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Damped harmonic oscillator
Real oscillators are always damped. The damped oscillator
shown in the figure consists of a mass m, a spring of
constant k and a vane submarged in a liquid. The liquid
exerts a damping force which in many cases is proportional
to the velocity (with opposite sign):
Fb  b
dx
dt
b – damping constant (22)
In this case the equation of motion can be written as
ma  kx  b
dx
dt
(23)
After rearrangement we have
Introducing the substitutions:
d 2x
dx
2




x0
0
dt 2
dt
k
2
 0
m
The solution of (25) for a small damping is:
x  Ae  t cos t 
where
  02   2
Figure from HRW,2
(24)
d 2x
dx
m 2 b
 kx  0
dt
dt
b

m
one gets
(25)
(26)
1
Damped harmonic oscillator, cont.
Solution (26) can be regarded as a cosine function with a
time dependent amplitude At   Ae .t Time t = τ, after
which the amplitude decreases e1/2 times is called the
average lifetime of oscillations or the time of relaxation.
1
At 
Ae t
1


 e  e2
for  
  t  
At    Ae
2
The angular frequency ω of the damped oscillator is less
than that of undamped oscillator ωo. For the small damping,
i.e. for ωo>> β, solution (26) can be approximated by
(27)
x  Ae   t cos0 t 
The amplitude for the
damped oscillator
decreases exponentially
with time.
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