APPLIED MECHANICS.] MECHANICS 7(57 curve at P, cutting OX in T ; FT=FY x secant obliquity, and this is to be a constant quantity ; hence the curve is that known as the On that piu turns an arm, carrying at a point P a tracing-point, pencil, or pen. Should the pen have a nib of two jaws, like those of an ordinary drawing-pen, the plane of the jaws must pass through PT. Then, while T is slid along the axis from towards X, P will be drawn after it from R towards C along the tractory. This curve, being an asymptote to its axis, is capable of being in definitely prolonged towards X ; but in designing pivots it should stop before the angle PTY becomes less than the angle of repose ot the rubbing surfaces, otherwise the pivot will be liable to stick in its bearing. The moment of friction of "Schiele s anti-friction pivot, as it is called, is equal to that of a cylindrical journal of the radius OR = PT the constant tangent, under the same pressure. Ill Friction of Teeth. Let N be the normal pressure exerted between a pair of teeth of a pair of wheels ; s the total distance through which they slide upon each other ; n the number of pairs of teeth which pass the plane of axis in a unit of time ; then nfSs (63) is the work lost in unity of time by the friction of the teeth. The sliding s is composed of two parts, which take place during the approach and recess respectively. Let those be denoted by s 1 and s.,, so that s = s 1 + s. 2 . In sect. 55 the velocity of sliding at any instant has been given, viz., = c(a 1 + o 2 ), where u is that velocity, c the distance TI at any instant from the point of contact of the teeth to the pitch-point, and a lt a, the respective angular velocities of the wheels. Let v be the common velocity of the two pitch-circles, r v r. 2 their radii ; then the above equation becomes / 1 1 u = cv 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 - 3 4- J --r-3 r ( 66 ). To apply this to involute teeth, let Cj be the length of the approach, c 2 that of the recess, M X the mean velocity of sliding during the approach, u. 2 that during the recess ; then also, let 6 be the obliquity of the action ; then the times occupied by the approach and recess are respectively ,-r<jst> UCOS0 f, finally, for the length of sliding between each pair of teeth, which, substituted in equation 63, gives the work lost in a unit of time by the friction of involute teeth. This result, which is exact for involute teeth, is approximately true for teeth of any figure. For inside gearing, if 1 be the less radius and r z the greater, _ is to be substituted for 1- . 1 112. 2 Friction of Cords and Belts. A flexible band, such as a cord, rope, belt, or strap, may be used either to exert an effort or a resistance upon a pulley round which it wraps. In either case the tangential force, whether effort or resistance, exerted between the band and the pulley is their mutual friction, caused by and propor tional to the normal pressure between them. Let Tj be the tension of the free part of the band at that side towards viich it tends to draw the pulley, or from which the pulley tends to draw it ; T 2 the tension of the free part at the other side ; T the tension of the band at any intermediate point of its arc of con tact with the pulley ; 6 the ratio of the length of that arc to the radius of the pulley ; d0 the ratio of an indefinitely small element of that arc to the radius ; F = Tj - T 2 the total friction between the band and the pulley ; d the elementary portion of that friction due to the elementary arc dQ ; / the coefficient of friction between the materials of the band and pulley. Then, according to a well-known principle in statics, the normal pressure at the elementary arc dQ is Td0, T being the mean tension of the band at that elementary arc ; consequently the friction on that arc is dV =fYdO. Now that friction is also the difference between the tensions of the band at the two ends of the elementary arc, or dT = d =/Trf0 ; which equation, being integrated throughout the entire arc of contact, gives the following formulas : . . (65). 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 a denote the arc of contact expressed in turns and fractions of a turn ; then = 6 2832 j e? 6 = number whose common logarithm is 2 7288/a 113. 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 follow ing side, which is necessary to bend*it into a curve fitting the pulley, and then to straighten it again. The following empirical formula 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, ?i = 48d 2 for white ropes and 35oP for tarred ropes ; r the effective radius of the pulley in inches ; T the tension in pounds. Then For white ropes, F- (0 "0012+0 001026+0 0012T) For tarred ropes, F- (0 > 006+0 001892n + > 00168T) (68). 114. Friction-Couplings. Friction is useful as a means of com municating 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 loost ly on 1 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 tlie 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 ha* 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 con trivances of the same class, but the two just mentioned may serve for examples. 115. 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 cir- cumstances, and with all materials, solid, liquid, and gaseous, arc comparatively recent achievements of Joule. The quantity of work which produces a British unit of heat (or so much heat as 1 elevates the temperature of one pound of pure water, at or near ordinary atmospheric temperatures, by one degree of Fahrenheit) is 772 foot-pounds. This constant, now designated as Joules equivalent," is the principal experimental datum of the science of thermodynamics. The heat produced by friction, when mode-rate in amount, is uselul
in softening and liquefying thick unguents ; but when excessive it