Structural Dynamics MCQ with answer and solution

Concept

In structural dynamics, a mode shape defines the relative pattern of motion during free vibration. The absolute amplitude of an eigenvector is arbitrary and not fixed until normalized.

Formula

([K] - ω² [M]) {φ} = {0}

Step-by-step solution

1. Free vibration gives the mode shape (eigenvector), which shows the relative displacement of different masses. 2. The ratio of displacement between different points is unique and fixed for each mode. 3. However, the absolute magnitude or amplitude of the eigenvector can be multiplied by any non-zero constant and still satisfy the equation. 4. Therefore, the absolute amplitude of the eigenvector remains undefined (or arbitrary). Answer: (D) Undefined

Explanation

Mode shapes give the relative ratios of displacements at different points. The exact value of amplitude is not fixed by the free vibration equation. Thus, the absolute amplitude of eigenvectors is undefined unless normalized.

Why the other options are wrong

A: The shape is unique, but the absolute amplitude can take any value. B: The amplitude is 1.0 only after specific mass or unity normalization, not by default. C: The amplitude cannot be defined without extra normalization rules.

Shortcut

Remember: Mode shapes give relative ratios, not exact values. Exact amplitude is always undefined/arbitrary.

Common mistake

Students often confuse the unique shape of a mode with a unique amplitude value.