‘9 
PISTON VELOCITY AND ACCELERATION DIAGRAMS 305 
~ but the velocity or acceleration of the piston for any position of the crank 
_ may be calculated by means of the formule now to be proved. 
Referring to Fig. 467, p. 300, it is readily seen that 
v _sin (9+) _ sin 6 cos } + cos @ sin $4. 04 cos # sin 
vV cos cos p cos 
. er, in : l 
But sin ¢=7 sin d=, andcos P= ,/1 sin’ p == ./n* — sin? 6. 
Therefore ~ * =sin 0+ sin 4 cata 6 =sin 0+ sin 20 % 
' 3 V Jn? — sin? 0 2 ,/n? — sin? 0 
For values of usual in direct-acting engines it will be sufficiently 
a ae in 2 
accurate to take ,/n®—sin? 0=n, then v=V(sin A+ a) approxi- 
mately. 
From this approximate expression for v the acceleration f is found 
as follows :— 
_dv_ (dd d0 2 cos 20 : 
1-4-WG cos O+ e ae V being constant, 
= v(" cos 64”. cos 20 =V"(cos 6+ on 
?. r n r n 
267. Angular Velocity of Connecting-rod.—The connecting-rod has 
a motion of translation along with the piston, and also an angular motion, 
_ the angle ¢ which it makes with the line of stroke changing from zero 
to a maximum, and back again to zero during one. stroke of the piston. 
¢ is evidently a maximum when the crank is perpendicular to the line of 
stroke, and it is zero when the crank is on the line of stroke. 
Referring to Fig 476, O is the instantaneous centre of the connect- 
ing-rod when in the position shown, and if BC represents V, the velocity 
of the crank pin, CD represents 
v, the velocity of the piston. 0! 
Imagine a velocity equal to v bane ' 
to be impressed on the connect- 
ing-rod in the direction~ CA. 
The point A will now be at A 
rest, and the connecting-rod will 
only have angular motion. The 
point B has now a velocity o ems 
which is the resultant of the 4% \ c 
velocities BC’ perpendicular to Fia. 476. 
BC, and=V=BC, and C’D’ 
parallel to CA, and=v=CD. This resultant will evidently be per- 
pendicular to AB and=BD. The angular velocity of AB in the 
given position is therefore equal to BD’/AB=BD/AB, and as AB is 
constant, the angular velocity is represented by BD. 
The foregoing result is also obvious when it is remembered that, at 
the instant considered, the connecting-rod is rotating about O, and its 
angular velocity about O is equal to BC’/OB=BC/OB=BD/AB, and 
this will also be the rate of change of the angle ¢. 
The angular velocity of AB may be plotted on the crank CB from 
the pole C, or on the piston or cross-head stroke as a base, but preferably 
U 
