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

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

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

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

5. 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

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

7. Overdamping – w' = If w' is imaginary: x(t) = C1 e–a t + C2 e–a tk 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

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

9. 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.

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. 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