Будь ласка, використовуйте цей ідентифікатор, щоб цитувати або посилатися на цей матеріал: http://ena.lp.edu.ua:8080/handle/ntb/42872
Назва: On free oscillations of a quadratic nonlinear oscillator
Автори: Olshanskiy, Vasyl
Olshanskiy, Stanislav
Slipchenko, Maksym
Приналежність: Petro Vasilenko Kharkiv National Technical University
Бібліографічний опис: Olshanskiy V. On free oscillations of a quadratic nonlinear oscillator / Vasyl Olshanskiy, Stanislav Olshanskiy, Maksym Slipchenko // Ukrainian Journal of Mechanical Engineering and Materials Science. — Lviv : Lviv Politechnic Publishing House, 2017. — Vol 3. — No 2. — P. 1–10.
Bibliographic description: Olshanskiy V. On free oscillations of a quadratic nonlinear oscillator / Vasyl Olshanskiy, Stanislav Olshanskiy, Maksym Slipchenko // Ukrainian Journal of Mechanical Engineering and Materials Science. — Lviv : Lviv Politechnic Publishing House, 2017. — Vol 3. — No 2. — P. 1–10.
Є частиною видання: Ukrainian Journal of Mechanical Engineering and Materials Science, 2 (3), 2017
Журнал/збірник: Ukrainian Journal of Mechanical Engineering and Materials Science
Випуск/№ : 2
Том: 3
Дата публікації: 19-жов-2017
Видавництво: Lviv Politechnic Publishing House
Місце видання, проведення: Lviv
Теми: quadratically nonlinear oscillator
soft and rigid elasticity characteristic
free oscillations
analytical solution
periodic elliptic functions
Кількість сторінок: 10
Діапазон сторінок: 1-10
Початкова сторінка: 1
Кінцева сторінка: 10
Короткий огляд (реферат): A free oscillations of a system with one degree of freedom, caused either by the initial deviation from the stable equilibrium position or by the initial velocity provided by the oscillator in this position was considered. Analytical solutions of the nonlinear Cauchy problem for a second-order differential equation were constructed. The solutions are expressed in terms of Jacobi's periodic elliptic functions relating to occultation of special functions. Compact equals are derived for calculating the displacements of the oscillator and the oscillation periods for various methods of motion perturbation and for various variants of the elastic characteristic. The restrictions on the initial excitations for an oscillator with a soft elastic characteristic are determined, when its free oscillations are possible. The existence of a solution of the nonlinear dynamics problem in elementary functions is established. The behavior of an oscillator with a soft characteristic of elasticity under conditions of its freezing are studied. It is shown that from the derived equals, as special cases, the results known in the theory of linear oscillators, as well as oscillators with a purely quadratic nonlinearity, without a linear component, follow when the solution of the problem can be expressed in terms of Ateb-functions. The aim of the work was to derive new calculation equals for determining the displacements of a mechanical system with one degree of freedom under conditions of free oscillations, in the absence of friction. To achieve this objective, the representation of the second integral of the differential equation of motion due to the incomplete elliptic integral of the first kind were used. Using the known tables of the indicated integral, examples of calculations are given in which the probability of the derived equals is confirmed. According to the results of the study, it is also established that in the case of a quadratic elasticity characteristic of the linear component, the motion of the oscillator is described by the periodic elliptic Jacobi function, both in providing it with an initial deviation from the stable equilibrium position, and giving it the initial velocity in this position. In the case of a soft elasticity characteristic, free oscillations are possible only with certain restrictions on the initial perturbations of the system.
URI (Уніфікований ідентифікатор ресурсу): http://ena.lp.edu.ua:8080/handle/ntb/42872
Власник авторського права: © Національний університет „Львівська політехніка“, 2017
© Olshanskiy V., Olshanskiy S., Slipchenko M., 2017
Перелік літератури: [1] A. A. Larin, Ocherki istorii razvitija terii mehanicheskih kolebanij [Essays on the History of Development of the Theory of Mechanical Oscillations]. Sevastopol', Ukraine: Veber Publ., 2013. (In Russian).
[2] K. V. Avramov and Ju.V. Mihlin, Nelinejnaja dinamika uprugih sistem. T. 1. Modeli, metody, javlenija [Nonlinear Dynamics of Elastic Systems. Vol. 1. Models, methods, phenomena]. Moscow, Russia: Institut komp'juternyh issledovanij Publ., 2010. (In Russian).
[3] Ju. A. Mitropol's'kij, Izbrannye trudy v 2-h tomah [Selected works in 2 volumes]. Kyiv, Ukraine: Naukova dumka Publ., 2012. (In Russian).
[4] P. Ya. Pukach, Yakisni metody doslidzhennia neliniinykh kolyvalnykh system [Qualitative methods for investigating nonlinear oscillation systems]. Lviv, Ukraine: Lviv Polytechnic Publ., 2014. (In Ukrainian).
[5] V. M. Shatohin, Analiz i parametricheskij sintez nelinejnyh silovyh peredach mashin [Analysis and parametric synthesis of nonlinear power transmission of machines]. Kharkiv, Ukraine: NTU “HPI” Publ., 2008. (In Russian).
[6] V. P. Olshanskiy and S. V. Olshanskiy, Metod VBK v raschetah nestacionarnyh kolebanij oscilljatorov [The VBK method in calculations of non-stationary oscillations of oscillators]. Kharkiv, Ukraine: Mіs'kdruk Publ.,2014. (In Russian).
[7] V. P. Olshanskiy, S. V. Olshanskiy and L. M. Tishchenko, Dynamika dysypatyvnykh ostsyliatoriv [Dynamics of dissipative oscillators]. Kharkiv, Ukraine: Mіs'kdruk Publ., 2016. (In Ukrainian).
[8] I. S. Gradshtejn and I. M. Ryzhik, Tablicy integralov, summ, rjadov i proizvedenij [Tables of integrals, sums, series, and products]. Moscow, Russia: Nauka Publ., 1962. (In Russian).
[9] A. Abramovic and I. Stigan, Spravochnik po special'nym funkcijam (s formulami, grafikami i matematicheskimi tablicami) [Handbook of special functions (with formulas, graphs and mathematical tables)]. Moscow, Russia: Nauka Publ., 1979. (In Russian).
[10] E. Janke, F. Jemde and F. Lesh, Special'nye funkcii [Special functions]. Moscow, Russia: Nauka Publ.,1977. (In Russian).
[11] V. P. Olshanskiy et al., “Pro kolyvannia ostsyliatora z kvadratychno-neliniinoiu zhorstkistiu” [“About fluctuation of oscillator with quadratic nonlinear stiffness”], Tekhnichnyi servis ahropromyslovoho, lisovoho ta transportnoho kompleksiv [Technical service of agriculture, forestry and transport systems], no. 8, pp. 177–185,2017. (in Ukrainian).
[12] G. Korn and T. Korn, Spravochnik po matematike (dlja nauchnyh rabotnikov i inzhenerov) [Handbook of Mathematics (for scientists and engineers)]. Moscow, Russia: Nauka Publ., 1974. (In Russian).
References: [1] A. A. Larin, Ocherki istorii razvitija terii mehanicheskih kolebanij [Essays on the History of Development of the Theory of Mechanical Oscillations]. Sevastopol', Ukraine: Veber Publ., 2013. (In Russian).
[2] K. V. Avramov and Ju.V. Mihlin, Nelinejnaja dinamika uprugih sistem. T. 1. Modeli, metody, javlenija [Nonlinear Dynamics of Elastic Systems. Vol. 1. Models, methods, phenomena]. Moscow, Russia: Institut komp'juternyh issledovanij Publ., 2010. (In Russian).
[3] Ju. A. Mitropol's'kij, Izbrannye trudy v 2-h tomah [Selected works in 2 volumes]. Kyiv, Ukraine: Naukova dumka Publ., 2012. (In Russian).
[4] P. Ya. Pukach, Yakisni metody doslidzhennia neliniinykh kolyvalnykh system [Qualitative methods for investigating nonlinear oscillation systems]. Lviv, Ukraine: Lviv Polytechnic Publ., 2014. (In Ukrainian).
[5] V. M. Shatohin, Analiz i parametricheskij sintez nelinejnyh silovyh peredach mashin [Analysis and parametric synthesis of nonlinear power transmission of machines]. Kharkiv, Ukraine: NTU "HPI" Publ., 2008. (In Russian).
[6] V. P. Olshanskiy and S. V. Olshanskiy, Metod VBK v raschetah nestacionarnyh kolebanij oscilljatorov [The VBK method in calculations of non-stationary oscillations of oscillators]. Kharkiv, Ukraine: Mis'kdruk Publ.,2014. (In Russian).
[7] V. P. Olshanskiy, S. V. Olshanskiy and L. M. Tishchenko, Dynamika dysypatyvnykh ostsyliatoriv [Dynamics of dissipative oscillators]. Kharkiv, Ukraine: Mis'kdruk Publ., 2016. (In Ukrainian).
[8] I. S. Gradshtejn and I. M. Ryzhik, Tablicy integralov, summ, rjadov i proizvedenij [Tables of integrals, sums, series, and products]. Moscow, Russia: Nauka Publ., 1962. (In Russian).
[9] A. Abramovic and I. Stigan, Spravochnik po special'nym funkcijam (s formulami, grafikami i matematicheskimi tablicami) [Handbook of special functions (with formulas, graphs and mathematical tables)]. Moscow, Russia: Nauka Publ., 1979. (In Russian).
[10] E. Janke, F. Jemde and F. Lesh, Special'nye funkcii [Special functions]. Moscow, Russia: Nauka Publ.,1977. (In Russian).
[11] V. P. Olshanskiy et al., "Pro kolyvannia ostsyliatora z kvadratychno-neliniinoiu zhorstkistiu" ["About fluctuation of oscillator with quadratic nonlinear stiffness"], Tekhnichnyi servis ahropromyslovoho, lisovoho ta transportnoho kompleksiv [Technical service of agriculture, forestry and transport systems], no. 8, pp. 177–185,2017. (in Ukrainian).
[12] G. Korn and T. Korn, Spravochnik po matematike (dlja nauchnyh rabotnikov i inzhenerov) [Handbook of Mathematics (for scientists and engineers)]. Moscow, Russia: Nauka Publ., 1974. (In Russian).
Тип вмісту : Article
Розташовується у зібраннях:Ukrainian Journal of Mechanical Engineering And Materials Science. – 2017. – Vol. 3, No. 2



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