Crowning Techniques

in Aerospace
Actuation Gearing
Anngwo Wang and Lotfi El-Bayoumy

(Copyright 2009 by ASME Proceedings of the ASME 2009 International Design Engineering Technical Conferences &
Computers and Information in Engineering Conference, August 30-September 2, 2009, San Diego, California, USA.)

Management Summary
One of the most effective methods in solving the edge loading problem due to excess misalignment and deflection
in aerospace actuation gearing is to localize tooth-bearing contact by crowning the teeth. Irrespective of the applied
load, if the misalignment and/or deflection are large enough to cause the contact area to reduce to zero, the stress
becomes large enough to cause failure. The edge loading could cause the teeth to break or pit, but too much crowning
may also cause the teeth to pit due to concentrated loading. In this paper, a proposed method to localize the contact
bearing area and calculate the contact stress with crowning is presented and demonstrated on some real-life examples
in aerospace actuation systems.

Introduction from wing bending or the deflection longitudinal direction and the contact
The high-lift system of an aircraft of the housing that supports the gears. is localized, but it will not be stabilized
composed of trailing and/or lead- Irrespective of the load, once the mis- unless the amount of crowning is opti-
ing edge flaps increases the lift dur- alignment and/or deflection cause mized.
ing takeoff, does flight controls during the contact area to vanish, the stress The purpose of this paper is to find
cruising and reduces the landing dis- becomes large enough to cause prob- an optimized crown so that the contact
tance when the airplane touches down. lems. pattern will not become too large and/
This flight control system is usually AGMA 2001-B88 (Ref. 2), pro- or sensitive to fall outside of the tooth
composed of power control units, vides a misalignment factor for straight surface, or too small to cause an exces-
torque tubes, bevel gearboxes, offset and helical gears, but it does not cover sive contact stress.
gearboxes, leading-edge rotary actua- crowned gears. AGMA2003-B97 (Ref. Leading-edge rotary actuators. A
tors, trailing edge rotary actuators and 5) has a crowning factor of 1.5 for all cross section of a typical leading-edge
leading-edge sector gears and pinions. bevel gears. rotary gear actuator is shown in Figure
The system also includes other protec- A way of localizing the gear con- 1, and the schematic of the compound
tive components such as torque lim- tact pattern from line contact to point stage is shown in Figure 2. There are
iters, slip clutches, no-back devices contact has been developed for reduc- three meshes on each one of the planet
and wing-tip brakes. Many of these ing noise and vibration by Litvin (Ref. gears. The center ring gear is usual-
components contain different types of 1). Using a parabolic function of the ly the output and the end ring gears
gears that are usually highly loaded to rotational relationship between the cut- are fixed to the structure. The reaction
increase the power-to-weight ratio. ter and the gear, one of the gears is forces from the ring gears on the com-
Deflection and misalignment crowned in both transverse and longi- pound planet gear bend the planet to a
between a pair of meshing gears can tudinal directions so that the piece-wise shape as shown in Figure 5. If the gears
become detrimental when the gears transmission error can be transformed are not crowned, the planets are edge-
are edge loaded—generating noise to a parabolic distribution. loaded, thereby reducing the overall
and high bending and contact stress- The traditional way of crowning is capacity of the actuator.
es. The deflection emanates from the by either plunging the cutter or chang- Trailing-edge rotary actuators. A
high loading and the misalignment ing the lead. The crowning is in the continued

www.geartechnology.com August 2010 GEARTECHNOLOGY 53

 cross section of a typical trailing-edge
rotary gear actuator is shown in Figure
3, and the gear schematic is shown in
Figure 4. The output consists of two
load paths from two end ring gears.
The sun gear driving the right end of
the planet gear and stiffness difference
between the right and left load paths
Figure 1—Typical leading-edge rotary Figure 2—Schematic of the output
actuator.
causes the compound planet gear to
stage of a typical leading-edge rotary
actuator. tilt. Thus, the planet gear loads—due
to meshing with the ring gears—have
to be considered, and the misalign-
ment from the two load paths needs to
be included for selecting the optimum
crown.
Leading-edge sector gears and
pinions. A typical leading-edge sector
gear and pinion is shown in Figure 6.
The pinion has to be crowned to allow
for wing bending if a spherical bearing
mount is not possible. This gear set is
exposed to the outside environment,
and the grease or dry film lubrication
may be compromised between mainte-
Figure 3—Typical trailing-edge rotary Figure 4—Schematics of the output
actuator. stage of a typical trailing-edge rotary nance servicing. The contact stress and
actuator. crowning radius have to be optimized,
so the risk of running dry is mitigated.
Torque tube splines. When mis-
alignment is relatively small, a
crowned spline can be used to trans-
mit torque between two components.
Figure 7 shows a typical crowned
spline. Similar to the sector and pin-
Figure 5—Typical compound planet crowning.
ion, the main purpose of the crowned
spline is to localize the contact to avoid
edge loading. It is usually a full crown.
Because of the large misalignment,
the contact area is considerably small.
Usually, bending or contact stress is
not an issue, but wear due to recip-
rocating rubbing on every revolution
becomes significant. To evaluate wear
Figure 6—Typical leading-edge sector gear and pinion. life, one can follow Dudley’s recom-
mendation (Ref. 4) to calculate the
contact stress.
Crowned bevel gears. A straight
bevel gear with crowned teeth is called
a Coniflex bevel gear. The curvature
from the cutting process is to provide
the needed misalignment capability.
Under light load, the contact pattern
should be located at the central toe of
the teeth (Fig. 8), and the length of the
Figure 8—Desired contact pattern bearing contact should not exceed 50%
Figure 7—Typical crowned splines. under light load. of the total face width.
54 GEARTECHNOLOGY August 2010 www.geartechnology.com
 Under operating load, the contact pinion and gear respectively, and E1
pattern should be located at the central and E2 are modulus of elasticity for
toe of the teeth (Fig. 9), and the length (5) pinion and gear respectively.
of the bearing contact should normally If a gear set is crowned, the crown
be 50–75% of the total face width. is usually on the pinion. The contact
There should be no edge loading where: stress calculated from Equation 3 con-
under any circumstances. The V-H Sc is the contact stress; Cp is the siders only the contact stress without
check is performed to validate that— elastic coefficient; Wt is the tangential crowning. As AGMA does not have
under simulated misalignment—the load; d is the operating pitch diameter an equation for the contact stress with
contact ellipse should always be within of pinion; F is the net face width; I is crowning, we propose using equations
the face width. the geometry factor for pitting resis- from Roark and Young (Ref. 3)—i.e.,
For the purpose of analysis, an tance; n1 and n2 are Poisson’s ratio for continued
equivalent spur gear can be used to
simulate the bevel gear.
From all the above applications in a
high-lift system, we can appreciate the
importance of gear crowning. How to
design, balance and position the con-
tact region is the subject of the next
section.
Optimization of Crowning Figure 9—Desired contact pattern under operating load.
The contact stress in a spur invo-
lute gear set is usually calculated at
the lowest point of single tooth contact
(LPSTC) of the pinion. The transverse
radii of curvature of the gear tooth pro-
files at this contact point are defined as
in AGMA standards (Ref. 2):

(1)

(2)
where:
r1 is the transverse radius of cur- Figure 10—Gear crowning. Figure 11—Gear crowning with mis-
vature of pinion at LPSTC; r2 is the alignment and deflection.
transverse radius of curvature of gear
at highest point of single tooth con-
tact (HPSTC); RO1 is the outside diam-
eter of the pinion; Rb1 is the base circle
radius of the pinion; N1 is the number
of teeth of the pinion; Cd is the cen-
ter distance of the gear set; fop is the
operating pressure angle; and -/+ is
for external and internal gear meshes,
respectively.
The contact stress in a spur gear set Figure 12—Derivation of crowning radius.
with no crowning and no misalignment
is defined in AGMA standards (Ref. 2):

(3)

where:

(4)
Figure 13—A triple planet gear.

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the contact stress for the general case gear pair; P is the normal load; a is tern lies within the tooth face at the
of two bodies in contact. The shape the semi-length of the instantaneous maximum misalignment condition.
of the instantaneous contact area is an contact ellipse in the face width direc-
ellipse, and the contact stress is calcu- tion; b is the semi-length of the instan- (13)
lated by the following equation. taneous contact ellipse in the profile
direction; a and b are geometrical coef- where:
(6) ficients (Ref. 3); R1 is the crowning q is the angular displacement from
radius of the pinion; and R2 would be the combination of misalignment and
where: infinite if the gear is not crowned. deflection. Depending on the amount
Gear crowning is specified using of misalignment and deflection, with
(7) the following equation, as shown in all the equations above, one can opti-
Fig. 10: mize the contact so that the contact
(8) stress is within the material allowable.
(12) Also, the contact ellipse is stabilized
(9) within the tooth face boundary with-
where: out edge loading. Depending on the
(10) R1 is crowning radius; Drop is the application, the optimization between
drop over the distance L 1, which is the variables a, R1, q and L1 can greatly
from the center of crown to the end of influence wear life.
(11) the tooth. The drop should include the Some gears have more deflection
deflection and misalignment. than misalignment, as in the case of
where: The general rule for a good crown the compound planet gears shown in
s c is contact stress for crowned design should be that the contact pat- Figure 5, and the contact area is wide.
Some gears are tilted by deflection, so
the center of the crown is not at the
center of the tooth face (called bias
crown). And some gears have more
misalignment than deflection, as in
the case of crowned splines, and the
contact ellipse is small compared to
the face width. Here, the center of the
crown is at the center of the tooth face
(called full crown).
A crowned spline has one more
limitation—the tooth thickness has to
Figure 14—End planet gear crowning.
be modified from the standard because
the minimum effective clearance is
zero. The tooth thickness Tmod is depen-
dent on R1 and q. From Figure 12, the
following equations can be derived:

(14)

(15)

(16)

Figure 15—Damaged gear without Figure 16—Good gear contact with (17)
crowning. crowning.
(18)

where:
C is the drop in the normal plane; G
is the total gage length; the gage from
Figure 17—Contact pattern of crowned sector pinion.
the center of the crown is G/2; and
56 GEARTECHNOLOGY August 2010 www.geartechnology.com
dT is the clearance between the space 1,000 lbf. From Fig 16 we can see that tact and bending stresses—is reduced.
width of the internal teeth and the tooth crowning has eliminated the pitting These proposed methods have been
thickness of the external teeth. problem, so that the full tooth is now successfully applied in finding the opti-
From the above equations, the fol- sharing the load. mum crown, so the crown radius is not
lowing relationship can be derived: Example 3. A sector and pinion too large to cause the contact pattern to
gear set in Figure 6 must accommodate fall outside the tooth surface—or too
the wing bending. Because of the envi- small,which would result in excessive
(19) ronmental exposure, the contact stress contact stress.
must be low enough that running the Although the method has been
Numerical Examples gears without re-grease is possible. For demonstrated here for spur gears, simi-
Example 1. A triple planet gear of a given misalignment, we would like lar approaches can be applied to heli-
a trailing-edge actuator is shown in to design a new crowning radius and cal, bevel or other types of gears.
Figure 13. The mating gears are all face width, so the stress is low enough References
internal gears. to eliminate the need for re-lubrication. 1. Litvin, F.L. et al. “Computerized Design and
The tangential load at both ends is The baseline design is regularly Generation of Low-Noise Gears with Localized
calculated as 3,600 lbf. lubricated, and maximum allowable Bearing Contact,” NASA Reference Publication
106880, ARL-TR-760, 1994.
The relative deflection under this misalignment is .0015 inch/inch. The 2. AGMA. “Fundamental Rating Factors and
load is .0015 inch, and the misalign- face width is 1.1 inch, crowning radius Calculation Methods for Involute Spur and
ment due to backlash and runout from 21.5 inch. The calculated contact stress Helical Gear Teeth,” ANSI/AGMA 2001–B88,
1988.
the ring gear and planet gear is .0023 is 312 ksi under the maximum operat- 3. Young, Warren C. Roark’s Formulas for
inch total. The face width on the end ing tangential force of 3,800 lbf. After Stress and Strain, Sixth Edition, McGraw Hill,
gear is 1.44 inch. The misalignment increasing the face width to 1.5 inch, 1989.
is .0008 inch/inch slope. The rela- the contact stress is reduced by only 4. Dudley, Darle W. “When Splines Need Stress
Control,” Product Engineering, 23, December,
tive slope under the load at point B in 8%. However, the increased face width 1957.
Figure 13 is .0011 inch/inch. The total comes with a weight penalty. One solu- 5. AGMA. “Rating the Pitting Resistance and
slope is .0019 inch/inch on point A. tion is to change to a material that has Bending Strength of Generated Straight Bevel,
Zero Bevel and Spiral Bevel Gear Teeth,” ANSI/
At point B, the slope is .0015/1.44 less higher allowable contact. The pinion AGMA 2003–B97, 1997.
.0011, and is equal to –.00006 inch/ shown in Figure 17 was tested for a no
inch. The total slope is .0008 –.00006 re-grease application. It is clear that Anngwo Wang is an engineering specialist
at MOOG Inc. Aircraft Group. He received
= .00074 inch/inch on point B. After although the contact pattern is local-
his doctorate from University of Illinois at
solving simultaneous equations, the ized—as a result of the higher contact Chicago in 1997. He is responsible for gear
crowning radius is 143 inch, and the stress—the initial lubrication eventual- design and analysis of the transmission in the
crowning center is .64 inch from loca- ly degrades and micropitting and rust- aircraft flight control system. He was awarded
tion B. A bias crown is shown in Fig. ing will soon follow. a NASA Tech Brief Award on “Software for
Local Synthesis of Spiral Bevel Gears.” He
14. A contact stress value of 239 ksi is
published, with professor Faydor Litvin, sev-
calculated. Compared to the baseline Conclusions eral technical papers based on his thesis enti-
design of the crowning radius of 126 In this paper a proposed method to tled, “Computerized Design and Generation
inch— crowning center is at the middle optimize the contact pattern and to cal- of Spiral Bevel Gears with Uniform and
of the end tooth and the contact stress culate the contact stress with crowning Tapered Teeth and Forged Straight Bevel
Gears.”
of 254 ksi—the contact stress is 6% is presented. Some real-life applica-
better. tions in aerospace actuation gearing Lotfi El-Bayoumy is the engineering manag-
Example 2. An offset gear—with with proposed crowning are demon- er of product integrity at MOOG Inc., Aircraft
one bearing very close to the one strated. Group. He received his doctorate from New
end and another support at the other Deflection and misalignment in a York University and has extensive experience
on jet engines, turbo machinery, airframe
end—is shown in Figure 15. Because gear set can be detrimental if the gears structures, hydraulic and pneumatic systems,
of excessive deflection, the gear is are edge loaded, generating noise and auxiliary power units, hydraulic power units,
edge-loaded and pitted, as shown. The high bending and contact stresses. engine control units, actuation systems, air-
face width is .80 inch. The total slope Deflection usually results from highly craft accessory drives, helicopter transmis-
including the deflection and misalign- loaded gears, and misalignment from sions, winches, hoists, cargo handling systems
and high lift systems. He was chair of the
ment is .0048 in/in. After solving the wing bending or deflection of the gear
Acoustics, Shock and Vibration Committee,
simultaneous equations, the crown- housing. SAE, and has produced an extensive list of
ing radius is 91 inch, and the crown- It is very important to have the publications covering a wide area of engineer-
ing center is at the end of the tooth. A right crowning, so the contact area is ing disciplines including vibration control,
contact stress of 266 ksi is calculated stabilized, and the possibility of edge turbo machinery, gearing, structural stability
and stress analysis.
under the maximum tangential load of loading—which leads to high con-
www.geartechnology.com August 2010 GEARTECHNOLOGY 57