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Design and Analysis of a Spiral Bevel Gear Matthew Brown April, 2009

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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

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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

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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 width Loads and bending moments are calculated based on a vectoral combination of two planes

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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 10 6 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: – A-A = 0.48 – B-B = 3.34

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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: – B-B =.87

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Hertz Stresses Must first calculate the geometry factor with an iterative procedure: Then calculate Hertz stresses using: Results: – Hertz stresses calculated = ksi – AGMA allowable stress = 250 ksi

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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

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Gear Life Calculations Life calculated using Miners 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

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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

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