Bulletin of TUIT: Management and Communication Technologies


Mathematical simulation of static and dynamic processes of strain taking into account of transverse bending under loading is presented in the paper in linear and geometrically nonlinear statements. An extensive analysis of research work carried out in universities and science centers all over the world is given, the relevance of the problem and the areas of solution application are emphasized. The mathematical correctness of the problem statement is shown. Variations of kinetic, potential energy, volume and surface forces are determined for mathematical models of static and dynamic processes of strain with taking into account of transverse bending of loaded rods in linear and geometrically nonlinear statements. Based on the theory of elastic strains and the refined theory of Vlasov-Dzhanelidze-Kabulov, and using the Ostrogradsky-Hamilton variation principle, a mathematical model of the statics and dynamics of the process of rod points displacement is developed for transverse bending in linear and geometrically nonlinear statements. Equations of a mathematical model with natural initial and boundary conditions in a vector form are given. A computational algorithm is developed for calculating the statics and dynamics of rods under loading in linear and geometrically non-linear statements using the central finite differences of the second order of accuracy. The strain processes when the rod is rigidly fixed at two edges are considered in linear and geometrically nonlinear statements. The calculation results obtained are given in the form of graphs. The propagation of longitudinal, transverse vibrations and the angle of inclination along the length of the rod was studied at different points of times. Linear and geometrically non-linear vibration results are analyzed and compared.

First Page


Last Page



[1] Li Z., Li J. Fundamental equations for dynamic analysis of rod and pipe string in oil-gas wells and application in static buckling analysis// Journal of Canadian Petroleum Technology. – 2002. – Vol. 41(5).– Pр. 44-53. [2] Hijmissen J.W., W.T. van Horssen. On aspect of damping for a vertical beam with tuned mass damper at the top // Nonlinear dynamics. - 2007. – Vol. 50(1). – Pр. 169-190. [3] Asghari M., Kahrobaiyan M.H., Ahmatian M.T. A nonlinear Timoshenko beam formulation based on the modified couple stress theory // International journal of Engineering Science. – 2010. - Vol. 48. – Pр: 1749-1761. [4] Zhu W., Chung J. Nonlinear lateral vibrations of a deploying Euler-Bernoulli beam with a spinning motion // International Journal of Mechanical Sciences. – 2015. – Vol. 90. – Pр. 200-212. [5] Mamandi A., Kargarnovin M.H. Nonlinear dynamic analysis of an inclined Timoshenko beam & subjected to a moving mass/ forse with beam’s weight included// shock and Vibration. – 2011. - Vol. 18(6). – Pр. 875-891. [6] Piovan M.T., Sampaio R.A study on the dynamics of rotating beams with functionally graded properties // Journal of Sound and Vibration. – 2009.– Vol. 327. – Pр. 134-143. [7] Hijmissen J.W., Van Horssen W.T. On transverse vibrations of a vertical Timoshenko beam // Journal of sound and vibration. – 2008. - Vol. 314. - Pр. 161 -179. [8] Anarova Sh.A., Nuraliyev F.M., Usmonov B.Sh., Chulliyev Sh.I. Numerical solution of the problem of spatially loaded rods in linear and geometrically nonlinear statements // International Journal of Engineering & Technology, 7 (4) (2018) 4563-4569. [9] Erof eev V I, Orehova O.I. Dispersion of a bending torsion wave propagating in a beam// Volga Scientific Journal. – 2011. – 4.1, № 2. – Pp. 715; 42, № 3. – С 20-26. (In Russian). [10] Erofeev V I 2012 Bending-torsional, longitudinally-bending and longitudinallytorsional waves in the rods // Bulletin of Scientific and Technical Development 5(57)(1) 3 -18. (In Russian). [11] Zinchenco A.S. Propagation of longitudinalbending and longitudinal-torsional waves in a rod // Scientific search . - 2012. - № 2 (6). - Pp. 38-40. (In Russian). [12] Kabulov V. K. 1966. Algorithmization in elasticity theory and deformation theory of plasticity. Tashkent: Fan, -394 p. (In Russian). [13] Mikhlin S. G. 1970. Variation methods in mathematical physics. -Moscow: Science, - 512p. (In Russian). [14] Erof eev V I, Kazhaev V V and Semerikova N L 2008 Nonlinear bending stationary waves in the beam of Tymoshenko 6(1) 348-358. (In Russian). [15] Nazirov Sh. A. 2013. Computational algorithms, realizing three-dimensional nonlinear -5 -4 -3 -2 -1 0 1 0 100 200 300 400 500 600 700 800 α lin. non-lin. 22 “Bulletin of TUIT: Management and Communication Technologies” Anarova Sh.A., Olimov M., Ismoilov Sh.M. 2018, 1 (43) mathematical model of the theory of elasticity and plasticity. Problems of Computational and Applied Mathematics: - Tashkent, Centrum RPPIAPK Issue-129. P. 9-22. [16] Novozhilov V. V 1948. Fundamentals of nonlinear theory of elasticity. - Leningrad: GITTL, - 211 p. (In Russian). [17] Samarsky A. A. 1971. Introduction to the theory of diffrence schemes. - Moscow: Nauka,-532 p. (In Russian). [18] Anarova SH.A., Yuldashev T. Mathematical model of nonlinear equations of rod vibrations under dynamic loading // UZB. Journal « Problems of computer science and energy. – Tashkent, 2014. - № 6.- Pp:. 36-42. [19] Anarova Sh.A., Safarov Sh.Sh. Mathematical software of fluctuations of bars with a spatial dynamic loding// Problems of Computational and Applied Mathematics – Tashkent 2016. - № 4. - Pp. 20-34. [20] Anarova Sh. A. Yuldashev, T. 2018. Derivation of Diffrential Equations of Oscillation of Rods in Geometrically Nonlinear Statement, Problems of Computational and Applied Mathematics. Tashkent, N2. - Pp. 72-105. (In Russian). [21] Vasidzu K. 1987. Variation methods in the theory of elasticity and plasticity: Translated from English. - Moscow: Mir, -542 p. (In Russian). [22] Anarova Sh.A. Algorithm of solution of geometrically nonlinear problem of rods with arbitrary mechanical geometrical characteristics // International Journal of Advanced Research in Science, Engineering and Technology Volume 4, Issue 11 (November, 2017). - Pр. 4796-4815. [23] Umbetqulova A.B., Hadjiyeva L.A.,Mamennikov B.B. On the analysis of nonlinear vibrations of drill rods with finite deformations // News of NAS RK – 2012. - № 1. –Pp:. 10-14. [24] Arvin H., Bakhtiari-Nejad F. Nonlinear modal analysis of a rotating beam // International Journal of Non-linear Mechanics. - 2011. - Pр. 1-15. [25] Hadjiyeva L.A Umbetqulova A.B., On the approximation of nonlinear oscillations of a compressed-twisted drill rod at finite strains // News of NAS RK. Physics and Mathematics Series. – 2014. - № 1 (293). –Pp:. 6975. (In Russian). [26] Anarova Sh.A., Nuraliev F.M., Dadenova G. Mathematical model of spatially loaded bars with account of torsion function and transverse shears // International Journal of Technical Research and Applications e-ISSN: 2320-8163, www.ijtra.com Volume 4, Issue 1 (January-February, 2016). - Pр. 22-32. [27] Mamandi A. Kargarnovin M.H. Farsi S. Nonlinear Vibration Solution for an Inclined Timoshenko Beam unde the Action of a Moving Force with Constant / Nonconstant Velocity // Journal of mathematical Sciences . - 2014.– Vol. 201(3) – Рp. 363-383. [28] Anarova Sh.A., Samidov M.N. Automation of the solution of geometrically nonlinear problems of rods with arbitrary mechanical and geometric. \\ Certificate of official registration of the program for electronic computers AIS RUz. № DGU 05459. 21. 06.2018 y.