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[Philosophical Magazine, Vol. xxxv. pp. 1—12, 1918.]
MODERN views respecting mechanical lubrication are founded mainly on the experiments of B. Tower*, conducted upon journal bearings. He insisted upon the importance of a complete film of oil between the opposed solid surfaces, and he showed how in this case the maintenance of the film may be attained by the dragging action of the surfaces themselves, playing the part of a pump. To this end it is " necessary that the layer should be thicker on the ingoing than on the outgoing side j-," which involves a slight displacement of the centre of the journal from that of the bearing. The theory was afterwards developed by 0. Reynolds, whose important memoir J includes most of what is now known upon the subject. In a later paper Sommerfeld has improved considerably upon the mathematics, especially in the case where the bearing completely envelops the journal, and his exposition^ is much to be recommended to those who wish to follow the details of the investigation. Reference may also be made to Harrison)], who includes the consideration of compressible lubricants (air).
In all these investigations the question is treated as two-dimensional. For instance, in the case of the journal the width—axial dimension—of the bearing must be large in comparison with the arc of. contact, a condition not usually fulfilled in practice. But MichellH has succeeded in solving the problem for a plane rectangular block, moving at a slight inclination over another plane surface, free from this limitation, and he has developed a system of pivoted bearings with valuable practical results.
It is of interest to consider more generally than hitherto the case of two dimensions. In the present paper attention is given more especially to the case where one of the opposed surfaces is plane, but the second not necessarily
* Proc. Imt. Mech. Eng., 1883, 1884.
t British. Association Address at Montreal, 1884; Scientific Papers, Vol. n. p. 344.
t Phil. Trans. Vol. 177, p. 157 (1886).
§ Zeitsehr.f. Math. t. 50, p. 97 (1904).
||  Camb. Trans. Vol. xxn. p. 39 (1913).
IF Zeitsehr.f. Math. t. 52, p. 123 (1905).g. Vol. xxvi. p. 256 (1888) ; Scientific Papers, Vol. in. p. 204. consider the phases represented by the factor eik (x~^'] in P. For the point 0, x = 0, r = p, and the exponential factor is e~ikt>. As in the ordinary theory of specular reflection, the same is true for every point in the plane AOA and therefore for the element of surface at A A whose volume is 27rRdRd£. For points in a plane