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: Mathematics 33A, 33B

Assignments & Grading:

Problem Sets - 20%
Projects - 5%
Attendance - 5%
Exam 1 Mid - 35%
Exam 2 Final - 35%

Textbook: Elementary Ordinary Differential Equations and Boundary Value Problems”. William E. Boyce & Richard C. DiPrima, 10th Edition,  Wiley 2012
ISBN-10: 1118157389
ISBN-13: 978-1118157381

Office Time TA / Tutor

Changyeob Shin (TA)
43-147
Office: Engineering IV
Office Hours: Wed 11:00 - 13:00
E-mail: shinhujune@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

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. Systems of linear equations

4.1 Review of matrices, eigenvalues, eigenfunctions
4.2 Homogeneous and no homogeneous linear systems with constant coefficients
4.3 Discussion on the function e^At
4.4 Systems of higher order Linear Equations and state space representation
4.5 Nonlinear Systems of Differential equations (Equilibrium points)
4.6 Concepts of stability and steady state Solutions
4.7 Special functions, Dirac Delta, Step function

5. Laplace Transform

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

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 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 (10th 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
Clapter 9 - Non Linear Diff. Eq. and Stability

 

Homework Problems - Color Code

Syllabus
Book

Chapter Sections
(Read)
Homework  

1a

9.27

Syllabus 1.1
Book Ch. 1

P. 7, Pr. 3
P. 8, Pr. 15-20
P. 9, Pr. 25
P. 16, Pr. 3
P. 24, Pr. 1-6
P. 25, Pr. 8, 12
Due date - Friday Week 2

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

HW 1 Solution



1b

9.29

Syllabus 1.2
Book Ch. 2




 

2a

10.4

Syllabus 1.3
Book Ch. 2


P. 40, Pr. 13, Pr. 16
P. 48, Pr. 1, Pr. 7
P. 50, Pr. 30
P. 62, Pr. 14
P. 63, Pr. 17
P. 101, Pr. 3 Pr. 7
P. 102, Pr. 30
Project 1
(*) Note: Problem 3 in project 1 refers to phase portrait which is similar to draw the direction field and integral curves of N vs t. You may refer session 2.5 Autonomous Equations and Population Dynamics on page 70 to 88 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 1 Solution

 

 

2b

10.6

Syllabus 1.4
Book Ch. 3


 

3a

10.11

Syllabus 1.4
Book Ch. 3

P. 173, Pr. 17
P. 184, Pr. 16, 20
P. 190, Pr. 1, 8, 16
P. 204, Pr. 11
P. 217, Pr. 6, 8
P. 218, Pr. 15
P. 219, Pr. 22 (*)
(*) see explanation at the bottom of P. 218
Project 2
Due date - Friday Week 4

HW 3 Solution
Project 2 Solution
Project 2 - Matlab (M - File)


 

3b

10.13

Syllabus 1.4
Book Ch. 3



 

4a

10.18

Syllabus 2
Book Ch. 4
P. 234 Pr. 24
P. 234 Pr. 39 a, b, c
P. 239 Pr. 3, 8, 12
P. 244 Pr. 10, 13, 14
Due date - Friday Week 5

HW 4 Solution


 

4b

10.20

Syllabus 2
Book Ch. 4




 

5a

10.25

Syllabus 3
Book Ch. 5

Prep for the midterm using practice exam

P. 40 Pr. 13
P. 133 Pr. 3
P. 184 Pr. 24
P. 218 Pr. 17
P. 239 Pr. 1
P. 244 Pr. 5

No Need to submite

Take Home Midterm
Due date - Friday Week 6

 

5b

10.27

Syllabus 3
Book Ch. 5



 

6a

11.1

Mid Term
Tue Week 6 - In Class




 

6b

11.3

Syllabus 3
Book Ch. 5


P. 280 Pr. 1
P. 280 Pr. 17
P. 286 Pr. 5
P. 294 Pr. 15
P. 294 Pr. 17
Due date - Friday Week 7

HW 5 Solution

 

7a

11.8

Syllabus 5
Book Ch. 6

P. 325 Pr. 9, 20, 22, 27
P. 334 Pr. 31
P. 340 Pr. 9
P. 341 Pr. 16
P. 348 Pr. 9
P. 348 Pr. 17
P. 355 Pr. 5
Project 3
Due date - Friday Week 8


HW 6 Solution
Project 3 Solution

 

7b

11.10

Syllabus 5
Book Ch. 6





 

8a

11.15

Syllabus 5
Book Ch. 6


P. 376 Pr. 4 a b c
P. 376 Pr. 6 a b c
P. 376 Pr. 8 a b c d
P. 377 Pr. 18
P. 377 Pr. 23
P. 388 Pr. 6
P. 388 Pr. 8
P. 388 Pr. 14
P. 388 Pr. 22

Due date - Friday Week 9


HW 7 Solution


 

8b

11.17

Syllabus 4
Book Ch. 7




 

9a

11.22

Syllabus 4
Book Ch. 7


P. 405 Pr. 17
P. 418 Pr. 23
P. 438 Pr. 18
P. 447 Pr. 3,4,5

Project 4
Due date - Friday Week 10


HW 8 Solution
Project 4 Solution


 

9b

11.24

Syllabus 4
Book Ch. 7





 

10a

11.29

Syllabus 7
Book Ch. 9


Prep for the Final using
practice exam

P. 280 Pr. 5
P. 286 Pr. 7
P. 294 Pr. 5

P. 348 Pr. 8
P. 348 Pr. 3
P. 447 Pr. 8

P. 447 Pr. 6
P. 447 Pr. 1


 

10b

 

Syllabus 7
Book Ch. 9



 

Exam 2 - Final

       

 

Class Notes

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
Class 20 : ***.


TA Sessions

TA Session 01
TA Session 02
TA Session 03
TA Session 04
TA Session 05
TA Session 06
TA Session 07
TA Session 08
TA Session 09
TA Session 10


Video Clips

Systems

Double Pendulum
Collapse of the Tacoma Narrows Bridge
Vibrations in a Steel Ruler
Car Suspention

CSeries Ground Vibration Test (GVT)
CH 650 E Ground Vibration Tests
Destructs this Chinook helicopter
Hydraulic shock absorbers
Braking a Glass
Wine glass resonance in slow motion
Vibration Tests On Pilots
SDOF Resonance Vibration Test
Plate Resonance Experiment
Taipei 101 - Earthquake 2002
Taipei 101 Damper ( Typhoon Soulik)

Laplace / Fourier Transforms
The Laplace Transform - A Graphical Approach
Fourier Transform, Fourier Series, and frequency spectrum
Introduction to the Fourier Transform

Beams
Tensile Testing
Beam - Finite element analysis of masonry arch bridge


Supplement References

First & Second Order Diff Eq.: ODE 1ODE 2
Laplace (Tables): Laplace Transform
Laplas Transform
Seriers Solution
System of First Order ODE

Book: Christian Constanda, Differential
Equations, A Primer for Scientists and Engineers

Book: Bill Goodwine, Engineering Diffrential Equations, Theory & Applications

The Theory of the Rainbow - Ariy' Diff Eq.

(*) Matlab

MATLAB Tutorial

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

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
Pandulum (Linear / Non Linear) Simulink Model - Class 2