Frequency response terminology: the frequency at which a system attains its maximum amplitude ratio in sinusoidal steady-state is called the __________ frequency.

Introduction / Context:
In frequency response analysis, engineers characterize systems by how their output amplitude and phase vary with sinusoidal input frequency. Several named frequencies are used for design and interpretation. This question targets the correct name for the frequency where the amplitude ratio peaks.

Given Data / Assumptions:

• Linear time-invariant system under sinusoidal excitation.
• Amplitude ratio (gain) plotted versus frequency (Bode or magnitude plot).
• We seek the term for the frequency where the magnitude attains a maximum.

Concept / Approach:
The peak in amplitude ratio is associated with resonance. For lightly damped second-order systems, the resonant frequency ω_r is slightly less than the undamped natural frequency ω_n, with ω_r = ω_n * sqrt(1 − 2ζ^2) when ζ < 1/√2. The “corner” (break) frequency marks slope change in first-order elements; “cross-over” is where loop magnitude crosses unity; “natural” frequency is a system parameter, not necessarily where the observed magnitude peaks in damped systems.

Step-by-Step Solution:
Identify the definition: maximum magnitude → resonance.Associate with lightly damped behavior and peaking in magnitude plots.Select the term “resonant frequency.”

Verification / Alternative check:
Examine a standard second-order Bode magnitude plot for ζ = 0.2–0.5; the peak appears near ω_r < ω_n, matching “resonant frequency.”

Why Other Options Are Wrong:
Corner frequency: Pertains to first-order roll-off points.Cross-over frequency: Where loop gain magnitude equals 1 (design criterion), not peak.Natural frequency: Parameter of the undamped system; damping shifts the peak.Nyquist frequency: Sampling theory term (half the sampling rate).

Common Pitfalls:
Confusing ω_n with ω_r; assuming the peak always occurs exactly at ω_n regardless of damping.

Final Answer:
Resonant frequency