Which damping case provides the fastest non-oscillatory approach to steady state?

• A. Underdamped
• B. Overdamped
• C. Critically Damped
• D. Undamped

Answer: (C)

The fastest non-oscillatory approach to steady state occurs when the system is critically damped. In a typical second-order model, the transient behavior depends on the damping ratio. If damping is less than critical, the system is underdamped and exhibits oscillations before settling. If it is greater than critical, the system is overdamped and returns to steady state without oscillating, but more slowly because there are two real decay modes and the slower one dominates. At the exact point of critical damping, the two modes merge into a single repeated real root, giving a response of the form (A + B t) e^{-ω_n t}. This returns to the final value without any overshoot and does so as quickly as possible compared with the overdamped cases. Hence, critically damped yields the fastest non-oscillatory approach to steady state.