Damped Harmonic Oscillators

The equation of motion for a damped harmonic oscillator would be equation (3) with F(t) = 0. Treated one dimensionally the equation becomes

 (4)

Equation (4) can have a solution in the form of

Substituting into (4) gives

 ∴

Apart from the solution when C = 0, the solution is a quadratic equation, with two roots.

To reduce the number of parameters in the equation to 2, we can first consider the undamped solution to (3) involving a mass on a spring affected by gravity, i.e. when b = 0 and F(t) = mg.

With x(0)=0 and x'(0)=0, This gives a solution of

Here, is the mean position of the mass, and the angular frequency of the oscillation is given by .

Applying this initial angular frequency ω to the damped solution, we can now begin to reduce the parameters. First, the equation is divided through by m:

This means the x term can be multiplied by ω2, and if we define another parameter then (4) can now be expressed as

which results in a general solution of

 (5)

where and .

This results in 3 different outcomes, depending on the value of ζ, called the damping factor:

If ζ<1, the system is under-damped, so it will oscillate around the mean value, with decreasing amplitude.

If ζ=1, the system is critically damped, meaning there will be no oscillation and the system will reach equilibrium quickly.

If ζ>1, the system in over-damped, and will reach equilibrium slowly with no oscillation.

Specific solutions to the equation of motion for each case can be found by solving (5). I will take each of the three cases listed individually and show how the solution could be obtained