BELT, ROPE, AND CHAIN GEARING 365 
of the diameters, say D,, then d, ae. Having fixed D, and d,, no 
N 
other diameters can be selected arbitrarily. The other diameters must 
not only satisfy the equations 4 Pa nd as “yr but they must be 
such that the same belt will be equally tight when pl placed on correspond- 
ing pairs of steps. This gives rise to two cases, (1) belt crossed, (2) belt 
For a crossed belt it was shown in the preceding Article that if 
the sum of the diameters of the pulleys is not altered, the length of the 
belt will be the same. Therefore, for a crossed belt, having fixed D, and. 
i D, +4, is known, and D, +d,, also D,+d, must be equal to D, ‘+d. 
e sum of a pair of diameters being known, and also the ratio of the 
one to the other, the diameters can be easily found. 
Coming now to the case where the belt is an open one, D, and d, are 
determined as before, then the length of the belt is 
D,-d 
t= *(D, +d,)+ Pr-O 2 
Then for D, and d,, 
D 
1=2(D, +d,) + 
Mr = 4)’ +26, 
but d= 57s, therefore 
2 
7 N De CE 5 rth 
a quadratic equation from which - 
Dom Ye +0d +003) Ft) 
where 7, = N 
N, . 
If N,=N, then D, =", 
; T 
The solution of the quadratic equation may be avoided, and a result . 
sufficiently accurate in most cases obtained, by first finding values for D, and 
d, on the assumption that the belt is crossed, that is, D,+d,=D, +d). 
Let the difference between the — af D, and d, thus found be equal 
to a, then approximately 5(D2 +d,)+2 ry. + 2e= 1, a simple equation from 
which D, +d, can easily be found. Use this more exact value of D,+d,, 
together with dy woe to find a more accurate value of D, —d,, which 
D, N 
eee Dy =4y) 
is to be substituted in the equation (D2 +d,) +\— +2c=1, The 
value of D, +d, found from this Sle equation is fio used, together 
with the equation noe to find D, and d,. 
810. Forms of Rims of Pulleys. —The rim of a pulley for a flat belt 
