Subtopic: Higher order linear ODEs with constant coefficients
Question Statement:
For the damped harmonic oscillator differential equation $\frac{d^2 y}{dt^2} + 2\zeta \omega_n \frac{dy}{dt} + \omega_n^2 y = 0$ (where $\omega_n > 0$), which of the following classifications is/are correct?
Options:
(A) If $\zeta > 1$, the system is overdamped and exhibits no oscillations
(B) If $\zeta = 1$, the system is critically damped with repeated real characteristic roots
(C) If $0 < \zeta < 1$, the system is underdamped and exhibits decaying sinusoidal oscillations
(D) If $\zeta = 0$, the system is undamped with pure imaginary characteristic roots
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
The characteristic equation is $m^2 + 2\zeta \omega_n m + \omega_n^2 = 0$, with discriminant $\Delta = 4\omega_n^2(\zeta^2 - 1)$:
1. $\zeta > 1 \implies \Delta > 0$: two distinct real negative roots, overdamped non-oscillatory decay (A is true).
2. $\zeta = 1 \implies \Delta = 0$: repeated real negative root $m = -\omega_n$, critically damped (B is true).
3. $0 < \zeta < 1 \implies \Delta < 0$: complex conjugate roots $-\zeta \omega_n \pm i \omega_n \sqrt{1-\zeta^2}$, underdamped oscillatory decay (C is true).
4. $\zeta = 0$: roots $\pm i \omega_n$, undamped simple harmonic motion (D is true).