17.9. Eigenvalues and Modes of Vibration

Try these first

1. 1.

   What is an eigenvalue?

   Hint

   Think about linear transformations.

2. 2.

   Define natural mode.

   Hint

   Relate it to oscillations in a string.

3. 3.

   What is the relationship between eigenvalues and natural frequency?

   • They are unrelated concepts.
   • Each eigenvalue corresponds to a specific natural frequency.
   • Eigenvalues are determined without considering natural frequencies.

   Hint

   Think about the mathematical representation.

4. 4.

   True or false: Resonance occurs when operational frequencies match natural frequencies.

   • True
   • False

   Hint

   Consider the implications of this in engineering.

5. 5.

   Given a fixed beam of length L, calculate its first three eigenvalues and corresponding natural frequencies.

   Hint

   Use the formula for eigenvalues and substitute n = 1, 2, 3.

6. 6.

   Discuss the implications of resonance in the design of tall buildings under seismic activity.

   Hint

   Consider examples from engineering practices.