Overdamped simple harmonic motion is a special case of damped simple harmonic motion
(1)
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in which
(2)
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Therefore
(3)
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(4)
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(5)
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where
(6)
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The general solution is therefore
(7)
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where and are constants. The initial values are
(8)
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(9)
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so
(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 , i.e.,
(12)
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the general solutions are
(13)
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(14)
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where
(15)
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(16)
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These give the identities
(17)
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(18)
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and
(19)
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(20)
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We can now use variation of parameters to obtain the particular solution as
(21)
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where
(22)
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(23)
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and the Wronskian is
(24)
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(25)
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These can be integrated directly to give
(26)
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(27)
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Integrating, plugging in, and simplifying then gives
(28)
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(29)
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where use has been made of the harmonic addition theorem and
(30)
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