UTAS Home › › Mathematics Pathways › Pathway to Engineering › COMPULSORY MODULES, 1-3
Complex numbers allow us to solve equations that have no ‘real solution’ and extend the number line into a 2 dimensional complex plane as shown below.
“think of Adam and Eve like an imaginary number, like the square root of minus one: you can never see any concrete proof that it exists, but if you include it in your equations, you can calculate all manner of things that couldn't be imagined without it.” Philip Pullman, The Golden Compass
Photo: The complex plane showing operations, and the Mandelbrot set.
Image by Kan8eDie (Own work) via Wikimedia Commons
Complex numbers are part of the national engineering unit MEM23004A Apply technical mathematics more details on unit content from training.gov.au
Required skill: to solve problems involving complex number quantities using the properties, operations and theorems of complex numbers.
There are no prerequisites required for this module; however, to complete the lessons and quiz below you should have an understanding of algebra and polynomial equations.
A complex numbers is a number that can be expressed in the form z = a + bi, where a and b are real numbers and i is the imaginary unit, satisfying i2 = -1
For Example, -3.5 + 2i is a complex number
Complex numbers are used by Electrical & Electronic Engineers to define the Alternating Current or AC concept of Impedance, and in Fourier analysis they are used in the processing of radio, telephone and video signals, see this page for more details.
Mechanical & Structural Engineers use complex numbers to analyse the vibration of structures in machines, buildings and bridges, the behaviour of fluid flow around aircraft, and that of wind around buildings and bridges, preventing failures such as the Tacoma Narrows Bridge (Please watch the following video).
Complex numbers are also the basis for the Mandelbrot set, an example of 2D fractal geometry as shown in the image below.
Image from Wikipedia licensed under the Creative Commons Attribution 2.5 Generic Licence
For more details on the Mandelbrot Set, watch this TED Talk by Benoit Mandelbrot (17min).
Examples of approximate fractals in the natural world include leaves, mountain ranges, clouds, broccoli and many others; see this page for more examples.
To get a better understanding of complex numbers review these lessons and quizzes from the following links:
Now go to the Interactive graphical addition, subtraction tool from geogebratube.org
Now take the interactive graphing quiz from Khan Academy to practice graphing complex numbers
Go to the Interactive graphical tool cartesian and polar coordinates from geogebratube.org to practice using the Argand Diagram and in exponential form
Use the interactive graphical tool from nrich.maths.org for practicing these ideas
All Khan Academy content is available for free at http://www.khanacademy.org/Example
Mechanical/Structural Engineering - Vibration
Machines and structures, such as a bridges or buildings are often subjected to varying loads or forces, which will cause the movement, or displacement, of a point on the machine or structure possibly causing vibration.
A weight and undamped spring are an example of simple harmonic motion
Image licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
Engineers may also need to create forced vibration to move bulk materials such as fine powders, gravel and rocks in industries such as mining, food processing and cement manufacturing see examples here.
Common method to create forced vibration by the rotation of an offset mass.
Image licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license from Brews ohare
x = XSinωt
Where, X = amplitude of single peak hence x varies between + X and - X, ω = frequency in radians/second, t = time in seconds.
Graphical representation of vibration by sine or cosine functions
Public domain image from the Wikimedia Commons
Differentiation of displacement with respect to time gives velocity, ν = ωΧCosωt
Differentiation of velocity with respect to time gives acceleration, a = -ω2ΧSinωt
These equations can be expressed in complex exponential form which simplifies many operations, especially when we have vibration in multiple planes in 3D space.
3D representation of the complex and real planes showing a complex function in exponential form ej*z resulting from the addition of a sine and cosine function as described below.
Image licensed from Qniemiec under the Creative Commons Attribution-Share Alike 3.0 Unported license.
Using Euler’s Equation, ei*ϑ = Cosϑ + iSinϑ where ϑ = angle in radians, e = base of the natural logarithm or "Euler's number" approximately = 2.71828.
If we multiply Euler’s Equation by Χ and substitute ϑ = ωt we get
Χeiωt = ΧCosωt + iΧSinωt
Note similarity to our equations for displacement and velocity above.
When we plot this equation on the complex plane or Argand diagram we get a vector of length Χ rotating at ω radians/sec.
Where displacement x = Real part of Χeiωt and velocity v = Imaginary part of Χeiωt
Image from Smith, J.O. Introduction to Digital Filters with Audio Applications,http://ccrma.stanford.edu/~jos/filters/, online book, accessed 10/01/14, copy permission https://ccrma.stanford.edu/~jos/copying.html
This image is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license from Gonfer
Problem:
For the design of a vibrating feeder we need to determine the forces in the system. We can approximate the motion of the feeder to simple harmonic motion represented by the sine or cosine function.
Our vibrating feeder design has a single peak amplitude Χ = 2mm or 0.002m at a frequency ω =18 cycles/sec. Calculate the single peak velocity and acceleration.
Worked Solution:
We need to convert ω from cycles/sec to radians/sec, 1 cycle or revolution = 2π radians, ω =18 x 2π
ω =36π radians/sec
Velocity, ν = ωΧCosωt, and peak velocity occurs when Cosωt=1 (see graph above)
ν = ωΧ = 36 π 0.002 = 0.23 m/sec
Acceleration, a = -ω2ΧSinωt, and peak acceleration occurs when Sinωt = 1
a = -ω2Χ = (36 π)2 0.002 = 25.58 m/sec2
Practice exercise:
If a new bridge design has a single peak amplitude Χ = 3mm or 0.003m at a frequency ω = 5 cycles/sec find the single peak velocity and peak acceleration?
Check your answer to the practice exercise here.
Example
Electrical/Electronic Engineering - Circuit Design
Electrical current measurements are important to people who want to use electricity in a creative way. Electrical current is used as a direct current (DC), the one directional flow of electric charge or alternating current (AC), where the flow of electric charge periodically reverses direction, see diagram below. When working with AC, complex numbers are needed.
Public domain image from the Wikimedia Commons
Electric current equations:
V= I x Z or Z = V/I
where, V = voltage, I = current and Z = impedance
Voltage: the difference in electrical charge between two points in a circuit, unit in volts, V
Current: the flow of electric charge, units are amperes or amps, A
Impedance: a measure of the opposition that a circuit presents to a current when voltage is applied.
The magnitude of complex impedance gives the ratio of the voltage amplitude to the current amplitude. In polar form Z = |Z|eiθ where θ =phase angle or difference between voltage and current, using complex notation Z = R + i X where, X = reactance and R = resistance, in DC circuits we only consider R units in ohms Ω.
In many applications the relative phase of the voltage and current is not critical so only the magnitude of the impedance is used or |Z|
Note: i is the imaginary unit, in electrical engineering j is used in place of i to avoid confusion with the symbol for electric current = i
Public domain image from the Wikimedia Commons
Problems:
The impedance in one part of a circuit shown above is 4 + 12i ohms, the impedance in another part of the circuit connected in series is 3 - 7i ohms. What is the total impedance in the circuit?
ZT = Z1 + Z2... = (4 + 12i ) + (3 - 7i )
ZT = 7 + 5i ohms
The voltage in a circuit is 45 + 10i volts and the impedance is 3 + 4i ohms, what is the current?
If the current in a circuit is 8 + 3i amps and the impedance is 1 - 4i ohms, what is the voltage?
I = 8 + 3i amps
Z = 1 - 4i ohms
V = I x Z
V = (8 + 3i ) x (1 - 4i )
V = (8 -32i + 3i - 12i2 ) = (8 - 32i + 3i + 12)
V = 20 - 29i volts
Practice exercises:
Note: For components connected in parallel, the voltage across each circuit element is the same; the ratio of currents through any two elements is the inverse ratio of their impedances.
Public domain image from the Wikimedia Commons
Check your answers to the practice exercises here.
Basic Interactions with Complex Numbers from Interactive Mathematics with solutions
Complex Numbers review from SOS Mathematics with solutions
These are some useful websites with information on Complex Numbers
Interactive Mathematics uses various math applets to enhance mathematics lessons.
Topics range from grade 8 algebra to college-level Forier and Laplace Transformations.
History of complex numbers.
When you have completed the lessons, verify your knowledge with this multiple choice Complex Numbers Quiz
If you need to repeat the quiz, click here (please note: you will not be able to repeat the first quiz)
PLEASE NOTE:
Photo: Solution to the differential equations for heat transfer in a pump casing model using finite element modelling software.
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license
Differential equations are part of the national engineering unit MEM23007A Apply calculus to engineering tasks. More details on unit content from training.gov.au
Required skill: identifying and solving simple first and second order differential equations.
The prerequisite for this module is MEM23004A Apply technical mathematics or equivalent. More details on this unit’s content from training.gov.au
First order ordinary differential equations are of the formsee solution details here
Second order ordinary differential equations are of the form
where P, Q, R and G are continuous functions, see solution examples here.
Second order ordinary differential equations are analogous to the equations of the conic sections circle, ellipse, parabola and hyperbola obtained by the intersection of a plane with a cone. The general equation of the conic section has the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, more details here.
Public domain image from the Wikimedia Commons
Second order ordinary differential equations also describe a wide range of real world phenomena such as electromagnetic, sound and fluid waves, heat transfer and forced vibrations.
Electrical & Electronic Engineers use differential equations to model electromagnetic waves to determine interference effects from electromagnetic fields (EMF) and to design electrical machines such as electric motors.
Simulation of an electric motor used in electric vehicles and a wide range of industries to drive pumps, fans and machines, courtesy of Ansys Inc.
Mechanical Engineers use differential equations to model heat transfer in systems such as power stations and building heating and cooling.
Aerospace Engineers use differential equations to model the flow of air around aircraft.
Structural Engineers use differential equations to analyse the vibration of springs and dampers in structures, reducing the effects of earthquakes on buildings. Please watch the following videos.
Model of circular drum vibration.
Public domain image from the Wikimedia Commons
Chemical Engineers use differential equations to model chemical reactions and fluid flow in processes such as oil and metals refining.
To get a better understanding of differential equations review these lessons and quizzes from the following links:
Now go to the Interactive graphical tool from geogebratube.org
All Khan Academy content is available for free at http://www.khanacademy.org/
Paul's Online Math Notes
MIT Open Courseware Differential equations
Interactive Maths exercises with solutions
SOS Mathematics exercises with solutions
From Mathematics Resources for Education and Industry site, registration and login required here, to complete at least 2 of the exercises below.
From the integralmaths.org ‘My home’ home page:
Select
1. Mathematics Resources for Level 3 Engineering
2. Find a resource by mathematical content
3. Calculus
Then find the following lessons on;
On completion of the exercises you may wish to review the solutions contained in;
When you have completed the lessons, verify your knowledge with this multiple choice 1st and 2nd Order Differential Equation Quiz
If you need to repeat the quiz, click here (please note: you will not be able to repeat the first quiz)
PLEASE NOTE:
An example of one of the many engineering applications of partial differential equations describes airflow, known as the Navier-Stokes equations. These equations are the basis for Computational Fluid dynamics or CFD used to model air flow around aircraft, ships and cars.
CFD airflow simulation around the 2009/2010 Basilisk Performance team's CO2 dragster entry for F1 in Schools which won Best Engineered Car. CFD was also used by the Australian Cold Fusion team who won the world championship in 2012.
Original Image courtesy of Symscape licensed under the Creative Commons Attribution 3.0 Unported license.
Partial differential equations are part of the national engineering unit MEM23007A Apply calculus to engineering tasks. More details on unit content from training.gov.au
Required knowledge: Partial differential equations.
The prerequisite for this module are MEM23004A Apply technical mathematics or equivalent. More details on this unit’s content from training.gov.au
A differential equation is partial if its solution depends on more than one variable and ordinary if its solution depends on a single variable.
For a function of two variable z = ƒ(x,y) we can keep one variable such as y fixed and treat our function ƒ as a function of x only, then calculate the derivative of f with respect to x (if it exists). Our new function is the partial derivative of ƒ with respect to x with the symbol .
Partial differential equations often model multidimensional systems such as systems in three dimensional spaces.
Electrical & Electronic Engineers use partial differential equations to model electromagnetic waves to determine interference effects from electromagnetic fields (EMF), and to design radio and microwave systems.
Simulation of microwave heating in an oven
Image from Adina R&D, Inc
Magnetic fields are used in the levitation of high speed trains, watch the video below on how high speed trains use magnetic fields.
They are also used in superconducting quantum levitation, watch this video on quantum levitation.
Mechanical Engineers use partial differential equations to model heat transfer in systems such as electronic products, vehicle brakes and building air conditioning.
Simulation of temperature and cooling air flow in an electronic product from Autodesk Simulation CFD.
Aerospace Engineers use partial differential equations to model the flow of air around aircraft prior to wind tunnel testing such as this Boeing 737 Max test, watch the video below.
Structural Engineers use partial differential equations to analyse the vibration of plates in structures, see the Resonance Experiment video below.
Structural vibration simulation from Autodesk
To get a better understanding of partial differential equations review these lessons:
For a uniformly heat conducting cylindrical rod, the absolute temperature T at time t is given by T(x,t). Then T satisfies the partial differential equation
where, δ is a constant called the diffusivity of the rod material.
For the three dimensional case we would have a function with respect to the x, y, z coordinate system T(x,y,z,t) and T would satisfy the partial differential equation
In general, partial differential equations are much more difficult to solve analytically than ordinary differential equations. When an analytical solution is not possible numerical methods using computer software are often applied, see solution details here.
When you have completed the lessons, verify your knowledge with this multiple choice Partial Differentiation Quiz
If you need to repeat the quiz, click here (please note: you will not be able to repeat the first quiz)
PLEASE NOTE:
Authorised by the Director, Centre for University Pathways and Partnership
2 May, 2018
Future Students | International Students | Postgraduate Students | Current Students
© University of Tasmania, Australia ABN 30 764 374 782 CRICOS Provider Code 00586B
Copyright | Privacy | Disclaimer | Web Accessibility | Site Feedback | Info line 13 8827 (13 UTAS)