Moment of Inertia Now that in class we've learned about Eigenvalues and Eigenvectors how can these be applied to Moment of Inertia? Why is moment of Inertia Important Output Process By using eigenvalues and eigenvectors, moment of inertia can be then calculated for rigid bodies. In order to calculate how much force is needed to accelerate or decelerate a machine, knowing the moment of inertia is necessary. For example when making a centrifuge, you want to be able to control the speed at which your rotor moves. Knowing the moment of inertia is helpful in calculating how much force the motor has to apply in order to keep it at a certain speed. Also, at high speeds a centrifuge has to be highly balanced so when designing the rotor, the center of mass has to be taken into account. This is just one of the many applications where moment of inertia can be used. Many machines use rotational motion Applications of Eigenvalues and Eigenvectors in Mechanical Engineering Carlos A. Diaz-Ruiz Rotating bodies depend on the moment of inertia because it defines their resistance to a change in angular velocity. It's the rotational equivalent of mass Image from: http://upload.wikimedia.org/wikipedia/commons/b/b4/Moment_of_inertia_rod_center.svg Depends on mass

distribution Also depends on location and orientation of rotation axis 1. Washer and Drier

2. Centrifuges

3. Wheels

4. Propellers The principal axes of a rigid body can be defined by the eigenvectors of the moment of inertia. Therefore, using eigenvalues is helpful in the calculation of moment of inertia Pictures taken from http://homepages.wmich.edu/~kamman/Me659PrincipalMomentsDirectionsInertia.pdf Works Cited http://homepages.wmich.edu/~kamman/Me659PrincipalMomentsDirectionsInertia.pdf

Even though Wikipedia was not used in direct citation, it was used in the initial steps of brainstorming possible applications:

http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

http://en.wikipedia.org/wiki/Moment_of_inertia https://encrypted-tbn3.gstatic.com/images?q=tbn:ANd9GcQKzXfKldJFx7KamQ4bd2L2xlhOI8QdsNGcnOg7kK8upcOIfNCVuw https://encrypted-tbn2.gstatic.com/images?q=tbn:ANd9GcT65W7GRmhLbgYTB-Ry4EzK16WWBYJCDyQB-HoZk4N5fMQFo0hzjw https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTwYXJPAcELzrZVUwpK_RZciEuIqkCk7uensm39NXDLR_BOPpKSIw https://encrypted-tbn1.gstatic.com/images?q=tbn:ANd9GcQjH7-OH2mQADE6sw9orb25V3JVP-BzRpHQ0vCVQVesJRk8qL8w https://encrypted-tbn3.gstatic.com/images?q=tbn:ANd9GcT9DLEIeOxbL24MgXWFgzkK2NncLXfXXoVdOSI8FIO6eoQESJ8x http://i.istockimg.com/file_thumbview_approve/8017392/2/stock-photo-8017392-wrench-and-bolt.jpg

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# Engineering Applications Eigenvalues

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