Overview: This article examines how complex numbers of the form $$ a+ bi $$ are used to describe the motion of an oscillating spring with damping.

Oscillating Springs

[Real World Complex Number Example]

When a mass is attached to the end of a spring and then the spring is stretched down and released, we expect the mass and spring to bob up and down. The bobbing eventually dies down and the spring-mass system comes to a rest (see figure belowFigure 1).
If we extract just the path indicated above, and plot it on coordinate axes we have the graph of a function (see Figure 2 below).

What's a Damped Oscillator?

This type of function is called a damped oscillator.

Oscillate means to move back and forth or up and down repeatedly.

Damp means that the oscillations will decrease due to some kind of friction, ie the spring will bounce up and down less and less until it eventually stops--this "slowing down" is damping.

And damped oscillators show up in lots of interesting and important areas of science and engineering. Some examples include electrical circuits, vibrations of charged particles (like electrons and protons), pendulums, Bungee jumping, mechanical vibrations, and shock absorbers on vehicles, to name just a few.

The Math behind Damped Oscillations

A damped oscillator function is constructed by multiplying exponential decay functions with sine and cosine functions (see figures below).

So, a basic function that describes a damped oscillator looks like this:

In the function, you will notice four parameters: a, b, c , and d. These are just numbers that control or describe different parts of the damped oscillator. The values of c and d are determined by the beginning height and speed of the oscillator. We won't be playing with those in this article.

Where Complex Numbers are needed

$ y = e^{\red a t}\cdot\Big[c\cdot\sin(\red b t)+d\cdot\cos(\red b t)\Big] $

The other two parameters however, are where complex numbers come into our discussion. The parameter a determines how quickly the oscillations damp out and b determines how fast the oscillations bounce up and down.

To find the values of a and b for a spring-mass system we have to solve a quadratic equation that looks like this:

$ mx^2 + rx + k = 0 $

where m represents the mass (in kilograms), k represents the stiffness of the spring, and r is a measurement of the things that cause the damping like air resistance and friction and such.

Complex Numbers are part of this real world solution

Let's do a quick example with actual numbers so you can see how this works. Suppose a 4-kilogram mass is attached to a spring with a stiffness measured at $$ k= 53 $$ and a damping of $$r = 8 $$. The quadratic equation we need to solve is

$ 4x^2 + 8x + 53 = 0 $

$ \text { which has a solution of } \\ \\ \\ \frac{-8\pm\sqrt{8^2-4(4)(53)}}{2(4)} \\ \frac{-8\pm\sqrt{64-16(53)}} 8 \\ \frac{-8\pm\sqrt{16(4-53)}} 8 \\ \frac{-8\pm\sqrt{16(-49)}} 8 \\ \frac{-8\pm\sqrt{-784)}} 8 \\ \frac{-8\pm 28 \red i} 8 \\ -\frac 8 8 \pm \frac{28 \red i} 8 \\ -1\pm 3.5 \red i $

The answers to this equation are complex numbers in the form a + bi . In this case, ($$ a=\blue{ -1} $$) and ($$ b = \red {3.5}$$) These are exactly the values we need for our damped oscillator function:

$ y = e^{\blue{-t}}\cdot\Big[c\cdot\sin(\red{3.5}t)+d\cdot\cos(\red{3.5}t)\Big] $

Remember, to get the values for c and d, we need information about position and speed. We also need calculus, so that part will have to be a discussion for a later time.

Here's a graph of the function we found above where the initial position was $$ y = -3$$ and the initial speed is 10 m/s.

Damped oscillators are only one area where complex numbers are used in science and engineering. Many of the real-world applications involve very advanced mathematics, but without complex numbers the computations would be nearly impossible. Even in this discussion I've had to skip all the math that explains why the complex numbers to the quadratic equation give us the necessary values of a and b. . These are things you will learn when you study calculus, differential equations, linear algebra and a little more physics. Nevertheless, complex numbers play a crucial role in our ability to study and understand the world around us.
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