between
2002 to 2005, when Dr. Stadtfeld established a gear research company in Germany and
taught gear technology as a professor at the University of Ilmenau, he returned to the Gleason
Corporation, where he holds today the position of vice president-bevel gear technology and
R&D. Dr. Stadtfeld has published more than 200 technical papers and eight books on bevel
gear technology. He holds more than 40 international patents on gear design and gear pro-
cess, as well as tools and machines.
Design. If two axes are positioned in
space—and the task is to transmit motion
and torque between them using some kind
of gears—then the following cases are com-
monly known:
• Axes are parallel → cylindrical gears
(line contact)
• Axes intersect under an angle → bevel
gears (line contact)
• Axes cross under an angle → crossed
helical gears (point contact)
• Axes cross under an angle (mostly 90°)
→ worm gear drives (line contact)
• Axes cross under any angle → hypoid
gears (line contact)
The axes of straight bevel gears, in most
cases, intersect under an angle of 90°. This
so-called shaft angle can be larger or smaller
than 90°; however, the axes always intersect,
which means they have at their crossing point
no offset between them (Author’s note: see
also previous chapter, “General Explanation
continued
GEARTECHNOLOGY September/October 2010 www.geartechnology.com48
of Theoretical Bevel Gear Analysis” on hypoid
gears). The pitch surfaces are cones that are
calculated with the following formula:
z
1
/z
2
= sing
1
/sing
2
∑ = g
2
= 90° –g
1
in case of ∑ = 90° →
g
1
= arctan (z
1
/z
2
) →
g
2
= 90° – g
1
where:
z
1
number of pinion teeth
z
2
number of gear teeth
g
1
pinion pitch angle
∑ shaft angle
g
2
gear pitch angle
Straight bevel gears are commonly
designed and manufactured with tapered teeth,
where the tooth cross section changes its size
proportionally to the distance of the crossing
Figure 1—Straight bevel gear geometry.
Figure 2—Tooth contact analysis of a straight bevel gear set.
www.geartechnology.com September/October 2010 GEARTECHNOLOGY 49
point between the pinion and gear axes. The
profile function of straight bevel gears is a
spherical involute, which is the direct analog
to the tooth profiles of cylindrical gears.
Figure 1 shows an illustration of a straight
bevel gear set and a cross-sectional drawing.
Straight bevel gears have no preferred driv-
ing direction. Because of the orientation of
the flanks during manufacture, the designa-
tions “upper” and “lower” flank are used. Per
definition, the calculation programs treat the
straight bevel pinion like a left-hand member
and the straight bevel gear like a right-hand
member. Consequently there is a drive side
and a cost side designation, which is for prop-
er definition rather than for implying better
suitability of torque and motion transmission.
Analysis. The precise mathematical func-
tion of the spherical involute will result in line
contact between the two mating flanks (roll-
ing without any load). In the case of a torque
transmission, the contact lines become contact
zones (stripes) with a surface-stress distribu-
tion that shows peak values at the two ends of
each observed contact line, where the contact
line is limited by the inner and outer end of
the tooth (toe and heel). In order to prevent
this edge contact, a crowning along the face
width of the teeth (length crowning) and in
profile direction (profile crowning) are intro-
duced into the pinion flanks, the gear flanks
or both. A theoretical tooth contact analysis
(TCA) previous to gear manufacturing can
be performed in order to observe the effect
of the crowning in connection with the basic
characteristics of the particular gear set. This
also affords the possibility of returning to the
basic dimensions in order to optimize them if
the analysis reveals any deficiencies. Figure 2
shows the result of a TCA of a typical straight
bevel gear set.
The two columns in Figure 2 represent the
analysis results of the two mating flank com-
binations (see also “General Explanation of
Theoretical Bevel Gear Analysis”). However,
the designation “drive” and “coast” are strictly
a definition rather than a recommendation.
The top graphics show the ease-off topogra-
phies. The surface above the presentation grid
shows the consolidation of the pinion and gear
crowning. The ease-offs in Figure 2 have a
combination of length and profile crowning,
thus establishing a clearance along the bound-
ary of the teeth.
Below each ease-off, the motion transmis-
sion graphs of the particular mating flank pair
are shown. The motion transmission graphs
show the angular variation of the driven gear
in the case of a pinion that rotates with a con-
stant angular velocity. The graphs are drawn
for the rotation and mesh of three consecutive
pairs of teeth. While the ease-off requires a
sufficient amount of crowning—in order to
prevent edge contact and allow for load-affect-
ed deflections—the crowning in turn causes
proportional amounts of angular motion varia-
tion of about 90 micro radians in this example.
At the bottom of Figure 2, the tooth con-
tact pattern is plotted inside of the gear tooth
projection. These contact patterns are calcu-
lated for zero-load and a virtual marking com-
pound film of 6 mm thickness. This basically
duplicates the tooth contact; one can observe
the rolling of the real version of the analyzed
gear set under light load on a roll tester, while
the gear member is coated with a marking
Figure 3—Contact line scan of a straight bevel gear set.
Figure 4—Rolling and sliding velocities of a straight bevel gear set along
the path of contact.
GEARTECHNOLOGY September/October 2010 www.geartechnology.com50
compound layer of about 6 mm thickness. The
contact lines extend in tooth length direction
as straight lines—each of which point to the
crossing apex point of face-pitch and root-
cone. The path of contact is oriented in profile
direction and crosses the contact lines under
about 90°.
The crowning reflected in the ease-off
results in a located contact zone inside the
boundaries of the gear tooth. A smaller tooth
contact area generally results from large ease-
off and motion graph magnitudes, and vice
versa.
Figure 3 shows eight discrete, potential
contact lines with their crowning amount along
their length (contact line scan). The length
orientation of the contact lines, caused by the
zero-degree spiral angle, results in a contact
line scan with horizontally oriented gap traces.
If the gearset operates in the drive direction,
then the contact zone (instant contact line)
moves from the top of the gear flank to the
root. There is no other utilization of the face
width than a contact spread under increasing
load.
The graph in Figure 4 illustrates the roll-
ing- and sliding-velocity vectors; each vector
is projected to the tangential plane at the point-
of-origin of the vector. The velocity vectors
are drawn inside the gear tooth boundaries
(axial projection of one ring gear tooth). The
points-of-origin of both the rolling- and slid-
ing-velocity vectors are grouped along the path
of contact, which is found as the connection of
the minima of the individual lines in the con-
Figure 5—Profile sliding and rolling in straight bevel gears.
Figure 6—Straight bevel gear cutting with disc cutter (top: lower flank, bot-
tom: upper flank).
www.geartechnology.com September/October 2010 GEARTECHNOLOGY 51
Figure 7—Force diagram for calculation of bearing loads.
tact line scan graphic (Fig. 3). Figure 4 shows
the sliding-velocity vectors with arrow tip, and
rolling-velocity vectors as plain lines. Contrary
to spiral bevel and hypoid gears, the directions
of both—sliding and rolling velocities—are
oriented in profile direction. The rolling veloc-
ities in all points are directed to the root, while
the sliding velocities point to the top above the
pitch line and to the root below the pitch line.
At the pitch line, the rolling velocity is zero,
just like in the case of cylindrical gears.
Straight bevel gears have properties very
similar to spur gears. The path of contact
moves from top to root (in the center of the
face width) and the contact lines are oriented
in face width direction (Fig. 2). Sliding- and
rolling-velocity vectors are pointing in profile
direction (Fig. 4), which will shift the con-
tact lines in Figure 4 exclusively in profile
direction. This means the crowning of the
contact lines has no significant influence on
the lubrication case (“General Explanation of
Theoretical Bevel Gear Analysis”), but only
the involute interaction will define the lubrica-
tion case and the hydrodynamic condition.
If the lubricant were presented, for exam-
ple, on the top of the gear tooth as in Figure
5, the sliding- and rolling-velocity directions
would result in Lubrication Case 2 as previ-
ously discussed in “General Explanation of
Theoretical Bevel Gear Analysis.” As the roll-
ing progresses below the pitch point, the slid-
ing velocity will change its direction and the
lubrication case becomes Case 3, which is very
unfavorable and reason to assure lubrication is
presented on both sides of the contact zone.
Manufacturing. The manufacturing pro-
cesses of straight bevel gears are planing with
two tool generators, milling with two inter-
locking disk cutters or milling with a single-
disk cutter (Gleason Coniflex). The planing
and interlocking disk cutter processes are out-
dated and typically performed on older, not
current mechanical machine tools. The single-
disk-cutter milling process was developed for
modern free-form machines. It enables the use
of carbide cutting tools in a high-speed, dry-
cutting process.
The blades of the circular cutter disk en-
velope an axial plane (or slight cone) on the
right side of the disk in Figure 6. This plane is
oriented in space and simulates one side of a
generating rack, analog to a cylindrical, gear-
generating rack. Due to the diameter of the cut-
ter disk, the root line of the straight bevel gear
cut shown in Figure 6 is curved, rather than
straight. The curve in the root is a side effect of
this particular process, and has never proven to
be of any disadvantage regarding the gear set
kinematics or strength. The left photo in Figure
6 shows the cutting of the lower flanks. The
opposite flanks of the same slots are cut with
the same tool in the upper position, as shown in
the right photo in Figure 6.
Hard finishing after heat treatment is possi-
ble by grinding with a permanent, CBN-coated
grinding wheel, which basically resembles the
geometry of the cutter disk. The geometry and
kinematics of the grinding process are identi-
cal to the cutting in Figure 6.
Application. Most straight bevel gears used
in power transmission are manufactured from
carburized steel and undergo a case hardening
to a surface hardness of 60 Rockwell C (HRC)
and a core hardness of 36 HRC. Because of
the higher pinion revolutions, it is advisable to
provide the pinion a higher hardness than the
ring gear (e.g., pinion 62 HRC, gear 59 HRC).
Regarding surface durability, straight bevel
gears are also very similar to spur gears. At the
pitch line, the sliding velocity is zero and the
rolling velocity, under certain loads, cannot
maintain a surface-separating lubrication film.
The result is pitting along the pitch line that
can destroy the tooth surfaces and even lead
to tooth flank fracture. However, it is possible
that the pitting can be stabilized if the damage-
causing condition is not often represented in
the duty cycle.
GEARTECHNOLOGY September/October 2010 www.geartechnology.com52
that are not ground or lapped after heat treatment
show the highest root strength with the lowest spi-
ral angles. This explains why—in those cases—
the straight bevel gear remains the bevel gear of
choice.
Straight bevel gears can operate with regular
transmission oil or, in the case of low RPMs, with
a grease filling. In case of circumferential speeds
above 10 m/min., a sump lubrication with regular
transmission oil is recommended. The oil level
has to cover the face width of the teeth lowest in
the sump. Excessive oil causes foaming, cavita-
tions and unnecessary energy loss. There is no
requirement for any lubrication additive. Because
the two kinds of flanks in a straight bevel gear
(upper and lower) are mirror images of each other,
there is no preferred operating direction, which is
advantageous for many industrial applications.
(Ed.’s Note: Next issue—“Zerol Bevel Gears.”)
The axial forces of straight bevel gears can
be calculated by applying a normal force vec-
tor at the position of the mean point at each
member (see also “General Explanation of
Theoretical Bevel Gear Analysis”). The force
vector normal to the transmitting flank is sepa-
rated in its X, Y and Z components (Fig. 7).
The relationship in Figure 7 leads to the
following formulas, which can be used to cal-
culate bearing force components in a Cartesian
coordinate system and assign them to the bear-
ing load calculation in a CAD system:
F
x
= –T / (A
m
• sing)
F
y
= –T • (cosg • sina) / (A
m
• sing • cosa)
F
z
= T • (sing • sina) / (A
m
• sing • cosa)
where: T torque of observed
member
A
m
mean cone distance
g pitch angle
a pressure angle
F
x
, F
y
, F
z
bearing load force
components
The bearing force calculation formulas are
based on the assumption that one pair of teeth
transmits the torque, with one normal-force
vector in the mean point of the flank pair. The
results are good approximations, which reflect
the real bearing loads for multiple-tooth mesh-
ing within an acceptable tolerance. A precise
calculation is, for example, possible with the
Gleason bevel and hypoid gear software.
Straight bevel gears have lesser axial forces
than spiral bevel gears. The axial force compo-
nent—due to the spiral angle—is zero. Zero-
spiral angle minimizes the face-contact ratio to
zero, but results in maximal tooth root thickness.
The tooth thickness counts squared in a
simplified root-bending-stress calculation
using a deflection beam analogy. The thick-
ness reduces by cos (spiral angle). The face-
contact ratio increases, simplified by tan (spi-
ral angle). Those formulas applied to a numeri-
cal example will always show an advantage
of the spiral angle in root-bending strength.
However, the crowning of real bevel gears
will always cause one pair of teeth to transmit
an over-proportionally high share of the load,
while the one or two additionally involved
tooth pairs will only share a small percentage
of the load. Finite element calculations can be
useful in finding the optimal spiral angle for
maximal root strength. As a rule, bevel gears
Corrections
The previous article in this series, “General
Explanations on Theoretical Bevel Gear
Analysis,” which appeared in the August 2010
issue of Gear Technology, contained two
errors. The corrected or clarified text is high-
lighted below.
The complete corrected version of the article
is available at http://www.geartechnology.com/
issues/0810
Corrections
Page 49, left middle paragraph:
6 mm instead of 6 mm. (The error appears twice
in this paragraph.)
Page 52, Formula 7:
Eliminate absolute value of Fn
Fn = Fx / (cosβ • cosa)
= –T / (A
m
• sing • cosβ • cosa) (7)
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