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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?
A. Damping increases the frequency.
B. Damping does not affect the frequency.
C. Damping decreases the frequency.
Damping Differential Equation
ma = –bv – kx
One general solution:
x(t) = Ae cos('t + )
–bt2m
where
' = km 4m2
b2–
Light Damping
x(t) = Ae cos('t + )
–bt2m
If ' > 0:
• Oscillates
• Frequency slower than undamped case
• Amplitude decreases over time
' = km 4m2
b2–
Critical Damping
If ' = 0:
x(t) = (C1 + C2t) e–at
• No oscillation
• If displaced, returns directly to equilibrium
' = km 4m2
b2–
Overdamping
• No oscillation
• If displaced, returns slowly to equilibrium
' = km 4m2
b2–
If ' is imaginary:
x(t) = C1 e–a t + C2 e–a t1 2
Energy in Damping
• Damping force –bv is not conservative
• Total mechanical energy decreases over time
• Power dE/dt = –bv2= F·v = –bv·v
Worksheet Problem
Your 1000-kg car is supported on four corners by identical springs with spring constant k = 10,000 N/m.
a) Find the natural frequency of oscillation of your car.
b) 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(dt)
Forced Oscillation
If no damping
If d = ', amplitude increases without bound
Resonance
If lightly damped:
greatest amplitude when d = '
Source: Young and Freedman, Fig. 13.28
Critical or over-damping (b ≥ 2 km):
no resonance