Critical damping is a special case of damped simple harmonic motion
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(1)
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in which
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(2)
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where 
is the damping constant. Therefore
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(3)
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In this case, 
so the solutions of the form 
satisfy
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(4)
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One of the solutions is therefore
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(5)
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In order to find the other linearly independent solution, we can make use of the identity
![x_2(t)=x_1(t)int(e^(-intp(t)dt))/([x_1(t)]^2)dt.](/images/equations/CriticallyDampedSimpleHarmonicMotion/NumberedEquation6.svg) |
(6)
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Since we have 
, 
simplifies to 
. Equation (6) therefore becomes
![x_2(t)=e^(-omega_0t)int(e^(-2omega_0t))/([e^(-omega_0t)]^2)dt=e^(-omega_0t)intdt=te^(-omega_0t).](/images/equations/CriticallyDampedSimpleHarmonicMotion/NumberedEquation7.svg) |
(7)
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The general solution is therefore
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(8)
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In terms of the constants 
and 
, the initial values are
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(9)
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(10)
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so
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(11)
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(12)
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The above plot shows a critically damped simple harmonic oscillator with 
, 
for a variety of initial conditions 
.
For sinusoidally forced simple harmonic motion with critical damping, the equation of motion is
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(13)
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and the Wronskian is
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(14)
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(15)
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Plugging this into the equation for the particular solution gives
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(16)
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 |  | ![A/((omega^2+omega_0^2)^2)[(omega_0^2-omega^2)cos(omegat)+2omegaomega_0sin(omegat)].](/images/equations/CriticallyDampedSimpleHarmonicMotion/Inline35.svg) |
(17)
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Applying the harmonic addition theorem then gives
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(18)
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where
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(19)
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