16.4. Free Vibration of Undamped MDOF Systems

Try these first

1. 1.

   What is natural frequency?

   Hint

   Think about how a swing oscillates.

2. 2.

   Define mode shape.

   Hint

   Consider how various points of a structure move together.

3. 3.

   What does ω² represent in the context of MDOF systems?

   • Damping Ratio
   • Eigenvalues
   • Stiffness

   Hint

   Think about the vibrations of the system.

4. 4.

   Mode shapes are orthogonal to mass and stiffness matrices. True or False?

   • True
   • False

   Hint

   What does orthogonality imply about two vectors?

5. 5.

   Given a system with a known mass matrix $M = [[1, 0], [0, 2]]$ and stiffness matrix $K = [[10, -5], [-5, 5]]$, compute the eigenvalues and eigenvectors.

   Hint

   Solve the characteristic equation.

6. 6.

   Discuss how the eigenvalue analysis of an MDOF system with two degrees of freedom differs from that of a single-degree system.

   Hint

   Consider the differences in analysis approaches.