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: matthewg@mit.edu, egilbert@mit.edu 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 TR = 9 38 9 67 67 33 = 0.065 = 1 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, 20- degree 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, 20- degree pressure angle, 45mm OD, 10mm thick nylon gear adequate to transmit 10 Nm of torque? D pitch = mod #teeth = 2 20 = 40mm
Example Force on Gear Teeth Is a mod 2, 20-tooth, 20- degree pressure angle, 45mm OD, 10mm thick nylon gear adequate to transmit 10 Nm of torque? D pitch = mod #teeth = 2 20 = 40mm F tangent = Torque D pitch /2 = 10Nm 20mm = 500N
Example Force on Gear Teeth Is a mod 2, 20-tooth, 20- degree pressure angle, 45mm OD, 10mm thick nylon gear adequate to transmit 10 Nm of torque? D pitch = mod #teeth = 2 20 = 40mm F tangent = Torque D pitch /2 = 10Nm 20mm = 500N F normal = F tangent cosφ = 50N cos(20 ) = 530N
Example Force on Gear Teeth Is a mod 2, 20-tooth, 20- degree pressure angle, 45mm OD, 10mm thick nylon gear adequate to transmit 10 Nm of torque? D pitch = mod #teeth = 2 20 = 40mm F tangent = Torque D pitch /2 = 10Nm 20mm = 500N F normal = F tangent cosφ = 50N cos(20 ) = 530N σ tooth = F normal A tooth = 530N 10mm 5mm = 10.6MPa 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 Transmit torque at 90 degree angle Rack and Pinion Rotary to linear motion Worm 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 Avoid: Aluminum on Aluminum F friction
What do these have in common? Bushings
Examples of sliding contact bearings Bushings
Friction coefficients Common materials for sliding contact bearings Materials 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 0.13 Steel Teflon 0.05-0.2 Teflon Teflon 0.04 0.04 http://www.engineeringtoolbox.com
Rotary sliding contact: efficiency vs diameter D outside d inside Axle Wheel Fundamentals (Slocum)
Rolling contact bearings Lower friction than sliding contact bearings Less wear Can achieve high speeds Fundamentals (Slocum)
F radial Radial Tapered roller F axial Thrust enginemechanics.tpub.com
Ball vs. roller bearings Fundamentals (Slocum) Ball bearings: Lower loads Higher precision Lower cost 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
Wheel Wheel Wheel Mounting bearings Groove for E-clip E-clip Shaft Bearing (fixed to ground) Small gap Shaft Threads Outer ring (fixed) Inner ring Shoulder Ball Nut Nut Shaft Shaft Threads Nut Nut
Hertz contact: point stresses F p 0 δ F = force δ = deflection p 0 = max stress E e = equivalent elastic modulus R e = equivalent radius of curvature Fundamentals (Slocum)
Hertz contact: line stresses F F = force δ = deflection p 0 = max stress E e = equivalent elastic modulus R e = equivalent radius of curvature L = contact length p 0 δ Fundamentals (Slocum)
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 m 2 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 m 2 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 Desired position Controller G(s) Amplifier Electromagnet Position measurement Hall effect sensor
Japanese magnetic levitation (Maglev) train
Magnetic Levitation (Maglev)
www.wsj.com
Maglev train: propulsion and levitation Propulsion Levitation www.dailytech.com 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