Critically Damped Simple Harmonic Motion

Critical damping is a special case of damped simple harmonic motion

in which

where is the damping constant. Therefore

In this case, so the solutions of the form satisfy

One of the solutions is therefore

In order to find the other linearly independent solution, we can make use of the identity

Since we have , simplifies to . Equation (6) therefore becomes

The general solution is therefore

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

and the Wronskian is

Plugging this into the equation for the particular solution gives

Applying the harmonic addition theorem then gives

where