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Outline Gear Theory Fundamental Law of Gearing Involute profile Nomenclature Gear Trains Loading Stresses

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Fundamental Law of Gearing functionally, a gearset is a device to exchange torque for velocity P=T the angular velocity ratio of the gears of a gearset must remain constant throughout the mesh pitch circle, pitch diameter d, pitch point gear pinion What gear tooth shape can do this? click here!

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Towards the Involute Profile A belt connecting the two cylinders line of action, pressure angle

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The Involute Profile

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Profile of the Involute Profile pressure angle, line of action, length of action, addendum

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Involute in Action video from

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Nomenclature Figure 11-8

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Pitches, Etc. circular pitch (mm, in.) base pitch (mm, in.) diametral pitch (teeth/in.) module (mm/teeth)

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Velocity Ratio pitches must be equal for mating gears, therefore

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Contact Ratio average number of teeth in contact at any one time = length of action divided by the base pitch, or, where C=center distance=(N g +N p )*1/p d *1/2

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Minimum # of Teeth minimum # of teeth to avoid undercutting with gear and rack

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Outline Gear Theory Fundamental Law of Gearing Involute profile Nomenclature Gear Trains Loading Stresses

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Simple Gear Trains

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Simple Gear Train

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Fine for transmitting torque between shafts in close proximity when m v does not need to be too large Use third gear (idler) only for directional reasons (not for gear reduction)

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Reverse on a Car

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Actual Manual Transmission

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Outline Gear Theory Fundamental Law of Gearing Involute profile Nomenclature Gear Trains Loading Stresses

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Loading of Gears W t =tangential (transmitted) load Wr=radial load W=total load (see board for figure or page 711 in book) Because T p is constant, should W t be static for a tooth?

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Gear vs. Idler

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Loading Example Given: pinion is driving gear with 50hp at 1500 rpm N p =20 m v =2 (a.k.a., m g =2) p d =4 /in. =20 degrees Find: Loading and effects of inputs on loading

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Outline Gear Theory Fundamental Law of Gearing Involute profile Nomenclature Gear Trains Loading Stresses

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Gear Failure Fatigue LoadingSurface Failure from contact b/n of teeth infinite life not possible failure is gradual from bending of teeth infinite life possible failure can be sudden Lewis, 1892, first formulation of gear tooth fatigue failure

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AGMA Gear Stress Formula many assumptions: see page 714… …including that the contact ratio 1

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Bending Strength Geometry Factor from tables on pgs Inputs pinion or gear number of teeth pressure angle long-addendum or full-depth tip loading or HPSTC JKvKmKaKsKBKIJKvKmKaKsKBKI

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Tip vs. HPSCT Loading

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Finding J for the Example pinion is driving gear with 50hp at 1500 rpm N p =20, N g =40 p d =4 /in. =20 degrees W t =420 lb. J p =0.34, J g =0.38

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Dynamic (Velocity) Factor JKvKmKaKsKBKIJKvKmKaKsKBKI to account for tooth-tooth impacts and resulting vibration loads from Figure or from Equations (pages ) Inputs needed: Quality Index (Table 11-7) Pitch-line Velocity V t = (radius)(angular speed in radians)

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Dynamic Factor JKvKmKaKsKBKIJKvKmKaKsKBKI V t,max =(A+Q v -3) 2

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Determining K v for the Example V t =(1500rev/min)(2.5 in)(1ft/12in)(2 rad/1rev)=1964 ft/min (well below V t,max ) therefore, Q v =10

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Load Distribution Factor K m JKvKmKaKsKBKIJKvKmKaKsKBKI to account for distribution of load across face 8/p d < F < 16/p d can use 12/p d as a starting point

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K m for the Example F=12/p d =3 in. K m =1.63

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Application Factor, K a JKvKmKaKsKBKIJKvKmKaKsKBKI to account for non-uniform transmitted loads for example, assume that all is uniform

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Other Factors JKvKmKaKsKBKIJKvKmKaKsKBKI to account for size to account for gear with a rim and spokes to account for extra loading on idler KsKs KBKB KIKI K s =1 unless teeth are very large K B =1 for solid gears K I =1 for non-idlers, K I =1.42 for idler gears

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Back to the Example

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Compared to What?? b is great, but what do I compare it to? similar concept to S e, but particularized to gears S fb´ Table Figure for steels Hardness S fb´

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Life Factor K L to adjust test data from 1E7 to any number of cycles

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Temperature & Reliability Factors K T =1 if T < 250°F see Equation 11.24a if T > 250°F

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Fatigue Bending Strength for Example Class 40 Cast Iron 450 rpm, 8 hours/day, 10 years T = 80°F Reliability=99% N=(450 rev/min)(60 min/hr)(8 hr/dy)(250 dy/yr)(10 yrs)=5.4E8 revs K L =1.3558N – =0.948 S fb = (0.948)13ksi = 12.3 ksi S fb ´ = 13ksi Table 11-20

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Safety Factor N bpinion =(12300 psi)/(3177psi)= 3.88 N bgear =(12300 psi)/(2842psi)= 4.33

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equal to 1 same as for bending Gear Failure Fatigue LoadingSurface Failure from contact b/n of teeth infinite life not possible failure is gradual from bending of teeth infinite life possible failure can be sudden Buckingham pioneered this work d= pitch diameter of smaller gear

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Geometry Factor I considers radius of curvature of teeth and pressure angle For example: p =0.691, g =1.87, I=0.095 top sign is for external gears, bottom for internal

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Elastic Coefficient considers differences in materials For example, C p =1960 psi

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Surface-Fatigue Strengths c is great, but what do I compare it to? same as for bending from Table 11-21

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Life Factor C L to adjust test data fro 1E7 to any number of cycles

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Hardness Ratio C H to account for pitting resistance when pinion is harder than gear, then gear is cold-worked only apply C H to the cold-worked gear when HB p =HB G, then C H =1

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Safety Factor for Loading stress is based on the square root of loading, therefore:

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Gear Design

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gear pinion Same for Pinion and Gear Different for Pinion and Gear p d (p c ),, F Power W, W t, W r V t N c d, N T,, N b

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Gear Design Strategy Given or Set: gear ratio Power,, T Properties of Gears to Determine: d p, d g, N p, N g p d, F -OR- Safety Factors (the others are given) WtWt Bending & Surface Stresses Safety Factors or p d and F

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