Underdamped Simple Harmonic Motion

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

in which

Since we have

it follows that the quantity

is positive. Plugging in the trial solution to the differential equation then gives solutions that satisfy

i.e., the solutions are of the form

Using the Euler formula

this can be rewritten

We are interested in the real solutions. Since we are dealing here with a linear homogeneous ODE, linear sums of linearly independent solutions are also solutions. Since we have a sum of such solutions in (9), it follows that the imaginary and real parts separately satisfy the ODE and are therefore the solutions we seek. The constant in front of the sine term is arbitrary, so we can identify the solutions as

so the general solution is

The initial values are

so and can be expressed in terms of the initial conditions by

The above plot shows an underdamped simple harmonic oscillator with , for a variety of initial conditions .

For a cosinusoidally forced underdamped oscillator with forcing function , so

define

for convenience, and then note that

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

where

and the Wronskian is

These can be integrated directly to give

Therefore,

where use has been made of the harmonic addition theorem and