Engineering 112Engineering 112Engineering Methods IIEngineering Methods II

Bob LeMasterBob LeMasterCollege of EngineeringCollege of Engineering

University of Tennessee MartinUniversity of Tennessee Martin

Course FormatCourse Format

Two two-hour instruction/laboratory sessions per week.

Session One: Introduction to Linear Algebra and Computer Programming - Matlab

Instructors: Bob LeMaster (2 Sections)Robert Benton ( 1 Section)

Session Two: Introduction to Computer Aided Design Software - AutoCAD

Instructor: Ed Wheeler (all sections)

How will my course grade be How will my course grade be determined?determined?

A grade will be assigned for each session by the instructor.The grade for each session will be combined into an overall course grade - equally weighted.You must pass both sessions to pass the course.

What is Linear Algebra?What is Linear Algebra?A set of mathematical rules which are used to

manipulate arrays of numbers.

The theoretical foundations were developed by mathematicians studying the properties of linear equations.

1.95.1x2.6x1.6x15.01.4x9.1x3x

11.23x4.5x2x

321

321

321

=++

=++

=++

=

1.915.011.2

xxx

5.12.61.61.49.13.03.04.52.0

3

2

1

BAx =

What is MATLAB?What is MATLAB?

A computer programming language developed to perform numeral computations encountered in

science and engineering operations.

MATrix LABoratory

Why do I need to know linear Why do I need to know linear algebra?algebra?

Systems of linear and nonlinear equations are encountered in every engineering discipline.

Controls

Circuits

Structures

Kinematics

Stress

Power Systems

IndustrialProcessSimulation Fluids

Finite Element Stress AnalysisFinite Element Stress Analysis(SSME Turbopump Example)(SSME Turbopump Example)

Deflection and stress analysis of the Space Shuttle Main Engine (SSME) Alternate Turbopump Development (ATD) High Pressure Oxidizer Turbopump (HPOTP) required the simultaneous solution of approximately 750,000 nonlinear equations using methods that you will be exposed to in this class.

Computational Fluid DynamicsComputational Fluid Dynamics(Space Shuttle Example)(Space Shuttle Example)

Calculation of temperature distributions required the simultaneous solution of thousands of nonlinear equations using techniques that you will be exposed in this course.

Why do I need to learn to Why do I need to learn to program in MATLAB?program in MATLAB?Computers dominate the industrial workplace.

The fundamental programming concepts learned using MATLAB will enable you to adapt to other programming environments.

You will be required to use MATLAB in future courses.

MATLAB is widely used in industry.

What will I be able to do at the What will I be able to do at the end of the course?end of the course?

Represent systems of equations using matrix notationCompute the Euclidean norm of column or row matricesPerform basic linear algebra operations symbolically, manually, and in MATLAB Solve systems of equations using Cramers ruleSolve systems of equations using MATLABFind the eigenvalues and eigenvectors of matricesDevelop computational software that employs structured programming control methodologyDisplay computed results in tabular and graphical formats

What will be expected of me?What will be expected of me?

Attend classAttendance will be takenIt is professional to be prompt and to have good attendance

Your future employers will expect you to have acquired these work habits

Attendance record will be taken into account for borderline grade situations

What will be expected of me?What will be expected of me?(Continued)(Continued)

Do homeworkYou will not be able to do well on exams without being proficient in solving problemsThere is no other way to become proficient at problem solving than to solve problemsKeep a notebook of all homework

Will aid in studying for examsWill aid review for EIT prior to graduationFacilitates grading

What will be expected of me?What will be expected of me?(Continued)(Continued)

Pass four examsEach exam will be worth 15% of session final grade.

Out of class assignments will be worth 20% of session final grade.

Pass Comprehensive FinalWill be worth 20 % of session one final grade.

Resource MaterialsResource Materials(Text)(Text)

Etter, D.M., D.C. Kuncicky, Introduction to MATLAB, Prentice-Hall.

Resource MaterialsResource Materials(Reference)(Reference)

Getting Started html document onMathworks web site

http:/www.mathworks.com/access/helpdesk/help/helpdesk.shtml

Click on Getting Started

Click on This manual in PDF

Open or save to disk

Print

Resource MaterialsResource Materials(Linear Algebra Reference Texts)(Linear Algebra Reference Texts)

Linear EquationsLinear Equations(Two Dimensions)(Two Dimensions)

y

x

y=mx+b

b

m

1

bmxy +=

One independent variable (x)One dependent variable (y)y-intercept (b)slope (m)

Linear EquationsLinear Equations(Three Dimensions)(Three Dimensions)

=

=

=

=++

nbintercept x

mb interceptx

lbintercept x

bnxmxlx

3

2

1

321

1x

2x

3x

Matrix NotationMatrix Notation

=

1.915.011.2

xxx

5.12.61.61.49.13.03.04.52.0

3

2

1

[ ]{ } { }BxA =

[ ]

{ } { }

=

=

=

9.10.152.11

,

1.56.26.14.11.90.30.35.40.2

B xxx

x

A

3

2

1

Symbolic FormatSymbolic Format

Matrix FormatMatrix Format

1.95.1x2.6x1.6x15.01.4x9.1x3x

11.23x4.5x2x

321

321

321

=++

=++

=++

Algebraic FormatAlgebraic Format

Matrix NotationMatrix Notation(Symbols)(Symbols)

[ ] Two dimensional array of numbers{ } Column array of numbers

Row array of numbers

Matrix NotationMatrix Notation(Size Designation)(Size Designation)

[ ] [ ] nm,or nm, Two dimensional array of numbershaving m rows and n columns{ } { }mor m Column array of m numbers

nor n Row array containing n numbers

m is used to designate the number of rowsm is used to designate the number of rowsn is used to designate the number of columnsn is used to designate the number of columns

Matrix NotationMatrix Notation(Text Books)(Text Books)

In many text books, it is standard practice to represent matrices with bold fonts.

[ ]{ } { }BxA =Is equivalent to

Ax=B

Matrix NotationMatrix Notation(Element Addressing)(Element Addressing)

[ ]

=

3,23,1

2,22,1

1,21,1

3,2

AAAAAA

A

column-j and row-i theat located matrix A ofElement A ji,

Matrix NotationMatrix Notation(Example)(Example)

[ ]

{ } { }

=

=

=

9.10.152.11

,

1.56.26.14.11.90.30.35.40.2

B xxx

x

A

3

2

1 15.0Bxx

1.6A9.1A

2

22

3,1

2,2

=

=

=

=

Reasons for Matrix NotationReasons for Matrix Notation

1) Enables many numbers to be represented as a single object2) Represents the object by a single symbol3) Allows mathematical operations to be performed with the symbols instead of arrays of numbers

Matrix Notation is a Mathematical Shortcut

Relationship Between Vectors Relationship Between Vectors and Matricesand Matrices

2x

1x

3x

Vectors are mathematical entities used to represent physical parameters.

Vectors have three characteristics: 1) magnitude, 2) orientation, and 3) sense.

Relationship Between Vectors Relationship Between Vectors and Matricesand Matrices

(Components)(Components)

1x

3x

kAjAiAA321 xxx

++=

ij

k

2x

The components of a vector may be written in matrix format.

321

3

2

1

xxx

x

x

x

AAAor AAA

Not all row or column matrices are vectors.

Matrix AdditionMatrix Addition

[ ] [ ] [ ]

ji,ji,ji, BAC

BAC

+=

+=

Note that the number of rows and columns in [A] and [B] must be equal.

n1,...,jm1,...,i

=

=

Matrix AdditionMatrix Addition(Example)(Example)

[ ]

[ ]

=

=

1.86.65.17.64.52.24.13.05.0

B

6.83.62.14.68.51.23.11.02.0

A

[ ]

=

6.82.102.72.120.134.32.70.40.7

C

ji,ji,ji, BAC +=

Properties of Matrix AdditionProperties of Matrix Addition

[ ] [ ] [ ] [ ][ ] [ ]( ) [ ] [ ] [ ] [ ]( )[ ] [ ] [ ][ ] [ ] 0A-A

A0ACBACBA

ABBA

=+

=+

++=++

+=+

[ ][ ] [ ]Ain valueof negative iselement each A-

zero are elements all0

Transpose of a MatrixTranspose of a Matrix[ ] [ ]

[ ].Ain columns and rows theinginterchangby obtainedmatrix (nxm) theis

Amatrix (mxn)an of A transposeThe T

[ ]

=

826243

A [ ]

=

822463

A T

[ ]

=

241

B [ ] 241B T =

Symmetric MatrixSymmetric Matrix

[ ] [ ]AA T =A matrix having real components is symmetric if

[ ]

=

413152321

A [ ]

=

413152321

A T

Note that symmetric matrices must be square(i.e. same number of rows as columns).

Diagonal MatrixDiagonal Matrix

A square matrix whose elements above and below the principal diagonal are all zero is called a diagonal matrix.

[ ]

=

300050001

A

Identity MatrixIdentity Matrix

A diagonal matrix whose elements on the principal diagonal are all 1 is called an identity matrix or unit matrix.

[ ]

=

100010001

I

AssignmentAssignment1. Download a copy of

Getting Started from The Math Works website.

[ ] [ ]

[ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ] [ ] [ ]( )symmetric? matrices theofany Are g)

BA&BAf)

B4Ae)2A&Cd)

[D]c)[C][D]b)[C]

[A][B] & [B][A] a)Find

.351

421D ,

403210

C

,1225

B ,4032

ALet .2

TTT

TT

++

+

++

=

=

=

=

Problem 2 should be done on engineering pad paper and turned in next class.