be taken at 206 = 3% feet per second, .910. 1.78 and H.P. = Vpf = .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, 1+0.65. 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 +0.657 × .910Vpf = .695 .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 = 2000pf, 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 = Sopf, or only 40% of that given by the English rule. F. A. Halsey (Clark's Pocket Book) gives a table calculated from the formula H.P. pfdx rpm. ÷ 850. Jones & Laughlins give H.P. = pfd X rpm. ÷ 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 = KW, 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 262pf, 200pf, and 139pf, respectively. W = Unwin also gives the following formula, based on the assumption that the pressure is distributed along the edge of the tooth: p = K1 VW, 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 f = 2p and the given values of K1 this reduces to W 200pf and W= respectively. = 139pf, Box, in his Treatise on Mill Gearing, gives H.P. = 12p2f Van in which n 9 1000 = number of revolutions per minute. This formula differs from the more modern formulæ in making the H.P. vary as p2f, instead of as pƒ, and in this respect it is no doubt incorrect. Making the H.P. vary as Vdn or as V, instead of directly as v, makes the velocity a factor of the working strength as in the Harkness and Lewis No v 1 which for different 300 600 900 1200 1800 2400 formulæ, the relative strength varying as or as velocities is as follows: Speed of teeth in ft. per min., v = 100 200 Relative strength = 1 .707 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. Cipfd × 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 C varies 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 × rpm. For v 100 ft. per min. C 7.4p; 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 LA FOR STRENGTH OF GEAR-TEETH. Safe working pressure per inch pitch and per inch of face, or value of c in formula W = cpf: v = 100 ft. v = 600 ft. per min. Lewis: Weak form of tooth, radial flank, 12 teeth... c 416 Harkness: Average tooth. Box: Tooth of 1 inch pitch.. 3 inches pitch.. 66 66 66 208 400 c = 1200 600 c = 347 184 Varions, 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 = 165 H.P.0015873pfd × 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. Thẹ 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; piten aiameter, 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 metal on metal may be taken at .15 to 20; for wood on metal, .25 to 30; and for wood on compressed paper, .20. The tangential driving force 7 may be as high as 80 lbs. per inch width of face of the driving surface, but this is accompanied by great pressure and friction on the journal-bearings. In frictional grooved gearing circumferential wedge-shaped grooves are cut in the faces of two wheels in contact. If P = the force pressing the wheels together, and N the normal pressure on all the grooves, P = N (sin af cos a), in which 2a = the inclination of the sides of the grooves, and the maximum tangential available force T = ƒN. The inclination of the sides of the grooves to a plane at right angles to the axis is usually 30°. Frictional Grooved Gearing.—A set of friction-gears for transmitting 150 H.P. is on a steam-dredge described in Proc. Inst. M. E., July, 1883. Two grooved pinions of 54 in. diam., with 9 grooves of 134 in. pitch and angle of 40° cut on their face, are geared into two wheels of 1271% in diam. similarly grooved. The wheels can be thrown in and out of gear by levers operating eccentric bushes on the large wheel-shaft. The circumferential speed of the wheels is about 500 ft. per min. Allowing for engine-friction, if half the power is transmitted through each set of gears the tangential force at the rims is about 3960 lbs., requiring, if the angle is 40° and the coefficient of friction 0.18, a pressure of 7524 lbs. between the wheels and pinion to prevent slipping. The wear of the wheels proving excessive, the gears were replaced by spurgear wheels and brake-wheels with steel brake-bands, which arrangement has proven more durable than the grooved wheels. Mr. Daniel Adamson states that if the frictional wheels had been run at a higher speed the result would have been better, and says they should run at least 30 ft. per second. HOISTING AND CONVEYING. Approximate Weight and Strength of Cordage. (Boston and Lockport Block Co.)-See also pages 339 to 345. Regular Mortise-blocks Single and Double, or Two Double Ironstrapped Blocks, will hoist about Working Strength of Blocks. (B. & L. Block Co.) Wide Mortise and Extra Heavy about Where a double and triple block are used together, a certain extra proportioned amount of weight can be safely hoisted, as larger hooks are used. Comparative Efficiency in Chain-blocks both in (Tests by Prof. R. H. Thurston, Hoisting, March, 1892.) No. 1 was Weston's triplex block; No. 3, Weston's differential; No 4, Weston's imported. The others were from different makers, whose names are not given All the blocks were of one-ton capacity. Proportions of Hooks.-The following formulæ are given by Henry R. Towne, in his Treatise on Cranes, as a result of an extensive experimental and mathematical investi gation. They apply to hooks of capaci- G = .75D. FIG. 164. H = 1.08A L .50A J = 1.20A K = 1.13A N= .85B-.16 The dimensions A are necessarily based upon the ordinary merchant sizes of round iron. The sizes which it has been found best to select are the following: |