From the Rolls-Royce experimental archive: a quarter of a million communications from Rolls-Royce, 1906 to 1960's. Documents from the Sir Henry Royce Memorial Foundation (SHRMF).
Mathematical analysis of the geometry of rollers using hyperboloid of revolution equations.
| Identifier | WestWitteringFiles\R\2October1927-November-1927\ 94 | |
| Date | 1st May 1927 guessed | |
| -4- to the axis of revolution, being at a constant (shortest) dis- tance a from that axis. It is easily shown that the axes of the rollers all lie on the hyperboloid of revolution of one sheet whose equation is:- x² + y² - z² tan² α = a², (I) the origin of co-ordinates being here at the centre of the neck and the axis of z being the axis of revolution. The two sheets which form the envelope of the rollers (and which give the required surfaces) are really far more complicated mathematical surfaces. To construct the radii r1 and r2 of their cross-section at distance z from the plane of symmetry. Let fig. 2 represent such a cross-section; O¹ is the point where the axis of revolu-tion meets the plane of section; NC is the projection on this plane of the axis of the roller, O¹N the projection of the shortest distance between the axis of the roller and the axis of revolution. The section of the roller itself is an ellipse, centre C, and of semi-axes c sec. α , c. If we now revolve this ellipse about O¹, it touches externally and internally two concentric circles of radii r1 and r2, which give the cross sections of the bearing surfaces required. [Diagram with labels: O', a, r1, r2, N, P, R, H, c, C, S, K, Q, <- z tan α, <- c sec α ->] Fig 2 FIG. 2. | ||