14.4.3 - Orthogonality Conditions
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Practice Questions
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What does it mean for two mode shapes to be orthogonal?
💡 Hint: Think about the geometric interpretation of orthogonality.
Can you name the two matrices that are crucial for defining orthogonality in mode shapes?
💡 Hint: These matrices play a role in structural dynamics.
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Interactive Quizzes
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What does the inner product of two orthogonal mode shapes equal?
💡 Hint: Visualize the geometric representation.
True or False: Orthogonality conditions imply that different mode shapes influence each other's responses.
💡 Hint: Think about their independence in modal analysis.
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Challenge Problems
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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.
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.
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