For a single-degree-of-freedom system under steady-state harmonic excitation, if the frequency ratio r = ω/ωₙ is very high (r → ∞), to what limiting value does the phase angle of the response tend for all practical damping ratios?
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A0°
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B90°
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C180°
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D360°
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E−90°
Answer
Correct Answer: 180°
Explanation
Introduction / Context
The phase angle between the steady-state response and the harmonic excitation determines whether the system moves in-phase or out-of-phase with the applied force. Understanding its limiting values helps in mount design and resonance avoidance.
Given Data / Assumptions
- Linear viscous damping with damping ratio ζ (light to moderate).
- Base or force excitation analyzed via standard SDOF models.
- Frequency ratio r = ω/ωn ≫ 1.
Concept / Approach
For force-excited SDOF systems, the phase angle φ is given by tanφ = (2ζr) / (1 − r²). As r increases far above 1, the denominator becomes large negative, so φ → 180° (i.e., the response is almost completely out of phase with the excitation force).
Step-by-Step Solution
1) Use tanφ = (2ζr)/(1 − r²).2) For r → ∞, (1 − r²) → −∞ while numerator grows linearly; thus tanφ → 0⁻ from negative side.3) The angle corresponding to a very small negative tangent in the second quadrant is φ → 180° (π radians).Verification / Alternative check
Plotting φ versus r shows φ ≈ 0° for r ≪ 1, φ = 90° at resonance (r ≈ 1), and φ → 180° as r → ∞.
Why Other Options Are Wrong
- 0°: low-frequency limit (mass follows force).
- 90°: near resonance.
- 360°: equivalent to 0°; not the high-frequency limit.
- −90°: not the conventional steady-state limit for this model.
Common Pitfalls
- Confusing displacement vs. acceleration phase; acceleration can be closer to in-phase with force at high r, but the standard response phase for displacement tends to 180°.
Final Answer
180°