BALANOING 419 
midway between A and B, and one in a plane Q midway between © and D. 
Find the weights of the balancing masses and the angular positions of the 
_ radii from the axis to their centres of gravity with respect to the reference 
— radius OX. 
‘4. Four masses, A, B, O, and D, weighing 70, 90, 120, and « Ibs, respectively, 
" revolve about an axis, each at a radius of 1 foot, in planes which are at equal 
intervals apart. Determine x and the angular positions of B, ©, and D in rela- 
_ tion to that of A in order that the masses may ce one another. 
} 350. Disturbing Forces due to Acceleration of Reciprocating 
Parts.—Consider the steam-engine mechanism shown in Fig. 687, where 
the connecting-rod is of the slotted bar type, which is the equivalent of 
_ the connecting-rod of infinite length. The resultant of the steam 
_ pressures on the cylinder ends is a force P, which tends to move the 
engine frame to the left. The 
+ 
_ resultant of the steam pressures ——— | Ry 
on the piston is a force Q, equal. & Z rR bly 
to P, which tends to move the call 9 7 fs sS 
piston to the right, J x 
If the piston is at rest,or 
VAM 
tr 
if it is moving with uniform J] a 
velocity, there will be a thrust iy. ei 
R on the crank pin equal to Q gh i 
and to P. If forces S and T, each equal to R, be applied to the crank 
_ shaft in the plane of revolution of the crank pin, as shown, then R and 
Twill form an effort couple whose moment is the turning moment on the 
_ shaft, and which will be balanced by the resistance couple. The remain- 
ing force 8 will be the thrust of the shaft on its bearings carried by 
_ the frame. Hence the frame is pushed to the right by a force S, and 
_ atthe same time it is pushed to the left bya force P, which is equal to §; 
there is therefore no tendency for the frame to move on its foundations. 
_ __- If, however, the piston is increasing or decreasing in speed, the force 
__R, and therefore the force §, will be less or greater than P, and there 
_ will therefore be a resultant force on the frame tending to move it to the 
left or right. The force tending to move the frame on its foundations is 
the difference between the forces P and §, and this is evidently the force 
necessary to accelerate the piston, and this force can be determined with- 
out any reference to the steam pressures within the cylinder. For the 
mechanism shown in Fig. 687, if the crank shaft is rotating with uniform 
velocity, the piston and the parts reciprocating with it have harmonic 
_ motion. When the piston is at a distance x from the middle of its stroke, 
_ the force in lbs. required to give the reciprocating parts the necessary 
] ‘acceleration is WV _ Wo% _ Wor cos ® (414 959, p, 298), where W 
; f J g 
is the weight of the reciprocating parts in lbs., V the velocity of the 
_ erank pin in feet per second, » the angular velocity of the crank in 
radians per second, r the radius of the crank or half the stroke of the 
_ piston in feet, and @ the inclination of the crank to the line of stroke 
ig. 687). 
: During the first half of a stroke the accelerating force on the piston varies 
_ uniformly from Ww*r/g to zero, and acts in the direction of the motion 
of the piston. During the second half of the stroke the accelerating force 
_ Varies from zero to Ww*/g, and acts in the opposite direction, that is, the 
YO 
