Introducing
Prezi AI.
Your new presentation assistant.
Refine, enhance, and tailor your content, source relevant images, and edit visuals quicker than ever before.
Trending searches
Loading content…
Loading…
Linear Algebra Applications to mechanical Engineering
67
Transcript
Example Cont.
Example: Simple Vibration Problem
- 2 masses 'm' attached to eachother and attached by springs with spring constant 'k' to fixed outer walls.
How does this Apply to Mechanical Engineering?
Finding Natural Frequency of a Large Body
- Finding Natural Frequencies
- Detecting cracks / broken parts of a body.
- Detecting the deformation of a body or structure.
1. Set up a Free Body Diagram for the system.
2. Solve the equations for X, and rearrange them into a matrix form.
3. Find the eigenvalues and eigenvectors.
4. Frequency = sqrt(-eigenvalue)
5. Modes (location of oscillation) = eigenvectors
Natural Frequency
Why is This Important?
- Definition: the frequency at which an object naturally oscillates when acted on by an external force.
- For a bridge or large structure, this outer force could be something as arbitrary as wind or rain.
- The reason a human cannot sense these frequencies from larger bodies is because they are beyond the 'human frequency range' (the human ear cannot detect it.)
- Larger bodies will almost always have multiple natural frequencies.
- So... how do we calculate these frequencies??
What are Eigenvalues and Eigenvectors???
- In 1940, the Tacoma Narrows Bridge in Washington (3rd largest suspension bridge in the world at the time) collapsed.
- What knocked it over?? The wind.
- The frequency of the wind blowing onto the bridge matched the natural frequency of the bridge. This created destructive interference of the waves, and caused the 4 month old structure to collapse.
- Therefore, natural frequencies must be found in order to create proper insulation to ensure the safety of large structures.
- If A is a square matrix, a vector (not equal to zero) C is an eigenvector of A if and only if there exists a number (can be real or complex) lambda where:
- (A*C) = (lambda*C)
- If the number lambda exists, then it is an eigenvalue of matrix A.
Eigenvector
Additional Uses for Natural Frequency Detection
- Cracks and deformities can be detected in bodies by observing a change in the eigenvalues, or natural frequencies of the object.
- Using eigenvalues and eigenvectors, small yet important cracks and their corresponding locations can be found and repaired.
- This allows for safer infrastructure with greater longevity.
- An eigenvector can also be defined as a vector that doesn't change its direction under the associated linear transformation.
- The eigenvector is merely strectched in its original direction, by a factor of the eigenvalue.
- (The blue vector in the transformation above).
Linear Algebra Applications to mechanical Engineering
Choose a template
Presentations from around the world
Learn more about creating dynamic, engaging presentations with Prezi