16.4.1. Eigenvalue Problem

Try these first

1. 1.

   Define an eigenvalue in the context of MDOF systems.

   Hint

   Think about how frequency relates to vibration.

2. 2.

   What do mode shapes represent?

   Hint

   Consider how an instrument would vibrate.

3. 3.

   What are eigenvalues in the context of vibrations?

   • Strings of a connected mass
   • Squared natural frequencies
   • Types of mode shapes

   Hint

   Consider how they affect the motion of objects.

4. 4.

   True or False: Mode shapes can be coupled in an MDOF system.

   • True
   • False

   Hint

   Think about the term 'decoupled'.

5. 5.

   Given the mass and stiffness matrices for a 2-DOF system, find the natural frequencies and mode shapes using the eigenvalue technique.

   Hint

   Breaking down the matrix determinant is key here.

6. 6.

   Analyze a structural model subjected to harmonics, using the eigenvalue approach to identify stability under dynamic loading.

   Hint

   Relate the natural frequencies to applied harmonic frequencies for insights on stability.