426 APPLIED MECHANICS 
axle carrying revolving masses equal to the portions of the reciprocating 
masses to be balanced in the wheels of that axle. The revolving masses — 
of the imaginary cranks are of course neglected. a ae 
Another way of proceeding is to find the separate masses necessary in 
the driving wheels to balance the reciprocating masses which are to be 
balanced and then divide these up into parts, which are placed in similar 
positions in the various wheels. For example, if A, and B, (Fig. 696) 
are the masses in the left-hand and right-hand driving wheels respectively 
which will balance, say, two-thirds ‘of . 
the reciprocating masses, and if equal 
portions of A, and B, be transferred 
to A, and B, in the left-hand and 
right-hand coupled trailing wheels, the 
radii from A, and B, being equal and 
parallel to the radii from A, and B, Fic. 696 
respectively, then this new distribution eae 
of balance weights will have the same effect horizontally in balancing 
the reciprocating parts, but vertically there will now be a less variation in 
the pressure on the rails per wheel. The balance weights thus found in 
the wheels to balance the reciprocating parts are then combined with the 
balance weights for the revolving masses, and the resultant balance weights 
determined. 
359. Complete Balancing of Reciprocating Parts having Harmonic 
Motion.—In Art. 351 it was shown that the disturbing forces due to 
the acceleration of the reciprocating parts are the same as those produced 
in the line of stroke by a revolving mass equal to that of the recipro- 
cating parts concentrated at the crank pin, but this revolving mass 
produces equal disturbing forces at right angles to the line of stroke. 
Now suppose a number of sets of reciprocating masses to be connected 
to the same number of cranks on a shaft. Next suppose that these re- 
ciprocating masses are removed, and masses equal to them are concentrated 
at their respective crank pins. The disturbing forces in the various lines 
of stroke will now be the same as before, but if the various imaginary 
revolving masses at the crank pins be of such magnitudes, and if their 
relative positions be such that they balance one another, then it is obvious 
that not only will the disturbing forces in the lines of stroke balance one 
another, but the disturbing forces in the directions at right angles to the 
lines of stroke will also balance one another. : 
The problem of the complete balancing of reciprocating masses there- 
fore reduces to that of balancing a number of revolving masses, a problem 
which was discussed in Art. 349, It must, however, be remembered that the 
revolving masses now being considered are imaginary, and that reczpro- 
cating masses can only be completely balanced by other reciprocating masses. 
A few cases will now be considered in illustration of the foregoing. 
First, take the case of a single-cylinder engine (Fig. 697), the piston 
being connected to a crank pin A revolving in a circle of radius 7. Let 
B and C be two other crank pins revolving in circles of radii r, and ry 
respectively, the planes of revolution of these pins being at distances 
6 and ¢ respectively from the plane of revolution of the crank pin A. 
Let w be the weight of the reciprocating masses connected to the crank 
pin A. Then, if the crank pins B and C be in the same plane with the 
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