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?

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/ω_n ≫ 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

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°