Structural Dynamics MCQ with answer and solution

Concept

Phase lag at resonance in forced vibrations

Formula

NONE

Step-by-step solution

1. The phase angle formula for steady-state forced vibration is tan(φ) = (2 × ζ × r) / (1 - r²), where r is frequency ratio. 2. At resonance, the excitation frequency equals natural frequency, making the frequency ratio r = 1. 3. Substituting r = 1 makes the denominator zero, giving tan(φ) = infinity, which corresponds to a phase lag of 90°. Answer: C

Explanation

For a harmonically forced single-degree-of-freedom system, phase lag describes the time delay between the exciting force and the structural displacement. At resonance, when the forcing frequency matches the system's natural frequency, this phase lag reaches exactly 90 degrees. This indicates that the peak displacement occurs one-quarter of a cycle after the peak force is applied.

Why the other options are wrong

A) A 0° phase lag occurs at very low excitation frequencies well below resonance. B) A 45° phase lag occurs at specific frequencies depending on damping, but not at resonance. D) A 180° phase lag occurs at excitation frequencies much higher than the natural frequency.

Shortcut

Phase lag at resonance (frequency ratio = 1) is always 90°.

Common mistake

Assuming the phase lag at resonance is 0 degrees or 180 degrees instead of 90 degrees.