Introduction to Harmonic Motion

Simple Harmonic Motion

Damped Harmonic Oscillators

Under damped systems

Over damped systems

Critically damped systems

Equations of motion for a damped pendulum

Applet

Over Damped Systems

For over-damped systems, ζ > 1, so is real. This means that both p1 and p2 are both negative. The general solution for this case is

for α = ωζ and β = , similar to the under-damped case. Note the lack of a complex term in the exponentials, so the system doesn't oscillate.

Placing the initial conditions of x(0)=1 and x'(0)=0, the solution will be

A plot of this function with ζ = 3 and ω = 2π shows that the system decays slowly:

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