766 MECHANICS [APPLIED MECHANICS. It is to be understood that the above-stated law of friction is only true for dry surfaces when the pressure is not sufficient to indent or abrade the surfaces, and for greased surfaces when the pressure is not sufficient to force out the unguent from between the surfaces. If the proper limit be exceeded, the friction increases more rapidly than in the simple ratio of the normal pressure. The limit of pressure for unguents diminishes as the speed increases. The following are some of its approximate values as in ferred from experience in railway locomotive and carriage axles : Velocity of rubbing in feet per second Intensity of normal pressure per Ib per square inch ) of surface ) 1 392 5 140 In pivots, the intensity of the pressure is usually fixed at about one ton per square inch. Unguents should be comparatively thick for heavy pressures, that they may resist being forced out, and comparatively thin for light pressures, that their viscidity may not add to the resistance. Unguents are of three classes, viz.: 1. Fatty: consisting of animal or vegetable fixed oils, such as tallow, lard, lard-oil, seal-oil, whale-oil, olive-oil. Drying oils, which absorb oxygen and harden, are obviously unfit for unguents. 2. Soapy : composed of fatty oil, alkali, and water. The best grease of this class should not contain more than about 25 or 30 per cent, of water ; bad kinds contain 40 or 50 per cent. The addi tional water diminishes the cost, but spoils the unguent. 3. Bituminous : composed of solid and liquid mineral compounds of hydrogen and carbon. 108. Work of Friction Moment of Friction. The work per formed 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 rota tion, 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, aud the angular velocity. The product of the force of friction by the distance at which it acts from the axis of rotation is called the moment 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/ the coefficient of friction. Then the moment of friction of du is fprdu ; the total moment of friction is ffpr. du ; and the work performed in a unit of r (" ) time in overcoming friction, when the angular velocity is a, is affpr.du. j 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 stated in sect. 107, anil also by conditions of strength and stillness. 109. Total Pressure betiveen Journal and Scaring. A. single piece rotating with an uniform velocity has four mutually balanced forces applied to it : (1) the etfort 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 <f>, 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 com pounded by the principles of statics into one force, and let this be called tJie given force. The directions of the effort and of the given force are cither 80. 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. 30, 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 <(> 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 $. 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 magni tude of that reaction and of the effort are then found by the principles of the equili brium of three forces already stated in Part I., sect. 7 (see also p. 702, 124). The work lost in overcoming the friction of the bearing is the same with 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. 110. 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 & pivot at the end of the shaft ; or, if that be impossible, against one or more collars, or rings projecting from the body of the shaft. The bearing of the pivot is called a step or footstep. Pivots require great hardness, and are usually made of steel. The fiat pivot 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 pres sure be uniformly distributed, which is the case when the rubbing surfaces of the pivot and its step are both true planes, the inten sity of the pressure is p- 7 >2 (60); and, introducing this value into equation 59, the moment of friction of the fiat pivot is found to be f/Nr ........ (61), or two-thirds of that of a cylindrical journal of the same radius under the same normal pressure. The friction of a conical pivot 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 a collar is given by the formula ,.3 _ r >3 where r is the external and / the internal radius. In the cup and ball pivot the end of the shaft and the step present two recesses facing each other, into which are fitted two shallow cups of steel or hard bronze. Between the concave spheri cal 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 men tioned 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 an oblique direction is to the rapidity of wear measured normally as the secant of the obliquity is to unity. Let OX (fig. 31) be the axis of a pivot, and let RFC 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