126 [Aug. 17, 
Briggs. | 
The Deviating Forces of an Unsymmetrically Balanced Ply-wheel. 
Mr. Briggs mentioned that he had not found in the text books of applied 
or practical Mechanics—Morin, Rankine, Weisbach, Fairbairn or others— 
any proper consideration had been given to the strains on the axis of a 
fly-wheel, which, correctly balanced with regard to the gravity of 
its masses, and also in the plane of rotation, yet without symmetry 
of position or mass of the balanced parts,is then accelerated or retarded to 
meet the usual requirements of a regulator of power. The fact that a 
fly-wheel must be balanced in one plane to run without vibratory effect at 
any given speed and when thus balanced the centrifugal forces of the parts 
will be in equilibrium and the axis permanent is fully stated by all recent 
writers, but the condition of permanency of axis when an unsymmetrically 
balanced fly-wheel gives out orabsorbs force has not been discussed. 
The following elementary case shows the proposition distinctly : Let it 
be supposed that a fly-wheel were formed of a pair of unequal weights at 
the extremities of arms (radii) of such length as will place the axis in the 
centre of gravity of the system, thus : 
M | m 
In pe (6) 
V ff, | R Vv 
Where M m = the’ masses andr r= the radii. Let V,, V, and v,,V, re- 
present the two velocities. The admitted energy from the change of ve- 
locities of the masses is thus expressed by the equation— 
F =[M (V,2—V¥) + m(v—v,) ] + 2g (1) 
But from the condition of balancing m = M; i= Vins and v, = Vin 
he i Re, Re : : 
B= (MW2=V,.) + WG V4G? VG) = 22 @) 
nr) R 2 9 Dh 
R= Mia Sen Oi Pl + 2g () 
Showing that the ratio of force given out by the two halves of the fly- 
wheel under any change of velocity, during any instant of time, will be 
unity, and the axis be in equilibrium, when 1 = R = rand in no other 
case, and the masses and velocities become equal in the same Case. 
This condition of unsymmetrical balancing of fly-wheels is by no means 
an unusual one. The castings of fly-wheels of steam engines and more 
especially of pulleys for transmission of force which act generally more or 
less as fly- wheels, are rarely of such uniformity as not to require balancing, 
—nearly always done on the rim of the wheel, regardless of point of in- 
equality, which is more frequently in the arms than in the rim. 
Perhaps the most striking instance is the case of the vertical blowing en- 
gine, where the whole weight of the pistons, crossheads and rods rests 
upon crank pins inserted in the arms of two fly-wheels at points from one- 
fourth to-one-third the radii of the rim, which weight is counteracted by 
a suitable load at the rim opposite the crank pins. It is then found that 
much less load is needed to give comparative steadiness of motion than 
