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2.72 Elements of Mechanical Design
Spring 2009
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2.72
Elements of Mechanical Design
Lecture 12:
Belt, friction, gear drives
Schedule and reading assignment
Quiz
‰
Bolted joint qualifying Thursday March 19th
Topics
‰
‰
‰
Belts
Friction drives
Gear kinematics
Reading assignment
• Read:
14.1 – 14.7
• Skim:
Rest of Ch. 14
© Martin Culpepper, All rights reserved
2
Topic 1: Belt Drives
© Martin Culpepper, All rights reserved
3
Belt Drives
Why Belts?
‰
‰
‰
‰
Torque/speed conversion
Cheap, easy to design
Easy maintenance
Elasticity can provide damping, shock absorption
Image by dtwright on Flickr.
Keep in mind
‰
‰
Speeds generally 2500-6500 ft/min
Performance decreases with age
Images removed due to copyright restrictions.
Please see:
Image by v6stang on Flickr.
http://www.tejasthumpcycles.com/Parts/primaryclutch/3.35-inch-harley-Street-Belt-Drive.jpg
http://www.al-jazirah.com.sa/cars/topics/serpentine_belt.jpg
© Martin Culpepper, All rights reserved
4
Belt Construction and Profiles
Many flavors
‰
‰
‰
Flat is cheapest, natural clutch
Vee allows higher torques
Synchronous for timing
Usually composite structure
‰
‰
Rubber/synthetic surface for friction
Steel cords for tensile strength
© Martin Culpepper, All rights reserved
5
Belt Drive Geometry
Driven Pulley
Slack Side
d2
ω1
d1
Driving
Pulley
© Martin Culpepper, All rights reserved
ω2
vbelt
Tight Side
6
Belt Drive Geometry
θ2
θ1
dspan
dcenter
© Martin Culpepper, All rights reserved
7
Contact Angle Geometry
θ2
d2
θ1
ω1 d1
ω2
dspan
dcenter
⎛ d2 −d1 ⎞
⎟⎟
θ1 = π −2sin ⎜⎜
⎝ 2dcenter ⎠
−1
© Martin Culpepper, All rights reserved
⎛ d2 −d1 ⎞
⎟⎟
θ2 = π +2sin ⎜⎜
⎝ 2dcenter ⎠
−1
8
Belt Geometry
θ2
d2
θ1
ω1 d1
ω2
dspan
dcenter
⎛ d2 −d1 ⎞
−⎜
⎟
⎝ 2 ⎠
2
dspan = d
2
center
© Martin Culpepper, All rights reserved
Lbelt = 4d
2
center
− (d2 −d1 ) + 1 (d1θ1 +d2θ2 )
2
2
9
Drive Kinematics
θ2
d2
θ1
ω1 d1
ω2
dspan
dcenter
d1
d2
vb = ω1 = ω2
2
2
© Martin Culpepper, All rights reserved
d1 ω2
=
d 2 ω1
10
Elastomechanics
Elastomechanics → torque transmission
‰
Kinematics → speed transmission
Link belt preload to torque transmission
‰
Proceeding analysis is for flat/round belt
Driven
Pulley
Slack Side
d2
ω1
d1
Driving
Pulley
© Martin Culpepper, All rights reserved
ω2
vbelt
Tight Side
11
Free Body Diagram
y
x
dS
F
μdN
F+dF
dN
d/2
•Tensile force (F)
•Normal force (N)
•Friction force (μN)
dθ
•Centrifugal force (S)
© Martin Culpepper, All rights reserved
12
Force Balance
y
dS
x
F
Using small angle approx:
μdN
F+dF
dN
d/2
dθ
dθ
dθ
ΣFy = 0 = −(F + dF ) − F
+ dN + dS
2
2
Fdθ = dN + dS
ΣFx = 0 = −μdN − F + (F + dF )
μdN = dF
© Martin Culpepper, All rights reserved
13
Obtaining Differential Eq
y
dS
x
F
Let m be belt mass/unit length
μdN
2
F+dF
dN
d/2
⎛d ⎞ 2
dS = m⎜ ⎟ ω dθ
⎝ 2⎠
Combining these red eqns:
2
dθ
⎛d ⎞
dF = μFdθ − μm⎜ ⎟ ω 2 dθ
⎝2⎠
2
dF
⎛d ⎞
− μF = − μm⎜ ⎟ ω 2
dθ
⎝2⎠
© Martin Culpepper, All rights reserved
14
Belt Tension to Torque
Let the difference in tension between the loose side (F2) and the
tight side (F1) be related to torque (T)
T
F1 − F2 =
d
2
F1
Solve the previous integral over contact angle and apply F1 and
F2 as b.c.’s and then do a page of algebra:
T
μθcontact
Te
+1
Ftension = μθcontact
de
−1
2e μθcontact
⎛d ⎞ 2
F1 = m⎜ ⎟ ω + Ftension μθcontact
e
+1
⎝2⎠
F2
2
2
2
⎛ d ⎞ 2
F2 = m⎜ ⎟ ω + Ftension μθcontact
e
+1
⎝2⎠
© Martin Culpepper, All rights reserved
Used to find stresses
in belt!!!
15
Practical Design Issues
Pulley/Sheave profile
‰
Which is right?
Manufacturer → lifetime eqs
‰
‰
Belt Creep (loss of load capacity)
Lifetime in cycles
A
B
C
Idler Pulley Design
‰
‰
Catenary eqs → deflection to tension
Large systems need more than 1
Idler
IDL
ALT
P/S
Water
pump
& fan
Idler
Crank
Images by v6stang on Flickr.
© Martin Culpepper, All rights reserved
Figure by MIT OpenCourseWare.
16
Practice problem
Delta 15-231 Drill Press
‰
‰
‰
‰
‰
‰
1725 RPM Motor (3/4 hp)
450 to 4700 RPM operation
Assume 0.3 m shaft separation
What is max torque at drill bit?
What size belt?
Roughly what tension?
© Martin Culpepper, All rights reserved
Images removed due to copyright restrictions. Please see
http://www.rockler.com/rso_images/Delta/15-231-01-500.jpg
17
Topic 2: Friction Drives
© Martin Culpepper, All rights reserved
18
Friction Drives
Why Friction Drives?
‰
‰
‰
Linear ↔ Rotary Motion
Low backlash/deadband
Can be nm-resolution
Images removed due to copyright restrictions. Please see
http://www.beachrobot.com/images/bata-football.jpg
http://www.borbollametrology.com/PRODUCTOS1/Wenzel/
WENZELHorizontal-ArmCMMRSPlus-RSDPlus_files/rsplus.jpg
Keep in mind
‰
‰
‰
‰
Preload → bearing selection
Low stiffness and damping
Needs to be clean
Low drive force
© Martin Culpepper, All rights reserved
19
Friction Drive Anatomy
Motor and
Transmission/Coupling
Drive Roller
Drive Bar
Concerned with:
•Linear Resolution
•Output Force
•Max Roller Preload
Backup
Rollers
•Axial Stiffness
© Martin Culpepper, All rights reserved
20
Drive Kinematics/Force Output
Kinematics found from no slip cylinder on flat
Δδ bar
d wheel
= Δθ ⋅
2
dwheel
Δθ
vbar
d wheel
= ωwheel
2
Δδ
Force output found from static analysis
‰
Either motor or friction limited
Foutput
© Martin Culpepper, All rights reserved
2Twheel
=
d wheel
where Foutput ≤ μFpreload
21
Maximum Preload
⎛ 1 −ν
1 −ν
⎜
Ee = ⎜
+
Ebar
⎝ Ewheel
2
wheel
2
bar
Variable Definitions
⎞
⎟⎟
⎠
−1
⎛
⎞
⎜ 1
1 ⎟
+
Re = ⎜
⎟
d
r
⎜ wheel
crown ⎟
2
⎝
⎠
−1
⎛ 3Fpreload Re ⎞
⎟⎟
acontact = ⎜⎜
⎝ 2Ee ⎠
For metals:
τ max =
3σ y
2
1
3
Shear Stress Equation
acontactEe ⎛ 1 + 2ν wheel 2
⎞
τ wheel =
+ ⋅ (1 +ν wheel ) ⋅ 2(1 +ν wheel ) ⎟
⎜
2πRe ⎝
2
9
⎠
Fpreload , max =
3
Re
2
16π 3τ max
⎞
⎛ 1+ 2ν wheel 2
3E ⎜
+ ⋅ (1+ ν wheel ) ⋅
2(1+ ν wheel ) ⎟
2
9
⎠
⎝
3
2
e
© Martin Culpepper, All rights reserved
22
Axial Stiffness
⎛
⎜
1
1
1
1
⎜
kaxial =
+
+
+
⎜ kshaft ktorsion ktangential kbar
⎜
2
d
wheel
⎝
⎞
⎟
⎟
⎟
⎟
⎠
−1
4ae Ee
ktangential =
(2 −ν )(1+ν )
k shaft =
k torsion =
k bar =
© Martin Culpepper, All rights reserved
3π Ed
4
shaft
3
4L
4
π Gd wheel
32 L
EA c , bar
L
23
Friction Drives
Proper Design leads to
‰
‰
Pure radial bearing loads
Axial drive bar motion only
Drive performance linked to motor/transmission
‰
‰
Torque ripple
Angular resolution
Images removed due to copyright restrictions. Please see
http://www.borbollametrology.com/PRODUCTOS1/Wenzel/WENZELHorizontal-ArmCMMRSPlus-RSDPlus_files/rsplus.jpg
© Martin Culpepper, All rights reserved
24
Topic 3:
Gear Kinematics
© Martin Culpepper, All rights reserved
25
Gear Drives
Why Gears?
‰
‰
‰
‰
Torque/speed conversion
Can transfer large torques
Can run at low speeds
Large reductions in small package
Image from robbie1 on Flickr.
Keep in mind
‰
‰
Requires careful design
Attention to tooth loads, profile
Image from jbardinphoto on Flickr.
Images removed due to copyright restrictions. Please see
http://elecon.nlihost.com/img/gear-train-backlash-and-contact-pattern-checking.jpg
http://www.cydgears.com.cn/products/Planetarygeartrain/planetarygeartrain.jpg
© Martin Culpepper, All rights reserved
26
Gear Types and Purposes
Spur Gears
‰
‰
‰
‰
Parallel shafts
Simple shape → easy design, low $$$
Tooth shape errors → noise
No thrust loads from tooth engagement
Helical Gears
‰
‰
‰
‰
Gradual tooth engagement → low noise
Shafts may or may not be parallel
Thrust loads from teeth reaction forces
Tooth-tooth contact pushes gears apart
Images from Wikimedia Commons, http://commons.wikimedia.org
© Martin Culpepper, All rights reserved
27
Gear Types and Purposes
Bevel Gears
‰
‰
Connect two intersecting shafts
Straight or helical teeth
Worm Gears
‰
‰
‰
Low transmission ratios
Pinion is typically input (Why?)
Teeth sliding → high friction losses
Rack and Pinion
‰
‰
Rotary ↔ Linear motion
Helical or straight rack teeth
Images from Wikimedia Commons, http://commons.wikimedia.org
Pinion, m2
F (t)
+
a
b
k
Figure by MIT OpenCourseWare.
© Martin Culpepper, All rights reserved
k
Rack, m1
Viscous damping, c
Rack & Pinion
28
Tooth Profile Impacts Kinematics
Want constant speed output
‰
‰
Conjugate action = constant angular velocity ratio
Key to conjugate action is to use an involute tooth profile
Output speed of gear train
“Real” involute/gear
“Ideal” involute/gear
ωout, [rpm]
Non or poor involute
time [sec]
© Martin Culpepper, All rights reserved
29
Instantaneous Velocity and Pitch
Model as rolling cylinders (no slip condition):
v v v v v
v = ω1 × r1 = ω2 × r2
ω1 r2
=
ω 2 r1
Model gears as two pitch circles
‰
Contact at pitch point
Pitch Circles Meet @ Pitch Pt.
r1
r2
ω2
r1
r2
ω2
ω1
ω1
v
© Martin Culpepper, All rights reserved
30
Instantaneous Velocity and Pitch
Meshing gears must have same pitch
-Ng = # of teeth, Dp = Pitch circle diameter
Diametral pitch, PD:
Circular pitch, PC:
© Martin Culpepper, All rights reserved
PD =
PC =
Ng
Dp
πD p
Ng
π
=
PD
31
Drawing the Involute Profile
•Gear is specified by
diametral pitch and
pressure angle, Φ
Pitch Point
Φ
Pitch Circle
Base Circle
Images from Wikimedia Commons, http://commons.wikimedia.org
DB/2
DP/2
Image removed due to copyright restrictions. Please see
http://upload.wikimedia.org/wikipedia/commons/c/c2/Involute_wheel.gif
DB = DP cos Φ
© Martin Culpepper, All rights reserved
32
Drawing the Involute Profile
Pitch Point
Pitch Circle
L3
L2
3
DB
Δθ
Ln = n
2
2
1
L1
Base Circle
DB/2
© Martin Culpepper, All rights reserved
33
Transmission Ratio for Serial Gears
Gear train
Power in: Tin y ωin
Transmission ratio for elements in series:
From pitch equation: P1 =
Power out: Tout y ωout
ωout
TR = ( proper sign ) ⋅
ωin
N1 N 2
=
= P2
D1 D 2
D1 N1 ω2
=
=
D 2 N 2 ω1
11
2
For Large Serial Drive Trains:
Productof drivingteeth
TR = ( proper sign) ⋅
Productof driven teeth
© Martin Culpepper, All rights reserved
34
Transmission Ratio for Serial Gears
TR = ( proper sign ) ⋅
Serial trains:
Example 1:
Product of driving teeth
Product of driven teeth
TR = ?
in
out
Example 2:
driven
drive
driven
drive
TR = ?
driven
drive
in
© Martin Culpepper, All rights reserved
out
35
Transmission Ratio for Serial Gears
Example 3: Integral gears in serial gear trains
‰
What is TR? Gear 1 = input and 5 = output
Product of driving teeth
TR = ( proper sign ) ⋅
Product of driven teeth
Gear - 1
N1 = 9
5
4
Gear - 2
N2 = 38
2
Gear - 3
N3 = 9
1
Gear - 4
N4 = 67
Gear - 5
N5 = 33
© Martin Culpepper, All rights reserved
3
36
Planetary Gear Trains
Planetary gear trains are very common
‰
Very small/large TRs in a compact mechanism
Terminology:
Ring
gear
Planet
Arm
Planet
gear
Planet
gear
Planet
Arm
ω
2
Sun
gear
© Martin Culpepper, All rights reserved
Planet
gear
37
Planetary Gear Train Animation
How do we find the transmission
ratio?
http://www.cydgears.com.cn/products/Planetarygeartrain/
planetarygeartrain.jpg
Train 1
Image removed due to copyright restrictions. Please see
Ring
gear
Arm
Planet
gear
Sun
Train 2
Ring
gear
Arm
Planet
gear
Sun
© Martin Culpepper, All rights reserved
38
Planetary Gear Train TR
Sun Gear
Planet Gear
If we make the arm
stationary, than this is
a serial gear train:
Ring Gear
ωra ωring − ωarm
=
= TR
ωsa ωsun − ωarm
N sun N planet
N sun
TR = −
⋅
= −
N planet N ring
N ring
Arm
ω pa
ω sa
=
ω planet − ω arm
ω sun − ω arm
= TR
N sun
TR = −
N planet
© Martin Culpepper, All rights reserved
39
Planetary Gear Train Example
Sun Gear
Planet Gear
Ring Gear
If the sun gear is the
input, and the ring
gear is held fixed:
0 − ωarm
ωra
=
= TR
ωsa ωsun − ωarm
Arm
N sun N planet
N sun
TR = −
⋅
=−
N planet N ring
N ring
TR
ωoutput = ωarm =
ωsun
TR −1
© Martin Culpepper, All rights reserved
40
Case Study: Cordless Screwdriver
Given: Shaft TSH (ωSH) find motor TM (ωSH)
‰
Geometry dominates relative speed (Relationship due to TR)
2 Unknowns: TM and ωM with 2 Equations:
‰
‰
Transmission ratio links input and output speeds
Energy balance links speeds and torques
© Martin Culpepper, All rights reserved
41
Example: DC Motor shaft
⎛
ω ⎞
⎟⎟
T(ω ) = TS ⋅ ⎜⎜1−
⎝ ω NL ⎠
P(ω) obtained from P(ω) = T(ω) y ω
T(ω):
Motor torque-speed curve
T(ω)
( 0 , TS )
Speed at maximum power output:
(ωNL , 0 )
2 ⎞
⎛
ω
⎟
P(ω ) = T(ω ) ⋅ ω = TS ⋅ ⎜ ω −
⎟
⎜
ω
NL
⎠
⎝
ω PMAX =
PMAX
ω NL
2
⎛ ω NL ⎞
= TS ⋅ ⎜
⎟
⎝ 4 ⎠
© Martin Culpepper, All rights reserved
ω
P(ω
)
Motor power curve
PMAX
(ωNL , 0 )
ωPMAX
ω
42
Example: Screw driver shaft
A = Motor shaft torque-speed curve
What is the torque-speed curve for the screw driver?
T(ω)
B
Train ratio = 1/81
A
C
ω
SCREW DRIVER SHAFT
TSH, ωSH
MOTOR SHAFT
TM, ωM
Electric
Motor
GT-1
GT-2
Screw Driver Shaft
System boundary
GEAR train # 1
© Martin Culpepper, All rights reserved
GEAR train # 2
43
Example: Screw driver shaft
P(ω)
D
C = Motor shaft power curve
What is the power-speed curve for the screw driver?
Train ratio = 1/81
C
E
ω
SCREW DRIVER SHAFT
TSH, ωSH
MOTOR SHAFT
TM, ωM
Electric
Motor
GT-1
GT-2
Screw Driver Shaft
System boundary
GEAR train # 1
© Martin Culpepper, All rights reserved
GEAR train # 2
44