Damped and Forced Oscillations Introducing non-conservative forces § 14.7–14.8

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

Poll Question How does damping affect the oscillation frequency? Damping increases the frequency. Damping does not affect the frequency. Damping decreases the frequency.

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

Light Damping x(t) = Ae cos(w't + f) – w' = If w' > 0: Oscillates –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

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

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

Energy in Damping Damping force –bv is not conservative Total mechanical energy decreases over time Power dE/dt = F·v = –bv·v = –bv2

Worksheet Problem Your 1000-kg car is supported on four corners by identical springs with spring constant k = 10,000 N/m. Find the natural frequency of oscillation of your car. Find the damping constant your shock absorbers must have in order to critically damp its vibrations.

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

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

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