Download Power Transmission Elements II: Gears and Bearings

Power Transmission Elements II:
Gears and Bearings
Lecture 3, Week 4
Announcements
• Lab 4 – need to finish by Friday
– Friday lab can get started today
• Project proposal
– Due at 23:59 tonight
– Email to us: [email protected], [email protected]
• Group project meetings tomorrow
• Kinematic couplings due tomorrow
Outline
• Gears
• Bearings
• Lab time
Gears
Rotating elements used to transmit torque or force
Common usage – convert motor’s high-speed lowtorque output to low-speed high-torque output
Gears in Nature
Wikipedia
Plant-hopper insect Issus has gear-like mechanism to synchronize legs when it jumps
Gear Shapes
Gear design goal:
Rolling contact with minimal backlash
Wikipedia
Gear Trains
Transmission ratio = #teeth of driving gear
# teeth of driven gear
What is the transmission
ratio here from red to
purple?
Fundamentals, Slocum
Gear Trains
Transmission ratio = #teeth of driving gear
# teeth of driven gear
What is the transmission
ratio here from red to
purple?
High torque
Low speed
Low torque
High speed
9 9 67
1
𝑇𝑅 =
∗
∗
= 0.065 =
38 67 33
16
Fundamentals, Slocum
Important Terminology
• Pitch Diameter – diameter of
circle where teeth make
contact
• Module (“mod”) – Pitch
diameter (mm) divided by
#teeth [mm/tooth]
• Pressure Angle – angle teeth
make contact at
• Pitch – 1/Module
[teeth/mm] (“tooth density”)
Fundamentals, Slocum
How do you choose gear sizes?
1. Choose number and diameter of gears based on
reduction ratio required and space constraints
2. Calculate forces at pitch diameter, and size
teeth to withstand this force (with some safety
factor)
3. Make sure meshing gears have the same
module and pressure angle so teeth mesh
correctly
Buying Gears
Mcmaster-carr.com
Example – Force on Gear Teeth
Is a mod 2, 20-tooth, 20degree pressure angle, 45mm
OD, 10mm thick nylon gear
adequate to transmit 10 Nm
of torque?
Example – Force on Gear Teeth
Is a mod 2, 20-tooth, 20degree pressure angle, 45mm
OD, 10mm thick nylon gear
adequate to transmit 10 Nm
of torque?
𝐷𝑝𝑖𝑡𝑐ℎ = 𝑚𝑜𝑑 #𝑡𝑒𝑒𝑡ℎ = 2 ∗ 20 = 40𝑚𝑚
Example – Force on Gear Teeth
Is a mod 2, 20-tooth, 20degree pressure angle, 45mm
OD, 10mm thick nylon gear
adequate to transmit 10 Nm
of torque?
𝐷𝑝𝑖𝑡𝑐ℎ = 𝑚𝑜𝑑 #𝑡𝑒𝑒𝑡ℎ = 2 ∗ 20 = 40𝑚𝑚
𝐹𝑡𝑎𝑛𝑔𝑒𝑛𝑡 =
𝑇𝑜𝑟𝑞𝑢𝑒
10𝑁𝑚
=
= 500𝑁
𝐷𝑝𝑖𝑡𝑐ℎ /2 20𝑚𝑚
Example – Force on Gear Teeth
Is a mod 2, 20-tooth, 20degree pressure angle, 45mm
OD, 10mm thick nylon gear
adequate to transmit 10 Nm
of torque?
𝐷𝑝𝑖𝑡𝑐ℎ = 𝑚𝑜𝑑 #𝑡𝑒𝑒𝑡ℎ = 2 ∗ 20 = 40𝑚𝑚
𝐹𝑡𝑎𝑛𝑔𝑒𝑛𝑡 =
𝐹𝑛𝑜𝑟𝑚𝑎𝑙
𝑇𝑜𝑟𝑞𝑢𝑒
10𝑁𝑚
=
= 500𝑁
𝐷𝑝𝑖𝑡𝑐ℎ /2 20𝑚𝑚
𝐹𝑡𝑎𝑛𝑔𝑒𝑛𝑡
50𝑁
=
=
= 530𝑁
𝑐𝑜𝑠𝜑
cos(20°)
Example – Force on Gear Teeth
Is a mod 2, 20-tooth, 20degree pressure angle, 45mm
OD, 10mm thick nylon gear
adequate to transmit 10 Nm
of torque?
𝐷𝑝𝑖𝑡𝑐ℎ = 𝑚𝑜𝑑 #𝑡𝑒𝑒𝑡ℎ = 2 ∗ 20 = 40𝑚𝑚
𝐹𝑡𝑎𝑛𝑔𝑒𝑛𝑡 =
𝐹𝑛𝑜𝑟𝑚𝑎𝑙
𝑇𝑜𝑟𝑞𝑢𝑒
10𝑁𝑚
=
= 500𝑁
𝐷𝑝𝑖𝑡𝑐ℎ /2 20𝑚𝑚
𝐹𝑡𝑎𝑛𝑔𝑒𝑛𝑡
50𝑁
=
=
= 530𝑁
𝑐𝑜𝑠𝜑
cos(20°)
𝜎𝑡𝑜𝑜𝑡ℎ =
𝐹𝑛𝑜𝑟𝑚𝑎𝑙
530𝑁
=
= 10.6𝑀𝑃𝑎
𝐴𝑡𝑜𝑜𝑡ℎ
10𝑚𝑚 ∗ 5𝑚𝑚
Nylon shear strength = 70 MPa
Yes – with safety factor 7
Planetary Gears
Attain high gear reduction in small space
Planet
Carrier
Planet
Sun
Ring
Planetary gears - animation
https://www.youtube.com/watch?v=UakeTEJlXGw
Other Gears
Bevel Gear
Rack and Pinion
Worm
Transmit torque at 90
degree angle
Rotary to linear motion
Transmit torque at 90
degree angle
Directindustry.com
Science.howstuffworks.com
How is this possible?
Regular octahedron
Irregular octahedron
http://www.thingiverse.com/thing:748202
Differentials
• Allows torque transmission to wheels rotating
at different speeds
https://www.youtube.com/watch?v=gIGvhvOhLHU
Bearings
• Allow relative motion
between components
• Constrain 1 or more DOFs
• Types:
– Contact
• Sliding
• Rolling
• Flexural
– Non-contact
• Fluid film
• Magnetic
Sliding contact bearings
• Better for low/moderate speeds
• Use a hard material and a soft material
– Lower friction coefficients
– Focuses wear on softer part
• Examples:
– Steel shaft on plastic bushing F
• Avoid:
– Aluminum on Aluminum
friction
What do these have in common?
Bushings
Examples of sliding contact bearings
Bushings
Friction coefficients
Materials
Common
materials for
sliding
contact
bearings
Friction coefficient
Material 1
Material 2
Clean and dry
Lubricated
Silver
Silver
1.4
0.55
Aluminum
Aluminum
1.05-1.35
0.3
Glass
Glass
0.94
0.1-0.6
Rubber
Asphalt
0.9
0.25-0.75
Steel
Polyethylene
0.2
0.2
Steel
Bronze - sintered
Steel
Teflon
0.05-0.2
Teflon
Teflon
0.04
http://www.engineeringtoolbox.com
0.13
0.04
Rotary sliding contact: efficiency vs
diameter
Doutside
dinside
Axle
Wheel
Fundamentals (Slocum)
Rolling contact bearings
• Lower friction than sliding contact bearings
• Less wear
• Can achieve high speeds
Fundamentals (Slocum)
Fradial
Radial
Tapered
roller
Faxial
Thrust
enginemechanics.tpub.com
Ball vs. roller bearings
Ball bearings:
•Lower loads
•Higher precision
•Lower cost
Fundamentals (Slocum)
Roller bearings:
•Higher loads
•Lower precision
•Higher cost
How to use choose bearings
• Calculate the expected forces; decide between a
radial vs. thrust bearing
• Calculate max RPM
• Select shaft diameter
Bearing for a 10mm shaft
Mounting bearings
Wheel
Groove for E-clip
Shaft
Bearing
(fixed to ground)
Wheel
E-clip
Small gap
Threads
Shaft
Inner ring
Shoulder
Nut Nut
Wheel
Outer ring (fixed)
Shaft
Threads
Shaft
Nut Nut
Ball
Hertz contact: point stresses
F
p0
F = force
δ = deflection
p0 = max stress
Ee = equivalent elastic modulus
Re = equivalent radius of curvature
Fundamentals (Slocum)
δ
Hertz contact: line stresses
F
F = force
δ = deflection
p0 = max stress
Ee = equivalent elastic modulus
Re = equivalent radius of curvature
L = contact length
Fundamentals (Slocum)
p0
δ
Non-contact bearings
Can achieve very high speeds with very little friction
Hydrodynamic
• Relies on relative motion to
cause hydrodynamic lift
• Part glides on thin fluid film
Hydrostatic
• Parts can be stationary
relative to each other
• High pressure fluid is
pumped through a gap
Hydrodynamic Bearings
• Fluid film prevents material-material contact
• Low cost, low friction, long life
• Less repeatable than rolling contact
Shaft
Low speed: material-material sliding
contact
Medium speed: mixed sliding contact
and viscous shear. Friction decreases
High speed: hydrodynamic life; no
mechanical contact
http://www.marinediesels.info
Fundamentals (Slocum)
www.duramaxmarine.com
Aerostatic and Hydrostatic Bearings
• Pressurized air or water flows into an
interface
• Maintains a gap of 0.005-0.1mm
• High accuracy
• Need a collection system for water
• Pressures:
– Air: 10 atm
– Water: 40-200 atm
F = load capacity
η = efficiency (25-40%)
P = fluid pressure
A = contact area
K = stiffness
h = air gap
Fundamentals (Slocum)
The power of air bearings
How much air
pressure to lift
the truck?
Assume mass = 2000 kg, contact area = 2 m2
https://www.youtube.com/watch?v=uRql1fjZQf0
F = load capacity
η = efficiency (25-40%)
P = fluid pressure
A = contact area
K = stiffness
h = air gap
The power of air bearings
How much air
pressure to lift
the truck?
P = ~ 4e4 Pa
Assume mass = 2000 kg, contact area = 2 m2
https://www.youtube.com/watch?v=uRql1fjZQf0
F = load capacity
η = efficiency (25-40%)
P = fluid pressure
A = contact area
K = stiffness
h = air gap
Magnetic bearings
• Is it possible to stably levitate a permanent
magnet with other permanent magnets?
?
Magnetic bearings
• Is it possible to stably levitate a permanent
magnet with other permanent magnets?
– Unfortunately, no. Proven by Earnshaw in 1842
• But, levitation is possible using:
– Diamagnetic materials, or…
– Electromagnets & closed loop control
Diamagnetism and paramagnetism
• Two types of materials:
– Paramagnetic: attracted by magnetic fields (ex: iron)
– Diamagnetic: repelled by magnetic fields
Diamagnetic levitation
of pyrolytic carbon
http://en.wikipedia.org/wiki/Diamagnetism
Levitation using electromagnets
hacknmod.com
zeltom.com
Closed loop position control
Controller
Desired
position
Amplifier
G(s)
Electromagnet
Position
measurement
Hall effect
sensor
Japanese magnetic levitation (Maglev) train
Magnetic Levitation (Maglev)
www.wsj.com
Maglev train: propulsion and levitation
Propulsion
www.dailytech.com
Levitation
http://www.hk-phy.org
Bearing types: rolling or sliding contact
•
•
•
•
Sliding contact
Higher friction
Lower cost
Lower complexity
“Bushings”
•
•
•
•
Rolling contact
Lower friction
Higher cost
Higher complexity
“Ball bearings”
•
•
•
•
Non-contact
Minimal friction
Nearly infinite
lifetime
Lower stiffness
Lower
repeatability
under varying
loads
How is this possible?
Regular octahedron
Irregular octahedron
http://www.thingiverse.com/thing:748202