- 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 called the 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 the Phil. Trans. pt. i. (1886). (See also Bearings.)
§ 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 a 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 flat 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 pressure be uniformly distributed, which is the case when the rubbing surfaces of the pivot and its step are both true planes, the intensity of the pressure is
and, introducing this value into equation 59, the moment of friction of the flat pivot is found to be
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—
| 23 ƒN | r 3 − r′3 | , |
| r 2 − r′2 |
where r is the external and r′ the internal radius.
 |
| Fig. 129. |
In the cup and ball pivot 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 an oblique direction 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 the tractory of 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 the Proc. 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
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 s1 and s2, so that s = s1 + s2. In § 45 the velocity of 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, α2 the 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
| u = cv | 1 | + | 1 | . |
| r1 | r2 |
To apply this to involute teeth, let c1 be the length of the approach, c2 that of the recess, u1, the mean volocity of sliding during the approach, u2 that during the recess; then
| u1 = | c1v | 1 | + | 1 | ; u2 = | c2v | 1 | + | 1 | |||
| 2 | r1 | r2 | 2 | r1 | r2 |
also, let θ be the obliquity of the action; then the times occupied by the approach and recess are respectively
| c1 | , | c2 | ; |
| v cos θ | v cos θ |
giving, finally, for the length of sliding between each pair of teeth,
| s = s1 + s2 = | c12 + c22 | 1 | + | 1 | ||
| 2 cos θ | r1 | r2 |
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 r1 be the less radius and r2 the greater, 1/r1 − 1/r2 is 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 T1 be the tension of the free part of the band at that side towards which it tends to draw the pulley, or from which the pulley tends to draw it; T2 the 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 − T2 the 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 difference
