Critically Damped Simple Harmonic Motion
المؤلف:
Papoulis, A
المصدر:
Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill
الجزء والصفحة:
...
11-6-2018
1183
Critically Damped Simple Harmonic Motion
Critical damping is a special case of damped simple harmonic motion
 |
(1)
|
in which
 |
(2)
|
where 
is the damping constant. Therefore
 |
(3)
|
In this case, 
so the solutions of the form 
satisfy
 |
(4)
|
One of the solutions is therefore
 |
(5)
|
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.](http://mathworld.wolfram.com/images/equations/CriticallyDampedSimpleHarmonicMotion/NumberedEquation6.gif) |
(6)
|
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).](http://mathworld.wolfram.com/images/equations/CriticallyDampedSimpleHarmonicMotion/NumberedEquation7.gif) |
(7)
|
The general solution is therefore
 |
(8)
|
In terms of the constants 
and 
, the initial values are
so
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
 |
(13)
|
and the Wronskian is
Plugging this into the equation for the particular solution gives
Applying the harmonic addition theorem then gives
 |
(18)
|
where
 |
(19)
|
REFERENCES:
Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, p. 528, 1984.
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