Design and Analysis of a Spiral Bevel Gear

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					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

   – 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 mid-
  point 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
• Loads and bending moments are
  calculated based on a vectoral
  combination of two planes
                   Fatigue Analysis
• Performed at the two most          • Margin of Safety
  critical sections of the gear
  shaft, sections A-A and B-B          calculated using
  shown previously
• Principle steady stress is
  calculated from vibratory
  bending, steady torsion, and
  normal stress, then converted
  into an equivalent vibratory       • Results:
  stress based on fatigue data at       – M.S. @ A-A = 0.48
  106 cycles
• Endurance limit of gear is            – M.S. @ B-B = 3.34
  modified for size effect factor,
  correlation factor, surface
  finish factor, and reliability
               Static Analysis
• Federal Aviation Administration requires static
  analysis be performed at 2X the endurance
  limit – analysis conducted at about 2.5X
• 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

• 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
• All analysis resulted in positive margins of
  safety and unlimited gear life in the intended
• All stress calculations are within the
  recommended allowable stress values
  published by the AGMA
• Design is safe for operation