Prof. Harkness, as a result of his investigation, found that all the formulæ on the subject might be expressed in one of three forms, viz.: in which Cis a coefficient, V velocity of pitch-line in feet per second, p = pitch in inches, and f = face of tooth in inches. From an examination of precedents he proposed the following formula for cast-iron wheels: He found that the teeth of chronometer and watch movements were subject to stresses four times as great as those which any engineer would dare to use in like proportion upon cast-iron wheels of large size. It appears that all of the earlier rules for the strength of teeth neglected the consideration of the variations in their form; the breaking strength, as said by Mr. Cooper, being based upon the thickness of the teeth at the pitchline or circle, as if the thickness at the root of the tooth were the same in all cases as it is at the pitch-line. Wilfred Lewis (Proc. Eng'rs Club, Phila., Jan. 1893; Am. Mach., June 22, 1893) seems to have been the first to use the form of the tooth in the con struction of a working formula and table. He assumes that in well-constructed machinery the load can be more properly taken as well distributed across the tooth than as concentrated in one corner, but that it cannot be safely taken as concentrated at a maximum distance from the root less than the extreme end of the tooth. He assumes that the whole load is taken upon one tooth, and considers the tooth as a beam loaded at one end, and from a series of drawings of teeth of the involute, cycloidal, and radial flank systems, determines the point of weakest cross-section of each, and the ratio of the thickness at that section to the pitch. He thereby obtains the general formula, W = spfy; in which W is the load transmitted by the teeth, in pounds; s is the safe working stress of the material, taken at 8000 lbs. for cast iron, when the working speed is 100 ft. or less per minute; p = pitch; f = face, in inches; y = a factor depending on the form of the tooth, whose value for different cases is given in the following table: SAFE WORKING STRESS, S, FOR DIFFERENT SPEEDS. The values of s in the above table are given by Mr. Lewis tentatively, in the absence of sufficient data upon which to base more definite values, but they have been found to give satisfactory results in practice. Mr. Lewis gives the following example to illustrate the use of the tables: Let it be required to find the working strength of a 12-toothed pinion of 1inch pitch, 21⁄2-inch face, driving a wheel of 60 teeth at 100 feet or less per -R minute, and let the teeth be of the 20-degree involute form. In the formula W spfy we have for a cast-iron pinion s = 8000, pf 2.5, and y =.078; and multiplying these values together, we have W 1560 pounds. For the wheel we have y = .134 and W 2680 pounds. = The cast-iron pinion is, therefore, the measure of strength; but if a steel pinion be substituted we have 820,000 and W 8900 pounds, in which combination the wheel is the weaker, and it' therefore becomes the measure of strength. = For bevel-wheels Mr. Lewis gives the following, referring to Fig. 168: D large diameter of bevel; d= small diameter of bevel; p= pitch at large diameter; n = actual number of teeth; f face of bevel; N= formative number of teeth = n y secant a, or the number corresponding to radius R; y = factor depending upon shape of teeth and formative number N; W = working load on teeth. d FIG. 163. W = spfy D3 - d3 3D2(D-d); or, more simply, W= =spfy D' which gives almost identical results when d is not less than 2% D, as is the case in good practice. In Am. Mach., June 22, 1893, Mr. Lewis gives the following formulæ for the working strength of the three systems of gearing, which agree very closely with those obtained by use of the table: in which the factor within the parenthesis corresponds to y in the general formula. For the horse-power transmitted, Mr. Lewis's general formula = 33,000 H.P. v spfyv 33,000' W = spfy, , may take the form H.P. = in which v = velocity in feet per minute; or since v dr X rpm. + 12 = .2618d X rpm., in which d diameter in inches and rpm. = revolutions per minute, Wv spfyXdx rpm. H.P. = = 33,000 =.000007933dspfy × rpm. It must be borne in mind, however, that in the case of machines which consume power intermittently, such as punching and shearing machines. the gearing should be designed with reference to the maximum load W, which can be brought upon the teeth at any time, and not upon the average horse-power transmitted. Comparison of the Harkness and Lewis Formulas.— Take an average case in which the safe working strength of the material, s = 6000, v = 200 ft. per min., and y = .100, the value in Mr. Lewis's table for an involute tooth of 15° obliquity, or a cycloidal tooth, the number of teeth in the wheel being 27. 910 V 3.167= 1.78, be taken at 200 603% feet per second, √1 +0.65V = and H.P. = 1.78Vpf = .571pfV, or about 52% of the result given by Mr. Lewis's formula. This is probably as close an agreement as can be expected, since Prof. Harkness derived his formula from an investigation of ancient precedents and rule-of-thumb practice, largely with common cast gears, while Mr. Lewis's formula was derived from considerations of modern practice with machine-moulded and cut gears. Mr. Lewis takes into consideration the reduction in working strength of a tooth due to increase in velocity by the figures in his table of the values of the safe working stress s for different speeds. Prof. Harkness gives expression to the same reduction by means of the denominator of his formula, 10.65V. The decrease in strength as computed by this formula is somewhat less than that given in Mr. Lewis's table, and as the figures given in the table are not based on accurate data, a mean between the values given by the formula and the table is probably as near to the true value as may be obtained from our present knowledge. The following table gives the values for different speeds according to Mr. Lewis's table and Prof. Harkness's formula, taking for a basis a working stress s, for cast-iron 8000, and for steel 20,000 lbs. at speeds of 100 ft. per minute and less: Comparing the two formulæ for the case of s = 8000, corresponding to a speed of 100 ft. per min., we have Harkness: H.P. 1+1+0.65V x .910Vpf = .695X.91 X 1%pf = 1.051pf In which y varies according to the shape and number of the teeth. For radial-flank gear with 12 teeth y=.052; 24.24pfy = 1.260pf; For 20° involute, 19 teeth, or 15° inv., 27 teeth y = .100; 24.24pfy =2.424pf; For 20° involute, 300 teeth y.150; 24.24pfy 3.636pf. Thus the weakest-shaped tooth, according to Mr. Lewis, will transmit 20 per cent more horse-power than is given by Prof. Harkness's formula, in which the shape of the tooth is not considered, and the average-shaped tooth, according to Mr. Lewis, will transmit more than double the horse power given by Prof. Harkness's formula. Comparison of Other Formulæ.-Mr. Cooper, in summing up his examination, selected an old English rule, which Mr. Lewis considers as a passably correct expression of good general averages, viz. X 2000pƒ, X= breaking load of tooth in pounds, p = pitch, f= face. If a factor of safety of 10 be taken, this would give for safe working load W = =200pf. George B. Grant, in his Teeth of Gears, page 33. takes the breaking load at 3500pf, and, with a factor of safety of 10, gives W = 350pf. Nystrom's Pocket-Book, 20th ed., 1891, says: "The strength and durability of cast-iron teeth require that they shall transmit a force of 80 lbs. per inch of pitch and per inch breadth of face." This is equivalent to W = =80pf, or only 40% of that given by the English rule. F. A. Halsey (Clark's Pocket Book) gives a table calculated from the formula pfd X rpm. H.P. Jones & Laughlins give H.P. pfd X rpm. 850. 550. These formulæ transformed give W = 128pf and W = = 218pf, respectively. Unwin, on the assumption that the load acts on the corners of the teeth, derives a formula p = KVW, in which K is a coefficient derived from existing wheels, its values being for slowly moving gearing not subject to much vibration or shock K.04; in ordinary mill-gearing, running at greater speed and subject to considerable vibration, K= .05; and in wheels subjected to excessive vibration and shock, and in mortise gearing, K = .06. Reduced to the form W Cpf, assuming that f = 2p, these values of K give W= 262pf, 200pf, and 139pf, respectively. Unwin also gives the following formula, based on the assumption that the pressure is distributed along the edge of the tooth: p = K1N where K1 = about .0707 for iron wheels and .0848 for mortise wheels when the breadth of face is not less than twice the pitch. For the case of ƒ = 2p and the given values of K1 this reduces to W: = 200pf and W = 139pf, respectively. 12p2f Van Box, in his Treatise on Mill Gearing, gives H.P. = = number of revolutions per minute. This formula differs from the more modern formulæ in making the H.P. vary as p2f, instead of as pf, and in this respect it is no doubt incorrect. Making the H.P. vary as Vdn or as Vo, instead of directly as v, makes the velocity a factor of the working strength as in the Harkness and Lewis No บ formulæ, the relative strength varying as or as 1 300 600 900 1200 1800 2400 .574 .408 .333 .289 .236 .204 Showing a somewhat more rapid reduction than is given by Mr. Lewis. For the purpose of comparing different formulæ they may in general be reduced to either of the following forms: H.P. = Cpfv, = H.P. C1pfd × rpm., W = cpf, in which p = pitch, f= face, d = diameter, all in inches; v = velocity in feet per minute, rpm. revolutions per minute, and C, C, and c coefficients. The formulæ for transformation are as follows: In the Lewis formula C varies with the form of the tooth and with the speed, and is equal to sy+33,000, in which y and s are the values taken from the table, and c = sy. In the Harkness formula Cvaries with the speed and is equal to .01517 910 √1+0.65 In the Box formula C varies with the pitch and also with the velocity, 12p Vd x rpm. and equals 1000v p = .02345 c = 33,000C = 774 For v 100 ft. per min. C77.4p; for v 600 ft. per minute c 31.6p. In the other formulæ considered C, C1, and c are constants. Reducing the several formulæ to the form W cpf, we have the following: COMPARISON OF DIFFERENT FORK LE FOR STRENGTH OF GEAR-TEETH. Safe working pressure per inch pitch and per inch of face, or value of c in formula W = cpf: Lewis: Weak form of tooth, radial flank, 12 teeth... Harkness: Average tooth.. Box: Tooth of 1 inch pitch.. 66 3 inches pitch. per min. v = 100 ft. v = 600 ft. per min. 208 400 600 ..... Various, in which c is independent of form and speed: Old English rule, c = 200; Grant, c = 350; Nystrom, c = 80; Halsey, c = 128; Jones & Laughlins, c = 218; Unwin, c = 262, 200, or 139, according to speed, shock, and vibration. The value given by Nystrom and those given by Box for teeth of small pitch are so much smaller than those given by the other authorities that they may be rejected as having an entirely unnecessary surplus of strength. The values given by Mr. Lewis seem to rest on the most logical basis, the form of the teeth as well as the velocity being considered; and since they are said to have proven satisfactory in an extended machine practice, they may be considered reliable for gears that are so well made that the pressure bears along the face of the teeth instead of upon the corners. For rough ordinary work the old English rule W = 200pf is probably as good as any, except that the figure 200 may be too high for weak forms of tooth and for high speeds. The formula W = 200pf is equivalent to H.P. = pfd X rpm. pfv 630 = 165 H.P..0015873pfd X rpm. = .006063pfv. Maximum Speed of Gearing.-A. Towler, Eng'g, April 19, 1889, p. 388, gives the maximum speeds at which it was possible under favorable conditions to run toothed gearing safely as follows: Prof. Coleman Sellers (Stevens Indicator, April, 1892) recommends that gearing be not run over 1200 ft. per minute, to avoid great noise. The Walker Company, Cleveland, O., say that 2200 ft. per min. for iron gears and 3000 ft. for wood and iron (mortise gears) are excessive, and should be avoided if possible. The Corliss engine at the Philadelphia Exhibition (1876) had a fly-wheel 30 ft. in diameter running 35 rpm. geared into a pinion 12 ft. diam. The speed of the pitch-line was 3300 ft. per min. A Heavy Machine-cut Spur-gear was made in 1891 by the Walker Company, Cleveland, O., for a diamond mine in South Africa, with dimensions as follows: Number of teeth, 192; pitch diameter, 30' 6.66"; face, 30"; pitch, 6"; bore, 27"; diameter of hub, 9' 2"; weight of hub, 15 tons; and total weight of gear, 6634 tons. The rim was made in 12 segments, the joints of the segments being fastened with two bolts each. The spokes were bolted to the middle of the segments and to the hub with four bolts in each end. Frictional Gearing.-In frictional gearing the wheels are toothless, and one wheel drives the other by means of the friction between the two surfaces which are pressed together. They may be used where the power to be transmitted is not very great; when the speed is so high that toothed wheels would be noisy; when the shafts require to be frequently put into and out of gear or to have their relative direction of motion reversed; or when it is desired to change the velocity-ratio while the machinery is in motion, as in the case of disk friction-wheels for changing the feed in inachine tools. Let P = the normal pressure in pounds at the line of contact by which two wheels are pressed together, T = tangential resistance of the driven wheel at the line of contact, f the coefficient of friction, V the velocity of the pitch-surface in feet per second, and H.P. = horse-power; then T may be equal to or less than ƒP; H.P, ≈ TV 550. The value of ƒ for |