15.3. Properties of Mode Shapes

Try these first

1. 1.

   What does orthogonality in mode shapes indicate?

   Hint

   Think about how angles relate to each other.

2. 2.

   Name one reason for normalizing mode shapes.

   Hint

   How can we measure contributions fairly?

3. 3.

   What is the orthogonality condition for mode shapes with respect to the mass matrix?

   • {ϕ}T[M]{ϕ} = 1 for i = j
   • {ϕ}T[M]{ϕ} = 0 for i ≠ j
   • {ϕ}T[M]{ϕ} = ω²

   Hint

   Think about how interactions could complicate motion.

4. 4.

   True or False: Normalization makes mode shapes unique in magnitude.

   • True
   • False

   Hint

   Consider if different modes could still exist in varying sizes.

5. 5.

   A three-story shear building shows interference between the first two mode shapes during an earthquake simulation. Discuss the potential implications and propose adjustments.

   Hint

   Reflect on the significance of independent modes.

6. 6.

   Consider a structure with two normalized mode shapes. If one mode shape contributes significantly more than the other during an analysis, how would you interpret this in terms of design?

   Hint

   Think about how mode shapes influence stress on materials.