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

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


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 for Engineers, Wei-Chau Xie, 2010, Cambridge University Press
ISBN-13 978-0-521-19424-2, ISBN-10 0-521-19424-5

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) 43-147
Office: Engineering IV
Office Hours: Wed 15:00 - 17:00
E-mail: shinhujune@ucla.edu

Haoran Wang (TA) 43-147
Office: Engineering IV
Office Hours: Mon 15:00 - 17:00
E-mail: haoranwang@ucla.edu


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

Color Color
Syllabus
Book

Week Sections
(Read)
Homework  

1a


Syllabus 1.1
Book Ch. 1

HW1
Plan A
P. 8, Pr. 1
P. 8-9, Pr. 11-16
P. 9, Pr. 21
P. 15, Pr. 3
P. 22, Pr. 1-4
P. 22, Pr. 9, 10

Plan B
P. 8, Pr. 1
P. 9, Pr. 21
P. 15, Pr. 3
P. 22, Pr. 1
P. 22, Pr. 10

Project 1 (Pendulum)


Due date - Friday Week 2

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

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



1b


Syllabus 1.2
Book Ch. 2




 

2a

 

Syllabus 1.3
Book Ch. 2


HW2
Plan A

P. 31, Pr. 2, Pr. 11
P. 38, Pr. 2, Pr. 5
P. 39, Pr. 25
P. 48, Pr. 10, Pr. 13
P. 68, Pr. 19
P. 75, Pr. 6, Pr. 20

Plan B
P. 31, Pr. 11
P. 38, Pr. 5
P. 39, Pr. 25
P. 68, Pr. 19
P. 75, Pr. 20

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.

Due date - Friday Week 3

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

 

 

2b

 

Syllabus 1.4
Book Ch. 3


 

3a

 

Syllabus 1.4
Book Ch. 3

HW3
Plan A

P. 109, Pr. 8 (real roots)
P. 125, Pr. 5 (complex roots)
P. 132, Pr. 1 (identical roots)
P. 141, Pr. 18 (Non Homo)
P. 146, Pr. 12 (Non Homo)
P. 168, Pr. 8,11,13,15,16
(MKC)

Plan B
P. 109, Pr. 8
P. 125, Pr. 5
P. 132, Pr. 1
P. 141, Pr. 18
P. 168, Pr. 8

Project 3 (Bungee Jumping)


Due date - Friday Week 4

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


 

3b

 

Syllabus 1.4
Book Ch. 3



 

4a

 

Syllabus 2
Book Ch. 4
HW4
Plan A

P. 180 Pr. 17
P. 180 Pr. 29 a, b, c
P. 184 Pr. 3, 9, 11
P. 188 Pr. 4, 9, 10


Plan B
P. 180 Pr. 17
P. 184 Pr. 3, 11
P. 188 Pr. 4, 9


Project 4 (Tacoma Nerrows)

Due date - Friday Week 5

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


 

4b

 

Syllabus 2
Book Ch. 4




 

5a

 

Syllabus 3
Book Ch.5

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


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


Project 5 - Due date - Friday Week 6

Project 5 Solution

Project 5 - Matlab (M-File)

 

5b

 

Syllabus 3
Book Ch. 5
Seriers Solution



 

6a

 

Syllabus 3
Book Ch. 5
Seriers Solution


HW6
Plan A

P. 218 Pr. 1
P. 218 Pr. 12
P. 223 Pr. 5
P. 229 Pr. 10
P. 230 Pr. 12


Plan B
P. 218 Pr. 12
P. 223 Pr. 5
P. 229 Pr. 10


Project 6 (Mid Term)

Due date - Friday Week 7

HW 6 Solution


 

6b

 

Mid Term
Thursday Week 6 - In Class


Take Home Midterm
(Project 6)
Due date - Friday Week 7




 

7a

 

Syllabus 4
Book Ch. 6
Laplace Transform


HW7
Plan A

P. 255 Pr. 7,15,19
P. 263 Pr. 22
P. 268 Pr. 4
P. 269 Pr. 11
P. 273 Pr. 7
P. 274 Pr. 13
P. 279 Pr. 5

Plan B
P. 255 Pr. 7,15,19
P. 263 Pr. 22
P. 268 Pr. 4
P. 269 Pr. 11
P. 273 Pr. 7
P. 274 Pr. 13
P. 279 Pr. 5

Project 7 (Bessel)

Due date - Friday Week 8


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

 

7b

 

Syllabus 4
Book Ch. 6
Laplace Transform





 

8a

 

Syllabus 4
Book Ch. 6
Laplace Transform


HW8
Linear Algebra Review

Plan A
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

Plan B
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)

Due date - Friday Week 9


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


 

8b

 

Syllabus 5
Book Ch. 7
System of ODE




 

9a

 

Syllabus 5
Book Ch. 7
System of ODE


HW9
Plan A

P. 318 Pr. 12
P. 328 Pr. 18
P. 344 Pr. 17
P. 351 Pr. 2,3,5


Plan B
P. 318 Pr. 12
P. 328 Pr. 18
P. 344 Pr. 17
P. 351 Pr. 2,3,5


Project 9 (Arms Races)

Due date - Friday Week 10


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


 

9b

 

Syllabus 5
Book Ch. 7
System of ODE





 

10a

 

Syllabus 6
Book Ch. 9


HW 10
Prep for the Final using
practice exam - No Need to submite

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


No Need to submite practice exam

 

10b

 

Syllabus 6
Book Ch. 9



 

Exam 2 - Final

       

 


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

Class 02 : First Order Diff Eq. - Linear
Human Population Growth - Crash Course Ecology #3
Overpopulation – The Human Explosion Explained
OVERPOPULATED - BBC Documentary
Human Population Through Time

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

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

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 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



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