Chapter 8

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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 (Berg Master 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
worm gear

• 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
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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?

• 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!
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BostonGear
Examples of off the shelf drives
Show slides
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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

www.gamweb.com/ power_series.htm
Types of Gear Boxes http//en.wikipedia.org/wiki/
Gear_box
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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.

www.torsen.com/products/ T-1.htm
How a manual transmission works
http//en.wikipedia.org/wiki/Manual_transmission
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How a differential works http//en.wikipedia.org/
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 gt 30,000
  hp)
• With planetary gear sets large gear ratios can
  be achieved (1001)
• 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.

Gear nG NG
Pinion nP NP
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Example

• Want a 31 reduction
• NP22 teeth
• What is NG?
• Solution
• VR 3 NG/NP
• NG 322 66 teeth

Figure 8-15, pg. 322
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n4, N4
n1, N1
Engine
Pump
Given n1 500 rpm, N1 20tN2 70t, N3 18t,
N4 54t Find n4
n2, N2
n3, N3
Example Double Speed Reducer

• Solution
• n2 500 rpm(20/70) 142.8 rpm
• n3 n2
• n4 142.8 rpm(18/54) 47.6 rpm
• Total reduction 500/47.6 10.5 (0r 10.51)

Torque?? Increases by 10.5!!Power?? Stays the
same throughout!
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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
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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)
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Gear Nomenclature

• N Number of teeth
• Use subscript for specific gear
• NPNumber of teeth on pinion (driver)
• NGNumber of teeth on gear (driven)
• NP lt NG (for speed reducer)
• NANumber 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(pD)/N

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Gear Nomenclature

• Gear Train Rule Pitch of two gears in mesh must
  be identical

PINION
p
DG
p
DP

P
NP
NG
GEAR
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Gear Nomenclature

• 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
DP
DG
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Gear Nomenclature
Figure 8-8
More Gear Nomenclature http//en.wikipedia.org/wi
ki/List_of_gear_nomenclature
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Gear Geometry

• Spur Gears
• Tooth Profile Conjugate shape
• Conjugate Profile
• Tooth is thicker at base, maximum moment
• s M/s
• Pressure Angle (f) - 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 f
Pressure Angle
F 14.5
F 20
F 25
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Figure 8-11
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Gear Nomenclature Example

• 8-1) Gear has 44 teeth, Æ20, full depth
  involute form diametral pitch Pd 12
• Pitch Diameter
• Circular Pitch

NG
44 teeth
3.667 inch



DG
12 t/in
Pd
DG
p
(p)
3.667in
.2617 in/t



Pc
NG
44 t
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Gear Nomenclature Example

• Addendum
• Dedendum

1
1
.0833 in
a



Pd
12 t/in
1.25
1.25
.1042 in

b


Pd
12 t/in
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Gear Nomenclature Example

• Clearance
• Whole Depth
• ht ab .1875 in
• Working Depth
• hk 2a .16667 in

.25
.25
.0208 in

c


Pd
12 t/in
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Gear Nomenclature Example

• Tooth Thickness
• Outside Diameter

PC
.2617in
.1309 in

t


2
2
N2
2.833 in

O.D.
DO


Pd
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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 NP decreases so does max NG
• 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
• 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!!

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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

nin


TV
(VR1)(VR2). . . .
nout
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Gear Train Alternate Solution
(VR1)(VR2)(VR3)
TV

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68
68
8.4

TV



25
22
30
ni
TV

nout
ni
1750 rpm
nout
208 rpm ccw



TV
8.4
Tout 8.4 Tin !! Lots of Torque
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YouTube Gear Animations

• Speed Reducers
• http//www.youtube.com/watch?v7LReoWPg_pMfeature
  related
•  
• http//www.youtube.com/watch?v1_jbZVBXjWcfeature
  related
•  
• Automotive Differential http//www.youtube.com/wa
  tch?viBLE0_Sjqw4featurerelated
• Manual Transmission http//www.youtube.com/watch?
  vMBmLJCeGu7ofeaturerelated
• Gear Cutting
• http//www.youtube.com/watch?vfps0OR1eF_sfeature
  related
• http//www.youtube.com/watch?vxF9CjluRFJ4feature
  related