# Resonant phenomena of elastic bodies that perform bending and torsion vibrations

2018;
: 65-73

Received: May 05, 2018
Revised: June 23, 2018
Accepted: June 26, 2018

A. Andrukhiv, B. Sokil, M. Sokil, "Resonant phenomena of elastic bodies that perform bending and torsion vibrations", Ukrainian Journal of Mechanical Engineering and Materials Science, vol. 4, no. 1, pp. 65-73, 2018.

Authors:
1
Lviv Polytechnic National University
2
Національна академія сухопутних військ імені гетьмана Петра Сагайдачного
3
Lviv Polytechnic National University

The method of study of the influence of torsional oscillations of one-dimensional models of nonlinear elastic bodies, along which moves with a constant velocity continuous flow of inelastic homogeneous medium, into bending, is developed. It is believed that information on torsional oscillations is known from empirical studies. Based on the latter, the refined model of the dynamics of the process of the investigated object is constructed. The latter is a boundary value problem for nonlinear nonautonomous differential equations with partial derivatives. The imposed restrictions on power factors and the main parameters of torsional oscillations allow for the analytical study of the dynamics of the process to use the basic ideas of the asymptotic integration of equations with partial derivatives. With their help, we obtain a two-parameter set of solutions that describe the determinant parameters of bending vibrations of an elastic body. It is established that for the considered elastic body there can be resonance oscillations, which are caused not only by external factors, but also by internal - torsional oscillations. Regarding the law of the change in the basic parameters of the dynamics of the bending motion of an elastic body, its rotation around the vertical axis reduces the frequency of its own flexural oscillations of the body, and even small torsional oscillations cause an additional periodic action on the transverse. In connection with the above bending vibrations of the elastic body, which performs complex oscillations (torsion and bending), resonances are possible both at the frequency of the external periodic perturbation and at the frequencies of the torsional oscillations (internal resonances). The amplitude of the transition through the resonance: a) at the basic frequency of external perturbation takes less value for elastic bodies of greater flexural rigidity and for higher values of the relative relative motion of the medium; b) at the frequency of torsional oscillations for larger valuesof the angular velocity takes more importance; c) with "fast" transition through resonance at the frequency of external or internal perturbation is less than with "slow". The obtained results can serve as the basis for the choice of operating parameters of elastic elements of machines that carry out complex oscillations.

[1] N. N. Bogoliubov, and Iu. A. Mitropolskii, Asimptoticheskie metody v teorii nelineinykh kolebanii [Asymptotic methods in the theory of nonlinear oscillations]. Moscow, Russia: Nauka Publ., 1974. [in Russian].

[2] D. Ye. Khaustov, “Umovy vnutrishnoho rezonansu viiskovoi husenychnoi tekhniky” [“Conditions of internal resonance of military tracked vehicles”], Systemy ozbroiennia i viiskova tekhnika [Systems of Arms and Military Equipment], no. 1 (45), pp. 73-76, 2016. [in Ukrainian].

[3] P. I. Ohorodnikov, V. M. Svitlytskyi, and V. I. Hohol, “Doslidzhennia zv’iazku mizh pozdovzhnimy i krutylnymy kolyvanniamy burylnoi kolony” [“Investigation of the connection between the longitudinal and torsional oscillations of the drill column”], Naftova haluz Ukrainy [The oil industry of Ukraine], no. 2, pp. 6-9, 2014. [in Ukrainian].

[4] I. A. Lukovskii, A. V. Solodun, and A. N. Timokha, “O vnutrennikh rezonansakh kolebanii zhidkostei v konicheskikh bakakh” [“On the internal resonances of oscillations of liquids in conical tanks”], Prykladna hidromekhanika [Applied hydromechanics], vol. 15, no. 2, pp. 46-52, 2013. [in Russian].

[5] P. Ya. Pukach, І. V. Kuzio, and M. B. Sokil, “Nelineinye izgibnye kolebaniia vrashchaiushchikhsia vokrug nepodvizhnoi osi tel i metodika ikh issledovaniia” [“Non-linear flexural vibrations of revolving bodies around a fixed axis and methods of their study”], Izvestiia vysshikh uchebnykh zavedenii. Gornyi zhurnal [News of the Higher Institutions. Mining Journal], no. 7, pp. 141-149, 2013. [in Russian].

[6] R. M. Rohatynskyi, Naukovo-prykladni osnovy stvorennia hvyntovykh transportno-tekhnolohichnykh mekhanizmiv [Scientific and applied foundations of creation of screw transport and technological mechanisms]. Ternopil, Ukraine: TNTU imeni Ivana Puliuia Publ., 2014. [in Ukrainian].

[7] I. V. Kuzio, Ye. V. Kharchenko, and M. B. Sokil, “Dynamichni protsesy u seredovyshchakh, yaki kharakteryzuiutsia pozdovzhnim rukhom, ta vplyv kraiovykh umov na amplitudu i chastotu yikh kolyvan” [“Dynamic processes in environments characterized by longitudinal motion and the effect of boundary conditions on the amplitude and frequency of their oscillations”], Vibratsii v tekhnitsi i tekhnolohiiakh [Vibrations in technique and technologies], no. 3 (48), pp. 53-56, 2007. [in Ukrainian].

[8] B. M. Hevko, and Yu. F. Pavelchuk, “Modeliuvannia kolyvan mekhanichnoi systemy pidvisnyi soshnyk: teoretychnyi analiz” [“Modeling of fluctuations of the mechanical system of a suspension hole: theoretical analysis”], Innovative solutions in modern science, no. 1 (10), pp. 1-9, 2017. [in Ukrainian].

[9] Y. A. Mitropol'skii, and B. I. Sokil, “On the application of Ateb-functions to the construction of an asymptotic solution of the perturbed nonlinear Klein-Gordon equation”, Ukrainian Mathematical Journal, no. 5 (50), pp. 754-760, 1998.

[10] B. I. Sokil, “Construction of asymptotic solutions of certain boundary-value problems for the nonautonomous wave equation”, Journal of Mathematical Sciences, no. 1 (96), pp. 2878-2882, 1999.

[11] V. I. Huliaiev, and O. I. Borshch, “Spiralni khvyli v zakruchenykh pruzhnykh trubchastykh sterzhniakh, shcho obertaiutsia z vnutrishnim potokom ridyny” [“Spiral waves in twisted elastic tubular rods, rotating with internal fluid flow”], Akustychnyi visnyk [ Acoustic Bulletin], vol. 10, no. 3, pp. 12-18, 2007. [in Ukrainian].

[12] Iu. A. Mitropolskii, “O postroenii asimptoticheskogo resheniia vozmushchennogo uravneniia Kleina–Gordona” [“On the construction of an asymptotic solution of the perturbed Klein-Gordon equation”], Ukrainskyi matematychnyi zhurnal [Ukrainian Mathematical Journal], vol. 47, no. 9, pp. 1209-1216, 1995. [in Russian].

[13] Iu. A. Mitropolskii, “O postroenii asimptoticheskogo resheniia vozmushchennogo uravneniia Kleina–Gordona” [“On the construction of an asymptotic solution of the perturbed Brezerton equation”], Ukrainskyi matematychnyi zhurnal [Ukrainian Mathematical Journal], vol. 59, no. 1, pp. 58-71, 1998. [in Russian].

[14] N. S. Pirogova, and P. A. Taranenko, “Raschetno-eksperimentalnyi analiz sobstvennykh i kriticheskikh chastot i form visokooborotnogo rotora mikrogazoturbinnoi ustanovki” [“Calculation-experimental analysis of intrinsic and critical frequencies and forms of a high-speed rotor of a micro-gas turbine unit”], Vestnik Iuzhno-Uralskogo gosudarstvennogo universiteta. Seriia «Mashinostroenie» [Bulletin of the South Ural State University. Series “Mechanical Engineering Industry”], vol. 15, no. 3, pp. 37-47, 2015. [in Russian].

[15] P. M. Senik, B. I. Sokil, “Ob opredelenii parametrov nelineinoi kolebatelnoi sistemy po amplitudno-chastotnoi kharakteristike” [“Non-linear flexural vibrations of revolving bodies around a fixed axis and methods of their study”], Matematicheskie metody i fiziko-mekhanicheskie polia [Mathematical methods and physicomechanical fields], no. 7, pp. 94-99, 1977. [in Russian].

[16] M. B. Sokil, and O. I. Khytriak, “Khvylova teoriia rukhu v doslidzhenni kolyvan hnuchkykh elementiv pryvodu ta transportuvannia z urakhuvanniam yikh pozdovzhnoho rukhu” [“Wave theory of motion in the study of oscillations of flexible elements of the drive and transportation in view of their longitudinal motion”], Viiskovo-tekhnichnyi zbirnyk [ Military Technical Collection], vol. 1, pp. 102-105, 2011. [in Ukrainian].

[17] I. M. Babakov, Teoriia kolebanii [Theory of oscillations]. Moscow, Russia: Nauka Publ., 1965. [in Russian].