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Chapter 8 – Kinematics of Gears

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Gears! Gears are most often used in transmissions to convert an electric motors high speed and low torque to a shafts requirements for low speed high torque: Speed is easy to generate, because voltage is easy to generate Torque is difficult to generate because it requires large amounts of current Gears essentially allow positive engagement between teeth so high forces can be transmitted while still undergoing essentially rolling contact Gears do not depend on friction and do best when friction is minimized Basic Law of Gearing: –A common normal (the line of action) to the tooth profiles at their point of contact must, in all positions of the contacting teeth, pass through a fixed point on the line-of-centers called the pitch point –Any two curves or profiles engaging each other and satisfying the law of gearing are conjugate curves, and the relative rotation speed of the gears will be constant

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Spur Gears Teeth are parallel to the axis of the gear Advantages Cost Ease of manufacture Availability Disadvantages Only works with mating gear Axis of each gear must be parallel

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Standard Spur Gears (Boston Gear Catalog)

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Helical Gears Teeth are at an angle to the gear axis (usually 10° to 45°) – called helix angle Advantages Smooth and quiet due to gradual tooth engagements (spur gears whine at high speed due to impact). Helical gears good up to speeds in excess of 5,000 ft/min More tooth engagement allows for greater power transmission for given gear size. Parallel to perpendicular shaft arrangement – Fig 8.2 Disadvantage More expensive Resulting axial thrust component

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Helical Gears Mating gear axis can be parallel or crossed Can withstand the largest capacity at 30,000 hp

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Worm Gears Gears that are 90° to each other Advantages Quiet / smooth drive Can transmit torque at right angles No back driving Good for positioning systems Disadvantage Most inefficient due to excessive friction (sliding) Needs maintenance Slower speed applications worm worm gear

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Bevel Gears Gear axis at 90°, based on rolling cones Advantages Right angle drives Disadvantages Get axial loading which complicates bearings and housings

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Spiral Bevel Gears Same advantage over bevel gears as helical gears have over spur gears!! Teeth at helix angle Very Strong Used in rear end applications (see differentials)

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Why Use Gears? 1. Reduce speed 2. Increase torque 3. Move power from one point to another 4. Change direction of power 5. Split power Generally this functionality is accomplished by many gears mounted in a gear box!

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Boston Gear Examples of off the shelf gearing

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Other Drives Splitter – One input with several outputs Right Angle – Transfers torque thru right angles, can be as simple as mating bevel gears power_series.htm Types of Gear Boxes:

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Other Drives Differentials Engines typically operate over a range of 600 to about 7000 revolutions per minute (though this varies, and is typically less for diesel engines), while the car's wheels rotate between 0 rpm and around 1800 rpm. Engine: higher speed, lower torque versus wheels. T-1.htm How a manual transmission works:

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How a differential works: rg/wiki/Differential_( mechanical_device)

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John Deere 3350 tractor cut in Technikmuseum Speyer Museum

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Gears vs Belts and Chains Gears are much more capable in terms of power rating (helical gear drives capable of > 30,000 hp) With planetary gear sets large gear ratios can be achieved (100:1) Gear applications include high torque and high speeds Can have multiple speed reductions by pairing different gears or gear trains (several gears in series)

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Gears used for Speed Reducer Recall the main purpose of mating/meshing gears is to provide speed reduction or torque increase. Pinion n P N P Gear n G N G

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Example: Want a 3:1 reduction N P =22 teeth What is N G ? Solution: VR = 3 = N G /N P N G = 3*22 = 66 teeth Figure 8-15, pg. 322

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Engine Pump n1, N1 n2, N2 n3, N3 n4, N4 Given: n1 = 500 rpm, N1 = 20t N2 = 70t, N3 = 18t, N4 = 54t Find: n4 Example: Double Speed Reducer Solution: 1.n2 = 500 rpm*(20/70) = rpm 2.n3 = n2 3.n4 = rpm*(18/54) = 47.6 rpm 4.Total reduction = 500/47.6 = 10.5 (0r 10.5:1) Torque?? Increases by 10.5!! Power?? Stays the same throughout!

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Pinion POWER n p Law of Kinematics Holds true if teeth have conjugate profile!! Fig 8-7 Line drawn perpendicular at point of contact always crosses centerline at same place then VR = n p /n G = constant DEMO!

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Spur Gear Nomenclature Pitch Circle(s) The circles remain tangent throughout entire engagement Pitch Diameter Diameter of pitch circle D P – Pitch of pinion D G – Pitch of gear (power gear or driving gear) (Driven gear)

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Gear Nomenclature N = Number of teeth Use subscript for specific gear N P =Number of teeth on pinion (driver) N G =Number of teeth on gear (driven) N P < N G (for speed reducer) N A =Number of teeth on gear A Circular Pitch, P is the radial distance from a point on a tooth at the pitch circle to corresponding point on the next adjacent tooth P=( D)/N

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Gear Nomenclature Gear Train Rule – Pitch of two gears in mesh must be identical DGDG NGNG = P DPDP NPNP

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Gear Nomenclature Diametral Pitch, (P d ) – Number of teeth per inch of pitch diameter *Two gears in mesh must have equal P d : *Standard diametral pitches can be found in Table 8-1 and 8-2 D N = PdPd DGDG NGNG = = PdPd DPDP NPNP

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Gear Nomenclature Figure 8-8 More Gear Nomenclature:

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Gear Formulas Courtesy of Boston Gear

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Gear Formulas Courtesy of Boston Gear (contd)

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Double Click On Image to Print PDF (will not work in presentation mode) Go to for the complete 18 page PDF on gearing Engineering Information

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Gear Geometry Spur Gears Tooth Profile – Conjugate shape Conjugate Profile Tooth is thicker at base, maximum moment σ = M/s Pressure Angle (φ) - angle between tangent and perpendicular line to gear tooth surface Allows constant velocity ratio between mating gears and smooth power transmission Conjugate profile Fillet Radius

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Force perpendicular at Φ = 14.5˚Φ = 20˚Φ = 25˚ Pressure Angle

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Figure 8-11

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Gear Nomenclature Example 8-1) Gear has 44 teeth, full depth involute form diametral pitch Pd = 12 Pitch Diameter Circular Pitch Pd NGNG inch = = DGDG 12 t/in 44 teeth = NGNG DGDG.2617 in/t == PcPc 44 t 3.667in =

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Gear Nomenclature Example Addendum Dedendum Pd in = = a 12 t/in 1 = Pd in= = b 12 t/in 1.25 =

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Gear Nomenclature Example Clearance Whole Depth ht = a+b =.1875 in Working Depth hk = 2*a = in Pd in= = c 12 t/in.25 =

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Gear Nomenclature Example Tooth Thickness Outside Diameter 2 PCPC.1309 in= = t in = Pd N in = = O.D.DODO =

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Gear Nomenclature Notes Clearance maybe a problem for small pinions driving large gears, therefore they wont mesh and will lock up (See Table 8-6) As N P decreases so does max N G If design necessatates small pinion, maybe able to increase clearance by undercutting gear tooth (See Figure 8-14)

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Summary of Gear Nomenclature: D P = Pitch diameter of pinion D G = Pitch diameter of gear N P = No. teeth (t) for pinion N G = No. teeth (t) or gear P d = diametral pitch = N/D = constant for meshing gears p = circular pitch = D/N = constant for meshing gears n P = speed of pinion (rpm) n G = speed of gear (rpm) VR = velocity ratio = n P /n G = N G /N P Power = constant across mating gears or series system: Pin = Pout Power in branched system is conserved: Pin = P A + P B + ….. Torque will change!!

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Conclusion: Total speed reduction = 1750/68 = 25.7 Torque increase = 25.7 Power = constant!!

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Gear Trains Train Value = TV = Product of the values for each gear pair in the train n in n out ==TV(VR1)(VR2)....

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Gear Train Alternate Solution = TV (VR 1 )(VR 2 )(VR 3 ) = TV 8.4 = * * nini n out = TV = nini n out = TV 208 rpm ccw = 1750 rpm 8.4 T out = 8.4 T in !! Lots of Torque

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YouTube Gear Animations: Speed Reducers: Automotive Differential: Manual Transmission: Gear Cutting:

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