Overdamped Simple Harmonic Motion
المؤلف:
Papoulis
المصدر:
A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill
الجزء والصفحة:
pp. 527-528
3-7-2018
1479
Overdamped Simple Harmonic Motion
Overdamped simple harmonic motion is a special case of damped simple harmonic motion
 |
(1)
|
in which
 |
(2)
|
Therefore
 |
(3)
|
where
 |
(6)
|
The general solution is therefore
 |
(7)
|
where 
and 
are constants. The initial values are
so
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)
|
the general solutions are
where
These give the identities
and
We can now use variation of parameters to obtain the particular solution as
 |
(21)
|
where
and the Wronskian is
These can be integrated directly to give
Integrating, plugging in, and simplifying then gives
where use has been made of the harmonic addition theorem and
 |
(30)
|
REFERENCES:
Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 527-528, 1984.
الاكثر قراءة في معادلات تفاضلية
اخر الاخبار
اخبار العتبة العباسية المقدسة
