Overdamped simple harmonic motion is a special case of damped simple harmonic motion
 |
(1)
|
with 
, 
,
and
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(2)
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Therefore
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(3)
|
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(4)
|
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(5)
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where
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(6)
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The general solution is therefore
 |
(7)
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where 
and 
are constants. The initial conditions are
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(8)
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(9)
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so
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(10)
|
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(11)
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The above plot shows an overdamped simple harmonic oscillator with 
, 
and three different initial
conditions 
.
For a cosinusoidally forced overdamped oscillator with forcing function 
for real 
, i.e.,
 |
(12)
|
the associated homogeneous equation has the linearly independent solutions
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(13)
|
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(14)
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where
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(15)
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(16)
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These give the identities
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(17)
|
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(18)
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and
 |  | ![1/4[beta^2-(r_1-r_2)^2]](/images/equations/OverdampedSimpleHarmonicMotion/Inline48.svg) |
(19)
|
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(20)
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We can now use variation of parameters to obtain the particular solution as
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(21)
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where
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(22)
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(23)
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and the Wronskian is
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(24)
|
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(25)
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These can be integrated directly to give
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(26)
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(27)
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Substituting and simplifying gives
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(28)
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(29)
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where use has been made of the harmonic addition theorem and
 |
(30)
|
Here 
denotes the principal complex argument, which
fixes the quadrant of the phase.
