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Vibrations in engineering and technology

space SCIENTIFIC JOURNALS OF VINNITSA NATIONAL AGRARIAN UNIVERSITY

Issue №: 1(96)

Published: 2020.09.01
DOI: 10.37128/2306-8744-2020-1


Description:
The journal deals with the problems of vibration technologies and machines, mathematical methods of vibration process studies, information on design and technological development, presents teaching and methodological aspects of teaching in the Higher School of Applied Sciences, where vibration machines and technologies are studied.

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ON OSCILLATIONS DESCRIBING A GENERALIZED DIFFERENTIAL RELAY EQUATION

DOI: 10.37128/2306-8744-2020-1-6
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Olshansky Vasiliy – Doctor of Physical and Mathematical Sciences, Professor of the Department Physic of Theoretical Mechanics Kharkov National Technical University of Agriculture named after Petr Vasilenko (St. Alchevskih 44, Kharkov, 61002, Ukraіne, e-mail: OlshanskiyVP@gmail.com).

Olshansky Stanislav – PhD in Physics and Mathematics, Kharkov National Technical University of Agriculture named after Petr Vasilenko (St. Alchevskih 44, Kharkov, 61002, Ukraіne, e-mail: stasolsh77@gmail.com).

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: tokarchyk@vsau.vіn.ua.

Annotation

The motion of an oscillatory system with one degree of freedom, described by the generalized Rayleigh differential equation, is considered. The generalization is achieved by replacing the cubic term, which expresses the dissipative strength of the equation of motion, by a power term with an arbitrary positive exponent. To study the oscillatory process involved the method of energy balance. Using it, an approximate differential equation of the envelope of the graph of the oscillatory process is compiled and its analytical solution is constructed from which it follows that quasilinear frictional self-oscillations are possible only when the exponent is greater than unity. The value of the amplitude of the self-oscillations in the steady state also depends on the value of the indicator. A compact formula for calculating this amplitude is derived. In the general case, the calculation involves the use of a gamma function table. In the case when the exponent is three, the amplitude turned out to be the same as in the asymptotic solution of the Rayleigh equation that Stoker constructed. The amplitude is independent of the initial conditions. Self-oscillations are impossible if the exponent is less than or equal to unity, since depending on the initial deviation of the system, oscillations either sway (instability of the movement is manifested) or the range decreases to zero with a limited number of cycles, which is usually observed with free oscillations of the oscillator with dry friction. These properties of the oscillatory system are also confirmed by numerical computer integration of the differential equation of motion for specific initial data. In the Maple environment, the oscillator trajectories are constructed for various values of the nonlinearity index in the expression of the viscous resistance force and a corresponding comparative analysis is carried out, which confirms the adequacy of approximate analytical solutions.

Keywords: quasilinear self-oscillations, generalized Rayleigh equation, energy balance method, numerical integration of the Cauchy problem.

List of references

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2. Andronov, A. A., Vit, A. A., Khaikin, S. E. (1959). [Teoriya kolebaniy] Theory of oscillations. Moscow, State. Publishing House of Phys.-Math. literature. [іn Russіan].

3. Babakov I. M. (2004). Teoriya kolebaniy [Theory of fluctuations]. Moscow, Drofa. [іn Russіan].

4. Bogolyubov, N. N., Mitropolsky, Yu. A. (1974). [Asimptoticheskiye metody v teorii nelineynykh kolebaniy] Asymptotic methods in the theory of nonlinear oscillations. Moscow, Nauka. [іn Russіan].

5. Panko, Ya. G. (1976). Osnovy prikladnoy teorii kolebaniy i udara [Fundamentals of the applied theory of oscillations and shock]. Leningrad, Mechanical Engineering. [іn Russіan].

6. Suryaninov, N. G., Dashchenko, A. F., Belous, P. A. (2000). [Teoreticheskiye osnovy dinamiki mashin] Theoretical foundations of the dynamics of machines. Odessa, OGPU. [іn Russіan].

7. Kosterin, Yu. I. (1960). [Mekhanicheskiye avtokolebaniya pri sukhom trenii] Mechanical self-oscillations during dry friction. Moscow, Publishing House of the Academy of Sciences of the USSR. [іn Russіan].

8. Lazarev, G. S. (1971). [Avtokolebaniya pri rezanii metallov] Auto-oscillations during metal cutting. Moscow, Higher School. [іn Russіan].

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11. Olshanskiу, V. P., Olshanskiу, S. V. & Tishchenko, L. N. (2015). Kolyvannya dysypatyvnykh ostsylyatoriv [Dissipative oscillators oscillations]. Kharkiv, Miskdruk. [іn Ukrainian].

12. Olshanskiу, V. P., Tishchenko, L. N. & Olshanskiу, S. V. (2016). Dynamika dysypatyvnykh ostsylyatoriv [Dynamics of Dissipative Oscillators]. Kharkiv, Miskdruk. [іn Ukrainian].

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About journal

Topics of the journal:

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

Key information:
ISSN (print): 2306-8744
DOI: 10.37128/2306-8744

The certificateof massmediaState registration:kv no 16643-5115 from 30.04.2010 .
Founder of the journal: Vinnytsia National Agrarian University
Chief editor
Kaletnik Hryhorii Mykolaiovych
Ph.D., Prof., academician of NAAS, VNAU (Vinnytsia)
Chief Editor Deputy
V.V.Adamchuk
DsCProf., NSC IMEG NAAS of Ukraine (Kyiv)
Executive Secretary
Solona Olena
Ph.D., Assoc., VNAU (Vinnytsia)
Members of the editorial board
Bulgakov V.M.- Doctor of Technical Sciences, Professor, Acad.NAAS, National University of Life and Environmental Sciences of Ukraine
Veselovskaya N.R.- Doctor of Technical Sciences, Professor, Vinnytsia National Agrarian University
Derevenko A.I.- Ph.D., Associate Professor, Vinnytsia National Agrarian University
Zinkovsky A.P.- Doctor of Technical Sciences, Professor, the Institute of Strength Problems named after H.S.Pysarenko, NAS of Ukraine
Iskovych-Lototskyi R.D.- Doctor of Technical Sciences, Professor, Vinnytsia National Technical University
Kupchuk I.M.- Ph.D, Senior Lecturer, Vinnytsia National Agrarian University
Nadutyi V.P. - Doctor of Technical Sciences, Professor, Geotechnical Mechanics Institute named after  PolyakovM.S. NAS of Ukraine
Matveyeyv V.V.  - Ph.D., Professor, Academician of the National Academy of Sciences of Ukraine, the Institute of Strength Problems named after H.S.PysarenkoNAS of Ukraine
Olshanskyi V.P.  - Ph.D, Professor, Kharkiv National Technical University of Agriculture named after Petro Vasylenko
PoleyvodaYu.A.  - Ph.D., Associate Professor, Vinnytsia National Agrarian University
Spirin A.V.- Ph.D., Associate Professor, Vinnytsia National Agrarian University
Tverdohlib I.V.  - Ph.D., Associate Professor, Vinnytsia National Agrarian University
Tokarchuk O.A.- Ph.D., Associate Professor, Vinnytsia National Agrarian University
Tsurkan O.V.- Ph.D.  Associate Professor., Vinnytsia National Agrarian University
 Jordan Todorov Maksimov - Doctor of Technical Sciences, Prof., Gabrovo Technical University (Bulgaria)
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/

History of journal:

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.