Handbook of industrial engineering equations, formulas ... · PDF fileHandbookof...

Handbook of

Industrial EngineeringEquations, Formulas,and Calculations

Adedeji B. Badiru

Olufemi A. Omitaomu

CRC PressTaylor Si Francis GroupBoca Raton London New York

CRC Press is an imprint of the

Taylor & Francis Group, an Informa business

Contents

Preface xxix

Authors ....xxxi

1 Computational Foundations of Industrial EngineeringEfficacy ofMathematical Modeling 1-1

Industrial Engineering and Computations 1-1

Definition and Applications .1-5

Orientation to STEM 1-6

IE Catchphrases 1-6

Span and Utility of IE 1-6

Heritage from Industrial Revolution 1-7

Historical Accounts 1-8

Chronology of Applications 1-10

Importance ofIE Calculations 1-14

Importance of Calculations Guide 1-16

Basic Queuing Equations 1-17

Queuing Birth-Death Processes ...1-21

Laws of Motion of Queuing Birth and Death 1-21

Queuing Birth-Death Law 1 1-21



Data Types for Computational Analysis 1-22

Nominal Scale 1-22

Ordinal Scale 1-22

Interval Scale 1-22

Ration Scale 1-23

Cardinal Scale 1-23

References 1-23

ix

x Contents

2 Basic Mathematical Calculations

Quadratic Equation 2-1

Overall Mean 2-2

Chebyshev's Theorem 2-2

Permutations 2-2

Combinations 2-2

Failure 2-3

Probability Distribution 2-3

Probability 2-3

Distribution Function 2-3

Expected Value 2-4

Variance 2-5

Binomial Distribution 2-5

Poisson Distribution i •2-5

Mean ofa Binomial Distribution .2-6

Normal Distribution 2-6

Cumulative Distribution Function 2-6

Population Mean 2-6

Standard Error ofthe Mean 2-6

t-Distribution 2-7

Chi-Squared Distribution 2-7

Definition ofSet and Notation 2-7

Set Terms and Symbols 2-8

Venn Diagrams 2-8

Operations on Sets 2-9

De Morgans Laws 2-9

Counting the Elements in a Set 2-10

Permutations 2-10

Combinations 2-11

Probability Terminology 2-11

Basic Probability Principles 2-11

Random Variable 2-12

Mean Value x or Expected Value u. 2-12

Series Expansions ,2-12

Mathematical Signs and Symbols 2-15

GreekAlphabets 2-16

Algebra 2-17

Laws ofAlgebraic Operations 2-17

Special Products and Factors 2-17

Powers and Roots 2-19

Proportion 2-19

Sum of Arithmetic Progression to n Terms 2-20

Sum of Geometric Progression to n Terms 2-20

Contents xi

Arithmetic Mean ofn Quantities, A 2-20

Geometric Mean of n Quantities, G 2-20

Harmonic Mean of n Quantities, H 2-20

Generalized Mean 2-20

Solution of Quadratic Equations 2-21

Solution of Cubic Equations 2-21

Trigonometric Solution ofthe Cubic Equation . 2-22

Solution of Quadratic Equations 2-23

Partial Fractions 2-23

Nonrepeated Linear Factors 2-23

Repeated Linear Factors 2-24

General terms 2-24

Repeated Linear Factors 2-25

Factors of Higher Degree 2-25

Geometry 2-25

Triangles 2-25

Right Triangle 2-26

Equilateral Triangle 2-26

General Triangle 2-26

Menelaus' Theorem 2-27

Ceva's Theorem 2-27

Quadrilaterals 2-27

Rectangle 2-27

Parallelogram 2-27

Rhombus 2-28

Trapezoid 2-28

General Quadrilateral 2-28

Theorem 2-28

Regular Polygon ofn Sides Each of Length b 2-28

Circle of radius r 2-28

Regular Polygon of n sides Inscribed in a Circle of Radius r 2-29

Regular Polygon of n Sides Circumscribing a Circle ofRadius r 2-29

Cyclic Quadrilateral 2-29

Prolemy's Theorem 2-29

Cyclic-Inscriptable Quadrilateral : 2-30

Sector of Circle ofRadius r 2-30

Radius of a Circle Inscribed in a Triangle of Sides a, b, and c 2-30

Radius of a Circle Circumscribing a Triangle ofSides a, b, and c 2-30

Segment of a Circle ofRadius r 2-30

Ellipse ofSemimajor Axis a and Semiminor Axis b 2-31

Segment of a Parabola 2-31

Planar Areas by Approximation 2-31

Solids Bounded By Planes 2-31

Cube 2-32

xii Contents

Rectangular Parallelepiped (or Box) 2-32

Prism 2-32

Truncated Triangular Prism 2-32

Pyramid 2-32

Frustum of a Pyramid 2-33

Prismatoid 2-33

Regular Polyhedra 2-33

Sphere of Radius r 2-34

Right Circular Cylinder of Radius r and Height h 2-34

Circular Cylinder of Radius r and Slant Height £ 2-35

Cylinder of Cross-Sectional Area A and Slant Height 1 2-35

Right Circular Cone ofRadius r and Height h 2-35

Spherical Cap ofRadius r and Height h 2-35

Frustum ofa Right Circular Cone ofRadii a and b and Height h 2-35

Zone and Segment of Two Bases 2-35

Lune 2-36

Spherical Sector 2-36

Spherical Triangle and Polygon 2-36

Spheroids 2-36

Ellipsoid 2-36

Oblate Spheroid 2-36

Prolate Spheroid 2-37

Circular Torus 2-37

Formulas from Plane Analytic Geometry 2-37

Distance d between Two Points 2-37

Slope m of Line Joining Two Points 2-37

Equation ofa Line Joining Two Points 2-37

Equation ofa Line in Terms of ^-intercept a 0 and^-intercept b ^ 0 2-38

Normal Form for Equation of a Line 2-38

General Equation of a Line 2-38

Distance From a Point {xi,y{) to the Line Ax + By + C = 0 2-38

Angle \[r between Two Lines Having Slopes w?i and m% 2-38

Area ofa Triangle with Vertices 2-39

Transformation of Coordinates Involving Pure Translation 2-39

Transformation of Coordinates Involving Pure Rotation 2-39

Transformation of Coordinates Involving Translation and Rotation 2-39

Polar Coordinates (r, 6) 2-40

Plane Curves 2-40

Catenary, Hyperbolic Cosine 2-40

Cardioid 2-40

Circle 2-40

Cassinian Curves 2-41

Cotangent Curve 2-41

Cubical Parabola 2-41

Contents xiii

Cosecant Curve. 2-41

Cosine Curve.: 2-41

Ellipse 2-41

Gamma Function 2-41

Hyperbolic Functions 2-41

Inverse Cosine Curve 2-42

Inverse Sine Curve 2-42

Inverse Tangent Curve 2-42

Logarithmic Curve 2-42

Parabola 2-42

Cubical Parabola 2-42

Tangent Curve 2-42

Distance d between Two Paints 2-43

Logarithmic Identities 2-46

Series Expansions 2-47

Limiting Values, 2-47

Inequalities 2-47

Polynomial Approximations 2-48

Exponential Function Series Expansion 2-49

Fundamental Properties 2-49

Definition of General Powers 2-49

Logarithmic and Exponential Functions 2-50

Periodic Property 2-50

Polynomial Approximations 2-50

Slopes 2-53

Trigonometric ratios 2-53

Sine Law 2r55

Cosine Law 2-56

Algebra 2-56

Expanding 2-56

Factoring 2-57

Roots of a Quadratic Equation 2-57

Law of exponents 2-57

Logarithms 2-58

3 Statistical Distributions, Methods, and ApplicationsDiscrete Distributions 3-1

Bernoulli Distribution 3-1

Beta Binomial Distribution 3-1

Beta Pascal Distribution 3-3


Discrete Weibull Distribution 3-3

Geometric Distribution 3-3

xiv Contents

Hypergeometric Distribution 3-3

Negative Binomial Distribution 3-4

Poisson Distribution 3-4

Rectangular (Discrete Uniform) Distribution 3-4

Continuous Distributions 3-4

Arcsin Distribution 3-4

Beta Distribution 3-5

Cauchy Distribution 3-5

Chi Distribution 3-5

Chi-Square Distribution ; 3-5

Erlang Distribution 3-5

Exponential Distribution 3-6

Extreme-Value Distribution 3-6

F Distribution 3-6

Gamma Distribution 3-7

Half-Normal Distribution 3-7

Laplace (Double Exponential) Distribution 3-7

Logistic Distribution 3-7

Lognormal Distribution 3-7

Noncentral Chi-Square Distribution 3-8

Noncentral F Distribution 3-8

Noncentral r-Distribution 3-8


Pareto Distribution 3-9

Rayleigh Distribution 3-9

f-Distribution 3-9

Triangular Distribution 3-10

Uniform Distribution 3-10

Weibull Distribution 3-10

Distribution Parameters 3-10

Average 3-10

Variance 3-11

Standard Deviation 3-11

Standard Error 3-11

Skewness 3-11

Standardized Skewness 3-11

Kurtosis 3-11

Standardized Kurtosis 3-11

Weighted Average 3-11

Estimation and Testing 3-11

100(l-ct)% Confidence Interval for Mean 3-11

100(1 —a)% Confidence Interval for Variance 3-11

100(1 — a)% Confidence Interval for Difference in Means 3-12

100(1 - a)% Confidence Interval for Ratio of Variances 3-12

Contents xv

Normal Probability Plot 3-12

Comparison of Poisson Rates 3-13

Distribution Functions—Parameter Estimation 3-13

Bernoulli 3-13

Binomial 3-13

Discrete Uniform 3-13

Geometric 3-13

Negative Binomial 3-13

Poisson 3-13

Beta 3-14

Chi-Square 3-14

Erlang 3-14

Exponential 3-14

F Distribution 3-14

Gamma 3-14

Log-Normal 3-15

Normal 3-15

Student's t 3-15

Triangular 3-15

Uniform 3-15

Weibull 3-16

Chi-Square Test for Distribution Fitting 3-16

Kolmogorov-Smirnov Test 3-16

ANOVA.... : 3-16

Notation 3-16

Standard Error (Internal) 3-17

Standard Error (Pooled) 3-17

Interval Estimates 3-17

Tukey Interval 3-17

Scheffe Interval 3-17

Cochran C-Test 3-17

BartlettTest 3-18

Hartley's Test 3-18

Kruskal-WallisTest 3-18

Adjustment for Ties, 3-18

Freidman Test 3-18

Regression 3-19

Notation 3-19

Regression Statistics 3-19

Predictions 3-20

Nonlinear Regression 3-21

Ridge Regression 3-21

Quality Control 3-22

For All Quality Control Formulas 3-22

xvi Contents

Subgroup Statistics 3-22

XBar Charts '. 3-22

Capability Ratios 3-23

R Charts 3-24

S Charts 3-24

C Charts 3-24

U Charts 3-24

P Charts 3-24

NP Charts 3-25

CuSum Chart for the Mean 3-25

Multivariate Control Charts 3-25

Time-Series Analysis : 3-25

Notation 3-25

Autocorrelation at Lag k 3-26

Partial Autocorrelation at Lag k 3-26

Cross-Correlation at Lag k 3-26

Box-Cox 3-26

Periodogram (computed using Fast Fourier Transform) 3-27

Categorical Analysis 3-27

Notation 3-27

Totals 3-27

Chi-Square 3-27

Fisher's Exact Test 3-28

Lambda 3-28

Uncertainty Coefficient 3-28

Somer'sD 3-29

Eta 3-29

Contingency Coefficient 3-30

Cramer's V 3-30

Conditional Gamma 3-30

Pearson's J? 3-30

Kendall's Taufc 3-30

TauC 3-30

Probability Terminology 3-30

Basic Probability Principles 3-31

Random Variable 3-31

Mean Value Jc or Expected Value [L 3-31

Discrete Distribution Formulas 3-32


Beta Binomial Distribution 3-32

Beta Pascal Distribution 3-32


Discrete Weibull Distribution 3-32


Contents xvii

Hypergeometric Distribution 3-33


Poisson Distribution : 3-34

Rectangular (Discrete Uniform) Distribution 3-34

Continuous Distribution Formulas 3-34

Arcsin Distribution 3-34


Cauchy Distribution 3-35

Chi Distribution 3-35

Chi-Square Distribution 3-35

. Erlang Distribution 3-35


Extreme-Value Distribution 3-36

F Distribution 3-36


Half-Normal Distribution 3-36

Laplace (Double Exponential) Distribution 3-37

Logistic Distribution 3-37

Lognormal Distribution 3-37

Noncentral Chi-Square Distribution 3-37

Noncentral F Distribution 3-38

Noncentral t-Distribution 3-38


Pareto Distribution 3-38

Rayleigh Distribution 3-39

t-Distribution 3-39


Uniform Distribution .' 3-39


Variate Generation Techniques 3-40

Notation 3-40

Variate Generation Algorithms 3-40

References 3-42

4 Computations with Descriptive Statistics

Sample Average 4-1

Application Areas 4-1

Sample calculations 4-1

Sample Variance 4-1


Sample Calculations 4-2

Sample Standard Deviation '. 4-2


Sample Standard Error of the Mean 4-3

xviii Contents


Skewness 4-3

Standardized Skewness 4-3

Kurtosis 4-4

Standardized Kurtosis • 4-4

Weighted Average 4-4

Estimation and Testing 4-4

100(1 - a)% Confidence Interval for Mean 4-4

100(1 - a)% Confidence Interval for Variance 4-4

100(1 - a)% Confidence Interval for Difference

in Means 4-4

100(1 - ct)% Confidence Interval for Ratio

of Variances 4-5

Normal Probability Plot '. 4-5

Comparison ofPoisson Rates 4-5

Distribution Functions and Parameter Estimation 4-5



Discrete Uniform Distribution 4-6



Poisson Distribution 4-6


Chi-Square Distribution 4-6

Erlang Distribution 4-6



F Distribution 4-7


Log-Normal Distribution 4-7



Uniform Distribution 4-8


Chi-Square Test for Distribution Fitting 4-8

Kolmogorov-Smirnov Test 4-9

ANOVA 4-9

Notation 4-9

Standard Error 4-9

Interval Estimates 4-9

Tukey Interval 4-10

Scheffe Interval 4-10

Cochran C.-test 4-10

Bartlett Test 4-10

Contents xix

'

Hartley's Test 4-10

Kruskal-WaUis Test .".4-11

Adjustment for ties 4-11

Freidman Test 4-11

Regression 4-12

Notation 4-12

Statistical Quality Control .....4-13

Subgroup Statistics 4-13

X-Bar Charts 4-14

Capability Ratios 4-14

R Charts 4-15

S Charts 4-15

C Charts 4-15

U Charts 4-15

P Charts 4-15

NP Charts 4-16

CuSum Chart for the Mean 4-16

Time-Series Analysis 4-16

Notation..., 4-16

Autocorrelation at Lag k 4-17

Partial Autocorrelation at Lag k . 4-17

Cross-Correlation at Lag k 4-17

Box-Cox Computation 4-17

Periodogram (Computed using Fast Fourier

Transform) 4-18

Categorical Analysis '. 4-18

Notation 4-18

Totals 4-18

Chi-Square 4-18

Lambda • 4-19

Uncertainty Coefficient 4-19

Somer's D Measure 4-20

Eta 4-20

Contingency Coefficient 4-21

Cramer's V Measure 4-21

Conditional Gamma 4-21

Pearson's R Measure 4-21

Kendall's Tau b Measure 4-21

TauC Measure 4-22

Overall Mean 4-22

Chebyshev's Theorem 4-22

Permutation 4-22

Combination 4-22

Failure 4-22

xx Contents

5 Computations for Economic AnalysisFundamentals of Economic Analysis 5-1

Simple Interest 5-1

Future value 5-1

Compound Interest 5-2

Continuous Compound Interest 5-2

Effective Rate 5-3

Present Value with Compound Interest 5-3

Annuities 5-4

Present value of annuity 5-4

Future value ofan annuity 5-4

Amortization ofLoans 5-5

Interest and Equity Computations 5-5

Equity Break-Even Formula 5-8

Sinking Fund Payment 5-9

Internal Rate ofReturn 5-9

Benefit-Cost Ratio 5-9

Simple Payback Period 5-9

Discounted Payback Period 5-10

Economic Methods of Comparing Investment Alternatives 5-10

Present Value Analysis 5-10

Annual Value Analysis 5-10

Internal Rate of Return Analysis 5-11

External Rate ofReturn Analysis 5-11

Incremental Analysis 5-11

Guidelines for Comparison of

Alternatives 5-12

Asset Replacement and Retention Analysis 5-12

Replacement Analysis Computation 5-14

Depreciation Methods 5-15

Depreciation Terminology 5-15

Depreciation Methods 5-16

Straight-Line (SL) Method 5-16

Declining Balance (DB) Method 5-16

Sums-of-Years' Digits (SYD) Method 5-17

MACRS Method 5-17

Effects of Inflation and Taxes 5-18

Foreign Exchange Rates 5-21

After-Tax Economic Analysis 5-21

Cost and Value Computations 5-22

Actual Cost ofWork Performed 5-23

Applied Direct Cost 5-23

Budgeted Cost for Work Performed 5-23

Contents xxi

Budgeted Cost for Work Scheduled 5-23

Direct Cost 5-23

Economies of Scale 5-23

Estimated Cost at Completion 5-23

First Cost 5-24

Fixed Cost 5-24

Incremental Cost 5-24

Indirect Cost 5-24

Life-Cycle Cost : 5-24

Maintenance Cost 5-24

Marginal Cost 5-24

Operating Cost 5-24

Opportunity Cost 5-25

Overhead Cost 5-25

Standard Cost 5-25

Sunk Cost 5-25

Total Cost ; 5-25

Variable cost 5-25

Cash-Flow Calculations 5-26

Calculations with Compound Amount Factor 5-26

Calculations with Present Worth Factor 5-27

Calculations with Uniform Series Present Worth Factor 5-27

Calculations with Uniform Series Capital Recovery Factor 5-28

Calculations with Uniform Series Compound Amount Factor 5-29

Calculations with Uniform Series Sinking Fund Factor 5-29

Calculations with Capitalized Cost Formula 5-30

Arithmetic Gradient Series•

5-31

Internal Rate of Return 5-32

Benefit-Cost Ratio Analysis 5-33

Simple Payback Period -5-33

Discounted Payback Period 5-34

Time Required to Double Investment 5-35

Effects of Inflation on Industrial Project Costing 5-36

Mild Inflation 5-40

Moderate Inflation 5-40

Severe Inflation 5-40

Hyperinflation 5-40

Break-Even Analysis 5-40

Profit Ratio Analysis 5-42

Project Cost Estimation 5-46

Optimistic and Pessimistic Cost Estimates 5-47

Cost Performance Index,

5-47

Cost Control Limits 5-48

Project Balance Computation 5-48

xxii Contents

6 Industrial Production Calculations

Learning Curve Models and Computations 6-1

The Average Cost Model 6-1

Computational Example 6-2


The Unit Cost Model 6-4

Productivity Calculations Using Learning Curves 6-5

Determining Average Unit Time 6-5

Calculation of the Learning Factor 6-5

Calculating Total Time for a Job 6-5

Time Required to Perform a Task for the wth Time 6-5

The Improvement Ratio 6-5


Solution 6-5

Computation for Improvement Goal in Unit Time 6-6

Computation ofMachine Output Capacities 6-6

Machine Utilization Ratio: Determining How Often a

Machine Is Idle 6-6

CalculatingNumber of Machines Needed to Meet Output 6-7

Alternate Forms for Machine Formula 6-7

Calculating Buffer Production to Allowfor Defects 6-8

Adjusting Machine Requirements to Allow for Rejects 6-9

Output Computations for Sequence of Machines 6-9

Calculating Forces on Cutting Tools 6-10

Calculating Pressure 6-11

Finding Required Horsepower 6-11

Machine Accuracy 6-12

Calculating the Goal Dimension 6-12

Drill Thrust 6-13

Drill Torque 6-14

Drill Horsepower 6-14

Calculating Speed and Rotation ofMachines 6-14

Shaft Speed 6-15

Surface Speed ......6-15

Tool Feed per Revolution 6-16

ToolFeedper Minute 6-16

Tool Feed per Tooth 6-17

Computation to Find the Volume of Material Removed 6-17

Time Rate for Material Removal in a Rectangular Cut 6-17

Calculation ofLinear Feed for Rectangular Material Removal 6-18

Takt Time for Production Planning 6-18

Production Crew Work Rate Analysis 6-19

Production Work Rate Example 6-20

Contents xxiii

Case ofMultiple Resources Working Together 6-20

Computational Examples 6-21

Calculation of Personnel Requirements 6-22

Calculation of Machine Requirements 6-23

References 6-23

7 Forecasting Calculations

Forecasting Based on Averages 7-1

Simple Average Forecast ; 7-1

Period Moving Average Forecast 7-2

Weighted Average Forecast 7-2

Weighted T-Period Moving Average Forecast 7-2

Exponential Smoothing Forecast 7-3

Regression Analysis • 7-3

Regression Relationships 7-3

Prediction 7-4

Control 7-4

Procedure for Regression Analysis 7-5

Coefficient ofDetermination 7-5

Residual Analysis 7-7

Time-Series Analysis 7-9

Stationarity and Data Transformation 7-10

Moving Average Processes 7-12

Autoregressive Processes 7-13

Forecasting for Inventory Control 7-15

Economic Order Quantity Model 7-15

Quantity Discount ; 7-16

.Calculation ofTotal Relevant Cost 7-16

Evaluation ofthe Discount Option 7-17

Sensitivity Analysis 7-19

Wagner-Whitin Algorithm 7-20

Notation and Variables 7-21

8 Six Sigma and Lean

Concept ofSix Sigma 8-1

Taguchi Loss Function 8-1

Identification and Elimination of Sources ofDefects 8-2

Roles and Responsibilities for Six Sigma 8-3

Statistical Techniques for Six Sigma .8-3

Control Charts 8-4

XandR-Charts 8-4

Data Collection Strategies 8-5

Subgroup Sample Size 8-5

Frequency of Sampling. •. 8-6

xxiv Contents

Stable Process 8-6

Out-of-Control Patterns 8-6

Calculation of Control Limits 8-7

Plotting Control Charts for Range and Average Charts 8-8

Plotting Control Charts for Moving Range and Individual

Control Charts 8-9

Case Example: Plotting ofControl Chart 8-9

Calculations 8-9

Trend Analysis 8-14

Process Capability Analysis for Six Sigma 8-18

Capable Process (Cp) 8-18

Capability Index (CPj,).. 8-19

Process Capability Example 8-20

Possible Applications of Process Capability Index 8-21

Potential Abuse of Cp and CP|. 8-22

Lean Principles and Applications 8-22

Kaizen of a Process 8-23

Lean TaskValue Rating System 8-24

9 Risk ComputationsCost Uncertainties 9-1

Schedule Uncertainties 9-2

Performance Uncertainties 9-2

Decision Tables and Trees 9-2

Reliability Calculations 9-7

General Reliability Definitions 9-8

Exponential Distribution Used as ReliabilityFunction ,....9-9

Reliability Formulas 9-10

General Reliability Definitions 9-10

Exponential Distribution Used as ReliabilityFunction 9-11

10 Computations for Project AnalysisPlanning 10-1

Organizing 10-1

Scheduling 10-1

Control 10-2

CPM Scheduling 10-2

Advantages for Communication 10-3

Advantages for Control 10-3

Advantages for Team Interaction 10-3

Activity Precedence Relationships,'. 10-4

Network Notation 10-4

Contents xxv

Forward-Pass Calculations 10-5

CPM Example 10-6

Backward-Pass Calculations 10-7

Calculation of Slacks 10-8

Calculations for Subcritical Paths 10-9

Plotting ofGantt Charts 10-10

Calculations for Project Crashing 10-13

Calculations for Project Duration Diagnostics 10-18

PERT Formulas 10-19

Activity Time Distributions 10-19

Project Duration Analysis 10-20

Simulation of Project Networks . 10-24

11 Product Shape and Geometrical Calculations

Equation of a Straight Line 11-1

Quadratic Equation 11-2

Conic Sections•

11-2

Case 1 of Conic Section: e — 1 (Parabola) 11-3

Case 2 ofConic Section: e < 1 (Ellipse) 11-3

Case 3 Hyperbola: e > 1 ....11-4

Case4e = l (Circle) 11-5

Conic Section Equation 11-6

Quadric Surface (Sphere). 11-6

Identities,

11-7

Trigonometry : 11-7

Law of Sines 11-7

Law of Cosines 11-8

Identities 11-8

Complex Numbers 11-10

Polar Coordinates 11-10

Euler's Identity 11-11

Roots 11-11

Matrices 11-11

Matrix Multiplication 11-11

MatrixAddition 11-11

Identity Matrix ..11-11

Matrix Transpose 11-11

Matrix Inverse 11-12

Determinants 11-12

Vectors 11-12

Gradient, Divergence, and Curl 11-13

Identities 11-13

Progressions and Series 11-14

Arithmetic Progression 11-14

xxvi Contents

Geometric Progression 11-14

Properties ofSeries 11-15

Power Series 11-15

Taylor's Series 11-16

Differential Calculus 11-16

The Derivative 11-16

Test for a Maximum 11-16

Test for a Minimum 11-16

Test for a Point ofInflection 11-16

The Partial Derivative 11-16

Curvature in Rectangular Coordinates 11-17

The Radius of Curvature 11-17

I/Hospital's Rule U-17

Integral Calculus 11-18

Derivatives and Indefinite Integrals , 11-18

Derivatives 11-18

Indefinite Integrals 11-20

Mensuration of Areas and Volumes 11-21

Nomenclature 11-21

Parabola 11-21

Ellipse 11-21

Circular Segment 11-22

Circular Sector 11-22

Sphere 11-22

Parallelogram 11-22

Regular Polygon (n Equal Sides) 11-23

Prismoid 11-23

Right Circular Cone 11-23

Right Circular Cylinder 11-23

Paraboloid ofRevolution 11-23

Centroids and Moments of Inertia 11-24

Difference Equations 11-24

First-Order Linear Difference Equation 11-25

Second-Order Linear Difference Equation 11-25

Numerical Methods 11-25

Newton's Method for Root Extraction 11-25

Newton's Method of Minimization 11-25

Numerical Integration 11-26

Euler's or Forward Rectangular Rule 11-26

Trapezoidal Rule 11-26

Simpson's Rule/Parabolic Rule {n Must Be

an Even Integer) 11-27

Calculation of Best-Fit Circle 11-27

Contents.

xxvii

12 General Engineering Calculations

Six Simple Machines for Materials Handling 12-1

Machine 1: The Lever 12-1

Machine 2: Wheel and Axle 12-1

Machine 3: The Pulley 12-2

Machine 4: The Inclined Plane 12-2

Machine 5: The Wedge 12-2

Machine 6: The Screw 12-2

Mechanics: Kinematics 12-3

Scalars and Vectors 12-3

Distance and Displacement 12-3

Acceleration 12-3

Speed and Velocity 12-3

Frequency 12-3

Period : 12-3

Angular Displacement 12-3

Angular Velocity 12-4

Angular Acceleration 12-4

Rotational Speed 12-4

Uniform Linear Motion 12-4

Uniform Accelerated Linear Motion 12-4

Rotational Motion 12-5

Uniform Rotation and a Fixed Axis 12-5

Uniform Accelerated Rotation about a Fixed Axis 12-5

Simple Harmonic Motion i 12-6

Pendulum 12-7

Free Fall 12-7

Vertical Project 12-7

Angled Projections 12-8

Horizontal Projection (a = 0) 12-8

Sliding Motion on an Inclined Plane 12-8

Rolling Motion on an Inclined Plane 12-9

Mechanics: Dynamics 12-10

Newton's First Law of Motion 12-10

Newton's Second Law ofMotion 12-10

Newton's Third Law of Motion 12-10

Momentum ofForce 12-10

Impulse of Force 12-11

Law of Conservation ofMomentum 12-11

Friction 12-11

General Law ofGravity 12-11

Gravitational Force 12-12

Centrifugal Force 12-12

Centripetal Force 12-12

xxviii Contents

Torque 12-12

Work 12-13

Energy 12-13

Conservation of Energy 12-14

Power• 12-14

Appendix A: Mathematical Patterns, Series, and Formulae

Number Sequence and Patterns A-l

Closed-Form Mathematical Expressions A-3

Derivation of the Quadratic Formula A-7

Appendix B: Measurement Units, Notation, and Constants

Common Notation B-l

Scientific Constants B-3

Numbers and Prefixes B-3

Appendix C: Conversion Factors

Area Conversion Factors C-l

Volume Conversion Factors C-2

Energy Conversion Factors C-2

Mass Conversion Factors C-3

Temperature Conversion Factors C-3

Velocity Conversion Factors C-4

Pressure Conversion Factors C-4

Distance Conversion Factors C-5

Physical Science Equations C-6

English and Metric Systems C-7

Household Measurement Conversion C-7

Appendix D: Factors and Tables

Formulas for Interest Factor D-l

Summation Formulas for Closed-Form Expressions D-3

Interest Tables D-3

Appendix E: Greek Symbols and Roman NumeralsGreek Symbols E-l

Roman Numerals E-2

Bibliography Bibliography-1

Index Index-1

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