Mechanical Power Transmission: Belts, Chains, and Ropes, Summaries of Theory of Machines

Download Mechanical Power Transmission: Belts, Chains, and Ropes and more Summaries Theory of Machines in PDF only on Docsity! Flexible Transmitting Elements Elements Of Mechanism Sixth Edition Flexible Connectors When the distance between the driving shaft and the driven shaft is too great (usually less than 6 ft) to be connected by gears, a flexible connector is used. If the wheel A, Fig. 12-1, is turning at a certain angular speed about the axis S, its outer surface will have a linear speed dependent upon the angular speed and the diameter of A. Flexible connectors may be divided into three general classes: 1. Belts made of leather, rubber, or woven fabrics are flat and thin, and require pulleys nearly cylindrical with smooth surfaces. Flat belts are used to connect shafts as much as 30 ft apart. Belts may be run economically at speeds as high as 4500 fpm. Belts are also made with V-shaped cross section to be used on grooved pulleys. V-belts are usually used for connecting shafts which are less than 15 ft apart. Speed ratios up to 7 to 1 and belt speeds up to 5000 fpm may be used. Pitch Surface and Line of Connection Figure 12-2 represents the edge view of a piece of a belt before being wrapped around the pulley. If it is assumed that there are no irregularities in the make-up of the belt the upper surface o is parallel to and equal in length to the surface i. Pitch Surface and Line of ConnectionWhen this same belt is stretched around a pulley, as in Fig. 12-3, the surface i is drawn firmly against the surface of the pulley while the The outer part of the belt must therefore stretch somewhat and the inner part compress. There will be some section between i and o which is neither stretched nor compressed, and the name neutral section may be given to this part of the belt. surface o bends over a circle whose radius is greater than that of the surface of the pulley by an amount equal to the belt thickness 2p. Speed Ratio and Directional Relation of Shafts Connected by a Belt That is, the angular speeds of the shafts are in the inverse ratio of the effective diameters of the pulleys, and this ratio is constant for circular pulleys. As the thickness of belts generally is small as compared with the diameters of the pulleys, it may be neglected. The speed ratio will then become which is the equation almost always used in practical calculations. ExiampLe 1. Assume that a shaft A makes 360 rpm. On A is a pulley 24 in. in diameter belted to a pulley 36 in. in diameter on another shaft B. To find speed of shaft B. From equation 2, Speed of A diameter of pulley on B Speed of B ~~ diameter of pulley on A When the known values are substituted, this equation becomes ee Speed of B24 Therefore, 24 Speed of B = 7 X 360 = 240 rpm ExampLe 2. Suppose that a shaft A making 210 rpm is driven by a belt from a 30-in, pulley on another shaft B which makes 140 rpm. To find the size of the pulley on A. Using the principle of equation 2, Speed of A diameter of pulley on B Speed of B diameter of puliey on A Therefore, 210 30 30 X 140 = 20 in, 140. « 210 un Then a 20-in. pulley is required on A. Kinds of Belts The material most commonly used for flat belts is leather. For some kinds of work, however, belts woven from cotton or similar material are used. When the belt is to be run in a place where there is much moisture, it may be made largely of rubber properly combined with fibrous material in order to give strength. Leather belts always should be run with the hair side against the pulleys, if possible. Kinds of Belts Leather belts are made by gluing or riveting together strips of leather cut lengthwise of the hide, near the animal's back. If single thicknesses of the leather are fastened end to end, the belt is known as a single belt and is usually about 3/16 in. thick. If two thicknesses of leather are glued together, flesh side to flesh side, the belt is known as a double belt and is from 5/16 in. to 3/8 in. thick. The manner of uniting the ends of the strips to form a belt, and of fastening together the ends of the belt to make a continuous band for running over pulleys, is very important. Power of Belting The amount of power which a given belt can transmit depends upon its speed, its strength, and its ability to adhere to the surf ace of the pulleys. The speed is usually assumed to be the same as the surface speed of the pulleys. The strength, of course, depends upon the width and thickness and upon the nature of the material of which the belt is made. The ability to cling to the pulley in order to run with little or no slipping depends upon the condition of the pulley surfaces and of the surface of the belt which is in contact with the pulleys, and upon the tightness with which the belt is stretched over the pulleys. Tension in a Belt If the belt sticks to the pulley B so that there is no slipping, the force T1, tends to cause the pulley B to turn as shown by the full arrow, and the force T2 tends to cause B to turn as shown by the dotted arrow. As soon as T1 becomes enough greater than T2 to overcome whatever resistance the shaft S offers to turning, the pulleys will begin to turn in the direction of the full arrow. The unbalanced force, then, which makes the driven pulley B turn is the difference between the tension T1 on the tight side of the belt and thetension T2 on the slack side of the belt. This difference in tensions is called the effective pull of the belt and is here represented by the letter E. Tension in a Belt From the above discussion it may be seen that the following equation holds true: Horsepower of a Belt The effective pull is the force in the belt which enables it to do work, it follows that the product of the effective pull by the speed of the belt in feet per minute will give the foot-pounds of work per minute that the belt performs, and this divided by 33,000 will give the horsepower which the belt transmits. If N is the rpm of S, and D the diameter of pulley A (in feet), the following equations express the horsepower of the belt. Approximate Formula for Calculating the Length of Belt Refer to Fig. 12-7 and let L represent the length of the belt: D, d, and C must be expressed in like linear units; if in feet, the resulting value of L will be in feet; if in inches, the value of L will be in inches. Exact Formulas for Length of Belt Connecting Parallel Axes It is necessary in designing certain pulleys, known as stepped pulleys and cone pulleys, to make use of an equation expressing exactly, or very nearly so, the belt length in terms of the diameters and the distance between centers of the pulleys. The crossed belt and the open belt must be considered separately. Exact Formulas for Length of Belt Connecting Parallel Axes Crossed BeltsLet D and d (Fig. 12-6) be the diameters of the connected pulleys; C the distance between their axes; and L the length of the belt. Angle θ is expressed in radians. Exact Formulas for Length of Belt Connecting Parallel Axes Open BeltsLet D and d be the diameters of the connected pulleys; C the distance between their axes; and L the length of the belt. Angle θ is expressed in radians. Exact Formulas for Length of Belt Connecting Parallel Axes Open Belts Exact Formulas for Length of Belt Connecting Parallel Axes Open Belts Stepped Pulleys If the belt is shifted to any other position, as that shown by dotted lines, Dx becomes the working diameter of the driving pulley and dx of the driven pulley. If nx represents the speed of S1 for this belt position Therefore, by properly proportioning the diameters of the different pairs of steps, it is possible to get any desired series of speeds for the driven shaft. Stepped Pulleys In designing such a pair of pulleys two things must be taken into account. First, the ratio of the diameters of the successive pairs of steps must be such as to give the desired speed ratios. Second, the sum of the diameters of any pair of steps must be such as to maintain the proper tightness of the belt for all positions. This second consideration makes the problem of design considerably more complicated. Speed Cones Sometimes, instead of stepped pulleys, pulleys which are approximately frustums of cones are used, as shown in Fig. 12-15. Here the working diameters of the pulleys, as Dx and dx for any belt position, are measured at the middle of the belt. Speed Cones When cone pulleys are used, a shipper must guide each part of the belt near the point where it runs on to the pulley otherwise the belt will tend to climb toward the large end of each pulley. Both shippers must be moved simultaneously when the belt is shifted. Belt Connections between Shafts Which Are Not ParallelNon-parallel shafts may be connected by a flat belt with satisfactory results, provided the pulleys are so located as to conform to a fundamental principle which governs the running of all belts, namely: The point where the pitch line of the belt leaves a pulley must lie in a plane passing through the center of the pulley toward which the belt runs. In other words, a belt leaving a pulley may be drawn out of the plane of the pulley, but when approaching a pulley its center line must lie in the mid plane of that pulley. This may be seen by a reference to Fig. 12-18. Belt Connections between Shafts Which Are Not ParallelHere the shafts S and T are intended to turn in the directions indicated by the arrows. Consider elevation A; the pitch line of the belt leaves the pulley M at the point a. If the pulley N is in such a position on the shaft T that a plane through the middle of its face contains the point a, the belt will run properly on to pulley N. XX is the trace of this plane and evidently contains point a. Similarly, in elevation B, the pitch line of the belt leaves the pulley N at b1 and M is so located on shaft S that a plane YY through the middle of its face contains b1. V-Belts Various advantages are claimed for this type of belting, such as high efficiency, silent operation, use on short center drives, high speed ratios, ability to absorb shock, and the fact that it is not affected by changes in direction of rotation or angular inclination of center line. V-belts should not be used where temperatures are very high, where there is likely to be an excess of oil, or in places where it is difficult to put on an endless belt. Flexible connectors may be divided into three general classes: 2. Ropes made of Manila, hemp, cotton, or wire are nearly circular in section and require either grooved pulleys or drums with flanges. Rope may be used for connecting shafts up to 100 ft apart and should operate at a speed of less than 600 fpm. Ropes and Cords Power is often transmitted by means of ropes running over pulleys, called sheaves, having grooved surfaces. For large amounts of power inside of buildings the ropes are made of hemp or similar material. For long-distance drives and drives which are exposed to the weather, wire ropes are used. For small amounts of power on machines, cords of cotton are common. Wire Ropes Wire rope is very suitable for the transmission of large powers to great distances, as for instance in cable and inclined railways. Its rigidness, great weight, and rapid destruction due to bending, however, unfit it for use in mill service, where the average speed of rope is about 4000 fpm. As the easiest way to break wire is by bending it, ropes made of it, by any method whatsoever, have proved unsatisfactory for drives of short centers and high speed unless the diameters of the sheaves are large enough to avoid bending the rope to strain it above the elastic limit. Wire Ropes Wire ropes will not support without injury the lateral crushing caused by the V-shaped grooves; hence it is necessary to construct the pulleys with grooves so wide that the rope rests on the rounded bottom of the groove, as shown in Fig. 12-35, which shows a section of the rim of a wire-rope pulley. The friction is greatly increased, and the wear of the rope diminished, by lining the bottom of the groove with some elastic material, as gutta-percha, wood, or leather, made up in short sections and forced into the bottom of the groove. Flexible connectors may be divided into three general classes: 3. Chains are composed of links or bars, usually metallic, jointed together, and require wheels, sprockets, or drums either grooved, notched, or toothed, to fit the links of the chain. Chains are usually used for connecting shafts which are less than 15 ft apart. The speed of the chain will depend upon the type of chain. Roller and silent chains may operate at speeds up to 2500 fpm. Chains Hoisting Chains. The most common form of hoisting chain (Fig. 12-36) consists of oval links and is called a coil chain. The form of sprocket used with such a chain is also shown in the figure. Another type of hoisting chain is known as the stud-link chain and is shown in Fig. 12-37. The stud-link chain will not kink or tangle so easily as the coil chain. Chains Conveyor Chains may be of the detachable or hook- joint type shown in Fig. 12-38, or of the closed-end pintle type illustrated in Fig. 12-39. The design of the sprocket teeth is largely empirical, care being taken to have the teeth so shaped and spaced that the chain will run onto and off the sprockets smoothly and without interference even after it has stretched or worn somewhat. Chains of this general class are often used for transmitting power at low speeds, as in agricultural machinery. They are usually made of malleable cast links and lack the smooth running qualities of the more carefully made chains. Chains Power-Transmission Chains. This class includes the three types known as block, roller, and silent. The chains are made of steel, accurately machined, with wearing parts hardened, and run on carefully designed sprockets. In the following discussion no attempt is made to give an exhaustive treatment of the subject, but merely to give some idea of the nature of the three types and some of the points which need to be considered in, their design. Chains Inverted Tooth Chains. Although roller chains on sprockets as now designed can run quietly at fairly high speeds, the inverted tooth chain, commonly known in the United States as the silent chain, is widely used when maximum quietness is desired and where it is necessary to transmit heavier loads than can be carried by roller chains of the same pitch. These chains have no rollers, but the links themselves are so shaped that they engage directly with the sprocket teeth. Like the roller chains, they adapt themselves to the sprocket after the pitch of the chain has increased because of wear. Chains Inverted Tooth Chains. Two examples will illustrate this type of chain. In selecting these examples no attempt has been made to illustrate the latest improvements, but merely to show the principle of action. Renold Inverted Tooth Chain. Morse Rocker-Joint Chain. Chains Renold Inverted Tooth Chain. Figure 12-45 shows a chain developed by Hans Renold. It consists of links C of a peculiar form with straight bearing edges a, b, which run over cut sprocket wheels with straight-sided teeth whose angles vary with the diameter of the wheel. Chains Morse Rocker-Joint Chain. This chain (Fig. 12-46) eliminates the sliding friction of the rivets as the chain bends around the sprocket. Instead of the ordinary pin bearing, a rocking bearing is provided at each joint. The following description, with slight changes, is taken from the catalog of the Morse Chain Co. Two pins are employed at each joint; the left- hand pin a is called the seat pin and the right-hand pin b the rocker. Each is securely held in its respective end of the link. Chains Morse Rocker-Joint Chain. The seat pin has a plane surface against which the edge of the rocker pin rocks or rolls when the chain goes on and off the sprockets. The joint is so designed that the pressure due to tension of driving will be taken on a flat surface when in between the sprockets. Design of Standard Sprocket Teeth for Roller Chains Figure 12-43 shows the construction for the outline of the approved standard sprocket teeth for a roller chain. Chain Length The open belt formula may be adapted to the cham as follows: Let D and d be the pitch diameters, in inches, of the sprockets having N arld n teeth respectively. Dp and dp the same diameters in pitches. P = pitch of chain; C = distance between centers of sprockets, in inches; Cp = center distance in pitches; L = length of chain in inches; Lp = length of chain in pitches. Angular Speed Ratio and Distance between Centers of Sprockets Since the pitch line of the chain lies on the driving sprocket as a part of a polygon it follows that, if this sprocket is turning at a constant angular speed, the speed with which the chain is drawn toward this sprocket varies from a maximum when the pitch line is in the position shown in full lines (Fig. 12-44) to a minimum for the dotted-line position. Angular Speed Ratio and Distance between Centers of Sprockets This variation of chain speed tends to cause a variation in angular speed of the driven sprocket. It is desirable to adjust the center distance so that the span of the chain (ab) between sprockets shall be equal to an integral number of pitches, thus reducing the angular speed variation to a minimum.