(52)The lengths ds, ds′ are proportional to the velocities of the points to whose paths they belong, and the proportions of those velocities to each other are deducible from the construction of the machine by the principles of pure mechanism explained in Chapter I.§ 90.Static Equilibrium of Mechanisms.—The principle stated in the preceding section, namely, that the energy exerted is equal to the work performed, enables the ratio of the components of the forces acting in the respective directions of motion at two points of a mechanism, one being the point of application of the effort, and the other the point of application of the resistance, to be readily found. Removing the summation signs in equation (52) in order to restrict its application to two points and dividing by the common time interval during which the respective small displacements ds and ds′ were made, it becomes P ds/dt = R ds′/dt, that is, Pv = Rv′, which shows that the force ratio is the inverse of the velocity ratio. It follows at once that any method which may be available for the determination of the velocity ratio is equally available for the determination of the force ratio, it being clearly understood that the forces involved are the components of the actual forces resolved in the direction of motion of the points. The relation between the effort and the resistance may be found by means of this principle for all kinds of mechanisms, when the friction produced by the components of the forces across the direction of motion of the two points is neglected. Consider the following example:—Fig. 126.A four-bar chain having the configuration shown in fig. 126 supports a load P at the point x. What load is required at the point y to maintain the configuration shown, both loads being supposed to act vertically? Find the instantaneous centre Obd, and resolve each load in the respective directions of motion of the points x and y; thus there are obtained the components P cos θ and R cos φ. Let the mechanism have a small motion; then, for the instant, the link b is turning about its instantaneous centre Obd, and, if ω is its instantaneous angular velocity, the velocity of the point x is ωr, and the velocity of the point y is ωs. Hence, by the principle just stated, P cos θ × ωr = R cos φ × ωs. But, p and q being respectively the perpendiculars to the lines of action of the forces, this equation reduces to Pp= Rq, which shows that the ratio of the two forces may be found by taking moments about the instantaneous centre of the link on which they act.The forces P and R may, however, act on different links. The general problem may then be thus stated: Given a mechanism of which r is the fixed link, and s and t any other two links, given also a force ƒs, acting on the link s, to find the force ƒtacting in a given direction on the link t, which will keep the mechanism in static equilibrium. The graphic solution of this problem may be effected thus:—(1) Find the three virtual centres Ors, Ort, Ost, which must be three points in a line.(2) Resolve ƒsinto two components, one of which, namely, ƒq, passes through Orsand may be neglected, and the other ƒppasses through Ost.(3) Find the point M, where ƒpjoins the given direction of ƒt, and resolve ƒpinto two components, of which one is in the direction MOrt, and may be neglected because it passes through Ort, and the other is in the given direction of ƒtand is therefore the force required.Fig. 127.This statement of the problem and the solution is due to Sir A. B. W. Kennedy, and is given in ch. 8 of hisMechanics of Machinery. Another general solution of the problem is given in theProc. Lond. Math. Soc.(1878-1879), by the same author. An example of the method of solution stated above, and taken from theMechanics of Machinery, is illustrated by the mechanism fig. 127, which is an epicyclic train of three wheels with the first wheel r fixed. Let it be required to find the vertical force which must act at the pitch radius of the last wheel t to balance exactly a force ƒsacting vertically downwards on the arm at the point indicated in the figure. The two links concerned are the last wheel t and the arm s, the wheel r being the fixed link of the mechanism. The virtual centres Ors, Ostare at the respective axes of the wheels r and t, and the centre Ortdivides the line through these two points externally in the ratio of the train of wheels. The figure sufficiently indicates the various steps of the solution.The relation between the effort and the resistance in a machine to include the effect of friction at the joints has been investigated in a paper by Professor Fleeming Jenkin, “On the application of graphic methods to the determination of the efficiency of machinery”(Trans. Roy. Soc. Ed., vol. 28). It is shown that a machine may at any instant be represented by a frame of links the stresses in which are identical with the pressures at the joints of the mechanism. This self-strained frame is called thedynamic frameof the machine. The driving and resisting efforts are represented by elastic links in the dynamic frame, and when the frame with its elastic links is drawn the stresses in the several members of it may be determined by means of reciprocal figures. Incidentally the method gives the pressures at every joint of the mechanism.§ 91.Efficiency.—Theefficiencyof a machine is the ratio of theusefulwork to thetotalwork—that is, to the energy exerted—and is represented byΣ · Ruds′=Σ · Ruds′=Σ · Ruds′=U.Σ · R ds′Σ · Ruds′ + Σ · Rpds′Σ · P dsE(53)Rubeing taken to represent useful and Rpprejudicial resistances. The more nearly the efficiency of a machine approaches to unity the better is the machine.§ 92.Power and Effect.—Thepowerof a machine is the energy exerted, and theeffectthe useful work performed, in some interval of time of definite length, such as a second, an hour, or a day.The unit of power, called conventionally a horse-power, is 550 foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour.§ 93.Modulus of a Machine.—In the investigation of the properties of a machine, the useful resistances to be overcome and the useful work to be performed are usually given. The prejudicial resistances arc generally functions of the useful resistances of the weights of the pieces of the mechanism, and of their form and arrangement; and, having been determined, they serve for the computation of thelostwork, which, being added to the useful work, gives the expenditure of energy required. The result of this investigation, expressed in the form of an equation between this energy and the useful work, is called by Moseley themodulusof the machine. The general form of the modulus may be expressed thus—E = U + φ (U, A) + ψ (A),(54)where A denotes some quantity or set of quantities depending on the form, arrangement, weight and other properties of the mechanism. Moseley, however, has pointed out that in most cases this equation takes the much more simple form ofE = (1 + A) U + B,(55)where A and B areconstants, depending on the form, arrangement and weight of the mechanism. The efficiency corresponding to the last equation isU=1.E1 + A + B/U(56)§ 94.Trains of Mechanism.—In applying the preceding principles to a train of mechanism, it may either be treated as a whole, or it may be considered in sections consisting of single pieces, or of any convenient portion of the train—each section being treated as a machine, driven by the effort applied to it and energy exerted upon it through its line of connexion with the preceding section, performing useful work by driving the following section, and losing work by overcoming its own prejudicial resistances. It is evident thatthe efficiency of the whole train is the product of the efficiencies of its sections.§ 95.Rotating Pieces: Couples of Forces.—It is often convenient to express the energy exerted upon and the work performed by a turning piece in a machine in terms of themomentof thecouples of forcesacting on it, and of the angular velocity. The ordinary British unit of moment is afoot-pound; but it is to be remembered that this is a foot-pound of a different sort from the unit of energy and work.If a force be applied to a turning piece in a line not passing through its axis, the axis will press against its bearings with an equal and parallel force, and the equal and opposite reaction of the bearings will constitute, together with the first-mentioned force, a couple whose arm is the perpendicular distance from the axis to the line of action of the first force.A couple is said to berightorleft handedwith reference to the observer, according to the direction in which it tends to turn the body, and is adrivingcouple or aresistingcouple according as its tendency is with or against that of the actual rotation.Let dt be an interval of time, α the angular velocity of the piece; then αdt is the angle through which it turns in the interval dt, and ds = v dt = rα dt is the distance through which the point of application of the force moves. Let P represent an effort, so that Pr is a driving couple, thenP ds = Pv dt = Prα dt = Mα dt(57)is the energy exerted by the couple M in the interval dt; and a similar equation gives the work performed in overcoming a resisting couple. When several couples act on one piece, the resultant of their moments is to be multiplied by the common angular velocity of the whole piece.§ 96.Reduction of Forces to a given Point, and of Couples to the Axis of a given Piece.—In computations respecting machines it is often convenient to substitute for a force applied to a given point, or a couple applied to a given piece, theequivalentforce or couple applied to some other point or piece; that is to say, the force or couple, which, if applied to the other point or piece, would exert equal energy or employ equal work. The principles of this reduction are that the ratio of the given to the equivalent force is the reciprocal of the ratio of the velocities of their points of application, and the ratio of the given to the equivalent couple is the reciprocal of the ratio of the angular velocities of the pieces to which they are applied.These velocity ratios are known by the construction of the mechanism, and are independent of the absolute speed.§ 97.Balanced Lateral Pressure of Guides and Bearings.—The most important part of the lateral pressure on a piece of mechanism is the reaction of its guides, if it is a sliding piece, or of the bearings of its axis, if it is a turning piece; and the balanced portion of this reaction is equal and opposite to the resultant of all the other forces applied to the piece, its own weight included. There may be or may not be an unbalanced component in this pressure, due to the deviated motion. Its laws will be considered in the sequel.§ 98.Friction. Unguents.—The most important kind of resistance in machines is thefrictionorrubbing resistanceof surfaces which slide over each other. Thedirectionof the resistance of friction is opposite to that in which the sliding takes place. Itsmagnitudeis the product of thenormal pressureor force which presses the rubbing surfaces together in a direction perpendicular to themselves into a specific constant already mentioned in § 14, as thecoefficient of friction, which depends on the nature and condition of the surfaces of the unguent, if any, with which they are covered. Thetotal pressureexerted between the rubbing surfaces is the resultant of the normal pressure and of the friction, and itsobliquity, or inclination to the common perpendicular of the surfaces, is theangle of reposeformerly mentioned in § 14, whose tangent is the coefficient of friction. Thus, let N be the normal pressure, R the friction, T the total pressure, ƒ the coefficient of friction, and φ the angle of repose; thenƒ = tan φR = ƒN = N tan φ = T sin φ(58)Experiments on friction have been made by Coulomb, Samuel Vince, John Rennie, James Wood, D. Rankine and others. The most complete and elaborate experiments are those of Morin, published in hisNotions fondamentales de mécanique, and republished in Britain in the works of Moseley and Gordon.The experiments of Beauchamp Tower (“Report of Friction Experiments,”Proc. Inst. Mech. Eng., 1883) showed that when oil is supplied to a journal by means of an oil bath the coefficient of friction varies nearly inversely as the load on the bearing, thus making the product of the load on the bearing and the coefficient of friction a constant. Mr Tower’s experiments were carried out at nearly constant temperature. The more recent experiments of Lasche (Zeitsch, Verein Deutsche Ingen., 1902, 46, 1881) show that the product of the coefficient of friction, the load on the bearing, and the temperature is approximately constant. For further information on this point and on Osborne Reynolds’s theory of lubrication seeBearingsandLubrication.§ 99.Work of Friction. Moment of Friction.—The work performed in a unit of time in overcoming the friction of a pair of surfaces is the product of the friction by the velocity of sliding of the surfaces over each other, if that is the same throughout the whole extent of the rubbing surfaces. If that velocity is different for different portions of the rubbing surfaces, the velocity of each portion is to be multiplied by the friction of that portion, and the results summed or integrated.When the relative motion of the rubbing surfaces is one of rotation, the work of friction in a unit of time, for a portion of the rubbing surfaces at a given distance from the axis of rotation, may be found by multiplying together the friction of that portion, its distance from the axis, and the angular velocity. The product of the force of friction by the distance at which it acts from the axis of rotation is called themoment of friction. The total moment of friction of a pair of rotating rubbing surfaces is the sum or integral of the moments of friction of their several portions.To express this symbolically, let du represent the area of a portion of a pair of rubbing surfaces at a distance r from the axis of their relative rotation; p the intensity of the normal pressure at du per unit of area; and ƒ the coefficient of friction. Then the moment of friction of du is ƒpr du;the total moment of friction is ƒ ∫ pr·du;and the work performed in a unit cf time in overcoming friction, when the angular velocity is α, is αƒ ∫ pr·du.(59)It is evident that the moment of friction, and the work lost by being performed in overcoming friction, are less in a rotating piece as the bearings are of smaller radius. But a limit is put to the diminution of the radii of journals and pivots by the conditions of durability and of proper lubrication, and also by conditions of strength and stiffness.§ 100.Total Pressure between Journal and Bearing.—A single piece rotating with a uniform velocity has four mutually balanced forces applied to it: (l) the effort exerted on it by the piece which drives it; (2) the resistance of the piece which follows it—which may be considered for the purposes of the present question as useful resistance; (3) its weight; and (4) the reaction of its own cylindrical bearings. There are given the following data:—The direction of the effort.The direction of the useful resistance.The weight of the piece and the direction in which it acts.The magnitude of the useful resistance.The radius of the bearing r.The angle of repose φ, corresponding to the friction of the journal on the bearing.And there are required the following:—The direction of the reaction of the bearing.The magnitude of that reaction.The magnitude of the effort.Let the useful resistance and the weight of the piece be compounded by the principles of statics into one force, and let this be calledthe given force.Fig. 128.The directions of the effort and of the given force are either parallel or meet in a point. If they are parallel, the direction of the reaction of the bearing is also parallel to them; if they meet in a point, the direction of the reaction traverses the same point.Also, let AAA, fig. 128, be a section of the bearing, and C its axis; then the direction of the reaction, at the point where it intersects the circle AAA, must make the angle φ with the radius of that circle; that is to say, it must be a line such as PT touching the smaller circle BB, whose radius is r · sin φ. The side on which it touches that circle is determined by the fact that the obliquity of the reaction is such as to oppose the rotation.Thus is determined the direction of the reaction of the bearing; and the magnitude of that reaction and of the effort are then found by the principles of the equilibrium of three forces already stated in § 7.The work lost in overcoming the friction of the bearing is the same as that which would be performed in overcoming at the circumference of the small circle BB a resistance equal to the whole pressure between the journal and bearing.In order to diminish that pressure to the smallest possible amount, the effort, and the resultant of the useful resistance, and the weight of the piece (called above the “given force”) ought to be opposed to each other as directly as is practicable consistently with the purposes of the machine.An investigation of the forces acting on a bearing and journal lubricated by an oil bath will be found in a paper by Osborne Reynolds in thePhil. Trans.pt. i. (1886). (See alsoBearings.)§ 101.Friction of Pivots and Collars.—When a shaft is acted upon by a force tending to shift it lengthways, that force must be balanced by the reaction of a bearing against apivotat the end of the shaft; or, if that be impossible, against one or morecollars, or ringsprojectingfrom the body of the shaft. The bearing of the pivot is called asteporfootstep. Pivots require great hardness, and are usually made of steel. Theflatpivot is a cylinder of steel having a plane circular end as a rubbing surface. Let N be the total pressure sustained by a flat pivot of the radius r; if that pressure be uniformly distributed, which is the case when the rubbing surfaces of the pivot and its step are both true planes, theintensityof the pressure isp = N / πr2;(60)and, introducing this value into equation 59, themoment of friction of the flat pivotis found to be2⁄3ƒNr(61)or two-thirds of that of a cylindrical journal of the same radius under the same normal pressure.The friction of aconicalpivot exceeds that of a flat pivot of the same radius, and under the same pressure, in the proportion of the side of the cone to the radius of its base.The moment of friction of acollaris given by the formula—2⁄3ƒNr3− r′3,r2− r′2(62)where r is the external and r′ the internal radius.Fig. 129.In thecup and ballpivot the end of the shaft and the step present two recesses facing each other, into which art fitted two shallow cups of steel or hard bronze. Between the concave spherical surfaces of those cups is placed a steel ball, being either a complete sphere or a lens having convex surfaces of a somewhat less radius than the concave surfaces of the cups. The moment of friction of this pivot is at first almost inappreciable from the extreme smallness of the radius of the circles of contact of the ball and cups, but, as they wear, that radius and the moment of friction increase.It appears that the rapidity with which a rubbing surface wears away is proportional to the friction and to the velocity jointly, or nearly so. Hence the pivots already mentioned wear unequally at different points, and tend to alter their figures. Schiele has invented a pivot which preserves its original figure by wearing equally at all points in a direction parallel to its axis. The following are the principles on which this equality of wear depends:—The rapidity of wear of a surface measured in anobliquedirection is to the rapidity of wear measured normally as the secant of the obliquity is to unity. Let OX (fig. 129) be the axis of a pivot, and let RPC be a portion of a curve such that at any point P the secant of the obliquity to the normal of the curve of a line parallel to the axis is inversely proportional to the ordinate PY, to which the velocity of P is proportional. The rotation of that curve round OX will generate the form of pivot required. Now let PT be a tangent to the curve at P, cutting OX in T; PT = PY ×secant obliquity, and this is to be a constant quantity; hence the curve is that known as thetractoryof the straight line OX, in which PT = OR = constant. This curve is described by having a fixed straight edge parallel to OX, along which slides a slider carrying a pin whose centre is T. On that pin 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 O 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 indefinitely prolonged towards X; but in designing pivots it should stop before the angle PTY becomes less than the angle of repose of 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.Records of experiments on the friction of a pivot bearing will be found in theProc. Inst. Mech. Eng.(1891), and on the friction of a collar bearing ib. May 1888.§ 102.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; thennƒNs(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 s1and s2, so that s = s1+ s2. In § 45 thevelocityof sliding at any instant has been given, viz. u = c (α1+ α2), where u is that velocity, c the distance T1 at any instant from the point of contact of the teeth to the pitch-point, and α1, α2the respective angular velocities of the wheels.Let v be the common velocity of the two pitch-circles, r1, r2, their radii; then the above equation becomesu = cv(1+1).r1r2To apply this to involute teeth, let c1be the length of the approach, c2that of the recess, u1, themeanvolocity of sliding during the approach, u2that during the recess; thenu1=c1v(1+1); u2=c2v(1+1)2r1r22r1r2also, let θ be the obliquity of the action; then the times occupied by the approach and recess are respectivelyc1,c2;v cos θv cos θgiving, finally, for the length of sliding between each pair of teeth,s = s1+ s2=c12+ c22(1+1)2 cos θr1r2(64)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 r1be the less radius and r2the greater, 1/r1− 1/r2is to be substituted for 1/r1+ 1/r2.§ 103.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 proportional to the normal pressure between them.Let T1be the tension of the free part of the band at that sidetowardswhich it tends to draw the pulley, orfromwhich the pulley tends to draw it; T2the tension of the free part at the other side; T the tension of the band at any intermediate point of its arc of contact with the pulley; θ the ratio of the length of that arc to the radius of the pulley; dθ the ratio of an indefinitely small element of that arc to the radius; F = T1− T2the total friction between the band and the pulley; dF the elementary portion of that friction due to the elementary arc dθ; ƒ 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 dθ is T dθ, T being the mean tension of the band at that elementary arc; consequently the friction on that arc is dF = ƒT dθ. Now that friction is also the differencebetween 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.T1= ƒθT2T1= eƒθT2F = T1− T2= T1(1 − e − ƒθ) = T2(eƒθ− 1)(65)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 themeantension remains unchanged. Its value is given by this formula—T1+ T2=eƒθ+ 122 (eƒθ− 1)(66)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 ofbrakecalled afriction-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 inturns and fractions of a turn; thenθ = 6.2832aeƒθ= number whose common logarithm is 2.7288ƒa(67)See alsoDynamometerfor 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.Thework lostin 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 = 48d2for white ropes and 35d2for tarred ropes; r theeffectiveradius of the pulley in inches; T the tension in pounds. ThenFor white ropes, F =n(0.0012 + 0.001026n + 0.0012T).rFor tarred ropes, F =n(0.006 + 0.001392n + 0.00168T).r(68)§ 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 arefriction-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 thefriction-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 orclutchrotates 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 theexactdetermination 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 theProc. 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 seeBearings.§ 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 correctlyrolling resistance. It is of the nature of acouple, resisting rotation. Itsmomentis found by multiplying the normal pressure between the rolling surfaces by anarm, 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 theangular velocityof the rolling surfaces relatively to each other. The following are approximate values of the arm in decimals of a foot:—Oak upon oak0.006 (Coulomb).Lignum vitae on oak0.004 ”Cast iron on cast iron0.002 (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 areciprocating 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 bestoredwhen the weight is lifted, andrestoredwhen 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—thusstoringthe periodical excess of energy andrestoringthat 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 isunbalanced, 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.3If 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 thecentrifugal force will cause strains, vibration and increased friction, and a tendency of the shafts to jump out of their bearings.§ 111.Centrifugal Couples of a Rotating Body.—Besides the tendency (if any) of the combined centrifugal forces of the particles of a rotating body toshiftthe axis of rotation, they may also tend toturnit out of its original direction. The latter tendency is calleda centrifugal couple, and vanishes for rotation about a principal axis.It is essential to the steady motion of every rapidly rotating piece in a machine that its axis of rotation should not merely traverse its centre of gravity, but should be a permanent axis; for otherwise the centrifugal couples will increase friction, produce oscillation of the shaft and tend to make it leave its bearings.The principles of this and the preceding section are those which regulate the adjustment of the weight and position of the counterpoises which are placed between the spokes of the driving-wheels of locomotive engines.(FromBalancing of Engines, by permission of Edward Arnold.)Fig. 130.§ 112.*Method of computing the position and magnitudes of balance weights which must be added to a given system of arbitrarily chosen rotating masses in order to make the common axis of rotation a permanent axis.—The method here briefly explained is taken from a paper by W. E. Dalby, “The Balancing of Engines with special reference to Marine Work,”Trans. Inst. Nav. Arch.(1899). Let the weight (fig. 130), attached to a truly turned disk, be rotated by the shaft OX, and conceive that the shaft is held in a bearing at one point, O. The force required to constrain the weight to move in a circle, that is the deviating force, produces an equal and opposite reaction on the shaft, whose amount F is equal to the centrifugal force Wa2r/g ℔, where r is the radius of the mass centre of the weight, and a is its angular velocity in radians per second. Transferring this force to the point O, it is equivalent to, (1) a force at O equal and parallel to F, and, (2) a centrifugal couple of Fa foot-pounds. In order that OX may be a permanent axis it is necessary that there should be a sufficient number of weights attached to the shaft and so distributed that when each is referred to the point O(1) ΣF = 0(2) ΣFa = 0
(52)
The lengths ds, ds′ are proportional to the velocities of the points to whose paths they belong, and the proportions of those velocities to each other are deducible from the construction of the machine by the principles of pure mechanism explained in Chapter I.
§ 90.Static Equilibrium of Mechanisms.—The principle stated in the preceding section, namely, that the energy exerted is equal to the work performed, enables the ratio of the components of the forces acting in the respective directions of motion at two points of a mechanism, one being the point of application of the effort, and the other the point of application of the resistance, to be readily found. Removing the summation signs in equation (52) in order to restrict its application to two points and dividing by the common time interval during which the respective small displacements ds and ds′ were made, it becomes P ds/dt = R ds′/dt, that is, Pv = Rv′, which shows that the force ratio is the inverse of the velocity ratio. It follows at once that any method which may be available for the determination of the velocity ratio is equally available for the determination of the force ratio, it being clearly understood that the forces involved are the components of the actual forces resolved in the direction of motion of the points. The relation between the effort and the resistance may be found by means of this principle for all kinds of mechanisms, when the friction produced by the components of the forces across the direction of motion of the two points is neglected. Consider the following example:—
A four-bar chain having the configuration shown in fig. 126 supports a load P at the point x. What load is required at the point y to maintain the configuration shown, both loads being supposed to act vertically? Find the instantaneous centre Obd, and resolve each load in the respective directions of motion of the points x and y; thus there are obtained the components P cos θ and R cos φ. Let the mechanism have a small motion; then, for the instant, the link b is turning about its instantaneous centre Obd, and, if ω is its instantaneous angular velocity, the velocity of the point x is ωr, and the velocity of the point y is ωs. Hence, by the principle just stated, P cos θ × ωr = R cos φ × ωs. But, p and q being respectively the perpendiculars to the lines of action of the forces, this equation reduces to Pp= Rq, which shows that the ratio of the two forces may be found by taking moments about the instantaneous centre of the link on which they act.
The forces P and R may, however, act on different links. The general problem may then be thus stated: Given a mechanism of which r is the fixed link, and s and t any other two links, given also a force ƒs, acting on the link s, to find the force ƒtacting in a given direction on the link t, which will keep the mechanism in static equilibrium. The graphic solution of this problem may be effected thus:—
(1) Find the three virtual centres Ors, Ort, Ost, which must be three points in a line.(2) Resolve ƒsinto two components, one of which, namely, ƒq, passes through Orsand may be neglected, and the other ƒppasses through Ost.(3) Find the point M, where ƒpjoins the given direction of ƒt, and resolve ƒpinto two components, of which one is in the direction MOrt, and may be neglected because it passes through Ort, and the other is in the given direction of ƒtand is therefore the force required.
(1) Find the three virtual centres Ors, Ort, Ost, which must be three points in a line.
(2) Resolve ƒsinto two components, one of which, namely, ƒq, passes through Orsand may be neglected, and the other ƒppasses through Ost.
(3) Find the point M, where ƒpjoins the given direction of ƒt, and resolve ƒpinto two components, of which one is in the direction MOrt, and may be neglected because it passes through Ort, and the other is in the given direction of ƒtand is therefore the force required.
This statement of the problem and the solution is due to Sir A. B. W. Kennedy, and is given in ch. 8 of hisMechanics of Machinery. Another general solution of the problem is given in theProc. Lond. Math. Soc.(1878-1879), by the same author. An example of the method of solution stated above, and taken from theMechanics of Machinery, is illustrated by the mechanism fig. 127, which is an epicyclic train of three wheels with the first wheel r fixed. Let it be required to find the vertical force which must act at the pitch radius of the last wheel t to balance exactly a force ƒsacting vertically downwards on the arm at the point indicated in the figure. The two links concerned are the last wheel t and the arm s, the wheel r being the fixed link of the mechanism. The virtual centres Ors, Ostare at the respective axes of the wheels r and t, and the centre Ortdivides the line through these two points externally in the ratio of the train of wheels. The figure sufficiently indicates the various steps of the solution.
The relation between the effort and the resistance in a machine to include the effect of friction at the joints has been investigated in a paper by Professor Fleeming Jenkin, “On the application of graphic methods to the determination of the efficiency of machinery”(Trans. Roy. Soc. Ed., vol. 28). It is shown that a machine may at any instant be represented by a frame of links the stresses in which are identical with the pressures at the joints of the mechanism. This self-strained frame is called thedynamic frameof the machine. The driving and resisting efforts are represented by elastic links in the dynamic frame, and when the frame with its elastic links is drawn the stresses in the several members of it may be determined by means of reciprocal figures. Incidentally the method gives the pressures at every joint of the mechanism.
§ 91.Efficiency.—Theefficiencyof a machine is the ratio of theusefulwork to thetotalwork—that is, to the energy exerted—and is represented by
(53)
Rubeing taken to represent useful and Rpprejudicial resistances. The more nearly the efficiency of a machine approaches to unity the better is the machine.
§ 92.Power and Effect.—Thepowerof a machine is the energy exerted, and theeffectthe useful work performed, in some interval of time of definite length, such as a second, an hour, or a day.
The unit of power, called conventionally a horse-power, is 550 foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000 foot-pounds per hour.
§ 93.Modulus of a Machine.—In the investigation of the properties of a machine, the useful resistances to be overcome and the useful work to be performed are usually given. The prejudicial resistances arc generally functions of the useful resistances of the weights of the pieces of the mechanism, and of their form and arrangement; and, having been determined, they serve for the computation of thelostwork, which, being added to the useful work, gives the expenditure of energy required. The result of this investigation, expressed in the form of an equation between this energy and the useful work, is called by Moseley themodulusof the machine. The general form of the modulus may be expressed thus—
E = U + φ (U, A) + ψ (A),
(54)
where A denotes some quantity or set of quantities depending on the form, arrangement, weight and other properties of the mechanism. Moseley, however, has pointed out that in most cases this equation takes the much more simple form of
E = (1 + A) U + B,
(55)
where A and B areconstants, depending on the form, arrangement and weight of the mechanism. The efficiency corresponding to the last equation is
(56)
§ 94.Trains of Mechanism.—In applying the preceding principles to a train of mechanism, it may either be treated as a whole, or it may be considered in sections consisting of single pieces, or of any convenient portion of the train—each section being treated as a machine, driven by the effort applied to it and energy exerted upon it through its line of connexion with the preceding section, performing useful work by driving the following section, and losing work by overcoming its own prejudicial resistances. It is evident thatthe efficiency of the whole train is the product of the efficiencies of its sections.
§ 95.Rotating Pieces: Couples of Forces.—It is often convenient to express the energy exerted upon and the work performed by a turning piece in a machine in terms of themomentof thecouples of forcesacting on it, and of the angular velocity. The ordinary British unit of moment is afoot-pound; but it is to be remembered that this is a foot-pound of a different sort from the unit of energy and work.
If a force be applied to a turning piece in a line not passing through its axis, the axis will press against its bearings with an equal and parallel force, and the equal and opposite reaction of the bearings will constitute, together with the first-mentioned force, a couple whose arm is the perpendicular distance from the axis to the line of action of the first force.
A couple is said to berightorleft handedwith reference to the observer, according to the direction in which it tends to turn the body, and is adrivingcouple or aresistingcouple according as its tendency is with or against that of the actual rotation.
Let dt be an interval of time, α the angular velocity of the piece; then αdt is the angle through which it turns in the interval dt, and ds = v dt = rα dt is the distance through which the point of application of the force moves. Let P represent an effort, so that Pr is a driving couple, then
P ds = Pv dt = Prα dt = Mα dt
(57)
is the energy exerted by the couple M in the interval dt; and a similar equation gives the work performed in overcoming a resisting couple. When several couples act on one piece, the resultant of their moments is to be multiplied by the common angular velocity of the whole piece.
§ 96.Reduction of Forces to a given Point, and of Couples to the Axis of a given Piece.—In computations respecting machines it is often convenient to substitute for a force applied to a given point, or a couple applied to a given piece, theequivalentforce or couple applied to some other point or piece; that is to say, the force or couple, which, if applied to the other point or piece, would exert equal energy or employ equal work. The principles of this reduction are that the ratio of the given to the equivalent force is the reciprocal of the ratio of the velocities of their points of application, and the ratio of the given to the equivalent couple is the reciprocal of the ratio of the angular velocities of the pieces to which they are applied.
These velocity ratios are known by the construction of the mechanism, and are independent of the absolute speed.
§ 97.Balanced Lateral Pressure of Guides and Bearings.—The most important part of the lateral pressure on a piece of mechanism is the reaction of its guides, if it is a sliding piece, or of the bearings of its axis, if it is a turning piece; and the balanced portion of this reaction is equal and opposite to the resultant of all the other forces applied to the piece, its own weight included. There may be or may not be an unbalanced component in this pressure, due to the deviated motion. Its laws will be considered in the sequel.
§ 98.Friction. Unguents.—The most important kind of resistance in machines is thefrictionorrubbing resistanceof surfaces which slide over each other. Thedirectionof the resistance of friction is opposite to that in which the sliding takes place. Itsmagnitudeis the product of thenormal pressureor force which presses the rubbing surfaces together in a direction perpendicular to themselves into a specific constant already mentioned in § 14, as thecoefficient of friction, which depends on the nature and condition of the surfaces of the unguent, if any, with which they are covered. Thetotal pressureexerted between the rubbing surfaces is the resultant of the normal pressure and of the friction, and itsobliquity, or inclination to the common perpendicular of the surfaces, is theangle of reposeformerly mentioned in § 14, whose tangent is the coefficient of friction. Thus, let N be the normal pressure, R the friction, T the total pressure, ƒ the coefficient of friction, and φ the angle of repose; then
ƒ = tan φR = ƒN = N tan φ = T sin φ
(58)
Experiments on friction have been made by Coulomb, Samuel Vince, John Rennie, James Wood, D. Rankine and others. The most complete and elaborate experiments are those of Morin, published in hisNotions fondamentales de mécanique, and republished in Britain in the works of Moseley and Gordon.
The experiments of Beauchamp Tower (“Report of Friction Experiments,”Proc. Inst. Mech. Eng., 1883) showed that when oil is supplied to a journal by means of an oil bath the coefficient of friction varies nearly inversely as the load on the bearing, thus making the product of the load on the bearing and the coefficient of friction a constant. Mr Tower’s experiments were carried out at nearly constant temperature. The more recent experiments of Lasche (Zeitsch, Verein Deutsche Ingen., 1902, 46, 1881) show that the product of the coefficient of friction, the load on the bearing, and the temperature is approximately constant. For further information on this point and on Osborne Reynolds’s theory of lubrication seeBearingsandLubrication.
§ 99.Work of Friction. Moment of Friction.—The work performed in a unit of time in overcoming the friction of a pair of surfaces is the product of the friction by the velocity of sliding of the surfaces over each other, if that is the same throughout the whole extent of the rubbing surfaces. If that velocity is different for different portions of the rubbing surfaces, the velocity of each portion is to be multiplied by the friction of that portion, and the results summed or integrated.
When the relative motion of the rubbing surfaces is one of rotation, the work of friction in a unit of time, for a portion of the rubbing surfaces at a given distance from the axis of rotation, may be found by multiplying together the friction of that portion, its distance from the axis, and the angular velocity. The product of the force of friction by the distance at which it acts from the axis of rotation is called themoment of friction. The total moment of friction of a pair of rotating rubbing surfaces is the sum or integral of the moments of friction of their several portions.
To express this symbolically, let du represent the area of a portion of a pair of rubbing surfaces at a distance r from the axis of their relative rotation; p the intensity of the normal pressure at du per unit of area; and ƒ the coefficient of friction. Then the moment of friction of du is ƒpr du;
the total moment of friction is ƒ ∫ pr·du;and the work performed in a unit cf time in overcoming friction, when the angular velocity is α, is αƒ ∫ pr·du.
the total moment of friction is ƒ ∫ pr·du;
and the work performed in a unit cf time in overcoming friction, when the angular velocity is α, is αƒ ∫ pr·du.
(59)
It is evident that the moment of friction, and the work lost by being performed in overcoming friction, are less in a rotating piece as the bearings are of smaller radius. But a limit is put to the diminution of the radii of journals and pivots by the conditions of durability and of proper lubrication, and also by conditions of strength and stiffness.
§ 100.Total Pressure between Journal and Bearing.—A single piece rotating with a uniform velocity has four mutually balanced forces applied to it: (l) the effort exerted on it by the piece which drives it; (2) the resistance of the piece which follows it—which may be considered for the purposes of the present question as useful resistance; (3) its weight; and (4) the reaction of its own cylindrical bearings. There are given the following data:—
The direction of the effort.The direction of the useful resistance.The weight of the piece and the direction in which it acts.The magnitude of the useful resistance.The radius of the bearing r.The angle of repose φ, corresponding to the friction of the journal on the bearing.
The direction of the effort.
The direction of the useful resistance.
The weight of the piece and the direction in which it acts.
The magnitude of the useful resistance.
The radius of the bearing r.
The angle of repose φ, corresponding to the friction of the journal on the bearing.
And there are required the following:—
The direction of the reaction of the bearing.The magnitude of that reaction.The magnitude of the effort.
The direction of the reaction of the bearing.
The magnitude of that reaction.
The magnitude of the effort.
Let the useful resistance and the weight of the piece be compounded by the principles of statics into one force, and let this be calledthe given force.
The directions of the effort and of the given force are either parallel or meet in a point. If they are parallel, the direction of the reaction of the bearing is also parallel to them; if they meet in a point, the direction of the reaction traverses the same point.
Also, let AAA, fig. 128, be a section of the bearing, and C its axis; then the direction of the reaction, at the point where it intersects the circle AAA, must make the angle φ with the radius of that circle; that is to say, it must be a line such as PT touching the smaller circle BB, whose radius is r · sin φ. The side on which it touches that circle is determined by the fact that the obliquity of the reaction is such as to oppose the rotation.
Thus is determined the direction of the reaction of the bearing; and the magnitude of that reaction and of the effort are then found by the principles of the equilibrium of three forces already stated in § 7.
The work lost in overcoming the friction of the bearing is the same as that which would be performed in overcoming at the circumference of the small circle BB a resistance equal to the whole pressure between the journal and bearing.
In order to diminish that pressure to the smallest possible amount, the effort, and the resultant of the useful resistance, and the weight of the piece (called above the “given force”) ought to be opposed to each other as directly as is practicable consistently with the purposes of the machine.
An investigation of the forces acting on a bearing and journal lubricated by an oil bath will be found in a paper by Osborne Reynolds in thePhil. Trans.pt. i. (1886). (See alsoBearings.)
§ 101.Friction of Pivots and Collars.—When a shaft is acted upon by a force tending to shift it lengthways, that force must be balanced by the reaction of a bearing against apivotat the end of the shaft; or, if that be impossible, against one or morecollars, or ringsprojectingfrom the body of the shaft. The bearing of the pivot is called asteporfootstep. Pivots require great hardness, and are usually made of steel. Theflatpivot is a cylinder of steel having a plane circular end as a rubbing surface. Let N be the total pressure sustained by a flat pivot of the radius r; if that pressure be uniformly distributed, which is the case when the rubbing surfaces of the pivot and its step are both true planes, theintensityof the pressure is
p = N / πr2;
(60)
and, introducing this value into equation 59, themoment of friction of the flat pivotis found to be
2⁄3ƒNr
(61)
or two-thirds of that of a cylindrical journal of the same radius under the same normal pressure.
The friction of aconicalpivot exceeds that of a flat pivot of the same radius, and under the same pressure, in the proportion of the side of the cone to the radius of its base.
The moment of friction of acollaris given by the formula—
(62)
where r is the external and r′ the internal radius.
In thecup and ballpivot the end of the shaft and the step present two recesses facing each other, into which art fitted two shallow cups of steel or hard bronze. Between the concave spherical surfaces of those cups is placed a steel ball, being either a complete sphere or a lens having convex surfaces of a somewhat less radius than the concave surfaces of the cups. The moment of friction of this pivot is at first almost inappreciable from the extreme smallness of the radius of the circles of contact of the ball and cups, but, as they wear, that radius and the moment of friction increase.
It appears that the rapidity with which a rubbing surface wears away is proportional to the friction and to the velocity jointly, or nearly so. Hence the pivots already mentioned wear unequally at different points, and tend to alter their figures. Schiele has invented a pivot which preserves its original figure by wearing equally at all points in a direction parallel to its axis. The following are the principles on which this equality of wear depends:—
The rapidity of wear of a surface measured in anobliquedirection is to the rapidity of wear measured normally as the secant of the obliquity is to unity. Let OX (fig. 129) be the axis of a pivot, and let RPC be a portion of a curve such that at any point P the secant of the obliquity to the normal of the curve of a line parallel to the axis is inversely proportional to the ordinate PY, to which the velocity of P is proportional. The rotation of that curve round OX will generate the form of pivot required. Now let PT be a tangent to the curve at P, cutting OX in T; PT = PY ×secant obliquity, and this is to be a constant quantity; hence the curve is that known as thetractoryof the straight line OX, in which PT = OR = constant. This curve is described by having a fixed straight edge parallel to OX, along which slides a slider carrying a pin whose centre is T. On that pin 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 O 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 indefinitely prolonged towards X; but in designing pivots it should stop before the angle PTY becomes less than the angle of repose of 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.
Records of experiments on the friction of a pivot bearing will be found in theProc. Inst. Mech. Eng.(1891), and on the friction of a collar bearing ib. May 1888.
§ 102.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
nƒNs
(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 s1and s2, so that s = s1+ s2. In § 45 thevelocityof sliding at any instant has been given, viz. u = c (α1+ α2), where u is that velocity, c the distance T1 at any instant from the point of contact of the teeth to the pitch-point, and α1, α2the respective angular velocities of the wheels.
Let v be the common velocity of the two pitch-circles, r1, r2, their radii; then the above equation becomes
To apply this to involute teeth, let c1be the length of the approach, c2that of the recess, u1, themeanvolocity of sliding during the approach, u2that during the recess; then
also, let θ be the obliquity of the action; then the times occupied by the approach and recess are respectively
giving, finally, for the length of sliding between each pair of teeth,
(64)
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 r1be the less radius and r2the greater, 1/r1− 1/r2is to be substituted for 1/r1+ 1/r2.
§ 103.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 proportional to the normal pressure between them.
Let T1be the tension of the free part of the band at that sidetowardswhich it tends to draw the pulley, orfromwhich the pulley tends to draw it; T2the tension of the free part at the other side; T the tension of the band at any intermediate point of its arc of contact with the pulley; θ the ratio of the length of that arc to the radius of the pulley; dθ the ratio of an indefinitely small element of that arc to the radius; F = T1− T2the total friction between the band and the pulley; dF the elementary portion of that friction due to the elementary arc dθ; ƒ 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 dθ is T dθ, T being the mean tension of the band at that elementary arc; consequently the friction on that arc is dF = ƒT dθ. Now that friction is also the differencebetween 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:—
F = T1− T2= T1(1 − e − ƒθ) = T2(eƒθ− 1)
(65)
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 themeantension remains unchanged. Its value is given by this formula—
(66)
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 ofbrakecalled afriction-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 inturns and fractions of a turn; then
θ = 6.2832aeƒθ= number whose common logarithm is 2.7288ƒa
(67)
See alsoDynamometerfor 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.
Thework lostin 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 = 48d2for white ropes and 35d2for tarred ropes; r theeffectiveradius of the pulley in inches; T the tension in pounds. Then
(68)
§ 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 arefriction-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 thefriction-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 orclutchrotates 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 theexactdetermination 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 theProc. 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 seeBearings.
§ 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 correctlyrolling resistance. It is of the nature of acouple, resisting rotation. Itsmomentis found by multiplying the normal pressure between the rolling surfaces by anarm, 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 theangular velocityof the rolling surfaces relatively to each other. The following are approximate values of the arm in decimals of a foot:—
§ 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 areciprocating 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 bestoredwhen the weight is lifted, andrestoredwhen 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—thusstoringthe periodical excess of energy andrestoringthat 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 isunbalanced, 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.3
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 thecentrifugal force will cause strains, vibration and increased friction, and a tendency of the shafts to jump out of their bearings.
§ 111.Centrifugal Couples of a Rotating Body.—Besides the tendency (if any) of the combined centrifugal forces of the particles of a rotating body toshiftthe axis of rotation, they may also tend toturnit out of its original direction. The latter tendency is calleda centrifugal couple, and vanishes for rotation about a principal axis.
It is essential to the steady motion of every rapidly rotating piece in a machine that its axis of rotation should not merely traverse its centre of gravity, but should be a permanent axis; for otherwise the centrifugal couples will increase friction, produce oscillation of the shaft and tend to make it leave its bearings.
The principles of this and the preceding section are those which regulate the adjustment of the weight and position of the counterpoises which are placed between the spokes of the driving-wheels of locomotive engines.
§ 112.*Method of computing the position and magnitudes of balance weights which must be added to a given system of arbitrarily chosen rotating masses in order to make the common axis of rotation a permanent axis.—The method here briefly explained is taken from a paper by W. E. Dalby, “The Balancing of Engines with special reference to Marine Work,”Trans. Inst. Nav. Arch.(1899). Let the weight (fig. 130), attached to a truly turned disk, be rotated by the shaft OX, and conceive that the shaft is held in a bearing at one point, O. The force required to constrain the weight to move in a circle, that is the deviating force, produces an equal and opposite reaction on the shaft, whose amount F is equal to the centrifugal force Wa2r/g ℔, where r is the radius of the mass centre of the weight, and a is its angular velocity in radians per second. Transferring this force to the point O, it is equivalent to, (1) a force at O equal and parallel to F, and, (2) a centrifugal couple of Fa foot-pounds. In order that OX may be a permanent axis it is necessary that there should be a sufficient number of weights attached to the shaft and so distributed that when each is referred to the point O
(1) ΣF = 0(2) ΣFa = 0