Asymptotic method in investigation of complex nonlinear oscillations of elastic bodies

2018;
: 58-67
https://doi.org/10.23939/ujmems2018.02.058
Received: October 11, 2018
Revised: November 23, 2018
Accepted: December 26, 2018
1
Lviv Polytechnic National University
2
Hetman Petro Sahaidachnyi National Army Academy
3
Lviv Polytechnic National University

The main provisions of the methodology of the study of complex oscillations of elastic bodies are outlined. Its main idea is as follows: a) on the basis of empirical studies, the change of the basic parameters of some forms of oscillation (usually smaller amplitude) is approximated by their analytical relations; b) these relationships are taken into account when constructing a mathematical model of the elastic body; c) for constructing and studying the solution of the obtained mathematical model of the process dynamics, the main ideas of the asymptotic integration of equations with partial derivatives are used.

Taken together, this allows us to obtain a two-parameter set of solutions that take into account the influence on the dynamics of the process of external and internal factors. The methodology is illustrated by the example of an elastic body, which simultaneously performs longitudinal and transverse vibrations. With the its aid  it  is established that resonant processes can exist in an elastic body not only by external actions, but also by the mutual influence of some forms of oscillation on others. 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 Yu. A. Mitropolskii, Asimptoticheskie metody v teorii nelineinykh kolebanii [Asymptotic methods in the theory of nonlinear oscillations]. Moscow, Russia: Nauka Publ., 1974. [in Russian].

[2] Yu. A. Mitropolskii, and B. I. Moseenkov, Asimptoticheskie resheniia uravnenii v chastnykh proizvodnykh [Asymptotic solutions of partial differential equations]. Kyiv, Ukraine: Vyshcha shkola Publ., 1976. [in Russian].

[3] P. Ya Pukach, I. V. Kuzio, and M. B. Sokil, “Nelineinye izgibnye kolebaniia vrashchaiushchikhsia vokrug nepodvizhnoi osi tel i metodika ikh issledovaniia” [“Nonlinear bending vibrations of bodies rotating around a fixed axis and the method of their investigation”], Izvestiia vysshikh uchebnykh zavedenii. Gornyi zhurnal [Vibrations in technique and technologies], no. 7, pp. 141-149, 2013. [in Russian].

[4] 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.

[5] 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.

[6] S. N. Vladimirov, A. S. Maidanovskii, S. S. Novikov, Nelineinye kolebaniia mnogochastotnykh avtokolebatelnykh sistem [Nonlinear oscillations of multi-frequency self-oscillatory systems]. Tomsk, Russia: Izdatelstvo Tomskogo universiteta, 1993. [in Russian].

[7] Ye. Kharchenko, and M. Sokil, “Bahatochastotni kolyvannia odnovymirnykh neliniino pruzhnykh rukhomykh seredovyshch ta metodyka pobudovy asymptotychnykh nablyzhen kraiovykh zadach, shcho yikh opysuiut” [“Multifrequency oscillations of one-dimensional nonlinearly elastic moving media and a method for constructing asymptotic approximations of boundary value problems describing them”], Mashynoznavstvo [Mechanical Engineering], no. 1, pp. 19-25, 2007. [in Ukrainian].

[8] M. B. Sokil, A. I. Andrukhiv, O. I. Khytriak, “Zastosuvannia khvylovoi teorii rukhu ta asymptotychnoho metodu dlia doslidzhennia dynamiky deiakykh klasiv pozdovzhno-rukhomykh system” [“Application of the wave motion theory and the asymptotic method for studying the dynamics of some classes of longitudinal and moving systems”], Visnyk Natsionalnoho universytetu "Lvivska politekhnika" [Bulletin of Lviv Polytechnic National University], no. 730, pp. 114-118, 2012. [in Ukrainian].

[9] M. Sokil, “Neliniini kolyvannia hnuchkykh trubchastykh til, vzdovzh yakykh rukhaietsia sutsilnyi potik seredovyshcha” [“Nonlinear oscillations of flexible tubular bodies, along which a continuous flow of medium moves”], Naukovyi visnyk NLTU Ukrainy [Scientific Bulletin of UNFU], vol. 24.10, pp. 351-356, 2014. [in Ukrainian].

[10] B. I. Sokil, and M. B. Sokil, “Vymusheni kolyvannia hnuchkykh trubchastykh til, vzdovzh yakykh rukhaietsia sutsilnyi potik seredovyshcha” [“Forced oscillations of flexible tubular bodies, along which a continuous flow of medium moves”], Visnyk Natsionalnoho universytetu "Lvivska politekhnika" [Bulletin of Lviv Polytechnic National University], no. 866, pp. 60-65, 2017. [in Ukrainian].

[11] A. S. Goldin, Vibratciia rotornykh mashin [Vibration of rotary machines]. Moscow, Russia: Mashinostroenie Publ., 2000. [in Russian].

[12] A. V. Goroshko, and V. P. Roizman, “Issledovanie dinamiki i snizhenie vibroaktivnosti turbonasosnogo agregata putem resheniia obratnykh zadach” [“Study of the dynamics and reduction of vibroactivity of the turbopump assembly by solving inverse problems”], Mashinostroenie i inzhenernoe obrazovanie [Mechanical Engineering and Engineering Education], no. 1, pp. 29-35, 2014. [in Russian].

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

[14] 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].

[15] A. Andrukhiv, B. Sokil, and 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.

[16] A. V. Goroshko, V. P. Royzman, A. Bubulis, and K. Juzėnas, “Methods for testing and optimizing composite ceramics-compound joints by solving inverse problems of mechanics”, Journal of Vibroengineering, vol. 16, issue 5, pp. 2178-2187, 2014.

[17] I. U. Albert, V. A. Petrov, and A. E. Skvortsova, “Analiz dinamicheskoi reaktcii konstruktivno-nelineinykh mekhanicheskikh sistem” [“Analysis of the Dynamic Response of Structural-Nonlinear Mechanical Systems”], Izvestiia VNIIG im. B. E. Vedeneeva [Proceeding of the VNIIG], vol. 241, pp. 38-59, 2002. [in Russian].

A. Andrukhiv, B. Sokil, M. Sokil, "Asymptotic method in investigation of complex nonlinear oscillations of elastic bodies", Ukrainian Journal of Mechanical Engineering and Materials Science, vol. 4, no. 2, pp. 58-67, 2018.