PISTON AND CRANK EFFORT DIAGRAMS 317 
eros head, Q, the thrust or pull on the connecting-rod, is equal to P/cos¢. 
At the crank pin the force Q produces a thrust or pull on the crank and 
a force T tangential to the path of the crank pin equal to Q sin (6 + ¢). 
. 3 Hence, T= ca = P(sin 6 + cos 6 tan ¢). 
Tf n is the ratio of the length of the connecting-rod to the radius of 
sin 0 
the crank, then tan d= Rr and therefore 
sin ™ cos 0 ‘ sin 20 
T =P {sin 6+ ag} -P sin 0+ 5 Jara Op. 
T is called the crank pin effort. The moment of T about C, namely, 
Tr, where r is the radius of 
the crank, is called the crank 
effort, but as r is constant, 
it follows that the crank 
effort is proportional to T. 
If Cd be made equal to P, 
and bd be drawn parallel 
to AB to meet Cd at d, Fig. 494, 
where Cd is perpendicular to the line of stroke, then 
Cd_ sin(O+¢) _ sin(0+¢) 
Cb sin(90°-¢) cosh” 
_sin(0+¢) © 
bu - =e 
must a equal to T. The construction for determining the crank effort 
from the piston or cross-head effort is therefore extremely simple, and if 
it be compared with the construction proved in Art. 260 for finding the 
piston velocity from the crank pin velocity, it will be seen that the con- 
structions are identical. In fact, the construction and formula for the 
- erank effort may be deduced at once from the construction and formula 
for piston velocity by the principle of work. 
277. Crank Effort Diagrams.—The construction of diagrams which 
____ Shall show the crank effort for any position of the crank will be readily 
. - understood by reference to Fig. 495. In the polar curves of crank effort, 
the effort found by the construction or by the formula of the preceding 
; 
3 
5 
Gd T 
» therefore A> Gh P: Hence, since Cd is equal to P, Cd 
Article is marked off, either on the crank from the centre of the crank shaft, 
or on the crank produced from the path of the crank pin. The most 
useful crank effort diagram is the ‘rectangular diagram,” in which the 
base is a straight line representing the circumference of the circle described 
by the crank pin, and the ordinates, perpendicular to that base, represent 
the crank effort. 
If the base of the rectangular diagram of crank effort be made equal 
to the circumference of the circle described by the crank pin, then, friction 
being neglected, the area of the crank effort diagram for one revolution 
will be equal to the sum of the areas of the piston effort diagrams, but 
practically all that is to be learned from the rectangular crank effort 
diagram can be learned from it, whatever be the length taken for the base, 
The principal use of the rectangular crank effort diagram is to show 
