On the Exact Solution of Non-Uniform Torsion for Beams with Axial Symmetric Cross-Section
In the traditional theory of non-uniform torsion the axial displacement field is expressed as the product of the unit twist angle and the warping function. The first one, variable along the beam axis, is obtained by a global congruence condition; the second one, instead, defined over the cross-section, is determined by solving a Neumann problem associated to the Laplace equation, as well as for the uniform torsion problem. So, as in the classical theory the warping function doesn’t punctually satisfy the first indefinite equilibrium equation, the principal aim of this work is to develop a new theory for non-uniform torsion of beams with axial symmetric cross-section, fully restrained on both ends and loaded by a constant torque, that permits to punctually satisfy the previous equation, by means of a trigonometric expansion of the axial displacement and unit twist angle functions. Furthermore, as the classical theory is generally applied with good results to the global and local analysis of ship structures, two beams having the first one an open profile, the second one a closed section, have been analyzed, in order to compare the two theories. Keywords—Non-uniform torsion, Axial symmetric cross-section, Fourier series, Helmholtz equation, FE method.
Nonuniform elastic torsion
Abstract Linear and nonlinear uniform torsion of prismatic members has been much studied, but even in the linear case nonuniform torques have been comparatively neglected. Thin-walled cross-sections have received useful approximate treatments by Goodier and others, but a general theory seems still to be lacking. The linear theory given here corresponds to standard theories for varying elastic flexure. The torque may vary continuously or discontinuously along the member but, as with flexure, we deliberately avoid specifying in detail how it is applied. The present analysis is for cross-sections with two axes of symmetry. The examples given include an important one illustrating the possible sensitivity of beams to eccentricity of loading.
Knocking in an Internal-Combustion Engine
The question remains open of the relation between the phenomena of knocking in the engine and the explosion wave. The solution of this problem is the object of this paper. The tests were conducted on an aircraft engine with a pyrex glass window in the cylinder head. Photographs were then taken of various combinations of fuels and conditions.