Damped Systems

Damped Harmonic Motion

Of all of the types of damped harmonic motion, the underdamped oscillators are the simplest to explain.  The "envelope" traps the motion of a the mass into a certain range, you can think of it this way.  When looking at simple harmonic motion of a mass, the amplitude of any function (position, velocity, or acceleration) is trapped in an envelope, but in the case of SHM the envelope stays the same, the motion is trapped between a positive maximun and a negative minium.  For damped harmonic motion the functions of motion still oscillates but the value of the amplitude slowly decreases until the amplitude goes to zero.  If you were to plot the amplitudes out with respect to time, you create this decay curve much like you would see with nuclear decay or decaying  capacitors.

Now why its called an under-damped system is that the object is allowed to oscillate back and forth over and over again. The red line inside the envelope is the position of the oscillated objects. The other thing about under-damped systems is that it takes a lot of time to come to a stop (the diagram really doesn't not show this well, the curve should got out a lot farther then it shows).

The nice thing about an underdamped system is that it takes a long time to dissipate the energy in the object.  Why this is a nice thing it allows a piece a equipment to slowly to come to a stop, like the wheels on your car, or mixing blades on a mixer.  You want the blades or the wheels to make more then just one rotation before stopping.

Critically damped systems are the ideal systems it brings the object to stop in the least amount time, the damping is such that functions for position, velocity, and acceleration that we normally see in simple harmonic motion with a decay curve, the only remnants of the oscillatory motion in the very beginning of the curve (the up and the beginning of the decline).

The nice thing about critically damped systems is it the most efficient way to bring a motion to a stop. That a swing door as, the under damp system allows the door to swing back and forth over and over again until it final coming to a stop.  With a critically damped the door closes as quick as possible, and if heated or cooled then the then there is as little temperuture change

Over-damped systems are ideal systems to bring an object to a stop in the longest time possible.  The damping is excessive and like the critically damped system the decay curve dominates and the oscillatory motion has completely disappeared (even more so then the critically damped system.

The nice thing about overdamped system is to give the maximum possible time to come to a stop.  Think of it as a car door or a hatch back, you want to give a person the greatest amount of time to get out of the way before the door closes