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