1919 .] 
Jenkinson.—Balancing of Locomotives. 
19 
BALANCING OF LOCOMOTIVES. 
By S. H. Jenkinson, Railway Department. 
The balancing of multi-cylinder engines is a branch of exact science that 
has been exhaustively treated, and is now on a firm and logical basis ; but 
the balancing of the ordinary locomotive is a problem that admits of no 
exact solution. It is a matter of compromise between three conflicting 
demands, and mathematics has to be blended with experience. For 
these reasons the problem has suffered somewhat in the application of 
the now so fashionable graphic methods to its solution. The work of 
Dalby,* * * § Sharp,f and AhronsJ in England, and of Henderson§ in America, 
has not been founded on the broad views that are necessary if formulae 
of universal application are to be derived ; and the writer has had to 
considerably extend and broaden their treatment of the subject before 
difficulties could be surmounted in certain special cases. The only dis¬ 
turbances considered in this paper are those that can be minimized by 
balance-weights rotating with the wheel. This restricts us to the inertia 
forces of parts rotating or reciprocating in unison with the crank, since 
the forces arising from all other causes, such as varying steam-load, torque, 
and drawbar-pull, are not in phase with the revolutions of the engine. 
These inertia forces are negligible at low speeds, but at high speeds they 
seriously and sometimes dangerously impair the smooth running of the 
locomotive. 
Unbalanced revolving masses (and such are the balance-weights, side- 
rods, crank-pins, crank-webs, or crank-bosses) exert centrifugal force along 
the instantaneous radii of their centres of gravity. This force is given by 
Centrifugal force in pounds = 
MU 
3-207 
MV 2 r 
gr / d 2 
where M = equivalent revolving mass at crank-radius in pounds, V = speed 
in miles per hour, r = crank-radius in inches, d = diameter of driving- 
wheel in inches. 
The equivalent mass is found by the formula 
r 
where = actual revolving mass in pounds, r 1 = radius of centre of 
gravity of Mj in inches. 
Unbalanced reciprocating masses exert horizontal forces given to the 
first approximation by 
Horizontal force of reciprocating mass in pounds 
MV 2 r , 1 
= 3-207 —| cos 6 + m co 2 0\ 
where 0 = instantaneous crank-angle, 
, • i -. crank-radius 
m = ratio less than 1 — - ^ - ry- • 
connectmg-rod length 
The exact treatment of the connecting-rod is not simple, but a useful 
approximation consists in considering -J of the mass supported by the 
* W. E. Dalby, The Balancing of Engines, 1901. 
t Archibald Sharp, Balancing of Engines, 1907. 
j E. L. Ahrons, The Internal Disturbances in and Balancing of Locomotives, 
Trans. Inst. Loco. Engrs., 29th paper, 1914. 
§ G. R. Henderson, Locomotive Operation, 2nd ed., 1907. 
