Constructive Solution of the 1918 Problem of Lord Rayleigh

Abstract

In 1918 Lord Rayleigh posed the problem of finding the profile (shape) of a one dimensional slider bearing which will support the greatest weight. In this paper we give, in closed form, the solution of this problem.

Optimal profiles for one dimensional slider bearings under technological constraints

“…Simple calculations shows that α opt ¼ 0:5357 and γ opt ¼ 0:7182 and F 2; max ; max ðα opt ; γ opt Þ ¼ 0:0687. This is already known [20]. Fig.…”

“…They are associated with different pressure boundary conditions [18,20]. Here the more general case of hybrid bearings is analyzed and Eqs.…”

“…Table 4 Constants to be used with Eqs. (19) and (20) for several lubricants [28]. Table 3 Γ 1 -1 Coefficient defined in Table 3 Γ 2 -1 Coefficient defined in Table 3 Γ…”

“…He showed that there exists a profile with a simple jump discontinuity which supports more weight than do profiles of the form 1 þ mx n or e βx with m, n and β positive [20]. This geometry eventually became known as the Rayleigh step bearing; it consists of two parallel surfaces -one having a rectangular cross-sectional dam.…”

“…Rayleigh tried to find the profile which supports the greatest weight but he only found a profile which caused the first variation of the weight functional to vanish. The problem has been fully solved in [20]. Using the calculus of variations, one can verify that the stepped bearing is the optimum unconstrained solution for a one-dimensional slider [10].…”