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CCCXX.

AN IMPROVED METHOD FOR FINDING THE DIAMETERS OF CONE AND STEP PULLEYS.

BY C. A. SMITH, PAWTUCKET, R. I.

(Member of the Society.)

THE diameters of cone and step pulleys must have certain relations to each other in order to make the belt run on them under the same amount of tension in all positions on the pulley. To determine these diameters correctly is the object of this paper.

The discussion will be divided in this paper into two parts for the convenience of the reader. The first will consist of brief rules, for the practical business man, for finding the correct diameters of cone and step pulleys. The second will give a brief history and analysis of the method.

It will be noticed that the order of the subjects is here reversed from that usually followed. When the rules and conclusions are interspersed all through the analysis it takes too much time for the hurried business man to boil the conglomeration down to the consistency required for his use; the consequence is that he loses much valuable matter because he has not the time to "dig for it." The endeavor will be to wait on the business man first and do the digging for him to some extent. The student and those wishing to know the why-wherefore-and-whence are referred to the latter part of this paper.

GRAPHICAL METHOD FOR CONE PULLEYS WITH OPEN BELT.

Case I. When the Greatest Belt Angle does not Exceed 18°.

1. Referring to Figs. 36 and 37: Lay off the center distance, C or EF, and draw the circles, D1 and d1, equal to the first pair of pulleys which are always previously determined by known conditions.

2. Draw the line HI tangent to the circles D, and d1.

3. From the point B, midway between E and F, erect BG perpendicular to EF.

4. Locate the point G by making BG=.314C; that is, multiply

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the center line, EF, by .314 and the product will be the height, BG, sought.

5. With G as a center draw a circle tangent to the line HI. Generally this circle will be on the outside of the belt line, HI, as shown in Fig. 36. When the center line, C, is short, and the

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first pulleys, D, and d1, are large, the circle will fall on the inside of the belt line, as shown in Fig. 37.

6. The belt line of any other pair of pulleys must be tangent to the circle G; hence any line as JK or LM drawn tangent to the

circle G will give the diameters D2, da or D, ds of the pulleys drawn tangent to these lines from the centers E and F.

7. To find any pair of diameters which will give any desired

D

d

velocity ratio, let r= ; that is, r is the ratio obtained by dividing the larger diameter by the smaller one, or by dividing the larger of the desired speeds of the shafts by the speed of the other one.

C

Locate the point K or M by making EK or FM=„_1; that is, subtract one from the desired ratio, r, and divide the center dis

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tance

Fig. 38.

by the remainder, the quotient will be the distance from E or F to a point K or M. From this point draw the line KJ or ML tangent to the circle G. The circles drawn tangent to this line with E and F as centers will give the desired velocity ratio, r, of the shafts E and F.

8. When the velocity ratio is near to unity, in which case the belt angle, A, is small, the point K falls so far away from E that it becomes very inconvenient, or impossible, to make use of this point K. In that case the following method will be available :Use a centrolinead, shown in Fig. 40, or improvise one as shown in Fig. 39, by fastening a piece of tracing material to a straight edge and drawing the lines OP and PN. To find the location o

C

the points N, O and P, make EN-E0-10(r-1); that is, one

tenth of the distance, EK, Figs. 36 and 37, as calculated above in

EN
10

rule 7. Now make EP- and draw the lines through the

points N, P, 0. On the diagram of the pulleys, as in Fig. 38, lay off the points and O from the small pulley, making EN and EO the same as those distances in Fig. 39 or 40. Now place the centrolinead, Fig. 39 or 40, upon the diagram, Fig. 38, so that the line NP coincides with the point N, Fig. 38, the line OP with the point 0, and the line or edge PQ tangent to the circle G as shown in Fig. 38. This gives the location of the belt line, PQ, to

Fig.39.

which the circles from the centers E and F are drawn tangent to give the pulley diameters of the desired ratio, r.

If one-tenth of EK, Figs. 36 and 37, should be too large to get on the drawing board, then any other convenient fractional part of EK may be taken for EN, Figs. 39 and 40, remembering, however, that the same fractional part of EN, Figs. 39 and 40, should be taken for EP. In other words, EN, Figs. 38, 39 and 40, must be a mean proportional between EK, Figs. 36 and 37, and EP, Figs. 39 and 40.

9. When the ratio, r, is unity, the belt line will be drawn parallel to the center line EF and tangent to the circle G, and its distance from the line EF will be half the diameter of the pulleys.

Case II. When the Greatest Belt Angle Lies Between
18° and 30°.

Not more than 18° of arc of the directing circle G, Figs. 36 and 37, should be used on each side of the vertical line BG. When the cone pulleys are so proportioned that the belt angle, A, be

comes greater than 18°, then the directing circle (or curve in this case) should be composed of two circular arcs on each side of the vertical line BG. We will only show the directing curve in this

E

Fig. 40.

of

case, somewhat distorted in Figs. 41, 42 and 43, for the purpose showing the different parts distinctly. Corresponding points and lines are designated by the same letters in all the diagrams for convenience in making comparisons. Description of one diagram applies equally to the others.

10. The process in this case is the same as in Case I., with the exception of a slight modification in rules 4 and 5. In addition

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to the point G, as located by rule 4, locate another point, m, Figs. 41, 42 and 43, by making Bm=.298 C. As seen in these diagrams the two points G and m may fall outside the belt line, HI, Fig. 41, corresponding to Fig. 36, or they may fall inside, Fig. 42,

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