Second-order underdamped step response: identify the single correct statement about how key response features depend on the damping ratio ζ.

Introduction / Context:
Second-order systems (mechanical mass–spring–damper, RLC circuits, many control loops) exhibit characteristic step responses governed by two parameters: natural frequency and damping ratio ζ. Understanding how overshoot, oscillation, and settling depend on ζ is essential for tuning and stability assessment.

Given Data / Assumptions:

• Underdamped regime: 0 < ζ < 1.
• Standard unit-step input and canonical second-order form.
• Common definitions: overshoot, decay ratio, rise/settling times.

Concept / Approach:
As damping decreases (ζ becomes smaller), energy dissipates more slowly, leading to larger oscillations around the final value. Consequently, percent overshoot increases as ζ decreases. The damped oscillation frequency is wd = wn * sqrt(1 − ζ^2), which is lower than the undamped natural frequency wn. Decay ratio relates successive peak amplitudes and is not the reciprocal of percent overshoot. “Response time” is not a standard term for the first crossing; “rise time” and “settling time” are the accepted metrics.

Step-by-Step Solution:
Recall overshoot dependence: smaller ζ ⇒ larger overshoot.Note oscillatory nature for 0 < ζ < 1 (non-monotonic).Remember wd = wn * sqrt(1 − ζ^2) < wn.Evaluate each statement vs. these facts and select the one that aligns.

Verification / Alternative check:
Bode/Nyquist interpretations also show more resonant peaking as damping decreases, consistent with larger time-domain overshoot.

Why Other Options Are Wrong:
a: The standard term is “rise time” for first reach/cross; “response time” is ambiguous.c: Decay ratio is the ratio of successive peaks, not 1/(percent overshoot).d: Underdamped implies oscillations; monotonic approach occurs for ζ ≥ 1 (critically/overdamped).e: Damped frequency wd is less than wn unless ζ = 0.

Common Pitfalls:
Confusing rise time with settling time; assuming higher damping always improves speed (it reduces overshoot but may lengthen rise time). Using percent overshoot without specifying its definition relative to final value can also cause inconsistency.

Final Answer:
Percent overshoot increases as the damping coefficient (i.e., damping ratio ζ) decreases.