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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

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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
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
slide4

Standard Spur Gears

(Berg Master Catalog)

helical gears
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 gears6
Helical Gears
  • Mating gear axis can be parallel or crossed
  • Can withstand the largest capacity at 30,000 hp
worm gears
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

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

slide11

BostonGear

Examples of “off the shelf” drives

Show slides

other drives
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

other drives13
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

slide14

How a differential works: http://en.wikipedia.org/wiki/Differential_(mechanical_device)

gears vs belts and chains
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
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
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

slide19

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 10.5:1)

Torque?? Increases by 10.5!!Power?? Stays the same throughout!

slide20

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
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
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 nomenclature24
Gear Nomenclature
  • Gear Train Rule – Pitch of two gears in mesh must be identical

PINION

p

DG

p

DP

=

P

NP

NG

GEAR

gear nomenclature25
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

gear nomenclature26
Gear Nomenclature

Figure 8-8

More Gear Nomenclature: http://en.wikipedia.org/wiki/List_of_gear_nomenclature

gear geometry
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

slide29

Pressure Angle

Force perpendicular at f

Φ = 14.5˚

Φ = 20˚

Φ = 25˚

gear nomenclature example
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

12 t/in

Pd

p

DG

(p)

3.667in

.2617 in/t

=

=

=

Pc

NG

44 t

gear nomenclature example32
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 example33
Gear Nomenclature Example
  • Clearance
  • Whole Depth

ht = a+b = .1875 in

  • Working Depth

hk = 2*a = .16667 in

.25

.25

c

=

=

.0208 in

=

Pd

12 t/in

gear nomenclature example34
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
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)
slide37

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!!
slide38

Conclusion:

  • Total speed reduction = 1750/68 = 25.7
  • Torque increase = 25.7
  • Power = constant!!
gear trains
Gear Trains
  • Train Value = TV = Product of the values for each gear pair in the train

nin

TV

=

=

(VR1)(VR2). . . .

nout

gear train alternate solution
Gear Train Alternate Solution

(VR1)(VR2)(VR3)

TV

=

30

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

youtube gear animations
YouTube Gear Animations:
  • Speed Reducers:
  • http://www.youtube.com/watch?v=7LReoWPg_pM&feature=related
  • http://www.youtube.com/watch?v=1_jbZVBXjWc&feature=related
  • Automotive Differential: http://www.youtube.com/watch?v=iBLE0_Sjqw4&feature=related
  • Manual Transmission: http://www.youtube.com/watch?v=MBmLJCeGu7o&feature=related
  • Gear Cutting:
  • http://www.youtube.com/watch?v=fps0OR1eF_s&feature=related
  • http://www.youtube.com/watch?v=xF9CjluRFJ4&feature=related