12.3. Natural Frequencies and Mode Shapes

Try these first

1. 1.

   What is a natural frequency?

   Hint

   Think about the default rhythm of a system.

2. 2.

   Define mode shape.

   Hint

   Consider it as how different parts of a system move together.

3. 3.

   What is the eigenvalue problem formed from the motion equation?

   • K - ω²M = 0
   • det(K - ω²M) = 0
   • M - K = 0

   Hint

   Look for the determinant condition.

4. 4.

   Natural frequencies directly relate to which concept?

   • True
   • False

   Hint

   Think about how vibrations are influenced.

5. 5.

   Consider a 2-DOF system defined with masses M1 and M2, find the natural frequencies if the stiffness matrix is given as K = [[30000, -10000], [-10000, 30000]].

   Hint

   Start by calculating the determinant for potential ω values.

6. 6.

   Discuss how changing the stiffness of one mass affects the mode shapes of this 2-DOF system.

   Hint

   Consider how frequency shifts impact vibration patterns.