14.4.3. Orthogonality Conditions

Try these first

1. 1.

   What does it mean for two mode shapes to be orthogonal?

   Hint

   Think about the geometric interpretation of orthogonality.

2. 2.

   Can you name the two matrices that are crucial for defining orthogonality in mode shapes?

   Hint

   These matrices play a role in structural dynamics.

3. 3.

   What does the inner product of two orthogonal mode shapes equal?

   • Equals one
   • Equals zero
   • Equals two

   Hint

   Visualize the geometric representation.

4. 4.

   True or False: Orthogonality conditions imply that different mode shapes influence each other's responses.

   • True
   • False

   Hint

   Think about their independence in modal analysis.

5. 5.

   A three-degree freedom system has the following mode shapes: ϕ_1, ϕ_2, and ϕ_3. Calculate the inner products to confirm orthogonality among them given specific values for the mass matrix.

   Hint

   Make sure to accurately multiply the mode shape vectors with the mass matrix, following the orthogonality condition.

6. 6.

   Explain how the lack of orthogonality in a system with multiple modes can lead to structural failure under seismic loads.

   Hint

   Think about how combined effects could amplify the response.