POST-IMPACT VIBRATIONS OF A SQUARE NONLINEAR DISPATIVE OSCILLATOR
Olshanskiy Vasyl Pavlovych – Doctor of Physical and Mathematical Sciences, Full Professor, Professor at the Department of Physics and Theoretical Mechanics Petro Vasilenko Kharkiv National Technical University of Agriculture, Alchevskikh str. 44, Kharkov, Ukraine; 61022; OlshanskiyVP@gmail.com тел.: (066) 0100955
Slipchenko Maksym Volodymyrovych – Candidate of Technical Sciences (Ph. D.), Associate Professor, Associate Professor at the Department of Physics and Theoretical Mechanics Petro Vasilenko Kharkiv National Technical University of Agriculture, Alchevskikh str. 44, Kharkov, Ukraine; 61022; Slipchenko_M@ukr.net, тел.: (066) 7120989
Spolnik Oleksandr Ivanovych – Doctor of Physical and Mathematical Sciences, Full Professor, Head of the Department of Physics and Theoretical Mechanics Petro Vasilenko Kharkiv National Technical University of Agriculture, Alchevskikh str. 44, Kharkov, Ukraine; 61022; firstname.lastname@example.org, (067) 764-84-64
Tokarchuk Oleksіі – PhD, Associate Professor of the Department of «Technologіcal Processes and Equіpment for Processіng and Food Productіons» of the Vіnnytsіa Natіonal Agrarіan Unіversіty (St. Solnyshchnaya, 3, Vіnnytsіa, Ukraіne, 21008, e-maіl: email@example.comіn.ua.
An oscillator damped by viscous linear resistance, due to the instantaneous increase in its mass after impact, can become a dissipative oscillatory system under the action of dry or positional friction. In the article describes the oscillations of a dissipative oscillator with an asymmetric quadratically nonlinear elastic characteristic and dry Coulomb friction, arising as a result of an inelastic vertical impact of a rigid body on it. In the article, the Cox model is used, which does not take into account local deformations of solid bodies subjected to impact. The paper establishes the dependences on the impact velocity and the values of other parameters at which the effect of asymmetry of the elastic characteristic of the system may appear or may not appear. The conditions are derived when the dynamic effect of asymmetry of the power characteristic is manifested in the system. It consists in the fact that the maximum displacement of the oscillator (oscillation range) in the direction of the shock pulse is less than the opposite extreme displacement (range) after the shock oscillations. The existence of such a critical value of the shock impulse is established, the excess of which leads to the loss of motion stability. The second integral of the oscillation equation describes the movement of the oscillator in time, expressed in terms of Jacobi elliptic functions. An approximate formula for their calculation is proposed. Formulas are also derived to determine the time to reach extreme deviations of the system from the equilibrium position. This time is expressed in terms of elliptic integrals of the first kind, which refer to the tabulated functions. Examples of calculations are considered, where, in addition to using the derived formulas, numerical computer integration of the original nonlinear differential equation of motion is carried out. A comparison of the results obtained for the displacement values of a quadratically nonlinear oscillator with dry friction expressed in terms of Jacobi elliptic functions and obtained by numerical integration is carried out. Good consistency of the calculation results in two ways confirmed the adequacy of the obtained analytical solutions of the nonlinear Cauchy problem.
1. Olshanskiy, V.P., Bogomolov, O.V., & Bogomolov, O.V. (2018) Pro peretvorennya udarom zadempfovanoyi mehanichnoyi systemy v oscylyator [About the transformation by a blow of a damped mechanical system into an oscillator]. Visnyk KhNTUSG: Suchasni napryamky texnologiyi procesiv pererobnyh i harchovyx vyrobnyctv – Bulletin of KhNTUSG: Particularly in the process of processing and food processin, 194, 18-30. [In Ukrainian].
2. Olshanskiy, V.P., Burlaka, V.V. & Slipchenko, M.V. (2018) Kolyvannya pruzhnoyi systemy z suhym tertyam pry udari [Oscillations of an elastic system with dry friction on impact]. Tehnichnyy servis agropromyslovogo, lisovogo ta transportnogo kompleksiv – Technical service of agro-industrial, forest and transport complexes, 12, 230-33. [In Ukrainian].
3. Olshansky, V., Bogomolov, O., Bogomolov, O. et al. (2019) Oscillations with positional friction under mechanical shock. Teka. Quarterly journal of agro-food industry. 19, 1, 49-58.
4. Olshanskiy, V.P., & Olshanskiy, S.V. (2018) Pro efekt nesymetriyi sylovoyi harakterystyky kolyvalnoyi systemy pry mehanichnomu udari [On the effect of asymmetry of the force characteristic of an oscillating system during mechanical impact]. Vibracyi v tehnici ta tehnologiyah – Vibrations in engineering and technology, 2 (89), 36-40. [In Ukrainian].
5. Korn, G & Korn, T (1974) Spravochnik po matematike (dlya nauchnyih rabotnikov i inzhenerov) [Mathematics handbook (for scientists and engineers)]. Moskva : Nauka. [In Russia].
6. Gradshteyn, I.S., & Ryzhik, I.M. (1962) Tablitsyi integralov, summ, ryadov i proizvedeniy [Tables of integrals, sums, series and products]. Moskva : Nauka. [In Russia].
7. Abramovitz, M., & Stegan, I. (1979). Tablitsyi integralov, summ, ryadov i proizvedeniy [Special functions reference (with formulas, graphs, and math tables)]. Moskva : Nauka. [In Russia].
8. Yanke, E., Emde, F., & Lesch, F. (1977) Spetsialnyie funktsii [Special functions]. Moskva : Nauka. [In Russia].
The journal "Vibrations in engineering and technology" presents materials on the following issues
• Theory of processes and machines
• Mechanical Engineering and materialprocessing
• Processing and food production
The certificateof massmediaState registration:kv no 16643-5115 from 30.04.2010 .
Founder of the journal: Vinnytsia National Agrarian University
Kind of publication: journal
Type of publication : Scientific
Publication status: Domestic
Year of founding:
Periodicity: 4 times a year
Extent: 18.75 nominal printed pages
ISSN: 2306-8744 (printed version), (online)
Language of edition : (mixed languages) Ukrainian, Russian, English
The scope of the distribution and the category of readers: national, foreign, teaching staff, scientists, businessmen.
Periodical is included in the list of scientific professional editions of Ukraine approved by the Order of Ministry of Education and Sciences of Ukraine from 21.12.2015 No. 1328.
The journal "Vibrations in engineering and technology" is included in the "Catalogue of periodicals of Ukraine".
Journal subscription can be executed in each post office department.
Subscription Index is 99720.
Old version of site: http://vibrojournal.vsau.edu.ua/
In June 1994 the 2nd International Scientific and Technical Conference "Application of vibrations for technological purposes" was organized on the basis of Vinnytsia State Agricultural Institute. Leading experts in this field, noting the significant contribution to the school of Vibration Engineering under the leadership of P. S. Bernyk, proposed to create a professional all-Ukrainian scientific and technical journal "Vibration in engineering and technology..The journal was foundedat Vinnytsia State Agricultural Institute and P.S. Bernyk was elected to be the chief editor .
For all these years (since 1994) theJournal "vibration in engineering and technology" published 94 issues wherestudy of vibration effects, the creation of progressive energy saving technologies and equipment for their implementation were highlighted.
Currently Kaletnik H.M PhD , professor, academician NAAS is the chief editor of the "Vibrations in engineering and Technology"
The journal "Vibration in Engineering and technology", which has no analogues on the territory of Ukraine, is well known abroad.