A second-order system has ωₙ = 10 rad/s and ζ = 0.6. A step input is applied. At what frequency does the step response oscillate?
A10 rad/s — that is the natural frequency of the system
B6 rad/s — the product ζ × ωₙ gives the oscillation frequency
C8 rad/s — the damped natural frequency ωd = ωₙ√(1 − ζ²) = 10√(1 − 0.36) ≈ 8 rad/s
DThe response does not oscillate because ζ = 0.6 is nearly critically damped
The natural frequency ωₙ is the frequency of oscillation only when there is *no damping* (ζ = 0). In an underdamped system (0 < ζ < 1), damping slows the oscillation: the actual frequency is the damped natural frequency ωd = ωₙ√(1−ζ²). For ζ = 0.6: ωd = 10√(1−0.36) = 10√0.64 = 10 × 0.8 = 8 rad/s. Confusing ωₙ with ωd is a frequent error — ωₙ appears in the transfer function and performance formulas, but ωd is what you observe in the actual oscillation. Note that ζ = 0.6 is well underdamped and will oscillate; critical damping (ζ = 1) would produce no oscillation.
Question 2 Multiple Choice
A control system specification requires settling within 2 seconds and less than 5% overshoot. A designer is choosing between ζ = 1.0 (critically damped) and ζ = 0.707 (underdamped). Which choice is likely better?
Aζ = 1.0 is always the best choice — it never overshoots and thus guarantees the <5% overshoot specification without any risk
Bζ = 0.707 is likely better — it produces only ~4.3% overshoot (within spec) and typically settles faster than the critically damped case at the same ωₙ, meaning it can meet both specs simultaneously
Cζ = 1.0 settles faster than ζ = 0.707 because it doesn't waste time in oscillation, so it better satisfies the settling time requirement
Dζ = 0.707 cannot be used because any underdamped system violates an overshoot specification by definition
The common misconception is that critically damped = best. ζ = 1.0 achieves the absolute minimum overshoot (zero), but it approaches the final value asymptotically and can actually have a *longer* settling time than a lightly underdamped system. ζ ≈ 0.707 produces only ~4.3% overshoot — within the <5% spec — while the response reaches and stays within the settling band faster. The settling time formula Ts ≈ 4/(ζωₙ) shows that at the same ωₙ, ζ = 0.707 settles faster than ζ = 1.0. ζ ≈ 0.707 is the near-optimal tradeoff point for many practical specifications.
Question 3 True / False
The natural frequency ωₙ is the frequency at which an underdamped second-order system actually oscillates in its step response.
TTrue
FFalse
Answer: False
ωₙ is the *undamped* natural frequency — the oscillation rate only when ζ = 0 (no damping). In any real underdamped system (0 < ζ < 1), the actual oscillation frequency is the damped natural frequency ωd = ωₙ√(1−ζ²), which is always less than ωₙ. For example, at ζ = 0.5, ωd = ωₙ√(1−0.25) = 0.866ωₙ — about 13% slower than ωₙ. At ζ = 0.9, ωd = ωₙ√(1−0.81) ≈ 0.436ωₙ — dramatically slower. Only as ζ → 0 does ωd → ωₙ. The peak time formula Tₚ = π/ωd correctly uses ωd for this reason.
Question 4 True / False
For a second-order system, increasing the damping ratio ζ while holding the natural frequency ωₙ constant will reduce overshoot but will generally increase the settling time.
TTrue
FFalse
Answer: True
Both settling time and overshoot are functions of ζ and ωₙ, but they trade off against each other as ζ changes at fixed ωₙ. Overshoot %OS = e^(−πζ/√(1−ζ²)) × 100 strictly decreases as ζ increases — more damping means less overshoot. Settling time Ts ≈ 4/(ζωₙ) at first decreases as ζ increases from 0 (more damping helps), but for ζ > 1 the system is overdamped and settles sluggishly — the formula no longer strictly applies, but the qualitative effect is that very high ζ is slow. Near ζ ≈ 0.707, settling time is near its minimum for a given ωₙ, which is why this value represents a practical optimum balancing both specifications.
Question 5 Short Answer
Explain why ζ ≈ 0.707 is often preferred over ζ = 1.0 (critically damped) in practical control system design, even though critical damping guarantees zero overshoot.
Think about your answer, then reveal below.
Model answer: Critically damped (ζ = 1.0) approaches the final value without overshooting, but does so asymptotically and slowly for the settling band. ζ ≈ 0.707 overshoots by only ~4.3% — within most engineering tolerances — but reaches and stays within the ±2% settling band faster at the same ωₙ. It represents the minimum settling time near-optimal tradeoff: fast response, acceptably small overshoot, and clean settling.
The numerical comparison: for a system with ωₙ = 10 rad/s, the 2% settling time at ζ = 1.0 is roughly Ts ≈ 6/ωₙ = 0.6 s (using the exact formula for critically damped), while at ζ = 0.707 it is Ts ≈ 4/(ζωₙ) = 4/(0.707×10) ≈ 0.57 s — faster. In the s-plane, ζ = 0.707 corresponds to poles at 45° from the negative real axis, which is the geometric sweet spot between poles that are too close to the imaginary axis (oscillatory) and poles that are too close to the real axis (slow). This value appears so often in control design that experienced engineers recognize it immediately.