The above formulæ are all based on the supposition that the arc of contact is 180° For other arcs, the transmitting power is approximately pro portional to the ratio of the degrees of arc to 180°. Some rules base the horse-power on the length of the arc of contact in Srw Sw πα and H P. = X x rpm. X 33000 33000 12 = a 180 we пла feet. Since L= 12 X 360 obtain by substitution H.P. = × LX rpm., and the five formulæ then take the following form for the several values of S: Su 16500 None of the handy formulæ take into consideration the centrifugal ten sion of belts at high velocities. When the velocity is over 3000 ft. per min ute the effect of this tension becomes appreciable, and it should be taken account of as in Mr. Nagle's formula, which is given below. Horse-power of a Leather Belt One Inch wide. (NAGLE.) Formula: H.P. = CVtw(S-.0122) + 550, 10 15 .51 .59 .63 .73 .84 1.05 1.18 15 1.69 1.94 .75 .88 1.00 1.16 1.321.66 1.77 2.242.57 20 1.00 1.17 1 32 1.54 1.752 192.31 2.79 3.19 25 1.23 1.43 1.61 1.88 2 16 2.69 2.86 30 1.47 1.72 1.93 2.25 2.58 3.223.44 35 1.69 1.97 2.22 2.59 2.96 3.70 3.94 40 1.90 2.222.49 2.90 3.32 4.15 4.44 45 2.09 2.45 2.75 3.21 3.67 4.58 4.89 50 2.27 2.65 2.98 3.48 3.98 4.97 5.30 55 2.44 2.84 3.19 3.72 4.26 5.32 5.69 60 2.58 3.01 3.38 3.95 4.51 5.61 6.02 65 2.71 3.16 3.55 4.14 4.745.92 6.32 70 2.81 3.27 3.68 4.29 4.91 6.14 6.54 75 2.89 3.37 3.79 4.425.05 6.31 6.73 80 2.94 3.43 3.86 4.50 5.15 6.44 6.86 85 2.97 3.47 3.90 4.55 5.20 6.50 6.93 90 2.97 3.473.90 4.55 5.20 6.50 6.93 100 The H.P. becomes a maximum at 87.41 ft. persec, = 5245 ft. p. min. 20 25 2.42 2.58 2.91 3.39 3.87 3.21 3.42 3.85 4.49 5.13 3.98 4.25 4.78 5.57 6.37 30 3.31 3.79 35 3.824 37 40 4.334.95 45 50 55 60 65 70 75 80 85 90 4.74 5.05 5.67 6.62 7.58 5.46 5.83 6.56 7.65 8.75 6.19 6.60 7.42 8.66 9 90 4.85 5 49 6.86 5.26 6 01 7.51 5.68 6.50 8.12 8.66 9.74 11.36 13 00 6.09 6.06 8.70 9.28 10.43 12 17 13.91 6.457.37 9.22 9.83 11.06 12 90 14.75 6.78 7.75 9 69 10.33 11 62 13.56 15.50 7.09 8.11 10.13 10.84 12.16 14.18 16.21 7.36 8.41 10 51 11.21 12.61 14.71 16.81 7.58 8.66 10.82 11.55 13.00 15.16 17.32 7.74 8.85 11.06 11.80 13.27 15.48 17.69 7.96 9.10 11.37 12.13 13 65 15.92 18.20 The H.P. becomes a maximum at 105.4 ft. per sec. = 6324 ft. per min. 7.32 8.43 9 70 10.98 8.02 9 02 10.52 12.03 In the above table the angle of subtension, a, is taken at 180°. Should it be.. 90° 100° 110° | 120° | 130° | 140° | 150° | 160° | 170° | 180° 200° Multiply above values by .65 .70.75.79.83.87.91.94.97 11.05 A. F. Nagle's Formula (Trans. A. S. M. E., vol. ii., 1881, p. 91. Tables published in 1882.) Taking S at 275 lbs. per sq. in. for laced belts and 400 lbs. per sq. in. for fapped and riveted belts, the formula becomes H.P. = CVtw(.50.00002182) for laced belts; H.P. CVtw(.727.00002182) for riveted belts. The following table gives a comparison of the formulæ already given for the case of a belt one inch wide, with arc of contact 180°. Horse-power of a Belt One Inch wide, Arc of Contact 180°. COMPARISON OF DIFFERENT FORMULA. Form. 5 Nagle's Form. Form. 1 Form. 2 Form. 3 Form. 4 dbl. belt 7/32" single belt H.P. H.P. = H.P. = H.P. =H.P. = wv พบ wv wv พบ Laced. Riveted 513 Width of Belt for a Given Horse-power.-The width of bell required for any given horse-power may be obtained by transposing the for mulæ for horse-power so as to give the value of w. Thus: Many authorities use formula (1) for double belts and formula (2) or (3) for single belts. 550 H.P. To obtain the width by Nagle's formula, w = or divide CVt(S.012V2)' the given horse-power by the figure in the table corresponding to the given thickness of belt and velocity in feet per second. The formula to be used in any particular case is largely a matter of judgment. A single belt proportioned according to formula (1), if tightly stretched, and if the surface is in good condition, will transmit the horse-power calculated by the formula, but one so proportioned is objectionable, first, because it requires so great an initial tension that it is apt to stretch, slip, and require frequent restretching and relacing; and second, because this tension will cause an undue pressure on the pulley-shaft, and therefore an undue loss of power by friction. To avoid these difficulties, formula (2), (3), or (4,) or Mr. Nagle's table, should be used; the latter especially in cases in which the velocity exceeds 4000 ft. per min. Taylor's Rules for Belting.-F. W. Taylor (Trans. A. S. M. E., xv. 204) describes a nine years' experiment on belting in a machine-shop, giving results of tests of 42 belts running night and day. Some of these belts were run on cone pulleys and others on shifting, or fast-and-loose, pulleys. The average net working load on the shifting belts was only 4/10 of that of the cone belts. The shifting belts varied in dimensions from 39 ft. 7 in. long, 3.5 in. wide, .25 in. thick, to 51 ft. 5 in. long, 6.5 in. wide, .37 in. thick. The cone belts varied in dimensions from 24 ft. 7 in. long, 2 in. wide, .25 in. thick, to 31 ft. 10 in. long, 4 in. wide, .37 in. thick. Belt-clamps were used having spring-balances between the two pairs of clamps, so that the exact tension to which the belt was subjected was accurately weighed when the belt was first put on, and each time it was tightened. The tension under which each belt was spliced was carefully figured so as to place it under an initial strain-while the belt was at rest immediately after tightening-of 71 lbs. per inch of width of double belts. This is equivalent, in the case of Raw-hide belts, Oak tanned and fulled belts, to 192 lbs. per sq. in. section; 66 66 66 to 253" 66 66 66 66 66 66 66 66 From the nine years' experiment Mr. Taylor draws a number of conclu. sions, some of which are given in an abridged form below. In using belting so as to obtain the greatest economy and the most satis. factory results, the following rules should be observed: A double belt, having an arc of contact of Or: The number of lineal feet of double- Or: A double belt 6 in. wide, running 4000 to 5000 ft. per min., will transmit.. The terms "initial tension," "effective pull," etc., are thus explained by Mr. Taylor: When pulleys upon which belts are tightened are at rest, both strands of the belt (the upper and lower) are under the same stress per in. of width. By "tension," initial tension," or "tension while at rest,' we mean the stress per in. of width, or sq, in. of section, to which one of the strands of the belt is tightened, when at rest. After the belts are in motion and transmitting power, the stress on the slack side, or strand, of the belt becomes less, while that on the tight side-or the side which does the pulling-becomes greater than when the belt was at rest. By the term "total load" we mean the total stress per in. of width, or sq. in. of section, on the tight side of belt while in motion. The difference between the stress on the tight side of the belt and its slack side, while in motion, represents the effective force or pull which is transmitted from one pulley to another. By the terms "working load," "net working load," or " effective pull," we mean the difference in the tension of the tight and slack sides of the belt per in. of width, or sq. in. section, while in motion, or the net effective force that is transmitted from one pulley to another per in. of width or sq. in. of section. The discovery of Messrs. Lewis and Bancroft (Trans. A. S. M. E., vii. 749) that the "sum of the tension on both sides of the belt does not remain constant," upsets all previous theoretical belting formulæ. The belt speed for maximum economy should be from 4000 to 4500 ft. per minute. The best distance from centre to centre of shafts is from 20 to 25 ft. Idler pulleys work most satisfactorily when located on the slack side of the belt about one quarter way from the driving-pulley. Belts are more durable and work more satisfactorily made narrow and thick, rather than wide and thin. It is safe and advisable to use: a double belt on a pulley 12 in. diameter or larger; a triple belt on a pulley 20 in. diameter or larger; a quadruple belt on a pulley 30 in. diameter or larger. As belts increase in width they should also be made thicker. The ends of the belt should be fastened together by splicing and cementing, instead of lacing, wiring, or using hooks or clamps of any kind. A V-splice should be used on triple and quadruple belts and when idlers are used. Stepped splice, coated with rubber and vulcanized in place, is best for rubber belts. For double belting the rule works well of making the splice for all belts up to 10 in. wide, 10 in. long; from 10 in. to 18 in. wide the splice should be the same width as the belt, 18 in. being the greatest length of splice required for double belting. Belts should be cleaned and greased every five to six months. Double leather belts will last well when repeatedly tightened under a strain (when at rest) of 71 lbs. per in. of width, or 240 lbs. per sq. in. section. They will not maintain this tension for any length of time, however. Belt-clamps having spring-balances between the two pairs of clamps should be used for weighing the tension of the belt accurately each time it is tightened. The stretch, durability, cost of maintenance, etc., of belts proportioned (A) according to the ordinary rules of a total load of 111 lbs. per inch of width corresponding to an effective pull of 65 lbs. per inch of width, and (B) according to a more economical rule of a total load of 54 lbs., corresponding to an effective pull of 26 lbs. per inch of width, are found to be as follows: When it is impracticable to accurately weigh the tension of a belt in tightening it, it is safe to shorten a double belt one half inch for every 10 ft. of length for (A) and one inch for every 10 ft. for (B), if it requires tightening. Double leather belts, when treated with great care and run night and day at moderate speed, should last for 7 years (A); 18 years (B). The cost of all labor and materials used in the maintenance and repairs of double belts, added to the cost of renewals as they give out, through a term of years, will amount on an average per year to 31% of the original cost of the belts (A); 14% or less (B). In figuring the total expense of belting, and the manufacturing cost chargeable to this account, by far the largest item is the time lost on the machines while belts are being relaced and repaired. The total stretch of leather belting exceeds 6% of the original length. The stretch during the first six months of the life of belts is 36% of their entire stretch (A); 15% (B). A double belt will stretch 47/100 of 1% of its length before requiring to be tightened (A); 81/100 of 1% (B). The most important consideration in making up tables and rules for the use and care of belting is how to secure the minimum of interruptions to manufacture from this source. The average double belt (A), when running night and day in a machine shop, will cause at least 26 interruptions to manufacture during its life, or 5 interruptions per year, but with (B) interruptions to manufacture will not average oftener for each belt than one in sixteen months. The oak-tanned and fulled belts showed themselves to be superior in all respects except the coefficient of friction to either the oak-tanned not fulled, the semi-raw-hide, or raw-hide with tanned face. Belts of any width can be successfully shifted backward and forward on tight and loose pulleys. Belts running between 5000 and 6000 ft. per min. and driving 300 H.P. are now being daily shifted on tight and loose pulleys, to throw lines of shafting in and out of use. The best form of belt-shifter for wide belts is a pair of rollers twice the width of belt, either of which can be pressed onto the flat surface of the belt on its slack side close to the driven pulley, the axis of the roller making an angle of 75° with the centre line of the belt. Remarks on Mr. Taylor's Rules. (Trans. A. S. M. E., xv., 242.) -The most notable feature in Mr. Taylor's paper is the great difference be tween his rules for proper proportioning of belts and those given by earlier writers. A very commonly used rule is, one horse-power may be transmitted by a single belt 1 in. wide running x ft. per min., substituting for x various values, according to the ideas of different engineers, ranging usually from 550 to 1100. The practical mechanic of the old school is apt to swear by the figure 600 as being thoroughly reliable, while the modern engineer is more apt to use the figure 1000. Mr. Taylor, however, instead of using a figure from 550 to 1100 for a single belt, uses 950 to 1100 for double belts. If we assume that a double belt is twice as strong, or will carry twice as much power, as a single belt, then he uses a figure at least twice as large as that used in modern practice, and would make the cost of belting for a given shop twice as large as if the belting were proportioned according to the most liberal of the customary rules. This great difference is to some extent explained by the fact that the problem which Mr. Taylor undertakes to solve is quite a different one from that which is solved by the ordinary rules with their variations. The prob. lem of the latter generally is, "How wide a belt must be used, or how nar row a belt may be used, to transmit a given horse-power?" Mr. Taylor's problem is: "How wide a belt must be used so that a given horse-power may be transmitted with the minimum cost for belt repairs, the longest life to the belt, and the smallest loss and inconvenience from stopping the machine while the belt is being tightened or repaired ?" The difference between the old practical mechanic's rule of a 1-in.-wide single belt, 600 ft. per min., transmits one horse-power, and the rule commonly used by engineers, in which 1000 is substituted for 600, is due to the belief of the engineers, not that a horse-power could not be transmitted by the belt proportioned by the older rule, but that such a proportion involved undue strain from overtightening to prevent slipping, which strain entailed too much journal friction, necessitated frequent tightening, and decreased the length of the life of the belt. Mr. Taylor's rule substituting 1100 ft. per min. and doubling the belt is a further step, and a long one, in the same direction. Whether it will be taken in any case by engineers will depend upon whether they appreciate the ex. tent of the losses due to slippage of belts slackened by use under overstrain, and the loss of time in tightening and repairing belts, to such a degree as to induce them to allow the first cost of the belts to be doubled in order to avoid these losses. It should be noted that Mr. Taylor's experiments were made on rather narrow belts, used for transmitting power from shafting to machinery, and his conclusions may not be applicable to heavy and wide belts, such as engine fly-wheel belts. MISCELLANEOUS NOTES ON BELTING. Formulæ are useful for proportioning belts and pulleys, but they furnish no means of estimating how much power a particular belt may be transmitting at any given time, any more than the size of the engine is a measure of the load it is actually drawing, or the known strength of a horse is a measure of the load on the wagon. The only reliable means of determining the power actually transmitted is some form of dynamometer. (See Trans. A. S. M. E., vol. xii. p. 707.) |