## CONTENTS

Preface ix

Chapter 1. Mathematics for Machinists and Metalworkers 1.1

1.1 Geometric Principles—Plane Geometry / 1.1

1.2 Basic Algebra / 1.7

1.2.1 Algebraic Procedures / 1.7

1.2.2 Transposing Equations (Simple and Complex) / 1.9

1.3 Plane Trigonometry / 1.11

1.3.1 Trigonometric Laws / 1.13

1.3.2 Sample Problems Using Trigonometry / 1.21

1.4 Modern Pocket Calculator Procedures / 1.28

1.4.1 Types of Calculators / 1.28

1.4.2 Modern Calculator Techniques / 1.29

1.4.3 Pocket Calculator Bracketing Procedures / 1.31

1.5 Angle Conversions—Degrees and Radians / 1.32

1.6 Powers-of-Ten Notation / 1.34

1.7 Percentage Calculations / 1.35

1.8 Temperature Systems and Conversions / 1.36

1.9 Decimal Equivalents and Millimeters / 1.37

1.10 Small Weight Equivalents:U.S. Customary (Grains and Ounces) Versus Metric

(Grams) / 1.38

1.11 Mathematical Signs and Symbols / 1.39

Chapter 2. Mensuration of Plane and Solid Figures 2.1

2.1 Mensuration / 2.1

2.2 Properties of the Circle / 2.10

Chapter 3. Layout Procedures for Geometric Figures 3.1

3.1 Geometric Constructions / 3.1

Chapter 4. Measurement and Calculation Procedures for Machinists 4.1

4.1 Sine Bar and Sine Plate Calculations / 4.1

4.2 Solutions to Problems in Machining and Metalworking / 4.6

4.3 Calculations for Specific Machining Problems (Tool Advance,Tapers, Notches and

Plugs, Diameters, Radii, and Dovetails) / 4.15

4.4 Finding Complex Angles for Machined Surfaces / 4.54

Chapter 5. Formulas and Calculations for Machining Operations 5.1

5.1 Turning Operations / 5.1

5.2 Threading and Thread Systems / 5.12

5.3 Milling / 5.22

5.4 Drilling and Spade Drilling / 5.38

5.5 Reaming / 5.61

5.6 Broaching / 5.63

5.7 Vertical Boring and Jig Boring / 5.66

5.8 Bolt Circles (BCs) and Hole Coordinate Calculations / 5.67

Chapter 6. Formulas for Sheet Metal Layout and Fabrication 6.1

6.1 Sheet Metal Flat-Pattern Development and Bending / 6.8

6.2 Sheet Metal Developments,Transitions, and Angled Corner Flange Notching / 6.14

6.3 Punching and Blanking Pressures and Loads / 6.32

6.4 Shear Strengths of Various Materials / 6.32

6.5 Tooling Requirements for Sheet Metal Parts—Limitations / 6.36

Chapter 7. Gear and Sprocket Calculations 7.1

7.1 Involute Function Calculations / 7.1

7.2 Gearing Formulas—Spur, Helical, Miter/Bevel, and Worm Gears / 7.4

7.3 Sprockets—Geometry and Dimensioning / 7.15

Chapter 8. Ratchets and Cam Geometry 8.1

8.1 Ratchets and Ratchet Gearing / 8.1

8.2 Methods for Laying Out Ratchet Gear Systems / 8.3

8.2.1 External-Tooth Ratchet Wheels / 8.3

8.2.2 Internal-Tooth Ratchet Wheels / 8.4

8.2.3 Calculating the Pitch and Face of Ratchet-Wheel Teeth / 8.5

8.3 Cam Layout and Calculations / 8.6

Chapter 9. Bolts, Screws, and Thread Calculations 9.1

9.1 Pullout Calculations and Bolt Clamp Loads / 9.1

9.2 Measuring and Calculating Pitch Diameters of Threads / 9.5

9.3 Thread Data (UN and Metric) and Torque Requirements (Grades 2, 5, and 8 U.S.

Standard 60° V) / 9.13

Chapter 10. Spring Calculations—Die and Standard Types 10.1

10.1 Helical Compression Spring Calculations / 10.5

10.1.1 Round Wire / 10.5

10.1.2 Square Wire / 10.6

10.1.3 Rectangular Wire / 10.6

10.1.4 Solid Height of Compression Springs / 10.6

10.2 Helical Extension Springs (Close Wound) / 10.8

10.3 Spring Energy Content of Compression and Extension Springs / 10.8

10.4 Torsion Springs / 10.11

10.4.1 Round Wire / 10.11

10.4.2 Square Wire / 10.12

10.4.3 Rectangular Wire / 10.13

10.4.4 Symbols, Diameter Reduction, and Energy Content / 10.13

10.5 Flat Springs / 10.14

10.6 Spring Materials and Properties / 10.16

10.7 Elastomer Springs / 10.22

10.8 Bending and Torsional Stresses in Ends of Extension Springs / 10.23

10.9 Specifying Springs, Spring Drawings, and Typical Problems and Solutions / 10.24

Chapter 11. Mechanisms, Linkage Geometry, and Calculations 11.1

11.1 Mathematics of the External Geneva Mechanism / 11.1

11.2 Mathematics of the Internal Geneva Mechanism / 11.3

11.3 Standard Mechanisms / 11.5

11.4 Clamping Mechanisms and Calculation Procedures / 11.9

11.5 Linkages—Simple and Complex / 11.17

Chapter 12. Classes of Fit for Machined Parts—Calculations 12.1

12.1 Calculating Basic Fit Classes (Practical Method) / 12.1

12.2 U.S. Customary and Metric (ISO) Fit Classes and Calculations / 12.5

12.3 Calculating Pressures, Stresses, and Forces Due to Interference Fits, Force Fits, and

Shrink Fits / 12.9

## PREFACE

This handbook contains most of the basic and advanced calculation procedures

required for machining and metalworking applications. These calculation procedures

should be performed on a modern pocket calculator in order to save time and

reduce or eliminate errors while improving accuracy. Correct bracketing procedures

are required when entering equations into the pocket calculator, and it is for this

reason that I recommend the selection of a calculator that shows all entered data on

the calculator display and that can be scrolled. That type of calculator will allow you

to scroll or review the entered equation and check for proper bracketing sequences,

prior to pressing “ENTER” or =. If the bracketing sequences of an entered equation

are incorrect, the calculator will indicate “Syntax error,” or give an incorrect solution

to the problem. Examples of proper bracketing for entering equations in the pocket

calculator are shown in Chap. 1 and in Chap. 11, where the complex four-bar linkage

is analyzed and explained.

This book is written in a user-friendly format so that the mathematical equations

and examples shown for solutions to machining and metalworking problems are not

only highly useful and relatively easy to use, but are also practical and efficient. This

the book covers metalworking mathematics problems, from the simple to the highly

complex, in a manner that should be valuable to all readers.

It should be understood that these mathematical procedures are applicable for:

Master machinists

Machinists

Tool designers and toolmakers

Metalworkers in various fields

Mechanical designers

Tool engineering personnel

CNC machining programmers

The gunsmithing trade

Students in technical teaching facilities