# Page:EB1911 - Volume 03.djvu/600

581
BEARINGS

the bottom of the engine framing, whence it is again fed to the pump through a strainer. The parts of an engine lubricated in this way must be entirely enclosed.

 Fig. 11.

Load on bearings.—The distribution of pressure over the film of lubricant separating the rubbing surfaces of a bearing is variable, being greatest at a point near but not at the crown of the brass, and falling away to zero in all directions towards the boundaries of the film. It is usual in practice to ignore this variation of pressure through the film, and to indicate the severity with which the bearing is loaded by stating the load per square inch of the rubbing surfaces projected on to the diametral plane of the journal. Thus the projected area of the surfaces of a journal 6 in. in diameter and 8 in. long is 48 sq. in., and if the total load carried by the bearing is 20,000 pounds, the bearing would be said to carry a load of 417 pounds per square inch. When a shaft rotates in a bearing continuously in one direction the load per square inch with which it is safe to load the bearing in order to avoid undue heating is much less than if the motion is intermittent. A table of a few values of the bearing loads used in practice is given in the article Lubricants.

Bearing Friction.—If W is the total load on a bearing, and if µ is the coefficient of friction between the rubbing surfaces, the tangential resistance to turning is expressed by the product µW. If v is the relative velocity of the rubbing surfaces, the work done per second against friction is µWv foot pounds. This quantity of work is converted into heat, and the heat produced per second is therefore µWv/778 British Thermal Units. The coefficient µ is a variable quantity, and bearing in mind that a properly lubricated journal is separated from its supporting brass by a film of lubricant it might be expected that µ would have values characteristic of the coefficient of friction between two metallic surfaces, merging into the characteristics properly belonging to fluid friction, according as the oil film varied from an imperfect to a perfect condition, that is, according as the lubrication is partial or complete, completeness being attained by the use of an oil bath or by some method of forced lubrication. This expectation is entirely borne out by experimental researches. Beauchamp Tower (“Report on Friction Experiments,” Proc. Inst. Mech. Eng., November 1883) found that when oil was supplied to a bearing by means of a pad the coefficient of friction was approximately constant with the value of 1⁄100, thus following the law of solid friction; but when the journal was lubricated by means of an oil bath the coefficient of friction varied nearly inversely as the load on the bearing, thus making µW = constant. The tangential resistance in this case is characteristic of fluid friction since it is independent of the pressure. Tower’s experiments were carried out at a nearly constant temperature. The later experiments of O. Lasche (Zeitsch. Verein deutsche Ingenieure, 1902, 46, pp. 1881 et seq.) show how µ depends upon the temperature. Lasche’s main results with regard to the variation of µ are briefly:—µW is a constant quantity, thus confirming Tower’s earlier experiments; µ is practically independent of the relative velocity of the rubbing surfaces within the limits of 3 to 50 ft. per second; and the product µt is constant, t being the temperature of the bearing. Writing p for the load per unit of projected area of the bearing, Lasche found that the result of the experiments could be expressed by the simple formula pµt = constant = 2, where p = the pressure in kilograms per square centimetre, and t = the temperature in degrees centigrade. If p is changed to pounds per square inch the constant in the expression is approximately 30. The expression is valid between limits of pressure 14 to 213 pounds per square inch, limits of temperature 30° to 100° C., and between limits of velocity 3 to 50 ft. per second.

 Fig. 12.

Theory of Lubrication.—After the publication of Tower’s experiments on journal friction Professor Osborne Reynolds showed (Phil. Trans., 1886, p. 157) that the facts observed in connexion with a journal lubricated by means of an oil bath could be explained by a theory based upon the general principles of the motion of a viscous fluid. It is first established as an essential part of the theory that the radius of the brass must be slightly greater than the radius of the journal as indicated in fig. 12, where J is the centre of the journal and I the centre of the brass. Given this difference of curvature and a sufficient supply of oil, the rotation of the journal produces and maintains an oil film between the rubbing surfaces, the circumferential extent of which depends upon the rate of the oil supply and the external load. With an unlimited supply of oil, that is with oil-bath lubrication, the film extends continuously to the extremities of the brass, unless such extension would lead to negative pressures and therefore to a discontinuity, in which case the film ends where the pressures in the film become negative. The minimum distance between the journal and the brass occurs at the point H (fig. 12), on the off side of the point O where the line of action of the load cuts the surface of the journal. To the right and left of H the thickness of the film gradually increases, this being the condition that the oil-flow to and from the film may be automatically maintained. With an unlimited supply of oil the point H moves farther from O as the load increases until it reaches a maximum distance, and then it moves back again towards O as the load is further increased until a limiting load is reached at which the pressure in the film becomes negative at the boundaries of the film, when the boundaries recede from the edges of the brass as though the supply of oil were limited.

In the mathematical development of the theory it is first necessary to define the coefficient of viscosity. This is done as follows:—If two parallel surfaces AB, CD are separated by a viscous film, and if whilst CD is fixed AB moves in a tangential direction with velocity U, the surface of the film in contact with CD clings to it and remains at rest, whilst the lower surface of the film clings to and moves with the surface AB. At intermediate points in the film the tangential motion of the fluid will vary uniformly from zero to U, and the tangential resistance will be F = µU/h, where µ is the coefficient of viscosity and h is the thickness of the film. With this definition of viscosity and from the general equations representing the stress in a viscous fluid, the following equation is established, giving the relations between p, the pressure at any point in the film, h the thickness of the film at a point x measured round the circumference of the journal in the direction of relative motion, and U the relative tangential velocity of the surfaces,

${\displaystyle \textstyle {{\it {d}} \over dx}(}$ h³ ${\displaystyle \textstyle {{\it {dp}} \over dx})}$ = 6µU ${\displaystyle \textstyle {{\it {dh}} \over dx}}$

(1)

In this equation all the quantities are independent of the co-ordinate parallel to the axis of the journal, and U is constant. The thickness of the film h is some function of x, and for a journal Professor Reynolds takes the form,

h = a {1 + c sin(θ − φ0)},

in which the various quantities have the significance indicated in fig. 12. Reducing and integrating equation (1) with this value of h it becomes

 dp 6RµUc {sin(θ − φ0) − sin(φ1 − φ0)} — = ————————————————— dθ ⁠⁠a²{1 + c sin(θ − φ0)}³

(2)

φ1 being the value of θ for which the pressure is a maximum. In order to integrate this the right-hand side is expanded into a trigonometrical series, the values of the coefficients are computed, and the integration is effected term by term. If, as suggested by Professor J. Perry, the value of h is taken to be h = h0 + ax², where h0 is the minimum thickness of the film, the equation reduces to the form

 dp ⁠6µU ⁠C – — = —————— + —————— dx (h0 + ax²)² (h0 + ax²)³

(3)

and this can be integrated. The process of reduction from the form (1) to the form (3) with the latter value of h, is shown in full in The Calculus for Engineers by Professor Perry (p. 331), and also the final solution of equation (3), giving the pressure in terms of x.