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


Under Damped Systems

For under-damped systems, ζ<1, and so ω is imaginary, so p1 and p2 are both complex. The general solution is then

where α = ωζ and γ = . This consists of a decaying exponential multiplied by an oscillation at the damped frequency γ.

If we place the initial conditions of x(0)=1 and x'(0)=0, then the solution will be


A plot of the first 5 seconds of a system with ζ = 0.1 and ω = 2π then looks like

Note that the damping factor is never negative. When it is zero, the system continues to oscillate undamped indefinitely, and when 0 < ζ < 1 the system oscillates with an exponentially decreasing amplitude, as shown above.

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