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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
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