Bionics Lab › Education > Classes > MAE 82 - Mathematics of Engineering

Classes

 

Instructor

Jacob Rosen
Office: Engineering IV Building, Room 37-146
Voice Office: 831.459.5302
e-mail: jacobrosen@ucla.edu
Anonymous Email (For the Subject Line use: MAE 82A)
Office Hours: Wed 2:00-3:00

MAE 82

Mathematics of Engineering

Course Summary:
Methods of solving ordinary differential equations in engineering. Review of matrix algebra. Solutions of systems of first- and second-order ordinary differential equations. Introduction to Laplace transforms and their application to ordinary differential equations. Introduction to boundary value problems. Nonlinear differential equations and stability

Prerequisite: (Matlab)


Table of Contents

Assignments & Grading
Textbooks
TA Office Hours
Syllabus
Book Notes / Problems (11th Edition)
Homework Problems / Assigned Reading
Class Notes
Video Clips (YouTube)
Supplement References
Matlab / Simulink


Assignments & Grading

Plan
Load

Content Grading

A

High


Textbook Pro. - Part 1 & 2
Project

Matlab Required: Yes


Problem Sets - 20%
Projects - 9%
Attendance - 5%
Exam 1 Mid - 33%
Exam 2 Final - 33%

B

Med


Textbook Pro. - Part 1
Project

Matlab Required: Yes


Problem Sets - 10%
Projects - 9%
Attendance - 5%
Exam 1 Mid - 38%
Exam 2 Final - 38%
C

Med

Textbook Pro. - Part 1 & 2

Matlab Required: No



Problem Sets - 20%
Attendance - 5%
Exam 1 Mid - 37.5%
Exam 2 Final - 37.5%
D

Low

Textbook Pro. - Part 1

Matlab Required: No


Problem Sets - 10%
Attendance - 5%
Exam 1 Mid - 42.5%
Exam 2 Final - 42.5%




Textbooks

Textbook (Required): Elementary Ordinary Differential Equations and Boundary Value Problems”. William E. Boyce & Richard C. DiPrima, 11th Edition,  Wiley 2017
ISBN: 978-1-119-32063-0

Textbook (supplement): Differential Equations with Boundary-Value Problems" Zill et al. 7th / 8th editions, Cengage Learning
ISBN-13: 978-1111827069 ISBN-10: 1111827060

Textbook (supplement): Differential Equations, A Primer for Scientists and Engineers, Christian Constanda

Textbook (supplement): Engineering Diffrential Equations, Theory & Applications, Bill Goodwine - Errata

Textbook (supplement): Differential Equations and Thier Applications, Martin Braun


TA Office Hours

Changyeob Shin (TA)
Office: Engineering IV
Office Hours: Mon 13:00 - 15:00; ENG IV 43-147
E-mail: shinhujune@ucla.edu

Yi-jui Chang (TA)
Office: Engineering IV
Office Hours: Wed 15:00 - 17:00; ENG IV 43-147
E-mail: michaelchang0928@gmail.com


Syllabus


1. First & Second order Linear Differential equations

1.1 Classifications of differential equations, linear/nonlinear, order

1.2 First order Linear Differential equations
1.2.1 Method of integrating factors
1.2.2 Variation of parameters

1.3. First Order non linear Differential Equations
1.3.1 Separable equations
1.3.2 Integrating factor
1.3.3 Change of variables and parametric solutions

1.4 Second order Linear Differential equations
1.4.1 Homogeneous equations with constant coefficients
1.4.2 Characteristic equation, distinct and repeated solutions
1.4.3 Nonhomogeneous equations, method of undetermined coefficients and variation of parameters
1.4.4 Applications to selected engineering problems

2. High order linear equations

2.1 Characteristic equation. Consider distinct and repeated solutions
2.2 Variation of Parameters (Wronskian)
2.3 Undetermined coefficients

[Mid Term Exam]

3. Series solution method

3.1 Power series solution for a linear equation (general n-th order) with constant coefficients
3.2 Introduction to the use of power series when coefficients are not constant
3.3 Series solution near an ordinary point
3.4 Series solution near a regular point
3.5 Error and accuracy of power series solution representation

4. Laplace Transform

4.1 Definition and convergence of Laplace Transform
4.2 Solution to initial value problems
4.3 Linear differential equations with discontinuous forcing term (Dirac delta, step fiunction)

5. Systems of linear equations

5.1 Review of matrices, eigenvalues, eigenfunctions
5.2 Homogeneous and no homogeneous linear systems with constant coefficients
5.3 Discussion on the function e^At
5.4 Systems of higher order Linear Equations and state space representation
5.5 Nonlinear Systems of Differential equations (Equilibrium points)
5.6 Concepts of stability and steady state Solutions
5.7 Special functions, Dirac Delta, Step function

[Final Exam]

6. Numerical Methods

Numerical Methods for first order equations
7.1 Introduce Euler Method
7.2 Runge-Kutta Method
7.3 Multistep Methods
7.4 Errors and stability


Book Notes / Problems (11th Edition)

Book Chapter - Notes

Clapter 1 - Intoduction
Clapter 2 - First Order Diff Eq.

Clapter 3 - Second Order Linear Diff. Eq.
Clapter 4 - Higher Order Linear Diff. Eq.
Clapter 5 - Series Solution
Clapter 6 - Laplace Transform
Clapter 7 - Systems of First Order Diff. Eq.
Clapter 8 - Numerical Methods
Clapter 9 - Non Linear Diff. Eq. and Stability

Book Problems (11th Edition)

Clapter 1 - Intoduction
Clapter 2 - First Order Diff Eq.

Clapter 3 - Second Order Linear Diff. Eq.
Clapter 4 - Higher Order Linear Diff. Eq.
Clapter 5 - Series Solution
Clapter 6 - Laplace Transform
Clapter 7 - Systems of First Order Diff. Eq.
Clapter 8 - Numerical Methods
Answers to Problems


Homework Problems / Assigned Reading

Week

Reading
Assignment
& Pre-Class

Class / Homework
Submission

1a


Book Ch. 1

Class Notes
Class 00 : Course Info.
Class 01 : Introduction.

Post Week Homework - HW1
Part 1
P. 8, Pr. 2
P. 8-9, Pr. 11-16
P. 9, Pr. 20
P. 15, Pr. 3
P. 22, Pr. 1
P. 22, Pr. 7

Part 2
P. 8, Pr. 3
P. 9, Pr. 19
P. 15, Pr. 4
P. 22, Pr. 3
P. 22, Pr. 8

Project 1 (Pendulum)


HW Due Date -
Friday Week 2 (Midnight)
Project Due Date -
Monday Week 3
(Midnight)

(*) Note 1:
Directional Field Example (Matlab)
(*) Note 2:
Vectorfiled2D Matlab Help

HW 1 Solution
Project 1 Solution
Project 1 - Matlab (M-File)

Preclass
None


1b


Book Ch. 2
Sections
2.1-2.4


Pre - Class
Pg. 58 Pr. 26

Class Notes
Class 02 : First Order Diff Eq. - Linear


Preclass

S1


 

Section Notes


None

2a

 

Book Ch. 2
Sections
2.5

Pre - Class
Pg. 67 Pr. 2


Class Notes
Class 02 : First Order Diff Eq. - Linear

Post Week Homework - HW2

Part 1

P. 31, Pr. 3
P. 38, Pr. 3
P. 39, Pr. 25
P. 48, Pr. 11
P. 68, Pr. 20
P. 75, Pr. 8


Part 2
P. 31, Pr. 12
P. 38, Pr. 6
P. 39, Pr. 25
P. 48, Pr. 14
P. 75, Pr. 21

Project 2 (Alle Effect)
(*) Note: Problem 3 in the project refers to phase portrait which is similar to draw the direction field and integral curves of N vs t. You may refer section 2.5 Autonomous Equations and Population Dynamics on page 58 to 66 of the textbook. The figures 2.5.3, 2.5.6, and 2.5.8 are good examples.

HW Due Date -
Friday Week 3 (Midnight)
Project Due Date -
Monday Week 4
(Midnight)

HW 2 Solution
Project 2 Solution
Project 2 - Matlab (M-File)

Preclass

 

2b

 

Book Ch. 2
Sections
2.6, 2.8


Pre - Class
Pg. 75 Pr. 1


Class Notes
Class 03 : First Order Diff Eq. - Non-Linear.

Preclass

Fri - HW1

S1


 

Section Notes


3a

 

Book Ch. 3
Sections
3.1-3.4

 

Pre - Class
Pg. 131 Pr. 7

Class Notes
Class 04 : Second Order Diff Eq. - Homogeneous

MathLets - POLES AND VIBRATIONS

Post Week Homework - HW3

Part 1

P. 109, Pr. 9 (real roots)
P. 125, Pr. 6 (complex roots)
P. 132, Pr. 2 (identical roots)
P. 141, Pr. 17 (Non Homo)
P. 146, Pr. 11 (Non Homo)
P. 168, Pr. 9
(MKC)

Part 2
P. 109, Pr. 11
P. 125, Pr. 7
P. 132, Pr. 8
P. 141, Pr. 21
P. 168, Pr. 8

Project 3 (Bungee Jumping)


HW Due Date -
Friday Week 4 (Midnight)
Project Due Date -
Monday Week 5
(Midnight)


HW 3 Solution
Project 3 Solution
Project 3 - Matlab (M - File)


Mon - Project 1

Preclass

3b

 

Book Ch. 3
Sections
3.7


Pre - Class
Pg. 158 Pr. 17




Preclass

Fri - HW2

4a

 

Book Ch. 3
Sections
3.5-3.6


Pre - Class
Pg. 141 Pr. 1

Post Week Homework - HW4

Part 1

P. 180 Pr. 16
P. 180 Pr. 29 a, b, c
P. 184 Pr. 1
P. 188 Pr. 3


Part 2
P. 180 Pr. 9
P. 184 Pr. 2
P. 188 Pr. 9


Project 4 (Tacoma Nerrows)

HW Due Date -
Friday Week 5 (Midnight)
Project Due Date -
Monday Week 6
(Midnight)

HW 4 Solution
Project 4 Solution
Project 4 - Matlab (M-File)


Mon - Project 2

Preclass

4b

 

Book Ch. 3
Sections
3.8


Pre - Class
Pg. 167 Pr. 4





Preclass

Fri - HW3

5a

 

Book Ch. 4
Sections
4.1-4.2


Pre - Class
Pg. 180 Pr. 2


HW 5
Prep for the midterm using practice exam - No Need to submit


P. 31 Pr. 1
P. 75 Pr. 1
P. 167 Pr. 4,5
P. 184 Pr. 12
P. 188 Pr. 3


Project 5 (Beam Buckling)
Problem 1 a-e p. 12
Problem 2 a,b p. 15
Problem 3 a p. 16

HW Due Date - No Need to submit
Project Due Date -
Monday Week 7
(Midnight)


Project 5 Solution

Project 5 - Matlab (M-File)

Mon - Project 3

Preclass

5b

 

Book Ch. 4
Sections
4.3-4.4


Pre - Class
Pg. 184 Pr. 2




Preclass

Fri - HW4

6a

 







Mon - Project 4

Preclass

6b

 

Mid Term
Thursday



In Class (Open Book & Open Notes)



7a

 

Book Ch. 5
Sections
5.1

Alternative Read

Series Solution (Zill)

Series Solution (Xie)

Pre - Class
Pg. 195 Pr. 7



Post Week Homework - HW6

Part 1

P. 218 Pr. 3
P. 218 Pr. 13
P. 223 Pr. 10
P. 229 Pr. 10
P. 230 Pr. 12


Part 2
P. 218 Pr. 20
P. 223 Pr. 12
P. 229 Pr. 13


Project 6 - None
HW Due Date -
Friday Week 8 (Midnight)

HW 6 Solution




Preclass

7b

 

Book Ch. 5
Sections
5.2-5.3

Alternative Read

Series Solution (Zill)

Series Solution (Xie)

Pre - Class
Pg. 204 Pr. 2






Preclass

8a

 

Book Ch. 5
Sections
5.5-5.6

Alternative Read

Series Solution (Zill)

Series Solution (Xie)

Pre - Class
Pg. 223 Pr. 1







Mon - Project 5

Preclass

8b

 

Book Ch. 6
Sections
6.1-6.2

Alternative Read

Laplace Transform

Pre - Class
Pg. 255 Pr. 8


Post Week Homework - HW7

Part 1

P. 255 Pr. 6,15,18
P. 263 Pr. 22
P. 268 Pr. 2
P. 269 Pr. 11
P. 273 Pr. 7
P. 274 Pr. 13
P. 279 Pr. 6

Part 2
P. 255 Pr. 19
P. 268 Pr. 6
P. 269 Pr. 12
P. 279 Pr. 9


Project 7 (Bessel)

HW Due Date -
Friday Week 9 (Midnight)
Project Due Date -
Monday Week 10
(Midnight)

HW 7 Solution
Project 7 Solution
Project 7 - Matlab (M-File)


Fri-HW6

Preclass

9a

 

Book Ch. 6
Sections
6.3-6.4

Alternative Read

Laplace Transform

Pre - Class
Pg. 268 Pr. 1






Mon -
Project 7

Preclass

9b

 

Book Ch. 6
Sections
6.5-6.6

Alternative Read

Laplace Transform


Pre - Class
Pg. 273 Pr. 1

Revew at Home - Book Ch. 7.1-7.4

Post Week Homework - HW8

Linear Algebra Review

(Review Only No Need To Submit)
P. 293 Pr. 4 a b c
P. 293 Pr. 5 a b c 6 a b c
P. 294 Pr. 8 a b c d
P. 294 Pr. 14
P. 297 Pr. 17
P. 303 Pr. 4,7,12,19



Project 8 (Mayfair Diner)

HW 8 Solution
Project 8 Solution
Project 8 - Matlab (M-File)


Preclass

Fri- HW7

10a

 

Book Ch. 7
Sections
7.5-7.8

Alternative Read

System of ODE


Pre - Class
Pg. 343 Pr. 1

Post Week Homework - HW9

Part 1 (*)

P. 318 Pr. 11 (Homo - Regular)
P. 328 Pr. 7 (Homo - Complex)
P. 344 Pr. 17 (Homo - Repated)
P. 351 Pr. 2,3,4
(Non Homo)
Note:
Sketck phase portraits for all solutions (See Video)


Part 2 (*)
P. 318 Pr. 12 (Homo - Regular)
P. 328 Pr. 8 (Homo - Complex)
P. 344 Pr. 18 (Homo - Repated)
P. 351 Pr. 1,5,8 (Non Homo
)
(*) Note: Sketck phase portraits for all solutions (See Video)

Project 9 (Arms Races)
Notes (Typos)
1. Q3a - second eq. - dy/dt
2. Q3b - y*=e+fx

Due date - Friday Week 10
HW Due Date -
Friday Week 10 (Midnight)
Project Due Date -
Friday Week 10
(Midnight)


HW 9 Solution
Project 9 Solution
Project 9 - Matlab (M-File)



Mon -
HW8
Project 8

Preclass

10b

 

Book Ch. 7
Sections
7.9

Alternative Read

System of ODE

Pre - Class
Pg. 351 Pr. 1





Preclass

Fri or Mon- HW9
Project 9

Exam 2 - Final

   

Prep for the Final using
practice Exam - No Need to submit

P. 218 Pr. 4
P. 218 Pr. 6
P. 223 Pr. 6
P. 229 Pr. 5
P. 268 Pr. 6
P. 273 Pr. 3
P. 273 Pr. 6
P. 351 Pr. 1
P. 351 Pr. 4
P. 352 Pr. 6
P. 352 Pr. 8

 

 


Class Notes

Post Class Slides (Annotated)

Class 00
: Course Info.
Class 01 : Introduction.
Class 02 : First Order Diff Eq. - Linear
Class 03 : First Order Diff Eq. - Non-Linear.
Class 04 : Second Order Diff Eq. - Homogeneous
Class 05 : Second Order Diff Eq. - Nonhomogeneous.
Class 06 : Second Order Diff Eq. - Nonhomogeneous - Force Vibration
Class 07 : High Order Diff. Eq. - Homogeneous.
Class 08 : High Order Diff. Eq. - Nonhomogeneous.
Class 09 : Series Soultion - Intro.
Class 10 : Series Soultion - Ordinary Point.
Class 11 : Series Soultion - Regular Singular Point.
Class 12 : Laplace - Intro & Diff Eq.
Class 13 : Laplace - Step Function.
Class 14 : Laplace - Impulse Function.
Class 15 : Laplace - Systems (S- Plane).
Class 15 Appendix : Linear Algebra - Review.
Class 16 : System of First Order Linear Diff. Eq. - Homogeneous.
Class 17 : System of First Order Linear Diff. Eq. - NonHomogeneous.
Class 18 : Numerical Approuch - First Order
Class 19 : Numerical Approuch - System

Pre Class Slides (Blank)

Class 01
Class 02
Class 03
Class 04
Class 05
Class 06
Class 07
Class 08
Class 09
Class 10
Class 11
Class 12
Class 13
Class 14
Class 15
Class 16
Class 17
Class 18
Class 19

TA Sessions

TA1 Session 01 , TA2 Session 01
TA1 Session 02 , TA2 Session 02
TA1 Session 03 , TA2 Session 03
TA1 Session 04 , TA2 Session 04
TA1 Session 05 , TA2 Session 05
TA1 Session 06 , TA2 Session 06
TA1 Session 07 , TA2 Session 07
TA1 Session 08 , TA2 Session 08
TA1 Session 09 , TA2 Session 09
TA1 Session 10 , TA2 Session 10


Video Clips (YouTube)

Class 01 : Introduction
e (Euler's Number) - Numberphile
The number e is everywhere
The number e explained in depth for (smart) dummies
How One Man Changed the High Jump Forever
An athlete uses physics to shatter world records (TED Ed)

Class 03 : First Order Diff Eq. - Linear
Human Population Growth - Crash Course Ecology #3
Overpopulation – The Human Explosion Explained
OVERPOPULATED - BBC Documentary
Human Population Through Time
A Magic Number? - American Scientist

Class 04 - Second Order Diff Eq - Homogeneous
Car Suspention
Hydraulic shock absorbers
Vibration Tests - Pilots
Taipei 101 - Earthquake 2002
Taipei 101 Damper ( Typhoon Soulik)
Boeing 787 Dreamliner wingflex
Wings of the Boeing 787
Two Towers (Model) with a damper
Trailer Sway

Class 05/06 - Second Order Diff Eq - Non Homogeneous
Collapse of the Tacoma Narrows Bridge
Vibrations in a Steel Ruler
Two Towres (Model) earthquake Protection
CSeries Ground Vibration Test (GVT)
CH 650 E Ground Vibration Tests
Chinook CH-47 Helicopter Ground Resonance Test
Braking a Glass
Wine glass resonance in slow motion
SDOF Resonance Vibration Test
Plate Resonance Experiment
Double Pendulum
Square wave
Fourier series

Class 07-08 High Order Diff. Eq. (Beams)
Tensile Testing
Beam - Finite element analysis of arch bridge Bending
Vibrations in Rotor | Resonance | Critical Speed | Whirling
Understanding Resonance Mode Shapes


Class 9-11 Series Solution
Taylor series - Wikipedia

Class 12-15 Laplace / Fourier Transforms
The Laplace Transform - A Graphical Approach
Fourier Transform, Fourier Series, and frequency spectrum
Introduction to the Fourier Transform
Impulse - Golf Ball
Football to the Face
Effects of a Concussion on Brain Cells - Medical Animation
Concussion / Traumatic Brain Injury (TBI)
What happens when an aircraft crashes

Class 16
Eigenvalues and eigenvectors (Wikipedia)
What is an Eigenvector?
Vibration Mode Shapes of a Tall Building
Sketching Phase Portraits



Supplement References


First & Second Order Diff Eq.: ODE 1ODE 2
Laplace (Tables): Laplace Transform
Laplace Transform
Seriers Solution
System of First Order ODE
The Theory of the Rainbow - Ariy' Diff Eq.


Matlab / Simulink


Matlab Tutorilas [PDF]


MATLAB Tutorial
Simulink
Simulink - Getting Started [PDF]
Simulink - User Guide [PDF]

MATLAB Books
Free access books for UCLA stdents (on campus or via VPN)
MATLAB Mathematical Analysis, César Pérez López
MATLAB Matrix Algebra, César Pérez López
MATLAB Graphical Programming, César Pérez López
MATLAB Differential Equations, César Pérez López

YouTube (Matlab)
The Complete MATLAB Course: Beginner to Advanced

YouTube (Simulink Tutorials)
Introduction to Simulink - Webinar
Simulink: Solving Second Order Differential Equation
Simulink: Solving Second Order Differential Equation (Part 1)
Simulink: Solving Second Order Differential Equation (Part 2)
Simulink: Solving a Set of three First Order Differntial Equations
Simulink: Solving a Set of two Second Order Differntial Equations
Simulink: Dyanmics Ferrari / Fiat (Laplace)

Access to MATLAB (UCLA)

1) Create SEASnet account
www.seasnet.ucla.edu > Accounts > Create Account

(After applying through website, you need to visit SEAS help desk for getting account)

2) Remote Connection
www.seasnet.ucla.edu > LABS > Seasnet Terminal Server > Terminal Server (Remote Desktop Connect) > Follow instructions


Matlab (Scripts) / Simulink (Models)

Intro to Matlab - Matlab Script

 


Paradoxes & Dilemas

The Prisoner's Dilema

Wikipedia - The Prisoner's dilemma
YouTube - The Prisoner's Dilemma
YouTube - The Iterated Prisoner's Dilemma - Cooperation

The Trolley's Problem

Wikipedia - The Trolley Problem - Wikipedia
YouTube - The Trolley Problem
YouTube - TED-Ed
YouTube - 10 Insane Trolley Problems

Gabriel's Horn

Wikipedia - Gabriel's Horn
YouTube- Gabriel's Horn - The Math

Selective Attention - The Black Gorila

YouTube - Experiment

Wisdon of the Crowd

YouTube - Experiment
YouTube - Explanation
The wisdom of crowds | Karl Mattingly | TEDxBrighton
The Wisdom of Crowds
The wisdom of the crowd (with Professor Marcus du Sautoy)
Statistics

Möbius strip
Wikipedia - Möbius strip
J.S. Bach - Crab Canon on a Möbius Strip
Topology of Möbius Cuts
Unexpected Shapes (Part 1) - Numberphile
Unexpected Shapes (Part 2) - Numberphile
Mobius Strip II Animated Movie
M C Escher Documentary
MC Escher, Images of Mathematics