11.2 The Harmonic Oscillator (aka a Spring)

There is another fairly simple physical system that is of fundamental importance in many areas. It describes the motion of a spring in one dimension.

A spring is a devise that has an equilibrium position, and when moved from that position, it pulls itself back toward it.

Suppose we have an object with mass m attached to such a spring. We can model this system by claiming that there is a force on this object proportional to its displacement from the equilibrium position. If we call that position the origin and denote the displacement from it by x, the model equation for this system is

Here k is the proportionality constant which is proportional to the strength of the spring.

There are several additional features we can add to this model.

First we can try to model the effect of friction. This we do by adding an additional (small) force that tends to oppose the motion of the object and is proportional to its velocity. This term adds a factor to the right-hand side of the equation.

Why add this particular term?

Because this is the simplest term that I can think of that does the right sort of thing: opposes the motion by an amount proportional to it.

Finally, we can imagine that the spring is hooked up with some other devices, and these devices provide additional forces on it. Imagine. For example, that the additional force on our object behaves like the sine function of a constant, , multiplied by t. This is a useful assumption, because if we can find the behavior of this spring for all values of , (and we can) we can, by some magic tricks, figure out how this system will respond to any forcing function.

The general model of a forced and damped (this means friction is considered) spring is then described by the equation

where A is a constant.

Notice the nature of this model: the first term on the right is the basic term which describes a spring. The second term is one aimed at causing the motion to slow down, being proportional to the velocity and directed opposite to it. The third term is an external forcing term independent of x or of , the velocity.

To construct it, you think of each of the factors that you expect to affect the spring, construct the simplest possible term to describe it, and add it in on the right here.

How do I know that this model is any good?

A priori, you don't know it. When you work out the solutions for x(t) that emerge from it, you can see if they describe real phenomena. This particular model is perhaps the most successful one ever. It predicts many interesting real and important phenomena, that describe not only the motion of springs but of many other important systems as well.

There is more than one solution?

The behavior of our object on one end of a spring depends on how we start it off, in other words, it depends on its initial position and velocity. Similarly, the solution to the model equation will depend not only on the terms in the equation, but on these initial conditions as well. There will be a different solution for each pair of initial values.

What phenomena are predicted by this model?

In general the motion of our object can be divided into two terms: a transient one, which is not unlike the motion that would occur if there were no forcing function (that is, if A = 0) and a steady state term which has the same frequency as the forcing term.

The transient term dies out eventually (if there is a second term here so that we have f > 0) and the steady state motion can then be characterized by two numbers: One is the ratio between the amplitude (which means size) of the steady state response to the amplitude, A, of the forcing function, and the second is the phase angle between the forcing function and the response.

The first of these numbers, which represents the relative strength of the response of the system to the forcing function, exhibits the phenomenon called resonance. There is a natural frequency at which the spring will oscillate if left alone. When the forcing function has its frequency near that natural frequency there is a big response, and that response gets smaller as a function of in the model, as moves away from the natural value here.

The transient term can oscillate, or if f is large enough it will be "critically damped" and will simply die off.

This is a qualitative picture of what this model predicts. And these phenomena are real.

The following applet describes the motion of this kind of system.

You will see in the next part of this course how you yourself can work out the consequences of this model to create for yourself what this applet shows, by creating an appropriate spreadsheet.