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Bike Wheels, Rotational Inertia, and EnergyAdam Kenvarg, Joel Rosenberg, and James RegulinskiAutodesk Sustainability Workshop 2013 AutodeskThis lesson plan should be useful to high school engineering/technology, physics, or math teachers.

The lesson aims to teach students the basic concept of moment of inertia, and how it relates to the energy of a spinning body, all in the context of a bicycle wheel. It is designed to be completed in one to two standard class sessions.

Within Inventor students will quickly alter a CAD model and take data that will be used in additional calculations about energy. A real-world application a flywheel bicycle is included to add authenticity.

More challenging math calculations are included.1Which of These is Easier to Spin?http://img.tradeindia.com/fp/1/001/012/187.jpghttp://www.wigglestatic.com/product-media/5360073219/shimano_WHR501R.jpg?w=1100&h=1100&a=7AluminumStainless Steel

2013 AutodeskHere are two bike wheels made from two different materials. What might make one easier to spin? (Mass, and distribution of mass.)

This lesson will explore the factors that affect their rotation.2What 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 AutodeskMoment of inertia is probably a new concept for most students.3What is Rotational Inertia? IISaid 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 straight-line velocity. 2013 AutodeskStudents should recognize that pushing something heavy (more mass) is harder than pushing something light. They can think of the greater mass resisting the change in motion.

For rotational systems, the moment of inertia plays that role.4Axis of rotationThe moment of inertia is almost always different for each different axis of rotation.

https://webspace.utexas.edu/cokerwr/www/index.html/RI.htmSolid cylinder (or disk) about central axisSolid cylinder (or disk) about central diameter

2013 AutodeskStudents might know the term axis from math class, as in the x-axis on a graph. Here, it refers to an imaginary line in space used as a reference around which the object rotates. IT MATTERS WHICH AXIS IS USED!

Point out the same cylinder, but different axis, in the two images, and how the equations are different. You might also point out that the length L doesnt show up in the moment of inertia equation for the central axis (left), but it does in the central diameter (right) equation.5

More Moment of Inertia Values 2013 AutodeskHere are equations for numerous other shapes, taken from: https://webspace.utexas.edu/cokerwr/www/index.html/RI.htm

The main idea is that we can model the behavior of an object by using a close approximation to its shape, and that the moment of inertia is different for different shapes and axes.6https://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.gifNote: the red sphere is a hollow shell

Which object will finish first?(All are equal mass and radius) 2013 AutodeskThe animation is on the next slide.

Before showing it, ask students to guess what order they think the objects will finish in, assuming they are all the same mass and radius.7Go!https://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.gif

Note: the red sphere is a hollow shell 2013 AutodeskThe animation should play automatically in presentation mode. If not, it can be viewed here (along with detailed math):https://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.ogv

The results are shown frozen on next slide.8Resultshttps://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.gifNote: the red sphere is a hollow shell

Which object has the highest moment of inertia?Which has the lowest? 2013 AutodeskThe order of finishing is:1) Orange solid sphere, 2) Blue disk, 3) Red hollow sphere, 4) Green ring.

ANSWERS: Highest moment of inertia? Green ring (lower).Lowest moment of inertia? Orange solid sphere (winner)9Moment of Inertia

Remember, the higher the moment of inertia of an object, the greater its resistance to a change in the rotational velocity. Thats 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 AutodeskThese are the takeaways for moment of inertia.10Classic Moment of Inertia DemoFigure 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 AutodeskThis is a demo that you could do in class. See http://www.exploratorium.edu/snacks/momentum_machine/ for more info.11When 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

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:SC06_2005_Porsche_Carrera_GT_wheel.jpg

http://en.wikipedia.org/wiki/File:Fourbats.jpg

2013 AutodeskHere are two examples of where someone might want a lower moment of inertia.12When Do You Want a High Moment of Inertia?Flywheels intended to smooth power variations in mechanical systems and store energy

Tightrope walkers sticks to slow the rate of rotation as they tip back and forth

Juggling clubs prevents slight mistakes by the jugglers from greatly altering the spin of the clubs. NOTE: You can buy both fast-spinning and slow-spinning clubs, i.e., ones with lower and higher moments of inertia.

http://en.wikipedia.org/wiki/File:Samuel_Dixon_Niagara.jpg

https://en.wikipedia.org/wiki/File:MassesDeMalabarisme.png

https://en.wikipedia.org/wiki/File:Volin.jpg 2013 AutodeskHere are three examples of where higher moment of inertia is useful.

The key idea is to get students to appreciate how resistance to change in motion can affect the purpose to which a design is put.13How 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.

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)http://en.wikipedia.org/wiki/File:Flybrid_Systems_Kinetic_Energy_Recovery_System.jpg

http://www.engadget.com/gallery/beacon-powers-stephentown-ny-flywheel-plant/4187482/#!slide=590081

2013 AutodeskAs energy storage devices, flywheels are massive objects that have high moments of inertia. The higher the moment, and the faster it spins, the more energy it stores.14A Bike With a Flywheel (2:58 video)http://www.gizmag.com/flywheel-bicycle-regenerative-braking/19532/picture/139996/http://www.sciencefriday.com/video/08/12/2011/boost-your-bike.html

2013 AutodeskClick the link to watch a 2:58 video of a college senior who built a bike with a flywheel for recovering energy from the back wheel.

NOTES: Video can be downloaded from that site. And here is a version with Spanish subtitles: http://www.sciencefriday.com/video/08/18/2011/una-bicicleta-con-volante-de-inercia.html15Calculating EnergyThe 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 AutodeskThis is the start of the more physics/math heavy section.16Angular velocity of WheelAn average bike rider can pedal around once every second (1 revolution per second).

1 revolution = 6.28 radians (= 2)

Average rider pedal speed = 6.28 radians / secondNOTE: 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 AutodeskWe assume a bike rider pedals about 60 times a minute, or once per second.

MATH NOTE: Students might know radians from geometry class as a measure of angle. One full circle, 360, is equivalent to 2 radians. Since 2 is around 6.28, one revolution = 6.28 radians. Radians are dimensionless, so it is just a number (e.g. no equivalent to meters). More info: https://en.wikipedia.org/wiki/Radian

SECOND MATH NOTE: The word per, which is often taken to mean divide, can also be interpreted as for every, e.g. 25 radians every second.17Energy Stored in Bike WheelPlug 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 = 1,000,000 mm2

0.000001 m2= 1 mm2

http://www.homeschoolmath.net/teaching/g/area/100-square-mm.gif 2013 AutodeskThis is math is fairly straightforward, and sets up the equation for the worksheet.

INVENTOR NOTE: The moments of inertia from Inventor are given in kg*mm2. To convert to m2, multiply by 0.000001 (10-6).MATH INTERPRETATION: One square millimeter equals one-millionth (0.000001) of a square meter. The image shows a square centimeter, and the relationship to square millimeters. Just picture a million instead of 100 to see the relationship.18Open Bike_Rim_For_Rotation.ipt

Find the moment of inertia using iProperties for variations on the rim.

Follow the instruction on the handoutLets Explore This Using an Example

2013 AutodeskBreak for Inventor sessions.19

Back To the Flywheel6.8 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.http://www.gizmag.com/flywheel-bicycle-regenerative-braking/19532/picture/139993/ 2013 AutodeskThis is another extension activity that can be done to practice some algebra calculations, and build some intuition about flywheel design.

The flywheel in the video is 6.8 kg. The quote is from a Scientific American article, just to give an idea of how popular science articles discuss these kinds of things.

Scientific American story: http://blogs.scientificamerican.com/observations/2011/06/24/a-bike-that-uses-its-brakes-for-a-speed-boost-and-other-student-engineer-inventions-video20Moment of Inertia of FlywheelThe 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 = 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 2013 AutodeskWe can estimate the moment of inertia for the flywheel by modeling it as a ring, and calculate it with the given mass (6.8 kg) and an estimate of its dimensions (10 diameter, which is about 1/3 of the diameter of a 27-inch bike wheel, based on a guess from the picture).21Energy and Angular Velocity of FlywheelLets 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 =w2 0.0451 kg*m2

26.8 radians / s= w(4.25 revolutions / s)

2013 AutodeskThe 32.5 J estimate is based on the Full Bicycle Wheel setup analysis. Here we are solving for angular velocity, to get an estimate of how fast the flywheel spins as the wheel stops spinning, assuming all of the energy goes to the flywheel.22SummaryMoment 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 AutodeskThe main ideas here are that energy can be stored and transferred, and in some cases reused. And in the case of something spinning, its amount of stored energy depends on its rotational speed, mass, and shape/mass distribution.23Autodesk 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.