3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Module 3 Angular Measurements, Thread Metrology, and Optics Page 1 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 3.1 Angular Units 3.1.1 Exploration: Angles If you had to measure the opening above, how would you go about it? What instruments would you use? What units would you measure it in? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ Page 2 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 3.1.2 Dialog: Angles An angle may be defined as the opening between two lines which meet at a point. An angle is generated by simply moving a line in an arc around a point. By this method, a complete circle can be made. It is from such a circle that the units of angular measurement have been derived. A circle is divided into 360 parts, each part is called a degree (°) of arc. Each degree of arc, like an hour of time, is divided into 60 parts called minutes (‘), and each minute is divided into 60 seconds (“), Table 3.1a. Table 3.1a: Conversions for Degrees, Minutes, and Seconds 1 Circle = 360 Degrees (°) 1 Degree (°) = 60 Minutes (′) 1 Minute (′) = 60 Seconds (″) 1 of a Circle 360 1 Degree (°) 1 Minute (′) = 60 1 1 Minute (′) 1 Second (″) = 60 1 Degree (°) = Sometimes degrees are expressed as a decimal. This is most often encountered when using a calculator or computer for angular computation. Therefore, it is important to be able to understand how to convert between decimal degrees, and degrees of arc and back again. The procedure for converting between expressions is outlined below. Converting Decimal Degrees to Degrees, Minutes, and Seconds 1. 2. 3. 4. The whole number of the decimal will remain as degrees. Multiply the decimal part of the degrees by 60′ to obtain minutes. If the number of minutes is not a whole number, multiply the decimal portion by 60″ in order to obtain seconds. Round if necessary. Combine degrees, minutes, and seconds. Example: Express 78.2356° in degrees, minutes, and seconds. Degrees = 78 Multiply 0.2356 by 60′ to obtain minutes. 60′ x 0.2356 = 14.136′ Multiply 0.136 by 60″ to obtain seconds. 60″ x 0.1360 = 8.16″ Round to whole seconds. 8.16” ⇒ 8” Combine degrees, minutes, and seconds. 78°+14′+8″ = 78°14′8″ Converting Degrees, Minutes, and Seconds to Decimal Degrees 1. 2. 3. 4. 5. Whole number degrees will remain as degrees in the decimal. Divide the seconds by 60 in order to obtain a decimal minute. Add the decimal minute to the given number of minutes. Divide the sum of the minutes by 60 in order to obtain the decimal degrees. Add the decimal degrees to the given number of degrees. Round to four decimal places. Page 3 3.0 Angular Measurement, Thread Metrology, and Optics Example: Express 78°14′8″ in decimal degrees. Degrees = 78 Divide the seconds by 60 to obtain the decimal minute. Add the decimal minute to the given minutes. Divide the sum of the minutes by 60 to obtain decimal degrees. Add the decimal degrees to the given degrees. Measurement and Quality 8″ ÷ 60 = 0.1333′ 0.1333′ + 14′ = 14.1333′ 14.1333′ ÷ 60 = 0.2356° 78° + 0.2356° = 78.2356° Knowing that 360° makes up a circle, then one-quarter the circumference of a circle, or one quadrant, represents 90°. Ninety degrees is referred to as a right angle(Figure 3.1a). One-half the circumference represents 180° and is called a straight angle. An angle smaller than 90° is called an acute angle, whereas an angle larger than 90° and smaller than 180° is called an obtuse angle, Figure 3.1a. x<90° 90° x>90° x x ACUTE ANGLE RIGHT ANGLE OBTUSE ANGLE Figure 3.1a: Right, acute, and obtuse angles. When the sum of two angles equals 90°, they are called complimentary angles: when their sum equals 180° they are known as supplementary angles. Thus, the complement of an acute angle of 30° (Angle A) is an angle of 60° (Angle B), but the supplement of a 30° (Angle A) angle is an obtuse angle of 150°(Angle C), Figure 3.1b. B A A Angle A compliment of Angle B = 90° Angle A supplement of Angle C = 180° C Figure 3.1b: Complimentary and Supplementary Angles Summary: Right Angle – the basic unit in angular measurement. By definition, it is the angle between two lines which intersect so as to make the adjacent angles equal, Figure 3.1c. Right Angle=90° 90° Acute Angle – an angle between 0° and 90°. Obtuse Angle – an angle between 90° and 180°. Figure 3.1c: Right Angle Page 4 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 3.1.3 Application: Angles Answer the following exercises on angles and conversions. 1.) How many degrees are in a circle? ______________________________ 2.) How many minutes are in a circle? ______________________________ 3.) Express the following degrees, minutes, and seconds as decimal degrees. Round to 1 decimal place. 34° 23′ 15″ 146° 57′ 23″ 289° 3′ 53″ 314° 32′ 4″ _______________ _______________ _______________ _______________ 3° 2′ 1″ 180° 5′ 16″ 87° 45′ 30″ 67° 44′ 43″ _______________ _______________ _______________ _______________ 4.) Express the following decimal degrees as degrees, minutes, and seconds. Round to the nearest whole second. 45.2345° 267.6784° 12.3843° 4.3123° _______________ _______________ _______________ _______________ 321.0022° 180.4321° 27.3482° 1.6785° _______________ _______________ _______________ _______________ 5.) What is the complimentary angle of 23°? The supplementary angle? Complimentary __________ Supplementary __________ 6.) What is the supplementary angle of 154°? ____________________________________ 3.1.4 Dialog: Angle Unit Systems The angular units that are most used in engineering are derived from the Inch System. In the Inch System, the basic unit is degrees(°) as described in Section 3.1.1. In this system a full circle is divided into 360°, where 1°=60 minutes of arc (1°=60’) 1 minute=60 seconds of arc (1’=60”) and a right angle=90° In the theoretical treatment of angles, the SI System is used most frequently. In the SI system, the Radian is the basic unit of angular measurement, where the Radian is equal to the length of an arc on a full circumference of a circle that is equal to the radius of the circle, Figure 3.1d. From geometry we know that the circumference of a circle(C) is proportionally to twice the radius(R). The following equation converts radians into equivalent degrees. Page 5 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Circumference C=2πR Arc length R = radius R Arc length R = 1 radian 2π rad in a circle R R 360° ≅ 57°,17’,44” ≈ 57.2958° rad = 2πR 1° = ≈ 206264” π 180 1′ of arc = 1′′ 1 radian=1arc length on the circumference. rad = 1.745329 ×10 −2 rad π 10800 of arc = Figure 3.1d: 1 Radian = Radius of Circle. rad = 2.908882 × 10 − 4 rad π 648000 rad = 4.848137 × 10 −6 rad 3.1.5 Dialog: Angle Arithmetic When computing precise angular measurements it is sometime necessary to add and subtract angles in degrees, minutes, and seconds. Examples Angle Arithmetic Addition a.) b.) c.) Subtraction a.) b.) c.) 35° + 27° = 62° 3° 15’ + 7° 49’ = 10° 64’ = 10° 60 + 4’ = 11° 4’ 265° 15’ 52” + 10° 55’ 17” 275° 70’ 69” = 275° 60’ + 10’ 60”+9” = 276° 11’ 9” 15° -8° = 7° 15° 3’ = 14° 63’ - 6° 8’ = 6° 8’ 8° 55’ 39° 18’ 13” = 38° 77’ 73” - 17° 27’ 52” = 17° 27’ 52” 21° 50’ 21” Page 6 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 3.1.6 Application: Angle Arithmetic Solve the following problems and write your answers in the space provided. Addition (a) 45° + 37° = __________ (b) 28°56′ + 7°34′ = __________ (c) 48°15′ + 67°51′ = __________ (d) 235 °25′ + 34°43′ = __________ (e) 34°45′15′′ + 57°7 ′23′′ = __________ (f) 126 °23′14′′ + 201°34′31′′ = __________ (g) 98°51′ 27′′ + 24°17′40′′ = __________ (h) 158 °32′34′′ + 36°56′27′′ = __________ Subtraction (i) 38° − 21° = __________ (j) 52°12′ − 34°3′ = __________ (k) 3°47 ′ − 1°57 ′ = __________ (l) 277 °12′ − 143 °31′ = __________ (m) 45°23′12′′ − 32°17 ′3′′ = __________ (n) 278 °56′23′′ − 167°34′21′′ = __________ (o) 342 °2′23′′ − 123 °34′45′′ = __________ (p) 157 °45′43′′ − 23 °43′47 ′′ = __________ 3.1.7 Dialog: Triangles The triangle is a geometric figure that is widely used in engineering and science. Knowledge of triangles is important in manufacturing. The geometric principles are needed in engineering design, manufacturing setup, and quality inspection. There are four types of triangles: scalene, isosceles, equilateral, and right triangle, Figure 3.1e. SCALENE ISOSCELES RIGHT EQUILATERAL Figure 3.1e: Types of triangles. Page 7 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality For every triangle, the sum of the internal angles is equal to 180°, Figure 3.1f. a° c° b° a + b + c = 180° Figure 3.1f: Sum of internal angles = 180° One of the most useful triangles in engineering and manufacturing is the Right triangle. The right triangle has a unique property, in that the sum of the squares of the two sides is equal to the square of the hypotenuse. The hypotenuse is defined as the side opposite the right angle and is always the longest side. This property is known as the Pythagorean Theorem and is expressed as: x2 + y 2 = h2 Using this formula, if we know the length of two sides of a right triangle we can always solve for the length of the third side. Example of The Pythagorean Theorem If h=4.5 and y=2 then x=? h = 4.5 y=2 x=? x2 + y 2 = h2 x 2 + 2 2 = 4.5 2 x 2 + 4 = 20.25 x 2 = 16.25 x = 16.25 x = 4.03 Page 8 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality As introduced previously, the sum of the internal angles of a triangle equal 180°. Therefore, the same rules for angles apply for complimentary and supplementary angles, Figure 3.1g. Complimentary angles ⇒ a + b = 90 Suplimentary angles ⇒ a + b + d = 180° o y h b ⇒ a + c = 180° d c a x Figure 3.1g: Angles in a Right Triangle Another use of right triangles is the calculation of trigonometric functions. These functions are important because we are able to use them to calculate angles in relationship to the ratios of the sides of triangles, Figure 3.1h. f b g c e h a d Figure 3.1h: If angle a = angle e, then side f has the same relationship to side h as side b has to side d. Trigonometric Functions Triangles are used to solve many problems in production, inspection, and machine set-up. You have been introduced to The Pythagorean Theorem in solving for the lengths of the sides in a right triangle; however, what if we only knew the length of one side? How would we determine the length of the unknown sides? We can solve this problem using trigonometric functions. There are six trigonometric functions. The following table lists the six functions and the common abbreviations for each. Function Abbreviation sine sin cosine cos tangent tan cotangent cot secant sec cosecant csc Looking at a right triangle we can define the sides as the hypotenuse, opposite, and adjacent sides, Figure 3.1i. The hypotenuse is defined as the side opposite the right angle and is always the longest side. The opposite side is the side across from the observed angle. The adjacent side is the side next to the observed angle. The adjacent and opposite sides will change depending on which angle is being observed, Figure 3.1i. Understanding these concepts are important to understanding the formulas for the trigonometric functions. Page 9 3.0 Angular Measurement, Thread Metrology, and Optics hypotenuse Measurement and Quality hypotenuse opposite adjacent adjacent opposite Figure 3.1i: Naming the sides of a triangle. Using the definitions established above, we will define the basic trig functions as follows. sin = opp. hyp. cos = adj. hyp. tan = opp. adj. csc = hyp. opp. sec = hyp. opp. cot = adj. opp. The three basic trig functions are sin, cos, and tan. We will not focus on the other three because they are simply the inverse of the first three. In relation to a right triangle, the formulas below can be used to solve for either the length of the a side or an interior angle provided other information is known. y h x cos a = h y tan a = x x cot a = y sin a = x h y cos b = h x tan b = y y cot b = x sin b = b h y a x In the triangle below, what would be the length of side x if we know that a= 30° and h= 4? We can see that side x is adjacent to angle a; therefore, we can use the cos function to determine x. 4 30° x Page 10 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality x 4 x = 4 cos 30 ο cos 30 ο = To solve for x we will need to use a calculator or a table of trigonometric functions. Using the table on the following page we see that the cos of 30° is equal to 0.8660. Therefore, we can solve for x as: x = 4(0.8660) = 3.464 Table 3.1b: Trigonometric Page 11 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Oblique Triangles Oblique triangles are triangles that do not contain a 90° angle. There are two methods for determining angles and the lengths of the sides of an oblique triangle. The first method is to simply break down the triangle into one or more right triangles (Figure 3.1j). From there, angles and side lengths can be calculated using the methods above. Figure 3.1j: An oblique triangle can be broken own into one or more right triangles. Sometimes, breaking down an oblique triangle can be difficult and cumbersome. Therefore, the second method is to use the Law of Sines or the Laws of Cosines. Law of Sines In any triangle, the sides are proportional to the side of the opposite angles. a b c = = sin A sin B sin C a b C B A c This formula may be used only if either: • Two angles and any one side are known. • Any two sides and the angle opposite one of the given sides are know. Example: 7 b = ο sin 65 sin 35 ο b= ( 7 sin 35ο sin 65ο ) 7 b 65° b = 4.43 Page 12 35° 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Law of Cosines If two sides and the included angle are known, then the following formulas are used. a 2 = b 2 + c 2 − 2bc(cos A) b 2 = a 2 + c 2 − 2ac(cos B ) c 2 = a 2 + b 2 − 2ab(cos C ) a b C B A c Example: To solve for c we would use the following. c 2 = 8 2 + 2 2 − 2(8)(2)(cos135ο ) ( c = 8 + 2 − 2(8)(2 ) cos135 2 2 ο 8 2 ) 135° c = 9.52 c If all three sides of the triangle are known, then the following formula is applicable. b2 + c2 − a2 cos A = 2bc a b cos B = a +c −b 2ac 2 cos C = a +b −c 2ab 2 2 2 2 2 C B A c Example: To solve for A we would use the following. b2 + c2 − a2 cos A = 2bc 22 + 92 − 42 cos A = 2(2)(9) 7.9 3.5 A 8.7 A = 33.03ο Page 13 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 3.1.8 Application: Triangles Solve the following inspection and drawing interpretation problems using triangles. 1.) The following diagram shows a bolt circle where the bolt holes are evenly spaced. If the diameter of the bolt circle is equal to 5.200 inches, what is the distance between A and B? (hint: solve using right triangles) ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ 5.200 DIA B A 2.) A ¾” diameter pin is used to inspect a groove. Determine x if the sides of the groove are equal. ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ ____________________________________ 0.75 IN. DIA. x 32.5° 8.25 IN. 3.5 IN. Page 14 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Bibliography Smith, Robert D.. Mathematics for Machine Technology, 5th Edition. Thomson Delmar Learning, 2004. Page 15 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 3.2 Measuring Angles with Gage Blocks, Plain Protractors, Sine Bars, and Vernier Protractors 3.2.1 Introduction Angular measurement is one of the most important activities in engineering and science. The use of transits and levels to lay out boundary lines, highways, and railroads is typical of the importance of precision angle measurement. The relation of the stars and their approximate distances are computed in astronomy by means of angular measuring devices. Ships and planes are able to navigate confidently beyond the sight of land, primarily because precise angle measurements are possible. Angular measurements are so common and so essential in the manufacture of interchangeable parts, jigs, dies, and fixtures that a basic knowledge of angles and their measurement is indispensable to successful manufacturing. Among the tools most commonly used for industrial angular measurement are the protractor, bevel (vernier) protractor, universal angle gage blocks, sine bar, squares, and levels. For the purpose of this course, we will focus on only protractors, angle gage blocks, and the sine bar. However, there are a number of angular measuring devices that you may encounter in practice. Angular Measurement Instruments The following section gives you an overview of some of the angular measuring devices used in industry. We will not be going into detailed use of all these instruments, but it is good to have some familiarity of what may be encountered in practice. Protractor Photo from www.Starrett.com The protractor is designed for draftsmen, civil engineers, and is particularly valuable for drawing any number of radial lines at any desired angle from a common center. Universal Bevel Protractor (Vernier Protractor) Page 16 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Photo from www.Starrett.com The universal bevel protractor is primarily used to measure angles using a vernier scale. Sine Bar A sine bar is a steel bar that has a cylinder at either end to form a hypotenuse of a triangle to make computation easy when making comparison measurements. Sine Block / Sine Plate Page 17 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality A sine block is a wide sine bar. They usually have tapped holes for the attachment of parts and a stop to prevent parts from sliding off. A sine plate is a sine block with an attached base. Angle Gage Blocks Photo from www.Starrett.com Angle gage blocks provide a fast accurate measurement of any angle by creating combinations of angles. Autocollimator Page 18 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality The autocollimator is a precision optical instrument for measuring very small angular displacement over a significant distance. They can be used for evaluating alignment of machine surfaces, surface plate flatness, squareness of one surface to another, straightness of shafts and a variety of other orientation measurements. Diagram of Autocollimator Principle Line diagram of typical autocollimator. Page 19 3.0 Angular Measurement, Thread Metrology, and Optics Straight Square Hardened steel device that is used to determine whether an angle is a right angle. Photo from www.Starrett.com Page 20 Measurement and Quality 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Cylindrical Square A cylindrical square is used as a master for testing a straight square. A direct reading cylindrical square is used for determining perpendicularity errors over a part length. Level Photo from www.Starrett.com Levels are useful measuring instruments. Levels are commonly used for machine alignment and setup. A level uses fluid filled tubes whose bubble is affected by gravity. A precision level has divisions that can be used for measurement. Page 21 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 3.2.2 Exploration: Angle Measurement with Protractor Materials: Simple Protractor, Paper, and a Pencil Parts: Wood Shims, Door Stop Wedges In this activity, we will begin our study of angular measurement by measuring some simple material using a protractor. Using the protractor determine the angle of both a shim and a doorstop wedge. Fill in your results in the table below. When complete, answer the following questions. Angle (degrees) Shim Door Stop Questions: 1.) How easy was it to measure the shim? the door wedge? What were some problems? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ________________ 2.) What are some sources of error in measuring with the protractor provided? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ____________________________________ 3.2.3 Dialog: Angular Measurement with the Plain Protractor The simplest angular measuring instrument is the plain protractor. They can be found in elementary classrooms as well as a machine shop. The discrimination on a plain protractor is usually limited to 1° increments. Therefore, it typically only used in layout, but can be used in inspection when the accuracy and precision is not an issue. Figure 3.2a: Plain Protractor Page 22 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Photo from www.Starrett.com A semi-circular protractor, like the one below, is graduated from 0° to 180°. There are usually two scales so that readings can be observed from left to right, Figure 3.2b. Figure from pg 264, Mathematics for Machine Technology, 5th Edition, Robert D. Smith. Figure 3.2b: Plain protractor To read a protractor, place the vertex of the angle to be measured at the center point of the base of the protractor, Figure 3.2c. The angle vertex is the point at which the two sides meet. In Figure 3.2c, the angle is rotated from the right and we can see that it is less than 90°. Therefore, we would read the scale that has a zero reading on the right side of the protractor. If the angle were from the left, we would read the scale that starts with zero on the left. Finally, read the point at which the angle crosses the appropriate scale. 90° Figure 3.2c: Measuring angles from the vertex. Some Angle α Example: Location of Vertex Angle. Base Page 23 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure from pg 264, Mathematics for Machine Technology, 5th Edition, Robert D. Smith. Measure angle α. We see that the angle is from the left. Therefore, we use the outer scale of the protractor. The side of the angle crosses the outer scale at 140°. Therefore, the angle is 140°. 3.2.4 Application: Angular Measurement with the Plain Protractor Materials: Plain Protractor Using a plain protractor, measure the following angles. When completed, answer the corresponding questions. 1.) __________ 2.) __________ 3.) __________ 5.) __________ 4.) __________ 6.) What is the precision of the protractor? ____________________________ 7.) What types of error are encountered when measuring with a plain protractor? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ____________ 8.) What are some ways in which we could reduce the error associated with the measurements? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ____________ Page 24 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 3.2.5 Exploration: Measuring Difficult Angles Materials: Plain protractor. Using the plain protractor measure the following angles. (1) (2) 2.) __________ 1.) __________ 3.) What if any difficulties did you encounter in measuring these angles? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ____________ 2.) What would have help to make the measurements easier? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ____________ 3.2.6 Dialog: Universal Bevel Protractor The bevel protractor shown in Figure 3.2d is a precision angle-measuring instrument equipped with a Vernier scale capable of measuring to 5’ (minutes) of angular arc. It consists of a base plate, and an adjustable blade, attached to a circular plate containing a vernier scale. An attachment can be added near the top of the protractor to make it possible to inspect acute angles; for this reason, it is called the acuteangle attachment. Page 25 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality ACUTE ANGLE BLADE Figure from pg 266, Mathematics for Machine Technology, 5th Edition, Robert D. Smith. Figure 3.2d: Universal Bevel Protractor A knurled-headed pinion may be inserted in a hole at the back of the base plate whenever fine adjustments are required. One side of the tool is flat permitting its being laid flat upon the paper or work. Figure 3.2e shows a portion of the main scale and the complete vernier scale. The sales are designed so that 12 divisions on the right or left vernier scale equal 23 divisions, on the main scale. Each vernier division is thus 5’ (minutes) shorter than two spaces on the main scale. Figure from pg 267, Mathematics for Machine Technology, 5th Edition, Robert D. Smith. Figure 3.2e: Vernier scale on bevel protractor To find the smallest reading of which the vernier protractor is capable, apply the rule of the least count of verniers, i.e., divide 60 minutes, the value of one main scale division, by the number of divisions on the Page 26 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality vernier scale. Because the vernier scale contains 12 divisions, the smallest reading possible is 60/20, or 5 minutes. Therefore, a vernier protractor may he read to 5’ of angular arc. The vernier scale of the protractor reads left and right from a center zero. When measuring with the protractor, the readings on the vernier scale are taken either to the right or left, according to the position of the zero of the vernier scale in relation to the zero on the main scale. For example, in Figure 3.2e, the righthand vernier scale would be used because the zero on the vernier is to right of the zero on the main scale. If the vernier zero were to the left, the left-hand vernier is used. Thus the inspector always reads the vernier whose numbers lie in the same direction as the numbers on the main scale. Reading the Vernier Bevel Protractor To read the vernier protractor accurately, the following rules should be observed. 1. 2. 3. Determine the direction in which the zero of the vernier lies from the zero on the main scale and select the left or right-hand vernier accordingly. Read directly the number of whole degrees between the zero of the main scale and the zero of the vernier scale. All reading is done from the zero on the vernier scale. Add to this reading the number of minutes represented by the line on the vernier scale which coincides exactly with a line, on the main scale. It must be remembered that the reading of a vernier protractor always represents a base-and-blade relationship. Acute angles can be measured directly from the scale because the main sale is divided into four quadrants of 90° each; however, obtuse angles are checked indirectly by subtracting the protractor reading from 180° or by adding the complement of the reading to 90°. For example, if an angle of 120° is measured, the vernier reading is 60°. To arrive at 120°, it is necessary to subtract 60° from 180°. The reading also can be made by adding 30°, the complement of 60° to 90°. The protractor can be used with the angle at the end of the scale by adding or subtracting the angle from the vernier protractor reading. Measurement Error An important point to remember is that the bevel protractor does not measure the angle on the part, it measures the angle between its own parts. Therefore, the closer that you can establish contact with the protractor blade and the part feature, the more accurate that your reading will be. By improperly contacting the blade base with the part you can create blade contact error. Just as with the other length measuring instruments in Module 2, it is important to clean the protractor and the surfaces that are being measured to allow for proper contact with the part surface. To determine if the blade or base is in full contact with the part surface, look for “leaks” of light from in between the blade and the surface. The following checklist is taken from Dotson, Harlow, and Thompson’s Fundamentals in Dimensional Metrology to help increase the reliability of the measurements taken with the bevel protractor. Mechanical considerations: 1. Can both the base and the blade reach their respective surfaces unobstructed? 2. Is the instrument being over constrained causing erroneous errors? 3. Do burrs, dirt, or excessive roughness interfere with intimate contact? Positional considerations (in yz plane): 1. Is vertical axis of the instrument parallel to the plane of the angle? 2. Is the horizontal axis of the instrument parallel to the plane of the angle? Page 27 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Observational considerations: 1. Is the reading the compliment of the angle being measured? 2. Is the reading the supplement of the angle being measured? 3. Does parallax error exist? 4. Are you conscious of bias? Setting the Vernier Protractor The operation of setting the vernier protractor is the reverse of that of reading it. For example, if it is required to set the instrument to 12°15’, the operation is as follows. 1. Move the vernier by means of the blade until the 12°mark of the main scale is opposite the zero of the vernier scale. 2. Move the blade carefully until the tenth line of the appropriate vernier coincides with a line of the main scale. For fine adjustments the knurled pinion provided with the tool may be used. Care of the Bevel Protractor 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Wipe off dust and oil. Examine for visual signs of damage. Run fingers along base and blade to detect burrs. Check that instrument is moving freely. Allow instrument to normalize (temperature). Determine that the instrument is calibrated. Avoid excessive handling to minimize heat transfer. Avoid work near heated surfaces. Do not slide along abrasive surfaces. Clean thoroughly before and after use. 3.2.7 Application: Angular Measurement with the Bevel Protractor Read the values on the following protractor scales. Page 28 3.0 Angular Measurement, Thread Metrology, and Optics 1.) __________________ Measurement and Quality 6.) __________________ 7.) __________________ 2.) __________________ 8.) __________________ 3.) __________________ 9.) __________________ 4.) __________________ 5.) __________________ Figure from pg 269, Mathematics for Machine Technology, 5th Edition, Robert D. Smith. Materials: V-Block, Bevel protractor Measure the angles on the following figures using the bevel protractor (determine to nearest minute of arc). Page 29 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 10.) ____________________ 11.) ____________________ 12.) Using the bevel protractor, measure the angle (to the nearest minute of arc) of the V-block and answer the questions below. V-Block Angle =____________ Questions: Page 30 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 13.) Were you able to measure the angle of the V-block? What difficulties did you encounter? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ________________ 14.) What are some of the sources of error in your measurement? (hint: think of error with both the instrument, the environment, and the scale.) ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ________________________ 3.2.8 Exploration: Angular Measurement with Angular Gage Blocks Materials: Set of Angular Gage Blocks In this activity, we will look at quickly measuring the angles below using the angular gage blocks. 1.) __________ 2.) __________ 3.) __________ Page 31 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 4.) __________ Questions: 1.) How easy was it to measure the angles above? What were some problems? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ________________ 2.) What might be some sources of error in measuring with the angular gage blocks? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ________________ 3.) What is the discrimination of the angular gage block set? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ____________ 3.2.9 Dialog: Angular Measurement with Angular Gage Blocks The angular gages shown in Figure 3.2f are precision ground tools which can be used for a large variety of angular measurements. These gages may be obtained in 10 block sets, which consist of three triangles, four blades graduated in degree increments, and three blades, graduated in minutes, Figure 3.2g. All gages are made of tool steel, hardened and ground to such precision that the variation from the exact angle of any Page 32 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality required combination will not exceed 1 minute. The universal angle members are fixed in combination for convenient handling by an ingenious clamping device. Three Triangles 30° 60° 45° 45° 15° 75° 90° 90° 90° Four Degree Blades 83° 84° 96° 85° 86° 94° 87° 88° 92° 89° 90° 91° 97° 95° 93° 90° Three Minute Blades 90° 5’ 90°10’ 89°50’ 90°15’ 90°20’ 89°40’ 90°25’ 90°30’ 89°35’ 89°55’ 89°45’ 89°30’ Figure 3.2g: Typical combinations in a angular gage block set. With a complete set of universal angle gages, 2160 combinations can be made ranging from 0° to 180° in intervals of 5 minutes. The large number of combinations can be achieved due to the ability to add and subtract angle combination. This concept will be shown in the example below. Universal angle gages may be used to make both solid and open angles. Angle gages constitute an improvement in the means of measuring and laying out angles because they can be applied to the work without obstructions. Example: The use of both the 41° and the 9° block can create two different angles. 9 50° 32° 9 41 41 The most common set that one may encounter in a tool room is the 16 block set, Figure 32h. This set consists of 5 blocks for both minutes and seconds of arc. These five blocks are usually in increments of 1, 3, 9, 27, and 41. The remaining six blocks are in 1, 3, 5, 1, 30, and 45 degree increments. Page 33 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Photo from Starrett Catalog Online, www.Starrett.com Figure 3.2h: 16 bock angle gage set. The 16 block set forms all angles from 0° to 90° with a discrimination of 1 second. These sets can form up to 356,400 angle combinations and are used in set-up and calibration. Example: If we wanted to recreate an angle of 46° 17′ 30″ with a 16 block set we would achieve this as follows: Degrees +41 +9 -3 -1 46° Minutes +27 -9 -1 Seconds +27 +3 17′ 30″ 3.2.10 Application: Angular Measurement with Angular Gage Blocks Materials: Universal Angle Gage Block Set Page 34 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 1.) We are looking to set-up a precision grinder to grind an part surface to an angle of 57° 15′ 30″. Using your universal angle gage set, and create the angle using the least number of blocks. Write down the combination of block in the table below. Block Number 1 2 3 4 Total Degrees Minutes Seconds 2.) We are looking to set-up an angle to use for a comparison measurement of 34° 00′ 45″. Using your universal angle gage set, and create the angle using the least number of blocks. Write down the combination of block in the table below. Block Number 1 2 3 4 Total Degrees Minutes Seconds 3.) We are looking to set-up an angle to use for a comparison measurement of 14° 55′ 15″. Using your universal angle gage set, and create the angle using the least number of blocks. Write down the combination of block in the table below. Block Number 1 2 3 4 Total Degrees Minutes 3.2.11 Exploration: Angular Measurement with the Sine Bar Materials: Sine Bar, Set of Gage Blocks, Surface Plate Page 35 Seconds 3.0 Angular Measurement, Thread Metrology, and Optics 5″ Measurement and Quality Gage Blocks Sine Bar X θ Surface Plate Figure from pg 395, Mathematics for Machine Technology, 5th Edition, Robert D. Smith. 1.) Determine the elevation X needed for the gage blocks to create the angle θ = 30° for the sine bar. List the gage blocks needed to meet the height X. (hint: use the right triangle highlighted in the figure) Block Number 1 2 3 4 5 6 Total Block Size Recreate the set-up with the sine bar, gage blocks, and surface plate. 2.) Repeat for θ = 42° Block Number 1 2 3 4 5 6 Total Block Size Recreate the set-up with the sine bar, gage blocks, and surface plate. 3.) Repeat for θ = 20° Block Number 1 2 3 Block Size Page 36 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 4 5 6 Total Recreate the set-up with the sine bar, gage blocks, and surface plate. 3.2.12 Dialog: Angular Measurement with the Sine Bar A sine bar or sine plate is used to measure angles that have been cut into a part, or to position parts on an angle to cut. A sine bar or plate is a steel bar that has two cylinders built into the base. The line through the center of the cylinders can be used to form the hypotenuse of a triangle, Figure 3.2i. Hypotenuse of Triangle Sine Plate Figure from pg 395, Mathematics for Machine Technology, 5th Edition, Robert D. Smith. Figure 3.2i: The imaginary line that is created between the cylinder centers can be thought of as the hypotenuse of a right triangle. Sine bars or plates usually come in one of two standard distances between the cylinders, 5 in. (12.7cm) or 10 in. (25.4cm). With one of the cylinders setting on a surface, you can set the bar or plate to a desired angle by raising the second cylinder, Figure 3.2i. Therefore, the problem become a simple exercise in determining the height that the second cylinder needs to be raised to coo respond to the desired angle. This can be accomplished by using the sine as see in Section 3.1, however, with the sine bar or plate the hypotenuse always remains the same.. To accurately achieve the desired height on the second cylinder, we can use a set of gage blocks. Also to ensure accuracy, the sine bar is designed to be used with a true surface such as a surface plate to reduce measurement error. Example: Figure from pg 395, Mathematics for Machine Technology, 5th Edition, Robert D. Smith. Right Triangle 37° 45′ Page 37 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Determine the height of the gage block stack (X) needed to create an angle of 37° 45′ using a 5 inch sine bar. To solve for the height X we look at the right triangle formed by the centers of the cylinders, the base plate and the gage block stack. The sin of an angle is equal to the opposite side divided by the hypotenuse of the angle. Hypotenuse Opposite Side 5 in. X 37° 45′ X 5 X sin 37.75ο = 5 ο 5 × sin 37.75 = X X = 3.061inches sin 37 ο45′ = The previous example used a 5 inch sine bar, but the same calculation would hold with any size sine bar. The elevations for angles up to 55 degrees can be read directly from a table of sine bar constants. Such tables are found in books like the Machinery’s Handbook. The use of these tables eliminates the need to perform trigonometric calculations. However, most tables only discriminate down to minutes of arc. If discrimination to seconds is needed it is better to calculated the required elevation. Sin Bar and Error Because the sine function is the result of dividing itself by the hypotenuse, and the hypotenuse is always constant, as the angle approaches 0°, the sine of the angle changes rapidly, Figure 3.2j. If we look at the sine of 3° for a 10 inch bar we see that it is 0.05233. If we than go to 2°, it is 0.03489, or about two thirds the value at 3°. If we look at decreasing the angle to 1°, we have 0.1745, or half that at 2°. Therefore, the sine of the angle changes rapidly near 0°. Page 38 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 15-84, pg. 478 from Dotson, Harlow, and Thompson. Fundamentals of Dimensional Metrology, 4th Edition Figure 3.2j: The sine increases slowly as it approaches 90°, and changes rapidly near 0°. The opposite is true as the angle approaches 90°. The sine of 90° is 1.000 and the sine of 89° is 0.9998, or nearly 1. Therefore, the sine changes slowly as it approaches 90°. This becomes important because when using a sine bar for measuring steep angles, the height change between angles is very little. When setting up smaller angles the height change is much greater, therefore more precise. For example, at 80°, a 10 inch sine bar requires only a 0.0005 inch height change to change it one minute. At 10°, the same one minute change needs a 0.0028 inch height change. Anything that disturbs the measurement at 80° will have one fifth the effect than at 10°. Therefore, when measuring angle of greater than 80°, it is best to measure the compliment of the angle to achieve the best results. Sine Blocks, Sine Plates, and Sine Tables Sometimes the terms sine bar, sine plate, and sine table are used interchangeably. However, usually the difference in the application. A sine block is a wide side bar or bock that have an end block tied to the end to prevent parts from falling off, Figure 3.2k. Page 39 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.2k: Sine Block Figure 15-80, pg. 477 from Dotson, Harlow, and Thompson. Fundamentals of Dimensional Metrology, 4th Edition A sine plate is typically a sine block with an attached base, Figure 3.2l. Finally, a sine table is a sine plate that is capable of being used as an integral part of a machine device. Figure 3.2l: Sine Plates are free standing with their own base. Figure 15-824, pg. 477 from Dotson, Harlow, and Thompson. Fundamentals of Dimensional Metrology, 4th Edition 3.2.13 Application: Angular Measurement with the Sine Bar Materials: Sine Bar, Set of Gage Blocks, Surface Plate Page 40 3.0 Angular Measurement, Thread Metrology, and Optics 10″ Measurement and Quality Gage Blocks Sine Bar X θ Surface Plate Figure from pg 395, Mathematics for Machine Technology, 5th Edition, Robert D. Smith. 1.) Determine the elevation X needed for the gage blocks to create the angle θ = 30° for the sine bar. List the gage blocks needed to meet the height X. (hint: use the right triangle highlighted in the figure) Block Number 1 2 3 4 5 6 Total Block Size Recreate the set-up with the sine bar, gage blocks, and surface plate. 2.) Repeat for θ = 42° 47’ Block Number 1 2 3 4 5 6 Total Block Size Recreate the set-up with the sine bar, gage blocks, and surface plate. 3.) Repeat for θ = 23° 15′ 30″ Block Number 1 2 3 4 Block Size Page 41 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 5 6 Total Recreate the set-up with the sine bar, gage blocks, and surface plate. 4.) Did you have any difficulty in setting up the angles in questions 1-3? If so, why? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ____________________________________ 5.) Determine the angle θ from the following gage block stack height. θ 6.345” θ = ______ Figure from pg 395, Mathematics for Machine Technology, 5th Edition, Robert D. Smith. 6.) Determine the angle θ from the following gage block stack height. Page 42 3.0 Angular Measurement, Thread Metrology, and Optics θ Measurement and Quality 2.464” θ = ______ Figure from pg 395, Mathematics for Machine Technology, 5th Edition, Robert D. Smith. 7.) Determine the angle θ from the following gage block stack height. θ 0.5323” θ = ______ Figure from pg 395, Mathematics for Machine Technology, 5th Edition, Robert D. Smith. Page 43 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Bibliography Smith, Robert D.. Mathematics for Machine Technology, 5th Edition. Thomson Delmar Learning, 2004. Dotson, Connie, Rodger Harlow, and Richard Thompson. Fundamentals of Dimensional Metrology, 4th Edition. Thomson Delmar Learning, 2003. Page 44 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 3.3 Thread Metrology 3.3.1 Exploration: Gaging and Inspection of Screw Threads Equipment 1. 6—inch rule 2. Screw thread pitch gage 3. Thread plug gages 4. Screw thread micrometer 5. Thread measuring wire set 6. Outside micrometer 7. Thread Standards Procedure Take 10 different bolts (and their nuts) from your instructor and determine by two separate methods: A. The major diameter B. The minor diameter C. The pitch D. The pitch diameter E. The angle of the thread F. The depth of the thread G. The lead H. The complete description (utilizing the appropriate thread standards). Page 45 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Answers: 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b A. B. C. D. E. F. G. H. A. B. C. D. E. F. G. H. 3.3.2 Dialog Importance of Threaded Fasteners Today’s manufactured products are complex. With very few exceptions, products are not made from a single part or component. In fact, they are assemblies comprised of multiple machined or otherwise fabricated component parts. For example, a typical automobile contains over 15,000 parts while a modern airliner requires millions of parts! Assemblies require the use of many types of joining methods, including threaded fasteners, rivets, adhesive bonds, welded joints, soldered joints, brazed joints, “snap” fits, and other methods. Often, the integrity of the joint is as important as the component parts themselves in the achievement of the products end-use performance requirements. Because of their wide application and versatility, threaded fasteners are without a doubt the most important joining methods employed in manufacturing. In this Section, threaded fasteners and the methods of measuring their dimensional characteristics will be explored. Page 46 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality There are many types of threaded fasteners employed in manufacturing, including: • Bolts • Nuts • Machine and Cap Screws • Set Screws The interchangeability of threaded parts is critical to efficient manufacture of products. Take, for example, the assembly of a valve cover to an engine block. The valve cover is joined to the block using eight threaded bolts screwed into threaded (“tapped”) holes in the block. Imagine the difficulty during assembly if all eight tapped holes had slightly different size and/or shaped threads requiring you (the assembler) to make a custom mating threaded bolt for each of the eight locations! We will, therefore explore in more detail the various standardized thread forms and classes which have been developed to ensure interchangeability of threaded fasteners throughout industry. Because interchangeability can only be ensured by precise measurement and inspection of threads, the key measurement techniques for these forms will be discussed. Threads Threads are an extremely important mechanical device which derives its usefulness from the inclined plane. The inclined plane is one of the six simple machines. A thread is a helical groove that is formed on the outside or inside of a cylinder. Threads may be either left-handed (LH) or right-handed (RH), but most threads are right-handed. If LH is not present in the thread definition, it is inferred to be right-handed (RH). Figure 3.3a: Thread helix. Internal threads are helical grooves produced in the walls of a hole while external threads are produced on the outside diameter of a cylindrical rod. Bolts, screws and set screws have external threads. Nuts and thrust nuts have internal threads. Threads are precision mechanical features and are produced by a variety of manufacturing processes using specialized cutting tools. Production of precise threads is one of the most common yet difficult machining processes requiring exacting process control. Taps are the most common cutting tools used to produce internal threads in holes. Internal threads are either cut or formed (rolled) into the walls of the hole by turning a properly sized tap. Powered machines such as drill presses, milling machines, or computer numerically controlled (CNC) machining centers may be used to turn the tap. Holes are also tapped by hand. Additionally, internal threads may be produced on a lathe by rotating the workpiece around a specially-shaped thread-cutting tool. Page 47 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.3b: Left - Threads being produced in a hole (tapping) using a powered machine to rotate a spiral tap. Below Left – Tapping by hand in a vise using a hand tap. Tap is rotated. Below Right – Producing internal threads on a lathe using a thread-cutting tool. The workpiece rotates while the cutting tool is fed in. External threads are most commonly produced on a lathe or computer-controlled turning center by feeding a specially shaped thread-cutting tool into a rotating workpiece. External threads may also be cut or formed (rolled) into the walls of the cylinder by turning a properly sized die. Dies are most commonly turned by hand. Figure 3.3c Upper Left – typical die used to produce external threads on a cylindrical workpiece by hand. Below Left – Thread-cutting tool holder and insert for use in lathe to produce external threads. Page 48 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.3d: Left – External threads being produced on a cylindrical workpiece using a lathe and a thread-cutting tool. Note that the workpiece rotates while the tool is fed in. Below Left – Threading by hand in a vise using a die. The die is turned and the workpiece remains stationary. While threads appear on threaded fasteners such as bolts, screws and nuts, they are also used for a variety of other applications. These include threads for adjustment purposes, such as the spindle on a micrometer. Power screws are another familiar application of threads. Power screws and thrust nuts are devices used in machinery to change angular motion into linear motion and usually to transmit power. Applications include the lead screws and thrust nuts of lathes and the screws for vises, presses and jacks. Figure 3.3e: Threaded adjustment screw in a vise. Page 49 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Thread Forms - General Because of the variety of applications in which threads are used, their helical grooves take several shapes with specific and even spacing. These specific shapes are referred to as forms and their dimensional requirements are the basis for interchangeability throughout the world. Some of the most common thread forms are: • Unified Thread Form (UNC, UNF, UNS) • ISO or SI Metric Form • American National Standard Taper Pipe Thread • Acme Threads • Square Thread Thread forms are specified by specific dimensional requirements for certain physical features. These features include: • Thread angle – normally 60o for machine bolts, nuts and screws • Thread depth – distance between the root and the crest • Pitch – 1/TPI • Pitch diameter – a theoretical diameter on a perfect thread where the distance between flanks is ½ the thread pitch, or P/2 Figure 3.3f: Basic thread terminology. Pitch diameter is commonly measured to determine thread form compliance. Page 50 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Thread Forms – Unified Thread The first attempts to obtain interchangeability by the standardization of screw threads were made in 1841 with the adoption of the Whitworth thread system in Great Britain. In 1868, the United States Standard (Sellers) system of screw threads was adopted by the United States Navy. The Sellers system included a standard gage for bolts, nuts and screws. Both the Whitworth and the Sellers systems underwent a long period of development and refinement, emerging eventually as the Unified Screw Thread System, or simply the Unified Thread Form. As far as the study of threaded fasteners is concerned, the Unified Thread Form is the most common and the most critical. The Unified Thread Form was adopted to help standardize manufacturing in the United States, Great Britain and Canada. This form has a 60o thread angle and is divided into the following three series: • UNC National Coarse • UNF National Fine • UNS National Special Unified coarse and unified fine refer to the number of threads per inch (TPI) of length on standard threaded fasteners. According to the standard, a specific diameter of bolt (or nut) will have a specific number of threads cut per inch of length. For example, a ½-inch diameter UNC bolt will have 13 TPI while a ½-inch diameter UNF bolt will have 20 TPI. These two bolts would be identified as follows: • ½ in – 13 UNC Coarse • ½ in – 20 UNF Fine Note that the ½ inch is the major diameter of the bolt and 13 (or 20) is the number of threads cut per inch of length (TPI). Note that pitch is simply 1/TPI. The specific dimensional requirements of this thread form will be covered later in this Section. Figure 3.3g: Illustration of threads per inch (TPI) and pitch. Unified National Special Threads are identified in the same manor as coarse and fine except that the number of threads per inch may vary for a specific diameter. For example, a ½-inch diameter UNS bolt may have 12, 14 or 18 threads per inch. These forms are less common than UNC or UNF, but are sometimes found in specialized applications. “LH” after the identification indicates a left-handed thread. If LH is not present, the thread is assumed to be right-handed. Classes of Thread Fits – Unified Thread Form Some thread applications can tolerate loose threads while other applications require tight threads. This degree of tolerance is called the class of thread fit. Unified Thread fits are classified as 1A, 2A, 3A or 1B, 2B, 3B. The A symbol applies to an external thread while the B symbol indicates an internal thread. Each Page 51 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality class of fit has a specific tolerance on the major diameter and the pitch diameter and are described as either “loose”, “regular” or “interference”: • Class 1 – “Loose” • Class 2 – “Regular” • Class 3 – “Interference” The thread fit notation is added to the thread size and threads per inch: • ½ in – 13 UNC 2A LH Coarse with class of fit = 2, left-handed thread • ½ in – 20 UNF 3B Fine with a class of fit = 3, right-handed thread Classes 1A and 1B have the greatest amount of manufacturing tolerance (“loose”). They are used when ease of assembly is desired and a loose thread is not objectionable. Class 2 fits (“regular”) are used on the largest percentage of threaded fasteners and are appropriate for the majority of general-purpose mechanical joining applications. Class 3 fits (“interference”) have the least amount of manufacturing tolerance and will be very tight when assembled. Page 52 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Standard Series of Threaded Fasteners – Unified Thread Form Threaded fasteners, including all common bolts and nuts, range in size from quite small machine screws up through quite large bolts. This range of sizes is known as the Standard Series of UNC and UNF Thread Fasteners. Below a diameter of ¼-inch, threaded fasteners are given a number. Above a size 12 (0.215 inch), the fastener size is expressed as the fractional size of the major diameter up to about 4 inches. A partial listing of the Standard Series of UNC and UNF Threaded Fasteners is displayed in Figure 3.3h . The sizes listed are common to all types of machines, automobiles and other mechanisms. More complete listings can be found in most engineering and machining handbooks. UNC UNF Size Major Diameter (in.) Threads per Inch Size Major Diameter (in.) Threads per Inch 1 .072 64 0 .059 80 2 .085 56 1 .072 72 3 .098 48 2 .085 64 4 .111 40 3 .098 56 5 .124 40 4 .111 48 6 .137 32 5 .124 44 8 .163 32 6 .137 40 10 .189 24 8 .163 36 12 .215 24 10 .189 32 1/4 .248 20 12 .215 28 5/16 .311 18 1/4 .249 28 3/8 .373 16 5/16 .311 24 7/16 .436 14 3/8 .373 24 1/2 .498 13 7/16 .436 20 9/16 .560 12 1/2 .498 20 5/8 .623 11 9/16 .561 18 3/4 .748 10 5/8 .623 18 7/8 .873 9 3/4 .748 16 1 .998 8 7/8 .873 14 1 .998 12 Figure 3.3h: Partial Table of Standard Series of UNC and UNF Threaded Fasteners Page 53 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Thread Forms – ISO Metric Thread The United States is one of the few countries in the world which continues to use the English (“Customary”) system of measurement based on the inch. Most other countries use the metric, or SI (system internationale), system based on the meter. As production of goods has become more global, the lack of consistency between US manufactured products and globally produced products has become a major problem with regard to interchangeability. In an effort to better align US manufacturing with global standards, the American National Standards Institute (ANSI) adopted an International Standards Organization (ISO) thread form known as the ISO 68 Metric Thread, or simply the “ISO Metric Form”. Like the Unified Thread Form, the ISO Metric form includes a 60o thread angle. Rather than specifying threads per inch, however the ISO Metric Form specifies the thread pitch. Metric thread notation takes the following form: • M 10 x 1.5 Where M 10 is the major diameter in millimeters (10 mm) and 1.5 indicates the thread pitch in millimeters (1.5 mm). Classes of Thread Fits – ISO Metric Thread Form As discussed earlier, Unified Thread fits are classified as 1A, 2A, 3A or 1B, 2B, 3B. For metric threads, fits are described by indicating the amount of tolerance on both the thread pitch and the major diameter (for external threads) or the minor diameter (for internal threads): • Numbers – amount of tolerance allowed o • • The smaller the number the smaller the amount of tolerance allowed Lower Case Letters – position of thread tolerance in relation to its basic diameter (external threads) o ”e” - large allowance o “g” - small allowance o “h” - no allowance Upper Case Letters – position of thread tolerance in relation to its basic diameter (internal threads) o “E” - large allowance o “G” - small allowance o “H” - no allowance The fit classes 6H/6g are usually assigned to general-purpose applications. They are comparable to the 2A/2B fits of the Unified National Forms. A designation of 4H5H/4h5h is approximately equal to the Unified National Form classes 3A/3B. A partial listing of the ISO Metric Threads is displayed in Figure 3.3i . More complete listings can be found in most engineering and machining handbooks. Page 54 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Major Diameter (mm) Pitch (mm) Major Diameter (mm) Pitch (mm) 1.6 .35 16.0 2.00 2.0 .40 20.0 2.5 2.5 .45 24.0 3.0 3.0 .50 30.0 3.5 3.5 .60 36.0 4.0 4.0 .70 42.0 4.5 5.0 .80 48.0 5.0 6.3 1.00 56.0 5.5 8.0 1.25 64.0 6.0 10.0 1.50 72.0 6.0 12.0 1.75 80.0 6.0 14.0 2.00 90.0 6.0 100.0 6.0 Figure 3.3i: Partial Table of Standard ISO Metric Threads A complete discussion of geometries of various thread forms is beyond the scope of this Section. Some basic dimensional information for common thread forms is given in the following pages. For a comprehensive discussion of thread form geometries, consult an applicable reference, including the Machinery’s Handbook. Page 55 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Dimensional Requirements – Unified National Threads • Thread Angle = 60o • Thread Depth (d) = 0.6134 * Thread Pitch (P) Where P = 1/TPI • P/2 Flat at Crest or Root = Pitch/8 A rounded root and crest are desirable, but not required. Pitch diameter is generally the physical characteristic checked to determine thread compliance. (See Figure 3.3j ) Pitch Diameter Figure 3.3j Dimensional Requirements – ISO Metric Threads • Thread Angle = 60o • Flat at Crest = Pitch/8 • Flat at Root = Pitch/4 A rounded root and crest are desirable, but not required. Page 56 3.0 Angular Measurement, Thread Metrology, and Optics Dimensional Requirements – Square Threads Dimensional Requirements – Modified Square Threads Dimensional Requirements – Acme Threads Page 57 Measurement and Quality 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Basic Methods of Thread Metrology for External Threads It is noted here that it is important to ensure that threads are clean and undamaged prior to performing any type of thread inspection. Evaluation of external threads generally includes the following characteristics: • Pitch diameter • Thread pitch • Major diameter • Thread angle The simplest method for checking the acceptability of an external thread is to try it with the mating part for fit. The fit is determined solely by “feel” with no measurement involved. Even without measurement, the fit can be subjectively categorized as “loose”, “medium” or “close” by this qualitative method. However, without actual measurement of the threads’ geometry, it is not possible to determine the degree of interchangeability with other fasteners of the same form and pitch. The most common qualitative (“good” versus “bad”) methods of thread metrology include: • Thread ring gages • Thread roll snap gages • Thread comparator micrometer • Thread pitch gages Thread ring gages are fixed gages used to check the pitch diameter compliance of specific external thread forms. They are used in pairs – a “GO” and a “NO-GO” gage. The NO-GO gage can be easily identified by a groove in the knurling on the outside of the ring gage. Acceptability of the thread is determined as follows: • The GO gage should enter the thread fully, and • The NO-GO gage should not exceed more than 1-1/2 turns on the thread being checked. Figure 3.3k: Thread ring gage used to evaluate external threads. Each gage checks a specific thread form and size. Thread roll snap gages may also be used to check the compliance of external threads. These tools combine the GO and NO-GO gages into a single gage allowing these two evaluations to be performed in a single pass. The actual pitch diameter is compared to a preset dimension on the roll gage. The first set of rolls is the “GO” and has multiple ribs to simulate the GO check of the stand-alone ring gage. The second Page 58 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality set of rolls is the NO-GO and contains only 2 ribs, simulating the NO-GO check of the stand-alone ring gage. Page 59 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.3l: NO-GO rolls Thread roll snap gage used to evaluate external threads. Both GO and NO-GO checks are made with a single snap gage. Gage shown is not adjustable and includes an integral thread plug gage. GO rolls Because the approach to the thread roll snap gage is in a radial direction, the gage can be applied to workpieces which are mounted between centers on a lathe. This cannot be done with individual ring gages which require an axial approach. Another advantage of the snap roll gage is the ability to check both rightand left-handed threads with the same gage. This cannot be done with stand-alone ring gages. Like individual ring gages, some snap roll gages are fixed for a specific form and size and cannot be adjusted. Other gages have rolls which are mounted on eccentric pins and can be adjusted for various forms and sizes. Thread setting plug gages with GO and NO-GO members are used as masters for setting the roll snap gage. Figure 3.3m: Thread plug gages can be used to set adjustable snap roll gages. The thread comparison micrometer may also be used to evaluate external threads. Contrary to what might be expected for a micrometer instrument, the thread comparator micrometer does not actually measure the pitch diameter of a thread. Instead, it is used to make a comparison with a known standard. The micrometer is first set to the actual value of the threaded part and then compared to a reading taken from the corresponding thread plug gage. Figure 3.3n: The thread comparator micrometer. This instrument is used for comparison to a standard. Page 60 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Like the other qualitative methods described so far in this Section, the results obtained from the thread comparison micrometer merely describe how the actual thread geometry compares to a known standard. The thread checked is only known to be either “good” or “bad”. No measurement values are taken which could quantify the external thread’s compliance to its thread form. Thread pitch gages are fixed gages that can be used to quickly determine the pitch (threads per inch) of an externally threaded part. Like the other basic methods already described in this Section, thread pitch gages do not actually measure the thread pitch. Each gage represents a specific thread pitch. The gage is physically laid against the threads to verify the thread pitch. Pitch gages are available for various thread forms. Figure 3.3o: Thread pitch gages. The major diameter of an external thread is the distance between the opposing crests of the thread. The major diameter normally checked on a GO/NO-GO basis with a ring gage. A micrometer may also be used when actual qualitative data is required. Ring gages are unthreaded hardened steel cylinders that are accurate in inside diameter. They are used to check outside diameters, such as the major diameter of threaded fasteners, on an accept (GO) or reject (NOGO) basis. Figure 3.3p: Ring gage. With accurate inside diameters, ring gages can be used to check the major diameter of screw threads. Page 61 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality There are two basic kinds of ring gages: • • Single Purpose. Separate GO and NO-GO rings. Progressive. GO and NO-GO are on the same ring. The GO is used first. The ring gage is slid onto the threaded workpiece and then allowed to fully engage the workpiece (if it will). The ring is not forced. Acceptability of the major thread diameter is determined as follows: • • If the GO goes on and the NO-GO does not, the major thread diameter is acceptable. If the NO-GO goes on, the thread is undersized. Ring gages will not indicate how much a diameter is out of tolerance. That must be determined using another measuring instrument such as a micrometer. Advanced Methods of Thread Metrology for External Threads As discussed previously, it is this assurance of interchangeability that is critically important to the effective and efficient use of fasteners in industry. Thus, it is imperative that quantitative methods of thread metrology be employed to assess the compliance of threads to their specifications in order to ensure interchangeability. In the balance of this Section basic quantitative methods of thread metrology will be discussed, including: • Three-wire measurement with standard micrometer • Thread measurement micrometer • Optical comparator It is noted here that it is important to ensure that threads are clean and undamaged prior to performing any type of thread inspection. Page 62 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality The Three-Wire Method of measuring pitch diameter is relatively quick and accurate and provides important measurement data about the actual thread form. No specialized measurement equipment is required. To perform a three-wire measurement, three wires of the same specified diameter are required. Any wire size that permits the wire to extend beyond the crest of the thread can be used. However, using the BestSize Wire helps to reduce measurement error. Therefore, to ensure accuracy and limit error, a “best wire size” should be selected for the measurement. For Unified Threads, the best wire size is calculated as follows: Best-Size Wire = 0.57735 * thread pitch The best-size wire is that which will contact the thread flanks halfway along their length, thereby contacting the pitch line directly. Figure 3.3q: Best-Size Wire. “Best Wire” contacts thread flank at the pitch diameter. Note that the wire must extend above the thread’s crest. __________________________________ For example, the pitch diameter of a ¼-20 UNC thread is to be measured using the Three-Wire Method: 1. Calculate Thread Pitch (P): Thread Pitch = 1/Threads per Inch = 1/20 = 0.050 inch 2. Find the Best-Size Wire: Best-Size Wire = 0.57735 * Pitch = 0.57735 * 0.050 = 0.0288 inch diameter wire __________________________________ Although any wire can be used, it should be recognized that the diameter of general-purpose wire can vary greatly over its length. Also, general-purpose wire is not normally hardened, therefore the wire is subject to deformation during measurement. To reduce the error introduced when the diameter of the wires are not equal, special kits are available which contain a selection of precision, hardened commonly used thread measuring wires. Thread measuring wires are normally ground to precision diameters and hardened to a Knoop Hardness number of 630. For each size, three wires of the same diameter are provided which are typically held to Page 63 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality within 0.00001 inch (“Grade A”). Lower tolerance wires are also available, but result in lower accuracy measurements. The surface finish of the wires is generally held to 2 micro-inches (µin). Page 64 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.3r: Typical thread measuring wire kit. There are three wires of the same size. The diameter of the three wires is generally held to within 0.00001 inch. A typical thread measuring wire kit can accommodate a range of threads, such as from 3 to 48 threads per inch. Some kits are also provided with a table indicating the best-wire size for various threads in lieu of making the best-size wire calculation. A partial listing of best-size wires for Unified threads is depicted in Figure 3.3s. Page 65 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Threads per Inch (TPI) Pitch Best-Size Wire Diameter Recommended Maximum Wire Diameter Recommended Minimum Wire Diameter 80 0.012500 0.00722 0.01263 0.00631 72 0.013889 0.00802 0.01403 0.00702 64 0.015625 0.00902 0.01579 0.00789 56 0.017857 0.01031 0.01804 0.00902 48 0.020833 0.01203 0.02105 0.01052 44 0.022727 0.01312 0.02296 0.01148 40 0.025000 0.01443 0.02526 0.01263 36 0.027778 0.01604 0.02807 0.01403 32 0.031250 0.01804 0.03157 0.01579 28 0.035714 0.02062 0.03608 0.01804 24 0.041667 0.02406 0.04210 0.02105 20 0.050000 0.02887 0.05052 0.02526 18 0.055556 0.03208 0.05613 0.02807 16 0.062500 0.03608 0.06315 0.03157 14 0.071429 0.04124 0.07217 0.03608 13 0.076923 0.04441 0.07772 0.03886 12 0.083333 0.04811 0.08420 0.04210 11 0.090909 0.05249 0.09185 0.04593 10 0.100000 0.05774 0.10104 0.05052 9 0.111111 0.06415 0.11226 0.05613 8 0.125000 0.07217 0.12630 0.06315 7 0.142857 0.08248 0.14434 0.07217 6 0.166667 0.09623 0.16839 0.08420 5 0.200000 0.11547 0.20207 0.10104 4 0.250000 0.14434 0.25259 0.12630 Figure 3.3s: Best, Maximum and Minimum Wire Sizes for Measuring External Unified Threads Page 66 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality The step-by-step procedure for the Three-Wire Method: 1. Calculate the Best-Size Wire as previously discussed. Acquire three wires from the thread measuring wire kit with diameter closest to the calculated best-size wire diameter. 2. Place the wires onto the threads to be measured – two on one side and one on the directly opposite side of the threads. 3. Using a micrometer with flat-faced anvils, measure the dimension over the wires. This measurement is labeled “M”. Good contact pressure is required to ensure the wires are firmly seated in the thread form. 4. Using measurement “M”, calculate the actual thread pitch diameter using the following formula: Pitch Diameter = M + (0.86603 * Pitch) – (3* wire size used) Figure 3.3t: Thread pitch diameter being measured using the Three-Wire Method. Note that two wires are placed on one size and a single wire is placed directly opposite. An elastic band may be helpful in holding the wires in place. All three wires are the same diameter. Measurement “M” is made across the three wires. Page 67 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality __________________________________ For example, a ¼-20 UNC external thread is measured using the three-wire method. The best wire size is calculated as Best-Size Wire = 0.57735 * Pitch = 0.57735 * (1/20) = 0.0288 inches The measuring wire kit contains wires of 0.029 inches in diameter. After the wires are set in place, the measurement across the three threads (M) is found to be 0.280 inches. The thread pitch diameter is calculated as: Pitch Diameter = M + (0.86603 * Pitch) – (3* wire size used) = 0.280 + (0.86603 * 0.050) – (3* 0.029) = 0.280 + 0.0433 – 0.087 = 0.2363 inches This calculated value of pitch diameter is then compared against the tolerance limited specified on the drawing or in the specification for the thread form and class of fit. If within the stated tolerance limits, the thread would be found to be acceptable. Exhaustively comprehensive tables of tolerances for various thread forms can be found in references, including the Machinery’s Handbook. In our example, the specification for ¼-20 UNC external thread is: Size Class ¼-20 UNC External 1A 2A 3A Pitch Diameter, Maximum 0.2164 0.2164 0.2175 Pitch Diameter, Minimum 0.2108 0.2127 0.2147 The pitch diameter of the evaluated thread is not in compliance with any applicable specification, and therefore its interchangeability cannot be confirmed. __________________________________ Another option for measuring pitch diameter is the Thread Micrometer. Unlike the thread comparison micrometer, the thread micrometer allows direct measurement of the thread pitch diameter. The thread micrometer is equipped with a double-V-shaped fixed anvil designed to fit over the thread. This anvil has sufficient clearance to prevent it from bearing on the top of the thread. The movable anvil is a 60degree cone, enabling it to enter the space between two threads. The cone-shaped anvil is slightly rounded at the tip so as to avoid bearing on the bottom of the thread and can be used on any 60-degree thread form. One of the disadvantages of this method is that a special set of fixed anvils is required for each thread form to be measured. To avoid the possibility of using the incorrect anvil set, the set-up of the thread micrometer is generally checked against a thread plug gage of known pitch diameter prior to taking the measurement. Page 68 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Thread micrometers are also available in sets with a capacity of 1 inch and each micrometer covering a range of threads. Typically these sets have four thread micrometers with the following typical ranges: • • • • No. 1 No. 2 No. 3 No. 4 8 to 14 threads per inch 14 to 20 threads per inch 22 to 30 threads per inch 32 to 40 threads per inch Figure 3.3u: Thread micrometer being used to directly measure the pitch diameter. Note that a special set of anvils is required for each thread form to be measured. The final quantitative method for measuring external threads is the use of the Optical Comparator. Optical comparators project a greatly magnified profile of the object being evaluated onto a screen. Various templates or patterns in addition to graduated scales can be placed on the screen and used to directly measure the projected shadow of the part. Electronic and digital readouts may also be used to increase reliability and discrimination. The optical comparator is particularly useful for measuring the geometry of screw thread forms, gears, and formed cutting tools. Figure 3.3v: Screw thread forms projected for evaluation using optical comparators. Page 69 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality When equipped with a digital read-out device (DRO), the optical comparators can be used to directly measure the following geometric features of external thread forms: • • • Thread pitch diameter Thread depth Helix angle One of the advantages of the optical comparator is the fact that it requires little or no modification or specialized tooling/fixturing to evaluate threads of different forms (Unified, square, acme, ISO metric, pipe threads, etc.). The final attribute of external threads to be addressed is the thread angle. Thread angle is the angle formed between the flanks of the threads. All Unified Threads and ISO Metric threads are 60-degree thread forms. Figure 3.3w: Thread angle is formed between the flanks of two adjacent threads. Thread angles are generally checked by comparing to a standard, such as standard fixed angle gages. For all 60-degree thread forms, a center gage may be used. Center gages are also used to verify the threading tools which are used to cut 60-degree external threads. Figure 3.3x: Center Gage. The nose of the center gage is 60 degrees and may be used to verify thread angle of Unified and IOS Metric thread forms. Page 70 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Basic Methods of Thread Metrology for Internal Threads It is noted here that it is important to ensure that threads are clean and undamaged prior to performing any type thread inspection. Evaluation of internal threads generally includes the following characteristics: • Pitch diameter • Minor diameter • Depth of threads in the hole (if not tapped through) The simplest method for checking the acceptability of an internal thread is to try it with the mating part for fit. The fit is determined solely by “feel” with no measurement involved. Even without measurement, the fit can be subjectively categorized as “loose”, “medium” or “close” by this qualitative method. However, without actual measurement of the threads’ geometry, it is not possible to determine the degree of interchangeability with other fasteners of the same form and pitch. For most applications, however pitch diameter of internal thread forms is checked with a precision Thread Plug Gage. For internal threads, the thread plug gage is analogous to the thread ring gage used for external threads. Thread plug gages are available in a variety of sizes, with the size generally stamped on the handle. Each plug gage checks a specific thread size and class of fit. The longer thread gage is the “GO” gage and the shorter thread gage is the “NO-GO” gage. NO-GO Figure 3.3y: Precision thread plug gage used for checking internal threads. GO The longer gage is the “GO” gage and the shorter gage is the “NO-GO” gage. The NO-GO gage is made to a slightly larger dimension than the pitch diameter for the class of fit that the gage checks. To check an internal thread, both the GO and the NO-GO ends of the plug gage should be tried in the threaded hole. If the part’s pitch diameter is within the range or tolerance of the gage: • • The GO end should turn in flush to the bottom of the internal thread, and The NO-GO end should just start into the hole and become snug with no more than three turns. Thread plug gages are precision instruments and should be cared for and protected accordingly. Under no circumstances should a plug gage be forced into the hole. Page 71 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality The minor diameter of an internal thread is the distance between the opposing crests of the thread. The minor diameter normally checked with a plug gage. Plug gages are unthreaded hardened steel pins that are accurate in outside diameter. The pins are mounted on a handle to facilitate handling during use. They are used to check inside diameters, such as drilled or tapped holes, on an accept (GO) or reject (NO-GO) basis. There are three basic kinds of plug gages: • • • Single Purpose. A GO or NO-GO member only (not both) Double End. Both a GO and NO-GO member, on opposite ends of the handle. Most common. Progressive. GO and NO-GO are on the same side. The GO is the first part used. Figure 3.3z: Plug gage being used to evaluate compliance of internal diameter, including minor thread diameter. The plug gage is inserted into the threaded hole at a slight angle, rotated to a 90-degree (vertical) position, and then allowed to fully enter the hole (if it will) under its own weight. The pin is not forced. Acceptability of the minor thread diameter is determined as follows: • • If the GO goes in and the NO-GO does not, the minor thread diameter is acceptable. If the NO-GO goes in, the thread is oversized. Plug gages will not indicate how much a diameter is out of tolerance. That must be determined using another measuring instrument such as a small hole gage and a micrometer. Plug gages will also not detect out-of-roundness, taper, or barrel-shaped holes. Page 72 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Small hole gages are generally used to measure inside diameters up to 0.400 inches. The gage has two rounded contact stems, a spreader, and an adjustment knob. They usually come in sets of four, each with a range of about 0.100 inches. After adjustment to the size of the inside diameter, the gage is removed and measured directly with a micrometer. As such, small hole gages are considered transfer-type gages. Figure 3.3aa: Typical small hole gage set. Small hole gages are used as follows: • • • • Select the gage that has the range for the diameter that is to be measured. Place the gage into the hole with the adjustment knob “loose”. Move the gage up and down for a short distance, tighten the adjustment knob until the contact points of the stems rub in the hole (against the crests of the threads) Measure the resulting diameter of the gage using a flat anvil micrometer. Take care not to overtighten the micrometer. Figure 3.3ab: Small hole gage and micrometer used to measure inside diameters, including minor diameter of threaded holes. For diameters above 0.400 inches, a telescoping gage is required. Telescoping gages operate in essentially the same manner as small hole gages. They can measure internal diameters from 5/16 to 6 inches. Figure 3.3ac: Telescoping gage and micrometer used to measure inside diameters, including minor diameter of threaded Page 73 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality The final aspect of thread metrology to be discussed is the measurement of threaded hole depth. Although many holes are tapped “through” to the bottom of the hole, often threads are only cut to a specified depth. The measurement of the threaded hole depth can be done using various methods. The most commonly used method is the Turn Method. The Turn Method uses a thread plug gage’s accurate “lead” to verify the depth of threads. The “lead” is the distance the thread plug gage will travel linearly with one full turn of the gage. It is also the distance between two adjacent threads – or the pitch! The following formula is applied: Depth of Thread Hole = Number of Turns * Lead _________________________________ For example, 3/8-16 thread is to be tapped into a hole to a depth of 0.625 inches. A 3/8-16 thread plug gage is turned into the threaded hole until the gage bottoms out. The gage is turned 9 full turns. Is the tapped hole of proper depth? The thread is 16 threads per inch, therefore: Pitch = 1/TPI = 1/16 = 0.0625 inch = “lead” Nine full turns of the gage would result in a linear motion of: turns * lead = 9 * 0.0625 = 0.5626 inches The hole has not been tapped to its specified depth. 10 turns would have been necessary to reach the specified depth of 0.625 inches. __________________________________ Threaded hole depth is generally specified from the top of the hole. Therefore, it is noted that the depth of any countersinks, chamfers, or counterbores at the top (entry) of the tapped hole generally must be ADDED to the linear distance obtained from the Turn Method to calculate the threaded hole depth. The complete formula for threaded hole depth is therefore: Depth of Thread Hole = X + (Number of Turns * Lead) Where X = depth of any chamfer, countersink, or counterbore at entry of thread Page 74 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 3.4 Optical Theory of Light Waves 3.4.1 Exploration There is no exploration exercise for this module. 3.4.2 Dialog Introduction to Light and Optics A key to precise metrology is to base systems on a phenomenon that is extremely precise, unchanging, and is measured in extremely small units. Such is light. Light is today believed to be a form of energy or electrical oscillations – quite possibly energy given off as electrons encircle the neutron. Sir Isaac Newton thought light to be a particle, as there are behaviors which would lead one to think that. In the 1800’s, it was determined that light – because of its constant properties, could be an extremely useful tool. As one may recall from early physical science class, there are different types of light – some visible and some not visible. They are all forms of electromagnetic radiation. Electromagnetic radiation is classified by wavelength. The breakdown is radio, microwave, infrared light, visible light, ultraviolet light, x-rays and gamma rays. Even though infrared and ultraviolet are considered light, neither can be detected by the human eye. We will concentrate here on visible light. Optics is the study of light and interaction of light and matter. One may recall that there are basic colors that make up visible white light from the acronym “ROY G BIV”, those colors being Red, Orange, Yellow, Green, Blue, Indigo and Violet. Each of those colors of visible light has a unique and constant wave length. Those wavelengths (excluding indigo – which is narrow and often included with violet) are listed below. In Figure 3.4a below, note that “nm” in the Meter column is nanometer, which is 1 billionth of a meter or 10-9 meters. To give you an idea of size, a human hair is roughly 100,000 nm in diameter. This would mean that these wavelengths are about 200 times smaller than the diameter of a hair. Also note that the wavelengths and frequencies below are given in ranges and not an exact number. Color Wavelength (meters) Wavelength (inches) Frequency Red Orange Yellow Green Blue Violet 780-622 x 10-9 (~701 nm) 622-597 x 10-9 (~610 nm) 597-577 x 10-9 (~587 nm) 577-492 x 10-9 (~535 nm) 492-455 x 10-9 (~473 nm) 455-390 x 10-9 (~423 nm) 27.5 – 24 x 10-6 24 – 23.2 x 10-6 23.2 – 22.4 x 10-6 22.4 – 19.7 x 10-6 19.7 – 18.1 x 10-6 18.1 – 15.7 x 10-6 ~433 x 1012 ~493 x 1012 ~512 x 1012 ~565 x 1012 ~635 x 1012 ~714 x 1012 Figure 3.4a: Light Spectrum Wave Lengths (ignoring indigo) and Frequencies from Fundamentals of Dim. Metrol. by Dotson – p 164 Page 75 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Wave Theory In the study of Physics, light often behaves as either a particle or wave, depending on circumstance. At about the same time as Newton, Christian Huygens stated that light was made up of waves vibrating up and down perpendicular to the direction that light travels. His was the successful theory of light wave motion in three dimensions. Huygens suggested that light wave peaks form surfaces like the layers of an onion. In a vacuum or other uniform medium, the light waves are spherical, and these wave surfaces advance or spread out as they travel at the speed of light. This theory explained why light shining through a pin hole or slit spreads out rather than going in a straight line – also known as diffraction. In addition to diffraction and interference, light also is governed by reflection, refraction and dispersion, and these will be explained in the next few sections. We first introduce some terminology associated with waves. Imagine that we drop a rock in the middle of a still pond and watch the waves emanating out from the center. From above, the wave crests might appear as in Figure 3.4b below. Figure 3.4b: Wave Crests as Seen From Above from http://theory.uwinnipeg.ca/physics/light/node3.html We assume the wave pattern is regular, and consider the following characteristics of these waves: • The wavelength ( λ ) is the distance between neighboring crests or troughs. • The speed ( v ) is the rate at which the crests (or troughs) move forward. • The Period ( T ) is the time that elapses between passing crests (or troughs) and can be expressed in terms of the speed and wavelength: T= λ/v • The frequency ( f ) is the number of crests (or troughs) that pass by per unit time and is equal to the inverse of the period: f=1/T Page 76 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Using the expression for T above we get the useful expression: v=fλ • A ray is a line drawn from one wave crest to another which intersects each crest at right angles, as in Figure 3.4c below. For light waves, the rays always point in the direction of the motion. Rays therefore provide a useful representation for describing the motion of light waves. Figure 3.4c: Rays from http://theory.uwinnipeg.ca/physics/light/node3.html Note: Wave crests coming from a point source (if you drop a rock in the middle of a still pond) give rise to circular waves as shown in Figure 3.4b. If one has very many point sources close together and in a straight line, they give rise to plane waves, whose crests all lie in a straight line as seen below in Figure 3.4d. Page 77 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.4d: Plane Waves Formed by Many Point Sources from http://theory.uwinnipeg.ca/physics/light/node3.html Huygens’ theory better described the early experiments. When considered a wave, light is characterized by a velocity (speed of light – a constant), a wavelength, and a frequency. In a classical sine wave diagram below (Figure 3.4e), the wavelength would be the distance from peak to peak – represented by the Greek letter lambda (λ). The frequency, or f (measured in Hertz or Hz), would be the number of cycles – or peak to peak movements, occurring in a given time (usually seconds). Thus f = 1 / T as given earlier. T is the time period, or λ / v (velocity), where for light, v = c (the speed of light). Thus frequency is also equal to the speed of light over the wavelength, or f = c / λ. The speed of light in a vacuum is 299,792.458 kilometers per second (approximately 300,000 km/sec) or 186,285 miles per second. Figure 3.4e: Classical Sine Wave Diagram from http://en.wikipedia.org/wiki/Light Diffraction and Interference Page 78 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Diffraction is the process by which light waves traveling through a small hole, slit or around a boundary will spread out. As the wave front travels through the slit, a new wave front radiates from the slit, and was the basic experiment conducted by Huygens. Diffraction would appear to bend the light the light rays as they pass an edge. Huygens’ principle states that all points along a wave front act as if they were point sources. When a wave comes against a barrier with a small opening, all but one of the effective point sources are blocked, and the light coming through the opening behaves as a single point source, so that the light emerges in all directions instead of just passing straight through the slit. Another better understanding of diffraction would be to pretend that at the points where the light rays strike the object, a new set of waves is emitted, which spreads out in all directions. Simple diffraction is shown below in Figure 3.4f. For sizeable diffraction effects to occur, the width of the opening must be of the same size or less than the wavelength of the light used. Figure 3.4f: Diffraction Shown Through Single Slit from http://homepages.tig.com.au/~flavios/diffrac.htm Diffraction actually limits the resolving power of microscopes and other magnifying devices. If the viewed object is smaller than the wavelength of light used, then the light diffracts around the object and severely distorts the image. Because of this, microscopes using visible light have a resolving power of only about 600 nm (about 10- 6m), but X-rays, whose wavelength is about 0.1 nm (10- 10 m) have a resolving power four times smaller. Diffraction was better understood and the concept of interference introduced, when wave theory was further developed via the two-slit experiment, as shown in Figure 3.4g below. Page 79 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.4g: Interference Phenomenon - Double Slit from http://homepages.tig.com.au/~flavios/diffrac.htm Interference is caused by waves overlapping with each other, causing either a cancellation of the wave at that point, or an amplification of the wave at that point. The above diagram shows the phenomenon of light interference. A screen with a double slit - equally separated, is illuminated with a bright light. The two slits cause diffraction of the light waves, and now two sets of wave fronts are produced, which will eventually overlap. At the point where the waves overlap, there will be either constructive interference (the bright areas) or destructive interference (the dark areas). This experiment was originally conducted by Thomas Young. In the experiment, as shown in Figure 3.4h below, light rays pass through two slits, separated by a distance d and strike a screen a distance, L , from the slits. Figure 3.4h: Double-slit Diffraction Trigonometry from http://theory.uwinnipeg.ca/physics/light/node9.html If d << L (<< meaning much less than) then the difference in path length r1 - r2 traveled by the two rays is approximately: r1 - r2 d sin θ Page 80 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality where the angle θ is approximately equal to the angle that the rays make relative to a perpendicular line joining the slits to the screen. If the rays were in phase when they passed through the slits, then the condition for constructive interference at the screen is: d sin θ = m λ , where m = ±1, ±2, ... whereas the condition for destructive interference at the screen is: d sin θ = (m + 1/2) λ , where m = ±1, ±2, ... When y - the distance from the interference fringe to the point of the screen opposite the center of the slits, is much less than L (y << L), one can use the approximate formula sin θ y/L so that the formulas specifying the y - coordinates of the bright and dark spots, respectively are y Bm = (m λ L) / d {bright spots} y Dm = ((m + ½) λ L) / d {dark spots} B D where m is the sequential ith number of the band (as in Figure 3.4g) seen from center and y m or y m could be considered as distance from center for the mth band, and the spacing between the dark spots would be ∆y=λL/d If d << L , then the spacing between the interference can be large even when the wavelength of the light is very small (as in the case of visible light). This provides a method for (indirectly) measuring the wavelength of light. Also, the above formulas assume that the slit width is very small compared to the wavelength of light, so that the slits behave essentially like point sources of light. Based on the sine wave diagram and interference phenomenon, one can now see graphically what happens during interference. Where constructive interference of the waves occur, or where two waves interfere but are in synch with each other (0 degrees out of phase), the amplitudes (or wave heights as shown above) double while the frequency and wavelength remain the same. In practical terms, an increase in amplitude results in an increase in the light’s intensity. This is shown in figure 3.4i below. It produces seen bright areas, as shown in figure 3.4g above. The doubling assumes that the originating light is coming from the same direction. If light were to come from different angles, trigonometry would dictate the resulting amplitude. Page 81 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 1 x Ampl. 1 x Ampl. 2 x Ampl. Figure 3.4i: Constructive Interference from http://homepages.tig.com.au/~flavios/diffrac.htm Where destructive interference occurs, two waves will actually cancel each other out to give a null point with zero amplitude. Where destructive interference of the waves occurs, or where two waves interfere but one’s peaks correspond to the other’s trough (180 degrees out of phase), the amplitudes become zero while the frequency and wavelength become indeterminate. This is shown in Figure 3.4j below. It produces seen dark areas, as shown in Figure 3.4g above. Figure 3.4j: Destructive Interference from http://homepages.tig.com.au/~flavios/diffrac.htm Page 82 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality If two identical waves of wavelength λ start out in phase, travel at the same speed for a distance of r1 and r2 respectively, where r1 > r2 , the crests of the one wave will be behind the crests of the other by a distance of r1 - r2 . The condition for constructive interference when the waves recombine is r1 - r2 = m λ , where m = 1,2, .... whereas the condition for destructive interference is r1 - r2 = (m + ½) λ , where m = 1,2, .... Reflection Reflection is the return of a wave of light from a surface it strikes into the medium through which it has traveled. The law of reflection states that the angle of reflection (the angle between the reflected ray and the normal, or line perpendicular to the surface at the point of reflection) is equal to the angle of incidence (the angle between the incident ray and the normal). A graphic description is shown below in Figure 3.4k. Note that the surface must be relatively smooth to allow reflection to occur, such as the mirror in the figure. This is also known as specular reflection. In fact, reflection of light may occur whenever light travels from a medium of a given refractive index into a medium with a different refractive index. In the most general case, a certain fraction of the light is reflected from the interface, and the remainder is refracted. In diffuse reflection, light bounces off in all directions due to the microscopic irregularities of the interface (or one with a poor surface finish); this is a common phenomenon, applicable for all non-shiny objects that are not black. Figure 3.4k: Reflection from http://www.shomepower.com/dict/r/reflection.htm When light is reflected off a denser medium with higher index of refraction, crests get reflected as troughs and troughs get reflected as crests. The wave is said undergo a 1800 change of phase on reflection. The net Page 83 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality effect of the phase change is that the reflected ray “jumps ahead” by half a wave length. Thus, if one of the rays undergoes an odd number of phase changes, the conditions for constructive and destructive interference must be modified: r1 - r2 = (m + ½) λ , where m = 1,2, .... {constructive} r1 - r2 = m λ , where m = 1,2, .... {destructive} The above formulas are valid when an odd number of phase changes occur. If an even number of phase changes occurs, then the original unmodified formulas must be used. The phase change does not happen when light is reflected off a less dense medium (for total internal reflection). Refraction Refraction is the turning or bending of any wave such as light or sound, when it passes from one medium into another of different density. Density is the ratio of a material’s mass to its volume. Materials possess an index of refraction n, which is the ratio of the speed of light c in a vacuum to the speed of light in that material (also known as phase velocity), or v. It is given by the formula: n = c / v. The denser the material, the slower the speed of light is in that material. Thus n in a vacuum equals 1 and for all other materials – including air, n > 1. Also, the frequency of light does not change when it passes from one medium to another one. Previously, we gave the formula f = v / λ. Rearranging this formula yields v = λ f. Since f does not change when passing from one medium to another when v does, the wavelength λ must in order to satisfy the formula. This allows us to write the index of refraction formula in terms of wavelengths: n = λ0 / λ where λ0 is the wavelength of light in a vacuum and λ is the wavelength of light in the medium. This change of speed and wavelength at the boundary between the two materials will cause the light to change direction. If θ1 is the angle of the light ray relative to the normal of the surface in medium 1, and θ2 is the angle relative to the normal in medium 2, then: n2 / n1 = v1 / v2 = λ1 / λ2 = sin θ1 / sin θ2 This relationship is shown graphically in figure 3.4l below, where the dashed line is the normal to the surfaces of the mediums. Page 84 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Medium 1 Medium 2 Figure 3.4l: Refraction from http://theory.uwinnipeg.ca/physics/light/node5.html The relationship between these angles is also known as Snell’s Law. It is similar to the relationship presented above, and gives the relation between the angles θ1 and θ2: n1 sin (θ1) = n2 sin(θ2) Note that, for the case of θ1 = 0° (i.e., a ray perpendicular to the interface) the solution is θ2 = 0° regardless of the values of n1 and n2. In other words, a ray perpendicular entering a medium perpendicular to the surface is never bent. It does not matter whether the light ray is moving from medium 1 to 2 or medium 2 to 1. In the figure above, θ1 is the angle of incidence and θ2 is the angle of refraction. If the direction was reversed, θ2 would be the angle of incidence, etc. Normally though, one designates the angle, medium, etc. of incidence with the subscript 1. An example of refraction is looking into a bowl of water. Air has a refractive index of just over 1, and water has a refractive index of about 1.3. If you look at a straight object, such as a ruler, which is placed at a slant and partially in the water, the object appears to bend at the water's surface. This is due to the light rays from the object being bent as they move from the water to the air. This causes water to appear shallower than it really is. Page 85 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.4m: Refraction, Ruler and Water from http://en.wikipedia.org/wiki/Refraction In Figure 3.4m above, the dark rectangle represents the actual position of a pencil sitting in a bowl of water. The light rectangle represents the apparent position of the pencil. Notice that the end (X) looks like it is at (Y), a position that is considerably shallower than (X). One can have a situation where there is total internal reflection. For a light ray which is passing from a denser material to a less dense material, there is a critical angle of incidence θC for which the refraction angle is 90 degrees. For any greater angles of incidence, light cannot pass through the boundary between them and is reflected within the denser material. For light passing from medium 1 to 2, the critical angle is determined by: sin θC = (n2 / n1) sin 900 = n2 / n1 or θC = arcsin (n2 / n1) since sin 900 equals 1, where n2 is the index of refraction for the less dense medium and n1 is the index of refraction for the denser medium. By looking at the formula, one must note that n1 has to be greater than n2 for there to be total internal reflection so that n2 / n1 results in a number between 0 and 1. This is because the value for the sine of any angle - sin θC in this case – must be between 0 and 1. If n2 > n1, then a number larger than 1 would result, and a sine value cannot exceed 1. Dispersion Dispersion of light is related to refraction. The velocity of light in a medium and its corresponding index of refraction depends on the wavelength of the light. In general, n varies inversely with wavelength. The value is greater for shorter wavelengths. This causes light inside materials to be refracted by different amounts according to its wavelength, and since the colors that make up white light have different wavelengths (as shown previously), we are able to see rainbows and the effects through prisms. Rainbows are actually caused by the combination of dispersion inside the raindrops and the total internal reflection of light from the back of those raindrops. Generally, light toward the blue end of the spectrum, which has shorter wavelengths, has a higher index of refraction – and gets bent more – than light toward the red end with longer wavelengths. Figure 3.4n below shows indices of refraction for different wavelengths of light through glass. Page 86 3.0 Angular Measurement, Thread Metrology, and Optics Color Measurement and Quality Wavelength Index of Refraction Blue 434 nm 1.528 Yellow 550 nm 1.517 Red 700 nm 1.510 Figure 3.4n: Indices of Refraction for Certain Light Wavelengths Through Glass from http://theory.uwinnipeg.ca/physics/light/node6.html Fresnel’s Equation The Fresnel equations, deduced by Augustin-Jean Fresnel, describe the behavior of light when moving between media of differing refractive indices. When light moves from a medium of a given refractive index n1 into a second medium with refractive index n2, both reflection and refraction of the light may occur. Figure 3.4o: Fresnel Diagram from http://en.wikipedia.org/wiki/Fresnel_equations In Figure 3.4o above, an incident light ray PO strikes at point O the interface between two media of refractive indices n1 and n2. Part of the ray is reflected as ray OQ and part refracted as ray OS. The angles that the incident, reflected and refracted rays make to the normal of the interface are given as θi, θr and θt, respectively. The relationship between these angles is given by the law of reflection and Snell's law. The Page 87 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality fraction of the incident light that is reflected from the interface is given by the reflection coefficient R, and the fraction refracted by the transmission coefficient T. The Fresnel equations may be used to calculateR and T in a given situation. The calculations of R and T depend on polarization of the incident ray. See the section below for a discussion on polarization. If the light is polarized with the electric field of the light perpendicular to the plane of the diagram above (s-polarized), the reflection coefficient is given by: where θt can be derived from θi by Snell's law. If the incident light is polarized in the plane of the diagram (p-polarized), the R is given by: . The transmission coefficient in each case is given by Ts = 1 - Rs and Tp = 1 - Rp. If the incident light is unpolarized (containing an equal mix of s- and p-polarizations), the reflection coefficient is R = ( Rs + Rp ) / 2. At one particular angle for a given n1 and n2, the value of Rp goes to zero and a p-polarized incident ray is purely refracted. This is known as Brewster's angle, and is shown in Figure 3.4p below. As discussed earlier, when moving from a denser medium into a less dense one (i.e. n1 > n2), above an incidence angle known as the critical angle all light is reflected and Rs=Rp=1 - also known as total internal reflection. Figure 3.4p: Brewster and Critical Angle Example from http://en.wikipedia.org/wiki/Fresnel_equations Page 88 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality When the light is at near-normal incidence to the interface (θi ≈ θt ≈ 0) , the reflection coefficient is given by: . Note that reflection by a window is from the front side as well as the back side, and that the latter also includes light that goes back and forth a number of times between the two sides. The total is 2R/(1+R). Polarization (Advanced Concept) Because light is a form of electromagnetic energy, it is subject to polarization. Magnets have positive and negative poles. Thus polarization is a property of waves, such as light and other electromagnetic radiation. Unlike more familiar wave phenomena such as waves on water or waves propagating on a string, electromagnetic waves are three dimensional, and it is this higher dimensional nature that gives rise to the phenomenon of polarization. Take the case of a simple plane wave, which is a good approximation to most light waves. The plane of the wave is perpendicular to the direction the wave is propagating in. Simply because the plane is two dimensional the electric vector in the plane at a point in space can be decomposed into two orthogonal components. Call these the x and y components (following the conventions of Analytic geometry). For a simple harmonic wave, where the amplitude of the electric vector varies in a sinusoidal manner, the two components have exactly the same frequency. However, these components have two other defining characteristics that can differ. First, the two components may not have the same amplitude. Second, the two components may not have the same phase; that is they may not reach their maxima and minima at the same time in the fixed plane we are talking about. Consider first the special case where the two orthogonal components are in phase. In this case the direction of the electric vector in the plane, the vector sum of these two components, will always fall on a single line in the plane. We call this special case linear polarization. The direction of this line will depend on the relative amplitude of the two components. This direction can be in any angle in the plane, but the direction never varies. Now consider another special case, where the two orthogonal components have exactly the same amplitude and are exactly ninety degrees out of phase. In this case one component is zero when the other component is at maximum or minimum amplitude. Notice that there are two possible phase relationships that satisfy this requirement. The x component can be 900 ahead of the y component or it can be 900 behind the y component. In this special case the electric vector in the plane formed by summing the two components will rotate in a circle. We call this special case circular polarization. The direction of rotation will depend on which of the two phase relationships exists. We call these cases right hand circular polarization or left hand circular polarization, depending on which way the electric vector rotates. All the other cases - where the two components are not in phase and either do not have the same amplitude or are not 900 out of phase are called elliptical polarization because the sum electric vector in the plane will trace out an ellipse. Linear polarized light is also described as being the sum of two circularly polarized beams of equal amplitude and opposite rotation. This formulation is helpful in understanding some of the observed phenomena of polarized light, such as optical rotation and circular dichroism. Page 89 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality The linear and circular cases are limiting cases of elliptical polarization. In the first case one of two the axes of the ellipse has zero length, and we have linear polarization. In the second case the two elliptical axes are equal and we have one of the two circular polarizations. In optical work the ellipse in the plane is usually characterized by an azimuth and an ellipticity. The azimuth angle, α, is the angle between the major semi-axis of the ellipse and the X axis. The ratio of the two semi-axes is called the ellipticity. The arc-tangent of this ratio, β, is also commonly used. Ellipticity is used in preference to the more common geometrical concept of eccentricity. An ellipticity of zero corresponds to linear polarization and an ellipticity of 1 corresponds to circular polarization. For circular polarization, it is also useful to consider how the direction of the electric vector varies along the direction of propagation at a single instant of time. While in the plane the vector rotates in a circle (as time advances), along the propagation axis (at one instant) the tip of the electric vector describes a helix. The pitch of the helix is one wavelength, and the helix screw sense is either right-handed or left-handed. Visualizing this spatial variation in the direction of the electric field is useful in understanding how circularly polarized light can interact differently with helical molecular conformations, depending on whether the electric field and the molecule helix sense are the same or opposite. This is part of the phenomenon of circular dichroism. Polarization of visible light can be observed using a polarizing filter (the lenses of Polaroid® sunglasses will work). While viewing through the filter, rotate it, and if linear or elliptically polarized light is present the degree of illumination will change. The blue sky is polarized because of the nature of the scattering phenomenon that produces the color. An easy first phenomenon to observe is at sunset to view the horizon at a 90° angle from the sunset. Common sources of light, such as the Sun and the electric light bulb emit what is known as unpolarized light. More specialized sources, such as certain kinds of discharge tubes and lasers, produce polarized light. The difference between these two types of light is caused by the behavior of the electromagnetic fields that make up the light. Light is a transverse wave made up of an interacting electric field E and a magnetic field B. The oscillations of these two interacting fields cause the fields to self-propagate in a certain direction, at the speed of light. In most cases, the directions of the electric field, the magnetic field, and the direction of propagation of the light are all mutually perpendicular. That is to say, both the E and B fields oscillate in a direction at right angles to the direction that the light is moving, and also at right angles to each other. In optics, it is usual to define the polarization in terms of the direction of the electric field, and disregard the magnetic field since it is almost always perpendicular to the electric field.. If the direction of oscillation of the electric field E is fixed, the light wave is said to be linearly polarized. The direction of polarization is arbitrary with respect to the light itself. It is usual to label the two linear polarization states in accordance with some other external reference. For example, the terms horizontally and vertically polarized are generally used when light is propagating in free space. If the light is interacting with a surface, such as a mirror, lens or some other interface between two media, the terms s- and ppolarized are used. For example, consider the following: | / | / | / | / | / | / | / |/ ============================= Page 90 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.4q: Reflecting Light and Polarization from http://en.wikipedia.org/wiki/Polarization In Figure 3.4q above, a light ray is reflecting off a mirror at some angle. If the electric field of the light is oscillating perpendicular to the plane of the diagram, the light is termed s-polarized. If it is oscillating in the plane of the diagram, it is termed p-polarized. Other terms used for s-polarization are sigma-polarized and sagittal plane polarized. Similarly, p-polarized light is also referred to as pi-polarized and tangential plane polarized. The polarization fate of these two components differs during reflection from a dielectric surface. If the direction of the electric field E is not fixed, but rotates in a circle as the light propagates, the light is said to be circularly polarized. Two possible independent circular polarization states exist, termed lefthand or right-hand circularly polarized depending on whether the electric field is rotating in a counterclockwise or clockwise sense, respectively, when looking in the direction of the light propagation. Reflection of circularly polarized light by a mirror reverses the sense of polarization. Individual photons are inherently circularly polarized; this is related to the concept of spin in particle physics. If the light consists of many incoherent waves with randomly varying polarization, the light is said to be unpolarized. It is possible to convert unpolarized light to polarized light by using a polarizer. One such device is Polaroid® sheet. This is a sheet of plastic with molecules that are arranged such that they absorb any light passing through it which has an electric field oscillating in a given direction; this has the effect of linearly polarizing the light. Other devices can split an unpolarized beam into two beams of orthogonal linear polarization. They are generally constructed from certain arrangements of prisms and optical coatings. The four mechanisms that can be used to produce polarized light from unpolarized light are dichroism, birefringence, dielectric reflection, and scattering. The common Polaroid® sheet is a dichroic polarizer. The angle of polarization of linearly polarized light can be rotated using a device known as a half-wave plate. Similarly, linear polarization can be converted to circular polarization and vice versa with the use of a quarter-wave plate. A quarter-wave plate is constructed from a birefringent material - that is, in the plane of the plate there are two orthogonal axes and light passing through it propagates at a different speed along one axis than on the other. The thickness of the plate is adjusted so that the net difference in propagation speed is one quarter of a wavelength. If this plate is oriented so that the fast axis is forty five degrees to the direction of linear polarization then the light emerging from the other side will have two components of equal amplitude and a 900 phase difference - creating circular polarization. Rotating the quarter wave plate 900 in the plane will reverse the sense of circular polarization. Birefringence can be created by straining a normally uniform material. A properly arranged and controlled mechanical oscillator coupled to a strain-free window can convert linearly polarized light of a single color impinging on the window into alternating left and right hand circularly polarized light emerging from the other side. In other words, the window can operate as an oscillating quarter wave plate. If this light is then passed through a material which has a circular dichroism at that color, the emerging light will have an amplitude modulation that varies with the frequency of the oscillator driving the quarter wave plate. This amplitude variation can be detected and used to measure the amount of circular dichroism exhibited. This amplitude will depend on the intrinsic property of the material, and upon the amount of material the light passed through, which in turn depends on the concentration of the absorbing substance and its thickness. Although the phenomenon measured this way is delta-absorption, the results are customarily reported in degrees of ellipticity through a simple algebraic conversion. Page 91 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Coherence (Advanced Concept) Coherent waves are all of the same frequency (monochromatic) and all the waves are in phase (that is, all the waves are "up" at the same point). Laser light is coherent. A red light can be monochromatic but still be incoherent because the waves are in random phase. Actually, an incoherent wave would have some dispersion, although it might be quite narrow. It can't be too narrow, or it would be coherent for all intents and purposes. White light is incoherent both because the phase of the waves are random and because white light is made of many different frequencies simultaneously. Figure 3.4r shows series of sine waves coherent and incoherent. Figure 3.4r: Coherent and Incoherent Waves from http://en.wikipedia.org/wiki/Coherence_(physics) Page 92 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Light and Optics in Metrology There are four prime applications of light and optics in metrology: 1. 2. 3. 4. Using the known wavelength of light as a length standard in itself Using the phenomenon of light to measure things, or interferometry Using light to visually enlarge an object for more precise examination and measurement, or magnification Using light rays to establish references like lines and planes, or alignment In number 1 above, one should recall that the wavelengths of colors that make up light in figure 3.4a were provided in ranges and not exact numbers. Even though this is true, the units are so small that it makes light useful to measure things even less than 1/10,000 of an inch! 3.4.3 Application: Light Calculation Experiments Materials: Pencil or pen, paper, and calculator The following are calculation exercises utilizing the fundamentals in this module. Questions 1. Two light pulses are emitted simultaneously inside a vacuum chamber and hit a screen directly in front of them 20 meters away. If one light pulse passes through 6.2 m of ice on its way to the screen, enters and exits perpendicular to the ice surface, what is the time difference between the arrivals of the two pulses at the screen? (The index of refraction of ice is 1.309 and use 3 x 108 m/s for speed of light) 2. A light ray in a vacuum chamber of wavelength λ = 589 nm is incident on glass with an angle of incidence of 30o . The index of refraction of this glass is 1.52. a) What is the angle of refraction? b) What is the wavelength of the light inside the glass? c) What is the speed of the light inside the glass? Page 93 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 3. Light passes through a flat slab of glass. The angle of incidence of the light onto the glass is 30 o . What is the angle with which the light emerges on the other side of the slab? 4. Monochromatic light goes through two thin slits 0.03mm apart. It is found that the second bright fringe on a screen 1.2 m away is 4.5 cm from the center. What is the wavelength and color of the light? (This problem shows that is possible to measure very short wavelengths using the double slit experiment quite accurately as long as L/d is very large) 5. White light approaches one side of a glass prism at an angle of incidence of 40 o as shown in the diagram below. The angle of the prism is 60 o at each corner. Refer to Figure 3.4n for indices of refraction. Page 94 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality a) At what angles does red light emerge on the other side? b) At what angle does blue light emerge? Assume the air has an index of refraction of 1. Page 95 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Solutions 1. Forget the 20m distance to the screen; we are only interested in the time difference through 6.2 m of a vacuum vs. 6.2 m of ice. Also t (time) = D (dist) / v (velocity) and from Refraction section, nice = c / vice or vice = c / nice, so vice = 3 x 108 m/s / 1.309 = 2.292 x 108 m/s; tice = 6.2 m / 2.292 x 108 m/s = 2.705 x 10-8 sec. Since the refractive index of a vacuum is 1, tvac = 6.2 m / 3 x 108 m/s = 2.067 x 10-8 sec. The time difference is thus tice – tvac = 2.705 x 10-8 s – 2.067 x 10-8 s = .638 x 10-8 s or 6.38 x 10-9 sec 2. a) From Refraction section, n1 sin (θ1) = n2 sin(θ2) and 1 is for air – 2 is for glass; therefore sin (θ2) = n1/ n2 sin(θ1) = 1/1.52 = .6579 x sin 300 = .6579 x .5 = .329 and θ2 = arcsin .329 or 19.2 degrees. b) From Refraction section and Intro, n2 / n1 = λ1 / λ2 or λ2 = n1 / n2 x λ1 = (1/1.52) x (589 x 10-9 m) = .658 x (589 x 10-9 m) = 387.6 x 10-9 m or 387.6 nm c) From Refraction section v2 = c / n2 = 3 x 108 m/s / 1.52 = 1.97 x 108 m/s 3. First calculate the angle of refraction inside the glass. From Refraction section, sin (θ2) = n1/ n2 sin (θ1), then refraction angle of light leaving glass and entering air, sin (θ3) = n2/ n3 sin (θ2). Substitute sin (θ2) from the first equation into the second, yielding sin (θ3) = n2/ n3 (n1/ n2 sin (θ1)). The n2’s cancel leaving sin (θ3) = n1/ n3 sin (θ1). Since n1= n3 (both air), sin (θ3) = sin (θ1). Since the sin’s are equal, the angles θ3 and θ1 are equal, and thus the ray emerges parallel to the incoming ray. B 4. From Young’s experiment under Diffraction and Interference and using the formula y m = (m λ L) / d , then rearranging so λ = y Bm d / m L = (4.5 cm) (.03 mm) / 2 (1.2 m) = (4.5 x 10-2 m) (.03 x 10-3 m) / 2 (1.2 m) = 0.135 x 10-5 m / 2.4 m = 1.35 x 10-6 m / 2.4 m = 0.558 x 10-6 m = 558 x 10-9 m = 558 nm. From the Intro section, this would correspond to green light. 5. From the Refraction section sin (θ2) = n1/ n2 sin (θ1) and from Figure 3.4n in glass nred = 1.51 and nblue = 1.528. The nair = 1. θ1 is 400, so for red, sin (θ2) = nair / nred sin 400 = 1/1.51 (.643) = 0 .426, or arcsin .426 = 25.20. For blue, sin (θ2) = nair / nblue sin 40 = 1/1.528 (.643) = .421, or 0 arcsin .421 = 24.9 . Next we calculate the angle of incidence θ3 on the far side using geometry. From the diagram above, we get the relationship (900 – θ2) + (900 – θ3) + 600 = 1800. Simplifying and Page 96 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality rearranging the equation yields θ3 = 600 – θ2. θ3 (red) = 600 – 25.20 = 34.80 and θ3 (blue) = 600 – 24.90 = 35.10. Finally we calculate the angle of refraction sin θ4 of the emerging rays. Sin (θ4) = n2/ n1 sin (θ3) and from Figure 3.4n in glass nred = 1.51 and nblue = 1.528. The nair = 1. For red, θ3 is 34.80, so sin (θ4) = nred / nair sin 34.80 = 1.51/1 (.571) = .862, or arcsin .862 = 59.50. For blue, sin (θ4) = nblue / nair sin 35.10 = 1.528/1 (.575) = .879, or arcsin .879 = 61.50. The blue light is bent 61.5 – 59.5 = 2 degrees more than red. Page 97 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 3.5 Use of Optical Flats 3.5.1 Exploration Materials: Bar gage or surface plate with attached dial indicator, Optical Flats set. Parts: Gage Block Set (3 Pieces) with desired surface to measure indicated Pick three gage block parts and measure flatness of indicated surface. Take 3 readings and record the results in the table below. Part Flatness (to nearest .000001”) Measurement 1 Measurement 2 Measurement 3 Avg. Measurement 1 2 3 Questions 1.) Were you able to take readings with the supplied instruments? Describe any difficulty you had. ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ____________________ 2.) Describe “flatness” in your own terms. How does it differ from “surface finish?” ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ________________ Page 98 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality 3.5.2 Dialog Introduction to Flatness Flatness is one of the 4 geometric tolerances of form – along with surface straightness, circularity and cylindricity. These are used to control shapes of finished items and specify how much a form or surface can deviate from a perfect shape. There is no other datum point or surface used with these form tolerances. Unlike a dimension or condition like perpendicularity, where one relates a surface or feature to another surface or feature, these tolerances of form relate only to themselves. Flatness is a condition whereby all elements of a surface lie in a given plane – over the entire extent of the surface. The tolerance zone is defined by two parallel planes within which the entire surface must lie. A visual description of this condition and part drawing callout example is found in Figure 3.5a below. Note that the flatness callout is a two-part symbol. It consists of the parallelogram shape followed by a number. The number is the entire tolerance zone for that surface. It is not a +/- figure, but rather a total, and is often referred to as TIR or “total indicator reading.” In terms of geometrical dimensioning and tolerancing (GD&T), it can also be stated that the flatness tolerance must be contained within the boundary of perfect form at maximum material condition (MMC). Figure 3.5a: Flatness Drawing Callouts and Visual Interpretation. Mfg Engr handbk (fig 9-13), Design Graphics (fig 19-77) For required flatnesses on the order of .001” to .025”, one could utilize various pieces of equipment and setups. A few examples are provided in the following figures. The entire surface of the part to be checked Page 99 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality must be supported and given a reading over the entire extent of its surface. The gage support piece must be of a known flatness that exceeds the objects being tested by a degree of perhaps 10 to 100. For instance, if one is checking flatness to within .001”, it would be beneficial if the gage support piece be flat to within .0001” to .00001”. This ensures the accuracy of the reading. The dial must be capable of discrimination to the degree required or better. For example, it would not serve much purpose to try to determine flatness to within .005” with a dial capable of discrimination only to within .01”. Some examples of flatness check set-ups can be found on the three Figures below. Figure 3.5b: Drawings from Mfg Engr handbk 9.6, fig 9-14 Page 100 3.0 Angular Measurement, Thread Metrology, and Optics Figure 3.5c: Drawings from Mfg Engr handbk 9.6, fig 9-15 Measurement and Quality Figure 3.5d: Drawings from Mfg Engr handbk 9.6, fig 9-16 Generally speaking, when one wants an extremely flat surface, that surface most likely will also have a very fine texture, or surface finish, called out. In most cases, the surface finish – measured within millionths of an inch (.000001”), or micro-inches, will be much finer than the degree of flatness required. One could realistically see a surface have a 32 finish callout (.000032” arithmetic average deviation from the mean plane of the surface) and also have a requirement for flatness of perhaps .001” to .002”. A few examples of where surface flatness would be an issue are: 1) Machine tool axis ways – in this case, two very flat pieces of hardened stainless steel may be sliding over one another, riding on a film of lubricant. Surface finish required could be a 4, or extremely fine. To get maximum contact over the total travel range and reduce the risk of creating micro chips of material with constant movement, both pieces would need to be extremely flat. The lubricant would actually collect in the micro peaks and valleys of the basic surface finish deviations. Machine ways are usually “scraped” to achieve the extreme flatness, which is done after any basic machining and heat treatment. Heat treatment can cause materials to warp slightly, which would lead to non-flat conditions for a surface. 2) Pump mating surfaces – Many fluid and air pumps have extremely small clearances designed internally. Use of a gasket or compound for sealing might affect these clearances. If both mating surfaces have an extremely fine surface finish and flatness, and then are drawn together with bolts all around, there is little chance for lubricant or air leakage. The degree of flatness and surface finish required would generally be determined by design testing, and would likely relate to internal pressures that exist. 3) The measuring faces of gage blocks – Since these are to used to measure other objects, accuracy and precision are very important. One may even use a known excellent-condition gage block to check the condition of a questionable one. 4) Surface plates – Since these are used to perform other precision measurements, and are a reference surface, the known condition of the plate is important for supplying precise measurements. 5) Refrigerator Seals – Like pump surfaces, the degree of flatness will dictate the efficiency of the system they are installed in. The Need for Optical Flats Page 101 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality When flatness requirements of surfaces fall below .001”, a difficulty is presented to measure that degree of flatness. Dial indicators and the fineness of the measurement stylus (including its diameter, tip shape, and durability) begin to be impractical below .001”. Some smaller objects can still be measured to within a few .0001” as long as the dial will discriminate to as little as .0001”. Extremely fine surface finishes and the achievement of high degrees of flatness come at additional cost, as it typically involves several additional processes and inspections, and involves additional equipment and time. It should be noted that flatness, if not specified on a drawing, is controlled by the basic dimension given for the feature. When called out on a drawing, the degree of flatness will be smaller than the dimensional tolerance for the feature. Objects like quality gage blocks, due to their function, require super fine surface finishes and flatness on the measuring surface. Their flatness requirements are in the neighborhood of millionths of an inch (.000001” to .00001”). Micrometer anvil and spindle faces, as well as other polished or lapped parts (lapping is an example of a micro-finishing operation) mentioned above, require flatness in the millionths of an inch range. Under these circumstances, dial indicators are no longer practical, as mentioned in the last section. One requires special equipment. Optics can now be brought into use. Because the surfaces of the objects mentioned above have such a fine finish (often 2 to 16 micro-inches), they are actually light reflective – similar to a mirror back plate). As long as the surface is light reflective, optical flats can be used. Optical flats come in a variety of diameters (1” to 12”) and thicknesses (1/4” to 1”), but one will most likely see flats from 1” to 3” diameter (25 to 75 mm). Flats are generally made of Pyrex glass or fused quartz, but could be made from inexpensive glass to very expensive sapphire. Optical flats are available in various degrees of accuracy, from a reference grade that is flat to within 1 micro-inch, to a master grade that is flat to within 2 micro-inches, to a working grade that is flat to within 4 micro-inches, and to a commercial grade that is flat to within 8 micro-inches. Optical flat manufacturers may finish one or both surfaces for measurement, and indicate the finished surface with an arrow on the edge of the flat. There are also optical parallels that have two measuring faces that must be parallel to each other. For an additional cost, one can purchase flats with a coated surface. It is usually a thin film of titanium oxide and reduces the amount of light lost by reflection. The fringe bands generally appear much clearer if using coated flats. Page 102 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.5e: Typical Optical Flats From http://www.globalspec.com/FeaturedProducts/Detail?ExhibitID=8123&deframe=1 http://images.google.com/imgres?imgurl=ekimstech.co.kr/tesa/i122.jpg&imgrefurl=http://ekimstech.co.kr/tesa-i.htm&h=337&w=300&sz=25&tbnid=Wz8kcs400J:&tbnh=114&tbnw=102&prev=/images%3Fq%3Doptical%2Bflats%26start%3D40%26 hl%3Den%26lr%3D%26ie%3DUTF-8%26oe%3DUTF-8%26sa%3DN Rather than making a more expensive and delicate instrument with higher amplification or definition/precision to measure near-perfect flatness, optical flats are relatively inexpensive and reasonably durable. Because light has a wavelength much less than 1 micro-inch, the wavelength of light is known and constant, and due to the controlled thickness of the optical flat, when we place the near-perfectly flat surface of an optical flat on another reflective surface, we can visually see light bands - which correspond to the distance separating the two surfaces. Optical Flat Working Principles Optical flats operate on the principles provided in the previous section, 3.4 - Optical Theory of Light Waves. Within that section, we introduced light as a wave, rays, and the representation of a light ray as a sine wave. Figure 3.5f below shows the three with a pictorial representation. Page 103 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.5f: Representation of Wave Fronts, Rays, and Light as a Sine Wave (Fund. Of Dim. Metr. By Dotson – fig 13-7 p341) The wave front in the picture is real, but the ray and sine wave are imaginary representation to demonstrate how we see light behave in actual experiments. Also as discussed previously, merging waves that are inphase add their amplitudes and create brighter areas. Those that merge completely out of phase (by 180o) cancel and create dark areas. Those that merge at degrees in between create “gray areas.” When duplicating the famous double-slit experiment, we see these light, dark and in-between fringe patterns (bands). The fringe patterns become useful in measuring distances. A reiteration of the principle of interference and the resultant reinforcement, cancellation and partials between the two is shown below in Figure 3.5g below. An enlargement of the view is in Figure 3.5h. Figure 3.5g: Fringe Patterns from Interference Phenomenon (Fund. Of Dim. Metr. By Dotson – fig 13-9 p342) Page 104 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.5h: Fringe Pattern Enlargement (Fund. Of Dim. Metr. By Dotson – fig 13-10 p342) There are rays of light from the source - to the bottom (working or flat) surface of the optical flat – and back to the eye, and there are rays from the light source – to the reflective surface of the work piece – and back to the eye. Based on known factors, the air gap between the two can be determined. A simplified diagram of the concept is in figure 3.5i below. Figure 3.5i: Optical Flat Principle (Fund of Metr fig 13-11) In the above figure, as one reduces the air gap between the bottom surface of the optical flat and the working surface, the ac ray approaches the position of the bc ray. The surface must become flatter for this to happen. The de reflection ray from the top of the optical flat surface will not be seen, nor will internally-reflected light, and refractions will cancel out. The reflections merely reduce the amount of total light energy seen in the resultant fringe patterns. The air gaps between optical flat and work surface can be measured because there is a difference in the lengths of the two paths as shown below in Figure 3.5j. This figure shows two views from the side when light goes from a source, to point x where it first enters the top of the optical flat, through the optical flat with a bottom surface S, reflects off the work surface R, back through the optical flat to top surface c, and then to the observer. The optical flat surface has gotten closer to the work surface in the right-hand view, and thus there is a change in the air gap as shown. As can be seen in the right-hand view, the change in total path xac roughly doubles given the change in air gap (down and back). This changes the phase relationship as well. Also, path xbc (to and from the work surface) is always longer than xac (to and from the bottom surface of the optical flat) – unless the work surface was actually perfectly flat. Page 105 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.5j: Path Length Differences (Fund. Of Dim. Metr. By Dotson – fig 13-12 p343) The change in phase relationship is shown below in Figure 3.5j. In this view, the two reflected paths are in phase. The light paths are also shown as nearly vertically-moving sine waves – so to speak. Consider the left hump of the sine wave as the trough – or minimum amplitude or energy, and the hump to the right as the peak – or maximum amplitude or energy. The maximum phases and minimum phases are opposites. When observing the resultants bands, one would see them combined as a bright band. The band is the result of an air gap distance in this case of 1-1/2 wavelengths – which depends on the wavelength of the particular light being used. Figure 3.5k: Paths and Phases (Fund. Of Dim. Metr. By Dotson – fig 13-13 p343) In this next view, we replace the observed point at the top of the optical flat c as in Figure 3.5j, and instead call it point E. The path xbc is now XBE. If we reduce the air gap from the 1-1/2 wave separation down to zero in 1/4-wavelength increments, as shown in the sequence of views below in Figure 3.5l, the path difference is twice the wavelength, or 4 times the wavelength in terms of 1/2-wavelengths. Page 106 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.5l: Band Changes by Wave Path Differences (Fund. Of Dim. Metr. By Dotson – fig 13-14 p344) Going from the first view to the second, we have decreased the air gap from 1-1/2 wavelengths to 1-1/4 wavelengths, have reduced the total path distance from 6 to 5 half-wavelengths, and thus have reduced the path by a total of 1/2 wavelength. The new path difference causes the two reflected paths to oppose each other. The old xac path from Figure 3.5j, which is now depicted as XAE, has not changed, but the old xbc path (now XBE) has changed. The new path difference in the second sequence causes the two reflected paths to now oppose each other. A maximum energy point on one path’s sine wave is now opposite the minimum energy point on the other wave. The waves cancel all along their length, so the observer doesn’t see much light – or a dark band now appears. As we further reduce the gap by another 1/4 wavelength in each subsequent view, we continue to alternate from observed light, then dark, then light … bands. One can now see that even-numbered half-waves of path difference result in bright bands and oddnumbered half-waves result in dark bands. Also, the separation from one dark band to the next (or one light band to the next, but it is easier to use the dark bands) represents 1/2-wavelength. If we could raise the flat slowly so that we could count the fringes, each time a dark band passed, we had raised the flat by another 1/2-wavelength. If we started where the two surface were able to touch perfectly, or zero air gap, the first dark band begins at 1/4-wave separation, and then every addition of 1/2wavelength thereafter (at 3/4, 1-1/4, 1-3/4 …). Measurements with Optical Flats Page 107 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality In order to work, the external lighting at the inspection station should be monochromatic (relatively one wavelength) – such as lighting using helium gas. If it were not, refraction and dispersion of the various color wavelengths would result and create confusing fringe patterns. Recall some of the principles and problems presented in the previous section. As mentioned earlier, some optical flats are coated in order to block out any other light sources and corresponding wavelengths. One also needs the part to be measured, an optical flat, some clean paper or camel hair brush (doesn’t shed), and a rigid work surface. Because precision measurement involves cleanliness, one must remove any dust from the surface of a flat or the surface to be measured. Temperature is also an issue for precise measurement at this minute level. Parts and flats should probably both have about an hour allowed for stabilization to the ambient temperature of the room. Handling the flat with warm human hands, because of the material it is made from, will generally not affect findings. White light is made up of a combination of colored light, but not all white light uses colored light in the same combination. As shown in the last module, each color of light has its own distinct wavelength. This is repeated in a condensed view from the previous module as Figure 3.5m below. Color Wavelength (meters) Red Orange Yellow Green Blue Violet 780-623 x 10-9 (~701 nm) 622-598 x 10-9 (~610 nm) 597-578 x 10-9 (~587 nm) 577-493 x 10-9 (~535 nm) 492-456 x 10-9 (~474 nm) 455-390 x 10-9 (~423 nm) Figure 3.5m: Light Spectrum Wave Lengths (ignoring indigo) and Frequencies from Fundamentals of Dim. Metrol. by Dotson – p 164 The visible spectrum of light – what we can see with our eyes – are wavelengths from about 390 nm to 750 nm. When international standards were chosen by metrologists and scientists, krypton 86 gas was chosen through experiment. When excited electrically, it emitted light, and the steps could be repeated fairly easily. The standard was not necessarily practical under normal conditions though. For practical uses, one must consider other factors: (a) the definition of the fringe bands, (b) how easily one can see them, (c) cost, and (d) convenience. Helium gas light has proven to be very practical, all things considered. Its light is somewhere toward the middle of the spectrum (yellow light) and can be generated by a number of monochromatic lights. Examples of the light set-ups can be seen below in Figure 3.5n. Page 108 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.5n: Popular Types of Monochromatic Light Using Helium (Fund. Of Dim. Metr. By Dotson – fig 13-18 p348) In the above figure, Type A uses a reflex principle and is used for production inspection, Type B is a popular general-purpose type, and Type C is generally used in laboratories. For best results, one should view from directly above, or perpendicular to the flat. The reflex Type C above achieves that perpendicularity, and utilizes a beam-splitting mirror. The reflex light even allows the light source to be perpendicular to the measurement surface. For maximum clarity, one should have the measurement surface as close to the light source as practical. The reflex Type A set-up above allows this. In the Type B set-up, one is reasonably near perpendicularity from a viewing standpoint. Various sources recommend that one should view from a distance of about 10 times the diameter of the flat. This would dictate the overall setup one should arrange in the common Type B set-up above. Because the helium light has a wavelength of 587.6 nm (or 23.13 millionths of an inch), the 1/2-wavelength is 293.8 nm (or 11.57 millionths of an inch). In inches, this is .00001157” or 11.57 micro-inches. When viewed, the dark bands will form in intervals of this distance. Figure 3.5o below present themselves as farther away from a when viewing through shows errors that the observer moves perpendicular position the optical flat. Page 109 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.5o: Viewing Angles and Possible Errors (Fund. Of Dim. Metr. By Dotson – fig 13-20 p349) As shown in the figure, when viewed from 0o to 5o off perpendicular, there is almost no degree of error in readings. As one increases the viewing angle up to 30o or even 60o, one could have a degree of reading error of 15% to 100%! The gage blocks or other item to be measured should be wrung properly beforehand to a flat surface. Preparation for wringing is a process whereby the assumed relatively flat surface of the block is cleaned, and then rubbed against a commercially-available, flat granite or ceramic stone. It will remove small nicks or burrs without abrading the surface. The optical flat surface should also be cleaned using the camel hair brush. Because extremely flat surfaces will actually adhere to each other by air pressure between the surfaces, a gage block and optical flat can adhere to each other in much the same way gage blocks can adhere to each other. Wringing occurs when the two surfaces are overlapped in partial contact, and then slid in a centered position. One should be able to lift one object up with the other. If not, that could indicate a lack of cleanliness, one surface is in fact not extremely flat, or the surface finish is not sufficiently fine. In much the same way, when using an optical flat, rubbing one surface against the other (flat to gage block) will constitute wringing. Never leave the two in contact for an extended period of time. If an optical flat wrings to a gage block or other surface and left overnight, one might actually have to break the optical flat to remove it. If the two do not separate readily by sliding one off the other, soak both in a solvent and then use a wood block (not metal). Note also that excessive wringing can wear the surface of the optical flat. Another method of contact, and preferred by many, is to place a clean sheet of paper between the to-bemeasured surface and the optical flat. Once in place, slowly slide the paper from between the two until the two are in contact. 3.5.3 Application – First Use of Optical Flat Materials: Preferably monochromatic light source but not required, Optical flat, Cleaning solution, Camel hair brush, Piece of paper, Solid and smooth working surface, Small dowel rod, Magnifying glass Parts: Gage Block Set (with thicker and thinner blocks) Page 110 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality If one could really begin an air gap at zero, and if all phase shifts would cancel out, one would see a fringe pattern as seen Figure 1 below. However, there is almost always some degree of phase shift and it is extremely difficult to force two nearly flat objects into perfect contact. Figure 1: Theoretical Condition of Contact (upper view from Fig 13-15, Dim. Metrol. by Dotson, p 345 Clean a thicker gage block and optical flat as described earlier. Place the gage block on a solid, smooth surface with the correct surface up. Set up lighting and prepare to view as specified earlier. Now place the optical flat on top of the gage block with the marked working surface down, and then press down until fringes are seen. Do not perform any wringing. As you push against the optical flat, you should see fringes diminishing. Press down very tightly against the right edge of the gage block surface so you get intimate contact (see Figure 1 again). No matter how hard you try, you should just make the first dark fringe on the right broaden – as seen in Figure 2 below. Figure 2: Actual Condition of Contact (lower view from Fig 13-15, Dim. Metrol. by Dotson, p 345 When you now let up or completely release the pressure, air should rush back in between the parts and you should see many fringes. Many believe contact begins with a dark fringe, but you can see that it does not. Now clean a thin gage block and wring it (per the procedure above) to the working surface of the optical flat. If you now hold the flat under the light and apply pressure to the center of the thin gage block backside, it should bend. Fringes should move away from the point of pressure. The harder you push, the wider the dark fringe should spread. This seems to indicate that contact does start with a dark band again. Page 111 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Results will change if we change the method of applying pressure. Create the set-up seen in Figure 3 below. Figure 3: Set-up Change (lower view from Fig 13-16, Dim. Metrol. by Dotson, p 345 Place a smooth dowel rod onto the solid working surface. Now take the wrung gage block and optical flat, and place this with gage block on top of the rod. Apply pressure to the optical flat as shown in the figure. As you apply slight pressure, fringes should begin to depart rapidly and the remaining dark fringe should widen. As you increase the applied pressure, the dark fringe should split and move almost exactly 1/2 of a normal band width to either side of the bright area. If you now look at the bright band with a magnifying glass, you will see that it is different from an ordinary bright band. The difference is the result of the intimate contact between the centers of the two surfaces! Questions: 1.) Describe what you saw as you performed the steps. Did you see what was described? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ________________ 2.) State any conclusion for group discussion and review? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ____________________ 3.5.4 Dialog There are two methods for measuring flatness with optical flats – the air wedge method and contact method. The flat is held at a small angle relative to the measured surface under the air wedge method. Under the contact method, the flat is in full contact with the surface to be measured. Page 112 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Contact Method When surfaces are both concave and convex, and thus irregular, the contact method is considered best. After performing the basic steps mentioned in the previous section, and the two surfaces (optical flat and surface being checked) are considered in contact, press down on all sides of the optical flat until interference fringes show (pushing down on one side only tends to create an air wedge). Don’t move the optical flat around on the surface or it could scratch the surface. Assuming the light source and observation angle are done properly (nearly perpendicular), and fringes do not show, it indicates one of two things. Either the surface being checked is not of sufficient surface finish to reflect light, the work piece is in fact almost perfectly flat, or there is still sufficient dust or burrs between flat and work piece that present too much distance between the parts (much greater than wavelengths of light being used). If so, repeat the previous cleaning and preparation steps and repeat the process. There could also be a thin layer of moisture or oil causing the optical flat to actually wring too closely to the work piece surface. The optical flat should make contact at the high points of the work piece part and will actually show up as round bands. We will get into interpretation of bands later. Air Wedge Method The most common method is the air wedge method. It is actually an extremely small wedge being produced, perhaps less than one second of arc. The wedge is actually produced by making contact at one edge of the work piece surface. The other end is up at a small angle – basically on air. Technique will be discussed in more detail later. If the wedge angle is too great, one could run into a situation where the interference bands are too numerous – so as to present the appearance of there not being bands present at all. This sometimes is correctable – as will be discussed later. The concept that makes the air wedge method work is the parallel separation planes concept. Fringe bands actually form in the air separating optical flat and work piece. A visual description is shown below in Figure 3.5p. Figure 3.5p: Parallel Separation Planes Concept (Fund. Of Dim. Metr. By Dotson – fig 13-21 p349) One must imagine a set of planes that are all parallel to the working surface of the optical flat and are 1/2-wavelength apart. Intersections of the planes and the work piece create dark fringe lines. The number of fringes represents the separation between these surfaces in units of 1/2-wavelength. One must remember that the number of bands seen through an optical flat is a measure of height difference, and not of an absolute height. This is important to remember throughout the material. The following is a basic description of what is happening under the air wedge method. Page 113 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality After cleaning and wringing, further work with the wringing until a fringe pattern can be seen across the entire work piece. In this example, a gage block is the work piece. You should see something like the pattern below in Figure 3.5q. Figure 3.5q: Basic Air Wedge Method – Initial View (Fund. Of Dim. Metr. By Dotson – fig 13-22 p350) The pattern crosses the entire gage block surface. In this case, the 5 fringes indicate an air gap of five 1/2wavelengths separating the optical flat from block. Initially, one is not sure which end is the open end of the wedge (right end or left end). It is either the right or left end as these ends are the ones perpendicular to the direction of the bands seen. By applying some force to the optical flat at either end, we can find out. In Figure 3.5r below, force is applied to the left end. Figure 3.5r: Basic Air Wedge Method – Second View (Fund. Of Dim. Metr. By Dotson – fig 13-23 p350) After pressing down on the left end, there is little or no difference seen in the band pattern. This indicates that this must be the edge of contact. If we were to press against the right edge, as seen in Figure 3.5s below, we would note that the fringes have spread out and that only 3 bands now appear. Figure 3.5s: Basic Air Wedge Method – Third View (Fund. Of Dim. Metr. By Dotson – fig 13-24 p350) Because of the change that was noted, it is known now that we just pushed down on the open (wedge) end, and forced the two parts to close. Figure 3.5t below shows a side representation of what was seen originally in Figure 3.5q. Page 114 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.5t: Basic Air Wedge Method – Side View Representation of Initial View (Fund. Of Dim. Metr. By Dotson – fig 13-25 p350) One can always find the contact edge by this method. The height difference at the widest point is always equal to the number of fringes seen multiplied by the 1/2-wavelength of the light used. In this case it was helium light with a 1/2-wavelength of 11.6 micro-inches (µin.), times 5 equaling 58 µin. total. The fewer the bands – the narrower the wedge angle, and the more numerous the bands – the greater the angle. The following Figure 3.5u shows a case where there is a sharp drop-off from the basic surface. Figure 3.5u: A Surface with a Sharp Drop-off (Fund. Of Dim. Metr. By Dotson – fig 13-28 p351) In this view, the surface to the left is flat. The surface to the right has a drop-off from the main surface. This surface is in itself flat (although not flat with respect to the surface on the left) because the fringe lines seen remain straight. One can tell there is a drop-off from the main surface to the left because the change in distance between the fringe bands is evident. Interpreting Fringe Patterns Figure 3.5v below shows 9 different fringe patterns. Note that the contact edge in each case has been denoted by R. Page 115 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.5v: Fringe Patterns (Fund. Of Dim. Metr. By Dotson – fig 13-29 p352) The reference surface R in each case had been determined by the pressing method described previously. Pressing against this surface did not change the fringe pattern. View A is the basic result – bands remain straight, although the band widths vary a little. In view B, bands curve toward edge R. This surface is convex and high in the center. In C, the bands curve away from R, which indicates the surface is concave and low in the center. For the next few examples, note that R has changed with respect to the part orientation. In D, the surfaces drop off toward the outer edges because of the curvature direction toward R If the opposite end had been R, it would have shown that there was a rise at the side edges. The majority of the surface would have been indicated flat due to the straight portions of the fringe bands. In E, the surface is flat at the opposite end (straight bands) but increasingly convex toward the R edge. F shows a similar condition, but one where the surface is progressively lower toward the lower left-hand corner. Note that the bands turn toward the line of contact and get progressively further apart. In G, the surface is flat from lower right to upper left, but it is slightly concave because the bands curve very slightly away from the contact point. In H, the surface is flat in the direction that the bands run. The surface drops off toward the ends because the bands are widely spaced in the center and closer at the ends. View I shows two contact points marked R. They are high spots surrounded by lower areas. Measuring Resultant Patterns Once you know about recognizing basic patterns and what they mean, it is time to quantify – or determine the degree of flatness. Some things to note are as follows. Straight bands, evenly spaced, indicate flat surfaces, or at least straight surfaces. Curved lines indicate concave or convex contours in the surface. A comparison of the distance between the bands and curvature “heights” will indicate the degree of contour. In other words, in order to interpret the extent of band curvature from absolute straightness, it is necessary Page 116 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality to know the reference line or point from which the measurements can be expressed. Imaginary lines parallel to the line of contact are expressed and should pass through the center of bands. In other words, the amount that the bands curve with reference to the distance between them, indicates the amount of flatness error. In judging the amount of curvature, imagine a line drawn across the surface from one end of any band to the other end of that same band. If this line just touches the previous band the flatness error is 1 band. If it comes half-way between the two bands the error is 1/2 band. If the surface is out of flat two bands the line will just touch the second band, if it is out 3 bands it will just touch the third band and so on. In practice the imaginary line may be made real by aligning a piece of fine wire or thread across the face of the monochromatic lights diffusion screen with the ends of the band, or by use of a transparent straight-edge Because of air gap height difference one may encounter, even though the distance between bands might vary, the height difference from band to band is always 11.6 µin. (assuming helium light is used) or 1/2 wavelength, and is always counted from the line of contact. The extent of curvature is always measured against the distance between bands. As mentioned earlier, it is best to view through optical flats perpendicular to the surface. If the viewer is not actually able to look directly down at the object (or 90o to the surface) however, the distance between bands will not really be 11.6 µin. It will be somewhat more based on the angle viewed from. Figure 3.5w below gives adjusted values for distance between bands with helium light based on the angle viewed from. Effective Band Values Viewing Angle (degrees) Band Value (micro inches) 10 20 30 40 50 60 70 80 90 68.6 33.8 23.1 18.00 15.1 13.4 12.3 11.8 11.6 Figure 3.5w: Viewing Angle-Adjusted Band Widths (from http://www.vankeuren.com/howtomflat4.htm) Figure 3.5x below shows some basic patterns and illustrates how to measure the results seen. Note the edge of contact indicated by R again. Page 117 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Figure 3.5x: Measuring Band Patterns (Fund. Of Dim. Metr. By Dotson – fig 13-31 p353) View A shows a convex surface (high in middle) by about 1/3 of a band, or 11.6 µin. x .33 = 3.87 µin. In view B, the surface is concave (low in center) by the same amount. View C is convex by 1/2 band, or 5.8 µin. View D is also convex, but by 1 band or 11.6 µin. View E is again convex, but this time by 1-1/2 bands or 13.9 µin. View F is quite common, and shows flatness except at the edges, which drop off by 1/4 band or 2.9 µin. The surface in view G has two low troughs, while the center and edges are at the same height. The troughs are about 5/6 of a band or 9.7 µin. Most surfaces you might encounter will not be as uniform as the previous ones shown. Most change from one end to the other. In view H, the left portion is flat and then becomes increasingly convex to the right. At point (a), it is about 1/2 band (1/4 wavelength) or 5.8 µin. convex, where at point (b) it is about 1 band (1/2 wavelength) or 11.6 µin. convex. In view I, the surface is flat near the reference line but then rises at the right edge. The top right edge is about 2 bands or 23.2 µin. high where the lower right edge is about 31/2 bands or 40.6 µin. high. In view J, the surface has 2 high points with a trough between marked by line XY. There are also 4 convex bands on each side of the high points. The contact method was probably used here (irregular surface) and the trough is roughly 4-1/2 bands or 52.3 µin. low. In each example shown, we could have wrung the surfaces and the optical flat so that the fringe pattern was perpendicular to the pattern depicted. The resulting contour “map” would have been just as useful, but measurement might have more difficult. In a case like view A below in Figure 3.5y, we would have many Page 118 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality bands to count to make a measurement. This figure is actually a series of views of the same object shown previously in Figure 3.5u. The left edge is the edge of contact. Figure 3.5y: Band Spacing Dependent Upon Air Wedge (Fund. Of Dim. Metr. By Dotson – fig 13-32 p353) Again remember that the number of bands is a measure of height difference and not of absolute height. All three views of fringe patterns in the above figure show that the surface is straight along its length, but then drops off along the right end. The band spacing varies in each view. The angle of the air wedge in view B is smaller than the one in view A. The angle of the air wedge in view C is greater than the one in view A. We could calculate the amount of drop-off from any of these views. It would be easier to calculate, however, if we reoriented the pattern. That is done below in Figure 3.5z. Figure 3.5z: Reorienting the Fringe Pattern (Fund. Of Dim. Metr. By Dotson – fig 13-33 p354) View A is a repeat view of what was originally done. In view B, we have actually pressed down against what was the bottom edge in view A and thus made this edge the contact edge. The fringe pattern now appears quite different. This view actually makes it easier to determine the amount of drop-off. The spacing between bands is now greater. It is easier to compare the distance between those bands and the percent of a bandwidth of curvature for each band. In summary, if we know the contact point for any surface, the fringe pattern will show the surface conformation. If the elevation changes, it is easy to measure the change by using the bands that cross that area of the surface. We can measure the change by the amount that the bands deviate from straightness. Page 119 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality For convenience, below in Figure 3.5aa is a conversion table in metric and English units based on fractions and multiples of band widths. Figure 3.5aa: Bandwidth Conversion Table (Fund. Of Dim. Metr. By Dotson – fig 13-48 p361) 3.5.5 Application: Measuring Flatness with Optical Flats Materials: Preferably monochromatic light source but not required, Optical flat, Cleaning solution, Camel hair brush, Piece of paper, Solid and smooth working surface Parts: Gage Block Set Page 120 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality Gage Block Calibration Certificate: from Michiana Calibration Service Look over the calibration certificate for the gage block set supplied. In it, you should find a declaration for flatness when in new condition. 1. Convert the flatness figures into nanometers or micro-inches: _________________________________ 2. Select 3 gage blocks at random and record which blocks are being tested for actual flatness: ________________________ __________________________ __________________________ 3. Prepare for proper set-up of flatness inspection station with monochromatic light source, work surface, etc. 4. Clean gage blocks and optical flat properly 5. Perform flatness check of working surfaces for each block and record results in below chart. Data Chart Part 1 Surface 1 Surface. 2 Block in Spec (y/n) Part 2 Surface 1 Surface 2 Block in Spec (y/n) Part 3 Surface 1 Surface 2 Block in Spec (y/n) Questions: 1.) Were you able to accurately measure flatness for the gage blocks? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ Page 121 3.0 Angular Measurement, Thread Metrology, and Optics Measurement and Quality ______________________________________________________________________________________ ________________ 2.) Compare how close your measurements were to the specified values. Would you say that this set is still within flatness specs based on the sample? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ________________ 3.) Comment on the precision, accuracy, and repeatability of your measurements. Prepare to discuss your results with the group. ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ________________ Page 122