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Chapter 13

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Gears – General

Lecture Slides

The McGraw-Hill Companies © 2012

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Chapter Outline

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Types of Gears

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Spur

Helical

Bevel

Worm

Figs. 13–1 to 13–4

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Nomenclature of Spur-Gear Teeth

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Fig. 13–5

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Tooth Size

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Tooth Sizes in General Use

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Table 13–2

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Standardized Tooth Systems (Spur Gears)

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Table 13–1

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Standardized Tooth Systems

  • Common pressure angle φ : 20º and 25º
  • Old pressure angle: 14 ½º
  • Common face width:

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Conjugate Action

  • When surfaces roll/slide against each other and produce constant angular velocity ratio, they are said to have conjugate action.
  • Can be accomplished if instant center of velocity between the two bodies remains stationary between the grounded instant centers.

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Fig. 13–6

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Conjugate Action

  • Forces are transmitted on line of action which is normal to the contacting surfaces.
  • Angular velocity ratio is inversely proportional to the radii to point P, the pitch point.

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  • Circles drawn through P from each fixed pivot are pitch circles, each with a pitch radius.

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Fig. 13–6

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Involute Profile

  • The most common conjugate profile is the involute profile.
  • Can be generated by unwrapping a string from a cylinder, keeping the string taut and tangent to the cylinder.
  • Circle is called base circle.

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Fig. 13–8

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Involute Profile Producing Conjugate Action

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Fig. 13–7

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Circles of a Gear Layout

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Fig. 13–9

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Sequence of Gear Layout

  • Pitch circles in contact
  • Pressure line at desired pressure angle
  • Base circles tangent to pressure line
  • Involute profile from base circle
  • Cap teeth at addendum circle at 1/P from pitch circle
  • Root of teeth at dedendum �circle at 1.25/P from �pitch circle
  • Tooth spacing from circular pitch, p = π / P

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Fig. 13–9

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Relation of Base Circle to Pressure Angle

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Fig. 13–10

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Tooth Action

  • First point of contact at a where flank of pinion touches tip of gear
  • Last point of contact at b where tip of pinion touches flank of gear
  • Line ab is line of action
  • Angle of action is sum of angle of approach and angle of recess

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Fig. 13–12

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Rack

  • A rack is a spur gear with an pitch diameter of infinity.
  • The sides of the teeth are straight lines making an angle to the line of centers equal to the pressure angle.
  • The base pitch and circular pitch, shown in Fig. 13–13, are related by

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Fig. 13–13

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Internal Gear

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Fig. 13–14

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Example 13–1

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Example 13–1

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Example 13–1

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Contact Ratio

  • Arc of action qt is the sum of the arc of approach qa and the arc of recess qr., that is qt = qa + qr
  • The contact ratio mc is the ratio of the arc of action and the circular pitch.

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  • The contact ratio is the average number of pairs of teeth in contact.

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Contact Ratio

  • Contact ratio can also be found from the length of the line of action

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  • The contact ratio should be at least 1.2

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Fig. 13–15

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Interference

  • Contact of portions of tooth profiles that are not conjugate is called interference.
  • Occurs when contact occurs below the base circle
  • If teeth were produced by generating process (rather than stamping), then the generating process removes the interfering portion; known as undercutting.

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Fig. 13–16

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Interference of Spur Gears

  • On spur and gear with one-to-one gear ratio, smallest number of teeth which will not have interference is

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  • k =1 for full depth teeth. k = 0.8 for stub teeth
  • On spur meshed with larger gear with gear ratio mG = NG/NP = m, the smallest number of teeth which will not have interference is

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Interference of Spur Gears

  • Largest gear with a specified pinion that is interference-free is

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  • Smallest spur pinion that is interference-free with a rack is

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Interference

  • For 20º pressure angle, the most useful values from Eqs. (13–11) and (13–12) are calculated and shown in the table below.

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Minimum NP

Max NG

Integer Max NG

Max Gear Ratio

mG= NG/NP

13

16.45

16

1.23

14

26.12

26

1.86

15

45.49

45

3

16

101.07

101

6.31

17

1309.86

1309

77

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Interference

  • Increasing the pressure angle to 25º allows smaller numbers of teeth

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Minimum NP

Max NG

Integer Max NG

Max Gear Ratio

mG= NG/NP

9

13.33

13

1.44

10

32.39

32

3.2

11

249.23

249

22.64

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Interference

  • Interference can be eliminated by using more teeth on the pinion.
  • However, if tooth size (that is diametral pitch P) is to be maintained, then an increase in teeth means an increase in diameter, since P = N/d.
  • Interference can also be eliminated by using a larger pressure angle. This results in a smaller base circle, so more of the tooth profile is involute.
  • This is the primary reason for larger pressure angle.
  • Note that the disadvantage of a larger pressure angle is an increase in radial force for the same amount of transmitted force.

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Forming of Gear Teeth

  • Common ways of forming gear teeth
    • Sand casting
    • Shell molding
    • Investment casting
    • Permanent-mold casting
    • Die casting
    • Centrifugal casting
    • Powder-metallurgy
    • Extrusion
    • Injection molding (for thermoplastics)
    • Cold forming

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Cutting of Gear Teeth

  • Common ways of cutting gear teeth
    • Milling
    • Shaping
    • Hobbing

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Shaping with Pinion Cutter

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Fig. 13–17

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Shaping with a Rack

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Fig. 13–18

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Hobbing a Worm Gear

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Fig. 13–19

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Straight Bevel Gears

  • To transmit motion between intersecting shafts

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Fig. 13–3

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Straight Bevel Gears

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  • The shape of teeth, projected on back cone, is same as in a spur gear with radius rb
  • Virtual number of teeth in this virtual spur gear is

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Fig. 13–20

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Parallel Helical Gears

  • Similar to spur gears, but with teeth making a helix angle with respect to the gear centerline
  • Adds axial force component to shaft and bearings
  • Smoother transition of force between mating teeth due to gradual engagement and disengagement

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Fig. 13–2

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Parallel Helical Gears

  • Tooth shape is involute helicoid

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Fig. 13–21

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Parallel Helical Gears

  • Transverse circular pitch pt is in the plane of rotation
  • Normal circular pitch pn is in the plane perpendicular to the teeth

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  • Axial pitch px is along the direction of the shaft axis

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  • Normal diametral pitch

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Fig. 13–22

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Parallel Helical Gears

  • Relationship between angles

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Fig. 13–22

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Parallel Helical Gears

  • Viewing along the teeth, the apparent pitch radius is greater than when viewed along the shaft.
  • The greater virtual R has a greater virtual number of teeth N'

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  • Allows fewer teeth on helical gears without undercutting.

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Fig. 13–23

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Example 13–2

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Example 13–2

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Interference with Helical Gears

  • On spur and gear with one-to-one gear ratio, smallest number of teeth which will not have interference is

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  • k =1 for full depth teeth. k = 0.8 for stub teeth
  • On spur meshed with larger gear with gear ratio mG = NG/NP = m, the smallest number of teeth which will not have interference is

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Interference with Helical Gears

  • Largest gear with a specified pinion that is interference-free is

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  • Smallest spur pinion that is interference-free with a rack is

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Worm Gears

  • Common to specify lead angle λ for worm and helix angle ψG for gear.
  • Common to specify axial pitch px for worm and transverse circular pitch pt for gear.
  • Pitch diameter of gear is measured on plane containing worm axis

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Fig. 13–24

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Worm Gears

  • Worm may have any pitch diameter.
  • Should be same as hob used to cut the gear teeth
  • Recommended range for worm pitch diameter as a function of center distance C,

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  • Relation between lead L and lead angle λ,

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Standard and Commonly Used Tooth Systems for Spur Gears

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Table 13–1

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Tooth Sizes in General Use

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Table 13–2

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Tooth Proportions for 20º Straight Bevel-Gear Teeth

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Table 13–3

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Standard Tooth Proportions for Helical Gears

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Table 13–4

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Recommended Pressure Angles and Tooth Depths �for Worm Gearing

Table 13–5

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Face Width of Worm Gear

  • Face width FG of a worm gear should be equal to the length of a tangent to the worm pitch circle between its points of intersection with the addendum circle

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Fig. 13–25

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Gear Trains

  • For a pinion 2 driving a gear 3, the speed of the driven gear is

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Relations for Crossed Helical Gears

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Fig. 13–26

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Train Value

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Fig. 13–27

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Compound Gear Train

  • A practical limit on train value for one pair of gears is 10 to 1
  • To obtain more, compound two gears onto the same shaft

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Fig. 13–28

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Example 13–3

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Example 13–4

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Example 13–4

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Compound Reverted Gear Train

  • A compound gear train with input and output shafts in-line
  • Geometry condition must be satisfied

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Fig. 13–29

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Example 13–5

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Example 13–5

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Example 13–5

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Example 13–5

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Example 13–5

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Planetary Gear Train

  • Planetary, or epicyclic gear trains allow the axis of some of the gears to move relative to the other axes
  • Sun gear has fixed center axis
  • Planet gear has moving center axis
  • Planet carrier or arm carries planet axis relative to sun axis
  • Allow for two degrees of freedom (i.e. two inputs)

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Fig. 13–30

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Planetary Gear Trains

  • Train value is relative to arm

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Fig. 13–31

Fig. 13–30

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Example 13–6

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Fig. 13–30

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Example 13–6

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Example 13–6

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Force Analysis – Spur Gearing

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Fig. 13–32

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Force Analysis – Spur Gearing

  • Transmitted load Wt is the tangential load

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  • It is the useful component of force, transmitting the torque

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Fig. 13–33

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Power in Spur Gearing

  • Transmitted power H

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  • Pitch-line velocity is the linear velocity of a point on the gear at the radius of the pitch circle. It is a common term in tabulating gear data.

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Power in Spur Gearing

  • Useful power relation in customary units,

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  • In SI units,

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Example 13–7

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Fig. 13–34

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Example 13–7

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Example 13–7

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Force Analysis – Bevel Gearing

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Fig. 13–35

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Example 13–8

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Fig. 13–36a

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Example 13–8

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Example 13–8

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Fig. 13–36b

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Example 13–8

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Example 13–8

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Force Analysis – Helical Gearing

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Fig. 13–37

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Example 13–9

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Fig. 13–38

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Example 13–9

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Example 13–9

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Fig. 13–39

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Example 13–9

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Example 13–9

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Force Analysis – Worm Gearing

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Fig. 13–40

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Force Analysis – Worm Gearing

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Fig. 13–40

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Force Analysis – Worm Gearing

  • Relative motion in worm gearing is sliding action
  • Friction is much more significant than in other types of gears
  • Including friction components, Eq. (13-41) can be expanded to

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  • Combining with Eqs. (13-42) and (13-43),

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Worm Gearing Efficiency

  • Efficiency is defined as

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  • From Eq. (13–45) with f = 0 in the numerator,

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Worm Gearing Efficiency

  • With typical value of f = 0.05, and φn = 20º, efficiency as a function of helix angle is given in the table.

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Table 13–6

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Worm Gearing Efficiency

  • Coefficient of friction is dependent on relative or sliding velocity VS
  • VG is pitch line velocity of gear
  • VW is pitch line velocity of worm

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Fig. 13–41

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Coefficient of Friction for Worm Gearing

  • Graph shows representative values
  • Curve A is for when more friction is expected, such as when gears are cast iron
  • Curve B is for high-quality materials

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Fig. 13–42

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Example 13–10

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Fig. 13–43

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Example 13–10

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Example 13–10

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Example 13–10

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Example 13–10

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Example 13–10

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Fig. 13–44

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Example 13–10

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Example 13–10

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