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

Critically Damped Systems

A critically damped system has ζ = 1, so p1 = p2 = -ζω. This means that one of the solutions is

The other linearly independent solution can be found by using the reduction of order method. This gives the solution

Using the initial conditions x(0) = 1 and x'(0) = 0 as in the under- and over-damped cases, the solution is now

A plot of this solution for ω = 2π looks like

Note that a critically damped system converges to zero faster than any other without oscillating.

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