SOLUTION MANUAL FOR:
Application Of Math Principles To Engineering
Problems
11th Edition
By
Professor David A. Hullender Latest
Update.
1
,Preface
The Objectives Of This Notebook Are To Introduce Solution Techniques For Solving Typical
Equations Encountered In The Analysis And Simulation Of Engineering Systems. When
Obtaining Solutions To Equations, It Is Very Beneficial To Focus On The Common Sense
Details Associated With The Performance Of The Actual Engineering System For Which The
Equations Are Written. Consequently, Throughout This Text, Equations For Actual Engineering
Systems Are Utilized So That It Is Possible To Apply Common Sense To The Prediction And
Approximation Of The Solution To The Equations Before Actually Solving The Equations.
Knowing The Approximate Solution Provides Confirmation To The Actual Solution In The End.
From The Very Beginning Of The Notebook, Computer Algorithms In Matlab Will Be Utilized
To Obtain And Plot Solutions For Typical Equations Encountered When Solving Engineering
Problems. Having Experience With Matlab Prior To Reading This Notebook Is Not Necessary.
Specific Details And Examples Of Using Matlab For Solving Engineering Equations And For
Simulating Dynamic Systems Are Included In Order For The Reader Not Experienced With
Matlab To Be Able To Utilize The Software Effectively Without Having To Be Creative And
Start From Scratch.
At The End Of Several Of The Sections, A List Of Previous Quiz, Homework, And Exam
Problems Are Listed That Utilize Some Of The Concepts Introduced In That Particular Section.
The Solutions To Almost All Of The Problems Are Provided; However, It Is Very Important To
Test Your Understanding Of The Concepts By Trying To Work Each Problem Without Looking
At The Solution. Then And Only Then Will You Know Your Weaknesses; It Is Better To
Determine Your Weaknesses Prior To An Exam Instead Of During An Exam! Reviewing The
Solution To A Problem Before Trying To Work It Without Looking At The Solution Is A Total
Waste Of Time; In My Opinion, This Is The Number One Mistake Students Make When
Preparing For Exams. The Second Greatest Mistake Is Preparing For An Exam With Other
Students; You Are Not Learning What You Don’t Know And, Unfortunately, When You Take
An Exam, That Is When You Will Find Out What You Don’t Know.
Table Of Contents Page No.
Chapter 1 Basic Math Principles
1.1 Complex Numbers 5
1.2 Notation For Time Derivatives And Use Of The ‘D’ Operator 6
1.2.1 Notation For Time Derivatives 6
1.2.2 Use Of The ‘D’ Operator To Convert Differential Eqn’s To Algebraic Eqn’s 6
1.3 Eigenvalues Of A System 7
1.3.1 General Properties Of Eigenvalues 7
1.3.2 Using Matlab To Compute Eigenvalues 8
2
,1.4 Solving Simultaneous Algebraic Equations 9
1.4.1 Manually Solving Simultaneous Equations 9
1.4.2 Using Symbolic Math In Matlab To Solve Simultaneous Equations 12
Chapter 2. Solving Differential Equations
2.1 Analytical Solutions Of Differential Equations Using The Laplace Transform 16
2.1.1 Linearizing Nonlinear Differential Equations 16
Suggestions For Getting Straight Line Approximations 18
2.1.2 The Definition Of The Laplace Transform 21
2.1.3 Properties Of The Laplace Transform 21
2.1.4 Final Value Theorem 22
2.1.5 Initial Value Theorem 22
2.1.6 Converting Differential Eqn’s To Algebraic Eqn’s Using Laplace Transform 23
2.1.7 Common Inputs And Their Laplace Transforms 24
2.1.7.1 Step Functions And Constants 24
2.1.7.2 Pulse And Impulse Functions 25
2.1.7.3 Exponential Functions 26
2.1.7.4 Periodic Functions 26
2.1.8 Inverse Laplace Transform 26
2.1.8.1 Inverse Laplace Transform Using Partial Fractions 26
Obtaining Partial Fractions Using Matlab 29
2.1.8.2 Inverse Laplace Transform Using The Residue Theorem 29
Shortcut For Complex Poles 31
2.1.8.3 Using Matlab To Plot The Inverse Laplace Transform Without
Getting An Equation For The Inverse Laplace Transform 33
2.1.9 Modes Of A System 35
2.2 Numerical Solutions To Differential Equations Using Matlab 37
2.2.1 General Considerations 37
2.2.2 Using The ‘Impulse’ Command In Matlab 37
2.2.3 Using The ‘Step’ Command In Matlab 38
Dc Gain 38
2.2.4 Using The Command “Lsim” 39
Pulse Series Input 40
2.2.5 Using The Command “Initial” 41
2.3 Using Matlab To Get The System Response Using Numerical Integration 42
2.3.1 Example Problems Using Ode45 In Matlab 42
Deterministic Inputs 42
Random And Data Series Inputs 44
2.3.2 Example Simulation Using Options And Event Functions With Ode45 46
2.4 Using Matlab To Get The System Steady State Response For Periodic Inputs 47
2.4.1 Frequency Response Analysis 47
2.4.2 Examples Of Frequency Response Function Plots 47
2.5 Using Matlab To Get Lower Order Approximations For Transfer Functions 51
2.5.1 Examples Of Modal Approximation 52
2.5.2 Transfer Function Approximations Using Inverse Freq. Domain Analysis 55
Suggestions For Achieving Convergence With The Inverse Frequency Algorithm 64
3
, 2.6 Converting Linear Continuous Differential Eqn’s To Discrete Time Eqn’s 66
2.6.1 Converting 1st Order Differential Eqn’s Using Inverse Laplace Transform 66
2.6.2 Conversion Using The Continuous-To-Discrete Matlab Command, ‘C2d’ 67
Chapter 3. Expressing Differential Equations In State Variable Format
3.1 Representing An Nth Order System By N First Order Differential Eqn’s. 69
3.2 Simulation Diagram Approach When There Are Input Derivatives 70
3.3 General Matrix Format For State Variable Equations 71
3.4 State Variable Equations In Matrix Format If There Are Input Derivatives 72
3.4.1 Only 1st Order Input Derivatives 72
3.4.2 Format Options For Input Derivatives Higher Than 1st Order 73
3.4.2.1 Simulation Diagram Approach 73
3.4.2.2 Phase Variable Approach 74
3.4.2.3 Matlab Approach 74
3.4.2.4 Example Demonstrating The Different Methods 75
3.4.2.5 Example Of Phase Variable Approach When The Numerator And
Denominator Orders Are Equal 76
3.5 Maintaining Access To Natural Variables When Creating State Variable
Equations From Block Diagrams 76
3.6 Simulink 78
3.7 Using The Laplace Transform To Solve Partial Differential Equations 80
Chapter 4 Example Problems 109
Chapter 5 Solutions To Example Problems In Chapter 4 138
Chapter 6 Previous Quiz Problems And Solutions 195
Chapter 7 Previous Homework And Solutions 221
Chapter 8 Previous Exams And Solutions 330
Chapter 9 Matlab M-Files 420
4
Application Of Math Principles To Engineering
Problems
11th Edition
By
Professor David A. Hullender Latest
Update.
1
,Preface
The Objectives Of This Notebook Are To Introduce Solution Techniques For Solving Typical
Equations Encountered In The Analysis And Simulation Of Engineering Systems. When
Obtaining Solutions To Equations, It Is Very Beneficial To Focus On The Common Sense
Details Associated With The Performance Of The Actual Engineering System For Which The
Equations Are Written. Consequently, Throughout This Text, Equations For Actual Engineering
Systems Are Utilized So That It Is Possible To Apply Common Sense To The Prediction And
Approximation Of The Solution To The Equations Before Actually Solving The Equations.
Knowing The Approximate Solution Provides Confirmation To The Actual Solution In The End.
From The Very Beginning Of The Notebook, Computer Algorithms In Matlab Will Be Utilized
To Obtain And Plot Solutions For Typical Equations Encountered When Solving Engineering
Problems. Having Experience With Matlab Prior To Reading This Notebook Is Not Necessary.
Specific Details And Examples Of Using Matlab For Solving Engineering Equations And For
Simulating Dynamic Systems Are Included In Order For The Reader Not Experienced With
Matlab To Be Able To Utilize The Software Effectively Without Having To Be Creative And
Start From Scratch.
At The End Of Several Of The Sections, A List Of Previous Quiz, Homework, And Exam
Problems Are Listed That Utilize Some Of The Concepts Introduced In That Particular Section.
The Solutions To Almost All Of The Problems Are Provided; However, It Is Very Important To
Test Your Understanding Of The Concepts By Trying To Work Each Problem Without Looking
At The Solution. Then And Only Then Will You Know Your Weaknesses; It Is Better To
Determine Your Weaknesses Prior To An Exam Instead Of During An Exam! Reviewing The
Solution To A Problem Before Trying To Work It Without Looking At The Solution Is A Total
Waste Of Time; In My Opinion, This Is The Number One Mistake Students Make When
Preparing For Exams. The Second Greatest Mistake Is Preparing For An Exam With Other
Students; You Are Not Learning What You Don’t Know And, Unfortunately, When You Take
An Exam, That Is When You Will Find Out What You Don’t Know.
Table Of Contents Page No.
Chapter 1 Basic Math Principles
1.1 Complex Numbers 5
1.2 Notation For Time Derivatives And Use Of The ‘D’ Operator 6
1.2.1 Notation For Time Derivatives 6
1.2.2 Use Of The ‘D’ Operator To Convert Differential Eqn’s To Algebraic Eqn’s 6
1.3 Eigenvalues Of A System 7
1.3.1 General Properties Of Eigenvalues 7
1.3.2 Using Matlab To Compute Eigenvalues 8
2
,1.4 Solving Simultaneous Algebraic Equations 9
1.4.1 Manually Solving Simultaneous Equations 9
1.4.2 Using Symbolic Math In Matlab To Solve Simultaneous Equations 12
Chapter 2. Solving Differential Equations
2.1 Analytical Solutions Of Differential Equations Using The Laplace Transform 16
2.1.1 Linearizing Nonlinear Differential Equations 16
Suggestions For Getting Straight Line Approximations 18
2.1.2 The Definition Of The Laplace Transform 21
2.1.3 Properties Of The Laplace Transform 21
2.1.4 Final Value Theorem 22
2.1.5 Initial Value Theorem 22
2.1.6 Converting Differential Eqn’s To Algebraic Eqn’s Using Laplace Transform 23
2.1.7 Common Inputs And Their Laplace Transforms 24
2.1.7.1 Step Functions And Constants 24
2.1.7.2 Pulse And Impulse Functions 25
2.1.7.3 Exponential Functions 26
2.1.7.4 Periodic Functions 26
2.1.8 Inverse Laplace Transform 26
2.1.8.1 Inverse Laplace Transform Using Partial Fractions 26
Obtaining Partial Fractions Using Matlab 29
2.1.8.2 Inverse Laplace Transform Using The Residue Theorem 29
Shortcut For Complex Poles 31
2.1.8.3 Using Matlab To Plot The Inverse Laplace Transform Without
Getting An Equation For The Inverse Laplace Transform 33
2.1.9 Modes Of A System 35
2.2 Numerical Solutions To Differential Equations Using Matlab 37
2.2.1 General Considerations 37
2.2.2 Using The ‘Impulse’ Command In Matlab 37
2.2.3 Using The ‘Step’ Command In Matlab 38
Dc Gain 38
2.2.4 Using The Command “Lsim” 39
Pulse Series Input 40
2.2.5 Using The Command “Initial” 41
2.3 Using Matlab To Get The System Response Using Numerical Integration 42
2.3.1 Example Problems Using Ode45 In Matlab 42
Deterministic Inputs 42
Random And Data Series Inputs 44
2.3.2 Example Simulation Using Options And Event Functions With Ode45 46
2.4 Using Matlab To Get The System Steady State Response For Periodic Inputs 47
2.4.1 Frequency Response Analysis 47
2.4.2 Examples Of Frequency Response Function Plots 47
2.5 Using Matlab To Get Lower Order Approximations For Transfer Functions 51
2.5.1 Examples Of Modal Approximation 52
2.5.2 Transfer Function Approximations Using Inverse Freq. Domain Analysis 55
Suggestions For Achieving Convergence With The Inverse Frequency Algorithm 64
3
, 2.6 Converting Linear Continuous Differential Eqn’s To Discrete Time Eqn’s 66
2.6.1 Converting 1st Order Differential Eqn’s Using Inverse Laplace Transform 66
2.6.2 Conversion Using The Continuous-To-Discrete Matlab Command, ‘C2d’ 67
Chapter 3. Expressing Differential Equations In State Variable Format
3.1 Representing An Nth Order System By N First Order Differential Eqn’s. 69
3.2 Simulation Diagram Approach When There Are Input Derivatives 70
3.3 General Matrix Format For State Variable Equations 71
3.4 State Variable Equations In Matrix Format If There Are Input Derivatives 72
3.4.1 Only 1st Order Input Derivatives 72
3.4.2 Format Options For Input Derivatives Higher Than 1st Order 73
3.4.2.1 Simulation Diagram Approach 73
3.4.2.2 Phase Variable Approach 74
3.4.2.3 Matlab Approach 74
3.4.2.4 Example Demonstrating The Different Methods 75
3.4.2.5 Example Of Phase Variable Approach When The Numerator And
Denominator Orders Are Equal 76
3.5 Maintaining Access To Natural Variables When Creating State Variable
Equations From Block Diagrams 76
3.6 Simulink 78
3.7 Using The Laplace Transform To Solve Partial Differential Equations 80
Chapter 4 Example Problems 109
Chapter 5 Solutions To Example Problems In Chapter 4 138
Chapter 6 Previous Quiz Problems And Solutions 195
Chapter 7 Previous Homework And Solutions 221
Chapter 8 Previous Exams And Solutions 330
Chapter 9 Matlab M-Files 420
4
