# Chapter 8 – Kinematics of Gears

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Chapter 8 – Kinematics of Gears

Gears! Gears are most often used in transmissions to convert an electric motor’s high speed and low torque to a shaft’s 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

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

Standard Spur Gears (Boston Gear Catalog)

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

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 gear

Bevel Gears Gear axis at 90°, based on rolling cones Advantages

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)

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

Examples of “off the shelf” gearing
Boston Gear Examples of “off the shelf” gearing

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:

Other Drives Differentials www.torsen.com/products/ T-1.htm
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. The following description of a differential applies to a "traditional" rear- or front-wheel-drive car or truck: Power is supplied from the engine, via the transmission , to a drive shaft (British term: propeller shaft, more commonly abbreviated to "prop-shaft"), which runs to the differential. A spiral bevel pinion gear at the end of the propeller shaft is encased within the differential itself, and it meshes with the large spiral bevel ring gear (British term: crown wheel). (The ring and pinion may mesh in hypoid orientation, not shown.) The ring gear is attached to a carrier, which holds what is sometimes called a spider, a cluster of four bevel gears in a rectangle, so each bevel gear meshes with two neighbors and rotates counter to the third, that it faces and does not mesh with. Two of these spider gears are aligned on the same axis as the ring gear and drive the half shafts connected to the vehicle's driven wheels. These are called the side gears. The other two spider gears are aligned on a perpendicular axis which changes orientation with the ring gear's rotation. These two gears are just called pinion gears, not to be confused with the main pinion gear. (Other spider designs employ different numbers of pinion gears depending on durability requirements.) As the carrier rotates, the changing axis orientation of the pinion gears imparts the motion of the ring gear to the motion of the side gears by pushing on them rather than turning against them (that is, the same teeth stay in contact), but because the spider gears are not restricted from turning against each other, within that motion the side gears can counter-rotate relative to the ring gear and to each other under the same force (in which case the same teeth do not stay in contact). Thus, for example, if the car is making a turn to the right, the main ring gear may make 10 full rotations. During that time, the left wheel will make more rotations because it has further to travel, and the right wheel will make fewer rotations as it has less distance to travel. The side gears will rotate in opposite directions relative to the ring gear by, say, 2 full turns each (4 full turns relative to each other), resulting in the left wheel making 12 rotations, and the right wheel making 8 rotations. The rotation of the ring gear is always the average of the rotations of the side gears. This is why if the wheels are lifted off the ground with the engine off, and the drive shaft is held (preventing the ring gear from turning inside the differential), manually rotating one wheel causes the other to rotate in the opposite direction by the same amount. When the vehicle is traveling in a straight line, there will be no differential movement of the planetary system of gears other than the minute movements necessary to compensate for slight differences in wheel diameter, undulations in the road (which make for a longer or shorter wheel path), etc. T-1.htm How a manual transmission works:

How a differential works: http://en. wikipedia

John Deere 3350 tractor cut in Technikmuseum Speyer Museum

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 ratio’s 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)

Gears used for Speed Reducer
Recall the main purpose of mating/meshing gears is to provide speed reduction or torque increase. Gear nG NG Pinion nP NP

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

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

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

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

Gear Nomenclature N = Number of teeth
Use subscript for specific gear NP=Number of teeth on pinion (driver) NG=Number of teeth on gear (driven) NP < NG (for speed reducer) NA=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=(p*D)/N

Gear Nomenclature Gear Train Rule – Pitch of two gears in mesh must be identical p DG p DP P = NG NP

Gear Nomenclature N = D NG NP = = DG DP
Diametral Pitch, (Pd) – Number of teeth per inch of pitch diameter *Two gears in mesh must have equal Pd: *Standard diametral pitches can be found in Table 8-1 and 8-2 N Pd = D NG NP Pd = = DG DP

Gear Nomenclature Figure 8-8
More Gear Nomenclature:

Gear Formulas Courtesy of Boston Gear

Gear Formulas Courtesy of Boston Gear (cont’d)

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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

Gear Geometry Conjugate profile Fillet Radius 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

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

Figure 8-11

Gear Nomenclature Example
8-1) Gear has 44 teeth, Æ=20°, full depth involute form diametral pitch Pd = 12 Pitch Diameter Circular Pitch NG 44 teeth = = = 3.667 inch DG Pd 12 t/in p DG (p) 3.667in Pc = = = .2617 in/t NG 44 t

Gear Nomenclature Example
Addendum Dedendum 1 1 a = = = .0833 in Pd 12 t/in 1.25 1.25 b = = = .1042 in Pd 12 t/in

Gear Nomenclature Example
Clearance Whole Depth ht = a+b = in Working Depth hk = 2*a = in .25 .25 c = = = .0208 in Pd 12 t/in

Gear Nomenclature Example
Tooth Thickness Outside Diameter PC .2617in t = = = .1309 in 2 2 N+2 O.D. = DO = = 2.833 in Pd

Gear Nomenclature Notes
Clearance maybe a problem for small pinions driving large gears, therefore they won’t mesh and will lock up (See Table 8-6) As NP decreases so does max NG If design necessatates small pinion, maybe able to increase clearance by undercutting gear tooth (See Figure 8-14)

Summary of Gear Nomenclature:
DP = Pitch diameter of pinion DG = Pitch diameter of gear NP = No. teeth (t) for pinion NG = No. teeth (t) or gear Pd = diametral pitch = N/D = constant for meshing gears p = circular pitch = pD/N = constant for meshing gears nP = speed of pinion (rpm) nG = speed of gear (rpm) VR = velocity ratio = nP/nG = NG/NP Power = constant across mating gears or series system: Pin = Pout Power in branched system is conserved: Pin = PA + PB + ….. Torque will change!!

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

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

Gear Train Alternate Solution
TV = (VR1)(VR2)(VR3) 30 68 68 TV = 8.4 = * * 22 30 25 ni TV = nout ni 1750 rpm nout = = = 208 rpm ccw TV 8.4 Tout = 8.4 Tin !! Lots of Torque