A second-order system has damping ratio 0.5 and natural frequency 4 rad/s, then1.The system is underdamped and exhibits oscillatory response.2.The damped natural frequency is less than the natural frequency.3.The peak overshoot is zero for this system.4.The settling time decreases as damping ratio increases (within underdamped region).Which of the above statements is/are correct?

Correct Answer: C - 1, 2 and 4 only

Quick Concept Explanation

Second-order dynamics are governed by two independent energy-storage states and are described by natural frequency and damping ratio. With ζ=0.5 and ωn=4 rad/s, ωd=ωn√(1−ζ²)≈3.46 rad/s, so the response is underdamped and has nonzero overshoot. Key relation: Characteristic form: s²+2ζω_n s+ω_n²=0.

Exam focus: ζ<1 underdamped, ζ=1 critical, ζ>1 overdamped. Common trap: Overdamped responses do not sustain oscillations.

Statement-wise Verification

• Statement 1 - Correct.
  The system is underdamped and exhibits oscillatory response.
  An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response.
• Statement 2 - Correct.
  The damped natural frequency is less than the natural frequency.
  Second-order dynamics are governed by two independent energy-storage states and are described by natural frequency and damping ratio.
• Statement 3 - Incorrect.
  The peak overshoot is zero for this system.
  For ζ=0.5 the system is underdamped and has nonzero overshoot
• Statement 4 - Correct.
  The settling time decreases as damping ratio increases (within underdamped region).
  Second-order dynamics are governed by two independent energy-storage states and are described by natural frequency and damping ratio.

Core Concept

Second-order dynamics are governed by two independent energy-storage states and are described by natural frequency and damping ratio. An underdamped response oscillates with exponentially decaying amplitude, critical damping gives the fastest non-oscillatory response, and overdamping gives a slower non-oscillatory response.

Formula / Key Relationship

Characteristic form: s²+2ζω_n s+ω_n²=0.

Step-by-Step Check

With ζ=0.5 and ωn=4 rad/s, ωd=ωn√(1−ζ²)≈3.46 rad/s, so the response is underdamped and has nonzero overshoot.

Why the Other Options Are Wrong

Option A is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 1, 4. Option B is incorrect because it includes incorrect statement(s) 3 and omits correct statement(s) 2, 4. Option D is incorrect because it includes incorrect statement(s) 3.

Exam Shortcut / Approach

ζ<1 underdamped, ζ=1 critical, ζ>1 overdamped.

Common Mistake

Overdamped responses do not sustain oscillations.

Quick Revision

Damped Frequency Overshoot and Settling is linked with Second-Order System, Time Response and Control Systems. ζ<1 underdamped, ζ=1 critical, ζ>1 overdamped.