Applications

Applications of eigen values and eigen vectors

            Eigenvalues and eigenvectors play a crucial role in a wide range of applications across engineering and science. Fields like control theory, vibration analysis, electric circuits, advanced dynamics, and quantum mechanics frequently rely on these concepts.

One key application involves transforming matrices into diagonal form, a process that simplifies complex calculations. 

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Practical Applications

(i)                 Most structures from buildings to bridges have a natural frequency of vibration. It means all these structures have their own system of eigen vibrations and eigen frequencies. Now external forces like wind and earthquake may cause these structures to undergo vibrations. In case the frequency of these vibrations becomes equal to the natural frequencies of these structures, vibrations with large amplitudes are set up. So vibration analysis of these structures is done at the time of designing using eigen values and eigen vectors.


                         Tacoma bridge was built in 1940. The bridge would form small waves like the surface of water. Many people wanted to drive over this moving bridge.In 4 months oscillations become bigger.

 At one point one edge of road was 28 feet higher than other edge. The bridge crashed into water below. Oscillations were caused by frequency of wind being too close to natural frequency of bridge. The natural frequency of the bridge is the eigen value of smallest magnitude of a system that models the bridge. So eigen values are important to engineers to analyse structures.

 

 

(ii)               Eigen values can also be used to test for cracks or deformities in structural components used for construction. When a beam is struck, its natural frequencies (eigen values) can be heard or measured. If the beam "rings," then it is not flawed. A dull sound will result from a flawed beam because the flaw causes the eigen values to change.

          

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