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Chapter 8 – Kinematics of Gears

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1 Chapter 8 – Kinematics of Gears

2 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

3 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

4 Standard Spur Gears (Boston Gear Catalog)

5 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 Helical gears offer a refinement over spur gears. The leading edges of the teeth are not parallel to the axis of rotation, but are set at an angle. Since the gear is curved, this angling causes the tooth shape to be a segment of a helix. The angled teeth engage more gradually than do spur gear teeth. This causes helical gears to run more smoothly and quietly than spur gears. Helical gears also offer the possibility of using non-parallel shafts. A pair of helical gears can be meshed in two ways: with shafts oriented at either the sum or the difference of the helix angles of the gears. These configurations are referred to as parallel or crossed, respectively. The parallel configuration is the more mechanically sound. In it, the helices of a pair of meshing teeth meet at a common tangent, and the contact between the tooth surfaces will, generally, be a curve extending some distance across their face widths. In the crossed configuration, the helices do not meet tangentially, and only point contact is achieved between tooth surfaces. Because of the small area of contact, crossed helical gears can only be used with light loads. Quite commonly, helical gears come in pairs where the helix angle of one is the negative of the helix angle of the other; such a pair might also be referred to as having a right-handed helix and a left-handed helix of equal angles. If such a pair is meshed in the 'parallel' mode, the two equal but opposite angles add to zero: the angle between shafts is zero -- that is, the shafts are parallel. If the pair is meshed in the 'crossed' mode, the angle between shafts will be twice the absolute value of either helix angle. Note that 'parallel' helical gears need not have parallel shafts -- this only occurs if their helix angles are equal but opposite. The 'parallel' in 'parallel helical gears' must refer, if anything, to the (quasi) parallelism of the teeth, not to the shaft orientation. As mentioned at the start of this section, helical gears operate more smoothly than do spur gears. With parallel helical gears, each pair of teeth first make contact at a single point at one side of the gear wheel; a moving curve of contact then grows gradually across the tooth face. It may span the entire width of the tooth for a time. Finally, it recedes until the teeth break contact at a single point on the opposite side of the wheel. Thus force is taken up and released gradually. With spur gears, the situation is quite different. When a pair of teeth meet, they immediately make line contact across their entire width. This causes impact stress and noise. Spur gears make a characteristic whine at high speeds and can not take as much torque as helical gears because their teeth are receiving impact blows. Whereas spur gears are used for low speed applications and those situations where noise control is not a problem, the use of helical gears is indicated when the application involves high speeds, large power transmission, or where noise abatement is important. The speed is considered to be high when the pitch line velocity (that is, the circumferential velocity) exceeds 5000 ft/min.[3] A disadvantage of helical gears is a resultant thrust along the axis of the gear, which needs to be accommodated by appropriate thrust bearings, and a greater degree of sliding friction between the meshing teeth, often addressed with specific additives in the lubricant

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

7 Worm Gears Gears that are 90° to each other Advantages Disadvantage
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

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

9 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)

10 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!

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

12 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:

13 Other Drives Differentials 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:

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

15 John Deere 3350 tractor cut in Technikmuseum Speyer Museum

16 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)

17 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

18 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

19 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!

20 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


22 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)

23 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

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

25 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

26 Gear Nomenclature Figure 8-8
More Gear Nomenclature:

27 Gear Formulas Courtesy of Boston Gear

28 Gear Formulas Courtesy of Boston Gear (cont’d)

29 (will not work in presentation mode)
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

30 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

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

32 Figure 8-11

33 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

34 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

35 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

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

37 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)


39 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!!

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

41 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

42 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

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

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