364 APPLIED MECHANICS a 
art 2) = v(= + #). The length of belt in contact with the smaller 
pulley is Ses 2$)=a(% +4), where the + sign applies to the crossed 
belt (Fig. 554), and the — sign to the open belt (Fig. 555). The in- 
clination of the straight portions of the belt to the line of centres is ¢, 
and the length of each straight portion is ccos¢. Hence if / is the 
total length of the belt, 
1=D(Z+$)+d(5 4) +2c.cos g 
= (D+) + $(D £4) + 2c cos > 
D+ 
c 
sin d= 
» cosd=1 -2 sin? $, 
If ¢ is a small angle, then, approximately, ¢=sin ¢, and 
i) \ 
$sin d= sin 3 
Hence approximately 
1=(D-+d)+sin $(D +d) + 2e(1 — 2 sin? 4 
_t (Did)? 5, D+dP 
T de de bgieeaae Cy alae 
=7(D+a)+O2O , 2c, where the + sign applies to 
the crossed belt, and the — sign to the open belt. 
Referring to the crossed belt (Fig. 554), it is evident that if D+d 
is constant, and ¢ is fixed, ¢ will remain the same, and therefore J is 
constant. 
309. Stepped Pulleys.—Two or more pulleys of different diameters 
placed side by side form a stepped pulley. A stepped pulley is, however, 
generally ‘cast in one piece. 
A pair of stepped pulleys and one belt form a common arrangement 
for driving a shaft or spindle at different speeds from a shaft rotating at 
a fixed speed. Fig. 556 shows a pair of stepped pulleys mounted on shafts 
A and B, whose axes are parallel and at a distance c apart. Let the speed 
of A be N revolutions 
per minute, and let it ee REE oe, 
be required to make B o,[ dy 
rotate at N,, N,, or a 
N, revolutions ; per "alan : J 4d 
minute as may be ne- Ly J Cs 3 
cessary. Each pulley A 
will requirethree steps. Fie. 556. 
Let the diameters of 
the pulley on A be D,, D,, and Dg, and let the diameters of the pulley on 
B be d,, d,, and d,. The following equations can be stated at once, viz. — 
of Be ok and A value may now be selected for one 
ay on Pale 
1 N, D, 2 D, N; 
a 
