rotational inertia

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Bike Wheels, Rotational Inertia, and EnergyAdam Kenvarg, Joel Rosenberg, and James Regulinski

Autodesk Sustainability Workshop

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Which of These is Easier to Spin?

http://img.tradeindia.com/fp/1/001/012/187.jpg http://www.wigglestatic.com/product-media/5360073219/shimano_WHR501R.jpg?w=1100&h=1100&a=7

Aluminum Stainless Steel

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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.

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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 straight-line velocity.

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Axis of rotation

The moment of inertia is almost always different for each different axis of rotation.

https://webspace.utexas.edu/cokerwr/www/index.html/RI.htm

Solid cylinder (or disk) about central axis

Solid cylinder (or disk) about central diameter

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More Moment of Inertia Values

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https://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.gif

Note: the red sphere is a hollow shell

Which object will finish first?(All are equal mass and radius)

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Go!



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Results



Which object has the highest moment of inertia?

Which has the lowest?

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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.

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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/

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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

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

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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

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

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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.

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

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A 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



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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.

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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

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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 = 1,000,000 mm2

0.000001 m2 = 1 mm2

http://www.homeschoolmath.net/teaching/g/area/100-square-mm.gif

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Open Bike_Rim_For_Rotation.ipt Find the moment of inertia using

iProperties for variations on the rim. Follow the instruction on the handout

Let’s Explore This Using an Example

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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.”

http://www.gizmag.com/flywheel-bicycle-regenerative-braking/19532/picture/139993/

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Moment of Inertia of Flywheel

The moment of inertia for the flywheel, modeled as a ring, is:

I = 1/2 * M * (R12 + R2

2)

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

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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 = w2

0.0451 kg*m2

26.8 radians / s = w

(4.25 revolutions / s)

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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.

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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.

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