When is a system considered critically damped?

• A. When the characteristic equation has two distinct real roots
• B. When there are repeated real roots
• C. When imaginary roots occur
• D. When the response oscillates infinitely

Answer: (B)

A system is considered critically damped when it has repeated real roots in its characteristic equation. This condition arises in second-order linear systems when the damping ratio is exactly equal to one. In this scenario, the system returns to equilibrium as quickly as possible without oscillating.

In a critically damped system, the response does not overshoot the equilibrium position, which is crucial for applications requiring rapid stabilization. This is different from an underdamped system, where the roots would be complex and the system would exhibit oscillatory behavior. Conversely, an overdamped system has two distinct real roots, leading to slow convergence to equilibrium without oscillation.

Thus, the key defining characteristic of critical damping is the presence of repeated real roots, allowing for the most efficient return to the desired state without oscillation, thereby preventing any overshoot.