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
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
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