ELEMENTARY PRINCIPLES OF CONE PULLEYS AND BELTS

ELEMENTARY PRINCIPLES OF CONE PULLEYS AND BELTS  Everyone knows that cone pulleys are usually made with regular steps; that is, if it is one inch from one step to the next, it is also one inch from the second to the third, etc., the reason being that when the centers of the shafts on which the cones run are a fair disance apart, the belt will pass very nearly half way around that part of each cone on which it is running, and the length of the belt will consequently be approximately equal to twice the distance between the shafts, added to half the circumference of the grade of one of the cones on which it is running, and half the circumference of the grade of the other cone on which it is running. As the steps are even, the half circumference of any two grades of each cone will, when added together, produce the same result. For example, if we had two cones, the diameters of the several grades of which were 6, 8, 10 and 12 inches, it is evident that the sum of half the diameters taken anywhere along the cones, as they would be set up for work, would in every case be the same. If the diameters are the same, it follows that the circumference must also be the same, and, of course, that half the circumference must be the same, so that when the centers of the shafts are a fair distance apart, and the difference between the largest and smallest step of the cone not too great, the same belt will run equally well anywhere on the cone, because it runs so near half way around each grade of the two cones on which it is running, that the slight difference is within the practical limit of the stretch of the belt.  But when the shafts are near together, and when the difference between the largest and smallest step of the cone is considerable, the belt is not elastic enough to make up this difference. Fig. 1 shows a three-step cone, the grades being 4, 18, and 32 inches diameter, respectively, there being a difference of 14 inches on the diameter for each successive grade, and the step being therefore 7 inches in each case. Of course, it is not likely that such a cone as this would be made for practical use, but it is well to go to extremes when looking for a principle. Now, it is evident that two cones, even if like the one shown in the cut, were set up far enough apart, they would still allow the belt to run very nearly half way around each grade of the two cones, the angularity of the belt would be slight, and the length of belt would therefore still be as mentioned above.  But (again taking an extreme case) by reference to Fig. 2, which is intended to represent a belt running from the largest grade of one cone to the smallest grade of the other cone, we see that the belt runs three quarters of the way around the large pulley, and only one quarter of the way around the small one, the distance between the shafts in this case being 19% inches.  The length of this belt will evidently be equal to three quarters of the distance around the large pulley, plus one quarter the distance around the small pulley, plus the distances A and B, which we find to be each 14 inches. The circumference of a 32-inch diameter pulley is 100% inches, and the circumference of a 4-inch diameter pulley is 12% inches (near enough for our present purpose); three quarters of 100% is 75%, and one quarter of 12% is 3%; the length of a belt, then, to go around a 4-inch pulley and a 32-inch pulley, running at a distance of 19% inches apart, is 75% plus 3% plus 14 plus 14; total, 106% inches.  Now, let us take the middle cone, when the belt is running on two pulleys, both 18 inches diameter (see Fig. 3), and, of course, the same distance apart as before. The circumference of an 18-inch pulley is 56% inches, and half the circumference of two 18-inch pulleys is evidently the same as the whole circumference of one 18-inch pulley; the length of belt in this case will then evidently be 56% plus 19% plus 19%; total, 96 inches. It is therefore evident that a belt long enough to run on a 4- and 32-inch pulley, 19% inches apart, is 10% inches too long to run on two 18-inch pulleys 19% inches apart, and, of course, it is therefore 10% inches too long to run on the middle grades of such a cone as we have under consideration.  The thing to do, then, is to make the middle grades of these cones (or the two 18-inch pulleys) enough larger than IB inches diameter to just take up this 10% inches of belt, and if this were the only case we had to deal with, it would be very easy to settle it by saying that as half the circumference of two 18-inch pulleys is the same as the whole circumference of one 18-inch pulley, we should make the two 18-inch pulleys enough larger in diameter to make an additional circumference of 10% inches; and as 3% inches is nearly the diameter of a 10%-inch circumference pulley, by making the middle of both cones 18 plus 3% inches diameter (that is, 21% inches diameter) our trouble would be ended in this particular case. It is easy enough to see, by looking at Fig. 2, that the belt being obliged to go three quarters of the way around the large pulley, is what makes it so much too long to go around the two middle pulleys, where, of course, it goes but half way around each. But, of course, what we want is some way of calculating the diameters to turn any pair of cones, running at any distance apart.  If we were to draw these same 32- and 4-inch pulleys twice 19% inches apart, and then three times 19% inches apart, and so on, until we got them far enough apart so that the belt would practically run half way around each, and should calculate the diameter of the middle grade of the cone to fit each distance, we would probably formulate a rule that would work for any distance apart, with this particular cone; but as it is evident that the further apart the cones are to run, the nearer to the nominal diameter of 18 inches must the middle of the cones be turned, so also must it be evident that the less difference between the largest and smallest diameter of the cone, the less must also be the excess over nominal diameter of the middle of the cones.  Any method, then, of calculating such problems must take both of these things into consideration. The nominal diameter of the middle of any cone will be equal to half the sum of the diameters of the largest and smallest part respectively. This is almost self-evident, and no proof of it is necessary in this connection. What we want, then, is some way to find out how much larger than the nominal diameter to turn any one cone or cones to fit the conditions under which they are to run. The following formula is the result of a thorough investigation of this subject by Prof. Rankine, and has proved itself to be practically correct in the shop, as well as satisfactory to those mathematicians who are competent to criticise it.  STRENGTH OF COUNTERSHAFTS There is scarcely a shop in existence which has not had a more or less serious accident from a countershaft some time in its history. It may have been caused by a heavy pulley running very much out of balance, or the shaft may have been bent in the beginning. Possibly the shaft was too light, or too long between hangers. The latter is responsible for most of the trouble, and is the one with which this discussion is principally concerned.  There are two methods in vogue for turning cones and pulleys; one is to set the rough casting to run true on the inside, and the other on the outside. This latter method makes a cheaper and an easier job, but when turned, it requires an enormous amount of metal to balance it. And here is the source of considerable trouble. We may balance a large cone perfectly on straight edges, but that is a standing balance only; and when the cone is put in place and speeded up to several hundred revolutions per minute, it shakes, and shows that it is decidedly out of balance. The trouble is that we have not placed the balance weights directly opposite, or in the plane of the heavy portion of the cone. The result is that neither weight, when rotating, has its counter balance pulling in the same line, and, of course, the pulley is sure to be out of balance. All cones and all other pulleys which have a wide face should be set to run true on the inside before turning.  A certain countershaft failed because it had been welded near the center. The weld twisted and bent open, and someone was badly injured by the fall. A weld in machine steel is so very uncertain that it should never be trusted for such a purpose. The extra expense of a new shaft would not warrant the hazard of such a risk.