Damped Oscillation

Damped Oscillation

Think of a spring on a perfectly smooth table - no friction. The spring is fixed to one side of the table. At the other end of the spring, a piece of wood is attached. Pull the wood, and let go. The wood oscillates left and right. It will keep going - forever.

Suppose that we change this to a table that is less smooth, with a little bit of friction. Again, pull the wood and release. The wood moves left and right. After some time, the amplitude becomes smaller and smaller.

Think of an extreme case when the table is very rough. Pull and let go. Because there is so much friction, the wood moves very slowly. After a while, it reaches the rest position. However, it is so slow by then, that there is no momentum to take it across the rest position. It may just stop at the rest position.

There is an interesting situation that is exactly in between the above two cases. This is when there is just enough friction to prevent the wood from crossing the rest position. If there is a bit less of the friction, the wood could overshoot the rest position and oscillate a little. So if there is just enough friction for the wood not to oscillate, the wood returns to the rest position most quickly.

Many things can oscillate. When they oscillate, there would often be some friction. The effect of this friction is called damping. To damp an oscillation is to make the amplitudes smaller. The following terms are used to the describe the above examples:

No damping: The wood oscillates without friction, so the amplitude stays the same.
Under damping: There is a little bit of friction, so the amplitude decreases slowly.
Over damping: There is a lot of friction. The wood may take a long time to return to its rest position, and there is no oscillation.
Critical damping: There is just enough friction to prevent the wood from oscillating. The wood returns to the rest postion in the shortest time.

Critical damping has some important uses. One example is the door that closes by itself. This type of door may have a spring device attached to itself and the door frame. So if you open the door, the spring will close the door for you. If you have used such doors before, you may notice that some of these close very slowly, while others close with a bang.

Those that close very slowly are over damped - there is too much friction. Those that close with a bang are under damped. The door tries to oscillate, but is stopped by the door frame - with a bang. Wouldn't it be nice if the door can close as quickly as possible, but without the annoying bang? That is exactly what critical damping is.

Another example is the bumpy car. If you often ride in a car or bus, you would notice that some are very bumpy, whereas others are very shaky. If you are lucky, you will sit in one that gives you a smooth ride.

Usually, there is some kind of spring between the body of a car, and the axles of the wheels. The intention is to give you a smooth ride. If a wheel runs over a bump on the road, the bump would knock the wheel upwards. This compresses the spring. The spring then applies a sudden force of the car's body. If the car design is good, the spring will return to the original length quickly. This is crtical damping. The sudden force would only act for a short time, and the car body would hardly be disturbed.

In a badly designed car, or a very old one, there might be a lot of friction in the spring between the wheels and the body. If a wheel goes over a bump on the road, the spring is hardly compressed. So the whole body jerks up with the wheel - very bumpy. This is over damping.

Or, there may be very little friction. When the wheel goes over a bump, the spring is compressed. Then the body starts oscillating up and down for a long time. The car is very shaky. This is under damping.

For a smooth, comfortable ride, we really need the spring to be critically damped. After the bump. the spring would be compressed, but would return the to the normal length quickly and with no oscillation.

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