Damped and Forced Oscillations

1. Damped and Forced Oscillations Introducing non-conservative forces § 13.7–13.8

2. Damping Force Such as viscous drag v Drag opposes motion: F = –bv

3. CPS Question How does damping affect the oscillation frequency? Damping increases the frequency. Damping does not affect the frequency. Damping increases the frequency.

4. –bt 2m x(t) = Ae cos(w't + f) where w'= k – b2 m 4m2 Damping Differential Equation ma= –bv – kx General solution:

5. Light Damping –bt 2m x(t) = Ae cos(w't + f) k – b2 w' = m 4m2 • If w' > 0: • Oscillates • Frequency slower than undamped case • Amplitude decreases over time

6. k – b2 w' = m 4m2 Critical Damping If w' = 0: x(t) = (C1 + C2t) e–at • No oscillation • If displaced, returns directly to equilibrium

7. k – b2 w' = If w' is imaginary: x(t) =C1 e–at + C2 e–a t m 4m2 1 2 Overdamping • No oscillation • If displaced, returns slowly to equilibrium

8. = F·v = –bv·v Energy in Damping • Damping force –bv is not conservative • Total mechanical energy decreases over time = –bv2 • PowerdE/dt

9. Example Problem 13.xx Your 1000-kg car is supported on four corners by identical springs with spring constant k = 1000 N/m. Find the natural frequency of oscillation of your car. Find the damping constant your shock absorbers most have in order to critically damp its vibrations.

10. Forced Oscillation Periodic driving force F(t) = Fmax cos(wdt)

11. Forced Oscillation If no damping If wd = w, amplitude increases without bound

12. Critical or over-damping (b ≥ 2 km): no resonance Resonance If under-damped: greatest amplitude when wd = w Source: Young and Freedman, Fig. 13.28