the catenary of the driving side is not affected by the speed or by the diameter of the rope. The deflection of the rope between the pulleys on the slack side varies with each change of the load or change of the speed, as the tension equation (1) indicates. The deflection of the rope is computed for the assumed value of T ane ! by the parabolic formula S = PL2 +PD, S being the assumed strain Ton the driving side, and t, calculated by equation (1), on the slack side. The tension t varies with the speed. Horse-power of Transmission Rope at Various Speeds. Computed from formula (2), given above. 26.8 28.8 30.6 30.8 28.2 19.8 0 32.8 36.4 39.2 41.5 41.8 37.4 27.6 0 60 72 2 84 134 18 The following notes are from the circular of the C. W. Hunt Co., New York: it For a temporary installation, when the rope is not to be long in use, might be advisable to increase the work to double that given in the table. For convenience in estimating the necessary clearance on the driving and on the slack sides, we insert a table showing the sag of the rope at different speeds when transmitting the horse-power given in the preceding table. When at rest the sag is not the same as when running, being greater or the driving and less on the slack sides of the rope. The sag of the driving side when transmitting the normal horse-power is the same no matter what size of rope is used or what the speed driven at, because the assumption is that the strain on the rope shall be the same at all speeds when transmitting the assumed horse-power, but on the slack side the strains, and consequently the sag, vary with the speed of the rope and also with the horse power. The table gives the sag for three speeds. If the actual sag is less than given in the table, the rope is strained more than the work requires. This table is only approximate, and is exact only when the rope is running at its normal speed, transmitting its full load and strained to the assumed amount. All of these conditions are varying in actual work, and the table must be used as a guide only. Sag of the Rope between Pulleys. The size of the pulleys has an important effect on the wear of the ropethe larger the sheaves, the less the fibres of the rope slide on each other, and consequently there is less internal wear of the rope. The pulleys should not be less than forty times the diameter of the rope for economical wear, and as much larger as it is possible to make them. This rule applies also to the idle and tension pulleys as well as to the main driving pulley. The angle of the sides of the grooves in which the rope runs varies, with different engineers, from 45° to 60°. It is very important that the sides of these grooves should be carefully polished, as the fibres of the rope rubbing on the metal as it comes from the lathe tools will gradually break fibre by fibre, and so give the rope a short life. It is also necessary to carefully avoid all sand or blow holes, as they will cut the rope out with surprising rapidity. Much depends also upon the arrangement of the rope on the pulleys, especially where a tension weight is used. Experience shows that the increased wear on the rope from bending the rope first in one direction and then in the other is similar to that of wire rope. At mines where two cages are used, one being hoisted and one lowered by the same engine doing the same work, the wire ropes, cut from the same coil, are usually arranged so that one rope is bent continuously in one direction and the other rope is bent first in one direction and then in the other, in winding on the drum of the engine. The rope having the opposite bends wears much more rapidly than the other, lasting about three quarters as long as its mate. This difference in wear shows in manila rope, both in transmission of power and in coalhoisting. The pulleys should be arranged, as far as possible, to bend the rope in one direction. Speed of Rope, in feet per second. TENSION ON THE SLACK PART OF THE ROPE. Diameter of the Rope and Pounds Tension on the Slack Rope. 11⁄2 5% For large amounts of power it is common to use a number of ropes lying side by side in grooves, each spliced separately. For lighter drives some engineers use one rope wrapped as many times around the pulleys as is necessary to get the horse-power required, with a tension pulley to take up the slack as the rope wears when first put in use. The weight put upon this tension pulley should be carefully adjusted, as the overstraining of the rope from this cause is one of the most common errors in rope driving. We therefore give a table showing the proper strain on the rope for the various sizes, from which the tension weight to transmit the horse-power in the tables is easily deduced. This strain can be still further reduced if the horse-power transmitted is usually less than the nominal work which the rope was proportioned to do, or if the angle of groove in the pulleys is acute. With a given velocity of the driving-rope, the weight of rope required for transmitting a given horse-power is the same, no matter what size rope is adopted. The smaller rope will require more parts, but the weight will be the same. Miscellaneous Notes on Rope-driving.-W. H. Booth communicates to the Amer. Machinist the following data from English practice with cotton ropes. The calculated figures are based on a total allowable tension on a 134-inch rope of 600 lbs., and an initial tension of 1/10 the total allowed stress, which corresponds fairly with practice. The most usual practice in Lancashire is summed up roughly in the following figures: 134-inch cotton ropes at 5000 ft. per minute velocity = 50 H.P. per rope. The most common sizes of rope now used are 134 and 15% in. The imaximum horse-power for a given rope is obtained at about 80 to 83 feet per second. Above that speed the power is reduced by centrifugal tension. At a speed of 2500 ft. per minute four ropes will do about the same work as three at 5000 ft. per min. Cotton ropes do not require much lubrication in the sense that it is required by ropes made of the rough fibre of manila hemp. Merely a slight surface dressing is all that is required. For small ropes, common in spinning machinery, from 1⁄2 to 34 inch diameter, it is the custom to prevent the fluffing of the ropes on the surface by a light application of a mixture of black-lead and molasses,-but only enough should be used to lay the fibres,put upon one of the pulleys in a series of light dabs. Reuleaux's Constructor gives as the "specific capacity" of hemp rope in actual practice, that is, the horse-power transmitted per square inch of cross-section for each foot of linear velocity per minute, .004 to .002, the cross-section being taken as that due to the full outside diameter of the rope. For a 134-in. rope, with a cross-section of 2.405 sq. in., at a velocity of 5000 ft. per min., this gives a horse-power of from 24 to 48, as against 41.8 by Mr. Hunt's table and 49 by Mr. Booth's. Reuleaux gives formulæ for calculating sources of loss in hemp-rope transmission due to (1) journal friction, (2) stiffness of ropes, and (3) creep of ropes. The constants in these formulæ are, however, uncertain from lack of experimental data. He calculates an average case giving loss of power due to journal friction = 4%, to stiffness 7.8%, and to creep 5%, or 16.8% in all, and says this is not to be considered higher than the actual loss. Spencer Miller, in a paper entitled "A Problem in Continuous Rope-driving" (Trans. A. S. C. E., 1897), reviews the difficulties which occur in ropedriving, with a continuous rope from a large to a small pulley. He adopts the angle of 45° as a minimum angle to use on the smaller pulley, and recommends that the larger pulley be grooved with a wider angle to a degree such that the resistance to slipping is equal in both wheels. By doing this the effect of the tension weight is felt equally throughout all the slack strands of the rope-drive, hence the tight ropes pull equally. It is shown that when the wheels are grooved alike the strains in the various ropes may differ greatly, and to such a degree that danger is introduced, for while onehalf the tension weight should represent the maximum strain on the slack rope, it is demonstrated in the paper that the actual maximum strain may be even four or six times as great. In a drive such as is recommended, with a wide angle in the large sheave with the larger arc of contact, the conditions governing the ropes are the same as if the wheels were of the same diameter; and where the wheels are of the same diameter, with a proper tension weight, the ropes pull alike. It is claimed that by widening the angle of the large sheave not only is there no power lost, but there is actually a great gain in power transmitted. An example is given in which it is shown that in that instance the power transmitted is nearly doubled. Mr. Miller refers to a 250-horse-power drive which has been running ten years, the large pulley being grooved 60° and the smaller 45°. This drive was designed to use a 14-in. manila rope, but the grooves were made deep enough so that a %-in. rope would not bottom. In order to determine the value of the drive a common g-in. rope was put in at first, and lasted six years, working under a factor of safety of only 14. He recommends, however, the employment in continuous rope-driving of a factor of safety of not less than 20. The Walker Company adopts a curved form of groove instead of one with straight sides inclined to each other at 45°. The curves are concave to the rope. The rope rests on the sides of the groove in driving and driven pulleys. In idler pulleys the rope rests on the bottom of the groove, which is semicircular. The Walker Company also uses a "differential" drum for heavy rope-drives, in which the grooves are contained each in a separate ring which is free to slide on the turned surface of the drum in case one rope pulls more than another. A heavy rope-drive on the separate, or English, rope system is described and illustrated in Power, April, 1892. It is in use at the India Mill at Darwen, England. This mill was originally driven by gears, but did not prove successful, and rope-driving was resorted to. The 85,000 spindles and preparation are driven by a 2000-horse-power tandem compound engine, with cylinders 23 and 44 inches in diameter and 72-inch stroke, running at 54 revolutions per minute. The fly-wheel is 30 feet in diameter, weighs 65 tons, and is arranged with 30 grooves for 134-inch ropes. These ropes lead off to receiving-pulleys upon the several floors, so that each floor receives its power direct from the fly-wheel. The speed of the ropes is 5089 feet per minute, and five 7-foot receivers are used, the number of ropes upon each being proportioned to the amount of power required upon the several floors. Lambeth cotton ropes are used. (For much other information on this subject see “ Rope Driving," by J. J. Flather, John Wiley & Sons, 1895.) FRICTION AND LUBRICATION. Friction is defined by Rankine as that force which acts between two bodies at their surface of contact so as to resist their sliding on each other, and which depends on the force with which the bodies are pressed together. Coefficient of Friction.-The ratio of the force required to slide a body along a horizontal plane surface to the weight of the body is called the coefficient of friction. It is equivalent to the tangent of the angle of repose, which is the angle of inclination to the horizontal of an inclined plane on which the body will just overcome its tendency to slide. The angle is usually denoted by 0, and the coefficient by f. f = tan 0. Friction of Rest and of Motion.-The force required to start a body sliding is called the friction of rest, and the force required to continue its sliding after having started is called the friction of motion. Rolling Friction is the force required to roll a cylindrical or spherical body on a plane or on a curved surface. It depends on the nature of the surfaces and on the force with which they are pressed together, but is essentially different from ordinary, or sliding, friction. Friction of Solids.-Rennie's experiments (1829) on friction of solids, usually unlubricated and dry, led to the following conclusions: 1. The laws of sliding friction differ with the character of the bodies rubbing together. 2. The friction of fibrous material is increased by increased extent of surface and by time of contact, and is diminished by pressure and speed. 3. With wood, metal, and stones, within the limit of abrasion, friction varies only with the pressure, and is independent of the extent of surface, time of contact and velocity. 4. The limit of abrasion is determined by the hardness of the softer of the two rubbing parts. 5. Friction is greatest with soft and least with hard materials. 6. The friction of lubricated surfaces is determined by the nature of the lubricant rather than by that of the solids themselves. Law of Unlubricated Friction.-A. M. Wellington, Eng'g News, April 7, 1888, states that the most important and the best determined of all the laws of unlubricated friction may be thus expressed: The coefficient of unlubricated friction decreases materially with velocity, is very much greater at minute velocities of 0 +, falls very rapidly with minute increases of such velocities, and continues to fall much less rapidly with higher velocities up to a certain varying point, following closely the laws which obtain with lubricated friction. Friction of Steel Tires Sliding on Steel Rails. (Westing house & Galton.) Speed, miles per hour..... Adhesion, lbs. per ton (2240 lbs.) 10 15 25 38 45 50 0.110 .087 .080 .051 .047 .040 246 195 179 128 114 90 |