mcgraw hill machining and metalworking handbook 3rd ed r walsh d cormier mcgraw hill 2006 ww 9 pptx

1.3.1 Process capability When a process planner selects machines to perform a given ation on a part, he or she must know whether or not the machine oper-Modern Metalworking Machinery, To

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Modern Metalworking

Machinery, Tools, and Measuring Devices

Metalworking machinery, tools, and measuring instruments haveadvanced considerably over the past 50 years This chapter will showsome of the new machines, tools, and instruments used throughoutindustry today that allow us to produce parts faster and more accu-rately than was possible in the past The widespread use and imple-mentation of microprocessors to control the actions of metalworkingmachinery is evident in many of the photographs of modern equipmentshown in this chapter Photographs of other modern metalworking

machinery appear throughout this Handbook.

When a metal part is fabricated, the part blank either can come from

a near-net-shape manufacturing process or it can come in the form ofbars, rods, plates, etc Metal casting processes such as die casting,sand casting, and investment casting are the most common methods

of producing a part blank that is close to its final shape (i.e., near netshape) Recent years also have seen a flood of new solid freeform fab-rication (SFF) processes that are capable of directly producing near-net-shape functional metal parts without the need for molds, dies,etc (see Chap 10) In the case of near-net-shape processes, rough

1

1

machining of large amounts of stock is not necessary Instead, it isonly necessary to finish machine those features that are critical tothe function of a part.

1.1.1 Primary processes

Die casting. Small or medium-sized parts in nonferrous alloys such

as magnesium, aluminum, and zinc are injected under pressureinto a steel die A machining allowance of 0.25 to 0.5 mm (0.010 to0.020 in) for critical features is typical

Sand casting. Molten metal is cast into a packed-sand mold Partsweighing from just a few ounces to several tons can be sand cast Themost commonly sand-cast metals include irons, stainless steels,aluminum, and nickel alloys Since the surface of the cast part istextured, a machining allowance typically is provided for critical fea-tures Recommended machining allowances for a variety of metalsare provided in Table 1.1

Investment casting. Both ferrous and nonferrous metals may beinvestment cast into a single-use refractory ceramic mold High-temperature-reactive metals such as titanium typically are vacuuminvestment cast

Forging. Metals such as nonferrous alloys (e.g., aluminum, sium, and brass), steels, and nickel alloys are relatively easy to forge.The slugs are essentially hammered by a die such that the metaldeforms to the shape of the die Recommended machining allowancesfor a variety of metals are provided in Table 1.2

magne-Powder metallurgy. Metal powder is compacted by a die and thensintered to hold its shape The resulting parts are porous and option-ally are infiltrated to 100 percent density

Extrusion. A heated billet is forced through a die opening such thatthe length of the billet takes on the cross-sectional shape of the dieopening

1.1.2 Metal-cutting processes

CNC machining. The two most versatile machines in the modernmachining industry are the computer numerical control (CNC)

2 Chapter One

milling machine (Fig 1.1) and the CNC lathe (Fig 1.2) A key to theversatility of these machines is the automatic tool changer Verticalmachining centers (VMCs) such as the one shown in Fig 1.1 include

a carousel that holds many different cutting tools such as millingcutters, drills, reamers, and taps The automatic tool changerchanges cutting tools between machining operations without any

Modern Metalworking Machinery, Tools, and Measuring Devices 3

TABLE 1.1 Sand Casting Allowances for Each Side

Allowance, mm (in)Casting size, mm (in)* Drag and sides Cope surfaceGray iron Up to 150 (up to 6) 2.3 (3⁄32) 3 (1⁄8)

user intervention, thus allowing several machining operations to

be executed in a single workpiece setup Likewise, the CNC lathe

in Fig 1.2 incorporates an automatic tool changer that can switch

between tools that perform facing, knurling, grooving, boring, and

many other turning operations

Electric discharge machining (EDM). EDM comes in two forms—sinker

EDM and wire EDM Sinker EDM uses spark erosion to machine a

workpiece with a graphite or copper electrode whose shape is the

negative of the cavity being machined Wire EDM uses spark

ero-sion with a wire to cut two-dimenero-sional (2D) profiles

Laser machining. A powerful laser beam coupled with a CNC

motion-control system is used to cut 2D profiles in sheet or plate material

4 Chapter One

TABLE 1.2 Typical Machining Allowances for Forgings

Forging size: Projected area at parting line, mm (in)

Modern Metalworking Machinery, Tools, and Measuring Devices 5

Figure 1.2 CNC lathe

Figure 1.1 Vertical machining center

Complex, thin parts whose quantity does not warrant a hard dieare produced using this method.

Chemical milling. Large masses of metal may be removed effectively

in producing a part using the etching action of chemicals Very thinand delicate parts also may be produced with chemical milling oretching A tough photoresistive substance covers the parts of themetal that are not to be removed Printed circuit board production

is actually a chemical milling operation

Waterjet machining. A very high pressure jet of water, loaded withmicrofine abrasives, is used to cut the sheet or plate material ofmetal, plastic, glass, or other composition As is the case with lasermachining, waterjet machining is useful when the production vol-umes do not warrant a hard die The absence of a heat-affectedzone is advantageous as well Figure 1.3 shows a nested pattern ofsheet metal parts being waterjet machined Figure 1.4 shows acomplex geometric shape cut from plate

6 Chapter One

Figure 1.3 Waterjet machining operation (Image courtesy of OMAX

Corpo-ration, www.omax.com.)

1.1.3 Sheet metal parts fabrication methods

Hard dies. A die set is used to stamp out the part in flat pattern gressive dies also bend the part into the required shape after it isstamped in flat pattern This is the most common, economical methoddevised to mass produce large quantities of parts to great accuracy

Pro-Punch press. Large sheet metal parts may be made to accuratestandards using modern computer-controlled automatic multistationpunch presses Programmers write the direct numerical control(DNC) programs for these machines, which are then loaded intothe machine’s computer or controller The machine operator startsthe program and stands back to watch the machine go through thesequence of operations required to produce the finished part in flatpattern

Modern Metalworking Machinery, Tools, and Measuring Devices 7

Figure 1.4 Complex waterjet-machined plate (Image courtesy of OMAX

Corporation, www.omax.com.)

Roll forming. Flat strips of sheet metal are fed into the ing machine, where they progress through a set of sequenced rollers

roll-form-to produce a long sheet metal part of constant cross-sectionalshape

Hydropressing. A sheet metal flat-pattern part is placed on a set offorming dies, being located correctly with locator pins, and is thenpressed into shape by the action of the hydropress Many aircraftsheet metal parts are produced in this manner Lightening holesand shrink flutes are produced simultaneously with the part tocontrol the metal along curved surfaces

Hydraulic brakes. In this machine, a flat-pattern sheet metal part

is given flanges or webs to produce the finished part The modernbrakes have automatic back gauges and material-handling devices

to assist the operator in making the various bends and flangesrequired on the part

Hydraulic shears. The standard hydraulic shear cuts sheet metalaccording to the back gauge set by the machine operator and his orher accuracy in placing the sheet into the machine

The preceding section provided an overview of many types of working and machining processes In a production environment,parts typically are fabricated according to specifications on thecomputer-aided design (CAD) drawing using one or more of theaforementioned processes At certain points during the fabricationprocess, parts are inspected to verify that they satisfy the requiredgeometric and dimensional tolerances In some cases, 100 percent

metal-of the parts are inspected In many instances, however, it is cient to inspect a subset of parts using a statistical sampling scheme.This section describes some of the instruments used to performcomponent inspection

suffi-1.2.1 Coordinate measuring machines (CMMs)

CMMs are highly versatile inspection machines Although CMMs areavailable in numerous configurations, the typical CMM consists of

a probe that is positioned beneath a gantry Depending on the type

8 Chapter One

of CMM, the probe can be moved manually by the operator’s hand, or

it can be moved automatically via a motion-control system Theworkpiece being inspected is rigidly clamped to the CMM’s granitetable

In manual mode, the operator tells the computer which feature(s)

he or she is going to inspect, and the control computer will theninstruct the user as to what points need to be probed for a givenfeature To measure the distance between two faces, for instance, theoperator must touch the probe to at least three points on the firstface (i.e., three points define the plane) and one point on the secondface (i.e., the perpendicular distance from a point on the secondsurface to the plane defined by the three points on the first surface)

To measure the diameter of a hole, the user is prompted to touchthe probe to three or more points around the perimeter of the hole.For each feature, the CMM control computer prompts the user totouch the probe to the appropriate number of points for the featurebeing inspected

Fully automated CMMs are also available With automatedCMMs, the inspection planner starts with the geometric and/ordimensional tolerances (GD&Ts) specified in the CAD model by the

mechanical designer CMM software packages such as PC-DMIS

are now available that are capable of extracting GD&T tions from a CAD model Using this software, the inspection planneridentifies each feature in the CAD model to be inspected in a givensetup on the bed of the CMM The software then automatically gen-erates an inspection plan for that setup on the CMM The process

specifica-is very much like generating toolpaths for a computer numericalcontrol (CNC) milling machine In this case, the touch probe ratherthan a rotating cutting tool automatically follows the prescribedpath After the workpiece has been inspected, the CMM softwaregenerates an inspection report In many instances, companies willstore these inspection results in a central database for purposes oftraceability

Both rigid and touch probes are available on CMMs With a rigidprobe, the operator must press a button manually so that the CMM

can capture the x,y,z coordinates of the probe at that instant With

a touch probe, the probe automatically senses when it has touched

the part, and the x,y,z coordinates are sent to the control computer

immediately Motorized touch probes are also available that cantilt and swivel in order to inspect features that otherwise would not

be accessible in a given setup orientation

Modern Metalworking Machinery, Tools, and Measuring Devices 9

1.2.2 Handheld measurement and gauging devices

Definitions

Precision. For any measuring device, precision is an indication of

how much variation one will observe when one measures the samedimension on the same part using the same measuring device The

terms precision and repeatability are often used interchangeably.

The sample standard deviation of multiple measurements taken

on the same feature with the same device by the same operator is

an indicator of precision The smaller the standard deviation, thehigher is the precision

Accuracy. Accuracy is an indication of how close the measured

dimension is to the true value for that dimension Note that racy and precision are not the same thing A device can be highlyprecise but very inaccurate In other words, it can consistently givethe same wrong measurement

accu-Resolution. This is the smallest unit of measure that can be played by the measuring device If a digital caliper displays mea-

dis-surements to four decimal places, then the resolution is 0.0001 in

10 Chapter One

Figure 1.5 Digital micrometer

permit the operator to toggle between millimeters and inches asneeded.

Dial indicators. Dial indicators show linear displacement of a lus as it is moved across the surface of a part or vice versa (Fig.1.7) They can be used to measure features such as the roundness

sty-of a rotational part, the flatness sty-of a surface, or the depth sty-of ahole

Height gages. Height gauges measure the height of a feature, asthe name implies (Fig 1.8)

Modern handheld digital measurement devices can be interfacedwith computers on the shop floor for use with statistical processcontrol (SPC) programs Measurements collected from these devices

do much more than indicate whether any individual part is withinspecifications When the measurements for a succession of parts areplotted graphically, the machine operator can detect any nonrandomtrends in machine performance visually and then take correctiveaction if necessary

1.3.1 Process capability

When a process planner selects machines to perform a given ation on a part, he or she must know whether or not the machine

oper-Modern Metalworking Machinery, Tools, and Measuring Devices 11

Figure 1.6 Digital caliper

is capable of satisfying the tolerances specified for that part Theprocess capability study is used to determine whether or not this

is the case For a given feature, a target dimension is specifiedalong with upper and lower tolerance values For instance, thespecification

has a target value of 2.500 in, an upper specification limit (USL) of2.503 in, and a lower specification limit (LSL) of 2.497 in

Modern Metalworking Machinery, Tools, and Measuring Devices 13

CpK is one measure of process capability that provides an

indi-cation of both accuracy and precision:

where ␴ is usually estimated by S:

where USL⫽ upper limit on the tolerance

LSL⫺⫽ lower limit on the tolerance

X ⫽ process mean, or average value of a set of ments

measure-␴ ⫽ standard deviation of entire population of parts

S⫽ standard deviation of measurements from a

sam-pling of n parts When CpK≥ 1, then one can conclude that at least 99.73 percent ofthe parts produced will fall within the range specified by the LSLand USL In plain English, this means that the process is centeredsufficiently close to the target dimension value and that the spread

of measurements is smaller than the tolerance range for that

fea-ture If CpK < 1, then one can conclude that fewer than 99.73

per-cent of the parts produced will meet the design specifications Inthis case, the manufacturing engineer can consider alternativeprocesses, or he or she can work to improve the existing process inorder to get the defects to an acceptable rate

Example The width of a slot has a design specification of 2.500 ±0.003 in The slot width for each part in a batch of 30 parts has beenmeasured, and the average value of the 30 measurements is 2.501 in.The standard deviation of these 30 measurements is 0.0008

The CpK value of 0.833 indicates that the defect rate for this

process will be unacceptably high if this company is striving for99.73 percent acceptance rate (i.e., 6␴ manufacturing)

i i n

1

1

14 Chapter One

a certain amount of random variation in feature sizes that are

pro-duced The magnitude of this random variation is what determineswhether or not a machine is capable of meeting required tolerances

for a given part There also may be assignable variation present.

Assignable variability refers to variations that can be attributedspecifically to a particular cause For example, a metal chip may belodged beneath parallels supporting a part in a vise The chip willraise the height of the part, thus increasing the depth of cut beyondwhat was intended This is an assignable cause of variation thatcan be identified and eliminated Control charts are an extremelyvaluable tool They allow the machine operator to see graphicallyboth sudden and gradual shifts in the process

Many different types of control charts are available, and ested readers are encouraged to consult books dedicated to statisticalprocess control In its simplest form, a process control chart graphi-cally plots the measured dimensions for the last 20 parts (typically)

inter-to be measured The target value, USL, and LSL are also indicated

on the chart Much more sophisticated SPC tools are available, buteven this simple control chart allows a machine operator to detecteither sudden or gradual shifts in process performance For exam-ple, Fig 1.9 shows measurements of a feature for 20 parts Thenominal (target) measurement is 2.500 in, with an allowable toler-ance of ±0.003 in This chart clearly shows an upward nonrandomtrend in the size of this feature On seeing a nonrandom (i.e.,

Modern Metalworking Machinery, Tools, and Measuring Devices 15

Figure 1.9 Control chart indicating upward trend

assignable) cause of variation, the machine operator would knowthat he or she should stop the machine and investigate the rootcause of this variation before parts are produced outside specifica-tions The root cause could be a cutting tool that is shifting in itscollett or any number of other problems.

16 Chapter One

proce-this chapter and other subsections of proce-this Handbook, all the basic

and important aspects of these branches of mathematics will be ered in detail

an equation with five variables, shown in terms of R Solving for

G gives

Gd4
R8ND3(cross-multiplied)

Solving for d gives

Solving for D gives

Solve for N using the same transposition procedures just shown.

Solving a typical algebraic equation. An algebraic equation can besolved by substituting the numerical values assigned to the vari-ables, which are denoted by letters, and then finding the unknownvalue

Example:

(belt-length equation)

If C = 16, D = 5.56, and d = 3.12 (the variables), solve for L (by

substituting the values of the variables into the equation):

Most of the equations shown in this Handbook are solved in a

similar manner, i.e., by substituting known values for the variables

in the equations and solving for the unknown quantity using dard algebraic and trigonometric rules and procedures

stan-Ratios and proportions. If a/b
c/d, then

Quadratic equations. Any quadratic equation may be reduced to theform

ax2 bx  c
0 The two roots, x1and x2, equal

When a, b, and c are real, if b2– 4ac is positive, the roots are real and unequal If b2– 4ac is zero, the roots are real and equal If b2– 4ac

is negative, the roots are imaginary and unequal

2.1.2 Plane trigonometry

There are six trigonometric functions: sine, cosine, tangent, gent, secant, and cosecant The relationship of the trigonometricfunctions is shown in Fig 2.1 Trigonometric functions shown for

cotan-angle A (right-cotan-angled tricotan-angle) include

sin A
a/c (sine) cos A
b/c (cosine) tan A
a/b (tangent)

− ±b b − ac a

2

42

a b b

c d d

a b b

c d d

Mathematics for Machinists and Metalworkers 19

Figure 2.1 Right-angled triangle

cot A
b/a (cotangent) sec A
c/b (secant) csc A
c/a (cosecant) For angle B, the functions would become

sin B
b/c (sine) cos B
a/c (cosine) tan B
b/a (tangent) cot B
a/b (cotangent) sec B
c/a (secant) csc B
c/b (cosecant)

As can be seen from the preceding, the sine of a given angle isalways the side opposite the given angle divided by the hypotenuse

of the triangle, the cosine is always the side adjacent to the givenangle divided by the hypotenuse, and the tangent is always theside opposite the given angle divided by the side adjacent to theangle These relationships must be remembered at all times whenperforming trigonometric operations Also,

sin A
1/csc A cos A
1/sec A tan A
1/cot A

This reflects the important fact that the cosecant, secant, andcotangent are the reciprocals of the sine, cosine, and tangent,respectively This fact also must be remembered when performingtrigonometric operations

Also, in any right-angled triangle,

sin x
cos (90°  x)

cos x
sin (90° – x) (x is the given angle other than 90°)

20 Chapter Two

tan x
cot (90°  x)

Equivalent expressions. The following trigonometric expressionsare mathematically equivalent and may be used to advantage insolving many trigonometric problems It is wise to try to remember

as many of these expressions as possible, although they may be

referred to in this chapter of the Handbook as required.

Note: The choice of the ± sign is determined by which quadrant

the angle x is situated in (see “Signs and Limits of Trigonometric

Functions” below)

Signs and limits of the trigonometric functions. The following nate chart shows the sign of the function in each quadrant and itsnumerical limits As an example, the sine of any angle between 0and 90° will always be positive, and its numerical value will rangebetween 0 and 1, whereas the cosine of any angle between 90 and180° will always be negative, and its numerical value will rangebetween 0 and 1 Each quadrant contains 90°; thus the fourthquadrant ranges between 270 and 360°

coordi-tan sincos

cot cossinsin cos

of the triangle when certain values are known Refer to Fig 2.2 forillustrations of the trigonometric laws that follow.

The law of sines(see Fig 2.2)

The law of cosines(see Fig 2.2)

a2
b2+ c2– 2bc cos A

b2
a2+ c2– 2ac cos B

c2
a2+ b2– 2ab cos C

The law of tangents(see Fig 2.2)

With the preceding laws, the trigonometric functions for right-angledtriangles, the Pythagorean theorem, and the following triangle

tan2

2

a A

b B

c C a

b

A B

b c B

sinsin

sinsi

n

sinsin

C

a c

A C

=

22 Chapter Two

solution chart, it will be possible to find the solution to any planetriangle problem, provided the correct parts are specified.

The Solution of Triangles

Known: Any two sides Use the Pythagorean theorem to solve

unknown side; then use the trigonometricfunctions to solve the two unknown angles.The third angle is 90°

Known: Any one side and Use trigonometric functions to solve the twoeither angle that is unknown sides The third angle is 180°—

Known: Three angles and Cannot be solved because there are an

no sides (all triangles) infinite number of triangles that satisfy

three known internal angles

Known: Three sides Use trigonometric functions to solve the

two unknown angles

Known: Two sides and any Use the law of sines to solve the second

one of two nonincluded unknown angle The third angle is 180°—

the other sides using the law of sines orthe law of tangents

Known: Two sides and the Use the law of cosines for one side and the

Known: Two angles and Use the law of sines to solve the other

angle is 180°—the sum of two knownangles

Known: Three sides Use the law of cosines to solve two of the

unknown angles The third angle is180°—the sum of two known angles

Mathematics for Machinists and Metalworkers 23

Figure 2.2 Oblique triangle

The Solution of Triangles (Continued )

Known: One angle and one Cannot be solved except under certain

side (non-right triangle) conditions If the triangle is equilateral or

isosceles, it may be solved if the knownangle is opposite the known side

Finding heights of non-right-angled triangles The height x shown in Fig.

2.3 and Fig 2.4 is found from

Figure 2.3 Height of triangle x.

Figure 2.4 Height of triangle x.

The area when the three sides are known is (see Fig 2.6) (this holdstrue for any triangle)

A= s s a s b s c( − )( − )( − )

Mathematics for Machinists and Metalworkers 25

Figure 2.5 Triangles: (a) right triangle, (b) oblique triangle.

Figure 2.6 Triangle

(b) (a)

Converting angles to decimal degrees. Angles given in degrees, utes, and seconds must be converted to decimal degrees prior to find-ing the trigonometric functions of the angle on a handheld calculator.

min-Procedure: Convert 26°41′26′′ to decimal degrees

Degrees
26.000000 in decimal degrees

Minutes
41/60
0.683333 in decimal parts of a degree

Seconds
26/3600
0.007222 in decimal parts of a degree

The angle in decimal degrees is then

26.000000  0.683333  0.007222
26.690555°

Converting decimal degrees to degrees, minutes, and seconds

Procedure: Convert 56.5675 decimal degrees to degrees, minutes,

and seconds

Degrees
56 degreesMinutes
0.5675  60
34.05
34 minutesSeconds
0.05 (minutes)  60
3 secondsThe answer, therefore, is 56°34′3′′

Samples of solutions to triangles

Solving right-angled triangles by trigonometry Required: Any one side and angle A or angle B (see Fig 2.7) Solve for side a:

Solve for side b:

cos

cos

A b c b

c a

Then angle B
180°  (angle A  90°)
180°  123.162°
56.838° We now know sides a, b, and c and angles A, B, and C.

Solving non-right-angled triangles using the trigonometric laws.

Solve the triangle in Fig 2.8 given two angles and one side:

A
45°

B
109°

a
3.250

First, find angle C:

Angle C
180°  (angle A  angle B)

180°  (45°  109°)

180°  154°

26°

Mathematics for Machinists and Metalworkers 27

Figure 2.7 Solve the triangle

Figure 2.8 Solve the triangle

Second, find side b by the law of sines:

Third, find side c by the law of sines:

Solve the triangle in Fig 2.9 given two sides and one angle:

b B

B B

sin sin

2 509

26 276°

a A

c C c

c

sin sin

b B b

Second, find angle C:

Angle C
180°  (angle A  angle B)

180°  42.276°

137.724°

Third, find side c from the law of sines:

We may now find the altitude or height x of this triangle (see Fig 2.9).

Refer to Fig 2.4

(where angle C′
180° 137.724°
42.276°)

This height x also can be found from the sine function of angle C′

when side a is known, as shown below:

Both methods yield the same numerical solution of 1.051

Solve the triangle in Fig 2.9a given three sides and no angles.According to the preceding triangle solution chart, solving thistriangle requires use of the law of cosines Proceed as follows: First,

solve for any angle (we will take angle C first):

c C c

c

sin sin

Second, by the law of cosines, find angle B:

Converting degrees to radians. To convert from degrees to radians,

you must first find the degrees as decimal degrees If R represents

2.1.3 Important mathematical constants

procedures for triangles

There are four possible cases in the solution of oblique triangles:

Case 1: Given one side and two angles: a, A, B

Case 2: Given two sides and the angle opposite them: a, b, A or B Case 3: Given two sides and their included angle: a, b, C

Case 4: Given the three sides: a, b, c

All oblique (non-right-angle) triangles can be solved by use ofnatural trigonometric functions: the law of sines, the law of cosines,

and the angle formula: angle A  angle B  angle C
180° This

may be done in the following manner:

Case 1: Given a, A, and B, angle C may be found from the angle formula, and then sides b and c may be found by using the law of

sines twice

Case 2: Given a, b, and A, angle B may be found by the law of sines, angle C from the angle formula, and side c by the law of sines again Case 3: Given a, b, and C, side c may be found by the law of cosines, and angles A and B may be found by the law of sines used twice

or angle A from the law of sines and angle B from the angle formula Case 4: Given a, b, and c, the angles all may be found by the law of cosines, or angle A may be found from the law of cosines, and angles

B and C from the law of sines, or angle A from the law of cosines, angle B from the law of sines, and angle C from the angle formula.

Note: Case 2 is called the ambiguous case, in which there may be one solution, two solutions, or no solution, given a, b, and A.

Mathematics for Machinists and Metalworkers 31

■ If angle A < 90° and a < b sin A, there is no solution.

■ If angle A < 90° and a
b sin A, there is one solution—a right

triangle

■ If angle A < 90° and b > a > b sin A, there are two solutions—

oblique triangles

■ If angle A < 90° and a ≥ b, there is one solution—an oblique triangle.

■ If angle A < 90° and a b, there is no solution.

■ If angle A > 90° and a > b, there is one solution—an oblique

triangle

Special half-angle formulas. In case 4 triangles where only the three

sides a, b, and c are known, the sets of half-angle formulas shown

below may be used to find the angles:

32 Chapter Two

≥

2.1.5 Powers-of-10 notation

Numbers written in the form 1.875  105or 3.452  106are sostated in powers-of-10 notation Arithmetic operations on numbersthat are either very large or very small are processed easily andconveniently using the powers-of-10 notation and procedures Ifyou will note, on the handheld scientific calculator this process iscarried out automatically by the calculator If the calculatedanswer is larger or smaller than the digital display can handle, theanswer will be given in powers-of-10 notation

This method of handling numbers is always used in scientificand engineering calculations when the values of the numbers sodictate Engineering notation usually is given in multiples of 3,such as 1.246  103, 6.983  106, etc

How to calculate with powers-of-10 notation. Numbers with manydigits may be expressed more conveniently in powers-of-10 notation,

Multiplication, division, exponents, and radicals in powers-of-10notation are handled easily, as shown below:

(1.246  104)  (2.573  10–4)
3.206  100
3.206 (Note: 100
1)(1.785  107)  (1.039  10–4)
(1.785/1.039)  107–(–4)
1 718  1011

■ Exponents are algebraically added for multiplication

■ Exponents are algebraically subtracted for division

■ Exponents are algebraically multiplied for power raising

■ Exponents are algebraically divided for taking roots

Mathematics for Machinists and Metalworkers 33

2.2 Geometric Principles

In any triangle, angle A  angle B  angle C
180°, and angle A

180° (angle A  angle B), and so on (see Fig 2.10) If three sides of

one triangle are proportional to the corresponding sides of another

triangle, the triangles are similar Also, if a:b:c
a′:b′:c′, then angle A

angle A′, angle B
angle B′, angle C
angle C′, and a/a′
b/b′
c/c ′ Conversely, if the angles of one triangle are equal to the respec-

tive angles of another triangle, the triangles are similar and their

sides proportional; thus, if angle A
angle A′, angle B
angle B′, and angle C = angle C ′, then a:b:c
a′:b′:c′ and a/a′
b/b′
c/c′ (see

■ Equilateral triangle (see Fig 2.13) If side a
side b
side c, angles A, B, and C are equal (60°)

■ Right triangle (see Fig 2.14) c2
a2 b2and c
(a2 b2)1/2when

angle C
90° Therefore, a
(c2 b2)1/2and b
(c2 a2)1/2 Thisrelationship in all right angle triangles is called the Pythagoreantheorem

■ Exterior angle of a triangle (see Fig 2.15) Angle C
angle A  angle B.

■ Intersecting straight lines (see Fig 2.16) Angle A
angle A′, and angle B
angle B′.

Mathematics for Machinists and Metalworkers 35

Figure 2.13 Equilateral triangle

Figure 2.12 Isosceles triangle

■ Two parallel lines intersected by a straight line (see Fig 2.17) Alternate interior and exterior angles are equal: Angle A

angle A ′, and angle B
angle B′.

■ Any four-sided geometric figure (see Fig 2.18) The sum of all

inte-rior angles
360°; angle A  angle B  angle C  angle D
360°.

■ A line tangent to a point on a circle is at 90°, or normal, to a radial line drawn to the tangent point (see Fig 2.19).

36 Chapter Two

Figure 2.15 Exterior angle of a triangle

Figure 2.16 Intersecting straight lines

Figure 2.14 Right-angled triangle

Mathematics for Machinists and Metalworkers 37

Figure 2.17 Straight line intersecting two parallel lines

Figure 2.18 Quadrilateral (four-sided figure)

Figure 2.19 Tangent at a point on a circle

■ Two circles’ common point of tangency is intersected by a line drawn between their centers (see Fig 2.20).

Side a
a′; angle A
angle A′ (see Fig 2.21).

Angle A
1⁄2angle B (see Fig 2.22).

Angle A
angle B
angle C All perimeter angles of a chord are

equal (see Fig 2.23)

Angle B
1⁄2angle A (see Fig 2.24).

a2
bc (see Fig 2.25).

38 Chapter Two

Figure 2.20 Common point of tangency

Figure 2.21 Tangents and angles