Simple Harmonic Motion

A good example of simple harmonic oscillation is in a spring system.  You can see an example of this in the video at the bottom of the blog.  These systems are represented by the formula: a+2βv ̇+w0^2x=Acos(wt). 2βv represents the dampening force, Acos(wt) represents the driving force and w0^2x represents the spring force.  The w0^2 shows the natural frequency (the frequency the oscillator will continuously oscillate at without any external force) of the oscillator while x represents the displacement.  If 2βv = 0 and Acos(wt) = 0, the motion of the spring is represented by:

This graph has no change in displacement because no outside force is acting on the system.  If 2βv does not equal 0, but Acos(wt) = 0.  The oscillator will start to get slower and slower until it comes to a complete stop.  This is because the dampening resists the motion of the spring.  This is represented by:

As you can see, the spring oscillates for a couple of seconds then comes to a complete stop.  This is because there is no driving force to help keep the system in motion.  If both 2βv and Acos(wt) do not equal 0, then the displacement of the spring will be off in the beginning but eventually even out and reach simple harmonic motion.  This is shown by the following graph:

(video obtained by physics animation channel on youtube)