1

1GRASP

Nora Ayanian

University of Pennsylvania

Cams, Gears, Belts & Chains

SAAST Robotics 2008

2GRASP

Gears

SAAST Robotics 2007

3GRASP

Gears

• Transmit motion by means of successively engaging teeth

• Most Common: Rotating shaft to another rotating shaft

• Rotating shaft to a rack – translates motion to straight line

• Why gear up/down a motor?

• Increase speed – decrease torque

• Decrease speed – increase torque

• Types of Gears

• Spur gears – transmit rotary motion between parallel shafts

4GRASP

• Types of Gears

• Internal gear and pinion

• Rack and pinion – rack is spur gear with infinitely large pitch diameter

Gears: Types of Gears

5GRASP

Gears: Types of Gears

• Types of Gears

• Helical gears – transmit motion btw/ parallel or non-parallel shafts

6GRASP

Gears: Types of Gears

• Types of Gears

• Bevel gears – transmit motion between intersecting shafts

7GRASP

Gears: Types of Gears

• Types of Gears

• Bevel gears – transmit motion between intersecting shafts

• Skew bevel gears – connect shafts whose axes don’t intersect – straight teeth

• Hypoid gears – transmit motion between shafts whose axes don’t intersect

• Curved teeth

8GRASP

Gears: Types of Gears

• Types of Gears

• Bevel gears – transmit motion between intersecting shafts

• Skew bevel gears – connect shafts whose axes don’t intersect – straight teeth

• Hypoid gears – transmit motion between shafts whose axes don’t intersect

• Curved teeth

9GRASP

• Types of Gears

• Worm gears – transmit motion between nonparallel non-intersecting shafts

• Noncircular gears – non constant angular velocity ratios between input and

output

Gears: Types of Gears

2

10GRASP

Gears: Gear Tooth Nomenclature

Pitch circles of mating gears are tangent to each other

Circular pitch (CP) – distance along arc of pitch circle between neighboring teeth

Diametral pitch (P) - # of teeth on gear/inch of pitch diameter

Backlash – difference between tooth space and thickness of engaging tooth at pitch circles

P = N/D

CP = (π D)/N

N = # of teeth

D = pitch diameter (in)

11GRASP

Gears: Gear Trains

Desire: ω3 = ωout = 2700 rpm, counterclockwise input to machine

Standard motor output: ω1 = ω2 = ωin = -1800 rpm, clockwise

12GRASP

Gears: Gear Trains

Desire: ω3 = ωout = 2700 rpm, counterclockwise input to machine

Standard motor output: ω1 = ω2 = ωin = -1800 rpm, clockwise

Solution: Simple Gear train – spur gears

• Pitch velocities of mating gears are equal and functions of pitch radii and angular

velocity

VP2 = r2 ω2 = VP3 = - r3 ω3 � ω3 / ω2 = - r2 / r3 = - 30 / 20

ω3 = -3/2 ω2 = -3/2 (-1800) = 2700 rpm ccw

• Angular velocities inversely proportional to pitch radii � pitch diameter � # of teeth

P = N/D

D = N/P

2*r = N/P

r ≈ N

in

out

in

out

out

in

N

N

r

r==−

ω

ω

13GRASP

Gears: Gear Trains

• What if want larger angular velocity ratio, like 60:1?

• Space limitations prevent one set of gears w/ diameter ratio of 60:1

N2 = 60 N1

14GRASP

Gears: Gear Trains

• What if want larger angular velocity ratio, like 60:1?

• Space limitations prevent one set of gears w/ diameter ratio of 60:1

• Solution: Compound gear train

N2 = 60 N1

- ωin / ωout = rout / rin = Nout / Nin

15GRASP

Gears: Gear Trains

Choose: N3/N2 = 3, N5/N4 = 4, N7/N8 = 5 � = 60

General Rule:

gearsdriveronteethofnumbersofproduct

gearsdrivenonteethofnumbersofproductRatioGear

driven

driver ==ω

ω

driver drivers

drivendriven driven

16GRASP

Gears: Gear Trains

• Planetary gear train - allow some gear axes to rotate about others

• Sun gear, planet carrier (or arm), planet gears

• 2 DOF’s � 2 inputs

• e = gear ratio

• ωF = rpm of first gear in planetary gear train

• ωL = rpm of last gear in planetary gear train

• ωA = rpm of arm

AL

AFeωω

ωω

−

−=

35

32

ωω

ωω

−

−=e

2

3

4

5

17GRASP

Gears: Gear TrainsPlanetary Gear Train Example

In the figure below, the sun gear is the input, driven clockwise at 100 rev/min. The ring gear is held stationary by being fastened to the frame. Find the rev/min and direction of rotation of the arm.

ωF =

ωL =

AL

AFeωω

ωω

−

−=

gearsNproduct

gearsNproducte

driver

driven

driven

driver ==ω

ω + = same direction

- = opposite direction

18GRASP

Gears: Gear TrainsPlanetary Gear Train Example

In the figure below, the sun gear is the input, driven clockwise at 100 rev/min. The ring gear is held stationary by being fastened to the frame. Find the rev/min and direction of rotation of the arm.

ωF = ω2 = - 100 rpm

ωL = ω5 = 0

Unlock gear 5, and hold arm stationary �

AL

AFeωω

ωω

−

−=

A

Aeω

ω

−

−−=

0

100

gearsNproduct

gearsNproducte

driver

driven

driven

driver ==ω

ω + = same direction

- = opposite direction

430

80

20

30

42

54

5

2 −=⋅−=⋅

⋅−==

NN

NNe

ω

ω

rpmA

A

A 200

1004 −=⇒

−

−−=− ω

ω

ω

3

19GRASP

Gears: Homework Problem 1

20GRASP

Gears: Homework Problem 2

This gear train consists of miter gears (same-size bevel gears) having 16 teeth each, a 4-tooth right hand worm, and a 40-tooth worm gear. The speed of gear 2 is given as +200 rpm, corresponding to counterclockwise rotation about y-axis. What is the speed and direction of the worm gear (5)?

21GRASP

Belts & Chains

SAAST Robotics 2007

22GRASP

Belts & Chains

• Used in conveying systems and for transmitting power over

long distances

• Replacements for gears, shafts, bearings

• Simplifies design, reduces cost

• Elastic, flexible nature is good for absorbing shock loads and

vibrations

• Minimum distance between pulley axles required for proper operation

Crowned pulleys

Grooved pulleys / sheaves

Grooved pulleys / sheaves

Toothed wheels / sprockets

23GRASP

Belts & Chains

• Why crowned pulleys?• Crowning the drive pulley prevents

belts from wandering off

• Eliminates the need for flanging and belt guide rollers

• How does it work?

24GRASP

Belts & Chains

• Why crowned pulleys?• Crowning the drive pulley prevents

belts from wandering off

• Eliminates the need for flanging and belt guide rollers

• How does it work?• Minimizing belt tension

• Which way will the belt move?

• Your intuition is not always correct!

25GRASP

Belts & Chains

• Belt characteristics:

• Used for long center distances

• Except for timing belts, slip and creep present so velocity ratios between driver and driven shafts not constant or equal to ratio of pulley diameters

• Use of idler or tension pulley to avoid adjusting center distances –

needed with old or new belts

Open-belt drive – slack side

should be on top

Reversing drives - open

Reversing drives - crossed

Both sides of belt contact pulleys � these

drives cannot be used with V or timing belts

26GRASP

Belts & Chains

Quarter-twist belt drive Variable speed belt drive – flat belts

Variable speed belt drive – flat, V and round belts

27GRASP

Belts & Chains: Flat belt calculations

D = diameter of large pulley

d = diameter of small pulley

C = center distance

θ = angle of contact

L = Length of belt - sum of two arc lengths with twice the

distance between the beginning and end of contact:

)2

(sin2

)2

(sin2

1

1

C

dD

C

dD

D

d

⋅

−⋅+=

⋅

−⋅−=

−

−

πθ

πθ

[ ] )()(42

12/12

dD dDdDCL θθ ⋅+⋅+−−⋅=

4

28GRASP

Belts & Chains: Flat belt calculations

D = diameter of large pulley

d = diameter of small pulley

C = center distance

θ = angle of contact (same for each pulley)

L = Length of belt - sum of two arc lengths with twice the

between the beginning and end of contact:

)2

(sin21

C

dD

⋅

+⋅+= −πθ

[ ] )(2

)(42/12

dDdDCL ++−−⋅=θ

29GRASP

Belts & Chains: Timing Belts

Timing Belts

• Do not slip, transmit power at constant angular-velocity ratio

• No initial tension needed, 97-99% efficient

• No lubrication needed, quiet

• 5 standard inch-series pitches available

• Pitch lengths available in lengths of 6” to 180”

• Pulleys: pitch diameters 0.60” to 35.8”, groove numbers from 10 to 120

30GRASP

Belts & Chains: Roller Chains

Roller Chains

• No slippage or creep

• Constant angular velocity ratio

• Requires lubrication, can be noisy

• Ability to drive multiple shafts from single source of power

• Single, double, triple, and quad strands available

Double strand roller chain

31GRASP

Belts & Chains: Roller Chain Calculations

γ/2 = angle of articulation

• Rotation of link through this causes impact between roller and sprocket teeth, wear in chain joint

• Design system to reduce this angle as much as possible

p = chain pitch

γ = pitch angle

D = pitch diameter of sprocket

N = number of sprocket teeth

L = chain length

C = center distance

N1 = # of teeth on small sprocket

N2 = # of teeth on large sprocket

)sin(360

)sin()

2sin(

180

22

2

N

D

p

pD

NSince

pDor

o

o

=⇒=

==

γ

γγ

)(4

)(

2

2

2

2

1221

pC

NNNN

p

C

p

L

⋅⋅

−+

++

⋅=

π

32GRASP

Cams

Transform one motion into another

Cam – curved/grooved surface

Rotational motion of cam �Follower oscillation and/or translation

Designed for motion, path or function generation

Many types of cams:

Plate/disk cam; wedge cam - translating roller followers

33GRASP

Types of cams:

Cylindrical cam; conical cam - translating followers

Face cam; globoidal cam – oscillating follower

Cams: Types of Cams

34GRASP

Cams: Types of Followers

Types of Followers:

Translating flat-faced follower Translating roller follower Translating point follower

Oscillating flat-face follower Oscillating roller-follower

35GRASP

Types of Followers:

Translating double-roller follower and double-lobed cam

Cams: Types of Followers

Translating positive-return follower Oscillating spherical-face follower

36GRASP

Cams: Applications

Design cam geometry for follower

displacement according to graph

Requirements for tracer point:

• Rise off prime circle by L

• Remain for a while (dwell) @ L

• Return to prime circle

• Remain at rest in 2nd dwell

• Repeat

5

37GRASP

Cams: Applications

Stamping Mechanism

(1) Stamping plate

(2) Flexures

(3) Stop

(4) Springs

(5) Anvil

Goal: (1) to be cyclically depressed against (5) according to time-displacement curve (6)

38GRASP

Mechanical Design Lab 1

Mechanical Design Lab #1:

Mechanism Synthesis, Prototyping, and the World’s

Strongest Truck

39GRASP

Design Game

Goal: Build a mechanical “arm” that can pick up a golf ball and place it into the cup provided.

Resources:

• Straws – 15

• Index card – 4” x 6”

• Tape – 12”

• String – 30 cm

• Paper clips – 5

Rules of the Competition:

• Each team has 2 attempts to place the ball into the cup. The highest scoring attempt is used to compute the final score for the team.

• Reach is defined as the farthest distance away from the cup the arm is such that the ball is successfully dropped into the cup.

Scoring:

• 1 pt per ½” of reach for successfully placing the golf ball in the cup

• Teams of 3

• 15 Minutes