The factor [theta] of the product T[theta] is found by means of a
revolution counter. The power of a motor is measured by the rate at
which it works, and this is expressed by T[omega] = T2[pi]N/60 in
foot-pounds per second, or T2[pi]N/33,000 in horse-power units. The
latter is commonly referred to as the "brake horse-power." The
maintenance of the conditions of steadiness implied in equation (1)
depends upon the constancy of F, and therefore of the coefficient of
friction mu between the rubbing surfaces. The heating at the surfaces,
the variations in their smoothness, and the variations of the
lubrication make [mu] continuously variable, and necessitate frequent
adjustment of W or of the nuts. J.V. Poncelet (1788-1867) invented a
form of Prony brake which automatically adjusted its grip as [mu]
changed, thereby maintaining F constant.
The principle of the compensating brake devised by J.G. Appold
(1800-1865) is shown in fig. 1. A flexible steel band, lined with wood
blocks, is gripped on the motor fly-wheel or pulley by a screw A,
which, together with W, is adjusted to hold the brake steady.
Compensation is effected by the lever L inserted at B. This has a
slotted end, engaged by a pin P fixed to the framing, and it will be
seen that its action is to slacken the band if the load tends to rise
and to tighten it in the contrary case. The external forces holding
the brake from turning are W, distant R from the axis, and the
reaction, W1 say, of the lever against the fixed pin P, distant R1
from the axis. The moment of W1 may be positive or negative. The
torque T at any instant of steady running is therefore {WR +- W1R1}.
[Illustration: FIG. 1.]
Lord Kelvin patented a brake in 1858 (fig. 2) consisting of a rope or
cord wrapped round the circumference of a rotating wheel, to one end
of which is applied a regulated force, the other end being fixed to a
spring balance. The ropes are spaced laterally by the blocks B, B, B,
B, which also serve to prevent them from slipping sideways. When the
wheel is turning in the direction indicated, the forces holding the
band still are W, and p, the observed pull on the spring balance. Both
these forces usually act at the same radius R, the distance from the
axis to the centre line of the rope, in which case the torque T is (W
- p)R, and consequently the brake horse-power is
(W - p)R x 2[pi]N
-----------------.
33,000
When mu changes the weight W rises or falls against the action of the
spring balance until a stable condition of running is obtained. The
ratio {W/p} is given by e^{ mu[theta]}, where e = 2.718; mu is the
coefficient of friction and [theta] the angle, measured in radians,
subtended by the arc of contact between the rope and the wheel. In
fig. 2 [theta] = 2[pi]. The ratio W/p increases very rapidly as
[theta] is increased, and therefore, by making [theta] sufficiently
large, p may conveniently be made a small fraction of W, thereby
rendering errors of observation of the spring balance negligible. Thus
this kind of brake, though cheap to make, is, when [theta] is large
enough, an exceedingly accurate measuring instrument, readily applied
and easily controlled. It has come into very general use in recent
years, and has practically superseded the older forms of block brakes.
[Illustration: FIG. 2.]
It is sometimes necessary to use water to keep the brake wheel cool.
Engines specially designed for testing are usually provided with a
brake wheel having a trough-shaped rim. Water trickles continuously
into the trough, and the centrifugal action holds it as an inside
lining against the rim, where it slowly evaporates.
Fig. 3 shows a band-brake invented by Professor James Thomson,
suitable for testing motors exerting a constant torque (see
_Engineering_, 22nd October 1880). To maintain e^{ mu[theta]} constant,
compensation for variation of [mu] is made by inversely varying
[theta]. A and B are fast and loose pulleys, and the brake band is
placed partly over the one and partly over the other. Weights W and w
are adjusted to the torque. The band turns with the fast pulley if
[mu] increase, thereby slightly turning the loose pulley, otherwise at
rest, until [theta] is adjusted to the new value of [mu]. This form of
brake was also invented independently by J.A.M.L. Carpentier, and the
principle has been used in the Raffard brake. A self-compensating
brake of another kind, by Marcel Deprez, was described with
Carpentier's in 1880 (_Bulletin de la societe d'encouragement_,
Paris). W.E. Ayrton and J. Perry used a band or rope brake in which
compensation is effected by the pulley drawing in or letting out a
part of the band or rope which has been roughened or in which a knot
has been tied.
In an effective water-brake invented by W. Froude (see _Proc. Inst. M.
E._ 1877), two similar castings, A and B, each consisting of a boss
and circumferential annular channel, are placed face to face on a
shaft, to which B is keyed, A being free (fig. 4). A ring tube of
elliptical section is thus formed. Each channel is divided into a
series of pockets by equally spaced vanes inclined at 45 deg.. When A is
held still, and B rotated, centrifugal action sets up vortex currents
in the water in the pockets; thus a continuous circulation is caused
between B and A, and the consequent changes of momentum give rise to
oblique reactions. The moments of the components of these actions and
reactions in a plane to which the axis of rotation is at right angles
are the two aspects of the torque acting, and therefore the torque
acting on B through the shaft is measured by the torque required to
hold A still. Froude constructed a brake to take up 2000 H.P. at 90
revs. per min. by duplicating this apparatus. This replaced the
propeller of the ship whose engines were to be tested, and the outer
casing was held from turning by a suitable arrangement of levers
carried to weighing apparatus conveniently disposed on the wharf. The
torque corresponding to 2000 H.P. at 90 revs. per min. is 116,772
foot-pounds, and a brake 5 ft. in diameter gave this resistance. Thin
metal sluices were arranged to slide between the wheel and casing, and
by their means the range of action could be varied from 300 H.P. at
120 revs. per min. to the maximum.
[Illustration: FIG. 3.]
[Illustration: FIG. 4.]
Professor Osborne Reynolds in 1887 patented a water-brake (see _Proc.
Inst. C.E._ 99, p. 167), using Froude's turbine to obtain the highly
resisting spiral vortices, and arranging passages in the casing for
the entry of water at the hub of the wheel and its exit at the
circumference. Water enters at E (fig. 5), and finds its way into the
interior of the wheel, A, driving the air in front of it through the
air-passages K, K. Then following into the pocketed chambers V1, V2,
it is caught into the vortex, and finally escapes at the
circumference, flowing away at F. The air-ways k, k, in the fixed
vanes establish communication between the cores of the vortices and
the atmosphere. From {1/5} to 30 H.P. may be measured at 100 revs. per
min. by a brake-wheel of this kind 18 in. in diameter. For other
speeds the power varies as the cube of the speed. The casing is held
from turning by weights hanging on an attached arm. The cocks
regulating the water are connected to the casing, so that any tilting
automatically regulates the flow, and therefore the thickness of the
film in the vortex. In this way the brake may be arranged to maintain
a constant torque, not withstanding variation of the speed. In G.I.
Alden's brake (see _Trans. Amer. Soc. Eng._ vol. xi.) the resistance
is obtained by turning a cast iron disk against the frictional
resistance of two thin copper plates, which are held in a casing free
to turn upon the shaft, and are so arranged that the pressure between
the rubbing surfaces is controlled, and the heat developed by friction
carried away, by the regulated flow of water through the casing. The
torque required to hold the casing still against the action of the
disk measures the torque exerted by the shaft to which the disk is
keyed.
[Illustration: Fig. 5.]
_Transmission Dynamometers._--The essential part of many transmission
dynamometers is a spring whose deformation indirectly measures the
magnitude of the force transmitted through it. For many kinds of
spring the change of form is practically proportional to the force,
but the relation should always be determined experimentally. General
A.J. Morin (see _Notice sur divers appareils dynamometriques_, Paris,
1841), in his classical experiments on traction, arranged his
apparatus so that the change in form of the spring was continuously
recorded on a sheet of paper drawn under a style. For longer
experiments he used a "Compteur" or mechanical integrator, suggested
by J.V. Poncelet, from which the work done during a given displacement
could be read off directly. This device consists of a roller of radius
r, pressed into contact with a disk. The two are carried on a common
frame, so arranged that a change in form of the spring causes a
relative displacement of the disk and roller, the point of contact
moving radially from or towards the centre of the disk. The radial
distance x is at any instant proportional to the force acting through
the spring. The angular displacement, [theta], of the disk is made
proportional to the displacement, s, of the point of application of
the force by suitable driving gear. If d[phi] is the angular
displacement of the roller corresponding to displacements, d[theta] of
the disk, and ds of the point of application of P, a, and C constants,
then
xd[theta] a
d[phi] = --------- = -- P ds = C.P ds,
r r
_
/s2
and therefore [phi] = C | P ds;
_/s1
that is, the angular displacement of the roller measures the work done
during the displacement from s1 to s2. The shaft carrying the roller
is connected to a counter so that [phi] may be observed. The angular
velocity of the shaft is proportional to the rate of working. Morin's
dynamometer is shown in fig. 6. The transmitting spring is made up of
two flat bars linked at their ends. Their centres s1, s2, are held
respectively by the pieces A, B, which together form a sliding pair.
The block A carries the disk D, B carries the roller R and counting
gear. The pulley E is driven from an axle of the carriage. In a
dynamometer used by F.W. Webb to measure the tractive resistance of
trains on the London & North-Western railway, a tractive pull or push
compresses two spiral springs by a definite amount, which is recorded
to scale by a pencil on a sheet of paper, drawn continuously from a
storage drum at the rate of 3 in. per mile, by a roller driven from
one of the carriage axles. Thus the diagram shows the tractive force
at any instant. A second pencil electrically connected to a clock
traces a time line on the diagram with a kick at every thirty seconds.
A third pencil traces an observation line in which a kick can be made
at will by pressing any one of the electrical pushes placed about the
car, and a fourth draws a datum line. The spring of the dynamometer
car used by W. Dean on the Great Western railway is made up of thirty
flat plates, 7 ft. 6 in. long, 5 in. x 5/8 in. at the centre, spaced
by distance pieces nibbed into the plates at the centre and by rollers
at the ends. The draw-bar is connected to the buckle, which is carried
on rollers, the ends of the spring resting on plates fixed to the
under-frame. The gear operating the paper roll is driven from the axle
of an independent wheel which is let down into contact with the rail
when required. This wheel serves also to measure the distance
travelled. A Morin disk and roller integrator is connected with the
apparatus, so that the work done during a journey may be read off.
Five lines are traced on the diagram.
[Illustration: FIG. 6.]
In spring dynamometers designed to measure a transmitted torque, the
mechanical problem of ascertaining the change of form of the spring is
complicated by the fact that the spring and the whole apparatus are
rotating together. In the Ayrton and Perry transmission dynamometer or
spring coupling of this type, the relative angular displacement is
proportional to the radius of the circle described by the end of a
light lever operated by mechanism between the spring-connected parts.
By a device used by W.E. Dalby (_Proc. Inst. C.E._ 1897-1898, p. 132)
the change in form of the spring is shown on a fixed indicator, which
may be placed in any convenient position. Two equal sprocket wheels
Q1, Q2, are fastened, the one to the spring pulley, the other to the
shaft. An endless band is placed over them to form two loops, which
during rotation remain at the same distance apart, unless relative
angular displacement occurs between Q1 and Q2 (fig. 7) due to a change
in form of the spring. The change in the distance d is proportional to
the change in the torque transmitted from the shaft to the pulley. To
measure this, guide pulleys are placed in the loops guided by a
geometric slide, the one pulley carrying a scale, and the other an
index. A recording drum or integrating apparatus may be arranged on
the pulley frames. A quick variation, or a periodic variation of the
magnitude of the force or torque transmitted through the springs,
tends to set up oscillations, and this tendency increases the nearer
the periodic time of the force variation approaches a periodic time of
the spring. Such vibrations may be damped out to a considerable extent
by the use of a dash-pot, or may be practically prevented by using a
relatively stiff spring.
[Illustration: FIG. 7.]
Every part of a machine transmitting force suffers elastic
deformation, and the force may be measured indirectly by measuring the
deformation. The relation between the two should in all cases be found
experimentally. G.A. Hirn (see _Les Pandynamometres_, Paris, 1876)
employed this principle to measure the torque transmitted by a shaft.
Signor Rosio used a telephonic method to effect the same end, and
mechanical, optical and telephonic devices have been utilized by the
Rev. F.J. Jervis-Smith. (See _Phil. Mag._ February 1898.)
H. Frahm,[1] during an important investigation on the torsional
vibration of propeller shafts, measured the relative angular
displacement of two flanges on a propeller shaft, selected as far
apart as possible, by means of an electrical device (_Engineering_,
6th of February 1903). These measurements were utilized in combination
with appropriate elastic coefficients of the material to find the
horse-power transmitted from the engines along the shaft to the
propeller. In this way the effective horse-power and also the
mechanical efficiency of a number of large marine engines, each of
several thousand horse-power, have been determined.
[Illustration: FIG. 8.]
When a belt, in which the maximum and minimum tensions are
respectively P and p lb., drives a pulley, the torque exerted is (P -
p)r lb. ft., r being the radius of the pulley plus half the thickness
of the belt. P and p may be measured directly by leading the belt
round two freely hanging guide pulleys, one in the tight, the other in
the slack part of the belt, and adjusting loads on them until a stable
condition of running is obtained. In W. Froude's belt dynamometer (see
_Proc. Inst. M.E._, 1858) (fig. 8) the guide pulleys G1, G2 are
carried upon an arm free to turn about the axis O. H is a pulley to
guide the approaching and receding parts of the belt to and from the
beam in parallel directions. Neglecting friction, the unbalanced
torque acting on the beam is 4r{P - p} lb. ft. If a force Q acting at
R maintains equilibrium, QR/4 = (P - p)r = T. Q is supplied by a
spring, the extensions of which are recorded on a drum driven
proportionally to the angular displacement of the driving pulley; thus
a work diagram is obtained. In the Farcot form the guide pulleys are
attached to separate weighing levers placed horizontally below the
apparatus. In a belt dynamometer built for the Franklin Institute from
the designs of Tatham, the weighing levers are separate and arranged
horizontally at the top of the apparatus. The weighing beam in the
Hefner-Alteneck dynamometer is placed transversely to the belt (see
_Electrotechnischen Zeitschrift_, 1881, 7). The force Q, usually
measured by a spring, required to maintain the beam in its central
position is proportional to (P - p). If the angle [theta]1 = [theta]2
= 120 deg., Q = (P - p) neglecting friction.
When a shaft is driven by means of gearing the driving torque is
measured by the product of the resultant pressure P acting between the
wheel teeth and the radius of the pitch circle of the wheel fixed to
the shaft. Fig. 9, which has been reproduced from J. White's _A New
Century of Inventions_ (Manchester, 1822), illustrates possibly the
earliest application of this principle to dynamometry. The wheel D,
keyed to the shaft overcoming the resistance to be measured, is driven
from wheel N by two bevel wheels L, L, carried in a loose pulley K.
The two shafts, though in a line, are independent. A torque applied to
the shaft A can be transmitted to D, neglecting friction, without
change only if the central pulley K is held from turning; the torque
required to do this is twice the torque transmitted.
[Illustration: FIG. 9.]
The torque acting on the armature of an electric motor is necessarily
accompanied by an equal and opposite torque acting on the frame. If,
therefore, the motor is mounted on a cradle free to turn about
knife-edges, the reacting torque is the only torque tending to turn
the cradle when it is in a vertical position, and may therefore be
measured by adjusting weights to hold the cradle in a vertical
position. The rate at which the motor is transmitting work is then
T2[pi]n/550 H.P., where n is the revolutions per second of the
armature.
See James Dredge, _Electric Illumination_, vol. ii. (London, 1885);
W.W. Beaumont, "Dynamometers and Friction Brakes," _Proc. Inst. C.E._
vol. xcv. (London, 1889); E. Brauer, "Ueber Bremsdynamometer and
verwandte Kraftmesser," _Zeitschrift des Vereins deutscher Ingenieure_
(Berlin, 1888); J.J. Flather, _Dynamometers and the Measurement of
Power_ (New York, 1893). (W. E. D.)