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The orthonormality properties of mode forms for the vibration of a uniform beam are well known (1,2), and they are widely applied in the evaluation of coefficients for the general solution of beam vibrations. The same properties can be extended to a class of nonuniform beams. The orthogonality property is well known from the theorem of Strum-Liouville when the expression of the lineal density of the beam is considered as a weighting function. The normalization factors are evaluated here for different end conditions of the beam by using Rayleigh's technique.

A method is presented for the determination of an optimum shape of a convex shell of revolution with respect to volume and weight. The technique depends on selecting a multiparameter equation and varying the parameters to achieve a near optimum shape for prescribed failure criteria. As an illustration of the method, the first quadrant of the meridian (x/a)α + (y/b)β = 1 is selected. Here a, b, α, and β are positive constants not necessarily integers, with α and β equal to or greater than unity. Variations in shape are expressed in terms of the parameters b/a, α and β. The procedure is applied to the selection of a thin shell which will fit within the space defined by a circular cylinder of radius b and length 2a. The shell is optimized, in terms of α and β, with respect to volume and weight. The numerical iteration was performed by means of a digital computer.

The geometry of both the Archimedean and the logarithmic spiral, when formed into the third dimension, is considered. Changes in profile resulting from the choice of constant helix angle and from constant spacing between adjacent coils along the axis of the conic are presented. The axial and lateral load-deflection behavior of conical spiral springs is considered in terms of the above variables. Strain-energy methods are used to obtain the load-deflection relationships. Nonlinear effects due to compression of the large diameter coils are discussed.

With simple beam theory, solutions of normal functions for transverse vibration of a tapered beam are obtained in terms of generalized hypergeometric functions by the method of Frobenius. For the beams considered, the cross-sectional area and the area moment of inertia vary along the beam according to any two arbitrary powers of the longitudinal coordinate. The frequency equation is formulated, and the numerical results for many different tapered cantilever beams are presented.

Contains an entry for each author and co-author included in this publication.

Contains an entry for each author and coauthor included in this issue of the publication.