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- 1. Autodesk Sustainability Workshop Bike Wheels, Rotational Inertia, and Energy Adam Kenvarg, Joel Rosenberg, and James Regulinski © 2013 Autodesk
- 2. Which of These is Easier to Spin? Aluminum http://img.tradeindia.com/fp/1/001/012/187.jpg © 2013 Autodesk Stainless Steel http://www.wigglestatic.com/product-media/5360073219/shimano_WHR501R.jpg?w=1100&h=1100&a=7
- 3. What is Rotational Inertia? • Rotational inertia, often called moment of inertia, is the resistance of an object to a change in angular velocity. • It can be thought of as the rotational version of the role that mass plays as the resistance of an object to a change in straight-line velocity. © 2013 Autodesk
- 4. What is Rotational Inertia? II • Said another way, the higher the moment of inertia of an object, the greater its resistance to a change in the rotational velocity. • Similarly, the higher the mass of an object, the greater its resistance to a change in the straightline velocity. © 2013 Autodesk
- 5. Axis of rotation The moment of inertia is almost always different for each different axis of rotation. Solid cylinder (or disk) about central axis Solid cylinder (or disk) about central diameter https://webspace.utexas.edu/cokerwr/www/index.html/RI.htm © 2013 Autodesk
- 6. More Moment of Inertia Values © 2013 Autodesk
- 7. Which object will finish first? (All are equal mass and radius) https://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.gif Note: the red sphere is a hollow shell © 2013 Autodesk
- 8. Go! https://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.gif Note: the red sphere is a hollow shell © 2013 Autodesk
- 9. Results Which object has the highest moment of inertia? Which has the lowest? https://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.gif Note: the red sphere is a hollow shell © 2013 Autodesk
- 10. Moment of Inertia Remember, the higher the moment of inertia of an object, the greater its resistance to a change in the rotational velocity. That’s why the green ring has the HIGHEST moment of inertia (most resistance). Moment of inertia is determined by the amount of mass and its distance from the axis of rotation. © 2013 Autodesk
- 11. Classic Moment of Inertia Demo Figure skaters control their moment of inertia by moving the distribution of their mass – closer to the axis means faster rotation. A simple demo of this can be done by spinning someone in a chair while the person brings weights closer and farther from their body. http://www.exploratorium.edu/snacks/momentum_machine/ © 2013 Autodesk
- 12. When Do You Want a Low Moment of Inertia? Examples include all kinds of wheels – allows for faster acceleration and deceleration of wheels and therefore the vehicle http://en.wikipedia.org/wiki/File:SC06_2005_Porsche_Carrera_GT_wheel.j pg Baseball bats – lets the batter swing the bat around more quickly (this can be accomplished by “choking up” on the bat) http://en.wikipedia.org/wiki/File:Fourbats.jpg © 2013 Autodesk
- 13. When Do You Want a High Moment of Inertia? Flywheels – intended to smooth power variations in mechanical systems and store energy Tightrope walker’s sticks – to slow the rate of rotation as they tip back and forth http://en.wikipedia.org/wiki/File:Samuel_Dixon_Niagara.j pg https://en.wikipedia.org/wiki/File:Volin.jpg https://en.wikipedia.org/wiki/File:MassesDeMalabarisme.p ng © 2013 Autodesk Juggling clubs – prevents slight mistakes by the jugglers from greatly altering the spin of the clubs. NOTE: You can buy both “fast-spinning” and “slowspinning” clubs, i.e., ones with lower and higher moments of inertia.
- 14. How Does Moment of Inertia Relate to Energy? In F1 racing cars are now allowed to store energy from braking in flywheels in a “kinetic energy recovery system” (a.k.a. KERS). These flywheels allow the driver to later use the additional energy for a speed boost to overtake opponents. http://en.wikipedia.org/wiki/File:Flybrid_Systems_Kinetic_Energy_Recovery_System .jpg http://www.engadget.com/gallery/beacon-powersstephentown-ny-flywheel-plant/4187482/#!slide=590081 © 2013 Autodesk Some companies are developing large scale flywheels to help even out the power output from power plants. These flywheels obviate the need for additional power plants running as full-time backup (which is a huge waste of energy and a large contributor to CO2 emissions)
- 15. A Bike With a Flywheel (2:58 video) http://www.sciencefriday.com/video/08/12/2011/boost-your-bike.html http://www.gizmag.com/flywheel-bicycle-regenerative-braking/19532/picture/139996/ © 2013 Autodesk
- 16. Calculating Energy The energy, E, stored in a rotating object is related to its moment of inertia, I, and its angular velocity, w, by this equation: E = 1/2 * I * w2 The angular velocity, w, is the rate at which something rotates, in radians per second. © 2013 Autodesk
- 17. Angular velocity of Wheel An average bike rider can pedal around once every second (1 revolution per second). 1 revolution = 6.28 radians (= 2π) Average rider pedal speed = 6.28 radians / second NOTE: Radians are “dimensionless,” so w ≈ 6.28 / s EXAMPLE: For a “high gear” ratio of 44/11 = 4, every revolution of the pedal/front gear turns the rear gear/wheel four revolutions, so: w = 4 * (6.28 radians / second) ≈ 25 / s © 2013 Autodesk
- 18. Energy Stored in Bike Wheel Plug in the moment of inertia and angular velocity to find the energy stored in a bike wheel. Here is the start of the calculation: E = 1/2 * I * w2 = 0.5 * I * (25 / s)2 = I * (312.5 / sec2) INVENTOR NOTE: The units of moment of inertia are given in kg*mm2. To convert to kg*m2, multiply by 10-6 (= 0.000001): 1 m2 0.000001 m2 © 2013 Autodesk = 1,000,000 mm2 = 1 mm2 http://www.homeschoolmath.net/teaching/g/area/100-square-mm.gif
- 19. Let’s Explore This Using an Example Open Bike_Rim_For_Rotation.ipt Find the moment of inertia using iProperties for variations on the rim. Follow the instruction on the handout © 2013 Autodesk
- 20. Back To the Flywheel 6.8 kg flywheel from a Porche “The flywheel increases maximum acceleration and nets 10 percent pedal energy savings where speeds are between 20 and 24 kilometers per hour.” © 2013 Autodesk http://www.gizmag.com/flywheel-bicycle-regenerativebraking/19532/picture/139993/
- 21. Moment of Inertia of Flywheel The moment of inertia for the flywheel, modeled as a ring, is: I = 1/2 * M * (R12 + R22) If we assume R2= 5” = 0.127 m and R1= 4” = 0.102 m, with mass M = 6.8 kg, then for the flywheel: I © 2013 Autodesk = 0.5 * 6.8 kg * ((0.102m) 2 + (0.127m) 2 ) = 3.4kg * (0.0161 m2 + 0.0104 m2) = 0.0901 kg * m2 http://www.notechmagazine.com/2011/08/flywheel-bicycle.html
- 22. Energy and Angular Velocity of Flywheel Let’s say that the wheel has 32.5 Joules of energy. If we assume that all of that energy is transferred to the flywheel, we can calculate its angular velocity, w: E = 32.5 J = 1/2 * I * w2 = 0.5 * 0.0901 kg*m2 * w2 32.5 J 0.0451 kg*m2 = w2 26.8 radians / s = w (4.25 revolutions / s) © 2013 Autodesk
- 23. Summary Moment of inertia is determined by both the mass and shape of an object. Higher moment of inertia results from more mass further from the axis of rotation. The amount of energy a rotating object stores is determined by its moment of inertia (resistance to motion) and angular velocity. When a bike rider pedals, energy is transferred to the parts of the bike, including the front wheel. Some of that energy can be stored in a flywheel, and returned from the flywheel later. © 2013 Autodesk
- 24. Autodesk is a registered trademark of Autodesk, Inc., and/or its subsidiaries and/or affiliates in the USA and/or other countries. All other brand names, product names, or trademarks belong to their respective holders. Autodesk reserves the right to alter product and services offerings, and specifications and pricing at any time without notice, and is not responsible for typographical or graphical errors that may appear in this document. © 2013 Autodesk, Inc. All rights reserved.

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