Overdamped Simple Harmonic Motion

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

(1)

in which

(2)

Therefore

(3)

			(4)
			(5)

where

(6)

The general solution is therefore

(7)

where  and  are constants. The initial values are

			(8)
			(9)

so

			(10)
			(11)

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

			(13)
			(14)

where

			(15)
			(16)

These give the identities

			(17)
			(18)

and

			(19)
			(20)

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

(21)

where

			(22)
			(23)

and the Wronskian is

			(24)
			(25)

These can be integrated directly to give

			(26)
			(27)

Integrating, plugging in, and simplifying then gives

			(28)
			(29)

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.