Damped Oscillations
 Energy is lost to the environment
 Dissipative force (friction, air resistance, viscosity in a fluid)
Often the dissipative force can be modeled as D~v:
D  bv
Newton’s 2nd law:
F  ma  bv  kx
Equation of motion:
d 2 x b dx k

 0
2
dt
m dt m
2nd order differential equation with constant coefficients:
Small damping (underdamping)
Solution:
x(t )  Ae
 2bm t
cos(t  0 )
where

k
b2
 b 
2




0


m 4m 2
 2m 
2
Small damping decreasing frequency, increases period.
Critical Damping and Overdamping
Mass creeps back to equilibrium position w/o oscillating
about it.
x(t )   A  Bt  e
x(t )  Ae
 2bm t
 2bm t 
 2bm 
critical damping
2
02
 Be
 2bm t 
http://www.lon-capa.org/~mmp/applist/damped/d.htm
overdamping
 2bm 
2
02
Forced Vibrations and Resonance
Physics: System with damping constant b, natural frequency
(k/m)1/2, and forcing function F0cos(dt), where wd is the driving
frequency.
F  ma  bv  kx  F0 cos(d t )
d 2 x b dx k F0

  cos(d t )
2
dt
m dt m m
Amplitude of oscillation as a function of driving frequency:
F0
A

2
d


m
2 2
0
 b 
 d 
 m 
2