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PULLEY
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PULLEY , a See also:wheel, either fixed to a turning See also:axle or carried freely on a stationary one, the periphery of which is adapted to receive some See also:form of wrapping connector. A pulley carried on a rotating See also:shaft and connected to another pulley on a second shaft by an endless See also:band consisting of a See also:flat See also:belt, rope, See also:chain or similar connector serves for the transmission of See also:power from the one shaft. to the other and is known as a. See also:driving pulley; while combinations of pulleys or " sheaves," mounted in fixed or movable frames or " blocks," constitute mechanisms used to facilitate the raising of heavy weights. The word appears in See also:Mid. Eng. as pulley or Polley (See also:late), also as poleyne (Prompt. Parvul.). The first forms seem to be from the O. Fr. poulie, which itself is regarded as coming from the O. Eng. pullean, to pull. The See also:Low See also:Lat. forms polea, polegia, whence Span. polea and Ital. poleggia, are apparently from the Fr. poulie. The earliest form, poleyne, is represented in Fr. by poulain, literally a See also:colt, Low Lat. pullanus, pullus, the See also:young of any See also:animal, the See also:root of which is seen in See also:English " foal." Poulain was used of a rope to let casks down into a cellar or to raise heavy weights. The use of the name of an animal for a See also:mechanical See also:device is not uncommon, cf. " See also:crane," or " easel," from Du. exel, literally " little See also:ass." Driving pulleys are usually constructed of See also:cast See also:iron, and are of circular form, having a central See also:nave by which they are secured to the shaft by keys or other fastenings, and straight or curved arms connecting the nave to the rim, which latter is of a form adapted to the connector. Pulleys are usually cast in one piece, and the proportions of the various parts are designed to resist the unknown stresses due to contraction of the casting in cooling, in addition to the stresses to which pulleys are subjected in use. The rim is slightly wider than the belt, and is of such a See also:section as will suffice to resist the stress due to the pull of the belt, which is commonly taken as 8o lb per See also:inch of width for single belting and 140 lb per inch of width for See also:double belting. The rim is also subject to a centrifugal tension of amount wv2/g pounds per square inch of section, where w is the See also:weight in pounds of a length of one See also:foot of the pulley rim one square inch in section, and v is the velocity of the rim in feet per second. This stress amounts to 1043 lb per square inch, if the velocity is See also:loo ft. per second. The See also:combination of these stresses generally limits the rim velocity of cast-iron pulleys to 8o or 'co ft. per second. The dimensions of the nave depend to a large extent on the method of keying or otherwise securing the pulley to the shaft. The number of the arms is arbitrary, and they may be curved to diminish the liability to fracture from contraction in the cooling of the cast iron, but in other respects are preferably straight, since they are then lighter and stronger. The arms are elliptical in See also:cross-section, diminishing from the nave to the rim, and are usually designed as equally loaded cantilevers, fixed at the nave and See also:free at the rim. These assumptions are probably not nearly correct, and, as the stresses caused by the cooling of the casting are unknown, it is necessary to choose a low working stress of about one ton per square inch. The statical experiments of C. H. See also:Benjamin (See also:American Machinist, 1898) on cast-iron pulleys loaded by a belt to imitate the conditions in practice led him to the conclusion that the rim is usually not sufficiently rigid to load the arms equally, and that the ends of the arms are subjected to bending movements of opposite sign, that at the nave being almost invariably the greater. Pulleys are also built up of wrought iron and See also:steel, and can then be constructed entirely free from See also:internal stress; they are thus much lighter and stronger, and are not liable to See also:fly to pieces like cast iron if they break. Fig. i shows a built-up pulley having a cast-iron nave A, straight wrought-iron arms B, screwed therein and connected to a steel See also:plate-rim C by riveted ends, and also by screwed flanges D riveted on each See also:side to the rim. The pulley is in halves to facilitate fixing, and when in See also:place the sections C are joined by plates E, bolted or riveted to the rim. The two halves of the nave Pulleys are also made of See also:paper, See also:wood and other materials. Wooden pulleys are preferably made of See also:maple, the rim being formed of small sections morticed, pinned and glued together, with the See also:grain set in such directions that any warping of the material will leave the cylindrical form practically unaltered. Wooden pulleys are generally made in two halves, bolted together at the rim and nave, and are provided with wooden. spokes dovetailed into the rim and secured by keys. The pulley is secured to the shaft by conical keys, to give a frictional grip on both the shaft and the pulley; these keys may have their exterior surfaces See also:eccentric to the shaft, with corresponding recesses in the nave, so that the pulley and keys virtually form one piece. If the centre of gravity of a pulley is on the See also:axis of rotation, and the whole See also:mass is distributed so that the axis of inertia coincides with the axis of rotation, there can be no unbalanced force or unbalanced couple as the pulley revolves. The magnitude of the unbalanced force, for a mass of w pounds at a See also:radius of r feet and a velocity of v feet per second, is expressed by wv2/gr lb; and, since the force varies as the square of the velocity, it is necessary carefully to See also:balance a pulley See also:running at a high See also:speed to prevent injurious vibrations. This can be accomplished by attaching balance-weights to the pulley until it will remain stationary in all positions, when its shaft rests on two See also:horizontal See also:knife-edges in the same horizontal See also:plane, or, preferably, the pulley and shaft may be supported on See also:bearings resting on springs, and balanced by attached masses until there is no perceptible vibration of the springs at the highest speed of rotation. The rims of pulleys, See also:round which flat bands are wrapped, may be truly cylindrical, in which See also:case the belt will run in-differently at any See also:part of the pulley, or the rim may be swelled towards the centre, when the central See also:line of the band will tend to run in the diametral plane of the pulley. This self-guiding See also:property may be explained by the tendency which a flat band has, when running upon a conical pulley in a direction normal to its axis, to describe a See also:spiral path as it wraps on to the See also:surface because of the lateral stiffness of the material; the advancing side therefore tends to rise towards the highest part of the See also:cone. If two cones are placed back to back the belt tends to rise to the See also:ridge and stay there. In practice the pulley rim is curved to a radius of from three to five times its breadth, and this not only guides the belt, but allows the line of direction of the advancing side to deviate to a small extent, depending on the See also:elasticity of the material. Parallel shafts may be driven by flexible bands or connectors passing over pulleys, the central planes of which coincide, without any guiding arrangements for the belting. The shafts revolve in the same or opposite directions, according as the belt is open or crossed. Means of changing the relative speeds of rotation are furnished by pulleys of continuously varying See also:diameter, or by speed cones (see See also:MECHANICS: Applied). A See also:common arrangement for driving a See also:lathe spindle, in either direction at several definite speeds, is to provide a See also:counter-shaft on which are mounted two fixed pulleys and two loose pulleys to accommodate two driving belts from the See also:main shaft, one of which is open and the other crossed. The belts are moved laterally by the forks of a striking See also:gear pressing on the advancing sides of the belts, and the pulleys are arranged so that the belts either wrap round the loose pulleys, or can be shifted so that one wraps round a fixed pulley, while the other still re-mains on its loose pulley. See also:Motion in either direction is thereby obtained, and a considerable variation in the speed of rotation can be obtained by providing a cone pulley on the counter-shaft, which drives the cone pulley secured to the lathe II spindle by a See also:separate band. The dimensions of the pulleys are generally so arranged that the return motion of the lathe spindle is faster than the forward motion. An alternative arrangement consists in providing two loose pulleys on the counter-shaft, driven by open and crossed belts respectively, and arranging two clutches on the shaft, so that by the See also:movement of a sliding See also:block, controlled by See also:hand, one or other of the clutches can be put in gear. The proportions of cone pulleys for open or crossed belts may be determined by considering the expression for the See also:half length (1) of a belt wrapping round pulleys of radius 1'1 and See also:r2 respectively, and with centres distant c apart. The value of l may be easily shown to be (ri+r2)7r/2+(ri r2)a+c See also:cos a, where the See also:positive sign is to be taken for a crossed belt and the negative sign for an open belt. In determining the dimensions of corresponding drums of cone pulleys it is evident that for a crossed belt the sum of the radii of each pair remains a See also:constant, since the See also:angle a is constant, while for an open belt a is variable and the values of the radii are then obtained by solving the equations 7.1= 1/7r— c(a See also:sin a +cos a) + Zc sin a, r2=1/a— c(a sin a+cos_a) — lc sin a. The value of a is in See also:general small, and an approximate See also:solution may be obtained by substituting two or three terms of the expansions for sin a and cos a. This, however, leads to a troublesome numerical solution. An accurate geometrical solution by C. Culmann gives the linear equivalents of the above equations in the following manner. A rectangle See also:ABCD (fig. 2), with side AB =a c/2 and AD =c, is constructed, and the quadrant AEF is See also:drawn with centre D and radius DA. F B is the evolute of this circle, and for any radius DE at an angle a and corresponding tangent EG terminated by the evolute, the perpendicular distance of G from the line AD is c(cos a+a sin a). If now a line be drawn from A to the bisector H of the side BC, it will meet the See also:vertical through G in I and IJ=c(cos a+a sin a)/r. A circular arc, centre D and radius c/2, meets D E in K, and the perpendicular KL gives lc sin a. This distance is marked off from the point I in each direction, whereby the points M and N are obtained, the distance apart of which represents the value rl—r2. If now the value l/,r=OJ be marked off, and a horizontal line be drawn through the point 0, the line OM represents n—1—r2. Repeating this construction for all values of a between o° and 90°, we obtain a See also:curve BPC, which can be used for determining the ratios of corresponding drums of cone pulleys or of conical drums for open belts. The curve BPC is generally used with the abscissae spaced more conveniently for See also:practical applications, and a modification of the See also:diagram by J. F. See also:Klein (Journ. See also:Franklin Inst., vol. lxxix.) is often used instead. When pulleys are mounted on shafts which are parallel to one another, the band will retain its position, provided that its cen- tral line advances towards each pulley in the diametral plane of this latter. This See also:condition is fulfilled in the example shown by fig. 3, in which the central planes of each pulley pass through the points of delivery of the other pulley for the given direction of motion. If the motion is reversed the condition is no longer satisfied and the belt will leave the pulleys. In more compli- cated cases See also:guide pulleys must be used.. In the most general case for inclined pulleys, any two points may be chosen on the line of intersection of the diametral planes, and tangents drawn to the See also:pitch circles of the pulleys. Guide pulleys are set with their diametral planes in the planes containing corresponding pairs of tangents, and a continuous belt wrapped round these pulleys in due See also:order can then be run in either direction. The rims of pulleys for See also:hemp or other See also:ropes or cords are grooved, and the sides are usually either inclined at 45° or curved to give a sharper angle at the outside than at the bottom of the groove; in the latter case, as the rope wears it engages in a groove of greater angle and less effective grip. See also:Wire ropes are injured by the lateral crushing of the material, and in this case the grooves are wide enough to allow the rope to See also:rest on the rounded bottom, which is lined with See also:leather or wood to diminish the See also:wear and increase the See also:friction. In English practice there are as many separate endless ropes as there are pairs of grooves in the two pulleys to be connected, but in cases of American practice the rope is continuously See also:wound round the two pulleys, and the free end passes over a pulley mounted on a movable weighted See also:carriage to adjust the tension. It is of considerable importance that the effective radius of See also:action of the rope remain constant throughout each pulley, otherwise the wear on the rope becomes very See also:great and its See also:life is diminished. The grooves must be turned exactly alike, and the rope must be of the same diameter throughout to diminish slip. Pulleys may be detachably connected to a shaft by friction clutches, so that they may be thrown in and out of engagement at will. The section, fig. 4, shows a clutch for a rope-driven pulley A, which runs freely on a See also:bush B on the shaft, and is provided with an enlarged cylindrical nave or clutch See also:box C. A split See also:ring D, carried by the clutch and turning with it, can be thrust against the clutch box by right- and See also:left-handed screws E, so that a sufficient grip is obtained to cause the clutch and the pulley to turn as one piece. The engagement of the pulley and clutch is determined by a hand-controlled block F sliding FIG. 4. on the shaft, the movement of which is communicated to the right- and left-handed See also:screw shafts by links G connected to the levers H. The resistance to slipping of a flat belt on a pulley may be obtained by considering the See also:equilibrium of a small arc of the pulley surface subtending an angle dB at the centre, and having tensions T and T+dT at its extremities. Neglecting quantities of the second order, the pressure on the pulley is TdO, and the friction is µTdB where u is the coefficient of friction between the belt and the pulley. We have therefore dT =µTdB and dT/T =See also:tide. Integrating the expression for an angle of wrapping 0, we obtain the relation See also:log s T1/T2 = µB, where Tl and T2 are the end tensions. For leather belts on cast-iron pulleys the value of u may be taken as 0.4, giving a ratio of the tensions on the tight and slack sides of T1/T2=3.514, when the angle of wrapping is. 180°. For ropes in the grooves of cast-iron pulleys, where 4 is the inclination of the sides of the grooves, the value of the normal pressure is increased in the ratio of cosec =1. A usual value of u for hemp ropes on cast-iron pulleys is 0.3, and the exponential log ratio is therefore o.37r cosec 10 when B =7r. At high speeds the centrifugal tension of the belt or rope, of amount wv2/g, may be considerable, and must be subtracted from the end tensions. Pulley Blocks.—Frames or blocks containing pulleys or sheaves are used in combination for lifting heavy weights. There are usually two blocks, of which one A (fig. 5) is fixed, and the other B is movable, and a rope or chain, with one end secured to one of the blocks at C, passes round the sheaves in a continuous coil, leaving a free end D at which the effort is applied. In the arrangement shown there are three equal sheaves in each block, and each set turns on a See also:pin secured in the framing. The load, supported by the See also:lower See also:hook, is raised by hauling on the free end and, neglecting any slight obliquity of the plies of rope, the free end moves six times as fast x;;i.%1f~IL` 11 combination is not realized, because of the friction of the sheaves against the pin or shaft, and more important still is the stiffness of the rope, which requires See also:work to be done upon it to See also:bend it round the sheave and straighten it again. The effect of pin friction is See also:equivalent to diminishing the radius of the effort and increasing that of the resistance, For a single pulley of diameter D, turning on a fixed pin of diameter d, the relation of the effort E to the load W, where f is the coefficient of friction, is expressed by E/W = (D+fd)/(D—fd)= D I+2fd/D approximately. The resistance of the rope to bending causes an additional resistance, which experiment shows can be expressed in'the form Wd2/cD where c is a coefficient. Hence E = W(t+2fd/D+d2/CD) = kW for a single pulley. In a six-sheaved pulley tackle the re- lation between E and W may be expressed as W = E (I/k+I/k2+t/k3+t/k4+1/k&+t/k,) = E(k6 — I)/k6(k-I), and with a probable value of k =1.1 this gives W = 4.355 E instead of W=6E. If the free end of the rope is re- leased the weight will descend, and the tackle is then said to overhaul. The conditions which enable a pulley tackle to sustain a weight when the effort is removed may be examined, to a first approximation, if we assume that the internal friction acts in such a way as virtually to diminish FIG. 5.—Sheave the effort E and to increase the resistance R by Pulley Block. amounts proportional to the magnitude of each, and in addition to cause a loss M due to the weights of the parts themselves. We may therefore See also:express the relation in the form (t—a)E=(I+b)R+M, whence we obtain R/E=(t—a)/(t+ b+M/R). If now the See also:machine be reversed and R becomes an effort corresponding to a resistance E' then we have R(i —b) _ (t+a)E'+M, giving E'/R= (t —b—M/R)/(t+a). (I) If the load is self-sustaining E' is zero or negative, and hence b + M/R must be equal to or greater than unity, and therefore it is impossible for the ratio of R/E to rise to a greater value than (t —a)/2, and hence at least half the effort is wasted if the tackle Pulley Block. Pulley Block. has the valuable property of sustaining a load when the effort is removed. If, however, an artificial resistance can be introduced, to come into action only when the effort is removed, it is possible to obtain a, tackle of greater efficiency. As an example we may take the case where a See also:brake is provided offering a resistance, c R, proportional to the load sustained, and where the values a and b are small compared with unity. See also:Equation (t) becomes E/R= (I—b—c—M/R)/(t—a), and hence b+c+M/R is equal to or greater than unity when the load is self-sustained, and we thus obtain a relation between R and E in the form t —a/2 —c, which shows to a first approximation, that as c approaches unity a high efficiency is obtainable, while the self-sustaining power of the tackle is retained. In order to obtain a greater ratio of R to E, without using a large number et sheaves, various arrangements are used, of which the See also:Weston See also:differential pulley block is a typical example. The upper block carries a pair of chain pulleys A (fig. 6), secured together and of slightly different effective diameters D and d. An endless chain B, passing through guides C and D, encircles these pulleys and the single loose pulley E of the lower block, as indicated. With this arrangement a single revolution of the upper sheave causes the endless chain to See also:wind up the chain on one side by an amount aD, and to unwind an amount Ord on the other side, and in consequence the lower sheave is raised by a(D—d)/2. Hence, neglecting friction, ETD=zR7r(D—d), i.e. E='-,R(t—d/D). The value d/D usually lies between the limits Io/t t and 15/16, and, if a greater difference of E from R is required, a further mechanical See also:advantage can be obtained by employing a separate hand-wheel and chain, or by forming the upper sheave with an See also:annular See also:spur-wheel gearing with a pinion driven by a hand-wheel and chain, as in the See also:Tangye form of Weston pulley-block. The efficiency of the Weston pulley-block is less than 5o%, and it does not therefore overhaul. An objection to this form of block is the great length of the endless chain, which may See also:drag on the ground and pick up dirt and grit, and thereby interfere with the smooth working of the mechanism. Other forms, which do not require so lengthy a chain, sometimes employ an epicyclic See also:train to obtain the reduced velocity of the load. The See also:Moore and See also:Head block has two equal chain-wheels A, B, fig. 7, loosely mounted on an axle C, and provided with annular toothed gear-wheels which usually differ by one tooth. A spur pinion D, gearing with both wheels, is carried loosely upon an eccentric E forming part of the central pin, so that when this latter is turned by the hand-wheel F and chain G the axis of the pinion describes a circle the diameter of which equals the throw of the eccentric, and a small relative motion of the two sheaves takes place, depending on the number of the See also:teeth of the annular wheels. The motion obtained is divided between the two vertical parts of the chain H, which is wrapped round each sheave in opposite directions, with a free See also:loop I between, while the ends are attached to the lifting hook. This form is self-sustaining at all loads. In order to obtain a. self-sustaining pulley tackle, which will have an efficiency of more than 5o%, various arrangements are adopted, which during lifting automatically throw out of action a brake and cause it to come into action again when the effort is removed. A See also:worm-gear tackle of this description is shown in fig. 8, in which a worm A, operated by a hand-wheel B and chain C, drives the worm-wheel D, thereby coiling up a chain E, one end F of which is secured to the upper block, and the other end hangs loosely, after passing round the sprocket-wheel. The worm is of great pitch, so that if the effort were removed the weight would descend, did not the axial end thrust of the worm shaft • H throw into action a friction brake H, the resistance of which prevents motion downwards. In the brake shown, the cone I is pressed against a corresponding See also:recess in the ratchet-wheel J, which latter turns loosely in the casing and is provided with a pawl not shown in the figure; this pawl allows freedom of motion when the load is being raised. The frictional grip between the two surfaces prevents return motion of the worm shaft and the load remains suspended, but it may be lowered by turning the hand-wheel so as to overcome the friction brake. Various other arrangements of friction brakes have been devised to give a resistance proportional to the load. Blocks, for lifting very heavy weights, are sometimes provided with an electric motor for driving the FIG. 8.—Worm-gear Pulley worm. The worm-wheel shaft then Block with Automatic Brake. sometimes carries a spur-pinion gear- See also:ing with a spur-wheel on the lifting shaft, whereby a much greater mechanical advantage is obtained with a small loss by friction of the spur gearing. a~ the lower block carrying the weight, and in the See also:absence of friction and other resistances the mechanical advantage will be in the same ratio of the effort to the resistance. In practice the See also:Ave," full advantage of this or any other similar IC See also:Die Hebezeuge; A. See also:Ritter, Lehrbuch der• technischen Mechanik; J. Weisbach and G. Herrmann, The Mechanics of Hoisting Machinery; F. Reuleaux, Der Constructeur; A. B. W. See also:Kennedy, Mechanics of Machinery; J. See also:Perry, Applied Mechanics; W. E. Dalby, Balancing of Engines. (E. G. Additional information and CommentsThere are no comments yet for this article.
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