alto the formation of the teeth of nearly every type of gear in common use ; there-fore, a study of the subject of gear-teeth can best be begun by a study of the formation of the teeth of the spur gear. 6. The pitch cylinders of spur gears are the imaginary cylinders on which the teeth are constructed and that roll together with the same relative speed as the gears themselves. 7. The pitch circle of a is a circle that represents the pitch cylinder. The  pic page alt="http://antiquemachinery.com/images-2020/Machinery-Magazine-March-1896-vol-2-no-7-top-Cover-men-working-old-shop-lineshaft-belt-drive-1896%20(2).jpg" < Pic 3

alt="https://antiquemachinery.com/images-American-Machinist-Oct-29-1942/pg-1237-Armamet-Tank-Engines-Airplane-made-on-Coverted-Vintage-old-Antique-Equipment-Baker-Rock-Island-Veteran-Drill-new-heavy-Fixture-Machine-Shop-Machines.jpeg" text pg 3

 

GEAR CALCULATIONS. § 17 23.

A rack is a gear with a pitch circle having an infinite radius, that is, the pitch cylinder has become a plane, so that all the teeth are arranged in a straight line. 24. A pinion is a small spur gear. The term is used especially for small gears that mesh with racks. 
PROPORTIONS FOR GEAR-TEETH. 25. Gear-Teeth Based on Circular Pitch.-The relative proportions of gear-teeth are usually based on the circular pitch. It is customary to have the addendum, the whole depth, and the thickness of the tooth conform to some arbitrary part of the circular pitch. There is no uni-formly adopted standard, and gears made in different ways require different proportions. Cut gears require less back-lash and clearance than cast gears. This is because the teeth of cut gears are more uniform in size, regular in out-line, and truer in form than those of cast gears. Gears of large diameter require less backlash than those of small diameter. The proportions given in Table I are those in common use. 
TABLE I. 

GEAR-TOOTH PROPORTIONS. 
I 2 3 Addendum  3o C .3o C .3o C Root  40 C .40 C. .35 C Working depth of tooth .6o C .6o C .6o C Total depth of tooth... .7o C .70. C .65 C Clearance   10 C .10 C .05 C Thickness of tooth  45 C .475 C .485 C Width of space  55 C .525 C .515 C Backlash.  10 C .05 C .03 C 
4  
1P. 1.157=-P to 1.125 P 2 2. 157 . 157 4- P to . 125 ± 1.51 P to 1.57 P 1.63 P to 1.57 P . 12 P to o 
Recently some manufacturers have made gears with shorter teeth than those indicated in Table I. In some 

§ 17 GEAR CALCULATIONS.  page 9 
*cases the total depth of teeth was not more than five-tenths the circular, or 11 divided by the diametral, pitch. The width of tooth and space remained the same except that the backlash might be slightly reduced. The Brown & Sharpe Company make the clearance one-tenth the thickness of the tooth on the pitch line, and The Pratt & Whitney Company make it one-eighth the addendum. The proportions given in the foregoing table have been used successfully and will serve as an aid in deciding upon suitable dimensions. Column 1 is for rough cast gears where the teeth are very irregular, and consequently a large amount of backlash and clearance is required ; column 2 is for the better class of cast gears ; column 3 is for cut gears ; and column 4 is for diametral pitch for cut gears. C stands for circular pitch, and P for diametral pitch. 26. Proportions of Gear-Teeth Based on Diam-etral PitCh.—As the gears most often met with are cut gears of diametral pitch, it would seem natural to propor-tion gear-teeth with the diametral pitch as a basis. It is much simpler to calculate gears using diametral pitch than when using circular pitch ; hence, this system is coming into very general use for all classes of gearing. Column 4 of Table I gives the proportions of gear-teeth based on this system as used by the leading manufacturers in this country. 

RULES FOR SPUR-GEAR CALCULATIONS. 27. Relation Between Circular Pitch and Diam-etral Pitch.—The product of the circular pitch of a gear and the diametral pitch is always the constant number 3.1416. Hence the following rules : Rule.— To change circular pitch to diametral pitch divide 3.1416 by the circular pitch. EXAMPLE.-If the circular pitch is .3927 inch, what is the diametral pitch ?

 pg 10--page 11

10 GEAR CALCULATIONS. 
§ 17 SOLUTION.-Applying the rule just given, the diametral pitch is 3.1416 .3927 = 8. Ans. 
28. Rule.— To change diametral pitch to circular pitch, divide 3.1416 by the diametral pitch. EXAMPLE.-If the diametral pitch is 4, what is the circular pitch ? SOLUTION.-Applying the above rule, the circular pitch is 3.1416   = .7854 in. Ans. 4 29. Relation Between Pitch Diameter, Number of Teeth, and Diametral Pitch. — The relation between the pitch diameter, number of teeth, and diametral pitch is expressed in the following rules : Rule.— To find the number of teeth when the pitch diam-eter and the diametral pitch are known, multiply the pitch diameter by the diametral pitch. EXAMPLE.-If a wheel is 30 inches in diameter and 3 pitch, how many teeth has it ? SOLUTION.-Applying the rule just given, the number of teeth is 30 x 3 = 90. Ans. 30. Rule.--To find the pitch diameter when the num-ber of teeth and the diametral pitch are known, divide the number of teeth by the diametral pitch. EXAMPLE.-What is the pitch diameter of a 21-pitch gear having 20 teeth ? SOLUTION.-By applying the rule just given, we find the diameter to be 
20 = 8 in. Ans. 31. Rule.— To find the diametral Pitch when the number of teeth and the pitch diameter are given, divide the number of teeth by the pitch diameter. EXAMPLE.-If a gear contains 50 teeth and has a pitch diameter of 10 inches, what is its diametral pitch ?. SOLUTION.-Applying the rule, 
50 = 5 diametral pitch. Ans. 

17 GEAR CALCULATIONS. 
page 11 
32. Finding the Outside Diameter of a Gear-Blank.—The diameter to which the blank for a spur gear should be turned is equal to the outside diameter of the ' gear. By reference to Fig. 4 it is seen that the outside diameter, and, hence, the diameter of the blank, is equal to • the pitch diameter plus twice the addendum. With the diametral-pitch system, in which the addendum is equal to 1 divided by the pitch, the outside diameter may be calculated from the pitch diameter and the pitch by an application of the following rule : 
Rule.— To find the outside diameter, or the diameter of the blank, when the pitch diameter and the itch, multiply the quotient by 2, and add the product to the pitch diameter. 
EXAMPLE.-What should be the diameter of a gear-blank for a 
the following rule : 
Rule.— To find the outside diameter, or the diameter of the blank, when the diametral pitch and the number of teeth are known, add 2 to the number of teeth and divide the sum by the pitch. 
EXAMPLE.-A wheel is to have 48 teeth, 6 pitch ; to what diameter ust the blank be turned ? SOLUTION.—By the rule just given, the outside diameter is 48 + 2 = 8.333 in. Ans. 6 
34. Calculations Based on the Outside Diameter. -- The diameter of the blank and the pitch being 

Gear-Calculations-Mach-1908-ICS.html

Spur 

page 12 -13  GEAR CALCULATIONS. 
§ 17 
given, the number of teeth may be calculated from the fol-lowing rule :    Rule.—To find the number of teeth when the outside diameter of the blank and the diametral pitch are known, multiply the outside diameter by the pitch and subtract 2 from the product. EXAMPLE. —A gear-blank measures 10i inches in diameter and is to be cut 4 pitch. How many teeth should the gear-cutter be set to space ? SOLUTION.-By applying the rule just given, we find the number of teeth to be 
10i x 4 — 2 = 42 — 2 = 40 teeth. Ans. 

35. Rule.— To find the diametral pitch when the outside diameter and the number of teeth are known, add 2 to the number of teeth and divide the sum by the outside diameter. EXAMPLE.-It is required to select a cutter for a. gear having 54 teeth that is to mesh with the change gears of a lathe. One of the change gears, which has 64 teeth, measures 6.6 inches, outside diam-eter ; for what pitch should the cutter be selected ? SOLUTION.-Applying the rule given, the pitch of the change gear is found to be 64 ± 2 = 10 ; hence, this is the pitch of the cutter 6.6 
required. Ans. 
36. In applying the rule given in Art. 35, it will some-times be found that the result obtained does not correspond with any standard pitch number ; for example, a gear with 68 teeth measures 15196 inches in outside diameter. Apply-ing the rule to these values, the pitch would be 68+2  15.5625 = 4.4979+ ; this number is so near to 41, which is a stand-ard pitch, that it is evident that 44 is the pitch of the gear and that either the blank was not turned to the exact diameter called for by the pitch and number of teeth or that the exact diameter was not determined by the measurement. When a set of standard gear-cutters is available, the pitch of the gear can also be determined by trying different cutters until one is found that fits. 

§ 17 
GEAR CALCULATIONS. 
page 13 
A considerable difference between the value obtained by applying the rule and the nearest standard pitch will indi-cate that an uncommon pitch has been used. In general, however, it may be assumed that the pitch is the standard whose number agrees most nearly with the value obtained from an application of the rule. As far as practicable, the pitch and diameter should be so chosen that the number of teeth will correspond to a num-ber of divisions that can be readily obtained with the aid of the indexing mechanism of the machine in which the gear is to be cut. 

37. Relation Between Pitch Diameter, Number of Teeth, and Circular Pitch.—If any two of the fac-tors named are given, the other can be found by applying one of the following rules: Rule.—To find the diameter of the pitch circle when the number of teeth and the circular pitch are known, take the continued product of the number of teeth, the circular pitch, and .3183. 
EXAMPLE.-What is the pitch diameter of a gear-wheel that has '75 teeth and whose circular pitch is 1.625 inches ? SOLUTION.-Applying the rule, the diameter is found to be 
1.625 x 75 x .3183 = 38.79 in. Ans. 
38. Rule.—To find the circular pitch when the pitch diameter and the number of teeth are known, multiply the pitch diameter by 3.1416 and divide the product by the num-ber of teeth. EXAMPLE. —What is the circular pitch of a gear 32 inches in diam-eter and having 84 teeth ? SOLUTION.-Applying the rule just given, the circular pitch is found to be 
32 x 3.1416 = 1.1968 in. Ans. 84 
39. Rule.— To find the number of teeth when the pitch diameter and the circular pitch are known, multiply the pitchpage 12 BEVEL  GEAR CALCULATIONS. 

 12 GEAR CALCULATIONS.  page 14 GEAR–CUTTING. 
§ 18 
slide d that is moved parallel to the axis of the spindle a, and is fed automatically to the work and returned. The cutter is adjusted for depth by lowering the gear blank. A limited side adjustment is usually provided for the cutter to allow cutters of different thicknesses to be set central. 30. Automatic gear-cutting engines are often arranged so that they can be used for cutting approximately correct bevel gears. The slide that carries the cutter is then ar-ranged in such a manner that it can be set at the required angle to the axis of the spindle. 31. Change Geafing.—The gearing that revolves the shaft carrying the worm is, as a general rule, actuated by a so-called stop-shaft, which is provided with a suitable clutching mechanism operated by the cutter slide. This clutching mechanism is so arranged that it allows the stop-shaft to make exactly one revolution whenever the return-ing cutter slide unlocks it. The change gears that will produce a certain number of divisions are selected in ac-teeth in worm-wheel cordance with the ratio   In case of teeth to be cut simple gearing, this is the simple ratio that gears are to be selected for ; in case of compound gearing, it is the com-pound ratio, which is resolved into factors. The gears are selected in the same manner as is done in gearing a lathe for thread cutting or a milling machine for the cutting of helixes. In adjusting the gear-cutting engine, the tripping ar-rangement for the stop-shaft clutching mechanism must be set so that it will act only after the cutter on its return stroke is entirely clear of the gear. 
CUTTING BEVEL GEARS WITH FORMED CUTTERS. 32. Selecting the Cutter.-While bevel gears cut with a cutter of fixed curve can be only approximately cor-. rect, the comparative cheapness of this method ha`s led to its being largely used. The ordinary cutters made for spur 

§ 18 GEAR–CUTTING. 15 
wheels should never be used -for this purpose, as they will cut the teeth of the bevel gear entirely too thin at the small end. Special miter-gear and bevel-gear cutters are made for this purpose ; these cutters are of the involute form, but thinner than the standard cutters. They are numbered from 1 to 8, and cover the same range as the standard involute cutters. A bevel-gear cutter cannot be selected in the same manner as the ordinary spur-gear cut-ter, that is, directly in accordance with the number of teeth ,of the bevel gear. It is to be selected, instead, for a number of teeth that is calculated by one of the rules given below, the first of which is as follows: Rule. To find the number of teeth a bevel-gear cutter is to be selected for, divide the number of teeth of the bevel gear by the natural cosine of the center angle a d e, Fig. 6. EXAMPLE. — The center 4. angle of a bevel gear having 24 teeth is 53° 15'. What number of teeth should the cutter be selected for ? SOLUTION.—The cosine of 53° 15' is .59832. Applying the rule, we get  24 = 40 teeth. Referring to the Table of Standard Cutters, .59832 we find that for gears having between 35 and 54 teeth, a No. 3 cutter is to be used. Hence, use a No. 3 bevel-gear cutter. Ans. 33. When a drawing of the bevel gear is available, use the following rule: Rule. Measure the slant height of the back cone, as a b in Fig. 6; double: it and multiply by the diametral pitch. The product will be the number of teeth the cutter is to be selected for. EXAMPLE.—The slant height of the back cone being 5 inches, and the diametral pitch being 4, what number of bevel-gear cutter is to be used ? SOLUTION.—Applying the rule just given, we get 5 X 2 X 4= 40 teeth. Referring to the Table of Standard Cutters, it is seen that a No. 3 bevel-gear cutter is to be used. Ans. 
34. Setting the Machine.-The cutter having been selected, place it on its arbor; put the gear blank into the 50 —36 

 

ppage 16 GEAR CALCULATIONS, section § 17

their respective velocity ratios are fixed, multiply twice the distance between centers by the velocity ratio of the smaller gear and divide the product by the sum of the velocity ratios of the two gears. EXAMPLE.-The smaller of two gears is to run five times as fast as the larger. The distance between centers being 12 inches, what must be the pitch diameter of the larger gear ? SOLUTION.-By an application of the rule just given, the pitch diam-eter of the larger gear is found to be 2 X 12 X 5 = 20 in. Ans. 5 ± 1 
43. Rule.— To find the diameter of the smaller gear in a pair when the distance between centers of the two gears and their respective velocities are fixed, multiply twice the distance between centers by the velocity of the larger gear and divide the product by the sum of the velocities of the two gears. EXAMPLE.-Taking the last example again, what should be the diameter of the smaller gear ? SOLUTION.-Applying the rule just given, we have, as the diameter of the smaller gear, 
2 x 12 x 1  5 ± 1 = 4 in. Ans. 
44. Since the distance between centers is equal to the sum of the radii of the two gears, it follows that when the diameter of either gear has been calculated, the diameter of the other may be found as follows : Rule.—To find the diameter of one gear of a pair of gears, when the distance between centers is fixed and the diameter of the other gear is known, subtract the known diameter from twice the distance between centers. EXAMPLE.-The distance between centers being 8 inches, and the diameter of one gear being 4 inches, what is the diameter of the other gear ? SOLUTION.-Applying the rule just given, we obtain 2 8 — 4 = 12 in. Ans. 

• 
§ 17 
GEAR CALCULATIONS. 17 
LAYING OUT GEAR-TEETH. 
FORMS OF GEAR-TOOTH OUTLINES. 45. Constant Velocity Ratio.—In order that a tooth of the driving gear shall press on a tooth of the driven gear in such a manner as to produce a constant speed of turning while they are in contact, it is necessary that the curved outline of the teeth shall be constructed according to a cer-tain law. The principle involved is that the line ik, Fig. 6, 
N N 

FIG. 6. 

called the normal to the tooth curves at their point of tan-gency, that is, the point where they touch, must pass through the point of tangency of the pitch circles for every position in which the two teeth are in contact. When this condition is satisfied there will be a constant velocity ratio for the gears. In Fig. 6, c and d are two pitch circles with their respect-ive centers at a and b. The curve outlines e and f represent the teeth of two gears with these pitch circles. These curves are tangent to each other at g, and hj is a straight line tan-gent to both curves at the point g. A line perpendicular to this tangent line at g is the normal to both curves. Such a normal k i passes through the point of tangency g of the tooth curves and the point of tangency 1 of the pitch circles. The curves e and f are so designed that for every position in which they can be in contact their common normal passes 50-33

12

page 18 GEAR CALCULATIONS. 
§ 17 
through the point of tangency of the pitch circles. These curves therefore satisfy the condition for constant velocity ratio. 
46. Devices for Drawing Gear-Teeth.—An odon-tograph is an arrangement to facilitate the laying out of the curved outlines of gear-teeth. The term has been applied in a number of different forms, but its application to the templets for laying out the tooth curves seems to be ' the best, hence many persons define an odontograph as a templet for laying out gear-teeth. The term, however, has also been applied to tables that give the radii for the tooth-curve outlines of gears, though it would seem better prac-tice 'to call these odontograph tables. In some cases the templet has to be used in connection with a table, the tem-plet being simply a means for finding the centers from which to draw the tooth curves. 47. Tooth Curves in General Use.--As stated in Art. 45, the motion transmitted by one gear to another will be smooth and uniform only when the teeth of the gears are given definite forms. Theoretically, the number of forms that meet these conditions is large practically, how-ever, owing to the necessity of simplicity and ease of con-struction, this number is restricted to a few simple types, while the importance of uniformity has still further restricted the types in common use to two general systems, known as the involute, or single-curve, system and the cycloidal, or double-curve, system. Of these two systems, the involute has a number of important advantages, especially when used for cut gears, that are constantly bringing it into more extensive use,. and many of the best authorities on gearing urge its universal adoption to the exclusion of the cycloidal system. 
INVOLUTE SYSTEM. 
48.. Definition of an Involute.—Mathematically, an involute is the curve that would be drawn by a pencil point at the end of a thine band, that will not stretch, and 

§ 17 GEAR CALCULATIONS. 
page 19 
that is drawn tight while being unwound from a cylinder. For example, suppose such a band to be unwound from the cylinder in Fig. 7, beginning with the pencil point at a on the circumference. As the band is unwound, the pencil point traces the curved line a-1-2-3-4, etc., which line is a part of the involute of the circle that represents the circumference of the cylinder. With true in-volute teeth, the path followed by the point of cont act of the tooth curves of the teeth on the two gears is a straight line, called the line of action, that is tangent to the base circles of both gears. The faces of involute rack teeth are straight lines perpendicular to this line of action. In practice, the base circles are usually so chosen that the line of action makes an angle of 15° with a line that is tangent to both pitch circles at their point of tangency. As the faces of the rack teeth are perpendicular to the line of action, they make an angle of 75° with their pitch plane. In the approximate tooth curves frequently used in practice, the faces of the rack teeth are composed wholly or partially of curves. 

49. Base Circle for Involute Teeth.—The base circle in the involute system of gearing is the circle to which the involute that forms the outline of the tooth is drawn. The radius of the base circle is smaller than that of the pitch circle, the difference between the two being gen-erally found by multiplying the pitch diameter by a number that is constant in any given system, but varies somewhat with different systems. For most purposes, a difference 

 

Page  20 

GEAR-CUTTING. . § 18 
TEMPLET-PLANING PROCESS. 42. The Machine.—The principle of operation of a templet-planing machine intended for planing the teeth of bevel gears is shown in diagrammatic form in Fig. 10. The gear blank a is attached to the index spindle b, which carries an indexing wheel c at its other end. An arm d supports a longitudinally movable slide e which carries the pointed cutting tool f. The arm d is mounted on a univer-sal joint in such a manner that its center of rotation g coin-cides with the axis of rotation of the gear blank. The 

FIG. 10. 
cutting tool is adjusted in the slide e in such a manner that the line of motion of its cutting point passes exactly through the center of rotation of the arm. A pin h is fastened to the free end of the arm, and is in contact with a templet 1, which is shaped to conform to a correct tooth curve for the number of teeth contained in the bevel gear. 

I 
§ 18 GEAR-CUTTING. page 21 
The arm d remains stationary while the cutting tool is traversed through the gear blank. One side of a tooth is finished at a time by successive cuts converging toward the apex of the pitch cone, which point, by reason of the con-struction of the machine, is also the center of rotation g of the arm. After the tool f has cleared the blank on its re-turn stroke, the arm is moved slightly along the templet i, keeping the pin /bin contact with the templet ; the position of the arm and, hence, the formation of the tooth curves, is thus determined by the templet. The form of the templet is reproduced on a smaller scale by the planing tool on the tooth operated on, and any errors existing in the templet are reduced. 
43. It is obvious that a different templet will be re-quired for each number of teeth, at least theoretically. Owing to the small divergence in the shape of the tooth curves, one templet can be made to serve for several gears, however, just as is done with formed gear-cutters. One templet will answer for all pitches within the range of the machine ; different sizes of bevel gears are cut by varying the distance from the gear to the center of rotation of the arm d. In an actual machine, the templet is movably mounted on a quadrant having its center of curvature at g; this adapts the machine for different gears, since it allows the angle between the axis of the spindle and the line of motion of the tool to be changed to suit the number of teeth of the gear. 
44. Templet-Grinding Process. — A modification of the templet-planing process has recently been perfected by the Leland & Faulconer Company, Detroit, Michigan, who have substituted a corundum wheel for the planing tool and thus are enabled to finish the teeth of hardened-steel bevel gears to a correct shape. The fundamental principle under-lying this templet-grinding process does not differ in any essential particular from that explained in connection with Fig. 10. 

Page 22  
22 'GEAR CALCULATIONS. * § TABLE H. 
GRANT'S INVOLUTE ODONTO1RAPII TABLE. 
No. of Teeth. Divide by the Diametral Pitch. Multiply by the Circular Pitch. Face Radius. Flank Radius. Face Radius. Flank Radius. I0 2.28 .69 .73 .22 II 2.40 .83 .76 .27 12 2.51 .96 .8o 13 2.62 1.09 .83 .34 14 2.72 1.22 .87 .39 15 2.82 1.34 .90 .43 16 2.92 1.46 .93 .47 17 3.02 1.58 .96 .50 18 3.12 1.69 -99 .54 19 3.22 1.79 1.03 .57. 20 3.32 1.89 1.06 .6o 21 3.41 1.98 1.09 .63 22 3.49 2.06 1.11 .66 23 3.57 2.15 1.13 .69 -24 3.64 2.24 1.16 .71 25 3.71 2.33 1.18 .74 26 3.78 2.42 1.20 .77 27 3.85 2.50 1.23 .So 28 3.92 2.59 1.25 .82 29 3.99 2.67 1.27 .85 30 4.06 2.76 1.29 .88 31 4.13 2.85 1.31 .91 32 4.2o 2.93 1.34 -93 33 4.27 3.01 1.36 .96 34 4.33 3.09 1.38 .99 35 4.39 3.16 1.39 I.0I 36 4.45 3.23 1.41 1.03 37-40 4.20 1.34 41-45 4.63 1.48 46-51 5.06 1.61 52-60 5.74 1.83 61-70 6.52 2.07 71-90 7.72 2.46 91-120 9.78 3.11 121-180 13.38 4.26 181-360 21.62 6.88 
§ 17 
GEAR CALCULATIONS. page 23 
however, and especially with wheels having a small number of teeth, the curve so obtained differs considerably from the correct curve and, in these cases, more satisfactory results are .obtained by the method explained in the following articles. 
52. Grant's Involute Odontograph Table.- By this method, for all gears having fewer than 37 teeth, the curve is approximated by two circular arcs-one extending from the pitch circle to the addendum circle and the other from the pitch circle to the base circle-having different radii, the center of the arc for each being on the base circle. The lengths of the radii with which the two arcs are drawn are obtained by the following method : In Table II, which is taken from Grant's " Treatise on Gear-Wheels," are two sets of numbers, a part of each set being in two columns. The first set has the general heading Divide by the Diame-tral Pitch and the two columns in this set have the respect-ive headings Face Radius. and Flank Radius. This set is to be used with the diametral-pitch system. To find the radius F, Fig. 8, for the face of a tooth for a gear having less than 37 teeth, divide the number in the column headed Face Radius, opposite the number that corresponds with the number of teeth in the gear, by the diametral pitch. To find the radius G of curved part of the flank, divide the corresponding number in the column headed Flank Radius by the diametral pitch. The second set of two columns of numbers is headed Multiply by the Circular Pitch and is to be used with the circular-pitch system. It is used in the same manner as the first set, except that the numbers taken from the table are to be multiplied by the circular pitch. Applying this method to the diametral-pitch gear of Art. 50, in which the number of teeth is 24 and the pitch.3, we proceed as follows : To find the radius of the face, we look in the first column for the number 24 and in the same horizontal line in the column headed Face Radius, we find the number 3.64, which, divided by the pitch, gives us . 

Page 24 GEAR CALCULATIONS. 
3.64 3 = 1.21 inches as the radius F of the face. In the t same horizontal line and in the column headed Flank Radius, we find the number 2.24; this number divided by 3 gives us 2.24 3 = .75 inch, nearly, as the radius G of the flank. 53. Odontograph Table for Gears Having More Than 36 Teeth.—An inspection of Table II shows that for tears having more than 36 teeth there is but one column of figures under each of the respective headings of d4metral pitch and circular pitch. The reason is that the whole curve is •drawn with a single radius, whose length is deter-mined by the general method already explain-ed. It is con-
stant for all gears the numbers of whose teeth are included in the several pairs of numbers given in the column headed, No. of Teeth ; for instance, the length of the radius for all numbers of teeth from 37 to 40 is determined by the use of the numbers in the horizontal line in which these numbers 
occur. EXAMPLE.—What is the length of the radius for the curves of the teeth of a gear having 64 teeth, 1.473 circular pitch ? vor -ft SOLUTION.—Since the number of teeth lies between the num-bers 61-70 in the first column of the table, and the pitch is in the circu-lar-pitch system, we multiply the pitch by the number 2.07, which is found in the second set of figures at the right of the numbers 61-70 and in the same horizontal row. Performing the multiplication, we get 1.473 x 2.07 = 3.05, say 3 in. , as the length of the radius. Ans. • 54. Completing the Tooth Outline. With any of the foregoing methods of constructing the tooth outline, the flanks of the teeth are radial between the base circle and the working-depth circle ; this part of the outline is therefore made to coincide with the straight line from the center 0 of the pitch circle to the point where the curved p-ortion of the outline intersects the base circle, as is shown by the line 0 I in Fig. 8. A fillet from the working-depth circle connects the radial portion of the outline with the root circle and completes the outline of the tooth. Brown & Sharpe make the radius of this fillet equal to one-seventh the width of a space at the addendum circle. 

 - GEAR CALCULATIONS.   Page 25 , 
55. Minimum Number of. Teeth.— The smallest number of teeth that should be used in a cut gear whose v - teeth are laid out by the method of single-arc approxima-tion is 30 ; with a smaller number, the difference between the curve obtained by this method and the correct curve is so great as to cause the teeth to work unsatisfactorily. By using

Grant's odontograph table in the manner explained, it is possible to make satisfactory gears that have as few as 10 teeth. 
56. Grant's Rule for Rack Teeth.—The teeth of a rack that is to mesh with an involute gear of a given pitch may be laid out by the following method, which is known as " Grant's rule for rack teeth." First draw the addendum, pitch, and root lines, Fig. 9, making the dis-tances A and B each equal to 1 divided by the diametral 
14 -4 T8 — , t - - 8 
Addendum Line.  -a Pitch Line  O FIG. 9. 

WocfrinsPfpL/7 Line. Rood` 
pitch, and the distance C equal to one-eighth of A. On the pitch line lay off the pitch distances D, D, and divide them into the two parts t and s, corresponding, respectively, to the thickness of the teeth and the width of the spaces on the pitch line. Draw the sides of the teeth from the working-depth line to the line a a, which is drawn half-way between the pitch line and the addendum line, as straight lines making angles of 15° with lines that pass through the pitch points perpendicular to the pitch line. Draw the miter half of the addendum as a circular arc having a 

26 GEAR CALCULATIONS. 
§ 17. 
radius .R whose length is 2.1 divided by the diametral pitch, or .67 multiplied by the circular pitch. A fillet from the working-depth_ line to the root line completes the outline of each side of the tooth. 
CYCLOIDAL SYSTEM. 
57. Definition of a Cycloid. - In mathematics, a cyclohPis a path described by a point on the circumference of a circle as the circle rolls upon a straight line thus, the curve a b c, Fig. 10 (a), described by the point b as the cixcle j rolls along the line a c is called a cycloid-. The circle j is called a describing, or rolling, circle. When the describing circle, as h, Fig. 10 (b), rolls upon the out-side of another circle, as g, the curve e d described by any point 

(a) FIG. 10. 
on the describing circlerg§ e, is called an epicycloid. If the describing circle, as i, rolls on the inside of another circle, as g, the curve e f generated by any point of the describing circle, as e, forms what is called a hypo,.--ycloid.. If the circle i has a diameter just one-half the diameter of the circle g, the hypocycloid will be a straight line; or a diam-eter of the circle g. If the diameter of the circle i is less than half that of the circle g, the hypocycloid will have a curve as shown at e f, while if the diameter of i is more than half the diameter of g, the curve will extend to the left from 
the point e instead of to the right of e. 
- 111 58. Laying Out Cycloidal Teeth. -- The pitch, addendum, working depth, and root circles are -drawn, and the pitch, thickness of teeth, and width of spaces on the pitch circle are laid off as described for the involute system. 

§ 17 GEAR CALCULATIONS. ft, After this the outlines of the teeth may be drawn as theoretical curves, but the more common method is to draw the curves for the faces and flanks as circular arcs that agree very closely with the theoretical curves. 
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59. Grant's Cycloidal Odontograph Table.—One of the most accurate practical methods of laying out the approximate curves of cycloidal teeth by means of circular arcs has been devised by Mr. George B. Grant. The lengths of the radii of the arcs and the location of their centers are determined by the pitch and number of teeth of the , gear, in conjunction with a table of factors that apply to gears of all sizes froth a 10-tooth pinion to a rack. Any two gears with teeth of the. same pitch and length laid out by this method will work satisfactorily with each other. The base of the odontograph table is a describing circle whose diameter is equal to the radius of the 12-tooth pinion; a gear laid out by its use will therefore work satisfactorily with any gear having the same pitch and general tooth dimensions, and with theoretical cycloidal curves con-structed with a describing circle whose diameter is equal to the radius of the 12-tooth pinion. 
60. Use of Grant's Cycloidal Odontograph Table. The first step in the use of the odontograph table is the location of the circles on which lie the centers of the arcs, 
FIG. 11. 
Fig. 11. These circles are concentric with the pitch circle, and their distances from it are determined in the following manner : In Table III, which is taken from Grant's " Treatise 

 

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