Structural Mechanics of Engineering Constructions and BuildingsStructural Mechanics of Engineering Constructions and Buildings1815-52352587-8700Peoples’ Friendship University of Russia (RUDN University)2301110.22363/1815-5235-2020-16-1-62-75Research ArticleNew operational ratios and their application to non-stationary tasks for rods based on S.P. Timoshenko theoryZonenbergAlexander L.<p>chief specialist engineer, Residential and Public Buildings Construction Department</p>zonenberg@list.ruJoint Stock Company “TSNIIEP zhilishcha - Institute for Complex Design of Residential and Public Buildings”15122020161627527022020Copyright © 2020, Zonenberg A.L.2020<p>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.</p>transient wave processS.P. Timoshenko theoryrodstraveling wavesoperational ratioLaplace - Carson integral transformationBessel functionsпереходные волновые процессытеория С.П. Тимошенкостержнибегущие волныоперационное соотношениеинтегральное преобразование Лапласа - Карсонабесселевы функции[Kolsky H. Volny napryazheniya v tverdyh telah [Stress waves in solids]. Moscow: Izd-vo inostrannoi literatury Publ.; 1955. (In Russ.)][Timoshenko S.P. Kurs teorii uprugosti [Course in the Theory of Elasticity]. Kiev: Naukova dumka Publ.; 1972. (In Russ.)][Grigolyuk E.I., Selezov I.T. Neklassicheskie teorii kolebanij sterzhnej, plastin i obolochek [Nonclassical Theories of Vibrations of Bars, Plates and Shells]. Advances in Sciences and Engineering. Mechanics of Deforming Solids. Moscow: VINITI Publ.; 1973. (In Russ.)][Selezov I.T. O razvitii teorii Timoshenko poperechnyh kolebanij uprugih sterzhnej [On the development of the Timoshenko theory of transversal oscillations of elastic rods]. Journal of Machinery Manufacture and Reliability. 2016;45(1):13–20.][Su Yu-Chi, Ma Chien-Ching. Theoretical analysis of transient waves in a simply-supported Timoshenko beam by ray and normal mode methods. International Journal of Solids and Structures. 2001;48(3–4):535–552.][Su Yu-Chi, Ma Chien-Ching. Transient wave analysis of a cantilever Timoshenko beam subjected to impact loading by Laplace transform and normal mode methods. International Journal of Solids and Structures. 2012;49(9): 1158–1176.][Wang X.Q., So R.M.C. Timoshenko beam theory: A perspective based on the wave-mechanics approach. Wave Motion. 2015;57:64–87.][Abramyan A.K., Indeitsev D.A., Postnov V.A. Running and Standing Waves of Timoshenko Beam. Mechanics of Solids. 2018;53(2):203–210.][Slepyan L.I., Yakovlev Yu.S. Integral'nye preobrazovaniya v nestacionarnyh zadachah mekhaniki [Integral Transformations in Non-Stationary Problems of Mechanics]. Leningrad: Sudostroenie Publ.; 1980. (In Russ.)][Leonard R.W., Budiansky B. On traveling waves in beams. NACA Repts. 1954;(1173):389–415.][Dengler M.A. Transversale Wellen in Stäben und Platten unter stoßförmiger Belastung. Österr. Ing.-Arch. 1956;10(1):39–66.][Flügge W., Zajac E.E. Bending impact waves in beams. Ingenieur-Archiv. 1959;28(1):59–70.][Lurie A.I. Operacionnoe ischislenie i ego prilozheniya k zadacham mekhaniki [Operational Calculus and its Application to the Problems in Mechanics]. Moscow, Leningrad: Gostekhizdat Publ.; 1950. (In Russ.)][Ditkin V.A., Prudnikov A.P. Spravochnik po operacionnomu ischisleniyu [Handbook of operational calculations]. Moscow: Vysshaya shkola Publ.; 1965. (In Russ.)][Ditkin V.A., Prudnikov A.P. Operacionnoe ischislenie [Operational calculus]. Moscow: Vysshaya shkola Publ.; 1966. (In Russ.)][Doetsch G. Rukovodstvo k prakticheskomu primeneniyu preobrazovaniya Laplasa [Guide to the Applications of Laplace Transforms]. Moscow: Nauka Publ.; 1965. (In Russ.)][Efros A.M., Danilevsky A.M. Operacionnoe ischislenie i konturnye integraly [Operational Сalculus and Contour Integrals]. Kharkiv: Gos. nauch.-tekhn. izd-vo Publ.; 1937. (In Russ.)][Watson G.N. Teoriya besselevyh funkcij [А treatise on the theory of Bessel functions]. Part 1. Moscow: Izd-vo inostrannoi literatury; 1949. (In Russ.)][Fikhtengol'ts G.M. Osnovy matematicheskogo analiza [Foundations of mathematical analysis]. Vol. 2. Moscow: Nauka Publ.; 1964. (In Russ.)][Uflyand Ya.S. Rasprostranenie voln pri poperechnyh kolebaniyah sterzhnej i plastin [Wave propagation in rods and plates undergoing transverse vibrations]. Prikladnaya matematika i mekhanika [J. Appl. Math. Mech.]. 1948; 12(3):287–300. (In Russ.)][Sagartz M.J., Forrestal M.J. Bending stresses propagating from the clamped support of an impulsively loaded beam. AIAA Journal. 1972;10(10):1373–1374. (Publ. online 17 May 2012). https://doi.org/10.2514/3.6628]