New operational ratios and their application to non-stationary tasks for rods based on S.P. Timoshenko theory

Abstract

Relevance. In order to study transient wave processes of deformation in rods on the basis of S.P. Timoshenko theory, it is necessary to have accurate analytical solutions to non-stationary problems in general form. Each accurate solution within this analytical model is an accurate description of the real process, serves as a criterion in assessing the accuracy of approximate solutions. When using operational calculus to analyze traveling waves, it is the inverse Laplace - Carson transformation that poses the greatest difficulty. It follows from the published works that the available solutions to some private problems either have a structure that does not allow to judge the main features of the investigated process, or their efficiency in calculations is achieved only in some rather limited areas of coordinate and time. This problem, which requires resolution, determined the purpose of this article. The aim of the work. The article is devoted to the development of new operational ratios and their application to the construction of accurate analytical solutions to the non-stationary problems of S.P. Timoshenko's theory for rods in a general form, in a physically visible and convenient form for practical calculations. Methods. The work uses methods of function theory of complex variable, operational calculus based on the integral Laplace - Carson transformation, methods of structure dynamics. Results. In general form three types of non-stationary tasks for semi-infinite rod based on Timoshenko theory are formulated. New operational ratios have been obtained. Based on these ratios, a method of inverse transformation without using a general conversion formula has been developed. Solutions of problems are recorded in the form of integrals from Bessel functions and, unlike solutions available in the literature, clearly show the wave nature of the studied processes, have a visual and compact appearance. An example of calculation is reviewed.