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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.
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 super geek and would like to delve deep into the physical meaning of i, click Bonus Lesson to learn more…
Lesson Navigation
0. Home
1. Mapping Imaginary to Physical - Lesson 1
2. Complex Numbers and Dynamic Systems - Lesson 2
3. An Imaginary Number Application - Lesson 3
B. The Physical Link of the Imaginary Unit - Bonus Lesson
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December 16th, 2009 at 5:38 am
Excellent for using the digital projector for demonstration in the classroom.
April 25th, 2011 at 10:34 am
what’s a portager, John?
April 25th, 2011 at 10:34 am
WHATS A PORTAGER JOHN?!
April 25th, 2011 at 10:35 am
……………..danteng
April 25th, 2011 at 10:37 am
I demand a sacrifice! Bring exactly 17 liters of ketchup to the top of the largest pyramid of Tenochtitlan and I may spare your pitiful souls!
April 25th, 2011 at 10:38 am
when you carry a canoe over land inbetween lakes
April 25th, 2011 at 10:38 am
sigh…
April 25th, 2011 at 10:44 am
Dim-witted slugs! It has been exactly 10 minutes and 23 seconds (in Ry’leh time) and I still have not received my ketchup! I will give you 20 minutes before I bring doom to you all!
June 16th, 2011 at 8:14 am
In the example with the spring I multipied out the values using Excel and got:
1.499869255 - 0.698038244j
What I’m wondering is, have I introduced a rounding error using ? - “=PI()” - to 10 sig fig’s or is it technically correct?
June 16th, 2011 at 9:24 pm
Hi Stuart,
Your answer is likely correctly calculated. I only reported the real part out to 2 significant digits, and the imaginary part out to one significant digit. If you round your answers, to this accuracy, it’s the same answer. It is important to note, however, that the inputs provided (e.g. k, c, w, i) are only ranging between 1 and 3 significant digits of accuracy, so the answer will not be meaningful beyond a couple of significant digits or so (one could do an analysis to see exactly how many sig figs are valid, but a couple or so is ballpark). So in the practical world, folks don’t generally report answers out to 10 significant digits, mainly because the quality of the inputs being used to make the calculation is rarely that accurate. In my work as an engineer, I see a majority of cases where people make extremely important decisions on calculations that we know have at best one significant digit of accuracy! If you’ve got the answer within 30% - you’re golden!
December 1st, 2011 at 7:43 am
the ozmund the purple cabbage went to chocolate for a turnip without turnipicity so it could make cats with a soup laptop off the grid and in the sky of purple clouding blooms in the squishes squidward city or soapy spongebob town to make a keyboard of jelly in kumkwat flavor in a bottle of tropicana frito juice while knawing at the ankles of big business, hiding from the angry walmart while mr. krabs was falling apart.