12.4. Orthogonality of Mode Shapes

Try these first

1. 1.

   What does it mean for mode shapes to be normalized?

   Hint

   Think about why scaling matters in analysis.

2. 2.

   State the mass-orthogonality condition.

   Hint

   This should involve vectors and matrices.

3. 3.

   What is the mass orthogonality condition?

   • The dot product of two modes equals one.
   • The dot product with the mass matrix equals zero for different modes.
   • Mass shapes depend on frequency.

   Hint

   Think about how energy is distributed across modes.

4. 4.

   True or False: Stiffness orthogonality means that modes do not affect each other in terms of stiffness.

   • True
   • False

   Hint

   Recall how modes interact with stiffness matrices.

5. 5.

   Evaluate a 3-DOF system's eigenvalues and establish the orthogonality criteria across three mode shapes. Describe the implications of finding zero products in each case.

   Hint

   Follow the standard procedure for eigenvalue problems in dynamic systems.

6. 6.

   How would you apply the principle of orthogonality to a complex building structure with many floors and irregular shapes? Discuss the implications.

   Hint

   Consider modeling methods that normalize and decouple systems effectively.