14.3.2. Eigenvalue Problem

Try these first

1. 1.

   Define the term 'eigenvalue' in the context of structural dynamics.

   Hint

   Think about how this relates to how structures vibrate.

2. 2.

   What does the stiffness matrix represent?

   Hint

   Focus on the mechanical properties of materials.

3. 3.

   What does the matrix equation [K - ω²M]ϕ = 0 represent?

   • A description of static equilibrium
   • An eigenvalue problem
   • A method for calculating forces

   Hint

   Consider the type of analysis we are discussing.

4. 4.

   True or False: Eigenvectors provide information about how a structure vibrates at a specific frequency.

   • True
   • False

   Hint

   Think about how structures respond dynamically.

5. 5.

   Given a simple two-degree of freedom mass-spring system, derive the eigenvalue equation and find the natural frequencies if the stiffness matrix is K and mass matrix is M.

   Hint

   Consider the dimensional consistency of the derived matrices during transformation.

6. 6.

   Analyze a case where a structure experiences resonance due to a specific external forcing frequency. What mitigation strategies could be employed using eigenvalue analysis?

   Hint

   Think about how to avoid matching frequencies.