Modern Machine-Shop Practice Part 184

[Ill.u.s.tration: Fig. 2674.]

But when this provision cannot be carried out, pulleys to guide the direction of motion of the belt must be employed; thus in Fig. 2674 are an elevation and plan[39] of an arrangement of these guide or mule pulleys; A B is the intersection of the middle planes E E and F F of the pulleys P and P' to be connected by belt. Select any two points, A and B, on this line and draw tangents A C, B D to the princ.i.p.al pulleys.

Then C A C and D B D are suitable directions for the belt. The guide pulleys must be placed with their middle planes coinciding with the planes C A C, D B D, and the belt will then run in either direction.

[39] From Unwin's "Elements of Machine Design."

[Ill.u.s.tration: Fig. 2675.]

In Fig. 2675 is an arrangement of guide pulleys by which two pulleys not in the same plane are connected, while the arc of contact of the smaller pulley C is increased by the idlers or guide pulleys A B, while either C or D may be driven running in either direction.

[Ill.u.s.tration: Fig. 2676.]

In Fig. 2676 is shown Cresson's adjustable mule pulley stand, which is a device for carrying guide pulleys, and admitting of their adjustment in any direction. Thus the vertical post being cylindrical, the brackets can be swung around upon it and fastened in the required position by the set-screws shown. The brackets carrying the pulleys are also capable of being swung in a plane at a right angle to the axis of the guide pulleys, and between these two movements any desired pulley angle may be obtained. It is obvious that by moving the brackets along the cylindrical post their distance apart may be regulated.

When a belt is stretched upon two pulleys and remains at rest there will be an equal tension on all parts of the belt (that is to say, independent of its weight, which would cause increased tension as the points of support on the pulleys are approached from the centre of the belt between the two pulley shafts); but so soon as motion begins and power is transmitted this equality ceases, for the following reasons:--

[Ill.u.s.tration: Fig. 2677.]

In the accompanying ill.u.s.tration, Fig. 2677, A is the driving and B the driven pulley, rotating as denoted by the arrows; hence C is the driving and D the slack side of the belt. Now let us examine how this slackness is induced. It is obvious that pulley A rotates pulley B through the medium of the side C only of the belt, and from the resistance offered by the load on B, the belt stretches on the side C. The elongation of the belt due to this stretch, pulley A takes up and transfers to side D, relieving it of tension and inducing its slackness. The belt therefore meets pulley B at the point of first contact, E, slack and unstretched, and leaves it at F, under the maximum of tension due to driving B.

While, therefore, a point in the belt is travelling from E to F, it pa.s.ses from a state of minimum to one of maximum tension. This tension proceeds by a regular increment, whose amount at any given point upon B is governed by the distance of that point from E. The increase of tension is, of course, accompanied by a corresponding degree of belt stretch, and therefore of belt length; and as a result, the velocity of that part of the belt on pulley B is greater than the velocity of any part on the slack side of the belt; hence the velocity of the pulley is also greater than that of the slack side of the belt. In the case of pulley A the belt meets it at G under a maximum of tension, and therefore of stretch, but leaves it at H under a minimum of tension and stretch, so that while pa.s.sing from G to H the belt contracts, creeping or slipping back on the pulley, and therefore effecting a reduction of belt velocity below that of the pulley. To summarize, then, the velocity of the part of the belt enveloping A is less than that of A to the amount of the creep; hence the velocity of the slack side of the belt is that of A minus the belt creep on A. The velocity of the part of the belt on B is equal to that of the slack side of the belt plus the stretch of the belt while pa.s.sing over B; and it follows that if the belt or slip creep on one pulley is equal in amount to the belt stretch on the other, the velocities of the two pulleys will be equal.

[Ill.u.s.tration: Fig. 2678.]

Now (supposing the elasticity of the belt to remain constant, so that no permanent stretch takes place) it is obvious that the belt-shortening which accompanies its release from tension can only equal the amount of elongation which occurs from the tension; hence, no matter what the size of the pulleys, the creep is always equal in amount to the stretch, and the velocity ratio of the driven pulley will (after the increase of belt length due to the stretch is once transferred to the slack side of the belt) always be equal to that of the driving pulley, no matter what the relative diameters of the pulleys may be. In Fig. 2678, for example, are two pulleys, A and B, the circ.u.mference of A being 10 inches, while that of B is 20; and suppose that the stretch of the belt is an inch in a revolution of A (A being the driving pulley). Suppose the revolutions of a to be one per minute, then the velocity of the belt where it envelops A and B, and on the sides C and D, will be as respectively marked.

Thus the creep being an inch per revolution of A, the belt velocity on the side C will be nine inches per minute, and its stretch on B being an inch, the velocity of B will be ten inches per minute, which is equal to the velocity of the driving pulley.

It is to be observed, however, that since A receives its motion independently of the belt, its motion is independent of the creep, which affects the belt velocity only: but in the case of B, which receives its motion from the belt, it remains to be seen if stretch is uniform in amount from the moment it meets this pulley until it leaves it, for unless this be the case, the belt will be moving faster than the pulley at some part of the arc of contact.

[Ill.u.s.tration: Fig. 2679.]

Thus suppose P, Fig. 2679, represents a driven pulley, whose load is 1,000 pounds, and that from A to B, from B to C, from C to D, and from D to E, represent equal arcs of contact between belt and pulley, then arc A B will have on it the amount of stretch due to a pull of 250 pounds at B, diminishing to nothing at A. Arc C B will have on it the amount of stretch due to a pull of 500 pounds at C and 250 at B; arc D C will have on it the amount of stretch due to a load of 750 at D, and 500 at C; and arc D E will have the tension due to a load of 1,000 pounds at E, and 750 pounds at D. Suppose, then, that the amount of belt stretch is greater between B and C than it is between D and E, then the belt will travel faster between B C than between D E to an amount equal to the difference in stretch, and will at B C slip over the pulley to that amount; or if the friction of the belt at B C is sufficient to move the pulley in accordance with the stretch, then the belt must move the pulley at a greater velocity than the belt motion from D to E.

But since the friction of the belt is greatest at D E, it will hold the pulley with the greatest force, and hence the velocity of the belt and pulley will be uniform, or at least the most uniform, at D E.

Here arises another consideration, in that the stretch of the leather is not uniform, and the section of belt at C B may stretch more or less under its load than section C D does under its load, in which event the velocities of the respective belt sections cannot be uniform, and to whatever amount belt slip ensues the velocity of the driven wheel will be less than that of the driver.

Attention has thus far been directed to the relative velocities of the pulleys while under continuous motion. But let us now examine the relative velocities when the two pulleys are first put in motion.

Suppose, then, the belt and pulley to be at rest with an equal degree of tension (independently of the weight of the belt, as before) on both sides of the belt. On motion being imparted to the driving pulley, the amount of belt elongation due to the stress of the load on the driving pulley has first to be taken up and transferred to the slack side of the belt, and during such transfer a creep is taking place on the arc of belt contact on the driving pulley.

[Ill.u.s.tration: Fig. 2680.]

Furthermore, let it be noted that while under continuous motion the belt first receives full stress at point F, Fig. 2677; at starting it first receives it at point E, and there will be a period of time during which the belt stretch will proceed from E towards F, the pulley remaining motionless. The length of duration of this period will, in a belt of a given width, and having a given arc of contact on the driven pulley, depend on the amount of the load. Thus, referring to Fig. 2680, if the amount of the load is such that the arc of contact between the top and the point B is sufficient to drive the pulley, the pulley will receive motion when the belt stretch has proceeded from A to B; but if the load on the pulley be increased the belt stretch will require to proceed farther towards C.

At the top the stretch will proceed simultaneously with that of the driving side of the belt, between the points F G, Fig. 2677; but from the friction between the belt and pulley, the stretch of the part enveloping the pulley will be subsequent and progressive from F towards E, Fig. 2677.

It follows, then, that the velocity of the driven wheel will be less than that of the driver at first starting than when in continuous motion.

As the length of the belt is increased, the gross amount of stretch, under any given condition, increases, and hence the longer the belt, the greater the variation of velocity at first starting between the driven pulley and the driver.

From what has been said, it follows that when a mathematically equal velocity ratio is essential, belts cannot be employed, but the elasticity that disturbs the velocity ratio possesses the quality of acting as a cushion, modifying on one pulley any shocks, sudden strains, or jars existing on the other, while the longer the belt and less strained within the limit of elasticity, the greater this power of modification; furthermore in case of a sudden or violent increase of load, the belt will slide on the pulley, and in most cases slip off it, thus preventing the breakage of parts of the driving gear or of the machine driven that would otherwise probably ensue. Furthermore, belt connections are lighter and cheaper than gear-wheel or other rigid and positive connections, and hence the wide application of leather belts for the transmission of power, notwithstanding the slight variations of pulley velocity ratio due to the unequal elasticity of the various parts of the leather composing the belt.

[Ill.u.s.tration: Fig. 2681.]

The ends of belts are joined by two princ.i.p.al methods, the b.u.t.t and the lap joint. In b.u.t.t joints the holes are pierced near the ends of the belts, and the ends of the belt are brought together by means of a leather lace threaded through these holes. If the duty is light a single row of holes is all that is necessary. An example of this kind is shown in Fig. 2681, in which there are five holes on one side, and four on the other of the joint, the extra hole coming in the middle of its end of the belt. The lace is drawn half-way through this extra hole, and laced each way to the side and back again to the middle, the ends being tied on the outside of the belt, which does not come in contact with the pulley surface. By this means the lacing is double through all the holes, and if the knot should slip the slackness will begin at the middle of the belt and extend gradually towards the edges; whereas, if the lacing terminated at one side, and the knot or fastening should slip, all the tension would be thrown on one edge of the belt, unduly stretching it, and rendering it liable to tear. By this method of lacing the lace is not crossed on either side of the belt, which is desirable, because it is found in practice that a crossed lace does not operate so well as an uncrossed one.

[Ill.u.s.tration: Fig. 2682.]

If the power to be transmitted is so much as to render it desirable to have the strength of the laced joint more nearly approach that of the solid belt than is obtainable with a single row of holes, a double row is provided, as shown in Fig. 2682.

For belts of about 3 inches wide and over, these holes are made as follows: A, B, and C, D, E, about an inch apart and 5/8 inch from the line of joint; F, G, H, and I, J, being about 1/2 inch behind A, B, and C, D, E, respectively.

For thinner belts the holes may be closer together, and to the edges of the belt the exact distances permissible being closer together as the duty is lighter; but however narrow the width of the belt, it should contain at least two holes on each side of the joint. The sizes of these holes are an important element, since the larger the hole the more the belt is weakened. The following are the sizes of holes employed in the best practice:--

Width of Belt. Size of Punched Hole.

Up to 4 inches 1/4 inch.

From 4 to 8 inches 5/16 "

From 8 inches upwards 3/8 "

[Ill.u.s.tration: Fig. 2683.]

The holes are usually made round, but from the pliability of the lace, which enables it to adapt itself to the form of the hole to a remarkable degree, it is not unusual to preserve the strength of a belt by making an oblong hole, as in Fig. 2683 at A, or a mere slit, as at B, which, from removing less material from the belt, leaves it to that extent stronger.

[Ill.u.s.tration: Fig. 2684.]

[Ill.u.s.tration: Fig. 2685.]

The ends of the belt should be cut quite square, and at a right angle to the edges, so that when the two ends are drawn together by the lace the edges of the belt will remain straight, and not curved, as they would do if either end of the belt were not cut at a right angle. Suppose, for example, that the ends of a belt were cut aslant, as in Fig. 2684, when laced up the edge of the belt would come as in Fig. 2685.

[Ill.u.s.tration: Fig. 2686.]

The holes must be punched exactly opposite to each other, or lacing the belt will bring the edges out of fair, as shown in Fig. 2686, the tension of the lace drawing the holes opposite to each other, irrespective of where the edges of the belt will come. If some of the holes are opposite and others are not, the latter will throw the edges of the belt out of line to some extent, especially if the lace is first entered in the holes that are not opposite, because, in that case, drawing the lace tight at once throws the belt edges out, and the subsequent lacing has but a limited effect in correcting the error, unless, indeed, the majority of the holes are opposite, and but one or two are out of line.

The lace should be drawn sufficiently tight to bring the ends of the belt firmly together, and should be laced with an even tension throughout, and for a belt doing heavy duty should have its ends tied in a knot at the back, and in the middle of the belt.

The width of the lace is usually about as follows:--

Width of Belt. Width of Lace.

24 inches and over 1/2 inch 6 to 24 inches 3/8 "

2 to 4 inches 5/16 "

2 inches and less 1/4 "

Since belts are tightened by cutting a piece off one end (preferably the end which shows the holes most stretched), it is obvious that a b.u.t.t-joint possesses an advantage, because as less of the belt length is occupied by the holes they may be cut quite out and new ones punched, whereas, in some cases, the length of the belt occupied by the holes in a lap-joint is more than the length of belt required to be cut out to tighten it.

[Ill.u.s.tration: Fig. 2687.]

[Ill.u.s.tration: Fig. 2688.]

[Ill.u.s.tration: Fig. 2689.]