between the tensions of the band at the two ends of the elementary arc, or dT = dF = ƒT dθ; which equation, being integrated throughout the entire arc of contact, gives the following formulae:—
| hyp log. T1T2 = ƒθ | |
| T1T2 = eƒθ | |
| F = T1 − T2 = T1 (1 − e − ƒθ) = T2 (eƒθ − 1) |
When a belt connecting a pair of pulleys has the tensions of its
two sides originally equal, the pulleys being at rest, and when the
pulleys are next set in motion, so that one of them drives the other
by means of the belt, it is found that the advancing side of the
belt is exactly as much tightened as the returning side is slackened,
so that the mean tension remains unchanged. Its value is given by
this formula—
| T1 + T2 | = | eƒθ + 1 |
| 2 | 2 (eƒθ − 1) |
which is useful in determining the original tension required to enable a belt to transmit a given force between two pulleys.
The equations 65 and 66 are applicable to a kind of brake called a friction-strap, used to stop or moderate the velocity of machines by being tightened round a pulley. The strap is usually of iron, and the pulley of hard wood.
Let α denote the arc of contact expressed in turns and fractions of a turn; then
θ = 6.2832a
eƒθ = number whose common logarithm is 2.7288ƒa
See also Dynamometer for illustrations of the use of what are essentially friction-straps of different forms for the measurement of the brake horse-power of an engine or motor.
§ 104. Stiffness of Ropes.—Ropes offer a resistance to being bent, and, when bent, to being straightened again, which arises from the mutual friction of their fibres. It increases with the sectional area of the rope, and is inversely proportional to the radius of the curve into which it is bent.
The work lost in pulling a given length of rope over a pulley is found by multiplying the length of the rope in feet by its stiffness in pounds, that stiffness being the excess of the tension at the leading side of the rope above that at the following side, which is necessary to bend it into a curve fitting the pulley, and then to straighten it again.
The following empirical formulae for the stiffness of hempen ropes have been deduced by Morin from the experiments of Coulomb:—
Let F be the stiffness in pounds avoirdupois; d the diameter of the rope in inches, n = 48d 2 for white ropes and 35d 2 for tarred ropes; r the effective radius of the pulley in inches; T the tension in pounds. Then
| For white ropes, F = nr (0.0012 + 0.001026n + 0.0012T). | |
| For tarred ropes, F =nr (0.006 + 0.001392n + 0.00168T). |
§ 105. Friction-Couplings.—Friction is useful as a means of communicating
motion where sudden changes either of force or velocity
take place, because, being limited in amount, it may be so adjusted
as to limit the forces which strain the pieces of the mechanism
within the bounds of safety. Amongst contrivances for effecting
this object are friction-cones. A rotating shaft carries upon a cylindrical
portion of its figure a wheel or pulley turning loosely on it,
and consequently capable of remaining at rest when the shaft is
in motion. This pulley has fixed to one side, and concentric with
it, a short frustum of a hollow cone. At a small distance from the
pulley the shaft carries a short frustum of a solid cone accurately
turned to fit the hollow cone. This frustum is made always to turn
along with the shaft by being fitted on a square portion of it, or by
means of a rib and groove, or otherwise, but is capable of a slight
longitudinal motion, so as to be pressed into, or withdrawn from,
the hollow cone by means of a lever. When the cones are pressed
together or engaged, their friction causes the pulley to rotate along
with the shaft; when they are disengaged, the pulley is free to stand
still. The angle made by the sides of the cones with the axis should
not be less than the angle of repose. In the friction-clutch, a pulley
loose on a shaft has a hoop or gland made to embrace it more or less
tightly by means of a screw; this hoop has short projecting arms or
ears. A fork or clutch rotates along with the shaft, and is capable
of being moved longitudinally by a handle. When the clutch is
moved towards the hoop, its arms catch those of the hoop, and
cause the hoop to rotate and to communicate its rotation to the pulley
by friction. There are many other contrivances of the same class,
but the two just mentioned may serve for examples.
§ 106. Heat of Friction: Unguents.—The work lost in friction is employed in producing heat. This fact is very obvious, and has been known from a remote period; but the exact determination of the proportion of the work lost to the heat produced, and the experimental proof that that proportion is the same under all circumstances and with all materials, solid, liquid and gaseous, are comparatively recent achievements of J. P. Joule. The quantity of work which produces a British unit of heat (or so much heat as elevates the temperature of one pound of pure water, at or near ordinary atmospheric temperatures, by 1° F.) is 772 foot-pounds. This constant, now designated as “Joule’s equivalent,” is the principal experimental datum of the science of thermodynamics.
A more recent determination (Phil. Trans., 1897), by Osborne Reynolds and W. M. Moorby, gives 778 as the mean value of Joule’s equivalent through the range of 32° to 212° F. See also the papers of Rowland in the Proc. Amer. Acad. (1879), and Griffiths, Phil. Trans. (1893).
The heat produced by friction, when moderate in amount, is useful in softening and liquefying thick unguents; but when excessive it is prejudicial, by decomposing the unguents, and sometimes even by softening the metal of the bearings, and raising their temperature so high as to set fire to neighbouring combustible matters.
Excessive heating is prevented by a constant and copious supply of a good unguent. The elevation of temperature produced by the friction of a journal is sometimes used as an experimental test of the quality of unguents. For modern methods of forced lubrication see Bearings.
§ 107. Rolling Resistance.—By the rolling of two surfaces over each other without sliding a resistance is caused which is called sometimes “rolling friction,” but more correctly rolling resistance. It is of the nature of a couple, resisting rotation. Its moment is found by multiplying the normal pressure between the rolling surfaces by an arm, whose length depends on the nature of the rolling surfaces, and the work lost in a unit of time in overcoming it is the product of its moment by the angular velocity of the rolling surfaces relatively to each other. The following are approximate values of the arm in decimals of a foot:—
| Oak upon oak | 0.006 (Coulomb). |
| Lignum vitae on oak | 0.004 ” |
| Cast iron on cast iron | 0.002 (Tredgold). |
§ 108. Reciprocating Forces: Stored and Restored Energy.—When a force acts on a machine alternately as an effort and as a resistance, it may be called a reciprocating force. Of this kind is the weight of any piece in the mechanism whose centre of gravity alternately rises and falls; for during the rise of the centre of gravity that weight acts as a resistance, and energy is employed in lifting it to an amount expressed by the product of the weight into the vertical height of its rise; and during the fall of the centre of gravity the weight acts as an effort, and exerts in assisting to perform the work of the machine an amount of energy exactly equal to that which had previously been employed in lifting it. Thus that amount of energy is not lost, but has its operation deferred; and it is said to be stored when the weight is lifted, and restored when it falls.
In a machine of which each piece is to move with a uniform velocity, if the effort and the resistance be constant, the weight of each piece must be balanced on its axis, so that it may produce lateral pressure only, and not act as a reciprocating force. But if the effort and the resistance be alternately in excess, the uniformity of speed may still be preserved by so adjusting some moving weight in the mechanism that when the effort is in excess it may be lifted, and so balance and employ the excess of effort, and that when the resistance is in excess it may fall, and so balance and overcome the excess of resistance—thus storing the periodical excess of energy and restoring that energy to perform the periodical excess of work.
Other forces besides gravity may be used as reciprocating forces for storing and restoring energy—for example, the elasticity of a spring or of a mass of air.
In most of the delusive machines commonly called “perpetual motions,” of which so many are patented in each year, and which are expected by their inventors to perform work without receiving energy, the fundamental fallacy consists in an expectation that some reciprocating force shall restore more energy than it has been the means of storing.
Division 2. Deflecting Forces.
§ 109. Deflecting Force for Translation in a Curved Path.—In machinery, deflecting force is supplied by the tenacity of some piece, such as a crank, which guides the deflected body in its curved path, and is unbalanced, being employed in producing deflexion, and not in balancing another force.
§ 110. Centrifugal Force of a Rotating Body.—The centrifugal force exerted by a rotating body on its axis of rotation is the same in magnitude as if the mass of the body were concentrated at its centre of gravity, and acts in a plane passing through the axis of rotation and the centre of gravity of the body.
The particles of a rotating body exert centrifugal forces on each other, which strain the body, and tend to tear it asunder, but these forces balance each other, and do not affect the resultant centrifugal force exerted on the axis of rotation.[1]
If the axis of rotation traverses the centre of gravity of the body, the centrifugal force exerted on that axis is nothing.
Hence, unless there be some reason to the contrary, each piece of a machine should be balanced on its axis of rotation; otherwise the
- ↑ This is a particular case of a more general principle, that the motion of the centre of gravity of a body is not affected by the mutual actions of its parts.
