POWER TRANSMISSION ELEMENTS AND MECHANISMS 169 



FLAT -BELT LENGTHS AND PULLEY DLA.METERS 



The chart on the following page is used for the 

 calculation of belt lengths for open belt drives, 

 step cone pulley sizes for open belts, and pulley 

 diameters on V-belt drives. 



The length of a belt can be calculated with 

 sufficient accuracy for all engineering problems 

 from the formula 



rf2(n - 1)2 



L = 2C + ^ rf(n + 1) + 



4C 



(32) 



where L = belt length 



C = distance between pulley centers 



d = smaU pulley diameter 



n = speed ratio 



D = large pulley diameter 



D = nd 



Any type of graphical solution of the equation 

 is not simple in this form, because there are four 

 variables in it and they cannot be shown in a 

 simple chart. If the Eq. (32) is divided by C, it 

 will take the form as follows : 



^ = 2+^^(n + l)+g(n- 1)2 (33) 



For further simplification, let L/C = x and 

 d/C = y. The equation will then become 



X = 2 +-y{n 



1) + y\n - ly (34) 



Equation (34) contains only three variables, of 

 which n, the speed ratio, is usually known. The 

 equation can be plotted on ordinary coordinate 

 paper as in the accompanying chart. The fol- 

 lowing examples show how to use it. 



Belt Length for Open Drive 



Example. — Assume the small pulley diameter 

 d = 5 in., the speed ratio n = 4, and the dis- 

 tance between pulley centers C = 50 in. Then 

 d/C = ^io = 0.10. From d/C = 0.10 on the 

 chart, trace horizontally to the speed ratio 

 n = 4 and follow vertically downward to read 

 L/C = 2.81. Therefore 



L = C X 2.81 



L = 50 X 2.81 = 140.5 in. 



Substituting the numerical values given in this 

 example in the Eq. (32), the solution will be 



52(4 - 1)2 



L 



100-1-^5(4+1)+ ^^^^ 



= 140.375 in. 



Although there is '^i in. difference in the belt 

 length L as obtained from the chart figures and 



by Eq. (32), the chart values are close enough for 

 all ordinary belt length calculations. 



Calculation of Step Cone Pulley Drives 



Example. — A four-step cone pulley drive is 

 required with speed ratios n of 2, 3, 4, and 5. 

 Assume that one speed ratio, namely, n = 4, and 

 that the diameter of the small pulley d = Bin. is 

 the same as in the preceding example. Center 

 distance is C = 50 in., and the belt length is 

 L = 140.375 in. The value of L/C = 2.81 will 

 be the same in each instance. 



For the speed ratio n = 2, read vertically from 

 L/C = 2.81 to where this line intersects the ray 

 of the speed ratio 2. Follow horizontally to 

 d/C, and read 0.17. When d/C = 0.17, then 



d = 0.17 X 50 in. = 8.5 in. 

 D = 2 X 8.5 in. = 17 in. 



For the speed ratio n = 3, d/C = 0.126 is 

 obtained from the chart in a similar manner. 

 Therefore : 



d = 0.126 X 50 = 6.3 in. 

 Z) = 3 X 6.3 in. = 18.9 in. 



For the speed ratio n = 4, as in the preceding 

 example, d/C = 0.10 and d = 5 in. Then 



D = 4 X 5 in. = 20 in. 



For the speed ratio ?i = 5, d/C = 0.083 on 

 the chart so that 



d = 0.083 X 50 = 4.15 in. 

 D = 5 X 4.15in. = 20.75 in. 



In this instance, the steps of the driven pulley 

 ■\vill be 4.15, 5, 6.3, and 8.5 in. diameter, mating 

 with steps on the driving pulley of 20.75, 20, 

 18.9, and 17 in. diameter, respectively. 



Pulley Diameters for V-belt Drive 



Example. — If the pitch length L of an endless 

 V-belt is 120 in., the speed ratio n = 4, and the 

 distance between centers C = 40 in., find the 

 pitch diameters of the pullej^s. 



If L/C = 3, then d/C = 0.1216 is read at the 

 intersection of the lines L/C and speed ratio 

 M = 4. Therefore 



d = 0.1216 X 40 = 4.864 in. 

 D = 4.864 X 4 = 19.456 in. 



A V-belt manufacturer's catalogue is then con- 

 sulted to ascertain pulley outside diameters. 



