Damping ratio and system classification: when the damping coefficient (ξ) equals 1, how is the second-order system classified and what does it imply about transient response?

Introduction / Context:
The damping ratio ξ is a key parameter for second-order dynamics. It dictates how the system responds to a step input—whether it overshoots, oscillates, or returns smoothly to steady state. Correct classification is fundamental to tuning controllers and predicting transient behavior.

Given Data / Assumptions:

• Standard second-order model with natural frequency ω_n and damping ratio ξ.
• Input: step change; Output: typical process variable.
• Question case: ξ = 1.

Concept / Approach:
For a canonical second-order system: 0 < ξ < 1 → underdamped (oscillatory); ξ = 1 → critically damped (fastest non-oscillatory return, no overshoot); ξ > 1 → overdamped (no overshoot, slower). Thus, ξ = 1 denotes the critically damped condition that achieves the quickest possible settling without overshoot.

Step-by-Step Solution:
Identify ξ = 1 boundary between underdamped and overdamped.Recall transient traits: no oscillation, no overshoot, fastest aperiodic response.Select classification: critically damped.

Verification / Alternative check:
Compare step responses: ξ = 0.7 shows overshoot; ξ = 1 shows monotonic rise to setpoint; ξ = 2 is slower with larger rise time and no overshoot. Plots confirm the qualitative behavior.

Why Other Options Are Wrong:
Overdamped: Requires ξ > 1 and yields slower response than critical.Underdamped: Requires 0 < ξ < 1 and shows oscillation/overshoot.Highly fluctuating/sustained: Implies oscillatory or unstable dynamics, not ξ = 1.Aperiodic unstable: Not applicable; ξ relates to stable, real or complex-conjugate pole patterns.

Common Pitfalls:
Confusing ξ = 1 with ξ = 0 (undamped) or thinking critical damping is “slow”; in reality it is the fastest non-overshooting case.

Final Answer:
Critically damped