Imaginary Numbers

An Imaginary Number Application - Lesson 3

Vibration Application Revisited

In lesson 3, we go deeper into the engineering application of complex numbers by revisiting the vibration system that we saw at the intro. The challenge here is to describe the output (motion of mass) relative to the input (motion of wheel). Four scenarios are presented to cover various parts of the complex plane…

1. Positive Real Response
Output is synchronized with input; high points and low points occur simultaneously.

2. Negative Real Response
Output opposes input; output high point occurs simultaneously with input low point, and visa versa.

3. Negative Imaginary Response
A damper is added to physically obtain an imaginary component. The output lags input by exactly quarter cycle. When the input is peaked, the output is at zero and moving up.

4. Complex Response
Damper still present for imaginary component. The output is described by the complex number 1.5 - 0.7i, having a large real component with a small negative imaginary component. The output is just slightly lagging behind the input.

Vibration equation solved for Output/Input

Why The Overhead?

OK, we don’t go around describing the sizes of our photographs with complex numbers; we just use two numbers, length and width. Can’t we just use two numbers when we need to describe any two dimensional thing?

Well, in many cases we are interested in using equations to describe how two dimensional variables interact within a system, not just in isolation. Complex numbers allow us to solve equations via the well defined rules of complex algebra, automatically handling the relationships between both dimensions. If we were to use two separate real numbers instead of a single complex number, we could not easily manipulate the variables in our equations.

The equation to the right illustrates our point; rollover to see the analysis of scenario 4 in our vibration example. You will have to use a little complex algebra to get to the final result.

Summary

We just applied complex numbers to a vibration problem, demonstrating how they allow us to conveniently include two dimensional variables in equations, and solve for quantities of interest. This example is just a tiny part of the more general utility - using complex numbers to manipulate variables that are two dimensional in nature. Applications abound in the real world, touching our lives via design of popular features such as the vibrating ringer in our cell phones or the bass boosters in our MP3 players. More heavy duty applications include the design of missile guidance systems. Now this is some powerful stuff!!

Congratulations on finishing our basic tutorial, but we’ve just scratched the surface! If you are a teacher and would like to checkout our fabulous imaginary tools, click Browse Store below. If you are a super geek and would like to delve deep into the physical meaning of i, click Bonus Lesson to learn more…