McGraw-Hill Machining and Metalworking Handbook 3rd ed - R. Walsh_ D. Cormier (McGraw-Hill 2006) WW Part 2 potx

Find the natural sine of the second angle, and multiply this bythe length of the sine plate.. Mathematics for Machinists and Metalworkers 79where D finished countersink diameter A counte

Thus the number 52 is 57.58 percent larger than the number 33.

We also can say that 33 increased by 57.58 percent is equal to 52;that is, 0.5758 × 33  33
52 Now

Thus the number 52 minus 36.54 percent of itself is 33 We also cansay that 33 is 36.54 percent less than 52; that is, 0.3654  52 = 19and 52  19
33 The number 33 is what percent of 52? That is,33/52
0.6346 Therefore, 33 is 63.46 percent of 52

2.6 Decimal Equivalents and Millimeter Chart

(see Fig 2.79)

52 33

52 0 3654

Figure 2.79 Decimal equivalents and millimeters

Mathematics for Machinists and Metalworkers

2.7 Degrees and Radians Chart (see Fig 2.80)

2.8 Mathematical Signs and Symbols

(see Table 2.1)

2.9 Greek Alphabet (see Table 2.2)

2.10 Sine Bar and Sine Plate Calculations

height for an angle of 34°25′ using a 5-in sine bar

sin 34°25′
x/5 (34°25′
34.416667 decimal degrees)

Figure 2.80 Degrees to radians conversion chart

TABLE 2.1 Mathematical Signs and Symbols

⬵ Is congruent to or approximately equal to

∼ Is approximately equal to or is similar to

< and Is less than, is not less than

Is greater than, is not greater than

± Plus or minus, respectively

ⴟ Minus or plus, respectively

→ Approaches, e.g., as x→ 0

Less than or equal to

≥,  More than or equal to

∂ Curly “d,” partial differentiation

Π The product of terms, product

arc As in arcsine (the angle whose sine is)

TABLE 2.2 The Greek Alphabet

sin 34.416667°
x/5

x
5  0.565207

x
2.826 inSet the sine bar height with Jo-blocks or precision blocks to 2.826 in.From this example it is apparent that the setting height can befound for any sine bar length simply by multiplying the length ofthe sine bar times the natural sine value of the required angle.The simplicity, speed, and accuracy possible for setting sine barswith the aid of the pocket calculator render sine bar tables obso-lete No sine bar table will give you the required setting heightfor such an angle as 42°17′26′′, but by using the calculator proce-

Figure 2.81 (a) Sine bar (b) Sine bar.

Mathematics for Machinists and Metalworkers 73

dure, this becomes a routine, simple process with less chance forerrors

Method

1 Convert the required angle to decimal degrees

2 Find the natural sine of the required angle

3 Multiply the natural sine of the angle by the length of the sinebar to find the bar-setting height (see Fig 2.81)

 are known to find angles X, A, B, and C.

tan X= tan cos 

sin C = cos /cos X Angle B = 180°  (angle A  angle C)

D= true angle

tan D = tan  sin T

tan sintan

θ

sin (sin cos )

Figure 2.82 Finding the unknown angles

Mathematics for Machinists and Metalworkers

cos A = cos E cos G cos A = sin  sin T

apparent angle, 
true angle, and 
angle of rotation

See also Fig 2.83b.

tan 
K/L

tan
K/(L cos )

tan cos 
K/L K/L
tan 
cos  tan or

tan 
cos  tan and

tan
tan  /cos The three-dimensional relationships shown for the angles andtriangles in the preceding figures and formulas are of impor-tance and should be understood This will help in the setting ofcompound sine plates when it is required to set a compoundangle

90° to each other, proceed as shown in Fig 2.84

Example: First angle
22.45° Second angle
38.58° (see Fig.2.84) To find the amount the intermediate plate must be raised from

the base plate (X dimension in Fig 2.84b) to obtain the desired first

Mathematics for Machinists and Metalworkers 75

3 This natural sine is now multiplied by the length of the sine plate

to find the X dimension in Fig 2.84b to which the intermediate

plate must be set

4 Set up the Jo-blocks to equal the X dimension, and set them in

position between the base plate and the intermediate plate

Figure 2.83 True and apparent angles

Mathematics for Machinists and Metalworkers

Figure 2.84 Setting angles on a sine plate.

(a)

(b)

(c)

Mathematics for Machinists and Metalworkers 77

cos 38.58°
0.781738tan 22.45°
0.4131920.781738  0.413192
0.323008arctan 0.323008
17.900872°

sin 17.900872°
0.3073710.307371  10 in (for 10-in sine plate)
3.0737 in

Therefore, set X dimension to 3.074 in (to three decimal places).

To find the amount the top plate must be raised (the Y dimension

in Fig 2.84c) above the intermediate plate to obtain the desired

second angle,

1 Find the natural sine of the second angle, and multiply this bythe length of the sine plate

2 Set up the Jo-blocks to equal the Y dimension, and set them

in position between the top plate and the intermediateplate

sin 38.58°
0.6326070.632607  10 in (for l0-in sine plate)
6.32607

Therefore, set the Y dimension to 6.326 in (to three decimal places).

2.11 Solutions to Problems in

Machining and Metalworking

The following sample problems will show in detail the importance

of trigonometry and basic algebraic operations as they apply tomachining and metalworking By using the methods and proce-

dures shown in this chapter of the Handbook, you will be able to

solve many basic and complex machining and metalworkingproblems

solve for d Use the tangent function:

tan A
y/x

d
D  2y

Mathematics for Machinists and Metalworkers

where A
taper angle

D
outside diameter of rod

d
diameter at end of taper

x
length of taper

y
drop of taper

Example: If the rod diameter
0.9375 diameter, taper length

0.875
x, and taper angle
20°
angle A, find y and d

Countersink depths (three methods for calculating)

Method 1: To find the tool, travel y from the top surface of the part

for a given countersink finished diameter at the part surface:

Mathematics for Machinists and Metalworkers 79

where D
finished countersink diameter

A
countersink angle

y
tool advance from surface of part

0.5397, or 0.540

Method 2: To find the tool travel from the edge of the hole (Fig

2.87), where D
finished countersink diameter, H
hole ter, and A
1⁄2countersink angle, 41°,

diame-tan A
x/y

y
x/tan A or x/(1⁄2countersink angle)

First, find x from

D
H  2x

If D
0.875 and H
0.500,

0.875
0.500 + 2x 2x
0.375

x
0.1875

Now solve for y, the tool advance:

y
x/tan A

y= 0 938 2=41

0 469

0 869

.tan

°

Figure 2.86 Countersink depth

Mathematics for Machinists and Metalworkers

0.1875/tan 41°

0.1875/0.8693

0.2157, or 0.216 (tool advance from edge of hole)

Method 3: To find tool travel from the edge of the hole (Fig 2.88),

where D
finished countersink diameter, d = hole diameter, 
1⁄2

countersink angle, and H
countersink tool advance from edge of hole,

H
1⁄2(D – d) cotan  or(Remember that cotan 
1/tan  or tan 
1/cotan .)

angle α and length x First, find angle α from

Mathematics for Machinists and Metalworkers 81

Then solve triangle ABC for 1⁄2angle α:

0 6875

2 175 0 316092

Figure 2.88 Tool travel from the edge of the hole

Figure 2.89 Finding taper angle

Mathematics for Machinists and Metalworkers

Now the x dimension is found from

tan 1⁄2α
0.9375/x

x
0.9375/tan 1⁄2α

2.966 (side A′ B′ or length x)

plane right triangles contains a compound angle Each of theseright triangles contains one of the angles of the compound angle,

and one of these angles is called the true angle The true angle of inclination of a plane to a reference plane is called the dihedral angle Figure 2.90 shows a typical triangular pyramid in which all

four faces are right triangles Most of the solids encountered inactual practice may be reduced to this type of pyramid by drawing aplane of symmetry The unknown angle then may be calculatedfrom any two known angles that lie in adjacent faces of the pyramid.The trigonometric solutions for all the compound angles of Fig 2.90may be calculated from the following trigonometric relationships:

Figure 2.90 Compound angles

Mathematics for Machinists and Metalworkers 83

Solutions To Compound Angles

Chapter

3

U.S Customary and Metric (SI) Measures and Conversions

3.1 Conversions for Length, Pressure, Velocity,

Volume, and Weight

To convert from: to: Multiply by:

Velocity

Volume

Cubic inches Liters 0.0164

U.S Customary and Metric (SI) Measures and Conversions

3.2 Standard Conversion Table: Measures Are

Found from the Table

meter

units (Btu)

meter

Cubic centimeters 2.113 × 10−3 Pints (liq.)

second

minute

U.S Customary and Metric (SI) Measures and Conversions

Multiply: By: To obtain:

meter

meter

Horsepower 1.014 Horsepower (metric)

minute

meter

meter

U.S Customary and Metric (SI) Measures and Conversions

Multiply: By: To obtain:

minute

Meters 1.094 Yards

minute

U.S Customary and Metric (SI) Measures and Conversions

Multiply: By: To obtain:

centimeter

meter

centimeter

meter

meter

meter

Watts 44.26 Foot pounds per minute

minute

3.3 Temperature Systems and Conversions

There are four common temperature systems used in engineeringand design calculations: (°F) Fahrenheit, (°C) Celsius (formerly centi-grade), (°K) Kelvin, and (°R) Rankine The conversion equation forCelsius to Fahrenheit or Fahrenheit to Celsius is

This exact relational equation is all that you need to convert fromeither system Enter the known temperature, and solve the equationfor the unknown value

Example: You wish to convert 66°C to Fahrenheit

The other two systems, Kelvin and Rankine, are converted asdescribed here The Kelvin and Celsius scales are related by theequation

K ⫽ 273.18 ⫹ °CThus 0°C ⫽ 273.18 K Absolute zero is equal to –273.18°C

59

6632

U.S Customary and Metric (SI) Measures and Conversions

Example: A temperature of ⫺75°C = 273.18 ⫹ (⫺75°C) ⫽ 198.18 K.The Rankine and Fahrenheit scales are related by the equation

°R ⫽ 459.69 ⫹ °FThus 0°F ⫽ 459.69°R Absolute zero is equal to ⫺459.69°F

Example: A temperature of 75°F ⫽ 459.69 ⫹ (⫹75°F) ⫽ 534.69°R

3.4 Small Weight Equivalents: U.S Customary

(Grains and Ounces) versus Metric (Grams)

10 grains ⫽ 0.02286 ounces or 0.648 grams

100 grains ⫽ 0.2286 ounces or 6.48 grams

Example: To obtain the weight in grams, multiply the weight ingrains by 0.0648, or divide the weight in grains by 15.43

Example: To obtain the weight in grains, multiply the weight ingrams by 15.43, or divide the weight in grams by 0.0648

Chapter

4

Materials: Physical Properties,

Characteristics, and Uses

Materials, including metals, alloys, plastics, and other compositecompounds, are of prime importance to the mechanical designer, tooldesigner, machinist, and metalworker The most important charac-teristics of materials to those who design and manufacture partsare the physical and chemical properties and the various uses towhich the materials may be applied This chapter discusses a greatnumber of metals, alloys, plastics, and compounds, including elas-tomers Included are composition, physical properties, hardness,heat-treatment temperatures, and other characteristics useful fordesign and metalworking practices

In design and metalworking practices, sometimes a material defect

or failure occurs for unknown reasons With the information vided in this chapter and in other appropriate American Society forTesting and Materials (ASTM), Society of Automotive Engineers(SAE), American Iron and Steel Institute (AISI), and American GearManufacturers Association (AGMA) standards, you may decide tohave the material analyzed at a metallurgical and chemical labo-ratory to check the properties and compositions, or you may havethe material checked mechanically at your company to determine if

pro-it meets the requirements of the material standards listed herein.When producing the engineering drawings or specifications for aparticular part, the appropriate American standard material des-ignation or military/federal specification always must be indicated

Source: McGraw-Hill Machining and Metalworking Handbook

on the drawings or specifications For example, if you are preparing

a drawing for a mechanical spring, and you wish to use music wire

in its construction, you must indicate on the drawing that thematerial is to meet ASTM A-228 or appropriate SAE specifications.You also may request a certified material analysis data sheet fromthe supplier of the material, whether it comes from a mill, a foundry,

a forge, or a materials processing plant It is the responsibility ofmaterial suppliers to provide design engineers and purchasingdepartments with these analysis sheets when so directed, which isthe usual procedure

Chapter 7 of this Handbook lists the machining characteristics and

machine tool cutting and drilling speeds for many common and ular steels, alloys, other metals, plastics, and composites In thischapter you also will find master cross-reference tables of all currenthardness scales or systems such as Rockwell, Brinell, and Vickers.With these tables, you will be able to convert the hardness numbersfor all the common and currently used hardness measurement sys-tems relative to each other

pop-Materials specifications and characteristics or properties tables

throughout this and other chapters of this Handbook are extracted

from the latest standards of the ASTM, SAE, and AGMA

There are a tremendous number of different engineering materialsfor which the ASTM and SAE have listings: metallic, plastic, andcomposite Steels alone account for hundreds of different alloys forwrought products, castings, and forgings Although these alloys arelisted and have specifications for composition and physical proper-ties, they may not be readily available, except as special order “millrun” quantities A typical example would be austenitic stainless steelsheet in light gauges, cold-rolled to three-quarters hard, spring tem-per This material is listed in the various 300 series stainless steelsbut may only be available from stock in 301 grade, three-quartershard, spring temper If you require 304 grade in your design applica-tion, it may be available only in mill-run quantities of a minimum

of 500 to 2000 lb per single order

As a general rule, during the early design stages of a particularpart, alternate materials must be investigated Owing to limitedavailability of your chosen material, you may be forced to use anothermaterial that is stocked by the material vendors When your antic-ipated material quantities are large, you then will have control ofthe material type as well as its physical size and special character-istics, such as finish, temper, and gauge For example, some of thelarger companies that order hundreds of tons of hot-rolled sheetsteel per year may specify to the mill that they want to run light on

the minus side of a particular gauge The tolerance on hot-rolledsheet steel is such that when it is run on the minus side of the gaugelimits, many thousands of pounds of steel can be saved The same

is true for wrought-copper products such as bus bars and coppersheet, where the savings may be even more advantageous

The list of steel alloys is so large that some authorities and lating organizations have contemplated restricting the materialstandards to controlled listings, wherein redundant materials areeliminated The large listings of alloys, plastics, and compositescreate a stocking problem for distributors of mill products, plastics,and composite materials

regu-For designers, machinists, tool makers, and metalworkersemployed at small to medium-sized companies, a good approach to theaforementioned materials problems is to obtain the stock materialscatalogs available from the large distributors and manufacturers such

as Ryerson, Vincent, Atlantic, Alcoa, Reynolds, Bethlehem Steel, U.S.Steel, Anaconda, General Electric, Dupont, Monsanto, and others.The design and fabrication data for a great number of materials are

listed in this and other chapters of this Handbook, together with

typ-ical uses and applications for these materials

4.1 Steels

This section lists carbon and alloy steels, as well as the stainlesssteels, in their wrought form, that is to say, in the hot-rolled, coldrolled, or cold-drawn forms The usual shapes are sheets, plates, bars

or strips, rounds, hexagons, tube, pipe, and structural configurations(beams, angles, channels, tees, square and rectangular tubes, andzees) Cast irons and steels and other casting materials are listed

in Chap 12

When carbon is added to iron in small quantities, carbon steel isproduced Besides carbon, a number of metallic elements can beadded to iron to give the characteristics inherent in the varioustypes of steels The usual alloying elements are

■ Aluminum, which controls grain size in the steel

■ Boron, which improves hardenability

■ Chromium, which increases response to heat treatment as well

as toughness (Chromium is used in stainless steels alone or withnickel.)

■ Columbium, which is used in 18-8 stainless steels and weldingelectrodes

Materials: Physical Properties, Characteristics, and Uses

■ Copper, which controls atmospheric corrosion and increases yieldstrengths

■ Lead, which greatly improves machinability

■ Manganese, which imparts strength and response to heattreatment

■ Molybdenum, which increases depth of hardness and toughness

■ Nickel, which increases strength and toughness but is not tive in improving hardenability

effec-■ Phosphorus, which is present in all steels and increases yieldstrength

■ Silicon, which improves tensile strength and can improve enability

hard-■ Sulfur, which improves machinability but is detrimental to forming properties

hot-■ Tellurium, which improves machinability in leaded steels

■ Titanium, which is added to 18-8 stainless steels to preventcarbide precipitation

■ Tungsten, which is used in good tool steels, making a fine-grainstructure when used in small amounts (When used in amountsfrom 17 to 20 percent, it produces a high-speed steel that retainshardness at high temperatures.)

■ Vanadium, which is used to improve the shock strength of steelsand retards grain growth even after hardening from high tem-peratures

Age hardening A process of aging that increases hardness andstrength Age hardening usually follows rapid cooling or coldworking

Annealing A process involving heat and cooling, usually applied

Case hardening A process of hardening a ferrous alloy so thatthe case, or surface layer, is much harder than the interior of thepart The typical case-hardening processes are carburizing andquenching, cyaniding, carbonitriding, nitriding, induction harden-ing, and flame hardening Cases of Rockwell C 55 to 60 are readilyobtained in medium- to high-carbon steels.

Ductility The property of a material that allows it to be drawn out

of shape before fracturing by stress

Elastic limit The maximum stress a material is capable of taining without a permanent set or deformation

sus-Fatigue The tendency for a metal to break after repeated orcyclic loadings that are below the ultimate tensile strength Also

known as fatigue-endurance limit.

Flame hardening A process of hardening a ferrous alloy byheating it above the transformation range by means of a flameand then cooling as required

Hardenability The property of a ferrous alloy that determinesthe depth and distribution of hardness induced by quenching

Induction hardening A process of hardening a ferrous alloy byheating it above the transformation range by means of electricalinduction and then cooling as required

Killed steel Steel that is deoxidized with silicon or aluminum inorder to reduce the oxygen content so that no reaction occursbetween carbon and oxygen during solidification

Martensite An unstable constituent in quenched steel formedwithout diffusion only during cooling below a certain tempera-ture Martensite is the hardest of the transformation products ofaustenite

Nitriding A process of case hardening in which a ferrous alloy,usually of special composition, is heated in an atmosphere ofammonia or in direct contact with a nitrogenous material to pro-duce surface hardening by absorbing nitrogen without quenching

Normalizing Heating a steel part of heavy section to a ature l00°F above the critical range and then cooling in still air

temper-Pickeling Chemical or electrochemical removal of surface scaleand oxides

Quench hardening Heating a steel within or above the formation range and cooling at a rate faster than the critical rate

Materials: Physical Properties, Characteristics, and Uses

to increase the hardness substantially Usually involves theformation of martensite.

Solution heat treatment A process in which an alloy is heated to

a suitable temperature, held at this temperature long enough forcertain constituents to enter into solid solution, and then cooledrapidly to hold the constituent in solution The metal is left in asupersaturated state that is unstable and subsequently mayexhibit age hardening

Spheroidizing Any process of heating and cooling that duces a round or globular form of carbide in steels

pro-Strain hardening An increase in hardness and strength caused by

plastic deformation at temperatures lower than the tion range

recrystalliza-Stress relieving A process of reducing residual stresses in ametal part by heating the part to a suitable temperature andholding this temperature for a sufficient time This process isapplied to relieve stresses induced by casting, quenching, nor-malizing, machining, cold working (i.e., springs), or welding

Temper A condition produced in a metal or alloy by mechanical

or thermal treatment and having characteristic structure andmechanical properties In addition to the annealed temper,conditions produced by thermal treatment are the solution heat-treated temper and the heat-treated and artificially aged temper

Yield strength The stress at which a material exhibits a fied limited deviation from proportionality of stress to strain Inmost steels, there is a proportionality between the amount ofstress that produces a certain amount of strain This phenomenon

speci-is known as Hooke’s law When a material passes its yield point, a

lesser amount of stress produces a greater amount of strain untilthe ultimate strength point is reached, where the material breaks

die steels: Physical properties, compositions, heat

treatment, and uses

Tables 4.1 through 4.20 contain the numbering system for cation of the various types of steels given in the SAE and UnifiedNumbering System (UNS) designations Figure 4.1 shows propertiesand heat treatments for carbon, alloy, and stainless steel Table 4.21

identifi-is an approximate equivalent hardness number table for Brinellhardness numbers for steel, with cross-references to other hardness

designation systems Table 4.22 is an approximate equivalenthardness number table for Rockwell C hardness numbers for steel,with cross-references to other hardness designation systems.

and readily available steels that are usually stocked by suppliers andwhich have vast applications in industry Tables 4.1 through 4.20show the physical properties and heat-treatment processes for agreat number of American standard steels The following list of appli-cations will prove useful to many designers and mechanical engi-neers, as well as to machinists, tool engineers, tool and die makers,and other metalworkers throughout industry The following list is notall-inclusive but rather is indicative of the much-used and readilyavailable types of carbon, alloy, and stainless steels

(Text continued on page 150.)

Figure 4.1 Brinell hardness numbers indicating approximate tensilestrengths of steels

Materials: Physical Properties, Characteristics, and Uses

Source: McGraw-Hill Machining and Metalworking Handbook