Design and Analysis of a Spiral
Bevel Gear
Matthew Brown
April, 2009
Gear Theory and Design
• Recommended design methodology published by American
Gear Manufacturing Association (AGMA)
• Gear teeth primarily designed for two factors:
– Resistance to pitting caused by Hertzian contact stresses
• Accounts for contact pressure between two curved surfaces and
therefore considers load sharing between adjacent gear teeth as well
as load concentration that may result from uncertainties in
manufacturing
– Bending strength capacity based on cantilever beam theory
• Accounts for compressive stresses at the tooth roots caused by the
radial component of the tooth load; the non-uniform moment
distribution of the load resulting from the inclined contact lines on the
gear teeth; stress concentration at the tooth fillet; load sharing
between adjacent contacting teeth; and lack of smoothness due to
low contact ratio
Material Selection and Processing
• In this application, spiral bevel gear materials are
limited to only those which are easily carburized and
case-hardened in order to provide high wear resistance
and high load carrying capacity
• SAE 9310 Steel selected
• Material processing:
– Heat treatment
• Convert weaker grain structures to stronger ones
– Tempering
• Relieve brittleness and internal strains prior to machining
– Carburization
• Adds high hardness and strength at surface and toughens core to
withstand impact stress
Bevel Gear Loading
• Torque application to a bevel gear
induces tangential, radial, and
separating loads assumed to act
as point loads applied at the midpoint of the gear tooth
• Reaction loads are a result of the
tapered roller bearings that
support the gear shaft and
counteract the gear loads
• Loads are primarily a function of
torque, pitch diameter, pitch
angle, pressure angle, and face
width
• Loads and bending moments are
calculated based on a vectoral
combination of two planes
Fatigue Analysis
• Performed at the two most
critical sections of the gear
shaft, sections A-A and B-B
shown previously
• Principle steady stress is
calculated from vibratory
bending, steady torsion, and
normal stress, then converted
into an equivalent vibratory
stress based on fatigue data at
106 cycles
• Endurance limit of gear is
modified for size effect factor,
correlation factor, surface
finish factor, and reliability
factor
• Margin of Safety
calculated using
• Results:
– M.S. @ A-A = 0.48
– M.S. @ B-B = 3.34
Static Analysis
• Federal Aviation Administration requires static
analysis be performed at 2X the endurance
limit – analysis conducted at about 2.5X
(590HP)
• Similar process as fatigue analysis except the
Margin of Safety is calculated by:
• Results:
– M.S. @ B-B = .87
Hertz Stresses
• Must first calculate the geometry factor with
an iterative procedure:
• Then calculate Hertz stresses using:
• Results:
– Hertz stresses calculated = 180.6 ksi
– AGMA allowable stress = 250 ksi
Bending Strength Capacity
• Must first calculate the geometry factor with
an iterative procedure:
• Then calculate bending strength in gear teeth
using:
• Results:
– Bending stresses calculated = 31.5 ksi
– AGMA allowable stress = 40 ksi
Gear Life Calculations
• Life calculated using Miner’s rule:
– The portion of useful fatigue life used up by a number of
repeated stress cycles at a particular stress is proportional to the
total number of cycles in the overall fatigue life of the part.
• Five maneuvers (1.53%) of anticipated helicopter flight
spectrum produce damage
• Damage accumulation calculations performed for both high
cycle fatigue and GAG (low cycle fatigue) for both Bending
Life and Durability Life – all calculations result in unlimited
life
Conclusion
• All analysis resulted in positive margins of
safety and unlimited gear life in the intended
application
• All stress calculations are within the
recommended allowable stress values
published by the AGMA
• Design is safe for operation