[Footnote: Read before Section G of British Association.]
By Professor W.C. UNWIN.
[Illustration: Fig. 1.]
In the ordinary strap dynamometer a flexible band, sometimes carrying
segments of wood blocks, is hung over a pulley rotated by the motor, the
power of which is to be measured. If the pulley turns with left-handed
rotation, the friction would carry the strap toward the left, unless the
weight, Q, were greater than P. If the belt does not slip in either
direction when the pulley rotates under it, then Q-P exactly measures
the friction on the surface of the pulley; and V being the surface
velocity of the pulley (Q-P)V, is exactly the work consumed by the
dynamometer. But the work consumed in friction can be expressed in
another way. Putting [theta] for the arc embraced by the belt, and [mu]
for the coefficient of friction,
Q/P = [epsilon]^{[mu]^{[theta]}},
or for a given arc of contact Q = [kappa]P, where [kappa] depends only
on the coefficient of friction, increasing as [mu] increases, and _vice
versa_. Hence, for the belt to remain at rest with two fixed weights, Q
and P, it is necessary that the coefficient of friction should be
exactly constant. But this constancy cannot be obtained. The coefficient
of friction varies with the condition of lubrication of the surface of
the pulley, which alters during the running and with every change in the
velocity and temperature of the rubbing surfaces. Consequently, in a
dynamometer in this simple form more or less violent oscillations of the
weights are set up, which cannot be directly controlled without
impairing the accuracy of the dynamometer. Professors Ayrton and Perry
have recently used a modification of this dynamometer, in which the part
of the cord nearest to P is larger and rougher than the part nearest to
Q. The effect of this is that when the coefficients of friction
increase, Q rises a little, and diminishes the amount of the rougher
cord in contact, and _vice versa_. Thus reducing the friction,
notwithstanding the increase of the coefficient. This is very ingenious,
and the only objection to it, if it is an objection, is that only a
purely empirical adjustment of the friction can be obtained, and that
the range of the adjustment cannot be very great. If in place of one of
the weights we use a spring balance, as in Figs. 2 and 3, we get a
dynamometer which automatically adjusts itself to changes in the
coefficient of friction.
[Illustration: FIG.2 FIG.3]
For any increase in the coefficient, the spring in Fig. 2 lengthens, Q
increases, and the frictional resistance on the surface of the pulley
increases, both in consequence of the increase of Q, which increases the
pressure on the pulley, and of the increase of the coefficient of
friction. Similarly for any increase of the coefficient of friction, the
spring in Fig. 3 shortens, P diminishes, and the friction on the surface
of the pulley diminishes so far as the diminution of P diminishes the
normal pressure, but on the whole increases in consequence of the
increase of the coefficient of friction. The value of the friction on
the surface of the pulley, however, is more constant for a given
variation of the frictional coefficient in Fig. 3 than in Fig. 2, and
the variation of the difference of tensions to be measured is less. Fig.
3, therefore, is the better form.
A numerical calculation here may be useful. Supposing the break set to a
given difference of tension, Q-P, and that in consequence of any cause
the coefficient of friction increases 20 per cent., the difference of
tensions for an ordinary value of the coefficient of friction would
increase from 1.5 P to 2 P in Fig. 2, and from 1.5 P to 1.67 P in Fig.
3. That is, the vibration of the spring, and the possible error of
measurement of the difference of tension would be much greater in Fig. 2
than in Fig. 3. It has recently occurred to the author that a further
change in the dynamometer would make the friction on the pulley still
more independent of changes in the coefficient of friction, and
consequently the measurement of the work absorbed still more accurate.
Suppose the cord taken twice over a pulley fixed on the shaft driven by
the motor and round a fixed pulley, C.
For clearness, the pulleys, A B, are shown of different sizes, but they
are more conveniently of the same size. Further, let the spring balance
be at the free end of the cord toward which the pulley runs. Then it
will be found that a variation of 20 per cent. in the friction produces
a somewhat greater variation of P than in Fig. 3. But P is now so much
smaller than before that Q-P is much less affected by any error in the
estimate of P. An alteration of 20 per cent. in the friction will only
alter the quantity Q-P from 5.25 P to 5.55 P, or an alteration of less
than 6 per cent.
[Illustration: FIG. 4]
To put it in another way, the errors in the use of dynamometer are due
to the vibration of the spring which measures P, and are caused by
variations of the coefficient of friction of the dynamometer. By making
P very much smaller than in the usual form of the dynamometer, any
errors in determining it have much less influence on the measurement of
the work absorbed. We may go further. The cord may be taken over four
pulleys; in that case a variation of 20 per cent. in the frictional
coefficient only alters the total friction on the pulleys 1¼ percent. P
is now so insignificant compared with Q that an error in determining it
is of comparatively little consequence.
[Illustration: FIG. 5]
The dynamometer is now more powerful in absorbing work than in the form
Fig. 3. As to the practical construction of the brake, the author thinks
that simple wires for the flexible bands, lying in V grooves in the
pulleys, of no great acuteness, would give the greatest resistance with
the least variation of the coefficient of friction; the heat developed
being in that case neutralized by a jet of water on the pulley. It would
be quite possible with a pulley of say 3 feet diameter, and running at
50 feet of surface velocity per second, to have a sufficiently flexible
wire, capable of carrying 100 lb. as the greater load, Q. Now with these
proportions a brake of the form in Fig. 3 would, with a probable value
of the coefficient of friction, absorb 6 horse power. With a brake in
the form Fig. 4, 8.2 horse power would be absorbed; and with a brake in
the form Fig. 5, 8.8 horse power would be absorbed. But since it would
be easy to have two, three, or more wires side by side, each carrying
its load of 100 lb., large amounts of horsepower could be conveniently
absorbed and measured.
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