Geodesic
Calculation of the natural frequencies of the transverse of cable oscillations at the area of application of insulation
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 1, pp. 70-77.

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Researches the transverse vibrations of the cable in the area where the insulation is applied to it. The considered mathematical model takes into account a wide range of factors affecting the oscillations: longitudinal motion, variable bending stiffness, environmental resistance, cable tension. The object belongs to a wide range of one-dimensional objects with moving boundaries. Moving boundaries complicate the description of such objects. The article introduces new variables that stop the boundaries. In this paper, using the Galerkin method, a fourth-order algebraic equation is obtained, which makes it possible to obtain two first natural frequencies of cable oscillations. The considered methods of statement and solution of the problem allow to solve the problems arising in the study of oscillations of objects with moving boundaries. Results can be used to ensure reliable operation of the technological installation for the manufacture of cables.
Keywords: oscillations of objects with moving boundaries, boundary value problems, resonant properties, natural frequencies.
Mots-clés : cable oscillations 
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V. N. Anisimov; V. L. Litvinov. Calculation of the natural frequencies of the transverse of cable oscillations at the area of application of insulation. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 1, pp. 70-77. http://geodesic.mathdoc.fr/item/SVMO_2019_21_1_a5/

[1] V. N. Anisimov, V. L. Litvinov, “Mathematical models of nonlinear longitudinal-cross oscillations of object with moving borders”, Vestn. Samar. Tekhn. Univ., Ser. Fiz.-Mat. Nauki, 19:2 (2015), 382–397 (In Russ.) | DOI | Zbl

[2] V. N. Anisimov, V. L. Litvinov, “Comparative analysis of linear and nonlinear models describing oscillations of systems with moving boundaries”, Proceedings of the X All-Russian Conference on Solid Mechanics (September 18–22, 2017, Samara, Russia), Samara State Technical University, Samara, 2017, 35–39 (In Russ.)

[3] O. A. Goroshko, G. N. Savin, Introduction in mechanics of one dimensional deformable bodies of variable length, Naukova Dumka, Kiev, 1971, 270 pp. (In Russ.)

[4] A. I. Vesnitskii, Vaves in systems with moving boundaries and loads, Fizmatlit, Moscow, 2001, 320 pp. (In Russ.)

[5] S. M. Sahebkar, M. R. Ghazavi, S. E. Khadem, M. H. Ghayesh, “Nonlinear vibration analysis of an axially moving drill string system with time dependent axial load and axial velocity in inclined well”, Mech. and Mach.Theory, 46:5 (2011), 743–760 | DOI | Zbl

[6] V. N. Anisimov, V. L. Litvinov, “Transverse vibrations of a rope moving in the longitudinal direction”, News of the Samara Scientific Center of the Russian Academy of Sciences, 19:4 (2017), 161–165 (In Russ.)

[7] W. D. Zhu, Y. Chen, “Theoretical and experimental investigation of elevator cable dynamics and control”, Trans. ASME. J. Vibr. And Acoust., 128:1 (2006), 66–78 | DOI

[8] A. I. Vesnitsky, A. I. Potapov, “Transverse oscillations of ropes in mine hoists”, Dynamics of systems, 7 (1975), 84–89 (In Russ.)

[9] L. O. Kechedgiyan, N. A. Pinchuk, A. M. Stolyar, “A problem of mathematical physics with moving boundary”, Vest.Vuzov North-Kaukaz. Region.Natural Sciences, 1 (2008), 22–27 (In Russ.) | Zbl

[10] A. A. Lezhneva, “Bending vibration of beam of variable length”, Izv. Acad. Nauk USSR. Mechanic of solidstate, 1 (1970), 159–161 (In Russ.)

[11] W. D. Zhu, N. A. Zheng, “Exact response of a translating string with arbitrarily varying length under general excitation”, Trans. ASME. J. Appl.Mech., 75:3 (2008), 031003 | DOI

[12] V. N. Anisimov, V. L. Litvinov, “Investigation of lateral vibrations of a rope moving in the longitudinal direction”, “Mathematical Modeling, Numerical Methods and Program Complexes” named after E.V. Voskresensky, Proceedings of the VIII International Scientific Youth School-Seminar, Middle Volga Mathematical Society, Saransk, 2018, 120–125 (In Russ.)

[13] V. N. Anisimov, V. L. Litvinov, “Calculation of natural frequencies of transverse vibrations of a viscoelastic rope moving in the longitudinal direction and having flexural rigidity”, Mathematical modeling and boundary value problems, Proceedings of the Fifth All-Russian Scientific Conference with international participation, v. 1, Samara State Technical University, Samara, 2008, 38–42 (In Russ.)

[14] V. S. Tikhonov, A. A. Abramov, “Transverse oscillations of a flexible thread of variable length in the flow”, Vestnik MGU. Ser. 1, 5 (1993), 45–48 (In Russ.) | Zbl

[15] K. I. Ragulsky, “Questions of dynamics of precision tape drive mechanisms”, Dynamics of cars, Nauka Publ., M., 1971, 169–-177 (In Russ.)

[16] Yu. P. Samarin, “About one nonlinear task for the wave equation in one-dimensional space”, Applied mathematics and mechanics, 26:3 (1964), 77–80 (In Russ.)

[17] V. I. Erofeev, D. A. Kolesov, E. E. Lisenkova, “Investigation of wave processes in a one-dimensional system lying on an elastic-inertial base, with a moving load”, Bulletin of Scientific and Technical Development, 62:6 (2013), 18–29 (In Russ.)

[18] Kotera Tadashi, “Vibration of a string with time-varying length”, Bulleten Japan Sosietyof Mechanical Engineers, 21:162 (1978), 1677–1684 | DOI | MR

[19] V. N. Anisimov, V. L. Litvinov, I. V. Korpen, “Use of the Kantorovich-Galerkin method for solving boundary value problems with conditions on moving boundaries”, News of the Russian Academy of Sciences. Solid mechanics, 2 (2018), 70–77 (In Russ.)

[20] Hu Ding, Li-Qun Chen, “Galerkin methods for natural frequencies of high-speed axially moving beams”, J. Sound and Vibr., 329:17 (2010), 3484–3494 | DOI