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Overdamped Simple Harmonic Motion


SHOOverdamped

Overdamped simple harmonic motion is a special case of damped simple harmonic motion

 x^..+betax^.+omega_0^2x=0,
(1)

in which

 beta^2-4omega_0^2>0.
(2)

Therefore

 D=beta^2-4omega_0^2>0.
(3)
x_1=e^(r_-t)
(4)
x_2=e^(r_+t),
(5)

where

 r_+/-=1/2(-beta+/-sqrt(beta^2-4omega_0^2)).
(6)

The general solution is therefore

 x=Ae^(r_-t)+Be^(r_+t),
(7)

where A and B are constants. The initial values are

x(0)=A+B
(8)
x^.(0)=Ar_-+Br_+,
(9)

so

A=x(0)-(r_-x(0)-x^.(0))/(r_--r_+)
(10)
B=(r_-x(0)-x^.(0))/(r_--r_+).
(11)

The above plot shows an overdamped simple harmonic oscillator with omega=0.3, beta=0.075 and three different initial conditions (A,B).

For a cosinusoidally forced overdamped oscillator with forcing function g(t)=Ccos(omegat), i.e.,

 x^..+betax^.+omega_0^2x=Ccos(omegat),
(12)

the general solutions are

x_1(t)=e^(r_1t)
(13)
x_2(t)=e^(r_2t),
(14)

where

r_1=1/2(-beta+sqrt(beta^2-4omega_0^2))
(15)
r_2=1/2(-beta-sqrt(beta^2-4omega_0^2)).
(16)

These give the identities

r_1+r_2=-beta
(17)
r_1-r_2=sqrt(beta^2-4omega_0^2)
(18)

and

omega_0^2=1/4[beta-(r_1-r_2)^2]
(19)
=r_1r_2.
(20)

We can now use variation of parameters to obtain the particular solution as

 x^*=x_1v_1+x_2v_2,
(21)

where

v_1=-int(x_1(t)g(t))/(W(t))
(22)
v_2=int(x_2(t)g(t))/(W(t))
(23)

and the Wronskian is

W(t)=x_1x^._2-x^._1x_2
(24)
=(r_2-r_1)e^((r_1+r_2)t).
(25)

These can be integrated directly to give

v_1=-C/(r_2-r_1)(omegasin(omegat)-r_2cos(omegat))/(e^(r_2t)(r_2^2+omega^2))
(26)
v_2=C/(r_2-r_1)(omegasin(omegat)-r_1cos(omegat))/(e^(r_1t)(r_2^2+omega^2)).
(27)

Integrating, plugging in, and simplifying then gives

x^*(t)=C(cos(omegat)(r_1r_2-omega^2)-sin(omegat)omega(r_1+r_2))/((r_1^2+omega^2)(r_2^2+omega^2))
(28)
=C/(sqrt(beta^2omega^2+(omega^2-omega_0^2)^2))cos(omegat+delta),
(29)

where use has been made of the harmonic addition theorem and

 delta=tan^(-1)((betaomega)/(omega^2-omega_0^2)).
(30)

See also

Critically Damped Simple Harmonic Motion, Damped Simple Harmonic Motion, Simple Harmonic Motion, Underdamped Simple Harmonic Motion

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References

Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, pp. 527-528, 1984.

Cite this as:

Weisstein, Eric W. "Overdamped Simple Harmonic Motion." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OverdampedSimpleHarmonicMotion.html

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