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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
Queuing Birth-Death Law 2 1-21
Queuing Birth-Death Law 3 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
Binomial 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
Normal Distribution 3-9
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
Bernoulli Distribution 3-32
Beta Binomial Distribution 3-32
Beta Pascal Distribution 3-32
Binomial Distribution 3-32
Discrete Weibull Distribution 3-32
Geometric Distribution 3-33
Contents xvii
Hypergeometric Distribution 3-33
Negative Binomial Distribution 3-33
Poisson Distribution : 3-34
Rectangular (Discrete Uniform) Distribution 3-34
Continuous Distribution Formulas 3-34
Arcsin Distribution 3-34
Beta Distribution 3-35
Cauchy Distribution 3-35
Chi Distribution 3-35
Chi-Square Distribution 3-35
. Erlang Distribution 3-35
Exponential Distribution 3-35
Extreme-Value Distribution 3-36
F Distribution 3-36
Gamma 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
Normal Distribution 3-38
Pareto Distribution 3-38
Rayleigh Distribution 3-39
t-Distribution 3-39
Triangular Distribution 3-39
Uniform Distribution .' 3-39
Weibull Distribution 3-40
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
Application Areas 4-1
Sample Calculations 4-2
Sample Standard Deviation '. 4-2
Application Areas 4-2
Sample Standard Error of the Mean 4-3
xviii Contents
Application Areas 4-3
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
Bernoulli Distribution 4-5
Binomial Distribution 4-5
Discrete Uniform Distribution 4-6
Geometric Distribution 4-6
Negative Binomial Distribution 4-6
Poisson Distribution 4-6
Beta Distribution 4-6
Chi-Square Distribution 4-6
Erlang Distribution 4-6
Exponential Distribution 4-7
Application Areas 4-7
F Distribution 4-7
Gamma Distribution 4-7
Log-Normal Distribution 4-7
Normal Distribution 4-8
Triangular Distribution 4-8
Uniform Distribution 4-8
Weibull 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
Computational Example 6-3
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
Computational Example 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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