Second-order systems – Match characteristic equations to damping type List I (Characteristic equation) A. s^2 + 15 s + 56.25 B. s^2 + 5 s + 6 C. s^2 + 20.25 D. s^2 + 4.55 s + 42.25 List II (Damping classification) Undamped Underdamped Critically damped Overdamped Choose the correct mapping.

Introduction / Context:
Damping classification follows from the roots of the characteristic equation. The sign and discriminant of the quadratic determine whether the system is underdamped, critically damped, overdamped, or undamped (pure oscillation).

Given Data / Assumptions:

• Standard second-order form s^2 + 2ζω_n s + ω_n^2 for positive parameters.
• Real coefficients and physical systems (stable if coefficients are positive).

Concept / Approach:

Use the discriminant Δ = b^2 − 4ac for s^2 + bs + c. If Δ > 0 → two distinct real roots (overdamped). If Δ = 0 → repeated real root (critical). If Δ < 0 → complex conjugate (underdamped). If b = 0 and c > 0 → purely imaginary roots (undamped sinusoid).

Step-by-Step Solution:

Verification / Alternative check:

Root plots confirm the classification: A has a repeated negative real root; B two distinct negative reals; C has pure imaginary roots; D complex conjugate with negative real part.

Why Other Options Are Wrong:

Any mapping that contradicts the discriminant test is incorrect.

Common Pitfalls:

Forgetting that b = 0 with positive c yields undamped oscillation; miscomputing the discriminant by sign errors.

Final Answer:

A-3, B-4, C-1, D-2.