JC-NRLF MACHINE SHOP MATHEMATICS m THE LIST IN MEMOR1AM FLOR1AN CAJORI *-/ . MACHINE-SHOP MATHEMATICS BY GEOKGE WENT WORTH t DAVID EUGENE SMITH AND HERBERT DRUERY HARPER GINN AND COMPANY BOSTON NEW YORK CHICAGO LONDON ATLANTA DALLAS COLUMBUS SAN FRANCISCO COPYRIGHT, 1922, BY GINN AND COMPANY ENTERED AT STATIONERS' HALL ALL RIGHTS RESERVED 422.4 Tgfte gtfrenatum GINN AND COMPANY PRO- PRIETORS BOSTON U.S.A. PEEFACE Purpose of the Work. This work has been prepared to meet the needs of students who expect to become machinists, either in the special line of automobile construction or in the more general lines of the machine shop. It is therefore strictly limited in scope to the needs of those who are entering upon this kind of work, and it treats only of such topics as experience has shown are demanded by the practical machinist who is determined to advance in his vocation. Work Presupposed. The student is supposed to have covered the work laid down in the authors' " Fundamentals of Practical Mathematics," or its equivalent, and therefore to be familiar with the use of whole numbers, common fractions, decimals, per cents, proportion, and the common tables of measure as applied to practical problems. While it is not an absolute essential that the student should have mastered the slide rule or should be thoroughly acquainted with the elements of trigonometry and with the metric system, it is desirable that he should have at least a fair working knowledge of these subjects. Topics Considered. A glance at the Contents will show the topics considered, the relative amount of attention given to each, and the sequence in which they are taken up. In general it may be said that the choice of topics and the time allotted to each are conditioned by the actual needs of the student, while the sequence is based chiefly upon the question of relative diffi- culty, although due attention has been given to the dependence of one topic upon another. The first thing that is needed is a knowledge of the measuring instruments actually used in the iii iv PREFACE machine shop, since without this knowledge the rest of the work is meaningless ; the second topic relates to speeds and feeds, this being the first thing that the student meets in the use of a machine ; the third topic, tapers and taper turning, follows natu- rally, and so on throughout the book. The authors believe that they have succeeded here, as in their earlier book in this field, in eliminating nonessentials, in empha- sizing the great principles, and in presenting the matter in a new but perfectly natural form, with definite and valuable applica- tions which initiate the student into the actual work of the shop. They hope that their efforts will meet with the approval of teachers and students alike. Acknowledgment. The authors wish to express their thanks to the following manufacturers who have given permission to use the illustrations shown on the pages mentioned: South Bend Lathe Works, South Bend, Ind., p. 8 ; K. K. LeBlond Machine Tool Co., Cincinnati, Ohio, p. 62 ; Cincinnati Milling Machine Co., Cincinnati, Ohio, p. 63 ; Brown and Sharpe Manufacturing Co., Providence, K. L, pp. 70, 81, 91, 99, 105, and 115 ; Meisel Press Manufacturing Co., Boston, Mass., pp. 97, 103, and 112 ; Niles- Bement-Pond Co., New York, N.Y., p. 110; Hoeffer Manufactur- ing Co., Freeport, 111., p. 118 ; Ohio Machine Tool Co., Kenton, Ohio, p. 124 ; Lynd-Farquhar Co., Boston, Mass., p. 126 ; Garvin Machine Co., New York, N.Y., p. 128; and the Putnam Machine Works, Fitchburg, Mass., p. 132. Other well-known manufacturers supply similar machines of a high degree of precision, and they would, no doubt, have been equally willing to give permission to use their illustrations had it been requested. CONTENTS CHAPTER PAGE I. MEASURING INSTRUMENTS 1 II. SPEEDS AND FEEDS . . . . 9 III. TAPERS AND TAPER TURNING 21 IV. SCREW THREADS 35 V. INDEXING AND SPIRAL CUTTING 61 VI. GEARS . 85 VII. REVIEW PROBLEMS 117 TABLES FOR REFERENCE 137 INDEX 159 MACHINE-SHOP MATHEMATICS CHAPTER I MEASURING INSTRUMENTS Measuring Lengths. In measuring short distances we take each distance with a pair of dividers, transfer it to a steel ruler, and then read off the length. Since, however, rulers are seldom graduated beyond g 1 ^", this method is not accu- rate enough for fine work. For finer work a diagonal scale, which applies the principle of parallel lines to measuring lengths, is sometimes used by draftsmen. In the scale here shown the distance from the vertical line 1 to is 1". From the ver- tical line 1 to the point where diagonal line cuts horizontal line 8, the distance is 1.08" ; 8 6 4 2 1 / PART OF A DIAGONAL SCALE to the point where di- agonal line 5 cuts horizontal line 6, the distance is 1.56"; and so on, so that by this method we can readily measure to 0.01". For the work required of a skilled machinist, however, a higher degree of accuracy than is obtainable by either of the two methods given above is necessary. A method of obtain- ing this greater precision by the application of the principle of the screw thread is explained on page 2. 1 MEASURING INSTRUMENTS Caliper. A caliper is an instrument for measuring thick- ness or diameters. One of the most common forms is a pair of compasses with legs curved inward for measuring outside diameters and the thickness of plates. The legs may also curve outward for measuring inside diameters. Micrometer Caliper. A caliper of the type shown in this figure is called a micrometer caliper, or simply a micrometer. Reading : 0.300" (numbered graduation on E) .025" (unnumbered graduation on E) .003" (thimble reading on F) 0.328" (reading of setting) MICROMETER CALIPER A, anvil ; B, screw ; (7, frame ; D, lock nut ; E, barrel ; F, thimble ; G, ratchet stop In this instrument the principle, which is also used in other types of measuring instruments, is that of a screw with 40 threads to the inch. At each revolution, therefore, the screw moves lengthwise -fa", or 0.025". If by means of marks on the instrument we can tell when the screw has made ^ of a revolution, we shall know when it has moved lengthwise ^ of -fa", or 0.001". In the caliper shown above the thimble is attached to the screw and moves along the barrel. When the screw is against the anvil, the on the thimble coincides with the on the barrel. The barrel is graduated to -fa", the numbered lines being 0.1" apart. Each revolution of the screw opens or closes the caliper ? y, or 0.025", and therefore moves the thimble one graduation on the barrel. Each of the 25 gradu- ations on the thimble indicates of 0.025", or 0.001".- MICROMETER CALIPER 3 Directions for Reading a Micrometer. In reading a microm- eter caliper of the type shown on page 2 proceed as follows : Count each numbered graduation on the barrel as 0.100", each unnumbered graduation as 0.026", and each graduation on the thimble as 0.001". The total is the reading of the setting. MICROMETER READINGS The reading of the figure at the left is 0.125"; that of the one at the right is 0.188" The student should carefully examine the above illustration and the one given on page 2 and should verify the readings. It is only by actually using the instrument, however, that he will acquire the necessary facility in reading it. Exercises. Reading the Micrometer Caliper 1. If the thimble of a micrometer caliper is given five revolutions, how far has the caliper opened or closed ? 2. What is the diameter of a piece of stock when the read- ing of the numbered graduations on the barrel is 7 and the thimble reading is 22 ? 3. Reduce ^' f to a decimal to the nearest 0.001" and tell how you would set the caliper to have this reading. 4. How many turns must you give the thimble to measure a thickness of 4" ? o 5. If, when the screw is against the anvil, the thimble is given | of a turn, what is the reading of the micrometer? 6. How many turns must you give the thimble to have a reading of 0.375"? of 0.261"? of "? of 0.955"? 4 MEASURING INSTRUMENTS Vernier Caliper. Calipers are also made with a device known as a vernier, by the aid of which we can read measurements to 0.001". Such a caliper is illustrated in the figure below. VERNIER CALIPER A, beam ; B, sliding head ; C, clamp ; D, solid jaw ; E, sliding jaw ; F, vernier The vernier is based upon the principle that although the eye cannot easily decide upon a fractional part of a small space, it can readily tell when two lines appear to coincide. On beam A each unnumbered division is -fa", or 0.025", while on the vernier F each unnumbered division is 0.024". Hence 25 divisions on F are together equal to 24 divisions on A, and each division on F is 0.001" shorter than each division on A. If, therefore, on F is slightly to the right of a mark on A, but 1 on F coincides with a mark on A, then we see that is 0.001" to the right of the mark at its left. In the above illustration, line 2 (indicated by the arrow) on F coincides with a hue on A. The reading of the scale on A is 1.000" + 0.500" 4- 0.025", or 1.525". Adding the 0.002" from F, we have 1.527" as the reading of this setting. For the clear understanding of this work the student should have a vernier caliper in his hands, setting it for various widths of objects to be measured and reading off the measurements. VERNIER CALIPER 12345 Reading the Vernier. The detail of part of the scale of a vernier caliper is shown below and illustrates more clearly than the figure on page 4 the method of reading the vernier caliper. Since the smaller of the numbered divisions on A represent 0.1", we first see that we have 0.8" plus two unnumbered divisions of ~~~,, T ^ r^r-^i, . PART OF THE SCALE OF A 0.020" each, or 0.850" in all. VERNIER CALIPER We also see that line 14 (in- A> scale on beam; F% vernier scale dicated by the arrow) on the vernier scale coincides with a line on the scale on the beam. We therefore add 0.014" to 0.850", the result being 0.864", the reading of this setting. Micrometer Caliper with a Vernier. By combining the vernier and the micrometer we can obtain readings to 0.0001". In the figure here shown a ver- nier of ten divisions is marked on the barrel A and occupies the same space as nine divisions on the thimble B. The microm- eter reading is 0.137", accord- ing to the explanation given on page 3. Since line 5 (indicated by the straight arrow at the left on the barrel A) of the vernier coincides with a line on the thimble, we add 0.0005" to 0.137" and the reading of this setting is 0.1375". Thus, by the aid of a relatively simple device we can find the diameter of a rod or the thickness of a sheet of metal to a higher degree of accuracy than would seem possible if one did not know the ingenious mathematical principle involved in the vernier. p A MICROMETER CALIPER WITH A VERNIER j, barrel ;#, thimble 6 MEASURING INSTRUMENTS Exercises. Verniers and Micrometers 1. How would you set the vernier micrometer in the second figure on page 5 to read 0.4843"? 2. What part of 1" is indicated by the sixteenth line from the zero mark on the beam of a vernier caliper? 3. What are the readings on the beam scale and on the vernier scale of a vernier caliper for a thickness of J"? 4. How would you set a vernier micrometer to read 0.5781"? to read 0.8704"? 5. What are the readings on the barrel, the thimble, and the vernier of a ten-thousandths micrometer for a setting of 0.1093"? for a setting of 0.2789"? 6. How far does the screw of a micrometer move at each revolution of the thimble ? 7. How would you set a vernier micrometer to read ^" ? 8. A machinist takes a micrometer caliper which is fully closed and gives the thimble three revolutions. What is the opening of the caliper? 9. How would you set a ten-thousandths micrometer to read 0.1878" ? to read 0.6593" ? 10. Through what part of a revolution must the thimble be turned to move the screw of a micrometer caliper 0.004" ? to move the screw 0.017"? 11. Which, if either, gives the more accurate measurement, a vernier caliper or a micrometer caliper? State fully the reason for your answer. 12. Find the distance between the anvil and the screw of a vernier micrometer when the reading of the numbered graduations on the barrel is 4, the thimble reading is 6, and the vernier reading is 3. VERNIERS AND MICROMETERS 13. Find the distance as in Ex. 12 when the reading of the numbered graduations on the barrel is 7, that of the un- numbered graduations is 3, the thimble reading is 4, and the vernier reading is 6. 14. Find the reading of a micrometer caliper for each of the settings which are represented in the following figures: 15. In the following table let A represent the reading of the numbered graduations on the barrel of a vernier microm- eter, B the reading of the unnumbered graduations on the barrel, C the thimble reading, and D the vernier reading. Copy the table and insert the number representing the reading of each setting in the blank spaces in the column marked E\ A P> C 7) E 6 2 19 1 6 2 9 3 1 9 7 1 20- 4 5 1 7 21 1 8 3 18 6 6 3 3 3 3 3 8 MEASURING INSTRUMENTS Gaging the Depth of a Cut. The principle of the micrometer screw is used on various machines in gaging the depth of cut. The picture below shows a graduated collar which is attached to the cross-feed screw of a lathe. For convenience let x = the number of inches in the endwise movement of the feed screw for each rotation through one division on the collar ; N= the number of divisions on the collar ; L = the number of inches in the lead of the screw ; that is, the distance that the screw advances in a single turn. We then have the formula L GRADUATED COLLAR ON CROSS FEED A, cross slide ; B, graduated collar ; C', cross-feed screw Exercises. Graduated Feed Screws 1. Find the distance that a cross-slide feed screw with a lead of 0.2" moves when the collar, which has 100 divisions, is turned through five divisions. 2. The table of a milling machine is moved by a double- thread feed screw the lead of which is ". Into how many divisions should the collar of the feed screw be divided in order that the table shall advance 0.001" when the collar is turned through one division ? 3. The screw that raises the knee of a milling machine is a single-thread screw which moves the knee 0.001" when the collar, which has 100 divisions, is turned through one division. Find the lead of the screw. CHAPTER II SPEEDS AND FEEDS Cutting Speed. The cutting speed of a machine is always given in feet per minute (F. P.M.), but the expression has somewhat different meanings for different machines. In turning work on a lathe it means the number of linear feet, measured on the surface of the work, which passes the edge of the cutting tool in one minute. On a planer it means the rate in F.P.M. at which the work passes the tool. On a shaper it means the rate in F.P.M. at which the tool passes the work. On a milling machine it means the surface speed of the cutter ; that is, the speed of a point on the rim of the cutter. The cutting speed is not always the same, even on the same machine, but depends upon the following conditions: 1. The kind and quality of the material. 2. The kind and quality of the cutting tool. 3. The depth of the cut. 4. The feed of the tool, that is, the distance traveled side- ways by the tool in one revolution of the work on a lathe, or a similar distance in other types of machines. 5. The lubrication. Thus, on a milling machine the cutting speed may be 30 F.P.M. for steel, 50 F.P.M. for cast iron, and 90 F.P.M. for brass. On such a machine a heavy flow of oil is necessary on the cutter when milling steel, and a failure to provide for it not only changes the speed but materially affects the life of the cutter. 9 10 SPEEDS AND FEEDS Cutting Speeds of Lathes. The following table gives the cutting speed in F.P.M. for roughing each of the materials specified with lathe tools of carbon steel or of high-speed steel : MATERIAL CARBON STEEL HIGH-SPEED STEEL Annealed tool steel . 20 45 Machine steel . . . 30 65 Wrought iron . . . Cast iron ..... Brass 30 40 90 65 90 190 On finishing cuts the speed is 50 % greater than it is in roughing. Letting C be the cutting speed in F.P.M., R the number of R. P. M. of the work, D the diameter of the work in inches, and taking 3 as the value of TT, we have these formulas : 12 " TTD The factor 12 is necessary for reducing inches to feet. Exercises. Cutting Speeds of Lathes 1. If a tool of high-speed steel is used, at how many R. P. M. should a cast-iron pulley 18" in diameter be roughed? Use the formula for R, cancel, and give the result to the nearest unit. 2. Find the speed of the finishing cut in boring the hub of a cast-iron pulley to a diameter of Zf~' with a carbon- steel tool; with a high-speed steel tool. 3. Find the speed at which a lathe should run to rough a brass bushing 1|" in diameter with a high-speed steel tool. 4. Some machine-steel axles are roughed with a carbon-steel tool in an axle lathe which has a fixed speed of 75 R.P.M. What is the diameter of each axle ? PLANERS 11 Cutting Speeds of Planers. Planers are usually so belted as to give only one cutting speed. This speed is usually about 25 F.P.M. and is used for brass, cast iron, and steel. Only the direct stroke is a cutting stroke, the return stroke being more rapid, usually two or three times as rapid as the cutting stroke. It is customary to say that the return stroke is then 2 to 1 or 3 to 1 ; and to speak of the planer as a 2-to-l, or a 3-to-l, planer. In setting up the work on a planer a distance of V is always allowed at each end of the work for the tool to clear. Exercises. Cutting Speeds of Planers 1. How long will it take to make one cut the length of a cast-iron block 14' 10" long, the cutting speed of the planer being 25 F.P.M. and the return stroke being 3 to 1 ? Allowing V at each end of the work for the tool to clear, the total forward movement is 15'. Hence the forward movement will take ^, or |, of a minute. How long will it take to make the cut? 2. A planer to be used for planing cast iron is belted to give a cutting speed of 45 F.P.M. If the return speed is 90 F.P.M., how many strokes per minute will the planer make when the length of the stroke is 2' 6"? when it is 6' ? when it is 8' 9" ? when it is 12' 6" ? 3. If a 2-to-l planer makes six strokes per minute and the length of each stroke is 3' 4", what is the cutting speed ? 4. How long will it take to make 36 cuts on an iron cast- ing 43" long, the cutting speed ,of the planer being 45 F. P. M. and the return stroke being 2 to 1 ? 5. On a casting 1' 7 fr long a 3-to-l planer is making 12 strokes per minute. What is the cutting speed ? 6. In Ex. 5 how many strokes per minute would be re- quired to have the same cutting speed on a 2-to-l planer? 12 SPEEDS AND FEEDS Cutting Speeds of Milling Machines. The table below gives the cutting speeds in F.P.M. for milling each of the following materials on a milling machine with carbon-steel cutters : Annealed tool steel Machine steel 30 40 Cast iron Brass 45 90 For high-speed steel cutters the speeds are 100% higher. In finding the cutting speeds of milling cutters we use the formulas given on page 10, letting D be the diameter of the cutter in inches instead of the diameter of the work. Exercises. Cutting Speeds of Milling Machines 1. Find the cutting speed of a side-facing cutter 6" in diameter running at 30 R.P.M. Use the formula for C given on page 10, and give the result to the nearest unit. Use cancellation whenever possible. 2. Find the cutting speed of a milling cutter 1J" in diam- eter running at 85 R.P.M. 3. Find the number of R.P.M. for milling a brass casting with a high-speed steel cutter 4" in diameter. 4. Find the cutting speed of a milling cutter 2f" in diameter running at 55 R.P.M. 5. How many R.P.M. should a carbon-steel cutter 3^" in diameter make in milling a square on a machine-steel shaft ? 6. Babbitt metal can be milled at a cutting speed of 120 F.P.M. Find the number of R.P.M. for a 41-inch cutter when milling a piece of this metal. 7. Using a carbon-steel cutter 5" in diameter, find the number of R.P.M. for milling a cast-iron lathe bed. DRILLS 13 Speeds of Drills. In the case of drills the number of R.P.M. for the drill is found by dividing a certain number called a constant, because it is always the same for the given material, by the diameter D of the drill. For example : For machine steel, R. P. M. = 125 For cast iron, R. P. M. = For brass, R.P.M. =--- In these cases the constants are 100, 125, and 225 respectively. The depth of cut for small drills is usually from 0.002" to 0.005" for each revolution, while for large drills it is from 0.005" to 0.020". Exercises. Speeds of Drills 1. Find the number of R.P.M. for a J-inch drill when drilling a cast-iron lathe bed. 2. How fast should you run a drill J^" in diameter when drilling brass bearings ? 3. When centering a machine-steel shaft with a ^-inch drill, how fast should the drill rotate ? 4. If a lathe has a fixed speed of 250 R.P.M., what is the largest diameter of drill that should be used on cast iron ? 5. Find the number of R.P.M. for a #30 drill when drilling machine steel. f A #30 drill has a diameter of 0.1285". 6. Find the number of R.P.M. for a l T 3 g-inch drill when drilling cast-iron surface plates. 7. Find to the nearest g\" the largest diameter of drill that should be used in drilling brass at 350 R.P.M. 14 SPEEDS AND FEEDS Exercises. Review 1. Find the number of R.P.M. of a lathe for roughing the outside of the brass cylinder, which is shown in the blueprint on page 15, with a tool of carbon steel. 2. Find the speed of the finishing cut in boring the hole in the same cylinder with a high-speed tool. 3. Find the number of R.P.M. for drilling the center hole in the spur-gear blank. 4. Using a carbon-steel tool, find the number of R.P.M. for roughing the face R of the spur-gear blank. 5. If the spur-gear blank in Ex. 4 were made of machine steel instead of cast iron and if the tool were of high-speed steel, should the speed of the lathe be increased or should it be decreased when making the cut? How much should be the increase or decrease? 6. What should be the number of R.P.M. for drilling the hole in the machine-steel wheel shown in the blueprint ? 7. Find the number of R.P.M. for a finishing cut over the outside of the steel wheel with a tool of carbon steel. 8. Find the number of R.P.M. for drilling the l^g-inch hole in the cast-iron piston. 9. Using a high-speed tool, find the. number of R.P.M. for roughing the outside of the piston. 10. Using a carbon-steel tool, find the number of R.P.M. for finishing the piston-ring grooves shown in the blueprint. 11. Find the number of R.P.M. for milling a brass casting with a high-speed cutter 7J" in diameter. 12. If a drill press is belted to make 365 R.P.M., find to the nearest fa" the largest diameter of drill that can be used when drilling cast iron. REVIEW EXERCISES 15 16 SPEEDS AND FEEDS Cutting Speed of Shapers. There are two leading kinds of shapers, namely, geared shapers and crank shapers. The cutting speed of a geared shaper is found in the same way as that of a planer, as explained on page 11. In the case of a crank shaper, however, the number of strokes per minute depends upon the speed of the cone pulley which drives the shaper. In this kind of shaper the return stroke has about twice the speed of the forward stroke ; that is, it is said to be about 2 to 1. For example, if a crank shaper is making 50 R. P. M., if the length of the stroke is I/, and if the return stroke is twice as rapid as the forward stroke, what is the cutting speed of the shaper ? Since the return stroke is twice as fast as the forward one, it takes half as long to make it. Hence the forward stroke takes and the re- turn stroke takes ^ of the time required for one revolution. In 1 min. there are 50 forward strokes of V each and 50 return strokes of the same length, the 50 forward strokes taking of a minute and the 50 return strokes taking ^ of a minute. Hence the cutting speed is 50' in min., or f of 50' in 1 min. That is, the cutting speed is 75 F.P.M. Exercises. Cutting Speeds of Shapers 1. The ram of a crank shaper makes 85 strokes per minute, and the length of the cut is 2". Find the cutting speed. In problems dealing with shapers take the ratio of speeds as 2 to 1, as given above, unless otherwise stated. 2. A crank shaper is cutting at the rate of 40 F.P.M. on a 14-inch cut. Find the number of R.P.M. 3. If a crank shaper makes 18 R.P.'M. and the length of the cut is 9", what is the cutting speed ? 4. If a crank shaper is cutting a 4-inch casting at the rate of 50 F.P.M., what is the number of R.P.M.? 5. Solve Ex. 4 for a cutting speed of 40 F.P.M. FEEDS 17 Expression of Feeds. Feeds can be expressed in several different ways, the following being the most common: 1. By the number of revolutions which the work makes while the tool is advancing 1". That is, we may speak of a feed as 75 revolutions to the inch. 2. By the number of inches that the tool advances along the work at each revolution or stroke. In the above case we may express the feed as J^" per revolution. Standard Feeds. The following are certain standard feeds in common use, each being expressed in inches per revolution : MATERIAL LATHE PLANER MILLING MACHINE DRILLS Annealed tool steel A to A V<1 0.002 to 0.015 0.002 to 0.015 Machine steel or wrought iron AM VtoJ 0.005 to 0.025 0.002 to 0.015 Cast iron A to A iVtoi 0.005 to 0.075 0.005 to 0.020 Brass A to A A toj 0.002 to 0.125 0.005 to 0.020 Exercises. Feeds 1. A milling cutter revolves 85 times while the table is moving V 1 . What is the feed in inches per revolution? 2. In 1 min. a lathe tool moves 1J" while the work makes 250 revolutions. What is the feed in revolutions per inch? 3. A shaper makes 15 strokes while the table moves 1J". What is the feed in inches per stroke ? 4. A drill making 244 R.P.M. has a feed of 0.008". How long will it take it to drill through a plate 2J-" thick ? 18 SPEEDS AND FEEDS Exercises. Review 1. The bench block shown in the blueprint on page 19 is made of cast iron and fy 1 ! was left on the surfaces marked / for finishing to the dimensions given. If a roughing cut is taken with a 1-inch feed and a finishing cut with a ^-inch feed, how long will it take to plane the top of the casting on a 3-to-l planer with a cutting speed of 25 F.P.M.? In such problems find the results to minutes, calling any fraction a whole minute, unless otherwise directed. 2. The die-shoe dimensions given in the blueprint are for the rough machine-steel casting. Taking one cut with a 0.015-inch feed and using a carbon-steel side-milling cutter having a 3-inch face and a diameter of 4", how long will it take to machine the entire surface of the shoe ? It will require six cuts on the top of the shoe, five on the bottom, one at each end, and one on each long side. 3. The multiple-spindle drill table is made of cast iron. Find the time required to plane the top on a 2-to-l planer with a cutting speed of 25 F. P. M., giving it one roughing cut with a ^g-inch feed and one finishing cut with a j^-inch feed. 4. Using a 2-to-l planer with a cutting speed of 25 F.P.M. and a ^g-inch feed, how long will it take to make one cut over the two surfaces marked / on the machine-steel angle guide ? 5. How many R.P.M. should be made by a carbon-steel T-slot cutter 1-J." in diameter when milling the T-slots in the cast-iron milling-machine table ? 6. What should be the speed of a 4-inch high-speed cutter when milling the 4-inch groove in the milling-machine table ? 7. Calculate the time required to take a single cut over the top of the milling-machine table, using a ^V-kich feed on a 3-to-l planer with a cutting speed of 25 F.P.M. REVIEW EXERCISES 19 20 SPEEDS AND FEEDS 8. Find the time required to drill through a cast-iron plate l T y thick with a {-inch drill and a feed of 0.008". In this case give the result to the nearest second. 9. A machine-steel cylinder 4' 9" long is to be turned in a lathe to a diameter of 6|". Using a ^-inch feed and a high- speed tool, how long will it take to make the roughing cut ? 10. In Ex. 9, using a ^-inch feed, how long will it take to make the finishing cut? 11. Using a carbon-steel cutter, how many R.P.M. should a y 9 g-inch end mill make while cutting the slots in a cast-iron milling-machine table ? 12. Using a J^-inch feed and a high-speed tool, how long will it take for a roughing cut over the face of a cast-iron pulley which has a diameter of 28" and a face 15" wide ? 13. If a 2-to-l planer makes eight strokes of 5' 6" per minute, what is the cutting speed of the planer ? 14. Some machine-steel axles 9' 2" long are turned down to 3 1" in diameter. If two cuts are made with a high-speed tool, the roughing cut with a J^-inch feed and the finishing cut with a ^L-inch feed, how long. should it take to finish 75 axles? First find the time to the next full minute for each axle. 15. How many R.P.M. must a 2-to-l crank shaper make to attain a cutting speed of 35 F.P.M. on a 7J-inch casting? 16. How much time is needed to make a single cut with a J^-inch feed over one side of an iron casting 22" wide and 8' 4" long on a planer whose forward speed is 35 F.P.M. and whose return speed is 60 F.P.M.? 17. Find the number of R.P.M. for drilling a cast-iron block |" thick with a drill J" in diameter. Find the feed necessary to drill through the block in 25 sec. CHAPTER III TAPERS AND TAPER TURNING Taper. The difference in diameter of a piece for a unit of length is called the taper of the piece. The piece may be round like a lathe center or flat like a taper gib. The taper is usually stated by giving the difference in diameter in inches for a foot of length, as, for example, \" per foot. Common Tapers. The tapers most frequently used are: Morse : approximately 0.625" per foot (all sizes) Brown and Sharpe: 0.500" per foot (all except #10) Jarno : 0.600" per foot (all sizes) Symbols. The following symbols or abbreviations are used in connection with tapers: T.P.I. = taper per inch D = larger diameter T.P.F. = taper per foot d = smaller diameter T.P.L. = taper in any length 1 = length of the taper All the measurements are always stated in inches. Formulas. The following formulas may be referred to in solving the exercises on tapers: T.P.F. //XT. P. F T.P.I. = _ D 12(D-d) //XT.P.FA V 12 / T.P.F. = ^- d 7xT.P.F. 12 (D-d) T .P. L . = ___ '=TRFT 21 22 TAPERS AND TAPER TURNING Exercises. Tapers 1. Find the T.P.F. of the taper collar shown in the blue- print on page 23. 2. If the taper of the end mill is 0.625" per foot, what is the length of the tapered part ? 3. If the T.P.F. of the milling-machine arbor is 0.500", find the diameter of the larger end of the taper. 4. The T.P.F. of the lathe center is T 9 g". Find the length A. 5. Find the T.P.F. of the flat drill shown in the blueprint. 6. Find the T.P.F. of a boring bar which is tapered 16" of its length to diameters of 1|" and J" respectively. 7. The shank of a drill is 31" long, D= 0.421", and d = 0.375". Find the T.P.F. 8. A drill collet has a T.P.F. of 0.602" and d = 4". Find o the diameter at a point 3|" from the smaller end. 9. The following formulas, in which N is the number of the taper in the table, D is the diameter of the larger end in inches, d is the diameter of the smaller end in inches, and I is the length of the taper in inches, are used in a J arno taper : Copy the following table and fill the columns, using decimals N D d I N /; d / N j) XFTArr a / TAPER 2. By using the taper attachment shown below, which causes the tool to feed transversely at the same time that it TAPER ATTACHMENT The graduations on the ends E, E are in T.P.F., and C is attached to the cross slide of the lathe. The graduations at one end are usually in tenths and at the other end in eighths feeds longitudinally, thus turning a taper. This attachment can be used for turning either ex- ternal or internal tapers. In using a taper attachment neither the tail- stock offset nor the distance between centers need be considered. 3. By using the compound rest, one form of which is here shown. Since the tool can be set at any COMPOUND REST The base G is graduated in degrees desired angle, very steep tapers can be cut with the compound METHODS OF GRADUATING THE BASE OF A COMPOUND REST The line A-B represents the center line of the lathe rest. The base of the rest is usually graduated in degrees in one of the four ways shown in the figure above. The method of setting the compound rest is described on page 28. 26 TAPERS AND TAPER TURNING Exercises. Offset Method 1. Find the T.P.F. of the drill-holder handle shown in the blueprint on page 27. 2. In Ex. 1 how far should the tailstock center be offset in order to turn the handle to the required taper ? 3. The crowned pulley is to be turned on a 10-inch arbor. What distance should the tailstock center be offset? 4. Find the T.P.F. of the crowned pulley. 5. If the thimble is to be turned on a 6J-inch mandrel, find the T.P.F. and the distance which the tailstock center should be offset in turning the taper. 6. In turning the taper reamer, what should be the offset of the tailstock center? What is the T.P.F.? 7. In turning the taper spindle, what should be the offset of the tailstock center? 8. A taper 15" long with diameters of 1.234" and 0.984" respectively is to be turned on a shaft 17" long. Find the offset for the tailstock center. 9. Copy the following table of measurements of Morse stand- ard taper reamers and supply the missing numbers : No. D d I T.P.F. 1 2 0.369" 0.741 0.252" 0.369 2 1 " 2i| 0.600" 0.602 3 0.979 0.778 0.602 4 1.280 1.020 0.623 5 1.790 1.475 6 6 2.559 2.116 8J 7 2.750 12 0.625 OFFSET METHOD 27 28 TAPERS AND TAPER TURNING Angles. A steep taper is usually referred to as an angle, and in designating the angle we generally give either the included angle or the angle with the center axis. In the figure below the angle DAC is the included angle and the angle a is the angle with the center axis. The following formula is used in finding a, the angle with the center axis : D-d --- The included angle is found by multiplying a by 2. For example, if D = O.S75", d = 0.250", and I = 1", we have tana = ^ 7 = |-=1 = J x x = 1 = 0.2500. From the table on page 146 we find that, to the nearest minute, 0.2500 = tan 14 2'. Therefore a, the angle with the center axis, is 14 2', and 2 a, the included angle, is 28 4'. If the taper per foot is given, the above formula becomes T.P.F tan a = 24 Setting the Compound Rest. To turn a given angle the compound rest shown on page 25 is swiveled from its zero position, which is perpendicular to the center line of the lathe, so that the axis of the feed screw of the rest shall be parallel to the side of the angle. Thus, to turn angle a in this figure the rest is swiveled so that the graduated base D is set at the complement of a ; that is, at 90 30, or 60. To turn angle >, which is measured from the perpendicular to the center axis, the rest is set direct ; that is, at 60. COMPOUND REST 29 Exercises. Use of the Compound Rest Find the included angles for the following tapers per foot : 1. 0.500". 2. 0.602". 3. T 9 g ". 4. 0.750". 5. A drill collet is to be bored taper by the aid of a compound rest. If it is required that D 1 Jg ", d = 1|", and I = 4J", find the angle made with the center axis. Find the angles made with the center axis in these tapers : 6T) Ql" /7 11" 7 1 H T) 91 It fl 11^7 Q" . JJ t>2 , a ^ , t i . o. JL* = z> , a = ig- , 6 = o . 10. In turning the tapers A and B on the taper bearing shown in this figure, at what angles should the compound rest be set? B First find the angle which _ each taper makes with the center axis of the bearing. 11. Find the T. P.F. of taper A on the bevel-gear blank here shown. At what angle should the L, 1 1// J compound rest be set to cut the taper? 12. If in a taper socket D = 0.435", d = 0.300", and I = 2f ", find the angle made with the axis. 13. If in the taper shank of a drill D = 0.475", d= 0.388", and Z=lf", find the included angle. 14. A counter bore has a taper shank in which D = 0.416", d = 0.312", and Z = lf". Find the included angle. 15. A bevel-gear blank is to be turned to an included angle of 134. Find the T.P.F. of the blank and the angle at which to set the compound rest. 53" "T 3" 1 30 TAPERS AND TAPER TURNING Exercises. Review 1. The center punch shown in the blueprint on page 31 is to be turned in a chuck with the aid of a compound rest. Find the angle which each taper makes with the axis. 2. The rotary oiler shown in the blueprint is used to oil the ways of a planer. Find the included angle. 3. In the drill socket shown in the blueprint find the T.P.F. and the angle made with the axis. 4. In the speed-lathe center shown in the blueprint find the T.P.F. and the included angle. 5. A milling-machine arbor 18" long is to be tapered 4" on one end to 0.500" per foot. Find the offset of the tail- stock center for turning the required taper. 6. A reamer has a T.P.F. of 0.625" and its diameters are 0.500" and 0.750" respectively. Find I 7. If the maximum offset of the tailstock center of a lathe is iy, what is the greatest T.P.F. that can be cut on a shaft 6" long ? on a shaft 2' 3" long ? on a shaft 3' 6" long ? 8. A taper bushing has diameters of 2|" and 1|", respec- tively, and a T.P.F. of 0.500". Find I 9. In the taper shank of a cutter which has a T.P.F. of 0.600", d = If" and I = 3|". Find 2). In each of the following find the angle made with the axis : 10. D = 0.354", d = 0.279", and 1= 3f". 11. D = 1.125", d= 0.900", and Z = 4i". 12. /> = 1.231", d = 1.020", and Z = 4 T y. 13. I>= 2.494", d= 2.116", and l = l\". 14. D= 1.2888", d = 1.0446", and /=5jJ". 15. D = 3.976", d = 2.165", and I = 3|". REVIEW EXERCISES 31 32 TAPERS AND TAPER TURNING Measuring Tapers with a Sine Bar. An instrument known as a sine bar is often used to measure the angle of a taper. MEASURING TAPERS WITH A SINE BAR P, scraped surface plate ; 7?, R, plugs ; S, hardened-steel sine bar ; T, taper plug gage ; U, straight edge ; V, vernier height gage The taper to be measured is placed on the straight edge U, which is parallel to the surface plate P, and the sine bar S, which has two plugs .#, R set 10" apart, is clamped along the taper. Then r, the difference in height in inches between the plugs, is found by means of the height gage F. Letting A be the included angle, we have the following formulas: sinA = 10 r = 10 sin A For example, in the above figure r = 0.525", and we have v r 9'i sin .4 = = ^- = 0.0525 ; whence A = 3 V. Therefore the included angle of the taper plug gage is 3 1'. Testing Tapers. To test a taper for a given angle, the difference in height r of the plugs is found from the second formula, and bar S is set to this distance by means of the height gage. The taper is then tested between bars S and U. For example, what should be the difference in height of the plugs for testing a taper which is to have an included angle of 26 30'? We have r = 10 sin A = 10 x 0.4462 = 4.462. Hence the difference in height of the plugs should be 4.462". MEASURING TAPERS 33 Measuring Tapers with Disks. The angle of a taper may also be measured by means of two disks of unequal diameters. 1 MEASURING TAPERS WITH DISKS 7?, J?, hardened-steel edges ; D, d, disks of different diameters ; I, distance between centers of disks The disks are placed as shown above, and the straight edges R, R, which are made of hardened steel and carefully ground, are adjusted so that the tangent lines form the taper. Taking a as the angle with the center axis, D as the larger diameter, d as the smaller diameter, and I as the dis- tance between the centers, as shown in the figure below, we have sin a = whence sin a = I D-d 21 Angle a can then be found from a table of sines, and from it we can find 2 a, the included angle of the taper. Furthermore, from the formula for sin a we have so that, given D, d, and the angle with the axis, we can find L 34 TAPERS AND TAPER TURNING Exercises. Measuring Tapers Find the difference in height of the plugs for setting a sine bar to test each of the following tapers per foot : 1. 0.600". 3. 0.623". 5. f". 7. 1^". 9. 2". 2. 0.602". 4. 0.630". 6. ". 8. If". 10. 2|". 11. If the disk diameters are 1" and 2" respectively and if Z, the center distance, is 4", what is the included angle of the taper? 12. If the disk diameters are 1" and 1.5" respectively and if Z = 4", what is the T.P.F.? the included angle? the angle formed with the axis ? 13. If the taper is 0.5" per foot and the disk diameters are 0.75" and 1" respectively, what is I? 14. The figure and table below show certain dimensions of Morse standard tapers, all the measurements being given in inches. Copy the table and fill in the blank columns marked Z>, c?, and " Angles with Axis." T.P.F. OF OUTER HOLE T.P.F. OF INNER HOLE A B c E D d ANGLES WITH Axis 0.602 0.600 31 2J_ 0.938 0.475 .623 .630 .626 .602 .602 .623 H 5 I sf 1.231 1.748 2.494 0.700 0.938 1.231 .626 .630 n 5 i 2.494 1.748 CHAPTER IV SCREW THREADS Thread. When a uniform spiral groove is cut around a cylindric bar, the material left by the cutting tool forms a projection which is called a thread, or a screw thread. The diameter of the cylindric bar on which the screw thread is cut is called the outside diameter of the thread. 1*1 LONGITUDINAL SECTION OF A SCREW, SHOWING A CROSS SECTION OF THE THREADS Do, outside diameter; D r , root diameter; d, depth; p, pitch The perpendicular distance from the top of the groove to the bottom is called the depth of the thread, and twice this depth is called the double depth of the thread. The bottom of the groove is called the root of the thread, and the diameter of the screw measured at the bottom of the groove is called the root diameter. The distance from the center of one thread to the center of the next thread is called the pitch of the screw thread. The advance that the screw makes in one revolution is called the lead of the screw. 35 36 SCREW THREADS Multiple Thread. A thread formed by cutting two or more uniform spiral grooves around a bar is called a multiple thread. A multiple thread is said to be a double thread, a triple thread, a quadruple -thread, and so on, according as the number of grooves thus cut is two, three, four, and so on. The pitch and the lead of a single thread are equal, but the lead of a double thread is twice the pitch. Similarly, in a triple thread the lead is three times the pitch, in a quadruple thread the lead is four times the pitch, and so on. MULTIPLE THREADS The left-hand figure shows a double thread and a single thread of the same lead, and the right-hand figure shows a triple thread and a single thread of the same lead. In each figure I shows the lead of each thread, p shows the pitch, and D r shows the root diameter Multiple threads are often used where machine parts are to be moved by the action of a screw. If, for example, a feed screw is desired which shall move the table of a machine \" per revolution, it would require a screw of large diameter in order to have a |-inch pitch and fulfill all the requisites for strength of thread. To keep the diameter of the screw to an appropriate size a double thread of ^-inch pitch might be used, and the desired lead of \" would be thus obtained. Hand of a Screw. A screw that advances when turned clockwise is called a right-hand screw. A screw that advances when turned counterclockwise is called a left-hand screw. The right-hand screw is the common form, and, unless otherwise stated, a right-hand thread is always understood in speaking of a screw. MULTIPLE THREAD 37 Exercises. Threads 1. A screw has 18 single threads to the inch. Find the pitch and the lead of the screw. 2. In the single-thread screw shown on page 35 suppose that the outside diameter is lyg" and that the depth of the thread is -f^ ff . Find the root diameter. It is evident from the figure that the root diameter is equal to the outside diameter minus the double depth of the thread. 3. In the double-thread screw shown on page 36 suppose that there are 16 threads to the inch ; that is, 8 double threads. Find the pitch and the lead of the screw. 4. If the root diameter of a double-thread screw is 1-|" and the depth of the thread is ^", find the outside diam- eter of the screw. 5. In the triple-thread screw shown on page 36 suppose that the pitch is -Jg-". What is the lead of the screw ? 6. The cross slide on a lathe is moved by a screw that has 16 threads per inch, and one revolution of the screw moves the slide \". How would you designate the thread? 7. A lead screw has a triple thread with 18 threads to the inch. Find the pitch and the lead. 8. When the feed screw on a milling machine makes one revolution it moves the table ^". If the screw has 8 threads to the inch, how would you designate the thread ? 9. How many revolutions must be made by a triple-thread feed screw of i-inch pitch in order to move the table which it controls a distance of 11" ? 10. When the screw that raises the knee on a milling machine makes one revolution it moves the knee 0.200". If the screw has 5 threads to the inch, how should the thread be designated ? SCREW THREADS Sharp V-Thread. The thread of which the cross section is an equilateral triangle is called a sharp V-thread. Since the cross section of this thread is an equilateral triangle, all the angles are 60. The depth of the thread is the altitude of the tri- angle, and the pitch is equal to the base, as is evident from the figure at the right. In a sharp V-thread the pitch and * P itch '> d > de P th the depth of the thread are found by the following formulas : SHARP Pitch = Depth = 1" number of threads to 1" 0.8660" number of threads to 1" = 0.8660 x pitch The table below shows the standard number of V-threads ( T) per inch for screws of the following diameters (^ ) : D a T Do T Do r D. 7' r 20 \l" 10 ir 5 2| /x 4 A 18 7 9 1* 4 2 3 H 1 16 i$ 9 2 41 3^ 'U A 14 1 8 2J 4^ 3 4 8* i 12 H 7 2* 4 1 8| 3i A 12 H 7 2f 4^ 8 4 8 i 1 11 it 6 2 4 3f 8} H 11 1} 6 2 5' 4 8j 3 1 10 if 5 2^ 4 4 3 Unless otherwise stated the diameter of a screw, as D in the table above, means the outside diameter. Since the sharp V-thread offers no advantage over the United States Standard thread described on page 40 and the sharp edges are easily injured, its manufacture is being discontinued. SHAKP V-THREADS 39 Exercises. Sharp V -Threads 1. Find the depth of a sharp V-thread with a pitch of J". In the following problems use the table and the formulas on page 38. Carry all computations involving decimals to four decimal places, but in the final result discard the fourth place, giving the dimension correct to the nearest 0.001". 2. Find the double depth of a sharp V-thread with 9 threads to the inch ; with 11 threads to the inch. 3. Find the depth of a sharp V-thread with 3J threads to the inch; with 4J threads to the inch. 4. Find the tap-drill size of a sharp V-thread tap which has an outside diameter of J". A tap is a tool for cutting internal threads, and the tap-drill size is the size of the drill for boring the hole to be tapped. In actual practice, to avoid breaking taps, the tap-drill size for ordinary work is generally taken a little larger than the root diameter of the thread on the tap, but in all problems in this book the tap^drill size is to be taken as equal to the root diameter. 5. A screw having 5 sharp V-threads to the inch has a root diameter of 1.4036". Find the outside diameter. 6. Find the tap-drill size for a sharp V-thread tap which has an outside diameter of 1|-". 7. If the root diameter of a sharp V-thread screw is 4.3072" and the outside diameter is 5", what is the depth of the thread and the number of threads per inch ? Find the double depth of a sharp V-thread, given that the pitch of each thread is as follows : s. T y. 9. ^". 10. i". 11. T y. 12. i". 13. Find the tap-drill size for a tap If" in diameter with 6 double sharp V-threads to the inch. 40 SCREW THREADS United States Standard Thread. A sharp V-thread flatted an equal amount at the top and bottom, as here shown, is known as the United States Stand- ard thread and is commonly des- ignated as the U. S. S. thread. S.A.E. Thread. A thread of the same shape as the U. S. S. thread, but which differs only in the number of threads per inch, bears the name of the Society of Automobile Engineers and is known as the S.A.E. thread. In each of these threads the following formulas are used: UNITED STATES STANDARD THREAD d, depth; /, flat; p, pitch Pitch = Depth = number of threads to 1" 0.6495" = 0. 6495 x pitch number of threads to 1" Flat (top and bottom) = | x pitch The table below shows standard U.S.S. and S.A.E. threads: U.S.S. THREAD S.A.E. THREAD Do T D T D T D T Do r r 20 V 8 2r 4| I" 28 i" ~s 14 A 18 11 7 2* 4 A 24 I 14 1 16 U 7 2| 4 3 f 24 U 12 T 7 TT 14 If 6 3 3^ T 7 * 20 1J 12 i 13 U 6 8J 8j i 20 ii 12 T 9 * 12 If 6i 3^ ?i T 9 * 18 U 12 i 11 *i 5 8| 3 1 18 . . i 10 if 5 4 3 U 16 i 9 2 n H 27 i i 16 In this table D stands for the diameter of the screw and T for the number of threads per inch. U.S.S. AND S.A.E. THREADS 41 Exercises. U.S.S. and S.A.E. Threads 1. Find from the formula on page 40 the depth of a U.S.S. thread with 8 threads to the inch. The U. S. S. thread was devised by William Sellers in 1869. It is also known as the Sellers thread and as the Franklin Institute thread. 2. Find the tap-drill size for a l|-inch S.A.E. thread. 3. Find the double depth of a U.S.S. thread of Jy-inch pitch; of y 1 ^ -inch pitch. 4. If the root diameter of a U.S.S. thread is 4.2551" and there are 2| threads to the inch, what is the outside diameter of the screw? 5. What is the root diameter of an S.A.E. thread ^" in diameter? |" in diameter? 6. Find the tap-drill size of a 31-inch U.S.S. thread. 7. If the root diameter of an S.A.E. thread of -J^-inch pitch is 0.9072", what is the outside diameter? Find the width of the point of a thread tool for cutting each of the following numbers of U. 8. S. threads to the inch : 8. 3. 9. 4|, 10. 7. 11. 13. 12. 18. 13. 20. The width of the point of the cutting tool is the same as the width of the flat at the bottom of the groove. 14. If the root diameter of a U.S.S. thread is 1.7113" and the outside diameter is 2", what is the depth of the thread ? 15. Find to the nearest ^" the proper size of drill for boring a hole which is to be tapped with a full 11-inch U.S.S. thread; with a full f-inch S.A.E. thread. A full thread is a thread cut to the depth given by the formula. For ordinary work a tap drill which will give about 75% of a full thread is generally used, and tables of tap drills often list the drill sizes in sixty- fourths of an inch which will give such a thread. 42 SCREW THREADS Whitworth Standard Thread. The thread of which the cross section is of the form here illustrated is called the Whitworth Standard thread. This thread was devised by Sir Joseph Whitworth in 1841 and, with slight modifications, is still the British standard. In such a thread the pitch, the depth, and the. radius are found by the following formulas : 1" WHITWORTH STANDARD THREAD (7, depth; p, pitch; r, radius of curvature Pitch = Depth = Radius = number of threads to 1" 0.6403" number of threads to 1" 0.1373" number of threads to 1" = 0.6403 x pitch = 0.1373 x pitch The following table shows the standard number of threads per inch for Whitworth threads of the following diameters: Do T DO T D 9 T A, r A" 60 A" 12 u" 7 111" U 48 f 11 1A 7 2 *J i 40 H 11 if 6 2i *1 A 32 3 ? 10 i* 6 2J 4 A 24 10 H 6 2i 4 & 24 I 9 1A 6 2| 3^ i 20 if 9 if 5 3 i A 18 1 8 itt 5 i 3i 1 16 1A 8 if 5 3^ 31 A 14 H 7 iH 5 3^ 3 * 12 iA 7 ii H 4 3 In this table D stands for the diameter of the screw and T for the number of threads per inch. WHITWOETH STANDARD THREADS 43 Exercises. Whit worth Threads 1. Find the depth of a Whitworth thread of ^-inch pitch. 2. Find the double depth of a Whitworth thread with 7 threads to the inch. Find the radius for the point of the tool for cutting a Whit- worth thread of which the pitch in each case is as follows : 3. 0.4". 4. i". 5. i". 6. i". 7. -s\"' 8. &" 9. Find the tap-drill size for a Whitworth thread of which the outside diameter is 4". o 10. If the root diameter of a Whitworth thread is 3.3231" and there are 3 threads to the inch, what is the outside diameter of the screw ? 11. If the root diameter of a Whitworth thread is 5.2377" and there are 2J threads to the inch, what is the outside diameter of the screw ? 12. Find the root diameter of a Whitworth thread with an outside diameter of Jg". 13. If the outside diameter of a Whitworth thread is 41" and the root diameter is 4.0546", what is the pitch ? 14. find the root diameter of a Whitworth thread with an outside diameter of 3". 15. If the diameter of a Whitworth screw is ^", how many threads are there to the inch ? What is the pitch ? the depth of thread ? the root diameter ? 16. If the root diameter of a Whitworth thread with 8 threads to the inch is 0.9024", what is the outside diameter? 17. If the diameter of a Whitworth screw is 3", how many threads are there to the inch ? What is the pitch ? the depth of thread ? the root diameter ? 44 SCREW THREADS Metric Threads. A metric thread is similar in shape to the U. S. S. thread shown on page 40, but the measurements are in millimeters (1 mm.= 0.0394"). Of the two standard metric threads in general use, the first is the Inter- national Standard thread, adopted at Zurich in 1898, and the second is the French Standard. These threads dif- fer but little, as will be seen by com- paring the measurements of the threads in the table below. The following formulas are used with these two threads: Depth = 0.6495 x pitch Flat (top and bottom) = | x pitch The following table shows standard pitches for both Inter- national Standard and French Standard metric threads: METRIC THREAD d, depth ; /, flat ; p, pitch INTERNATIONAL STANDARD FRENCH STANDARD Diameter of screw Pitch Diameter of screw Pitch mm. in. mm. mm. in. mm. 3 0.1181 0.55 3 0.1181 0.50 4 .1575 0.70 4 .1575 0.75 5 .1969 0.85 5 .1969 0.75 6 .2362 1.00 6 .2362 1.00 7 .2756 1.00 7 .2756 1.00 8 .3150 1.25 8 .3150 1.00 9 .3543 1.25 9 .3543 1.00 10 .3937 1.50 10 .3937 1.50 11 .4331 1.50 11 .4331 1.50^ 12 .4724 1.75 12 .4724 1.50* 14 .5512 2.00 14 .5512 2.00 16 .6299 2.00 16 .6299 2.00 18 .7087 2.50 18 .7087 2.50 20 .7874 2.50 20 .7874 2.50 22 .8661 2.50 22 .8661 2.50 METRIC THREADS 45 Exercises. Metric Threads 1. Find from the formula on page 44 the depth of a French Standard thread of 1-millimeter pitch. The introduction of French and Italian automobiles into this country and our recent expansion of foreign trade render computation in metric units necessary as well as desirable. In problems dealing with metric threads give the final results to the nearest 0.1 mm., unless otherwise specified. 2. Find the double depth of a French Standard thread of 3-millimeter pitch. 3. Find the root diameter of a 10-millimeter International Standard thread. 4. Find the pitch of an International Standard thread of which the outside diameter is 27 mm. and the root diameter is 23.10 mm. 5. Find the outside diameter of a French Standard thread of which the root diameter is 48.86 mm. and the pitch is 5.50 mm. 6. For particularly accurate work the formula for the depth of an International Standard thread is as follows: depth = 0.64952 X pitch. If the pitch is 4.500 mm., find the depth to the nearest 0.01 mm., using this formula. 7. In Ex. 6 if the depth is 0.974 mm., what is the pitch? 8. Find the width of the flat of an International Standard thread with an outside diameter of 11 mm. 9. If the root diameter of a French Standard thread is 23.10 mm. and the pitch is 3 mm., what is the outside diam- eter of the screw ? 10. If the outside diameter of a French Standard thread is 36 mm. and the root diameter is 30.80 mm., what is the pitch of the thread? 46 SCREW THREADS Square Thread. The thread of which the cross section is a square, as shown in the figure, is called a square thread. In theory the cross sections of a square thread and of the space between two successive threads are both squares. In practice the groove is slightly wider and deeper than the width of the thread. For our present purposes we shall take SQUARE THREAD the theoretical measurements. a, depth; /, flat; p, pitch In such a thread the pitch, the depth, and the flat are found from the following formulas: 1" " number of threads to 1" Depth = flat = I x pitch Square-Thread Tool. The tool used in cutting a square thread is shaped like a parting tool, or cutting-off tool, except that it has a side clearance, or ( / rake, which varies for the pitch and diameter of the thread. The distance that the tool travels around the work is the root cir- cumference of the thread ; the distance that it travels sideways is the pitch. Letting the pitch p and the root circumference C r be two sides of a right triangle, as in this figure, a is the angle for grind- ing the side clearance of the tool and is found by the use of tangents. For example, find the angle of clearance of the tool for cutting a square thread of ^-inch pitch on a shaft 2" in diameter. 0.25 0.25 tan a = = = = 0.0455. C r 3.1416 x (2 - 0.25) 5.4978 Therefore a = 2 36'. ^ SQUARE-THREAD TOOL a, angle for grinding the tool SQUARE THREADS 47 Exercises. Square Threads 1. From the formulas on page 46 find the pitch, the depth, the double depth, and the flat of a square thread with 8 threads to the inch. 2. If the outside diameter of a screw with square threads is 2|", and there are 8 threads to the inch, what is the root diameter of the screw ? 3. At what angle should the tool be ground for cutting the thread referred to in Ex. 2? 4. If a double square thread has a diameter of iy and a lead of J", what is the root diameter? 5. In cutting a square thread with 2 threads to the inch on a shaft 41" in diameter, what angle of side clearance should be' allowed for the tool? 6. If the double depth of a square thread is 0.0417", what is the number of threads to the inch ? 7. Find the angle for grinding the side clearance of the tool used in cutting a square thread having an outside diameter of 1|" and a pitch of -^g". 8. A triple square-thread screw of outside diameter 1|" has a lead of \". Find the root diameter of the thread. 9. Find the angle for grinding the side clearance of the tool for cutting the thread in Ex. 8. 10. Find the outside diameter of a square thread which has a root diameter of 3 T 5 g-" and a pitch of -J". 11. Find the angle for grinding the tool used in cutting the thread described in Ex. 10. What would be the angle if the thread were a double thread of the same lead? 12. If the root circumference of a square thread is 2.3562" and the pitch is J", what is the outside diameter? 48 SCKEW THREADS Acme Thread. The thread illustrated in the figure below is called an Acme thread. The nature of this thread will be understood from the illustration. Nut Owing to the ease of cutting and to the greater strength secured, the Acme thread has to a large extent replaced the square thread. From the formulas given below it will be seen that the depth of the Acme ACME THREAD screw thread differs by 0.0100" from that of the square thread. d ' depth; p ' pit( The following formulas are used in computing measure- ments of Acme screw threads and tap threads: number of threads to 1" Depth for screws =| x pitch + 0.0100" Depth for taps = | x pitch + 0.0200" Width of flat at root for screws = 0.3707 X pitch 0.0052" Width of flat at root for taps = 0.3707 x pitch - 0.0052" Width of flat at top for screws = 0.3707 x pitch Width of flat at top for taps = 0.3707 x pitch 0.0052" Width of space at top for screws = 0.6293 x pitch Width of space at top for taps = 0.6293 x pitch + 0.0052" Diameter of tap = diameter of screw + 0.0200" Root diameter for screws and taps =D (pitch + 0.0200") The symbol D in the last formula stands for " outside diameter." It is evident from the above formulas that the clearance at the bottom of an Acme thread between the nut and the screw is obtained by making the outside diameter of the nut thread 0.0200" larger than that of the screw thread. ACME THREADS 49 Exercises. Acme Threads 1. Find from the formula on page 48 the width of the flat at the root of an Acme screw thread having 2J- threads to the inch. 2. Find the depth of an Acme tap thread of ^-inch pitch. 3. Find the width of the flat at the top of an Acme screw thread having 12 threads to the inch. 4. Find the root diameter of an Acme tap thread having 4 threads to the inch and an outside diameter of 2J". 5. Find the outside diameter of an Acme screw thread having a root diameter of 3.4443" and 3J threads to the inch. 6. A boring bar with a diameter of 3|" is threaded with an Acme screw thread which has a double depth of 0.2422". Find the pitch of the thread. 7. In Ex. 6 find the width of the flat at the root. 8. Find the width of the flat at the top and also at the root of an Acme screw thread of |-inch pitch. 9. Find the root diameter of an Acme tap thread which has an outside diameter of 1-jV and 7 threads to the inch. 10. In an Acme screw thread of -j 3 g-inch pitch find the width of the flat at the top, the width of the flat at the root, and the width of the space at the top. 11. If the outside diameter of an Acme screw thread hav- ing 3 threads to the inch is ij", what is the root diameter ? the outside diameter of the corresponding tap thread ? 12. If the root diameter of an Acme screw thread is 3.2982" and the outside diameter is 4J", what is the pitch of the thread ? 13. Find the outside diameter of an Acme screw thread with 12 threads to the inch and a root diameter of 1.1467". 50 SCREW THREADS Briggs Pipe Threads. A Briggs pipe thread differs from the threads so far considered in that for a certain distance, A~B in figure III below, the outside of the pipe is tapered. The threads P, which are on the taper, are perfect threads, the in BRIGGS PIPE THREADS Figure I shows a cross section of a perfect standard thread as devised by Robert Briggs, a British engineer, in 1882 ; figure II shows a similar cross section of the modified Briggs thread ; figure III shows a longitudinal section of the complete thread, the letters in this figure being explained in the text next two threads /, not being on the taper, are imperfect on top, and the next four threads I' are imperfect at the bottom as well, owing to the chamfer of the threading die. The nominal inside diameter (N.I. D.) is the size under which the pipe is listed and differs considerably from the actual inside diameter (A.I.D.), which is the exact measure- ment, as is seen in the table on page 51. The actual outside diameter (A. O.D.) is measured at the large end of the taper, which has a T.P.F. of |" between points A and B. The following formulas are used with Briggs threads and show the differences between the standard and modified forms : 1" Pitch number of threads to 1" Depth (standard) = 0.8 x pitch Depth (modified) = 0.833 x pitch Flat (modified) = i x pitch The modified form of Briggs thread is more easily cut and hence has come into more general use. BRIGGS PIPE THREADS 51 Table of Briggs Threads. The table below shows the stand- ard measurements and the number of threads (T 7 ) per inch: N.I.D. A.I.D. A.O.D. T N.I.D. A.I.D. A.O.D. T i" 0.270" 0.405" 27 2" 2.067" 2.375" Hi .364 .540 18 21 2.468 2.875 8 I .494 .675 18 3 3.067 3.500 8 .623 .840 14 31 3.548 4.000 8 a .824 1.050 14 4 4.026 4.500 8 1 1.048 1.315 111 4| 4.508 5.000 8 1J 1.380 1.660 Hi 5 5.045 5.563 8 u 1.610 1.900 Hi 6 6.065 6.625 8 Exercises. Briggs Pipe Threads 1. Find the depth of a modified Briggs pipe thread hav- ing 27 threads to the inch. 2. Find the double depth of a modified Briggs pipe thread having 111 threads to the inch. 3. Find the width of the flat of a modified Briggs pipe thread with 8 threads to the inch. Find the root diameter at the large end of the taper of a Briggs Standard thread for each of the following sizes of pipes (N. /. Z>.) : 4. \". 5. \". 6. 1". 7. II". 8. 3". 9. 4". 10. In the Briggs Standard thread the length of the per- , 0.8 D + 4.8 feet thread, P in figure III on page 50, is N , where D is the A.O.D. and N is the number of threads per inch. Find the length of the perfect thread for a |-inch pipe. As in Ex. 10, find the length of the perfect thread in each case for pipes of the following sizes : 11. " . 12. V. 13. 1J". 14. 1J". 15. 2". 16. 31". 52 SCREW THREADS Measuring Threads. In addition to the outside diameter and the root diameter of a screw, there is also the pitch diameter, or the angle diameter, which is used in measuring threads. The pitch diameter is theoretically the outside diameter minus the single depth of the' thread; that is, it includes half the depth of the thread on opposite sides of the screw. The best means of measuring the pitch diameter of a thread is by the use of a special thread micrometer. In such an instrument the anvil is V-shaped so as to fit over the thread, and the screw, pointed to 60, is slightly rounded so as to enter the groove. In practice, however, the pitch diameter is not actually measured, but is tested by the use of the ordinary micrometer and of TESTING THE PITCH three wires of the same diameter placed as DIAMETER OF A TJ S S THREAD here shown. The wires used are hardened, M, M, micrometer jaws; tempered, accurately ground, and lapped w, w, w, wires to size. One wire is placed in the angle of the thread, and the other two wires are placed in adjacent angles on the opposite side. The micrometer is placed over the wires, and the reading is taken. By trigonometry it can be shown that if M is the measure- ment over the wires, D the outside diameter of the screw, p the pitch, C a certain constant, and D w the diameter of the wires, the following formula is true : The pitch diameter of the screw is correct (that is, it is equal to the theoretical pitch diameter) when the micrometer reading over the wires is equal to the result from the formula. MEASURING THEEADS 53 Illustrative Problems. 1. Find the correct measurements over wires of a -inch U. S. S. thread, the diameter of the measuring wires being 0.05773" and the constant, which is used for all U. S. S. threads, being 1.5155. We first refer to the table of U. S. S. threads on page 40, and find that when the outside diameter of the thread is " there are 10 threads to the inch. The pitch, therefore, is ^". Using the formula given on page 52, we find M, the measurement over the wires, as follows : M= D -pC+ 3 D w = 0.75 - 1.5155 10 + 3 x 0.05773 = 0.77164 Since a vernier micrometer gives readings only to 0.0001" we give the above result as 0.7716". If the reading of the caliper agrees with this result, the pitch diameter is correct. 2. Find the correct micrometer reading over wires for a T 5 g-inch sharp V-thread, the diameter of the wires being 0.035" and the constant for sharp V-threads being 1.732. For sharp V-threads we find on page 38 that when D = T 5 ^" the number of threads is 18, and hence the pitch is T V- We therefore have + 3 x 0.035 = 0.3125 - = 0.3213. TESTING THE PITCH The correct reading for a ^-inch sharp DIAMETER OF A SHARP V-thread is therefore 0.3213". V-THREAD Use of the Three-Wire Method. If, in cutting the sharp V-thread described above, it is desired to test the depth of cut, the wires are placed as shown in the figure and the mi- crometer reading is taken over them. If, for example, this reading is 0.01" more than 0.3213", the thread must be cut 0.01" deeper to be correct ; if the reading is less than 0.3213", the thread has been cut too deep. 54 SCKEW THREADS Exercises. Measuring Threads 1. Find the correct reading over wires for a 2 1 -inch Whitworth thread, the wires being 0.150" in diameter and the constant for this thread being 1.6008. The formula for the correct reading of a Whitworth thread differs from the one given on page 52. It is as follows : M = D -pC + 3.1657 D w For the pitch see the table on page 42. 2. Find the correct reading over wires for a 1^-inch U.S.S. thread, using wires 0.090" in diameter. 3. Find the correct reading over wires TESTING THE PITCH for a sharp V-thread If" in diameter, DIAMKTEBOFA WHIT- r - WORTH THREAD using wires 0.150" in diameter. 4. Find the correct reading over wires for a U.S.S. thread 3" in diameter, using wires 0.200" in diameter. 5. Copy the following table and in the proper space insert the correct measurement over wires for each thread : U. S. S. THREAD WHITWORTH THREAD Outside diameter Diameter of wires Measure- ment over wires Outside diameter Diameter of wires Measure- ment over wires IM 2 0.050" _7_" 16 0.04026" 1 0.06415 1 3 *8 0.09394 If 0.10497 2 0.12526 2 i 0.12830 8 i 0.17344 H 0.17623 3| 0.18789 REVIEW EXERCISES 55 Exercises. Review 1. Find the lead of a double-thread screw which has 12 threads to the inch; which has 20 threads to the inch. 2. Find the root diameter of an S.A.E. thread that is 11" in diameter. 3. Find the double depth of a Briggs Standard pipe thread for a 4 1 -inch pipe. 4. In an Acme screw thread with a pitch of T y find the width of the flat at the top, the width of the flat at the root, and the width of the space at the top. 5. What is the pitch of a square thread when the double depth is 0.2222"? How many threads are there to I" ? 6. If the root diameter of an International Standard thread is 5.70 mm. and the pitch is 1 mm., what is the outside diameter ? the width of the flat ? 7. At what angle should the thread tool be ground to cut a square thread with 6 threads to the inch and an out- side diameter of 1^"? 8. Find the tap-drill size of a l|-inch U.S.S. tap. 9. Find the root diameter of an Acme tap thread 2-|" in outside diameter with 5 threads to the inch. 10. Find the correct reading over wires for a sharp V-thread 11" in diameter, the diameter of the wires being 0.070". 11. The root diameter of a U.S.S. thread with 3J threads to the inch is 2.6288". Find the outside diameter. 12. The root diameter of a Whitworth Standard thread is 4.5343" and the outside diameter is 5". Find the pitch of the thread. 13. If the outside diameter of an Acme screw thread hav- ing 11 threads to the inch is 31", what is the root diameter ? 56 SCEEW THREADS Screw-Thread Cutting. The cutting of a thread in a lathe is a mechanical operation in which the selection of the proper change gears is a matter of greatest importance. In order to cut the required number of threads per inch, the number of revolutions of the work must be exactly equal to the number of threads to be cut during the time that the carriage carry- ing the tool rest moves LATHE HEADSTOCK SHOWING SIMPLE GEAR- exactly 1" along the bed ING FOR CUTTING THREA S of the lathe. Change A > lathe s P indle ; B > lead screw; c, stud shaft; E, intermediate gear gears are used so as to give the proper number of revolutions to the lead screw, which moves the carriage, gears of different sizes being placed on the stud shaft and on the lead screw according to the size of thread to be cut. For fine threads the carriage moves slowly, but for coarse threads it moves rapidly along the work. A simple-geared lathe has only one change of speed be- tween the stud shaft and the lead screw. Since the gear on the lathe spindle has the same number of teeth as the driven gear on the stud spindle, the two spindles have the same speed. The intermediate gear connecting the driving gear on the stud and the gear on the lead screw may be of any size since it has no effect on the rate of turning of the lead screw. The lead of the lathe can be found by counting the threads per inch on the lead screw, or by placing gears of the same number of teeth on the stud shaft and on the lead screw and then counting the number of revolutions of the lead screw while the carriage moves 1". SCKEW-THKEAD CUTTING 57 Illustrative Problem. If the lead screw of a lathe has 6 threads per inch and it is desired to cut 12 threads per inch on a screw, what change gears are required ? In this case, since the lead screw will revolve 6 times while the car- riage advances 1", and the work must revolve 12 times during the same period, change gears with a ratio of 6 to 12, or of 1 to 2, should be placed on the stud shaft and the lead screw. That is, expressed as a general proportion, which may be used in any problem requiring the use of simple gearing, we have the following : Number of teeth on outside stud gear Number of teeth on lead-screw gear _ number of threads per inch on lead screw number of threads per inch on work Each term of the ratio of the gears is multiplied by a number such that the result is a ratio whose terms correspond to the numbers of teeth on two of the gears with which the lathe is equipped. 4x6 24 In this case we have = 4 x 12 48 That is, a gear with 24 teeth should be placed on the end of the stud shaft, and a gear with 48 teeth on the lead screw. An intermediate gear of the proper size is used to connect the two gears. Variation of Lathes. The number of threads per inch on the lead screw varies on different makes of lathes, and this makes it necessary to have different sets of change gears. In our problems we shall use the following sets of gears : For a lead screw with 5 threads per inch : 25, 25, 30, 35, 40, 45, 50, 55, 60, 65, 69, 70, 80, 90, 100, 110, 120. FoY a lead screw with 6 threads per inch : 24, 24, 32, 40, 44, 48, 52, 56, 60, 64, 72, 110. For a lead screw with 8 threads per inch : 24, 24, 28, 32, 36, 40, 44, 48, 52, 56, 64, 69, 72, 80, 96. In the exercises use only gears which are shown above as belonging to the set for the lead screw specified in each problem. 58 SCREW THREADS Exercises. Screw-Thread Cutting 1. What change gears are required on a lathe with a lead screw of 1-inch pitch in cutting a screw thread of -Jg-inch pitch ? in cutting a thread with a pitch of 0.125" ? Find the change gears required on a lathe with a lead screw of A-inch pitch for cutting each of the following numbers of threads per inch: 2. 12. 3. 7. 4. 13. 5. 18. 6. 111. Find the change gears required on a lathe having a lead screw with 8 threads per inch for cutting each of the following num- bers of threads per inch : 7. 4. 9. 9. 11. 12. 13. 18. 15. 10. 8. 5. 10. 16. 12. 7. 14. 6. 16. 20. 17. What is the pitch of the thread that is cut on a lathe with a lead screw of J-inch pitch when there is a 25-tooth gear on the stud and a 40-tooth gear on the lead screw ? Find the change gears required on a lathe having a lead screw with 6 threads per inch for cutting each of the following num- bers of threads per inch : 18. 6. 19. 10. 20. 13. 21. 18. 22. 271. 23. Find the largest number of threads per inch that can be cut with simple gearing on a lathe which has a lead screw of ^-inch pitch ; of 1-inch pitch ; of ^-inch pitch. 24. Find the smallest number of threads per inch that can be cut with simple gearing on a lathe which has a lead screw of ^-inch pitch ; of 1-inch pitch ; of ^-inch pitch. 25. Find the change gears necessary to cut a thread with a lead of f " if the lead screw of the lathe has 5 threads per inch ; if the lead screw has 8 threads per inch. COMPOUND GEAKING 59 Compound Gearing. A compound gearing consists of two gears keyed together to revolve at the same rate and held in position by a bracket. The gears are introduced between the gear on the stud and that on the lead screw and are usually in the ratio of 2 to 1, but the general principle of oper- ation is the same with gears of any other ratio. Since the cutting of a very small or a very large number of threads per inch with simple gearing would require the use of gears with a very large num- ber of teeth, it is neces- sary to use compound gearing on a lathe. Illustrative Problem. Find the change gears required on a lathe with a lead screw of -^-inch pitch in cutting a screw thread with 30 threads per inch. Solving as for simple gearing, the ratio of the gears is found to be 3%-, and since a 25-T gear is the smallest one that goes with a lead screw of ^-inch pitch, we shall need a gear with 150 teeth if we use only simple gearing. To avoid this we split the ratio ^o i nto factors, thus : 5 _ 1 _ 1 (gear on stud) 1 (driving gear on compound) 30 6 3 (gear on lead screw) 2 (driven gear on compound) We therefore see that the gears on the stud and the lead screw must have the ratio of 1 to 3, such as 30 T to 90 T, while the gears on the com- pound must have the ratio of 1 to 2, such as 25 T to 50 T, the smaller gear being the driving gear and the larger gea'r being the driven gear. LATHE HEADSTOCK SHOWING COMPOUND GEARING FOR CUTTING THREADS A, lathe spindle; B, lead screw; C, stud shaft; D, first gear on compound, driven; E, second gear on compound, driving 60 SCREW THREADS Exercises. Screw-Thread Cutting 1. Find the change gears required on a lathe with a lead screw of i-inch pitch in cutting 25 threads to the inch. This example shows that the gears on the compound are not neces- sarily in the ratio of 2 to 1, the principle being the same for any ratio. Find the change gears required on a lathe with a lead screw of -inch pitch in cutting a thread with each of the following numbers of threads to the inch : 2. 28. 3. 7|. 4. 3|. 5. 32. 6. 42. Similarly, for a lead screw of ^-inch pitch, the numbers being : 7. 1. 8. 11. 9. 22. 10. 36. 11. 28. Similarly, for a lead screw of ^-inch pitch, the numbers being : 12. 2. 13. 23. 14. T 5 r 15. f. 16. 1^. Find the change gears required on a lathe having a lead screw with 5 threads to the inch in cutting a thread with each of the following leads : 17. T y. is. f". 19. f". 20. f. 21. T y. 22. Find the change gears required on a lathe having a lead screw with 6 threads to the inch in cutting a metric screw with a lead of 2 mm. First find the number of threads to the inch (1" = 25.4 mm.) corre- sponding to the given lead in millimeters. It will be found that a gear with 127 teeth is required to cut a metric thread on a lathe which has an English lead screw. In Exs. 22 and 23 assume that the lathe is equipped with such a gear in addition to those listed on page 57. 23. On a lathe having a lead screw with 6 threads to the inch, find the change gears required for cutting a metric thread with a lead of 3 mm.; with a lead of 3.5 mm. CHAPTER V INDEXING AND SPIRAL CUTTING Indexing. The process of dividing a circumference into equal parts is called indexing. Indexing is done by the aid of a milling-machine fixture known as the dividing head, spiral head, or index head, and which is described on page 63. The dividing head is ordinarily employed in obtaining equal divisions on a circumference, but may be used to obtain equal divisions of various kinds on other types of work. There are various kinds of indexing, direct, simple, angular, compound, and differential. These will be considered and explained in the following pages. Among the cases in which the machinist will make use of indexing are the following : in cutting spur, bevel, and spiral gears ; in cutting worm wheels ; in cutting plate and cylin- dric cams; and in making milling-machine cutters, counter bores, reamers, and twist drills. The dividing-head spindle may be set at any angle from about 5 below the horizontal to about 10 beyond the vertical. The swiveling block is usually graduated about its circumference in fourths of a degree, but some machines have a vernier that reads to ^. Work is held in a dividing head in a manner similar to that used on an engine lathe, the three following general methods being used : (1) by mounting it on centers, (2) by holding it on an arbor, and (3) by secur- ing it in a chuck. When the work is held on an arbor or mounted on centers it is necessary to use the dividing-head tailstock, which has an adjustable center capable of being raised or lowered. 61 62 INDEXING AND SPIRAL CUTTING Index Center. The index center shown below is used to obtain exact and rapid spacing on work that requires only a small number of divisions upon the periphery, as is the case when milling bolt heads and fluting reamers and taps. It is used only on machines that are not fitted with a dividing head, since the dividing head will do all that an index center can perform, and has many additional advantages. INDEX CENTER A, spindle ; B, index plate ; C, headstock ; D, footstock ; E, stop pin The index center consists of two main parts, the headstock C and the footstock D. The spindle A is moved freely by the index plate B, which is fastened to it. The spindle is locked at the desired division by means of the stop pin E, which is attached to the headstock. There is an even number of grooves, usually 24, 36, or 48, on the circumference of the index plate, and the number of divisions which can be made on the work is limited to the number of grooves and to factors of this number. Direct Indexing. It is seen from the above description that the index plate gives direct motion to the spindle, and the use of the instrument in this way is called direct indexing. DIVIDING HEAD 63 Dividing Head. Instead of connecting the index plate B directly to the spindle A, as in the index center, the index crank F of the dividing head here shown is connected to DIVIDING HEAD A, spindle ; B, index plate ; H, index pin shaft E, on which is cut a single-thread worm D. The worm meshes with the 40-tooth worm wheel C on the spindle, thus transmitting the motion from the crank. The index plate B is not fastened to shaft E and can be held stationary by the stop pin P. The index pin H is adjustable and, by the tension of a spring, is held in any one of the holes in the index plate. The sector arms /, / can be set to include any MECHANISM OF DIVIDING HEAD A, spindle; B, index plate; C, worm wheel ; D, single-thread worm ; E, shaft; F, index crank; H, index pin; J, /, sector arms ; P, stop pin desired number of holes, thus rendering it unnecessary to count the holes every time that the index crank is turned. In direct indexing the worm is thrown out of mesh, and the spindle is locked in the desired position by a special stop pin. 64 INDEXING AND SPIRAL CUTTING Simple Indexing. For simple indexing we make use of the dividing head already shown on page 63 and here represented again in detail. The worm wheel (7, which has 40 teeth, is driven by the single-thread worm Z>, on shaft E. Therefore every revolu- tion of the worm D, which corresponds to a single revo- lution of the index crank F, moves the worm wheel C ex- actly one tooth and moves the MECHANISM OF DIVIDING HEAD A, spindle ; B, index plate ; (7, worm Spindle A 1 which is attached wheel; D, single-thread worm; E, shaft; fn if 1 nf a rpvnlntinn F ' index Crank 5 H > index P in 5 7 * tO It, 7 01 a revolution. sector arms; P, stop pin Therefore 40 revolutions of the index crank F cause the spindle A, which carries the work, to make one revolution. Hence if 40 divisions are required on the work, one revolution of F must be made between each pair of cuts; if 20 divisions are required, two revolutions of F are necessary ; if 10 divisions are required, four revolutions of F are necessary ; and so on. Formula for Simple Indexing. Letting D be the number of divisions required and C the number of revolutions of the index crank, we have the following simple formula: For example, using the above instrument, find the indexing required to obtain 80 divisions. C = = = - D 80 2* We have Therefore any circle of holes in the index plate may be used pro- vided the number of holes is divisible by 2. In this case we might take a 16-hole circle and move the index crank 8 holes for each division. SIMPLE INDEXING 65 Index Plates. Different makes of dividing heads have dif- ferent sets of index plates. In our exercises we shall use plates which have circles with the number of holes as follows : BROWN AND SHARPE Plate 1 : 15, 16, 17, 18, 19, 20 Plate 2 : 21, 23, 27, 29, 31, 33 Plate 3: 37, 39, 41, 43, 47, 49 CINCINNATI MILLING MACHINE CO. One plate drilled on both sides, as follows: First side: 24, 25, 28, 30, 34, 37, 38, 39, 41, 42, 43 Second side: 46, 47, 49, 51, 53, 54, 57, 58, 59, 62, 66 Exercises. Simple Indexing 1. If a spur gear is to have 36 teeth, find the indexing required on a Cincinnati dividing head. Since C = f = 1^, we must have a circle with the number of holes divisible by 9, in this case 54. The indexing required is therefore one revolution of the index crank plus 6 holes on the 54-hole circle. 2. Find the indexing required on a B. & S. (Brown and Sharpe) dividing head for 7 divisions. Find the indexing on a B. $ S. dividing head for each of the following divisions : 3. 3. 5. 45. 7. 98. 9. 145. 11. 205. 4. 26. 6. 68. 8. 116. ' 10. 164. 12. 312. Find the indexing on a Cincinnati dividing head for each of the following divisions : 13. 5. 15. 17. 17. 44. 19. 70. 21. 92. 14. 9. 16. 23. 18. 55. 20. 85. 22. 110. 66 INDEXING AND SPIRAL CUTTING Angular Indexing. If the dividing head requires 40 turns of the index crank for one revolution of the spindle, that is, to turn the work through 360, one turn of the crank will produce a turning of -^ of 360, or 9. The process of indexing work by angles is known as angular indexing. Therefore, if one revolution of the index crank gives an angle of 9, two holes on the 18-hole circle or three holes on the 27-hole circle of a B. & S. dividing head will correspond to 1. Halves and thirds of a degree are indexed by taking one hole on the 18-hole circle or one hole on the 27-hole circle respectively. On a Cincinnati dividing head the 54-hole circle is used, six holes corresponding to 1, and so on. For example, using a B. & S. dividing head, find the indexing for an angle of 25. Since one turn of the index crank gives an angle of 9, the number of turns of the crank will be -^ 5 -, or 2. The indexing required is, there- fore, two revolutions of the crank plus 14 holes on the 18-hole circle. Exercises. Angular Indexing 1. Find the indexing on a B. & S. head for an angle of 11. 2. Find the indexing on a Cincinnati dividing head for an angle of Using a B. $ S. dividing head, find the indexing for each of the following angles : 3. 1. 5. 4J. 7. 13. 9. 33. 11. 44f. 4. 2J. 6. 8f. 8. 17i. 10. 40i. 12. 451. Using a Cincinnati dividing head, find the indexing for each of the following angles : 13. 3. 15. 11. 17. 23. 19. 56^. 21. 79|. 14. 51. 16. 19|. 18. 31 J. 20. 75. 22. 82 '. ANGULAR AND COMPOUND INDEXING 67 Compound Indexing. To obtain divisions which are beyond the range of simple indexing a device known as compound indexing is sometimes employed. In this method the index crank is first turned a definite amount in the regular way, and then the stop pin holding the index plate is disengaged and the index plate is turned either in the same direction or in the opposite direction. The indexing is thus compounded of two separate movements which are, in reality, two simple-indexing operations. Compound indexing should be used only when there is no other con- venient way of obtaining the division. This is because of the chances of error due to the fact that the holes must be counted when moving the index plate, as will be seen in the example below. The method has been largely superseded by the differential method explained on page 70. If, for example, the index crank of a B. & S. dividing head is turned 40 holes on the 18-hole circle, that is, two revo- lutions plus four holes, and then, the stop pin being disen- gaged, the index plate is turned 40 holes on the 19-hole circle, that is, two revolutions plus two holes, in the same direction, the two movements will cause the spindle holding the work to make ^ + Jg, or -f^ of a revolution. If, however, the index plate is turned in the opposite direction from the index crank, the spindle will make y 1 ^ Jg, or g^, of a revolution. Since the two revolutions of the index crank and the index plate are in opposite directions, they offset each other, and the same result is obtained by moving the index crank four holes on the 18-hole circle in one direction and the index plate two holes on the 19-hole circle in the opposite direction. Thus, by combining the two movements we can obtain 342 divisions with the index plates listed on page 65. With simple indexing we should have required a plate with a circle of 171 holes. 68 INDEXING AND SPIRAL CUTTING Illustration of Compound Indexing. For example, to find the indexing required on a B. & S. dividing head to obtain 147 divisions, we proceed as follows: 1. Find two factors of 147. That is, 147 = 3 x 49. 2. Try to find two circles on the same plate, one with 49 holes and the other with perhaps ten holes less. In this case let us try the 49-hole and 39-hole circles. 3. Factor the difference between the numbers of holes in the two circles. That is, 49 - 39 - 10 = 2 x 5. 4. Take the indicated product of these two sets of factors as the numerator of a fraction. That is, the numerator is (3 x 49) x (2 x 5). 5. Take for the denominator of this fraction (1) the fac- tors of 40, that is, of the number of turns of the index crank to one turn of the dividing-head spindle ; (2) the factors of the larger circle ; (3) the factors of the smaller circle. That is, (1) 40 = 2 x 2 x 2 x 5, (2) 49 = 7 x 7, (3) 39 = 3 x 13, and hence the denominator is (2 X 2 X 2 X 5) X (7 X 7) x (3 x 13). 6. Now cancel, and if all the factors in the numerator cancel out, the correct circles have been chosen. 7 _ , . , (48 x 3") X (2 X 0) In this case we have ^- E2 ? M - , (2 x 2 x x 0) x (/ x 7) x (JJ x 13) and therefore the correct circles have been chosen. If all the factors in the numerator do not cancel out, other numbers representing the numbers of holes in the circles of the index plates must be chosen until two circles are found that will permit the cancellation of all these factors. COMPOUND INDEXING 69 7. Take the product of the uncanceled factors in the denominator as the numerator of each of two fractions, called the indexing fractions, in which the numerator is the number of holes to be turned, and the denominator is the number of holes in the circle on which the number turned is counted. In this case the numerator of the indexing fractions is 2 x 2 x 13, or 52, and the fractions which give the desired divisions are + ^f and |, the signs + and being used to show that the two move- ments are to be made in opposite directions. That is, by turning 52 holes on the 39-hole circle in one direction and 52 holes on the 49-hole circle in the other direction, we shall have the proper movement of the spindle. This is seen from the fact that If any whole number is subtracted from each of two fractions, the difference of the fractions is not affected. In this case, therefore, we may subtract 1 from each fraction, and we have -l=49, and -!=& Hence we give the indexing fractions required to obtain 147 divisions by compound indexing as + ^f and ^. Exercises. Compound Indexing Using compound indexing on a B. $ S. dividing head, find the indexing fractions for each of the following divisions : 1. 69. 2. 51. 3. 99. 4. 217. 5. 294. 6. By the compound method, find the indexing fractions for cutting a spur gear 'with 77 teeth on a milling machine which is equipped with a B. & S. dividing head. 7. Find the fractions in Ex. 6 for a gear with 87 teeth. 8. Find the indexing fractions for cutting 129 teeth on the rim of a ratchet wheel, using compound indexing on a B. & S. dividing head. 70 INDEXING AND SPIRAL CUTTING Differential Indexing. The method of indexing known as differential indexing is the same in general principle as com- pound indexing. It differs, however, from the latter in that the index plate is revolved by a suitable gearing interposed between it and the dividing-head spindle, the stop pin holding the index plate being disengaged altogether. The movement DIVIDING HEADS GEARED FOR DIFFERENTIAL INDEXING The index plate of the dividing head shown at the left is connected to the spindle by simple gearing ; that of the dividing head shown at the right, by compound gearing. The letters show the following parts : P, stop pin ; R, gear on spindle (driving) ; S, gear on bevel-gear shaft, or worm, (driven) ; T, intermediate gear ; U, first gear on stud (driving) ; V, second gear on stud (driven) of the index plate takes place when the crank is turned, the index plate moving either in the same direction as the crank or in the opposite direction. Therefore, in using differential indexing to obtain a given number of divisions we have to find two factors, (1) the indexing fraction and (2) the change gears which will turn the index plate the required amount in the proper direction during the movements of the index crank. Change Gears. The change gears furnished with a B. & S. dividing head for differential indexing and spiral milling have the following numbers of teeth: 24 24 28 32 40 44 48 56 64 72 86 100 In solving the exercises on differential indexing use only these gears. DIFFERENTIAL INDEXING 71 Illustration of Differential Indexing. For example, let us consider the differential indexing on a B. & S. dividing head for 59 divisions. In simple indexing p^= Cj for 60 divisions the movement of the index crank is |$, or J, of a turn for each cut. If the crank is given | of a turn 59 times, it makes 391 turns, or | of a turn less than the 40 turns required for one revolution of the work. Hence the index plate must move in the same direction as the crank ^ of a B revolution while the work revolves once. We therefore have 59 x = 39^, and 40-39J = f. Hence the ratio of the gears is to 1, or 2 : 3, 16 x 2 ^ 32 16 x 3 ~48' and MECHANISM OF DIVIDING HEAD GEARED FOR DIFFERENTIAL INDEXING A, spindle; B, index plate; C, worm wheel; D, worm ; E, shaft ; H, index pin ; P, stop pin ; ft, gear on spindle (driving) ; S, gear on bevel- gear shaft, or worm, (driven) ; !T, intermediate gear ; U, first gear on stud (driving) ; V, second gear on stud (driven) A 32-T gear (driving) is placed on the special differential-indexing center in the spindle of the dividing head, and a 48-T gear (driven) is placed on the bevel-gear shaft which turns the index plate. An intermediate gear is used to make the index plate move in the same direction as the index crank, which is given of a turn for each cut ; that is, the indexing fraction may be given as -^f . 72 INDEXING AND SPIRAL CUTTING Exercises. Review 1. Find the differential indexing on a B. & S. dividing head for 175 divisions. For 180 divisions, T 4 g o- = Then 175 x f = 38f , and 40-38f = l^. Hence the required gearing must be in the ratio of 1 to 1, or 10 : 9. Since no two gears will give this ratio, we have JQ. 40 S y 5 D US' 54" Gears in the ratio of 8 (driving) to 9 (driven) are used on the spindle and bevel-gear shaft, and gears in the ratio of 5 (driving) to 4 (driven) are used on the stud. We therefore have 8_x_8 = 64 8 x 5 _ 40 8 x 9 ~ 72 ' 8 x 4 ~ 32 ' The arrangement of the gears is as follows : spindle, 64 T ; bevel- gear shaft, 72 T; first gear on stud, 40 T; second gear on stud, 32 T. With compound gearing an intermediate gear is not needed to make the index plate turn in the same direction as the index crank. The in- dexing fraction may be given as T 4 ? , which is equivalent to f of a turn. Find the differential indexing on a B. $ S. dividing head for each of the following divisions : 2. 53. 4. 107. 6. 161. 8. 199. 10. 271. 3. 67. 5. 112. 7. 173. 9. 214. 11. 321. Using a Cincinnati dividing head, find the simple indexing movement for each of the following divisions : 12. 116. 13. 145. 14. 205. 15. 245. 16. 392. Using a B. $ S. dividing head, find the indexing movement for each of the following angles : 17. 12. 18. 211. 19. 291. 20. 53|. 21. 66. 22. Using compound indexing on a B. &. S. head, find the indexing fractions for milling 273 slots on a feed disk. SPIRALS 73 Spirals. A winding cut made by a tool moving around a cone and at the same time advancing along its slant line at a uniform rate is called a spiral cut. Since the method of feeding the work is the same in the case of a straight reamer as in that of a taper reamer, it is customary to give the name " spiral cut " to the helical flutes of straight drills and reamers and also to the spiral teeth in mills, gears, and cutters. The dividing head is used not only for indexing but also for the milling of spirals. When a spiral is being milled the work is turned slowly by the dividing head, while the table of the milling machine feeds lengthwise. The lead of a spiral is figured in much the same way as the lead of a screw thread ; that is, it is the distance that the spiral advances in making one turn around the work. The spiral, however, is spoken of as having so many inches to one turn instead of as having so many threads to one inch, as is the case with a screw thread. In the process of milling a spiral, change gears are used to connect the dividing head with the feed screw on the table of the milling machine, so that the spindle of the dividing head rotates a definite amount in proportion to the distance that the table travels. The action of this gearing is similar to that used in cutting a thread in a lathe. Spirals are always cut on a universal milling machine, the table of which can be set at any desired angle to the spindle holding the milling cutter. The table is set at such an angle as will bring the milling cutter in line with the cut to be made on the work, as is explained on page 78. The angle at which the table is set has no effect on the lead of the spiral, which is determined by the change gears used. If the table were left in its normal position at right angles to the cutter spindle, a spiral of the same lead would be obtained, but the shape of the cut would be changed. 74 INDEXING AND SPIRAL CUTTING Spiral Milling. The figure below shows both the side view and the end view of a dividing head mounted on the table of a milling machine and arranged for spiral milling. The rotary movement of the dividing-head spindle A, which drives the work, is caused by the turning of the feed screw B, which also causes the table to move longitudinally. 40 DIVIDING HEAD CONNECTED BY COMPOUND GEARING WITH FEED SCREW OF MILLING MACHINE A, spindle ; B, feed screw ; (7, first gear on stud (driving) ; D, gear on bevel-gear shaft, or worm, (driven) ; E, second gear on stud (driven) ; F, gear on feed screw (driving) ; P, stop pin; R, bevel-gear shaft; T, teeth, as in 32 T The motion of the feed screw B is transmitted through the compound train of change gears F, E, (7, D to the bevel- gear shaft R, which has a set of bevel gears inside the divid- ing head connecting with the shaft on which the worm which drives the spindle is cut, as will be seen in the figures on pages 63 and 71. The gear on the feed screw is custom- arily spoken of as the " gear on screw," while that on shaft R is called the " gear on worm." When the dividing head is used for spiral milling, the stop pin P, which holds the index plate, is disengaged altogether. SPIKAL MILLING 75 Lead of a Milling Machine. If the change gears are in the ratio of 1 to 1 (that is, if gear D revolves once for each revo- lution of F), the lead of the milling machine is the distance that the table travels while the spindle revolves once. This distance is a constant used in figuring change gears and varies with different makes of machines. If R is the number of revolutions of the feed screw for one revolution of the spindle with 1-to-l gearing and F is the lead of the feed screw, Jf, the lead of the milling machine, is found as follows : M = RF For example, if with 1-to-l gearing R = 40 on a milling machine which has F = ", we have 3/^/^ = 40 x 4" = 10". Change Gears. Letting L be the lead of the spiral to be cut and M the lead of the milling machine, the ratio of driven to driving gears is found from the following formula : Gears driven L Gears driving ~~ M Letting T w represent the gear on worm (driven), T s the gear on screw (driving), S l the first gear on stud (driving), and S 2 the second gear on stud (driven), the above ratio is split into two factors as follows: Gears driven _ T Gears driving ~~ T s The terms of each of the ratios T w : T 8 and S 2 : S l are multiplied by a number which will give numbers in the list of gears on page 70. To find the lead of a spiral which will be cut with a given combination of change gears we have the following formula : _ M x product of driven gears product of driving gears 76 INDEXING AND SPIKAL CUTTING Illustrative Problems. 1. Find the change gears required on a milling machine, which has a lead of 10", for cutting a spiral with a lead of 36"- Using the formulas given on page 75, we have Gears driven _ L _ 36 _ T w ^ 2 _ 9 ^ 4 Gears driving ~ M ~ 10 ~~ T 8 * S l ~ 5 X 2 ' T w 9 8 x 9 72 , 5 2 4 16 x 4 64 whence _^- = _ = _, and -2 = - = __ = _. We therefore have the following arrangement of gears : gear on worm (driven), 72 T ; gear on screw (driving), 40 T ; first gear on stud (driving), 32 T ; and second gear on stud (driven), 64 T. 2. Find the change gears required on a milling machine with a lead of 10" for cutting a spiral with a lead of 12.80". Gears driven = L = 12.80 _ 128 = T w S^ _ _6 2 Gears driving M ~ 10 ~ 100 ~ T 8 * ^ ~~ 100 X 1 ' T w 64 5 2 28 x 2 56 Then - = > and - = = = T 8 100 S l 1 28 x 1 28 We therefore have the following arrangement of gears: gear on worm (driven), 64 T ; gear on screw (driving), 100 T ; first gear on stud (driving), 28 T ; and second gear on stud (driven), 56 T. 3. What is the lead of the spiral that will be cut on a milling machine which has a lead of 10" when the following arrangement of change gears is used : gear on worm, 86 T ; first gear on stud, 40 T; second gear on stud, 48 T ; and gear on screw, 44 T ? Using the formula given at the foot of page 75, we have 3 T _M X product of driven gears _ ffl x 86 x $fi _ 258 _ _ product of driving gears x 11 t 11 Hence the lead of the spiral that will be cut is 23.45". CUTTING SPIRALS 77 Exercises. Cutting Spirals 1. If a milling machine has a lead of 10", what change gears are needed to cut a spiral with a lead of 16" ? 2. If a milling machine has a lead of 10", what change gears are needed to cut a spiral with a lead of 22.50" ? 3. For milling a spiral the change gears on a milling machine which has a lead of 10" were placed as follows: a 64-T gear as the gear on worm, a 40-T gear as the gear on screw, a 32-T gear as the first gear on stud, "and a 56-T gear as the second gear on stud. What is the lead of the spiral which was cut with this arrangement of gears ? Find the change gears required on a milling machine with a lead of 10" for milling each of the following spiral leads : 4. 0.80". 7. 4.125". 10. 18.75". 13. 31.50". 5. 2.20". 8. 12.60". 11. 21.00". 14. 37.50". 6. 3.50". 9. 17.20". 12. 25.80". 15. 52.50". 16. Find the change gears required on a milling machine with a lead of 10" in cutting a spiral of one turn in 13J". 17. In milling a spiral the following arrangement of change gears was used : gear on worm, 86 T ; gear on screw, 56 T ; first gear on stud, 28 T ; second gear on stud, 48 T. If the lead of the milling machine was 20", what was the lead of the spiral which was milled ? 18. What is the lead of the spiral which will be cut on a milling machine with a lead of 10" when there is a 64-T gear on the bevel-gear shaft, or worm, a 32-T gear as the first gear on the stud, a 48-T gear as the second gear on the stud, and a 56-T gear on the feed screw ? 19. What would be the lead of the spiral in Ex. 18 if the gears on the stud were reversed? 78 INDEXING AND SPIRAL CUTTING Setting the Table. When cutting a spiral the table of the milling machine is set at the angle which the spiral cut is to make with the axis of the work, as explained on page 73. MILLING-MACHINE TABLE SET TO ANGLE FOR SPIRAL CUTTING The position of the table shown by A is for cutting a right-hand spiral, and that shown by B is for cutting a left-hand spiral As shown in the figures below, there is a fixed relation connecting the spiral angle #, the lead Z, and the circumfer- ence of the work C. i *1 If we let the base of a right triangle be equal to the lead , as shown in the second figure, and let the altitude be equal to the circumference C, the angle a formed by the hypotenuse and the base is the required angle at which to set the table of the milling machine, and is found by the use of tangents. SETTING THE TABLE 79 Illustrative Problem. Find the angle at which to set the table of a milling machine when cutting a spiral with a lead of 28.00" on a 3i-inch blank. Referring to the lower figures on page 78, in which C represents the circumference of the work and L the lead of the spiral, we see that tan a = L C 3.1416 x 31 Then tan a - = = 0.3647, and hence a = 20 2'. Since the scale by which angle a is set is usually graduated only to fourths of a degree, we give the above result as 20. Exercises. Setting the Table Find the angle at which to set the table of a milling machine to cut each of the following : 1. A spiral with a lead of 13.12" on a spiral-gear blank which is 2|" in diameter. In all problems dealing with setting the table of a milling machine give the results to the nearest J. 2. A spiral of 8.95-inch lead on a 1-inch twist drill. 3. A spiral with a lead of 15.75" on a milling cutter which is to be 4" in diameter. o 4. A spiral with a lead of 0.67" on a |-inch spindle. 5. A spiral with a lead of 67.45" on a spiral-gear blank which has a diameter of 5|". 6. If the angle of the spiral on a twist drill \\ n in diameter is 191, what is the lead? 7. If the table of a milling machine is set at an angle of 10^ for cutting a spiral with a lead of 68.57", what is the diameter of the work ? 80 INDEXING AND SPIRAL CUTTING 0.87 Cams. A rotating part of a machine that gives a recipro- cating or oscillating motion to another part is called a cam. The piece to which the motion is thus given is called the follower, and the contact between the two is called the line of contact. A cam may also have a reciprocating or oscillating motion. The distance that the follower is moved by the action of the cam is called the rise, or lead, of the cam. Thus, the cam shown in this figure has a constant rise in 0.87 of its circumference, and in one revolution imparts to the follower a movement of 0.150". Cams are made in many different shapes, and often have more than one lobe, but we shall consider only what are known as plate cams, or peripheral cams. Plate cams having a constant rise, such as are used on automatic screw machines, can be cut on a universal milling machine. Since these cams usually have shorter leads than can be obtained on a milling machine by any practical combination of change gears, it is necessary when cutting them to use the vertical-spindle attachment of the milling machine to hold the milling cutter. Such cams are required in great variety, each differing from the others by only a few thousandths of an inch. Since there are practically no duplications, it is impracticable to cut them by means of master cams. 0.6 0.4 A CAM OF ONE LOBE The lobe extends through 0.87 of the cir- cumference, or approximately 313, and the cam has a lead of 0.150" CAMS 81 Milling Cams. When the dividing head is set with its spindle at right angles to the table of the milling machine, the lead, which will be cut on a cam if milled for a complete revolution, is the same as the lead of the spiral cut with the same change gears. If, how- ever, the dividing head and the vertical-spindle attach- ment are both set at an angle, any required lead can be obtained, provided it is less than the spiral lead for which the milling ma- chine is geared. There- fore change gears, which will give a longer lead than that to be cut on the cam, are first selected from a table of spiral leads. The dividing head is then elevated from the horizon- tal a certain angle, which is found by the formula given on page 82, and the vertical-spindle attachment, which drives the milling cutter, is swiveled around from its vertical position an amount equal to the complement of the angle at which the dividing head is set. This places the milling cutter in line with the spindle of the dividing head so that the edges of the cam will be cut parallel with its axis. Since the graduations for elevating the dividing head are usually marked only in fourths of a degree, in all problems dealing with cams the angles are to be figured to the nearest ^. MILLING A SCREW-MACHINE CAM , milling cutter; D, vertical-spindle attachment ; W, cam being cut 82 INDEXING AND SPIRAL CUTTING Finding the Angle. If A is the angle for setting the divid- ing head, c the lead of the cam, or its rise if continued at the DIVIDING HEAD SET TO ANGLE FOR MILLING A CAM A, angle of dividing head ; B, cutter ; D, vertical-spindle attachment ; W, cam given rate for a complete revolution, and s the lead of the spiral for which the milling machine is geared, we have sin^ = -. s If the rise of the cam is given only for n, a definite num- ber of hundredths of the circumference, the formula becomes c ~ ns Illustrative Problems. 1. Find the angles at which to set the dividing head and the vertical-spindle attachment in milling a cam with a lead of 0.602", the milling machine being geared to cut a spiral with a lead of 0.67". We have and hence A = 63 58'. Calculating to the nearest , the dividing head is elevated 64, and the vertical-spindle attachment is set at 90 64, or 26, from the vertical. CUTTING PLATE CAMS 83 2. In Ex. 1 find the angles if the cam were to have a lead of 0.150" in 300, or in 0.83 of the circumference. We have sin A = = 5lL52 = 0.2697, ns 0.83 x 0.67 and hence A = 15 39'. The dividing head is elevated 15 f and the vertical-spindle attach- ment is swiveled around 90 15f , or 74^, from the vertical. Exercises. Cutting Plate Cams 1. Find the angle at which to elevate the dividing head when cutting a cam with a lead of 1.249", the milling machine being geared to cut a spiral with a lead of 1.333". 2. A milling machine is geared to cut a spiral with a lead of 1.860". At what angles should the dividing head and the vertical-spindle attachment be set for milling a cam with a lead of 1.801" ? 3. A cam having a rise of 0.634" in 0.75 of its circum- ference is to be milled on a milling machine geared to cut a spiral with a lead of 0.930". Find the angle at which to elevate the dividing head. 4. In Ex. 3 find the angle at which to set the vertical- spindle attachment if the given rise were to be obtained in 0.83 of the circumference of the cam. A milling machine being geared to cut a spiral with a lead of 0.916", find the angle at which to elevate the dividing head for milling a cam with each of the following leads : 5. 0.669". 8. 0.789". 11. 0.733". 14. 0.898". 6. 0.691". 9. 0.712". 12. 0.859". 15. 0.771". 7. 0.701". 10. 0.853". 13. 0.751". 16. 0.904". 84 INDEXING AND SPIRAL CUTTING Exercises. Review 1. If the feed screw on the table of a milling machine has a lead of 0.4" and it requires 40 revolutions of the feed screw with 1-to-l gearing to turn the dividing-head spindle once, what is the lead of the machine ? 2. Find the change gears required on a milling machine with a lead of 10" for milling a cutter with spiral teeth of 9.60-inch lead. 3. Find the angle at which to elevate the dividing head when milling a cam with a rise of 0.150" in 0.75 of the circumference, the machine being geared to cut a spiral with a lead of 0.67". Find the angle at which to set the table to cut a spiral with a lead of 13.760", given the diameter of each piece as follows : 4. 1". 6. \". 8. f". 10. 1". 12. |". 5. If". 7. If". 9. 21". 11. IJ". 13. 31". 14. Find the lead of the spiral which is cut on a milling machine with a lead of 10" when there is a 100-T gear on the worm, a 24-T gear as the first and a 64-T gear as the second gear on the stud, and a 28-T gear on the screw. 15. Find the angles at which to set the dividing head and the vertical-spindle attachment when milling a cam with a rise of 0.136" in 0.85 of the circumference, the milling machine being geared to cut a spiral with a lead of 0.67". Find the change gears required on a milling machine with a lead of 10" for milling each of the following spiral leads : 16. 0.90". 19. 3.75". 22. 12.375". 25. 27.50". 17. 1.44". 20. 6.60". 23. 13.125". 26. 31.25". 18. 2.45". 21. 8.25". 24. 16.875". 27. 42.00". CHAPTER VI GEARS Nature of Gears. If two wheels are mounted on parallel axes, and if the rims of the wheels are placed in close con- tact, when one is turned the other will also turn if there is sufficient friction. The surfaces thus placed in contact are called friction surfaces. The circumferences of these wheels are called pitch circles, and the diameters of these circles are called pitch diameters. To prevent slipping the friction surfaces are usually grooved so as to form teeth which mesh together. Wheels thus provided with teeth are known as gear wheels, or gears- A toothed wheel in which the teeth are cut parallel to the axis and at right angles to the sides is called a spur gear. In the spur gearing here shown the smaller gears are called pinions and the straight bar is called a rack. Besides spur gears there are gears on conical surfaces which transmit power to shafts at an angle to each other. Such gears are known as bevel gears. Gears having teeth on the inside of the rim are called internal gears. Teeth. It is evident that the teeth may have various shapes, but there are only two sys- tems in general use, the involute, or single-curve form of tooth, and the cycloidal, or double-curve form. The cycloidal form is being rapidly replaced by the involute form. 85 SPUR GEARING 86 GEARS Terms Used. The figure on page 87 illustrates the defini- tions of the following terms, which we shall use in connection with the work on spur gearing: Addendum. The height (a) of a tooth above the pitch circle. Center distance. The distance (^) c ) between the centers of the shafts to which the gears are fastened. Chordal pitch. The distance from the center of one tooth to the center of the next tooth, measured on a straight line; that is, on the chord of the pitch circle. Circular pitch. The distance (P c ) between the centers of two adjacent teeth, measured on the pitch circle. It includes the equivalent of one space and one tooth. Clearance. The distance (c) from the working-depth circle to the whole-depth circle to allow freer action of the gears. Dedendum. The depth (d) of a tooth between the pitch circle and the working-depth circle. Diametral pitch. The number of teeth per unit of the pitch diameter of the gear. Thus, if there are 48 teeth on a gear with a pitch diameter of 12", the diametral pitch of the gear is T f, or 4. Unless otherwise stated, the "pitch" of a gear means the " diametral pitch ." Face of a tooth. The working surface (Fa) outside the pitch circle and extending the width of the tooth. Flank of a tooth. The working surface (FF) inside the pitch circle and extending the width of the tooth. Outside circle. The circumference ( F) of the gear blank. Outside diameter. The diameter (I> ) of the outside circle. Pitch circle. The circumference of the friction cylinder. Pitch diameter. The diameter (Dp) of the pitch circle. Pitch line. The circumference line (X) of the pitch circle. Root. The total depth of a tooth inside the pitch circle, equivalent to the dedendum plus the clearance. TERMS USED 87 Thickness of tooth. The thickness () of the tooth, measured on the pitch line in the same manner as the circular pitch. TERMS USED IN SPUR GEARING , addendum ; c, clearance ; d, dedendum ; D c , center distance ; D , outside diam- eter; D p , pitch diameter; Fa, face; Fl, flank; P c , circular pitch ; t, thickness of tooth on pitch line ; U, whole-depth circle ; F, working-depth circle ; W, whole depth ; X, pitch circle ; F, outside circle Whole depth. The distance (JF) from the outside circle to the circle (U) formed by the bottom of the grooves. Working depth. The distance from the outside circle to the circle (F) formed by the top of the teeth of the mating gear. 88 GEARS Symbols. We shall use the following symbols in the for- mulas and computations for spur gearing: a = addendum P c circular pitch c = clearance P d = diametral pitch d = dedendum r = root D = outside diameter t = thickness of tooth D v = pitch diameter W = whole depth N= number of teeth W' ' = working depth Spur-Gearing Formulas. The following formulas used in practical work may be referred to in solving the exercises : 10 TT D P d P* 2 7 ~ 7T (N+2)a =Z> a P in which we find c from the formula c - and t from the formula t = 5-^- Since TT = 3.1416, 1 5708 we have = 1.5708, and hence t = - . If we substitute this value for t in the formula c = -, we shall then have c = ' , or c = -^ -- 10 r& -frf Using this value for c, we have Such a formula when once developed may be used in finding the whole depth of tooth for any given pitch. 12. How many teeth should there be on a gear having a pitch of 12 and an outside diameter of 3|"? 13. If _ZJ = 1|, what is the value of P c ? Write a statement of the meaning of the problem, not using the symbols. 14. If the outside diameter of a gear is 4" and the number of teeth is 46, what is the pitch ? 15. Find the number of teeth for a gear having a pitch of 14 and an outside diameter of 7f ". 16. Find the thickness of a tooth on an 8-pitch gear. 17. Find the whole depth of a tooth on an 11-pitch gear. 18. Find the pitch of a 75-tooth gear having an outside diameter of 19J". 19. The pitch diameter of a gear is 1.8182" and the pitch is 22. Find the number of teeth and the outside diameter. 20. The circular pitch of a gear is If". Find the working depth of a tooth. CENTER DISTANCE 91 Cutting Spur Gears. In cutting spur gears on a milling machine a cutter of the proper pitch is placed on the arbor of the machine, the table of which is set at zero. The center line on the cutter teeth must be central with the axis of the gear blank, which is held on an arbor between the dividing- head and tailstock centers. The gear is indexed by one of the methods described in Chapter V. Center Distance. In speaking of a pair of gears the smaller gear, if one is smaller than the other, is called the pinion, the other being called the gear. In finding the center distance D c of two spur gears in mesh, we use the formula CUTTING A SPUR GEAR ON A MILLING MACHINE in which N g and N p represent the number of teeth on the gear and the number of teeth on the pinion respectively, and P d is the diametral pitch. There is also a convenient rule used in finding the pitch diameters of two spur gears in mesh, given the center distance and the ratio v g : v p of the velocities of the gear and the pinion in lowest terms. This rule expressed by formulas is as fol- lows, D pg and D pp being the pitch diameters of the gear and the pinion respectively: D pg = 2v h x 92 GEARS Illustrative Problems. 1. Two gears in mesh have 90 teeth and 70 teeth respectively. If the pitch of the gears is 10, what is the center distance? 90 + 70 = 160 = 2 x 10 20 Therefore the center distance of the pair of gears is 8". 2. Find the pitch diameters of two gears that are 12" between centers and have velocities hi the ratio of 1 to 2. The smaller gear, or pinion, will obviously have the larger velocity, so that in the ratio of velocities 2 represents the velocity of the pinion and 1 represents that of the gear. D = 2 x 2 x -I?- = 2 x 2 x 4 = 16. Therefore the pitch diameters of the gear and the pinion are 16" and 8" respectively. Exercises. Spur Gears 1. Two gears in mesh have 36 teeth and 24 teeth respec- tively. If their pitch is 8, what is the center distance ? 2. Two gears in mesh have 128 teeth and 76 teeth respec- tively. If their pitch is 18, what is the center distance ? 3. If the ratio of the velocities of two gears in mesh is 4 to 5 and the center distance of the gears is 27", what are the pitch diameters? 4. If the center distance of two gears in mesh is 16" and the ratio of the velocities is 2 to 3, what are the pitch diameters of the gears ? 5. The ratio of the velocities of two gears in mesh is 3 to 4, and the center distance is 21". If the gears have a pitch of 10, what is the outside diameter of each gear? SPUE GEAKS 93 6. In Ex. 5 find the number of teeth on each of the gears. 7. If two 6-pitch gears in mesh have 132 teeth and 66 teeth respectively, what is the center distance ? 8. If the velocity ratio of two gears in mesh is 5 to 6 and the distance between the centers is 33", what are the pitch diameters of the gears ? 9. If the pitch in Ex. 8 is 6, find the outside diameters. 10. Find the number of teeth on each of the gears in Ex. 8. 11. If two 3-pitch gears in mesh have 182 teeth and 96 teeth respectively, what is the center distance ? 12. If the velocity ratio of two gears in mesh is 3 to 1 and the distance between the centers of the gears is 48", what are the pitch diameters? 13. If the pitch in Ex. 12 is 4, find the outside diameters. 14. Find the number of teeth on each of the gears in Ex. 12. 15. The center distance of two gears in mesh is 6". One gear has 48 teeth and its pitch is 16. Find the number of teeth and the outside diameter of the other gear. 16. If the center distance of two gears in mesh is 12", and the gears have a velocity ratio of 4 to 6, find the pitch diam- eters. If the pitch of the gears is 10, find the number of teeth on each gear and also find the outside diameters. 17. If the gears in Ex. 16 are cut on a milling machine which is equipped with a Cincinnati dividing head, find the indexing required in cutting each gear. 18. If the pitch diameters of two gears in mesh are 41" and 13J" respectively, find the velocity ratio and the dis- tance between the centers of the gears. 19. Two gears mesh together, one having 76 teeth and an outside diameter of 5", and the other 49 teeth and an outside diameter of 3". Find the center distance of the gears. 94 GEARS Comparative Sizes of Gear Teeth. In order that the student may have an idea of the comparative sizes of the gear teeth that he is using, several figures are shown on page 95. In some of these cases both P d and P c are given, but either may be found from the other by the formulas on page 88. Exercises. Gear Teeth Find the circular pitch of a gear when the diametral pitch in each case is as follows : 1. f. 3. 14. 5. 18. 7. 22. 9. 28. 11. 38. 2. 1J. 4. 16. 6. 20. 8. 24. 10. 32. 12. 46. Find the diametral pitch of a gear when the circular pitch in each case is as follows : 13. 0.0628". 15. 0.0873". 17. 0.1047". 19. 0.2856". 14. 0.0714". 16. 0.0924". 18. 0.1208". 20. 0.8976". 21. Find the thickness on the pitch line of a \-P d gear tooth; of a 1|-^J gear tooth. A \-P d gear tooth is a tooth on a gear which has a pitch of \. 22. Find the addendum of a 4-inch-^ gear tooth. A 4-inch-P c gear tooth is a tooth on a gear in which P c = 4". 23. Find the whole depth of a %\-P d gear tooth. 24. Find the thickness on the pitch line of a i|-inch-JJ gear tooth; of a l T 3 g-inch-P gear tooth. 25. If the pitch diameter of a spur gear is 2.8571" and there are 40 teeth, what is the pitch of the gear ? 26. Find the addendum of a 1J--ZJ gear tooth. 27. Find the clearance of a Jg-inch-p; gear tooth. 28. Find the whole depth of a 38-P, gear tooth. 29. Find the working depth of a |-inch-^ gear tooth. SIZES OF GEAR TEETH 95 36 P d 32 P d 28 P d 24 P d 20 P 16 P 14 P 2 P d P c = 0.2618" 10 / J d /> c = 0.3142'' 9 P d 7% -0.3491' 3 P d P c = 1.0472" COMPARATIVE SIZES OF GEAR TEETH 96 GEAKS Exercises. Review 1. Find the number of teeth on a spur gear having an outside diameter of 4.4" and a pitch of 10. 2. Find the pitch diameter of a spur gear that has 32 teeth and a pitch of 10. 3. In Ex. 2 find the outside diameter. 4. Find the pitch of a gear in which ^ = 6.2832". 5. Find the center distance of two 12-JJ gears in mesh that have 66 teeth and 54 teeth respectively. 6. Find the circular pitch of a gear that has a pitch of 34. 7. Find the whole depth of a tooth on a rack that has a linear pitch of 0.1745". When referring to a rack the term ** linear pitch " is used in place of the term " circular pitch." 8. Find the pitch diameter of a gear with 42 teeth and a circular pitch of 0.2618". 9. The velocity ratio of two gears in mesh is 3 to 5 and the center distance is 24". Find the pitch diameters. 10. If the pitch of the gears in Ex. 9 is 4, what is the number of teeth on each gear? 11. In Ex. 10 what is the outside diameter of each gear? 12. Find the pitch of a spur gear that has 80 teeth and an outside diameter of 16.4". 13. If a spur gear has a pitch of 36, what is the working depth of a tooth? 14. If the circular pitch of a gear is 0.1653", what is the addendum ? the thickness of tooth on the pitch line ? 15. If a gear has 140 teeth and an outside diameter of 10}", what is the pitch ? BEVEL GEARS 97 BEVEL GEARS Bevel Gears. Bevel gears are used for transmitting rotary motion from one shaft to another where the two shafts are not parallel to each other. While the two shafts are generally at right angles to each other, this is not essen- tial ; gears can be made to allow for any particular angle desired. As before, one of the two gears, the smaller, if there is a smaller, is called the pinion, the other one being then called the gear. The teeth of bevel gears are con- structed on imaginary friction cones in the same way that the teeth of spur gears are constructed on imaginary fric- tion cylinders. Such terms as diametral pitch, circular pitch, whole depth, pitch diameter, and outside diameter are used with the same general meaning as in the case of spur gears. Since the teeth of a bevel gear constantly become thinner as they approach the apex of the cone, in cutting bevel gears with a rotary cutter on a milling machine, it is not possible to obtain perfect finished teeth. The teeth are cut to approximate size by offsetting the cutter and rotating the blank, and the final thinning of the small end of the teeth is generally done by filing. At the large end of the teeth on a bevel gear the circular pitch, the thickness of tooth on the pitch line, and the whole depth of tooth are identical with those of a spur gear of the same size and pitch. Bevel gears are always figured in pairs, and we shall con- sider only such cases as those in which the shafts are at right angles to each other. The method of computing the essential measurements for cutting a pair of bevel gears is shown on page 101. 98 GEARS Terms Used. In addition to the terms used with spur gearing given on page 86, the following terms, illustrated in the figure below, are used in connection with bevel gears : Addendum angle. The angle (V) between the pitch line and the top line of a tooth extended to the vertex. TERMS USED WITH BEVEL GEARS a, addendum ; a', addendum angle ; (7, cutting angle ; d, dedendum ; d', deden- dum angle ; D , outside diameter ; D p , pitch diameter; F, width of face ; F', face angle ; K, pitch-cone angle ; P, pitch line ; /?, pitch-cone radius ; W, whole depth ; x, angular addendum ; Z, vertex Angular addendum. The straight-line distance (x) from the pitch circle to the outside circle. Twice the angular addendum is called the diameter increment. Cutting angle. The angle (C) between the bottom of a groove and the axis of the gear. This angle is the same as the angle at which the dividing head is set while cutting the teeth. Dedendum angle. The angle (d 1 ) formed by the bottom of a groove and the pitch line. TERMS USED 99 Face angle. The angle (/' 7 ') with the axis to which the gear blank is turned. Pitch circle. The circle formed by a cross section of the imaginary friction cone, the diameter of the circle being the pitch diameter. Since bevel gears are constructed on cones they may have several pitch circles, as, for example, a pitch circle at the larger end of the teeth and one at the smaller end. In speaking of the pitch of a bevel gear we always mean the number of teeth per unit of the diameter of the largest pitch circle ; that is, the diam- eter D p in the figure on page 98. Pitch-cone angle. The angle (7f) with the axis of the imag- inary friction cone. Pitch-cone radius. The radial distance (J) from the vertex to the largest pitch circle. This distance is measured from the vertex Z to the circle of which D p is the diameter in the figure on page 98. If the surface of the imag- inary friction cone were laid out flat it would be a sector of a circle. The pitch-cone radius is the radius of this circle, and is sometimes called the apex distance, or vertex distance. Pitch line. A line equivalent to the slant height of the imaginary friction cone ; more specifically, that part (P) of the line lying within the teeth. Also, as in spur gearing, the circumference of the greatest pitch circle ; that is, the circle measured by D p in the figure on page 98. Vertex. The apex (Z) of the cone on which the gear is cut. Width of face. The length (.F) of the teeth, usually cut so as to be about one fourth the pitch-cone radius. CUTTING A BEVEL GEAR ON A MILLING MACHINE 100 GEARS Symbols. In the formulas and in computations relating to bevel gears we shall use the following symbols : a = addendum F 1 = face angle a' = addendum angle K = pitch-cone angle C = cutting angle N = number of teeth d = dedendum P d = diametral pitch d' = dedendum angle R = pitch-cone radius D = outside diameter W= whole depth D p = pitch diameter x = angular addendum The subscript g for " gear " or p for " pinion " may be added to indi- cate that a certain symbol applies to the gear or to the pinion. Thus C g is the cutting angle of the gear and C p is that of the pinion. Bevel-Gear Formulas. The following formulas may be referred to in solving the exercises on bevel gears : All the formulas in which the subscripts g for " gear " and p for " pinion " are not used, apply to both the gear and the pinion. We may indicate that a formula applies to the gear or to the pinion by adding the proper subscripts, as in the following cases : D =j? D = T? ^d r d The result found from the formula for d is actually the dedendum plus the clearance, but in bevel gears we treat this as the dedendum. If the machinist uses formed cutters and the correct indexing, the thickness on the pitch line at the large end of a tooth will be correct. This measurement is usually checked with a table of chordal thicknesses. BEVEL-GEAR FORMULAS 101 Illustrative Problem. Make the calculations for the pair of -7J bevel gears shown below in which N g = 48 and N p = 24. (1) tan K g = a. = = 2.0000, and hence K g = 63 26'. JN p 24 K p = 90 - K g = 90 -63 26' = 26 34'. Hence D p of gear is 4" and Z) p of pinion is 2". = 4 > v 8Q44 Hence D of gear is 4.075" and D of pinion is 2.149". D p 4 ~ 2 sin K y ~ 2 x 0.8945 5 ' Hence R for both gear and pinion is 2.236". = 0.1798. Hence W g and W p are each 0.180". (5) W = = (6) a = = = 0.0833. Hence a g and a p are each 0.083". (7) d = = 0.0964. Hence d g and d p are each 0.096". (8) tan a' = - = ' = 0.0373, and a' (both gear and pinion) = 2 8'. ''' (9) tan d' = = ' = 0.0431, and ef (both gear and pinion) = 2 28 7 . -R 2.23o9 (10) C g = K g -d' = 63 26' - 2 28' = 60 58'. C p = K p -d' = 26 34' - 2 28' = 24 6'. (11) F' g = K g + a' = 63 26' + 2 8' = 6534 X . F' = K + a' = 2634' + 28 / = 2842 / . 102 GEAKS Exercises. Bevel Gears For a pair of level gears, if it is required that P d 9, N g = 36, and N p = 18, find each of the following : 1. The pitch diameters of both the gear and the pinion. 2. The pitch-cone angles of both the gear and the pinion. 3. The outside diameters of both the gear and the pinion. 4. The pitch-cone radius. 5. The whole depth of the teeth. 6. The addendum and the dedendum. 7. The addendum angle and the dedendum angle. 8. The cutting angles of both the gear and the pinion. 9. The face angles of both the gear and the pinion. 10. The thickness of tooth on the pitch line. This measurement at the large end of the teeth may be calculated from one of the spur-gearing formulas for t given on page 88, but it is not an essential measurement when the gear is cut with formed cutters. As a check, however, the thickness of tooth is usually measured with a special gear-tooth vernier caliper and compared with a table of chordal thicknesses of gear teeth, since the caliper measures the chordal thickness and not the thickness of tooth along the pitch circle. Make all the necessary calculations for cutting each of the following pairs of bevel gears : 11. Gear, 52 teeth; pinion, 36 teeth; pitch, 14. 12. Gear, 45 teeth ; pinion, 15 teeth ; pitch, 10. 13. Gear, 56 teeth ; pinion, 32 teeth ; pitch, 8. 14. Gear, 48 teeth; pinion, 12 teeth; pitch, 14. 15. Gear, 60 teeth; pinion, 18 teeth; pitch, 12. The elements required in making the calculations for cutting a pair of bevel gears are shown on page 101. SPIRAL GEARS 103 SPIRAL GEARS Spiral Gears. Gears in which the teeth are wound spirally around the axis instead of being cut parallel to the axis, as in a spur gear, are called spiral gears or helical gears. Although it is the common term, the name " spiral gear " is not, strictly speaking, correct, since the teeth are not wound around the axis in a spiral, but in a helix. Spiral gears are essentially cylinders with a succession of equally spaced spiral grooves cut on their periphery, and hence are more nearly like spur gears than bevel gears. Spiral gears may have their axes parallel, as in the case of spur gears, or the axes may be at any desired angle to each other. A tooth of a spiral gear is much like the thread of a screw, although it does not have the same cross section nor is it meant to do the same kind of work. In this connection it should be remembered that the meaning of the term " pitch " when applied to gears is different from that when the term is applied to screw threads. In the treatment of spiral gears we shall consider only those cases in which a variation in the center distance of the gears is permissible ; that is, those in which the center distance is said to be approximate. Uses of Spiral Gears. Spiral gears are used when high speed with smooth and noiseless action is desired. The absence of noise is due to the fact that the engagement of the teeth is gradual, for instead of striking a full line of contact at once, the teeth roll on each other with a sliding motion, one set engaging while the preceding set is still in contact. Thus, the smoothness of action of spiral gears is hardly impaired by wear, and these gears are used extensively in automobiles and many different types of machines. 104 GEARS Terms Used. In addition to the terms used in connection with spur gearing, as given on page 86, the following special terms are used with reference to spiral gears : Axial pitch. The distance from the center of one tooth to the center of the next tooth, measured parallel to the axis of the gear. Since it is made parallel to the axis, this meas- urement is similar to the pitch of a screw thread. Circular pitch. The distance from the center of one tooth to the center of the next tooth, measured on the pitch circle. This measurement is made at right angles to the axis of the gear as in spur gearing. On account of the different uses of the word " pitch " with spiral gears, this is sometimes called the SPIRAL-GEAR TEETH real circular pitch, and the diametral pitch corre- This shows the differ- sponding to this is called the real pitch. ence between the ways in which the normal Lead. The distance that the spiral on circular pitch (N) and , . , ., ., , . the axial pitch (A) are which the teeth are cut advances in a measured single turn. Normal circular pitch. The distance from the center of one tooth to the center of the next tooth, measured at right angles to the center line of the teeth. This distance is shorter than the real circular pitch. The section of the teeth which shows the normal circular pitch is called the true section. Normal diametral pitch. The diametral pitch corresponding to the normal circular pitch. The diametral pitch of the cutter used in cutting the teeth is the same as the normal pitch (normal diametral pitch) of the gear. Tooth angle. The angle measured at the pitch circle between the center line of the teeth and a line parallel to the axis. This angle is also called the spiral angle of the gear. SPIRAL GEARS 105 Cutting Spiral Gears. The teeth of spiral gears when cut on a milling machine are generally formed with ordinary involute spur-gear cutters, the cutter of the proper pitch and number for the given spiral gear being selected as described on page 107. The gear cutter is fastened firmly on the arbor of the milling machine. The dividing head and tailstock are set in position on the table of the machine with their centers in alignment and central with the cutter. The change gears needed to give the required spiral lead are placed in their proper positions, as shown on page 74, and the divid- ing head is set to index the number of teeth to be cut on the gear. The gear blank is pressed firmly on the mandrel and is mounted between the dividing-head and the tail- stock centers. The blank is brought under the cutter, and the table is set to the required tooth angle and clamped. The blank is then moved clear of the cutter, the knee is raised for the required depth of cut, and the vertical dial is set at zero. The machine is then started and the table moved by hand until the gear blank touches the cutter, when the power feed is thrown in. In returning the blank for another cut, the knee should be lowered until the blank clears the cutter, thus preventing the cutter from dragging through the cut just made. CUTTING A SPIRAL GEAR ON A MILLING MACHINE 106 GEARS Symbols. The following symbols are used in the formulas and computations for spiral gears: A = tooth angle P n = normal pitch D c = center distance r = ratio of gear to pinion D = outside diameter S = angle of shafts D p = pitch diameter T c = number for selecting L = lead of spiral gear cutter N = number of teeth W = whole depth In order to distinguish between the symbols for the gear and those for the pinion, we add the subscript g or p as in previous cases. Thus, A g is the tooth angle of the gear and A p is the tooth angle of the pinion. The ratio r of gear to pinion is the ratio of the number of teeth on the gear to the number on the pinion. If the numbers are the same, the ratio is commonly said to be equal. Spiral-Gear Formulas. The following formulas are used with spiral gears when the ratio is unequal, the center dis- tance is approximate, and the shafts are at any angle : When the subscripts g for "gear" and/> for "pinion" are not used, the formulas apply to both gear and pinion, and in distinguishing between the formulas for the gear and those for the pinion the proper subscripts may be added, as in the following cases : L g = ' Tfl) pg cot A g L p = irD pp cot A p In the above formula for N pt if A g = A p , we have , T _2D c P n cosA SPIRAL GEARS 107 Gear Cutters. In the Brown and Sharpe system of cutters for spur gears with involute teeth, eight different shapes of cutters, as shown in the table below, are made for each pitch. The cutters are num- bered according to the number of teeth to be cut on the gear, a #8 %-P d cutter, for example, being required for cutting 8-P z spur gears with 12 or 13 teeth. For cutting a spiral gear the proper number of cutter of the normal pitch of the gear is selected from the table according to the number found INVOLUTE GEAR by the formula for T c on page 106. CUTTER If with these cutters the correct indexing and the proper depth of cut are used, the thickness of tooth on the pitch line will be correct. NUMBER OF CUTTER NUMBER OF TEETH NUMBER OF CUTTER NUMBER OF TEETH 1 135 to rack 5 21 to 25 2 55 to 134 6 17 to 20 3 35 to 54 7 14 to 16 4 26 to 34 8 12 to 13 A #1 cutter is used in cutting gears with 135 or more teeth or in cutting the teeth on racks. Change Gears. The change gears required to cut the proper lead on a spiral gear may be found by the method described on page 75. Since the spiral leads as calculated from the formula on page 106 usually contain decimals which are difficult to factor, it is customary, in practice, to use tables of spiral leads, which show the leads obtainable with every possible arrangement of the change gears furnished with the dividing head. When an exact lead cannot be found in the table the nearest obtainable lead is generally used. 108 GEARS Illustrative Problem. Make the calculations for cutting a pair of spiral gears in which it is required that r = 4, P n = 6, D c (approximate) = 12", A g = 30, A p = 20, and S= 50. r 2D c P n cosA ff cosA p _ 2 x 12 x 6 x 0.8660 x 0.9397 _ 25 o p = rcoaA p + coaA ff 4 x 0.9397 + 0.8660 Such a result is taken to the nearest whole number, in this case 25. N g = rN p = 4 x 25 = 100. The gear and pinion have 100 teeth and 25 teeth respectively. no 9456 = p 6 x 0.9397 The pitch diameters of gear and pinion are 19.246" and 4.434". (3) D og = D pg + ^ = 19.2456 + | = 19.5789. * n D o P = D pp + - = 4.4340 + | = 4.7673. * n The outside diameters of gear and pinion are 19.579" and 4.767". (4) L g = irD pg cotA g = 3.1416 x 19.2456 x 1.7321 = 104.7262. L p = irD pp coiA p = 3.1416 x 4.4340 x 2.7475 = 38.2724. The leads of the spirals on gear and pinion are 104.73" and 38.27". 1QO _ 154 n ~ 301 cos 3 A p 0.9397 3 " 0.8298 " From the table on page 107 we see that a #1 Q-P (l cutter is required for the gear and a #4 6-P rf cutter for the pinion. The whole depth of tooth on both gear and pinion is 0.360''. (7) D c = J (D pg + D pp ) = $ (19.2456 + 4.4340) - 11.8398. The "exact" center distance of the gears is 11.840". SPIRAL GEARS 109 Exercises. Spiral Gears at Any Angle For a pair of spiral gears, if it is required that r = 2, P n = 10, D c (approximate) = 10", A g =30, A p = 20, and S= 50, find each of the following : 1. The number of teeth on the gear and on the pinion. 2. The pitch diameters of both the gear and the pinion. 3. The outside diameters of both the gear and the pinion. 4. The leads of the spirals on both the gear and the pinion. 5. The numbers of the two cutters required. 6. The whole depth of the teeth. 7. The exact center distance of the gears. For a pair of spiral gears, if it is required that r = 3, J^=4, D c (approximate) = 16' f , A g =40, A p =20, and S = 60, find each of the following : 8. The number of teeth on the gear and on the pinion. 9. The pitch diameters of both the gear and the pinion. 10. The outside diameters of both the gear and the pinion. 11. The leads of the spirals on both the gear and the pinion. 12. The numbers of the two cutters required. 13. The whole depth of the teeth. 14. The exact center distance of the gears. 15. Make the necessary calculations for cutting a pair of spiral gears, it being required that D c (approximate) = 1", r = l^, P n =S, A g =25, 4, = 15, and S=40. 16. Make the necessary calculations for cutting a pair of spiral gears, it being required that D c (approximate) = 1 2", r = 4, %=$, 4, = 30, 4, = 30, and S=60. In finding N p use the formula given at the foot of page 106. 110 GEARS Spiral Gears with Parallel Shafts. When the ratio of a pair of spiral gears is unequal, the center distance approxi- mate, and the shafts parallel, as shown in the illustration, the tooth angle of the gear is equal to the tooth angle of the pinion ; that is, A g = A p . For example, make the calculations for cutting a pair of spiral gears with parallel shafts, if it is required that D c (approximate) = 10", N g = 48, N p = 24, % = 4, and A= 24. CD ft, " 99 P n cosA 4 x 0.9135 24 = 13.1363. = 6.5681. 4 x 0.9135 The pitch diameters of gear and pinion are 13.136" and 6.568" respectively. (2) D og = = 13.1363 + f = 13.6363. = 6.5681 * = 7.0681. SPIRAL GEARS WITH PARALLEL SHAFTS The outside diameters of gear and pinion are 13.636" and 7.068". (3) L g = TrD pg cotA = 3.1416 x 13.1363 x 2.2460 = 92.6902. L p = irD pp cotA = 3.1416 x 6.5681 x 2.2460 = 46.3446. The leads of the spirals on gear and pinion are 92.69" and 46.34". N g 48 48 cos 3 .! ~ N n (4) T cg = TCP 0.9135 3 0.762.3 24 24 = 63.0. = 31.5. ooeM 0.9135 3 0.7623 From the table on page 107 we see that a #2 4-P d cutter is required for the gear and a #4 4-P d cutter for the pinion. (5) W = -* = 0.5393. The whole depth of tooth on both gear and pinion is 0.539". (6) D c = $ (D pg + D pp ) = $ (13.1363 + 6.5681) = 9.8522. The "exact" center distance of the gears is 9.852". SPIRAL GEARS 111 Exercises. Spiral Gears with Parallel Shafts 1. For a certain pair of spiral gears it is required that the shafts be parallel at an approximate center distance of 2^", that JV^=32, N p =20, P n =l2, and ^=18. Find the pitch diameters of both the gear and the pinion. 2. In Ex. 1 find the outside diameters of both the gear and the pinion. 3. In Ex. 1 find the leads of the spirals on both the gear and the pinion. 4. In Ex. 1 find the number of the cutter to be used for cutting the gear and the same for the pinion. 5. In Ex. 1 find the whole depth of the teeth. 6. In Ex. 1 find the exact center distance. For a pair of spiral gears, if it is required that the shafts be parallel, that D c (approximate) = 12-j-", r = 1-j, P n = 6, and A = 20, find each of the following : 7. The number of teeth on the gear and on the pinion. In finding N p use the formula given at the foot of page 108. 8. The pitch diameters of both the gear and the pinion. 9. The outside diameters of both the gear and the pinion. 10. The leads of the spirals on both the gear and the pinion. 11. The number of the cutter to be used for cutting the gear and the same for the pinion. 12. The whole depth of the teeth. 13. The exact center distance of the gears. 14. A spiral gear with a pitch diameter of 6.2121" has a spiral lead of 72.83". Find the angle at which the milling- machine table was set in cutting the gear. 112 GEAES Worm Gearing. When the difference between the velocities of two shafts is to be great, worm gearing is often employed. In this case the driver, which is called a worm, is a single- thread or multiple-thread screw, the threads of which mesh with a special form of spur gear known as a worm wheel. Uses of Worm Gearing. Worm gearing is generally used when the load is heavy and when smoothness of action and a large reduction in velocity are required in trans- mitting rotary motion from one shaft to another. Usually the two shafts are at right angles to each other. Terms Used. In general the terms and symbols which are used with spur gearing are also used in connection with worm gearing, together with the following addi- tions, which will be understood from the second figure on page 113: Face angle. The angle, usually arbitrarily selected, to which the ends of the teeth on the worm wheel are trimmed. This angle varies from about 55 to 65, depending upon the kind of worm used. The face angle of the wheel shown on page 113 is 60. Gashing angle (). The angle for swiveling the table of the milling machine when cutting the teeth on the worm wheel to correspond to the spiral angle of the thread on the worm. Linear pitch (JJ). This is identical with the circular pitch of the worm wheel and is used in speaking of the worm. Since the two terms are identical, we use the symbol P c for both. Throat diameter (!><). The diameter of the worm wheel measured at the center of the curved teeth. The throat diameter is equivalent to the outside diameter of a spur gear having the same number of teeth and the same pitch. WORM GEARING WORM GEARING 113 Worm-Thread Formulas. The following formulas are used with standard worm threads which have an angle of 29 : a = 0.3183P C W- 0.6866P C D p =D -2a W t = 0.3354 P c The formula for D gives the outside diameter for a single- thread worm; for a double- thread worm use 5^, and for a triple-thread worm use The width (JF 6 ) at the root of the point of the tool used in Worm-Wheel Formulas. The with worm wheels: STANDARD WORM THREAD a, addendum; P, pitch line; P c , linear pitch; W, whole depth of thread ; W b , width at root of thread ; W t , width at top of thread of the thread is the width cutting the worm thread. following formulas are used tanG= PC TTD 'pw n _J^ p _ IT CROSS SECTION OF WORM P~~~p~ ** = |Br AND WORM WHEEL * d * c D c , center distance ; D , outside D Q = D p -J- 4 a t = \P c ' diameter; D p , pitch diameter; D t , throat diameter In these formulas D pw is the pitch diameter of the worm and D p is the pitch diameter of the worm wheel. The depth of tooth on the worm wheel, when figured for gashing the teeth on a milling machine, is taken as the whole depth of the worm thread. This evidently makes no allow- ance for clearance, but when the teeth are hobbed, as ex- plained on page 115, an extra depth of tooth is given to the hob to make the necessary allowance. 114 GEARS Illustrative Problem. Make the calculations for cutting a worm and worm wheel with a velocity ratio of 40 to 1, the worm to be single-threaded with a linear pitch of 0.2618". _ TT _ 3.1416 _ (1)Pd -F c ~2618- The diametral pitch of the worm wheel is 12. (2) a = -L = =0.0833. The addendum of the worm wheel is 0.083". (3) Since the velocity ratio is 40 to 1, the wheel must have 40 teeth. _^_40 p ~ ~P~ d ~ 12 ~ The pitch diameter of the worm wheel is 3.333". (4) D t = D p + 2 a = 3.3333 + (2 x 0.0833) = 3.4999. The throat diameter of the worm wheel is 3.500". (5) D = D p + 4 a = 3.3333 + (4 x 0.0833) = 3.6665". The outside diameter of the worm wheel is 3.667". (6) t = \P C = % x 0.2618 = 0.1309. The thickness of tooth on the pitch line of the worm wheel is 0.131". (7) D = 4 P c = 4 x 0.2618 = 1.0472. The outside diameter of the worm is 1.047". (8) The addendum of the worm is the same as that of the worm wheel. D p = D -2a = 1.0472 - (2 x 0.0833) = 0.8806. The pitch diameter of the worm is 0.881". (9) D c = } (D p + D pw ) = i (3.3333 + 0.8806) = 2.1070. The center distance of the worm and worm wheel is 2.107". (10) W = 0.6866 P c = 0.6866 x 0.2618 = 0.1798. The whole depth of the worm thread is 0.180". (11) W b = 0.3095 P c = 0.3095 x 0.2618 = 0.0810. The width of the point of the worm-thread tool is 0.081". -p A Oftl Q (12) tan G = ^- = 3 1416 x . 8806 = 0.0946, and hence G = 5 24'. Since the angle for setting a milling-machine table is calculated to the nearest \, the gashing angle for the worm-wheel teeth is 5 \. WORM GEARING 115 Cutting Worm Wheels. When cutting worm wheels on a milling machine, the teeth are usually gashed first and then hobbed to give them their final shape. The gashing is done with an involute gear cutter of the proper number and pitch, the table of the machine being first adjusted to center the blank and cutter and then set to the gashing angle. After the teeth are gashed the milling-machine table is set at zero, and the cutter is re- placed by a hob. A hob is a tool shaped like a worm and grooved to form cutting teeth, the depth GASHING A WORM WHEEL ON ,. i . T . T T.^T A MILLING MACHINE of which is made a little greater than the depth of the worm thread to allow for clearance. Exercises. Worm Gearing From the data given, make the calculations for cutting each of the following sets of worm gearing : 1. Worm wheel, 30 teeth; worm, single-threaded with a linear pitch of 0.3927". 2. Velocity ratio, 20 to 1 ; worm, double-threaded with a linear pitch of 0.5236". Each revolution of this worm will move the worm wheel two teeth. 3. Worm wheel, 24 teeth; worm, single-threaded with a linear pitch of 0.2244". 4. Worm wheel, 96 teeth; worm, triple-threaded with a lead of 2.3562". 116 GEAKS Exercises. Gears 1. If, in a spur gear, D = 2.228" and P c = ^", what is the number of teeth? 2. If an 8-P d spur gear has 203 teeth, what is D ? 3. Find the whole depth of tooth for a |-JJ spur gear. 4. Find the circular pitch for a 32-T spur gear in which it is required that D p 16.55". 5. Two 8-JJ bevel gears are to have 80 teeth and 20 teeth respectively. Make the calculations for cutting the gears. 6. If a 20-inch spur gear has 198 teeth, what is the pitch ? 7. If the circular pitch of a spur gear is 2|", what is the addendum? the whole depth of tooth? 8. If it is required that a worm wheel is to have 20 teeth and the worm a single thread with a linear pitch of 0.3142", make the calculations for cutting the gearing. 9. If a 39-T spur gear has a pitch of 31, what is D ? 10. Make the necessary calculations for cutting a pair of spiral gears, it being required that D c (approximate) = 12", P n =6, ^=30, ^=30, =60, andr=3. 11. A lf-^J spur gear has a pitch diameter of 97.14". Find the outside diameter and the number of teeth. 12. Make the calculations for cutting a pair of bevel gears, it being required that the pitch be 5, the number of teeth on the pinion 17, and the velocity ratio 3 to 1. 13. A worm wheel is to have 60 teeth and is to mesh with a double-thread worm having a linear pitch of 1.0472". Make all the calculations for cutting the gearing. 14. Find the pitch diameters of two spur gears in mesh if the center distance between the gears is 12", and one gear has 40 teeth and the other 20 teeth. CHAPTER VII REVIEW PROBLEMS Exercises. General Applications Change the following tapers per inch to tapers per foot : 1. 0.0762". 2. 0.08175". 3. 0.0162". 4. 0.0542". 5. If the outside diameter of a spur gear is 4J" and the pitch is 18, what is the number of teeth ? 6. Find the tap-drill size of a l|-inch U. S. S. thread. 7. If a quadruple-thread screw has 16 threads per inch, what is the pitch and what is the lead ? 8. Find the change gears required on a lathe, which has a lead screw with 5 threads to the inch, when cutting a metric thread with a lead of 1.5 mm. Assume that the lathe is equipped with a 127-T gear. 9. Two 12-7J spur gears in mesh have 66 teeth and 40 teeth respectively. Find the center distance. 10. Find the pitch diameter of a spur gear having 37 teeth and a pitch of 10. 11. If a spiral lead of 27" is to be cut on a J-inch drill, to what angle should the milling-machine table be set ? 12. Find the indexing required on a B. & S. dividing head for milling a ratchet wheel with grooves 13J apart. 13. Find the outside diameter of a 96-T spur gear which has a pitch of 14. 117 118 REVIEW PROBLEMS Exercises. Drill Press 1. The part of the cast-iron column marked A in the figure below is 2' 4" long and was turned in a lathe to its present diameter of 6". Find how long it took to machine this casting if a roughing cut with a |-inch feed and a finishing cut with a ^-inch feed were taken with a high-speed tool. 2. The finished base B is 12" x 16" and was planed on a 3-to-l planer at 25 F.P.M., a roughing cut with a y^-inch feed and a finishing cut with a |-inch feed being taken. Find the length of time consumed in machining this casting. 3. The diameter of the driv- ing pulley C is 10 J" at the center of the crown and 10" at the edge, and the width of the face is 3". Find the T. P. F. of the pulley. 4. What was the tailstock offset if the pulley in Ex. 3 was turned on a 7J-inch arbor? 5. The diameters of the steps of the cone pulleys are 5", 7", 9", and 11" respectively, and pulley C makes 425 R.P.M. Beginning with the belt on the largest step of the driving pulley, find each of the four speeds of shaft S. 6. Spindle D is made of machine steel and has a diameter of llf" and a length of 28". Find the number of R.P.M. of the lathe for a roughing cut with a carbon-steel tool. DRILL PRESS DKILL PEESS 119 7. The spindle in Ex. 6 was turned from round stock, a roughing cut with a T l g-inch feed and a finishing cut with a 0.01 5-inch feed being taken with a high-speed steel tool. Find the time taken in machining the spindle. 8. If the adjusting screw E has a U. S. S. thread of Jg-inch pitch, what is the root diameter of the screw ? 9. Find the change gears required on a lathe, which has a lead screw with 5 threads per inch, in cutting screw E. 10. Bevel gear G is a 14-JJ gear with 51 teeth, and the pinion meshing with it has 17 teeth. Make all the necessary calculations for cutting this pair of gears. 11. If spur gear E has 71 teeth and a circular pitch of 0.3491", what is its outside diameter? 12. In Ex. 11 find the addendum and the whole depth of tooth of gear R. 13. Find the indexing required on a B. & S. dividing head in cutting gear E. 14. Worm W is a single-thread screw with a linear pitch of 0.2244", and meshes with a worm wheel having 25 teeth. Make the calculations for cutting the worm and wheel. 15. If the adjusting screw E has an Acme screw thread with a pitch of 0.1111", find the depth, the width at the top of the thread, and the number of threads per inch. 16. The circular table of the drill press is made of cast iron and has a diameter of 27" and a thickness of 14". o Find the time required for machining the circumference, a roughing cut with a feed of 0.05" and a finishing cut with a feed of 0.125" being taken with a high-speed steel tool. 17. Find the number of R.P.M. for boring the center hole in the circular table in Ex. 16 to a diameter of 1^" with a carbon-steel tool. 120 EEVIEW PROBLEMS Exercises. Indexing and Milling 1. Using a Cincinnati dividing head, find the indexing required in cutting the teeth on the milling cutter shown in the first figure below. 2. In cutting the teeth on -the milling cutter mentioned in Ex. 1 a high-speed steel cutter 1J" in diameter was used. At how many R. P. M. should this second cutter run in order to give a cutting speed of 55 F.P.M.? 3. If the spiral lead of the teeth is to be 26.25" and the diameter of the blank is 2J", at what angle should the table of the milling machine be set for cutting the spiral grooves in the milling cutter shown in this figure ? 4. If the lead of the milling machine in Ex. 3 is 10'-, what change gears should be used ? 5. If a milling cutter similar to the one in Ex. 3 is to be made with 24 teeth, find the indexing required on a B. & S. dividing head when cutting the grooves. 6. Using a milling machine equipped with a Cincinnati dividing head, find the indexing required in milling the teeth in the metal-slitting saw here shown. 7. If a metal-slitting saw similar to the one mentioned in Ex. 6 is to be made with 79 teeth, find the indexing required on a B. &. S. dividing head. 8. If the table of a milling machine is set at an angle of 7| for cutting a spiral with a lead of 62.50", what is the diameter of the work ? 9. If the lead of the milling machine in Ex. 8 is 10", what change gears should be used ? INDEXING AND MILLING 121 10. Find the different numbers from 1 to 100 that can be indexed by simple indexing with the 21-hole circle of a B. & S. dividing head. 11. Using a Cincinnati dividing head, find the indexing required in cutting the teeth on the side-milling cutter shown in this figure. 12. If the side-milling cutter in Ex. 11 is 4|" in diameter, what is the cutting speed when the cutter is driven at 21 R.P.M.? at 38 R.P.M.? Give such results to the nearest unit. 13. Using a Cincinnati dividing head, find the indexing required in milling the screw-slotting saw shown in this figure, the saw having 72 teeth. 14. Consider Ex. 13 if the saw is to have 89 teeth and the milling machine is equipped with a B. & S. dividing head. 15. Using compound indexing on a B. & S. dividing head, find the indexing fractions required for cutting a screw-slotting saw which is to have 93 teeth. 16. Find the change gears required on a B. & S. dividing head for indexing ^. It is necessary to index for 4 x 360, or 1440 divisions. In figuring the ratio of the change gears assume a movement of the index crank of one hole on the 33-hole circle for each division. 17. Find all the circles on a B. & S. dividing head that will index in common the numbers 3, 12, 15, and 60. 18. What is the lead of the spiral that will be cut on a milling machine which has a lead of 10" when there is a 56-T gear on the worm, a 32-T gear as the first gear on the stud, a 48-T gear as the second gear on the stud, and a 72-T gear on the feed screw ? 122 EEVIEW PROBLEMS Exercises. Tapers 1. Find the T.P.F. of taper A on the automobile pinion shaft shown in the blueprint on page 123. 2. In Ex. 1 find the tailstock offset for cutting taper A. 3. As in Ex. 1, find the T.P.F. of taper B. 4. In Ex. 3 find the tailstock offset for cutting taper B. 5. Find the T.P.I, of the gib key shown in the blueprint. In the following table supply each missing specification for 2-ball clamping levers of the type shown in the blueprint : L i D d T.P.F. TAILSTOCK OFFSET 6. 41" 2ft" 1" ft" 7. 5J 8ft H i 8. <* m I T 9 = fV', and d = 0.140", find the T.P.F. and the included angle of the taper. TAPERS 123 124 REVIEW PROBLEMS Exercises. Planer 1. The table A of the planer shown in the figure below is 4' wide and 18' long and was planed on a 2-to-l planer which had a cutting speed of 25 F.P.M. If a roughing cut with a ^ 3 2 -inch feed and a finishing cut with a T 7 g-mch feed were taken, how long did it take to machine the top of the table ? PLANER 2. The V-shaped ways B are planed to an angle of 50, and the width of the V at the top is 2 y y. Find the depth of the V. Use the formula given on page 28, the included angle being 50. 3. The spur gear C has a pitch diameter of 4" and a pitch of 12. Find the outside diameter of the gear. 4. In Ex. 3 find the number of teeth on the gear. 5. In Ex. 3 find the whole depth of a tooth. PLANER 125 6. In Ex. 3 find the indexing required in cutting the gear on a milling machine equipped with a B. & S. dividing head. 7. The cast-iron pulley D is 22" in diameter, and the face is 3" wide. Find the number of R.P.M. of a lathe for finish- ing the face with a high-speed steel tool. 8. In Ex. 7 how long will it take to make a single cut over the face with a ^ -inch feed ? 9. Find the depth of the thread on the feed screw E which has 6 U. S. S. threads to the inch. 10. In Ex. 9 find the change gears required on a lathe with a lead screw of ^-inch pitch when cutting the thread. 11. The 14-^ spur gear F has 57 teeth. Find the outside diameter of the gear. 12. In Ex. 11 find the thickness of tooth on the pitch line and also find the working depth of a tooth. 13. In Ex. 11 find the indexing required on a B. & S. dividing head in cutting the gear. Why will two intermediate gears be needed ? 14. If the crowned pulley G has a 6i-inch face and is T 5 g" larger in diameter at the center of the crown than at the edges, find the T.P.F. of the pulley. 15. If pulley G is turned on a Ill-inch mandrel, find the tailstock offset. 16. If the cast-iron pulley H has a diameter of 28 |-" and a 3-inch face, find the number of R.P.M. for roughing the face with a tool of carbon steel. 17. In Ex. 16 how long will it take to make a single cut over the face with a ^-inch feed ? 18. Find the number of R.P.M. of pulley H if it is belted to a 24-inch pulley which has a speed of 125 R.P.M. 126 REVIEW PROBLEMS Exercises. Drilling Machine 1. If motor A in the figure below makes 750 R.P.M., if gear 1 on the motor has 24 teeth, and if gear 2 has 56 teeth, what is the speed of the shaft to which gear 2 is fastened? WALL RADIAL-DRILLING MACHINE 2. If the gears in Ex. 1 were spur gears with a pitch of 14, what would be the respective outside diameters ? 3. What would be the center distance if gears 1 and 2 were spur gears which meshed together instead of being con- nected by the endless chain ? 4. Find the indexing required on a Cincinnati dividing head for cutting the gears in Ex. 3. 5. The 6-7J bevel gears B have 48 teeth and 16 teeth respectively. Make the calculations for cutting these gears. DRILLING MACHINE 127 6. The socket in spindle C is bored to a #4 Morse taper, which has a T.P.F. of 0.623". To what angle should a compound rest be set to bore this taper? 7. Shaft Z>, which is made of machine steel, is 10' 6" long and has a diameter of If". Find the number of R.P.M. of a* lathe for roughing the shaft with a high-speed steel tool. 8. In Ex. 7 find how long it will take to make a single cut over the shaft with a feed of 0.015". 9. The worm wheel G has 60 teeth and it meshes with a single-thread worm, the linear pitch of which is 0.1963". Make the calculations for cutting the worm and wheel. 10. If the weight H is made of cast iron and is approxi- mately 2|-" x 5|" x 8^-", find its approximate weight. The weight of cast iron may be taken as 450 Ib. per cubic foot. 11. If the diameter at the middle of the l|-inch face of the cast-iron wheel J is 4||" and that at the edge is 4|", what is the T.P.F. of the wheel? 12. Using the smaller diameter in Ex. 11, at what speed should the face be finished with a high-speed tool ? 13. In Ex. 11 find the time required for finishing 25 wheels of this type, making a single cut with a gL-mch feed over the face of each wheel. 14. Shaft D makes an angle of 22 with brace E and is perpendicular to shaft F. If the distance between the axis of shaft F and the vertex of the angle formed by Z> and E is 9' 8", find to the nearest 0.1" the length of E between the vertices of the angles formed by E with D and F. The ratio D : E is what function of the angle between them ? This example and Exs. 3-22 on page 134 may be omitted by those students who have not had the equivalent of Chapter IV (Trigonometry) in " Fundamentals of Practical Mathematics " in this series. 128 REVIEW PROBLEMS Exercises. Universal Milling Machine 1. In the universal milling machine shown below the diameter of the machine-steel arm A is 4J" and the length is 3' 2". Find the number of R.P.M. for a roughing cut with a high-speed tool. 2. Taking a roughing cut with a ^-inch feed and a finishing cut with a -j 5 ^ inch feed, how long will it take to machine the arm in Ex. 1 ? 3. If table B is wide and 39" long, how long will it take, using a Jg-inch feed, to plane the top of the table on a 2-to-l planer which has a cutting speed of 25 F.P.M.? UNIVERSAL MILLING MACHINE If spur gear C has an outside diameter of 8j" and a pitch of 12, find each -of the following : 4. Number of teeth. 6. Addendum and clearance. 5 1 . Circular pitch. 7. Whole depth of tooth. 8. In the taper hole* in the spindle D the large diameter is 1.260", the small diameter is 1.045", and the length of the taper is 5". Find the T.P.F. 9. In Ex. 8 find the included angle of the taper. 10. Screw E, which regulates the vertical adjustment of the table, has a double Acme screw thread of ^-inch lead. Find the double depth of the thread. UNIVERSAL MILLING MACHINE 129 11. In Ex. 10 find the change gears required on a lathe with a lead screw of 1-inch pitch when cutting the thread. In cutting a double thread the gears are figured as if for a single thread of the required lead. One groove is cut to the double-thread depth, the work is given a half turn, and then the other groove is cut. 12. If gear F has 44 teeth, find the indexing required on a B. & S. dividing head in cutting the gear. 13. If the diameters of the steps of the cone pulley H are 7", 8-f", 101", and 12" respectively, and the pulley is made of cast iron, find the number of R.P.M. needed in roughing the face of each step with a tool of carbon steel. 14. If each step of the cone pulley in Ex. 13 has a 21-inch face, find the time taken in making a roughing cut over the four faces with a -jL-inch feed.. 15. Beginning with the smallest step, a finishing cut was taken over the faces of the steps of the cone pulley in Ex. 13, the speeds of the lathe being as follows: 70 R.P.M., 60 R.P.M., 50 R.P.M., 45 R.P.M. Find the average cut- ting speed used in finishing the pulley. 16. If the diameter at the center of each step of the cone pulley in Ex. 13 is -fa 11 greater than that at the edge, find the T.P.F. of the face. 17. The cone pulley in Ex. 13 is driven by a similar cone pulley on a countershaft which makes 98 R.P.M., the 7^-inch steps being opposite the 12-inch steps. Beginning with the lowest speed, find the speed of the shaft to which cone pulley H is fastened for each position of the belt on the steps. 18. If the outside diameter of the nose of the spindle D is 2^-" and the nose is threaded with a sharp V-thread of ^-inch pitch, to what diameter should the collar G be bored? 19. If the lead of the milling machine is 10", what change gears are needed in cutting a spiral with a lead of 2.24"? 130 REVIEW PROBLEMS Exercises. Tapers 1. Find the T.P.F. of the taper on the automobile stub axle shown in the blueprint on page 131. 2. In Ex. 1 find the tailstock offset for turning the taper. In the following table supply each missing specification for automobile stub axles of the type shown in the blueprint : 3. 4. 5. L I D d T.P.F. TAILSTOCK OFFSET 11J" 3" if" 1.500" 9f IT"* H" 1.250 !<>A 2 T 9 e 1* 1.4 In the following table supply each missing specification for ball-crank handles of the type shown in the blueprint : 6. 7. 8. 9. L I D d T. P. F. TAILSTOCK OFFSET 3" If" |W * iV' 31 H & A 3| Hi M 44 4 8* H 4f In the following table supply each missing specification for bench-lathe drawback chucks of the type shown in the blueprint : 10. 11. 12. 13. / D d ANGLE WITH Axis T.P.F. 0.300" 0.500" 0.335" 0.850 0.650 12 1.156 0.950 2.125" 1.125 1.875 1.625 TAPERS 131 132 KEVIEW PROBLEMS Exercises. Engine Lathe 1. In the lathe here shown the cast-iron faceplate A has a diameter of 14", and the width of the face is 1|". Using a carbon-steel tool, at what speed should the face be finished ? ENGINE LATHE 2. In Ex. 1 how long will it take, using a gL-inch feed, to make a single cut over the face ? 3. If the lead screw B has an Acme screw thread with 8 threads to the inch, find the depth of the thread. In the screw described in Ex. 3 find each of the following : 4. The width of the space at the top of the thread. 5. The width of the flat at the root of the thread. 6. The width of the flat at the top of the thread. 7. The change gears required on a lathe, which has a lead screw with 5 threads per inch, in cutting the thread. ENGINE LATHE 133 8. Spur gear C is a 12-J^ gear with 96 teeth. Find the pitch diameter of the gear. 9. In Ex. 8 find the outside diameter of the gear. 10. In Ex. 8 find the whole depth of tooth. 11. The dead center D is 1.475" in diameter at its small end, 1.790" in diameter at its large end, and the length of the taper is 6". Find the T.P.F. 12. In Ex. 11 find the included angle of the taper. 13. In the cast-iron cone pulley E the diameters of the steps are 61", 7 T \", 8|", 91f", and 11" respectively, and the width of each face is 1|". Beginning with the smallest step, find the number of R.P.M. for finishing each face with a high-speed steel tool. 14. If the diameters at the centers of the steps of the pulley in Ex. 13 are ^V greater than those at the edges, find the angle which the faces make with the axis of the pulley. 15. If the cone pulley in Ex. 13 is turned on a 7^-inch arbor, find the offset of the tailstock center. 16. If the machine-steel feed rod F is {" * n diameter and 5' 8" in length, what is the number of R.P.M. needed in roughing the rod with a tool of carbon steel? 17. In Ex. 16 how long will it take to make a single cut over the rod with a I -inch feed ? o 18. If the spur gear G has 117 teeth and the pitch is 8, what is the outside diameter? 19. In Ex. 18 find the circular pitch and the thickness of tooth on the pitch line. 20. Using a B. & S. dividing head, find the indexing required in cutting the gear in Ex. 18. 21. A piece of cast iron 1^" in diameter is turned in the lathe at 175 R.P.M. Find the cutting speed used. 134 REVIEW PROBLEMS Exercises. General Applications 1. Find the angle a in the lathe center shown in the blueprint on page 135. 2. In Ex. 1 find the length L of the pointed end. 3. The three holes J, B, C are to be drilled on a milling machine in the jig plate shown in the blueprint. Find to the nearest 0.001" the vertical movement from A for drilling hole C. Find the lateral movement from C for drilling hole B. In connection with Exs. 3-22 see the note at the foot of page 127. 4. The holes in the cylinder head shown in the blueprint are equally spaced on the circle. Find the straight-line dis- tance between the centers of the holes. 5. Consider Ex. 4 if there are 9 holes on a 12-inch circle ; 13 holes on an 18-inch circle ; 21 holes on a 16-inch circle. 6. In the pulleys and belting shown in the blueprint find the angle a and also find the distance D between the centers of the pulleys. 7. Find the diameter of the round stock required for making hexagonal heads which are to be |" across the flats. Consider Ex. 7 for each of the following measurements across the flats : 8. If". 9. l T y. 10. 1J". 11. 111". 12. 2 T 3 /. Find in each case the largest square that can be milled on a round shaft in which the diameter is as follows : 13. T y. 14. ||". 15. l T y. 16. lif ". 17. 2 5 y. Find in each case the diameter of the round stock required in milling a square in which the length of side is as follows : 18. f". 19. 11". 20. l T y. 21. If". 22. 2|". GEKEKAL APPLICATIONS 135 136 REVIEW PROBLEMS Exercises. Miscellaneous Problems 1. A plug gage T 9 g" in diameter enters a hole with un- known taper a distance of 3^", and another plug gage 11" in diameter enters the same hole a distance of 2J". Find the T.P.F. of the hole. 2. Find the change gears required on a milling machine with a lead of 10" in cutting a spiral with a lead of 12.90". 3. Make the necessary calculations for cutting a pair of 16-^ bevel gears if the velocity ratio of the gears is to be 3 to 4, if the shafts are to be at right angles to each other, and if the pinion is to have 48 teeth. 4. If a milling machine is geared to cut a spiral with a lead of 0.67", find the angle at which the dividing head should be elevated in milling a cam which is to have a rise of 0.310" in 0.47 of the circumference. 5. The center of a 12-7J spur gear with 54 teeth is 8|" from the center of a mating gear. Find the number of teeth and the outside diameter of the second gear. 6. Make all the necessary calculations for cutting a pair of spiral gears if it is required that D c (approximate) = 12", r = 2, A g = 33, A p = 27, P n = 8, and S = 60. 7. Two spur gears are to mesh with a distance of 27" between centers, and the velocity ratio is to be 1 to 2. If the gears are to be cut 10 pitch, what should be the outside diameter and the number of teeth on each gear? 8. If an Acme screw thread with 3 threads per inch is cut on a 2^-inch shaft, what is the root diameter of the thread ? 9. The tailstock center of a certain lathe can be offset a maximum distance of 21" from the center line of the lathe. What is the greatest T.P.F. that can be cut on a piece of work 9" long? on a piece of work 17" long? TABLES FOR REFERENCE TABLES OF MEASURES LENGTH 12 inches (in. or ") = 1 foot (ft. or ') 3 feet = 1 yard (yd.) 5| yards, or 16| feet = 1 rod (rd.) 320 rods, or 5280 feet = 1 mile (mi.) SQUARE MEASURE 144 square inches (sq. in.) = 1 square foot (sq. ft.) 9 square feet = 1 square yard (sq. yd.) 30| square yards = 1 square rod (sq. rd.) 160 square rods = 1 acre (A.) CUBIC MEASURE 1728 cubic inches (cu. in*.) = 1 cubic foot (cu. ft.) 27 cubic feet = 1 cubic yard (cu. yd.) 128 cubic feet = 1 cord (cd.) WEIGHT 16 ounces (oz.) = 1 pound (Ib.) 2000 pounds = 1 ton (T.) LIQUID MEASURE 4 gills (gi.) = 1 pint (pt.) 2 pints = 1 quart (qt.) 4 quarts = 1 gallon (gal.) DRY MEASURE 2 pints (pt.) = 1 quart (qt.) 8 quarts = 1 peck (pk.) 4 pecks = 1 bushel (bu.) A bushel contains 2150.42 cu. in. (approximately l cu. ft.). 137 138 TABLES FOB REFERENCE METRIC LENGTH 1 kilometer (km.) = 1000 meters Meter (m.) 1 centimeter (cm.) =0.01 meter 1 millimeter (mm.) =0.001 meter In comparing the metric measures of length with those of our common system we usually think of 1 km. as 2 mi., of 1 m. as 39^", of 1 cm. as f", or 0.4", and of 1 mm. as ^V > or 0.04". For a higher degree of accuracy, however, we use the following approximate equivalents : 1 km. = 0.62 mi., 1 m. = 39.37", 1 cm. = 0.3937", and 1 mm. = 0.0394". To the machinist the millimeter is the most important of the metric measures on account of its use in connection with imported automobiles and with machines made for foreign trade. METRIC WEIGHT 1 metric ton (t.) = 1000 kilograms 1 kilogram (kg.) = 1000 grams Gram (g.) The following approximate equivalents may be used in compar- ing metric measures of weight with those of our common system : 1 1. = 2204.6 lb., 1 kg. = 2.2 lb., and 1 g. = 15.43 grains. METRIC CAPACITY 1 hektoliter (hi.) = 100 liters Liter (1.) 1 centiliter (cl.)= 0.01 liter The liter is approximately a quart. It is the volume of a cube that is 0.1 m., or about 4", on an edge. ANGLES AND ARCS 60 seconds (") = 1 minute (') 60 minutes = 1 degree () 90 degrees = 1 right angle 360 degrees = 1 circumference TABLES FOE, REFERENCE 139 DECIMAL EQUIVALENTS OF COMMON FRACTIONS FRACTION DKCIMAL FRACTION DECIMAL A 0.015625 ii 0.515625 1 38 .03125 ii .53125 3_ .046875 35 .546875 _i .0625 T 9 (f .5625 A .078125 37 .578125 3 32 .09375 Si .59375 TT 7 4 .109375 II .609375 1 8 .125 5 8 .625 o 9 4 .140625 ti .640625 5 T52 .15625 H .65625 1 1 04 .171875 43 4 .671875 T 3 (T .1875 11 .6875 H .203125 H .703125 V<>- .21875 If .71875 U .234375 ti .734375 1 .25 3 .75 4 4 iJ .265625 H .765625 _9 .28125 14 .78125 i j> .296875 H .796875 tV .3125 ii .8125 04 .328125 H .828125 U .34375 II .84375 H .359375 5 5 ?J4 .859375 3 8 .375 7 8 .875 If .390625 ft .890625 i a .40625 ^ .90625 2 7 04 .421875 H .921875 7 .4375 1 5 .9375 it .453125 1 li .953125 M .46875 U .96875 in .484375 11 .984375 2 .5 1 1. 140 TABLES FOR REFERENCE CONVENIENT RULES The circumference of a circle is ^ times the diameter. For a higher degree of accuracy, c = 3.1416 d. The diameter of a circle is -j^ of the circumference. For a higher degree of accuracy, d = 0.3183 c. The area of a circle is j^ of the square of the diameter. For a higher degree of accuracy, A = 0.7854 cP. The height of an equilateral triangle is 0.8660 of the side. The diagonal of a square is 1.4142 times the side. The diagonal of a square is also called the " long diameter " or the " distance across the corners." The long diameter of a regular hexagon is twice the side. The short diameter, or the perpendicular distance between parallel sides, of a regular hexagon is 1.7321 times the side. To convert Fahrenheit temperature into centigrade temper- ature subtract 32 from the Fahrenheit reading and take ^ of the result. Expressed as a formula, C = f (F 32). To convert centigrade temperature into Fahrenheit temperature take j? of the centigrade reading and add 32 to the result. Expressed as a formula, F = f C + 32. COMMON EQUIVALENTS In ordinary cases the following equivalents may be used: 1 gal. contains 231 cu. in., or 0.134 cu. ft. 1 cu. ft. contains 7| gal. 1 barrel (bbl.) contains 3l|gal., or 4|cu. ft. 1 cu. ft. of water weighs 62.425 Ib. (approximately 62 1 lb.). 1 gal. of water weighs 8.345 lb. (approximately s| lb.). 1 1. of water weighs 1 kg. TABLES FOE BEFEKENCE 141 NATURAL TRIGONOMETRIC FUNCTIONS Tables of four of the natural functions of an angle, the sine, cosine, tangent, and cotangent, for every 6', or 0.1, from to 90 are given on pages 142-149. The following examples illustrate the use of these tables : 1. Find sin 62 19'. On page 143 we look for 62 in the column at the left, and in the same line at the right under 18' we find that sin 62 18' = 0.8854. Under V in the column of differences at the right and in line with 62 we find that the difference for 1' is 1 (actually this is 0.0001). Since the column of differences is marked "+ Differences," this differ- ence is added to sin 62 18'. The addition is done mentally, only the result being written. Therefore sin 62 19' = 0.8855. 2. Find cot 5 56'. From page 148, cot 5 54' = 9.6768. Notice that after the 5 at the left the integral part is 11, but since two black numbers intervene, we decrease the 11 by 2. The cotangent is here changing so rapidly that we use a process commonly known as interpolation. By subtraction we find that the difference between cot 5 54' and cot 6 0' is 0.1624. Since 2' is ^ of 6', we take of 0.1624, and hence the difference for 2' is 0.0541. Since the column of differences is marked te Differences," this differ- ence is subtracted from cot 5 54'. Therefore cot 5 56'= 9.6227. 3. Find the angle of which the tangent is 0.5693. On page 146 we find that the tangent next smaller than this tangent is 0.5681, which is the tangent of 29 36', and by subtraction we see that the difference between 0.5693 and 0.5681 is 0.0012. In the column of differences we see that 12 is the difference to be added for 3'. Adding 3' to 29 36', we see that 29 39' is the angle of which the tangent is 0.5693. 142 NATURAL SINES. 0-45 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 + Differences 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' V 2' 3' 4' 5' ~o .0000 .0017 .0035 .0052 .0070 .0087 .0105 .0122 .0140 .0157 3 6 9 12 15 1 .0175 .0192 .0209 .0227 .0244 .0262 .0279 .0297 .0314 .0332 3 6 9 12 15 2 .0349 .0366 .0384 .0401 .0419 .0436 .0454 .0471 .0488 .0506 3 6 9 12 15 3 .0523 .0541 .0558 .0576 .0593 .0610 .0628 .0645 .0663 .0680 3 6 9 12 15 4 .0698 .0715 .0732 .0750 .0767 .0785 .0802 .0819 .0837 .0854 3 6 9 12 14 5 .0872 .0889 .0906 .0924 .0941 .0958 .0976 .0993 .1011 .1028 3 6 9 12 14 6 .1045 .1063 .1080 .1097 .1115 .1132 .1149 .1167 .1184 .1201 3 6 9 12 14 7 .1219 .1236 .1253 .1271 .1288 .1305 .1323 .1340 .1357 .1374 3 6 9 12 14 8 .1392 .1409 .1426 .1444 .1461 .1478 .1495 .1513 .1530 .1547 3 6 9 12 14 9 .1564 .1582 .1599 .1616 .1633 .1650 .1668 .1685 .1702 .1719 3 6 9 12 14 10 .1736 .1754 .1771 .1788 .1805 .1822 .1840 .1857 .1874 .1891 3 6 9 11 14 11 .1908 .1925 .1942 .1959 .1977 .1994 .2011 .2028 .2045 .2062 3 6 9 11 14 12 .2079 .2096 .2113 .2130 .2147 .2164 .2181 .2198 .2215 .2233 3 6 9 11 14 13 .2250 .2267 .2284 .2300 .2317 .2334 .2351 .2368 .2385 .2402 3 6 8 11 14 14 .2419 .2436 .2453 .2470 .2487 .2504 .2521 .2538 .2554 .2571 3 6 8 11 14 15 .2588 .2605 .2622 .2639 .2656 .2672 .2689 .2706 .2723 .2740 3 6 8 11 14 16 .2756 .2773 .2790 .2807 .2823 .2840 .2857 .2874 .2890 .2907 3 6 8 11 14 17 .2924 .2940 .2957 .2974 .2990 .3007 .3024 .3040 .3057 .3074 3 6 8 11 14 18 .3090 .3107 .3123 .3140 .3156 .3173 .3190 .3206 .3223 .3239 3 6 8 11 14 19 .3256 .3272 .3289 .3305 .3322 .3338 .3355 .3371 .3387 .3404 3 5 8 11 14 20 .3420 .3437 .3453 .3469 .3486 .3502 .3518 .3535 .3551 .3567 3 5 8 11 14 21 .3584 .3600 .3616 .3633 .3649 .3665 .3681 .3697 .3714 .3730 3 5 8 11 14 22 .3746 .3762 .3778 .3795 .3811 .3827 .3843 .3859 .3875 .3891 3 5 8 11 14 23 .3907 .3923 .3939 .3955 .3971 .3987 .4003 .4019 .4035 .4051 3 5 8 11 14 24 .4067 .4083 .4099 .4115 .4131 .4147 .4163 .4179 .4195 .4210 3 5 8 11 13 25 .4226 .4242 .4258 .4274 .4289 .4305 .4321 .4337 .4352 .4368 3 5 8 11 13 26 .4384 .4399 .4415 .4431 .4446 .4462 .4478 .4493 .4509 .4524 3 5 8 10 13 27 .4540 .4555 .4571 .4586 .4602 .4617 .4633 .4648 .4664 .4679 3 5 8 10 13 28 .4695 .4710 .4726 .4741 .4756 .4772 .4787 .4802 .4818 .4833 3 5 8 10 13 29 .4848 .4863 .4879 .4894 .4909 .4924 .4939 .4955 .4970 .4985 3 5 8 10 13 30 .5000 .5015 .5030 .5045 .5060 .5075 .5090 .5105 .5120 .5135 3 5 8 10 13 31 .5150 .5165 .5180 .5195 .5210 .5225 .5240 .5255 .5270 .5284 2 5 7 10 12 32 .5299 .5314 .5329 .5344 .5358 .5373 .5388 .5402 .5417 .5432 2 5 7 10 12 33 .5446 .5461 .5476 .5490 .5505 .5519 .5534 .5548 .5563 .5577 2 5 7 10 12 34 .5592 .5606 .5621 .5635 .5650 .5664 .5678 .5693 .5707 .5721 2 5 7 10 12 35 .5736 .5750 .5764 .5779 .5793 .5807 .5821 .5835 .5850 .5864 2 5 7 9 U2 36 .5878 .5892 .5906 .5920 .5934 5948 5962 .5976 .5990 .6004 2 5 7 9 12 37 .6018 .6032 .6046 .6060 .6074 6088 6101 .6115 .6129 .6143 2 5 7 9 12 38 .6157 .6170 .6184 .6198 .6211 6225 6239 .6252 .6266 .6280 2 5 7 9 11 39 .6293 .6307 .6320 .6334 .6347 6361 6374 .6388 .6401 .6414 2 4 7 9 11 40 .6428 .6441 .6455 .6468 .6481 6494 6508 .6521 .6534 6547 2 4 7 9 11 41 .6561 .6574 .6587 .6600 .6613 6626 6639 .6652 6665 .6678 2 4 7 9 11 42 .6691 .6704 .6717 .6730 .6743 6756 6769 .6782 .6794 .6807 2 4 6 9 11 43 .6820 .6833 .6845 .6858 .6871 6884 6896 .6909 .6921 .6934 2 4 6 8 11 44 6947 .6959 .6972 .6984 .6997 7009 7022 .7034 .7046 .7059 2 A 6 8 10 All the above sines are less than 1. NATURAL SINES. 45-90 143 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -f- Differences 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 1' 2' 3' 4' 5' 45 .7071 .7083 .7096 .7108 .7120 .7133 .7145 .7157 .7169 .7181 2 4 6 8 10 46 .7193 .7206 .7218 .7230 .7242 .7254 .7266 .7278 .7290 .7302 2 4 6 8 10 47 .7314 .7325 .7337 .7349 .7361 .7373 .7385 .7396 .7408 .7420 2 4 6 8 10 48 .7431 .7443 .7455 .7466 .7478 .7490 .7501 .7513 .7524 .7536 2 4 6 8 10 49 .7547 .7559 .7570 .7581 .7593 .7604 .7615 .7627 .7638 .7649 2 4 6 8 9 50 .7660 .7672 .7683 .7694 .7705 .7716 .7727 .7738 .7749 .7760 2 4 6 7 9 51 .7771 .7782 .7793 .7804 .7815 .7826 .7837 .7848 .7859 .7869 2 4 5 7 9 52 .7880 .7891 .7902 .7912 .7923 .7934 .7944 .7955 .7965 .7976 2 4 5 7 9 53 .7986 .7997 .8007 .8018 .8028 .8039 .8049 .8059 .8070 .8080 2 3 5 7 9 54 .8090 .8100 .8111 .8121 .8131 .8141 .8151 .8161 .8171 .8181 2 3 5 7 8 55 .8192 .8202 .8211 .8221 .8231 .8241 .8251 .8261 .8271 .8281 2 3 5 7 8 56 .8290 .8300 .8310 .8320 .8329 .8339 .8348 .8358 .8368 .8377 2 3 5 6 8 57 .8387 .8396 .8406 .8415 .8425 .8434 .8443 .8453 .8462 .8471 2 3 5 6 8 58 .8480 .8490 .8499 .8508 .8517 .8526 .8536 .8545 .8554 .8563 2 3 5 6 8 59 .8572 .8581 .8590 .8599 .8607 .8616 .8625 .8634 .8643 .8652 1 3 4 6 7 60 .8660 .8669 .8678 .8686 .8695 .8704 .8712 .8721 .8729 .8738 1 3 4 6 7 61 .8746 .8755 .8763 .8771 .8780 .8788 .8796 .8805 .8813 .8821 1 3 4 6 7 62 8829 .8838 .8846 .8854 .8862 .8870 .8878 .8886 .8894 .8902 1 3 4 5 7 63 8910 .8918 .8926 .8934 .8942 .8949 .8957 .8965 .8973 .8980 1 3 4 5 6 64 .8988 .8996 .9003 .9011 .9018 .9026 .9033 .9041 .9048 .9056 1 3 4 5 6 65 9063 .9070 .9078 .9085 .9092 .9100 .9107 .9114 .9121 .9128 1 2 4 5 6 66 9135 .9143 .9150 .9157 .9164 .9171 .9178 .9184 .9191 .9198 1 2 3 5 6 67 .9205 .9212 .9219 .9225 .9232 .9239 .9245 .9252 .9259 .9265 1 2 3 4 6 68 .9272 .9278 .9285 .9291 .9298 .9304 .9311 .9317 .9323 .9330 1 2 3 4 5 69 .9336 .9342 .9348 .9354 .9361 .9367 .9373 .9379 .9385 .9391 1 2 3 4 5 70 .9397 .9403 .9409 .9415 .9421 .9426 .9432 .9438 .9444 .9449 1 2 3 4 5 71 .9455 .9461 .9466 .9472 .9478 .9483 .9489 .9494 .9500 .9505 1 2 3 4 5 72 .9511 .9516 .9521 .9527 .9532 .9537 .9542 .9548 .9553 .9558 1 2 3 4 4 73 .9563 .9568 .9573 .9578 .9583 .9588 .9593 .9598 .9603 .9608 1 2 2 3 4 74 .9613 .9617 .9622 .9627 .9632 .9636 .9641 .9646 .9650 .9655 1 2 2 3 4 75 .9659 .9664 .9668 .9673 .9677 .9681 .9686 .9690 .9694 .9699 1 1 2 3 4 76 .9703 .9707 .9711 .9715 .9720 .9724 .9728 .9732 .9736 .9740 1 1 2 3 3 77 .9744 .9748 .9751 .9755 .9759 .9763 .9767 .9770 .9774 .9778 1 2 3 3 78 .9781 .9785 .9789 .9792 .9796 .9799 .9803 .9806 .9810 .9813 1 2 2 3 79 .9816 .9820 .9823 .9826 .9829 .9833 .9836 .9839 .9842 .9845 1 2 2 3 80 .9848 .9851 .9854 .9857 .9860 .9863 .9866 .9869 .9871 .9874 1 2 2 81 9877 .9880 .9882 .9885 .9888 .9890 .9893 .9895 .9898 .9900 1 2 2 82 9903 .9905 .9907 .9910 .9912 .9914 .9917 .9919 .9921 .9923 1 2 2 83 9925 .9928 .9930 .9932 .9934 .9936 .9938 .9940 .9942 .9943 1 1 2 84 9945 .9947 .9949 .9951 .9952 .9954 .9956 .9957 .9959 .9960 1 1 1 85 9962 .9963 .9965 .9966 .9968 .9969 .9971 .9972 .9973 .9974 1 1 1 86 9976 .9977 .9978 .9979 .9980 .9981 .9982 .9983 .9984 .9985 1 1 1 87 9986 .9987 .9988 .9989 .9990 .9990 .9991 .9992 .9993 .9993 1 1 88 9994 .9995 .9995 .9996 .9996 .9997 .9997 .9997 .9998 .9998 89 9998 .9999 .9999 .9999 .9999 1.000 1.000 1.000 1.000 1.000 The precise value of all sines except sin 90 is less than 1. 144 NATUKAL COSINES. 0-45 o 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Differences 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 1' 2' 3' 4' 5' ~0 1.000 1.000 1.000 1.000 1.000 1.000 .9999 .9999 .9999 .9999 1 .9998 .9998 .9998 .9997 .9997 .9997 .9996 .9996 .9995 .9995 2 .9994 .9993 .9993 .9992 .9991 .9990 .9990 .9989 .9988 .9987 3 .9986 .9985 .9984 .9983 .9982 .9981 .9980 .9979 .9978 .9977 1 1 1 4 .9976 .9974 .9973 .9972 .9971 .9969 .9968 .9966 .9965 .9963 1 1 1 5 .9962 .9960 .9959 .9957 .9956 .9954 .9952 .9951 .9949 .9947 1 1 1 1 .9945 .9943 .9942 .9940 .9938 .9936 .9934 .9932 .9930 .9928 1 1 1 2 7 .9925 .9923 .9921 .9919 .9917 .9914 .9912 .9910 .9907 .9905 1 1 2 2 8 .9903 .9900 .9898 .9895 .9893 .9890 .9888 .9885 .9882 .9880 1 1 2 2 9 .9877 .9874 .9871 .9869 .9866 .9863 .9860 .9857 .9854 .9851 1 1 2 2 10 .9848 .9845 .9842 .9839 .9836 .9833 .9829 .9826 .9823 .9820 1 1 2 2 3 11 .9816 .9813 .9810 .9806 .9803 .9799 .9796 .9792 .9789 .9785 1 1 2 2 3 12 .9781 .9778 .9774 .9770 .9767 .9763 .9759 .9755 .9751 .9748 1 1 2 3 3 13 .9744 .9740 .9736 .9732 .9728 .9724 .9720 .9715 .9711 .9707 1 1 2 3 3 14 .9703 .9699 .9694 .9690 .9686 .9681 .9677 .9673 .9668 .9664 1 1 2 3 4 15 .9659 .9655 .9650 .9646 .9641 .9636 .9632 .9627 .9622 .9617 1 2 2 3 4 16 .9613 .9608 .9603 .9598 .9593 .9588 .9583 9578 .9573 .9568 1 2 2 3 4 17 .9563 .9558 .9553 .9548 .9542 .9537 .9532 9527 .9521 .9516 1 2 3 4 4 18 .9511 .9505 .9500 .9494 .9489 .9483 .9478 9472 .9466 .9461 1 2 3 4 5 19 .9455 .9449 .9444 .9438 .9432 .9426 .9421 9415 .9409 .9403 1 2 3 4 5 20 .9397 .9391 .9385 .9379 .9373 .9367 .9361 9354 .9348 .9342 1 2 3 4 5 21 .9336 .9330 .9323 .9317 .9311 .9304 .9298 9291 .9285 .9278 1 2 3 4 5 22 .9272 .9265 .9259 .9252 .9245 9239 .9232 9225 .9219 .9212 1 2 3 4 6 23 9205 .9198 .9191 9184 .9178 9171 .9164 9157 .9150 .9143 1 2 3 5 6 24 9135 9128 .9121 .9114 .9107 .9100 .9092 9085 .9078 .9070 1 2 4 5 6 25 9063 9056 9048 9041 9033 9026 .9018 9011 9003 8996 1 3 4 5 6 26 8988 8980 8973 8965 8957 8949 .8942 8934 8926 8918 1 3 4 5 6 27 8910 8902 8894 8886 8878 8870 8862 8854 8846 8838 1 3 4 5 7 28 8829 8821 8813 8805 8796 8788 8780 8771 8763 8755 1 3 4 6 7 29 8746 8738 8729 8721 8712 8704 8695 8686 8678 8669 1 3 4 6 7 30 8660 8652 8643 8634 8625 8616 8607 8599 8590 8581 1 3 4 6 7 31 8572 8563 8554 8545 8536 8526 8517 8508 8499 8490 2 3 5 6 8 32 8480 8471 8462 8453 8443 8434 8425 8415 8406 8396 2 3 5 6 8 33 8387 8377 8368 8358 8348 8339 8329 8320 8310 8300 2 3 5 6 8 34 8290 8281 8271 8261 8251 8241 8231 8221 8211 8202 2 3 5 7 8 35 8192 8181 8171 8161 8151 8141 8131 8121 8111 8100 2 3 5 7 8 36 8090 8080 8070 8059 8049 8039 8028 8018 8007 7997 2 3 5 7 9 37 7986 7976 7965 7955 7944 7934 7923 7912 7902 7891 2 4 5 7 9 38 7880 7869 7859 7848 7837 7826 7815 7804 7793 7782 2 4 5 7 9 39 7771 7760 7749 7738 7727 7716 7705 7694 7683 7672 2 4 6 7 9 40 7660 7649 7638 7627 7615 7604 7593 7581 7570 7559 2 4 6 8 9 41 7547 7536 7524 7513 7501 7490 7478 7466 7455 7443 2 4 6 8 10 42 7431 7420 7408 7396 7385 7373 7361 7349 7337 7325 2 4 6 8 10 43 7314 7302 7290 7278 7266 7254 7242 7230 7218 7206 2 4 6 8 10 44 7193 7181 7169 7157 7145 7133 7120 7108 7096 7083 2 4 6 8 10 The precise value of all cosines except cos is less than 1. NATURAL COSINES. 45-90 145 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Differences 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 1' 2' 3' 4' 5' "45 .7071 .7059 .7046 .7034 .7022 .7009 .6997 .6984 .6972 .69591 2 4 6 8 10 46 .6947 .6934 .6921 .6909 .6896 .6884 .6871 .6858 .6845 .6833 2 4 6 8 11 47 .6820 .6807 .6794 .6782 .6769 .6756 .6743 .6730 .6717 .6704 2 4 6 9 11 48 .6691 .6678 .6665 .6652 .6639 .6626 .6613 .6600 .6587 .6574 2 4 7 9 11 49 .6561 .6547 .6534 .6521 .6508 .6494 .6481 .6468 .6455 .6441 2 4 7 9 11 50 .6428 .6414 .6401 .6388 .6374 .6361 .6347 .6334 .6320 .6307 2 4 7 9 11 51 .6293 .6280 .6266 .6252 .6239 .6225 .6211 .6198 .6184 .6170 2 5 7 9 11 52 .6157 .6143 .6129 .6115 .6101 .6088 .6074 .6060 .6046 .6032 2 5 7 9 12 53 .6018 .6004 .5990 .5976 .5962 .5948 .5934 .5920 .5906 .5892 2 5 7 9 12 54 .5878 .5864 .5850 .5835 .5821 .5807 .5793 .5779 .5764 .5750 2 5 7 9 12 55 .5736 .5721 .5707 .5693 .5678 .5664 .5650 .5635 .5621 .5606 2 5 7 10 12 56 5592 .5577 .5563 .5548 .5534 .5519 .5505 .5490 .5476 .5461 2 5 7 10 12 57 5446 .5432 .5417 .5402 .5388 .5373 .5358 .5344 .5329 .5314 2 5 7 10 12 58 5299 .5284 .5270 .5255 .5240 .5225 .5210 .5195 .5180 .5165 2 5 7 10 12 59 5150 .5135 .5120 .5105 .5090 .5075 .5060 .5045 .5030 .5015 3 5 8 10 13 60 5000 .4985 .4970 .4955 .4939 .4924 .4909 .4894 .4879 .4863 3 5 8 10 13 61 4848 .4833 .4818 .4802 .4787 .4772 .4756 .4741 .4726 .4710 3 5 8 10 13 62 4695 .4679 .4664 .4648 .4633 .4617 .4602 .4586 .4571 .4555 3 5 8 10 13 63 4540 4524 .4509 .4493 .4478 .4462 .4446 .4431 .4415 .4399 3 5 8 10 13 64 4384 4368 .4352 .4337 .4321 .4305 .4289 .4274 .4258 .4242 3 5 8 11 13 65 4226 4210 .4195 .4179 .4163 .4147 .4131 .4115 .4099 .4083 3 5 8 11 13 66 4067 4051 .4035 .4019 .4003 .3987 .3971 .3955 .3939 .3923 3 5 8 11 13 67 3907 3891 .3875 .3859 .3843 .3827 .3811 .3795 .3778 .3762 3 5 8 11 13 68 3746 3730 .3714 .3697 .3681 .3665 .3649 .3633 .3616 .3600 3 5 8 11 14 69 3584 3567 .3551 .3535 .3518 .3502 .3486 .3469 .3453 .3437 3 5 8 11 14 70 3420 3404 .3387 .3371 .3355 .3338 .3322 .3305 .3289 .3272 3 5 8 11 14 71 3256 3239 .3223 .3206 .3190 .3173 .3156 .3140 .3123 .3107 3 6 8 11 14 72 3090 3074 .3057 .3040 .3024 .3007 .2990 .2974 .2957 .2940 3 6 8 11 14 73 2924 2907 2890 .2874 .2857 2840 .2823 .2807 .2790 .2773 3 6 8 11 14 74 2756 2740 2723 .2706 .2689 2672 .2656 .2639 .2622 .2605 3 6 8 11 14 75 2588 2571 2554 .2538 .2521 2504 .2487 .2470 .2453 .2436 3 6 8 11 14 76 2419 2402 2385 .2368 .2351 2334 .2317 .2300 .2284 .2267 3 6 8 11 14 77 2250 2233 2215 .2198 .2181 2164 .2147 .2130 .2113 .2096 3 6 9 11 14 78 2079 2062 2045 .2028 .2011 1994 .1977 .1959 .1942 .1925 3 6 9 11 14 79 1908 1891 .1874 .1857 .1840 1822 .1805 .1788 .1771 .1754 3 6 9 11 14 80 1736 .1719 .1702 .1685 .1668 1650 .1633 .1616 .1599 .1582 3 6 9 11 14 81 1564 .1547 .1530 .1513 .1495 1478 .1461 .1444 .1426 .1409 3 6 9 12 14 82 1392 1374 .1357 .1340 .1323 1305 .1288 .1271 .1253 .1236 3 6 9 12 14 83 1219 1201 1184 .1167 .1149 1132 .1115 .1097 .1080 .1063 3 6 9 12 14 84 1045 1028 1011 .0993 .0976 0958 0941 .0924 .0906 .0889 3 6 9 12 14 85 0872 0854 0837 .0819 .0802 0785 0767 .0750 .0732 .0715 3 6 9 12 14 86 0698 0680 0663 .0645 .0628 0610 0593 0576 0558 0541 3 6 9 12 15 87 0523 0506 0488 0471 .0454 0436 0419 0401 0384 0366 3 6 9 12 15 88 0349 0332 0314 0297 .0279 0262 0244 0227 0209 0192 3 6 9 12 15 89 0175 0157 0140 0122 .0105 0087 0070 0052 0035 0017 3 6 9 12 15 All the above cosines are less than 1. 146 NATURAL TA^GEXTS. 0-45 e o 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 + Differences 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 1' 2' 3' 4' 5' 0.0000 .0017 .0035 .0052 .0070 .0087 .0105 .0122 .0140 .0157 3 6 9 12 15 1 0.0175 .0192 .0209 .0227 .0244 .0262 .0279 .0297 .0314 .0332 3 6 9 12 15 2 0.0349 .0367 .0384 .0402 .0419 .0437 .0454 .0472 .0489 .0507 3 6 9 12 15 3 0.0524 .0542 .0559 .0577 .0594 .0612 .0629 .0647 .0664 .0682 3 6 9 12 15 4 0.0699 .0717 .0734 .0752 .0769 .0787 .0805 .0822 .0840 .0857 3 6 9 12 15 5 0.0875 .0892 .0910 .0928 .0945 .0963 .0981 .0998 .1016 .1033 3 6 9 12 15 6 0.1051 .1069 .1086 .1104 .1122 .1139 .1157 .1175 .1192 .1210 3 6 9 12 15 7 0.1228 .1246 .1263 .1281 .1299 .1317 .1334 .1352 .1370 .1388 3 6 9 12 15 8 0.1405 .1423 .1441 .1459 .1477 .1495 .1512 .1530 .1548 .1566 3 6 9 12 15 9 0.1584 .1602 .1620 .1638 .1655 .1673 .1691 .1709 .1727 .1745 3 6 9 12 15 10 0.1763 .1781 .1799 .1817 .1835 .1853 .1871 .1890 .1908 .1926 3 6 9 12 15 11 0.1944 .1962 .1980 .1998 .2016 .2035 .2053 .2071 .2089 .2107 3 6 9 12 15 12 0.2126 .2144 .2162 .2180 .2199 .2217 .2235 .2254 .2272 .2290 3 6 9 12 15 13 0.2309 .2327 .2345 .2364 .2382 .2401 .2419 .2438 .2456 .2475 3 6 9 12 15 14 0.2493 .2512 .2530 .2549 .2568 .2586 .2605 .2623 .2642 .2661 3 6 9 12 16 15 0.2679 .2698 .2717 .2736 .2754 2773 .2792 .2811 .2830 .2849 3 6 9 13 16 16 0.2867 .2886 .2905 .2924 .2943 .2962 .2981 .3000 .3019 .3038 3 6 9 13 16 17 0.3057 .3076 .3096 .3115 .3134 .3153 .3172 .3191 .3211 .3230 3 6 10 13 16 18 0.3249 .3269 .3288 .3307 .3327 .3346 .3365 .3385 .3404 .3424 3 6 10 13 16 19 0.3443 .3463 .3482 .3502 .3522 3541 .3561 .3581 .3600 .3620 3 7 10 13 16 20 0.3640 .3659 .3679 .3699 .3719 3739 .3759 .3779 .3799 .3819 3 7 10 13 17 21 0.3839 .3859 .3879 .3899 .3919 3939 .3959 .3979 .4000 .4020 3 7 10 13 17 22 0.4040 .4061 .4081 .4101 .4122 4142 .4163 .4183 .4204 .4224 3 7 10 14 17 23 0.4245 .4265 .4286 .4307 .4327 434C .4369 .4390 .4411 .4431 3 7 10 14 17 24 0.4452 .4473 .4494 .4515 .4536 4557 .4578 .4599 .4621 .4642 4 7 11 14 18 25 0.4663 .4684 .4706 .4727 .4748 4770 .4791 .4813 .4834 .4856 4 7 11 14 18 20 0.4877 .4899 .4921 .4942 .4964 4986 .5008 .5029 .5051 .5073 4 7 11 15 18 27 0.5095 .5117 .5139 .5161 .5184 5206 .5228 .5250 .5272 .5295 4 7 11 15 18 28 0.5317 .5340 .5362 .5384 .5407 5430 .5452 .5475 .5498 .5520 4 8 11 15 19 29 0.5543 .5566 .5589 .5612 .5635 5658 .5681 .5704 .5727 .5750 4 8 12 15 19 30 0.5774 .5797 .5820 .5844 .5867 5890 .5914 .5938 .5961 .5985 4 8 12 16 20 31 0.6009 .6032 .6056 .6080 .6104 6128 .6152 .6176 .6200 .6224 4 8 12 16 20 32 0.6249 .6273 .6297 .6322 .6346 .6371 .6395 .6420 .6445 .6469 4 8 12 16 20 33 0.6494 .6519 .6544 .6569 .6594 6619 .6644 .6669 .6694 .6720 4 8 13 17 21 34 0.6745 .6771 .6796 .6822 .6847 6873 .6899 .6924 .6950 .6976 4 9 13 17 21 35 0.7002 .7028 .7054 .7080 .7107 7133 .7159 .7186 .7212 .7239 4 9 13 18 22 36 0.7265 .7292 .7319 .7346 .7373 7400 .7427 .7454 .7481 .7508 5 9 14 18 22 37 0.7536 .7563 .7590 .7618 .7646 7673 .7701 .7729 .7757 .7785 5 9 14 18 23 38 0.7813 .7841 .7869 .7898 .7926 .7954 .7983 .8012 .8040 .8069 5 9 14 19 24 39 0.8098 .8127 .8156 .8185 .8214 8243 .8273 .8302 .8332 .8361 5 10 15 20 24 40 0.8391 .8421 .8451 .8481 .8511 .8541 .8571 .8601 .8632 .8662 5 10 15 20 25 41 0.8693 .8724 .8754 .8785 .8816 8847 .8878 .8910 .8941 .8972 5 10 16 21 26 42 0.9004 .9036 .9067 .9099 .9131 .9163 .9195 .9228 .9260 .9293 5 11 16 21 27 43 0.9325 .9358 .9391 .9424 .9457 9490 .9523 .9556 .9590 .9623 6 11 17 22 28 44 0.9657 .9691 .9725 .9759 .9793 .9827 .9861 .9896 .9930 .9965 6 11 17 23 29 All tangents less than tan 45 are less than 1. NATURAL TANGENTS. 45- 90 147 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 + Differences 0' 6' 12' IB' 24' 30' 36' 42' 48' 54' 1' 2' 3' 4' 5' 45 1.0000 .0035 .0070 .0105 .0141 .0176 .0212 .0247 .0283 .0319 6 12 18 ~24 30 46 1.0355 .0392 .0428 .0464 .0501 .0538 .0575 .0612 .0649 .0686 6 12 18 25 31 47 1.0724 .0761 .0799 .0837 .0875 .0913 .0951 .0990 .1028 .1067 6 13 19 25 32 48 1.1106 .1145 .1184 .1224 .1263 .1303 .1343 .1383 .1423 .1463 7 13 20 26 33 49 1.1504 .1544 .1585 .1626 .1667 .1708 .1750 .1792 .1833 .1875 7 14 21 28 34 50 1.1918 .1960 .2002 .2045 .2088 .2131 .2174 .2218 .2261 .2305 7 14 22 29 36 51 1.2349 .2393 .2437 .2482 .2527 .2572 .2617 .2662 .2708 .2753 8 15 23 30 38 52 1.2799 .2846 .2892 .2938 .2985 .3032 .3079 .3127 .3175 .3222 8 16 24 31 39 53 1.3270 .3319 .3367 .3416 .3465 .3514 .3564 .3613 .3663 .3713 8 16 25 33 41 54 1.3764 .3814 .3865 .3916 .3968 .4019 .4071 .4124 .4176 .4229 9 17 26 34 43 55 1.4281 .4335 .4388 .4442 .4496 .4550 .4605 .4659 .4715 .4770 9 18 27 36 45 56 1.4826 .4882 .4938 .4994 .5051 .5108 .5166 .5224 .5282 .5340 10 19 29 38 48 57 1.5399 .5458 .5517 .5577 .5637 .5697 .5757 .5818 .5880 .5941 10 20 30 40 50 58 1.6003 6066 .6128 .6191 .6255 .6319 .6383 .6447 .6512 .6577 11 21 32 43 53 59 1.6643 6709 .6775 .6842 .6909 .6977 .7045 .7113 .7182 .7251 11 23 34 45 56 60 1.7321 7391 .7461 .7532 .7603 .7675 .7747 .7820 .7893 .7966 12 24 36 48 60 61 1.8040 8115 .8190 .8265 .8341 .8418 .8495 .8572 .8650 .8728 13 26 38 51 64 62 1.8807 8887 .8967 .9047 .9128 .9210 .9292 .9375 .9458 .9542 14 27 41 55 68 63 1.9626 9711 .9797 .9883 .9970 .0057 .0145 .0233 .0323 .0413 15 29 44 58 73 64 2.0503 0594 .0686 .0778 .0872 .0965 .1060 .1155 .1251 .1348 16 31 47 63 78 65 2.1445 1543 .1642 .1742 .1842 .1943 .2045 .2148 .2251 .2355 17 34 51 68 85 66 2.2460 2566 .2673 .2781 .2889 .2998 .3109 .3220 .3332 .3445 18 37 55 73 92 67 2.3559 3673 .3789 .3906 .4023 .4142 .4262 .4383 .4504 .4627 20 40 60 79 99 68 2.4751 4876 .5002 .5129 .5257 .5386 .5517 .5649 .5782 .5916 22 43 65 87 108 69 2.6051 6187 .6325 .6464 .6605 .6746 .6889 .7034 .7179 .7326 24 47 71 95 119 70 2.7475 7625 .7776 .7929 .8083 .8239 .8397 .8556 .8716 .8878 26 52 78 104 130 71 2.9042 9208 .9375 .9544 .9714 .9887 .0061 .0237 .0415 .0595 29 58 87 116 144 72 3.0777 0961 .1146 .1334 .1524 .1716 .1910 .2106 .2305 .2506 32 64 96 129 161 73 3.2709 2914 .3122 .3332 .3544 .3759 .3977 .4197 .4420 .4646 36 72 108 144 180 74 3.4874 5105 .5339 .5576 .5816 .6059 .6305 .6554 .6806 .7062 41 81 122 1631204 75 3.7321 7583 .7848 .8118 .8391 .8667 .8947 .9232 .9520 .9812 76 4.0108 0408 .0713 .1022 .1335 .1653 .1976 .2303 .2635 .2972 77 4.3315 3662 .4015 .4373 .4737 .5107 .5483 .5864 .6252 .6646 78 4.7046 7453 .7867 .8288 .8716 .9152 .9594 .0045 .0504 .0970 Use ordinary 79 5.1446 1929 .2422 .2924 .3435 .3955 .4486 .5026 .5578 .6140 interpolation. 80 5.6713 7297 .7894 .8502 .9124 .9758 .0405 .1066 .1742 2432 81 6.3138 3859 .4596 .5350 .6122 .6912 .7720 .8548 .9395 0264 82 7.1154 .2066 .3002 .3962 .4947 .5958 .6996 .8062 .9158 0285 83 8.1443 2636 .3863 .5126 .6427 .7769 .9152 .0579 .2052 3572 84 9.5144 6768 .8448 .0187 .1988 .3854 .5789 .7797 .9882 2048 85 11.430 11.66 11.91 12.16 12.43 12.71 13.00 13.30 13.62 13.95 86 14.301 14.67 15.06 15.46 15.89 16.35 16.83 17.34 17.89 18.46 87 19.081 19.74 20.45 21.20 22.02 22.90 23.86 24.90 26.03 27.27 88 28.636 30.14 31.82 33.69 35.80 38.19 40.92 44.07 47.74 52.08 89 57.290 63.66 71.62 81.85 95.49 114.6 143.2 191.0 286.5 573.0 The integral part of tangents in heavy-face type is 1 greater than preceding part. 148 NATUKAL COTANGENTS. 0-45 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Differences 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 1' 2'| 3' 4' 5' 00 573.0 286.5 191.0 143.2 114.6 95.49 81.85 71.62 63.66 1 57.290 52.08 47.74 44.07 40.92 38.19 35.80 33.69 31.82 30.14 2 28.636 27.27 26.03 24.90 23.86 22.90 22.02 21.20 20.45 19.74 3 4 19.081 14.301 18.46 13.95 17.89 13.62 17.34 13.30 16.83 13.00 16.35 12.71 15.89 12.43 15.46 12.16 15.06 11.91 14.67 11.66 Use ordinary interpolation. 5 11.430 .2048 .9882 .7797 .5789 .3854 .1988 .0187 .8448 .6768 6 9.5144 .3572 .2052 .0579 .9152 .7769 .6427 .5126 .3863 .2636 7 8.1443 .0285 .9158 .8062 .6996 .5958 .4947 .3962 .3002 .2066 8 7.1154 .0264 .9395 .8548 .7720 .6912 .6122 .5350 .4596 .3859 9 6.3138 .2432 .1742 .1066 .0405 .9758 .9124 .8502 .7894 .7297 10 5.6713 .6140 .5578 .5026 .4486 .3955 .3435 .2924 .2422 .1929 11 5.1446 .0970 .0504 .0045 .9594 .9152 .8716 .8288 .7867 .7453 12 4.7046 .6646 .6252 .5864 .5483 .5107 .4737 .4373 .4015 .3662 13 4.3315 .2972 .2635 .2303 .1976 .1653 .1335 .1022 .0713 .0408 14 4.0108 .9812 .9520 .9232 .8947 .8667 .8391 .8118 .7848 .7583 15 3.7321 .7062 .6806 .6554 .6305 .6059 .5816 .5576 .5339 .5105 41 81 122 163 203 16 3.4874 .4646 .4420 .4197 .3977 .3759 .3544 .3332 .3122 .2914 36 72 108 144 180 17 3.2709 .2506 .2305 .2106 .1910 .1716 .1524 .1334 .1146 .0961 32 64 96 129 161 18 3.0777 .0595 .0415 .0237 .0061 .9887 .9714 .9544 .9375 .9208 29 58 87 116 144 19 2.9042 .8878 .8716 .8556 .8397 .8239 .8083 .7929 .7776 .7625 26 52 78 104 130 20 2.7475 .7326 .7179 .7034 .6889 .6746 .6605 .6464 .6325 .6187 24 47 71 95 119 21 2.6051 .5916 .5782 .5649 .5517 .5386 .5257 .5129 .5002 .4876 22 43 65 87 108 22 2.4751 .4627 .4504 .4383 .4262 .4142 .4023 .3906 .3789 .3673 20 40 60 79 99 23 2.3559 .3445 .3332 .3220 .3109 .2998 .2889 .2781 .2673 .2566 18 37 55 73 92 24 2.2460 .2355 .2251 .2148 .2045 .1943 .1842 .1742 .1642 .1543 17 34 51 68 85 25 2.1445 .1348 .1251 .1155 .1060 .0965 .0872 .0778 .0686 .0594 16 31 47 63 78 26 2.0503 .0413 .0323 .0233 .0145 .0057 .9970 .9883 .9797 .9711 15 29 44 58 73 27 1.9626 .9542 .9458 .9375 .9292 .9210 .9128 .9047 .8967 .8887 14 27 41 55 68 28 1.8807 .8728 .8650 .8572 .8495 .8418 .8341 .8265 .8190 .8115 13 26 38 51 64 29 1.8040 .7966 .7893 .7820 .7747 .7675 .7603 .7532 .7461 .7391 12 24 36 48 60 30 1.7321 .7251 .7182 .7113 .7045 .6977 .6909 .6842 .6775 .6709 11 23 34 45 56 31 1.6643 .6577 .6512 .6447 .6383 .6319 .6255 .6191 .6128 .6066 11 21 32 43 53 32 1.6003 .5941 .5880 .5818 .5757 .5697 .5637 .5577 .5517 .5458 10 20 30 40 50 33 1.5399 .5340 .5282 .5224 .5166 .5108 .5051 .4994 .4938 .4882 10 19 29 38 48 34 1.4826 .4770 .4715 .4659 .4605 .4550 .4496 .4442 .4388 .4335 9 18 27 36 45 35 1.4281 .4229 .4176 .4124 .4071 .4019 .3968 .3916 .3865 .3814 9 17 26 34 43 36 1.3764 .3713 .3663 .3613 .3564 .3514 .3465 .3416 .3367 .3319 8 16 25 33 41 37 1.3270 .3222 .3175 .3127 .3079 .3032 .2985 .2938 .2892 .2846 8 16 24 31 39 38 1.2799 .2753 .2708 .2662 .2617 .2572 .2527 .2482 .2437 .2393 8 15 23 30 38 39 1.2349 .2305 .2261 .2218 .2174 .2131 .2088 .2045 .2002 .1960 7 14 22 29 36 40 1.1918 .1875 .1833 .1792 .1750 .1708 .1667 .1626 .1585 .1544 7 14 21 28 34 41 1.1504 .1463 .1423 .1383 .1343 .1303 .1263 .1224 .1184 .1145 7 13 20 26 33 42 1.1106 .1067 .1028 .0990 .0951 .0913 .0875 .0837 .0799 .0761 6 13 19 25 32 43 1.0724 .0686 .0649 .0612 .0575 .0538 .0501 .0464 .0428 .0392 6 12 18 25 31 44 1.0355 .0319 .0283 .0247 .0212 .0176 .0141 .0105 .0070 .0035 6 12 18 24 30 The integral part of cotangents in heavy-face type is 1 less than preceding part. NATUKAL COTANGENTS. 45-90 149 o 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Differences 0' 6' 12' 18' 24' 30' 36' 42' 48' 54' 1' 2' 3' 4' 5' "45 1.0000 9965 9930 9896 9861 9827 9793 9759 9725 9691 5 11 17 23 29 46 0.9657 9623 9590 9556 9523 9490 9457 9424 9391 9358 5 11 17 22 28 47 0.9325 9293 9260 9228 9195 9163 9131 9099 9067 9036 5 11 16 21 27 48 0.9004 8972 8941 8910 8878 8847 8816 8785 8754 8724 5 10 16 21 26 49 0.8693 8662 8632 8601 8571 8541 8511 8481 8451 8421 5 10 15 20 25 50 0.8391 8361 8332 8302 8273 8243 8214 8185 8156 8127 5 10 15 20 24 51 0.8098 8069 8040 8012 7983 7954 7926 7898 7869 7841 5 9 14 19 24 52 0.7813 7785 7757 7729 7701 7673 7646 7618 7590 7563 5 9 14 18 23 53 0.7536 7508 7481 7454 7427 7400 7373 7346 7319 7292 5 9 14 18 23 54 0.7265 7239 7212 7186 7159 7133 7107 7080 7054 7028 4 9 13 18 22 55 0.7002 6976 6950 6924 6899 6873 6847 6822 6796 6771 4 9 13 17 21 56 0.6745 6720 6694 6669 6644 6619 6594 6569 6544 6519 4 8 13 17 21 57 0.6494 6469 6445 6420 6395 6371 6346 6322 6297 6273 4 8 12 16 20 58 0.6249 6224 6200 6176 6152 6128 6104 6080 6056 6032 4 8 12 16 20 59 0.6009 5985 5961 5938 5914 5890 5867 5844 5820 5797 4 8 12 16 20 60 0.5774 5750 5727 5704 5681 5658 5635 5612 5589 5566 4 8 12 15 19 61 0.5543 5520 5498 5475 5452 5430 5407 5384 5362 5340 4 8 11 15 19 62 0.5317 5295 5272 5250 5228 5206 5184 5161 5139 5117 4 7 11 15 18 63 0.5095 5073 5051 5029 5008 4986 4964 4942 4921 4899 4 7 11 15 18 64 0.4877 4856 4834 4813 4791 4770 4748 4727 4706 4684 4 7 11 14 18 65 0.4663 4642 4621 4599 4578 4557 4536 4515 4494 4473 4 7 11 14 18 66 0.4452 4431 4411 4390 4369 4348 4327 4307 4286 4265 3 7 10 14 17 67 0.4245 4224 4204 4183 4163 4142 4122 4101 4081 4061 3 7 10 14 17 68 0.4040 4020 4000 3979 3959 3939 3919 3899 3879 .3859 3 7 10 13 17 69 0.3839 3819 3799 3779 3759 3739 3719 3699 .3679 .3659 3 7 10 13 17 70 0.3640 .3620 3600 .3581 .3561 .3541 3522 .3502 .3482 .3463 3 7 10 13 16 71 0.3443 .3424 3404 .3385 .3365 .3346 3327 .3307 .3288 .3269 3 6 10 13 16 72 0.3249 .3230 .3211 .3191 .3172 .3153 3134 .3115 .3096 .3076 3 6 10 13 16 73 0.3057 .3038 .3019 .3000 .2981 .2962 2943 .2924 .2905 .2886 3 6 9 13 16 74 0.2867 .2849 .2830 .2811 .2792 .2773 2754 .2736 .2717 .2698 3 6 9 13 16 75 0.2679 .2661 .2642 .2623 .2605 .2586 2568 .2549 .2530 .2512 3 6 12 16 76 0.2493 .2475 .2456 .2438 .2419 .2401 2382 .2364 .2345 .2327 3 6 12 15 77 0.2309 .2290 .2272 .2254 .2235 .2217 .2199 .2180 .2162 .2144 3 6 12 15 78 0.2126 .2107 .2089 .2071 .2053 .2035 .2016 .1998 .1980 .1962 3 6 12 15 79 0.1944 .1926 .1908 .1890 .1871 .1853 .1835 .1817 .1799 .1781 3 6 < 12 15 80 0.1763 .1745 .1727 .1709 .1691 .1673 .1655 .1638 .1620 .1602 3 6 12 15 81 0.1584 .1566 .1548 .1530 .1512 .1495 .1477 .1459 .1441 .1423 3 6 12 15 82 0.1405 .1388 .1370 .1352 .1334 .1317 .1299 .1281 .1263 .1246 3 6 12 15 83 0.1228 .1210 .1192 .1175 .1157 .1139 .1122 .1104 .1086 .1069 3 6 12 15 84 0.1051 .1033 .1016 .0998 .0981 .0963 .0945 .0928 .0910 .0392 3 6 12 15 85 0.0875 .0857 .0840 .0822 .0805 .0787 .0769 .0752 .0734 .0717 3 6 12 15 86 0.0699 .0682 .0664 .0647 .0629 .0612 .0594 .0577 .0559 .0542 3 6 12 15 87 0.0524 .0507 .0489 .0472 .0454 .0437 .0419 .0402 .0384 .0367 3 6 12 15 88 0.0349 .0332 .0314 .0297 .0279 .0262 .0244 .0227 .0209 .0192 3 6 12 15 89 0.0175 .0157 .0140 .0122 .0105 .0087 .0070 .0052 .0035 .0017 3 6 12 15 All cotangents greater than cot 45 are less than 1. 150 TABLES FOE REFERENCE POWERS AND ROOTS No. Squares Cubes Square Roots Cube Roots No. Squares Cubes Square Roots Cube Roots 1 1 1 1.000 1.000 51 2601 132 651 7.141 3.708 2 4 8 1.414 1.260 52 2704 140 608 7.211 3.733 3 9 27 1.732 1.442 53 2809 148 877 7.280 3.756 4 16 64 2.000 1.587 54 2916 157 464 7.348 3.780 5 25 125 2.236 1.710 55 3025 166 375 7.416 3.803 6 36 216 2.449 1.817 56 3136 175 616 7.483 3.826 7 49 343 2.646 1.913 57 3249 185 193 7.550 3.849 8 64 512 2.828 2.000 58 3364 195 112 7.616 3.871 9 81 729 3.000 2.080 59 3481 205 379 7.681 3.893 10 100 1000 3.162 2.154 60 3600 216 000 7.746 3.915 11 121 1331 3.317 2.224 61 3721 226 981 7.810 3.936 12 144 1728 3.464 2.289 62 3844 238 328 7.874 3.958 13 169 2197 3.606 2.351 63 3969 250 047 7.937 3.979 14 196 2744 3.742 2.410 64 4096 262 144 8.000 4.000 15 225 3375 3.873 2.466 65 4225 274 625 8.062 4.021 16 256 4096 4.000 2.520 66 4356 287 496 8.124 4.041 17 289 4913 4.123 2.571 67 4489 300763 8.185 4.062 18 324 5832 4.243 2.621 68 4624 314 432 8.246 4.082 19 361 6859 4.359 2.668 69 4761 328 509 8.307 4.102 20 400 8000 4.472 2.714 70 4900 343 000 8.367 4.121 21 441 9261 4.583 2.759 71 5041 357 911 8.426 4.141 22 484 10648 4.690 2.802 72 5184 373 248 8.485 4.160 23 529 12167 4.796 2.844 73 5329 389 017 8.544 4.179 24 576 13824 4.899 2.884 74 5476 405 224 8.602 4.198 25 625 15625 5.000 2.924 75 5625 421 875 8.660 4.217 26 676 17576 5.099 2.962 76 5776 438 976 8.718 4.236 27 729 19683 5.196 3.000 77 5929 456 533 8.775 4.254 28 784 21952 5.292 3.037 78 6084 474 552 8.832 4.273 29 841 24389 5.385 3.072 79 6241 493 039 8.888 4.291 30 900 27000 5.477 3.107 80 6400 512 000 8.944 4.309 31 961 29791 5.568 3.141 81 6561 531441 9.000 4.327 32 1024 32768 5.657 3.175 82 6724 551 368 9.055 4.344 33 1089 35937 5.745 3.208 83 6889 571 787 9.110 4.362 34 1156 39304 5.831 3.240 84 7056 592 704 9.165 4.380 35 1225 42875 5.916 3.271 85 7225 614 125 9.220 4.397 36 1296 46656 6.000 3.302 86 7396 636 056 9.274 4.414 37 1369 50653 6.083 3.332 87 7569 658 503 9.327 4.431 38 1444 54872 6.164 3.362 88 7744 681 472 9.381 4.448 39 1521 59319 6.245 3.391 89 7921 704 969 9.434 4.465 40 1600 64000 6.325 3.420 90 8100 729000 9.487 4.481 41 1681 68921 6.403 3.448 91 8281 753 571 9.539 4.498 42 1764 74088 6.481 3.476 92 8464 778 688 9.592 4.514 43 1849 79507 6.557 3.503 93 8649 804 357 9.644 4.531 44 1936 85184 6.633 3.530 94 8836 830 584 9.695 4.547 45 2025 91125 6.708 3.557 95 9025 857 375 9.747 4.563 46 2116 97336 6.782 3.583 96 9216 884 736 9.798 4.579 47 2209 103 823 6.856 3.609 97 9409 912 673 9.849 4.595 48 2304 110 592 6.928 3.634 98 9604 941 192 9.899 4.610 49 2401 117649 7.000 3.659 99 9801 970 299 9.950 4.626 60 2500 125000 7.071 3.684 100 10000 1 000 000 10.000 4.642 TABLES FOR REFERENCE 151 SIZES OF TWIST DRILLS WITH DECIMAL EQUIVALENTS SIZE DECIMAL EQUIVALENT SIZE DECIMAL EQUIVALENT SIZE DECIMAL EQUIVALENT SIZE DECIMAL EQUIVALENT I" 0.5000" 4 0.2500" #26 0.1470" #56 0.0465" ir .4844 E .2500 #27 .1440 #57 .0430 ji- .4688 D .2460 ft" .1406 #58 .0420 ir .4531 c .2420 #28 .1405 #59 .0410 i 7 ??" .4375 B .2380 #29 .1360 #60 .0400 ii" .4219 si" .2344 #30 .1285 #60^ .0390 z .4130 A .2340 r .1250 #61 .0380 ii" .4063 #1 .2280 #31 .1200 #62 .0370 Y .4040 #2 .2210 #32 .1160 #63 .0360 X .3970 ft" .2188 #33 .1130 #64 .0350 IT 6 4: .3906 #3 .2130 #34 .1110 #65 .0330 w .3860 #4 .2090 #35 .1100 #66 .0320 V .3770 #5 .2055 ft" .1094 ft" .0313 r .3750 #6 .2040 #36 .1065 #67 .0310 U .3680 ir .2031 #37 .1040 #68 .0300 |" .3594 #7 .2010 #38 .1015 #681 .0295 T .3580 #8 .1990 #39 .0995 #69 .0290 S .3480 #9 .1960 #40 .0980 #69 .0280 jr .3438 #10 .1935 #41 .0960 #70 .0270 R .3390 #11 .1910 ft" .0938 #71 .0260 Q .3320 #12 .1890 #42 .0935 #71 .0250 ir .3281 ft" .1875 #43 .0890 #72 .0240 p .3230 #13 .1850 #44 .0860 #73 .0230 .3160 #14 .1820 #45 .0820 #73^ .0225 ft" .3125 #15 .1800 #46 .0810 #74 .0220 N .3020 #16 .1770 #47 .0785 #74^ .0210 if" .2969 #17 .1730 ft" .0781 #75 .0200 M .2950 H" .1719 #48 .0760 #76 .0180 L .2900 #18 .1695 #49 .0730 #77 .0160 ft" .2813 #19 .1660 #50 .0700 sV' .0156 K .2810 #20 .1610 #51 .0670 #78 .0150 J .2770 #21 .1590 #52 .0635 #78^ .0145 I .2720 #22 .1570 A" .0625 #79" .0140 H .2660 .5 " .1563 #53 .0595 #79| .0135 ir .2656 #23 .1540 #54 .0550 #80 .0130 G .2610 #24 .1520 #55 .0520 . . . . . F .2570 #25 .1495 ft" .0469 . . . 152 TABLES FOR REFERENCE SPEEDS OF DRILLS DIAMETER OF DRILL WROUGHT IRON AND STEEL CAST IRON BRASS DIAMETER OF DRILL WROUGHT IRON AND STEEL CAST IRON BRASS ft" 1712 2383 3544 1ft" 72 108 180 1 855 1191 1772 U 68 102 170 ft 571 794 1181 1ft 64 97 161 397 565 855 U 58 89 150 T 5 ff 318 452 684 1ft 55 84 143 1 265 377 570 If 53 81 136 TV 227 323 489 1ft 50 77 130 4 183 267 412 14 46 74 122 Tff 163 238 367 l T 9 ff 44 71 117 i 147 214 330 if 40 66 113 133 194 300 38 63 109 1 112 168 265 13 37 61 105 it 103 155 244 1 T? 36 59 101 . 96 144 227 If 33 55 98 if 89 134 212 Hf 32 53 95 1 76 115 191 2 31 51 92 The speeds in the above table are given in R. P. M. MACHINE SCREW THREADS, OLD STANDARD SIZES NUMBER OF SCREW OUTSIDE DIAMETER THREADS TO 1" NUMBER OF SCREW OUTSIDE DIAMETER THREADS TO 1" 1 0.071" 64 12 0.221" 24 14 2 .081 .089 56 56 13 14 .234 .246 22 20 3 .101 48 15 .261 20 4 .113 36 16 .272 18 5 .125 36 18 .298 18 6 .141 32 20 .325 16 7 .154 32 22 .350 16 8 .166 32 24 .378 16 9 .180 30 26 .404 16 10 .194 24 28 .430 14 11 .206 24 30 .456 14 The basic form of this thread is the same as that of a U.S.S. thread. TABLES FOR REFERENCE 153 MACHINE SCREW THREADS, A.S.M. E. STANDARD SIZES NUMBEK OF SCREW MAXIMUM DIAMETER THREADS TOl" NUMBER OF SCREW MAXIMUM DIAMETER THREADS TO 1" 0.060" 80 12 0.216" 28 1 .073 72 14 .242 24 2 .086 64 16 .268 22 3 .099 56 18 .294 20 4 .112 48 20 .320 20 5 .125 44 22 .346 18 6 .138 40 24 .372 16 7 .151 36 26 .398 16 8 .164 36 28 .424 14 9 .177 32 30 .450 14 10 .190 30 - MACHINE SCREW THREADS, A.S.M.E. SPECIAL SIZES NUMBER OF SCREW MAXIMUM DIAMETER THREADS TOl" NUMBER OF SCREW MAXIMUM DIAMETER THREADS TO 1" 1 0.073" 64 9 0.177" 24 2 .086 56 10 .190 32 3 .099 48 10 .190 24 4 .112 40 12 .216 24 4 .112 36 14 .242 20 5 .125 40 16 .268 20 5 .125 36 18 .294 18 6 .138 36 20 .320 18 6 .138 32 22 .346 16 7 .151 32 24 .372 18 7 .151 30 26 .398 14 8 .164 32 28 .424 16 8 .164 30 30 .450 16 9 .177 30 The A.S.M.E. (American Society of Mechanical Engineers) standard specifies a maximum and a minimum outside diameter. The minimum diameter (not given in these tables) may be found by the formula 0.336 Minimum D n maximum D n 40 + number of threads to V 154 TABLES FOR REFERENCE U.S.S.-THREAD BOLTS AND NUTS M . AREA IN DIMENSIONS OF NUTS AND H ^ M ^ SQUARE BOLT HEADS H *"* ^ ^ INCHES M o H h 5 33 a 3 1 5 O k-3 fl o O H H OUTBID 8 a J 1 1 5 P 1 | ft 1 j" 20 0.185'' w 0.049 0.026 r 0.578 /x 0.707" 4~ 4 T 6 ?r 18 0.240 I 0.076 0.045 0.686 0.840 A H | 16 0.294 A 0.110 0.068 11 0.794 0.972 f T 7 s 14 0.345 H 0.150 0.093 it 0.902 1.105 A I! | 13 0.400 H 0.196 0.126 1.011 1.237 i 7 T(T ^ 12 0.454 *f 0.248 0.162 1.119 1.370 * ii 1 11 0.507 0.307 0.202 *A 1.227 1.502 17. 10 0.620 fi 0.442 0.302 4 1.444 1.768 ^ |" | 9 0.731 0.601 0.419 1.660 2.033 -7 1 ' 1 8 0.838 |5 0.785 0.551 15 1.877 2.298 1 T H 7 0.939 ^ 0.994 0.694 HI 2.093 2.563 H SI i^ 7 1.064 1-^. 1.227 0.893 2 2.310 2.828 11 if 6 1.158 !A 1.485 1.057 2 3 2.527 3.093 ]3 1 *\ 4 6 1.283 1.767 1.295 23 s 2.743 3.358 H iA if ty 1.389 Iff 2.074 1.515 2 A 2.960 3.623 If if 5 1.490 1^1 2.405 1.746 3.176 3.889 4 i| i& 5 1.615 1^ 2.761 2.051 2!-! 3.393 4.154 4 1 ), f; 2 4^ 1.711 !fl 3.142 2.302 3.609 4.419 2 'fV 2 i 41 1.961 3.976 3.023 3 4.043 4.949 2? 4 2* 4 2.175 2^1 4.909 3.719 3| 4.476 5.479 2 i 2| 4 2.425 2fi 5.940 4.620 4| 4.909 6.010 2| 21 3 3 i 2.629 2 T 7.069 5.428 4 5.342 6.540 3 2f?r 3^ 4 2.879 2 T | 8,296 6.510 5 5.775 7.070 3J 2 i 3 i 3 i 3.100 3 f i 9.621 7.548 51 6.208 7.600 2H 3f 3 3.317 11.045 8.641 6.641 8.131 32 4 3 3.567 3| 12.566 9.963 6^ 7.074 8.661 4 3 iV 4| 2| 3.798 3 fi 14.186 11.340 6 ^ 7.508 9.191 4 4 3^ < 4 2 | 4.028 4 TJ\ 15.904 12.750 6 & 7.941 9.721 4 i 3^ 4| 2 I 4.255 4 TV 17.721 14.215 7 i 8.374* 10.252 4 s| 5 2 | 4.480 4 T 9 ^ 19.635 15.760 7 f 8.807 10.782 5 3 yii 5^ 2^ 4.730 413 21.648 17.570 8 9.240 11.312 5| 4 5 i 2 i 4.953 ^A 23.758 19.260 8f 9.673 11.842 51 4 fc 5| 2 i 5.203 ^ A 25.967 21.250 8 10.106 12.373 5| 4 ; ^ ' 6 2 i 5.423 5 2 28.274 23.090 9 I 10.539 12.903 6 4 A TABLES FOR REFERENCE 155 TAP DRILLS FOR MACHINE-SCREW TAPS NUMBER OF TAP THREADS TO 1" NUMBER OF DRILL NUMBER OF TAP THREADS TO \" NUMBER OF DRILL 2 48 48 13 20 17 2 56 46 13 24 15 2 64 45 14 20 14 3 40 48 14 22 13 3 48 47 14 24 11 3 56 45 15 18 12 4 32 45 15 20 10 4 36 43 15 24 7 4 40 42 5 5 30 32 41 40 16 16 16 16 18 20 10 7 5 5 36 38 16 24 1 5 40 36 6 6 6 30 32 36 39 37 35 17 17 17 16 18 20 7 4 2 6 40 33 18 16 2 7 28 32 18 18 1 7 30 31 18 20 B 7 32 30 19 16 C 8 24 31 19 18 D 8 30 30 19 20 E 8 32 29 20 16 E 9 24 29 20 18 E 9 28 27 20 20 F 9 30 26 22 16 H 9 32 24 22 18 I 10 24 26 24 14 K 10 28 24 24 16 L 10 30 23 24 18 M 10 32 21 26 14 O 11 24 20 26 16 P 11 28 19 11 30 18 28 14 R 12 20 21 28 16 S 12 22 19 30 14 T 12 24 19 30 16 U 156 TABLES FOR REFERENCE TAP DRILLS FOR U.S.S. THREADS THREAD THREADS DIAMETER THREAD THREADS DIAMETER DIAMETER TOl" OF DRILL DIAMETER TOl" OF DRILL 20 0.191" 1" 8 0.854" A 18 .248 H 7 0.957 i 16 .302 u 7 1.082 TV 14 .354 If 6 1.179 1 13 .409 ll 6 1.304 A 12 .465 If H 1.412 i 11 .518 if 5 1.515 1 10 .632 17 5 1.640 7 9 .745 2 H 1.739 TAP DRILLS FOR SHARP V-THREADS THREAD DIAMETER THREADS TOl" DIAMETER OF DRILL THREAD DIAMETER THREADS TOl" DIAMETER OF DRILL i f/ 20 0.184" 1 8 0.832" A 18 .239 1| 7 0.932 1 16 .293 I* 7 1.057 7 TS 14 .345 1| 6 1.144 i 12 .399 H 6 1.269 I 9 * . 12 .453 if 5 1.347 f 11 .506 if 5 1.472 1 10 .618 Ml ^ 1.566 i 9 .728 2 Ji 1.691 TAP DRILLS FOR BRIGGS PIPE THREADS NOMINAL SIZE THREADS TOl" DIAMETER OF DRILL NOMINAL SIZE THREADS TOl" DIAMETER OF DRILL I" 8 27 0.328" 1$" HJ 1.719" 18 .453 2 111 2.188 I 18 .594 2 i 8 2.688 14 .719 3 8 3.313 a 4 14 .938 H 8 3.813 1 11J 1.188 4 8 4.313 H Vj, 1.469 H 8 4.813 The diameter of drill allows for reaming the hole before tapping. TABLES FOR REFERENCE 157 TOOTH PARTS OF DIAMETRAL-PITCH GEARS DIAMETRAL PITCH CIRCULAR PITCH THICKNESS OF T(X)TH ON PITCH LINE ADDENDUM WORKING DEPTH OF TOOTH ROOT OF TOOTH WHOLE DEPTH OF TOOTH 1 6.2832" 3.1416" 2.0000" 4.0000" 2.3142" 4.3142" a 4 4.1888 2.0944 1.3333 2.6666 1.5428 2.8761 1 3.1416 1.5708 1.0000 2.0000 1.1571 2.1571 1| 2.5133 1.2566 0.8000 1.6000 0.9257 1.7257 2.0944 1.0472 .6666 1.3333 .7714 1.4381 1| 1.7952 0.8976 .5714 1.1429 .6612 1.2326 2 1.5708 .7854 .5000 1.0000 .5785 1.0785 21 1.3963 .6981 .4444 0.8888 .5143 0.9587 2 5 1.2566 .6283 .4000 .8000 .4628 .8628 2| 1.1424 .5712 .3636 .7273 .4208 .7844 3 1.0472 .5236 .3333 .6666 .3857 .7190 3 S 0.8976 .4488 .2857 .5714 .3306 .6163 4 .7854 .3927 .2500 .5000 .2893 .5393 5 .6283 .3142 .2000 .4000 .2314 .4314 6 .5236 .2618 .1666 .3333 .1928 .3595 7 .4488 .2244 .1429 .2857 .1653 .3081 8 .3927 .1963 .1250 .2500 .1446 .2696 9 .3491 .1745 .1111 .2222 .1286 .2397 10 .3142 .1571 .1000 .2000 .1157 .2157 12 .2618 .1309 .0833 .1666 .0964 .1798 14 .2244 .1122 .0714 .1429 .0826 .1541 16 .1963 .0982 .0625 .1250 .0723 .1348 18 .1745 .0873 .0555 .1111 .0643 .1198 20 .1571 .0785 .0500 .1000 .0579 .1079 22 .1428 .0714 .0455 .0909 .0626 .0980 24 .1309 .0654 .0417 .0833 .0482 .0898 26 .1208 .0604 .0385 .0769 .0445 .0829 28 .1122 .0561 .0357 .0714 .0413 .0770 30 .1047 .0524 .0333 .0666 .0386 .0719 32 .0982 .0491 .0312 .0625 .0362 .0674 36 .0873 .0436 .0278 .0555 .0321 .0599 40 .0785 .0393 .0250 .0500 .0289 .0539 44 .0714 .0357 .0227 .0455 .0263 .0490 48 .0654 .0327 .0208 .0417 .0241 .0449 50 .0628 .0314 .0200 .0400 .0231 .0431 158 TABLES FOR REFERENCE TOOTH PARTS OF CIRCULAR-PITCH GEARS CIRCULAR PITCH DIAMETRAL PITCH THICKNESS OF TOOTH ON PITCH LINE ADDENDUM WORKING DEPTH OF To< )TH ROOT OF TOOTH WHOLE DEPTH OF TOOTH 4" 0.7854 2.0000" 1.2732" 2.5464" 1.4732" 2.7464" 3i 3 0.8976 1.0472 1.7500 1.5000 1.1140 0.9549 2.2281 1.9098 1.2890 1.1049 2.4031 2.0598 2 4 1.1424 1.3750 .8753 1.7506 1.0128 1.8881 2 , 1.2566 1.3963 1.2500 1.1250 .7957 .7162 1.5915 1.4323 0.9207 .8287 1.7165 1.5448 2 1.5708 1.0000 .6366 1.2732 .7366 1.3732 1J 1.6755 0.9375 .5968 1.1937 .6906 1.2874 If If 1.7952 1.9333 .8750 .8125 .5570 .5173 1.1141 1.0345 .6445 .5985 1.2016 1.1158 ** 2.0944 2.1855 .7500 .7187 .4775 .4576 0.9549 .9151 .5525 .5294 1.0299 0.9870 If* 2.2848 .6875 .4377 .8754 .5064 .9441 2.3936 .6562 .4178 .8356 .4834 .9012 H 2.5133 2.6456 .6250 .5937 .3979 .3780 .7958 .7560 .4604 .4374 .8583 .8154 11 2.7925 .5625 .3581 .7162 .4143 .7724 ^A 2.9568 .5312 .3382 .6764 .3913 .7295 i 3.1416 .5000 .3183 .6366 .3683 .6866 if 7 8 H 3 4 U f 3.3510 3.5904 3.8666 4.1888 4.5696 5.0265 .4687 .4375 .4062 .3750 .3437 .3125 .2984 .2785 .2586 .2387 .2189 .1989 .5968 .5570 .5173 .4775 .4377 .3979 .3453 .3223 .2993 .2762 .2532 .2301 .6437 .6007 .5579 .5150 .4720 .4291 5.5851 .2812 .1790 .3581 .2071 .3862 1 6.2832 .2500 .1592 .3183 .1842 .3433 TV 7.1808 .2187 .1393 .2785 .1611 .3003 3 8.3776 .1875 .1194 .2387 .1381 .2575 1. 9.4248 .1666 .1061 .2122 .1228 .2289 T 5 ff 10.0531 .1562 .0995 .1989 .1151 .2146 1 12.5664 .1250 .0796 .1591 .0921 .1716 3 16.7552 .0937 .0597 .1194 .0690 .1287 1 25.1327 .0625 .0398 .0796 .0460 .0858 TV 50.2655 .0312 .0199 .0398 .0230 .0429 INDEX PAGE Acme thread 48 Addendum 86 angle 98 angular 98 Angle 28, 138 addendum 98 with axis 28, 32, 33 clearance 46 cutting 98 dedendum 98 diameter 52 face 98, 112 gashing 112, 115 included 28, 32, 33 for milling cams .... 82 pitch-cone 99 spiral 73, 78, 104 tooth 104 Angular addendum 98 indexing 66 Axial pitch 104 Bevel gear . . . Briggs pipe thread . . 97 . 50, 156 Caliper micrometer . . with vernier vernier .... Cam . . . 2 . . 2 . . 5 4 80 Center distance . . . 86, 91, 103 Change gears 56, 57, 70, 73, 75, 107 PAGE Chordal pitch 86 Circle, outside 86 pitch 85, 86, 98, 99 Circular pitch .... 86, 97, 104 Clearance .... 46, 86, 113, 115 Compound gearing . . .59, 72, 75 indexing 67 rest 25, 28 Cutter, gear 107 Cutting angle 98 speeds ... 9, 10, 11, 12, 16 Decimal equivalents .... 139 Dedendum 86 angle 98 Depth, double 35 of thread 35 whole 87, 90, 97 working 87 Diagonal scale 1 Diameter, angle 52 increment 98 outside 35, 86, 97 pitch .... 52, 85, 86, 97, 99 root 35 throat ........ 112 Diametral pitch ... 86, 97, 104 Differential indexing .... 70 Direct indexing 62 Dividing head 61, 63, 73 Drills, sizes of . 151 speeds of 13, 152 159 160 INDEX Equivalents, common decimal . PACK 140 139 Face 86, 99 angle 98, 112 Feed 9, 17 Finishing speeds 10 Flank 86 Follower 80 Fractions, equivalents of ... 139 indexing 69,70 Franklin Institute thread . . 41 French thread 44 Friction surface . . . .85, 97, 99 Functions, tables of ... 141-149 Gaging depth of cut 8 Gashing angle 112, 115 Gearing, compound . . . 59, 72, 75 simple 56, 71 worm 112 Gears . 85 bevel 97 change . . 56, 57, 70, 73, 75, 107 cutters for 107 helical 103 internal 85 spiral 103, 110 spur 85, 86 teeth of ... 85, 94, 157, 158 Hob 115 Robbing 113, 115 Increment, diameter .... 98 Index center 62 head 61 plates 65 Indexing 61 angular 66 PAGE Indexing (continued) compound 67 differential 70 direct 62 fractions . . . . . . 69, 70 simple 64 Instruments, measuring ... 1 International thread .... 44 Involute teeth . . . ;-- ;- r . 85 sizes of . . ... . .94, 157, 158 Lathe, compound gearing for . 59 cutting speed of .... 10 simple gearing for ... 56 Lead of a cam 80 of a milling machine . . 75 of a screw 35 of a spiral .... 73, 75, 104 screw 56 Linear pitch 96, 112 Machine screw thread 152, 153, 155 Measures, common 137 metric . ., ,. 138 Measuring instruments ... 1 tapers . . 32, 33 threads 52 Metric measures 138 thread 44 Micrometer . 2 caliper $/ with vernier .... 5 reading a 8 Milling machine, cutting speed of 12 lead of 75 setting table of 78 Multiple thread 36 Normal circular pitch .... 104 diametral pitch 104 INDEX 161 PAGE Offset for tapers 24 Outside circle 86 diameter 35, 86, 97 Pitch 35, 86, 103 axial 104 chordal 86 circle 85, 86, 98, 99 circular 86, 97, 104 -cone alible 99 radius 99 diameter of a gear 85, 86, 97, 99 of a thread 52 diametral .... 86, 97, 104 line 86, 98, 99 linear 96, 112 normal 104 Pinion 85, 91, 97 Pipe thread 50, 156 Planer, cutting speed of ... 11 Problem material bevel gears .... 102, 116 calipers 3, 6 cams ........ 83, 84 cutting speeds 10, 11, 12, 14, 16, 18, 20 drill press 118 drilling machine .... 126 drills, speeds of . . 13, 14, 20 feeds 17, 18 gaging cuts 8 general applications .117, 134, 136 indexing . 65, 66, 69, 72, 120 lathes .... 10, 14, 20, 132 milling . . 12, 14, 18, 120, 128 planers 11, 18, 124 review 14, 18, 30, 55, 72, 84, 96, 117-136 shapers 16, 20 spiral gears . . 109, 111, 116 PAGE Problem material (continued) spirals '. 77, 79, 84 spur gears . 89, 92, 94, 96, 116 tapers . . 22, 26, 29, 30, 34, 122, 130 threads 37, 39, 41, 43, 45, 47, 49, 51, 54, 56, 58, 60 worm gearing . . . 115, 116 Rack 85, 96 Radius, pitch-cone 99 Ratio, velocity 91, 106 Rest, compound 25, 28 Review problems . 14, 18, 30, 55, 72, 84, 96, 117-136 Rise 80 Root 35, 86 diameter 35 Roughing speeds 10 Rules, convenient 140 S. A. E. thread 40 Scale, diagonal 1 Screw, hand of 36 lead of 35 thread 35 Sellers thread 41 Shaper, cutting speed of ... 16 Sharp V-thread 38 Simple gearing 56, 71 indexing 64 Sine bar 33 Speeds, cutting . 9, 10, 11, 12, 16 of drills 13, 152 surface 9 Spiral 73 angle 73, 78, 104 gears 103 with parallel shafts . 110 head 61 lead of 73, 75, 104 162 INDEX PAGE Spur gears 85, 86 Square thread 46 tool 46 Stud ! 56, 59 Symbols . . 21, 88, 100, 106, 112 Tables angles and arcs 138 Briggs pipe threads . . 51, 156 common fractions .... 139 measures 137 cosines 144 cotangents 148 cutting speeds . . . . 10, 12 decimal equivalents . . . 139 drills, sizes of 151 speeds of 152 feeds 17 machine screw threads 152, 153, 155 metric measures .... 138 threads 44 powers and roots .... 150 S. A. E. thread 40 sines 142 tangents 146 tap drills ..... 155, 156 tooth parts .... 157, 158 U. S. S. thread .... 40, 156 bolts and nuts . . . 154 V-thread 38, 156 Whitworth thread ... 42 Tailstock, offsetting the ... 24 Tap 39 -drill size . . 39, 41, 155, 156 Taper 21 attachment 25 measuring a 32, 33 testing a 32 turning a ...'.-... 24 PAGE Teeth, cycloidal 85 involute 85, 95 sizes of ... 94, 157, 158 Thread 35 Acme j . 48 cutting 56, 59 machine screw . 152, 153, 155 measuring 52 metric 44 multiple . .. 36 pipe ........ 50, 156 S. A. E 40 screw 35 square 46 U. S. S. . . . 40, 52, 154, 156 V- 38, 53, 156 Whitworth 42, 54 Throat diameter 112 Tooth angle 104 thickness of . 87, 100, 102, 107 Trigonometric functions ... 141 U. S. S. thread 40, 52, 154, 156 V-thread, sharp ... 38, 53, 156 Velocity ratio 91, 106 Vernier 4 caliper 4 micrometer caliper with . 5 reading a 5 Vertex of a bevel gear .... 99 Wheel, gear 85 worm ....... 63, 64, 112 Whitworth thread .... 42, 54 Whole depth 87, 90, 97 Working depth 87 Worm 63, 64, 73, 112 gearing 112 wheel 63, 64, 112 PRINTKD IN THE UNITED STATES OF AMERICA UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. 2lMar'50B6 LIBRARY USE BRARY USE mo LIBRARY USE SEP 8 1960 SEP 15 1960 NP ;*D Lp REC'D Lp- MAK1H ) -1 LD 21-100m-ll,'49(B7146sl6)476 .. O f ) * -f M306017 TI THE UNIVERSITY OF CALIFORNIA LIBRARY