Geometrical characteristics of flat-faced bodies of revolution

Abstract

Transition curves between flat faces and parallel middle bodies are developed as families of special polynomials termed cubic polynomials. The curves start and end with zero curvature to provide no discontinuities in curvature at the junction with the flat faces and the parallel middle bodies. Each cubic polynomial is a linear combination of independent polynomials controlled by adjustable parameters. Permissible ranges of the adjustable parameters are examined with respect to selected geometrical constraints such as inflection points. Nomenclature = polynomial coefficients = constant of integration = diameter of parallel middle body = diameter of flat face = integral defined in Eq. (31) — cubic polynomial ] = polynomial corresponding to k^ ] = polynomial corresponding to ki ] = polynomial corresponding to ko = curvature = rate of change of curvature with arc length = rate of change of curvature with arc length at x = I = rate of change of curvature with arc length at x = 0 = polynomial for restraining conditions = arc length = axial coordinate = axial length of forebody - normalized axial coordinate = radius = normalized radius = unspecified constant = adjustable conditions = unspecified constant = restraining conditions = unspecified constant = single differentiation with x = double differentiation with x = triple differentiation with x