FREE OSCILLATOR OSCILLATIONS IN THE PRESENCE OF QUADRATIC VISCOUS RESISTANCE AND DRY FRICTION
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
Zamrii Mykhailo – 4th year student of specialty 208 of Agroengineering, Faculty of Engineering and Technology of Vinnytsia National Agrarian University (str. Sonyachna, 3, Vinnytsia, 21008, Ukraine, e-mail: email@example.com).
The article is devoted to the derivation of formulas for calculating the ranges of free damped oscillations of a double nonlinear oscillator. Using the Lambert function and the first integral of the nonlinear differential equation of motion, formulas are derived for calculating the ranges of free damped oscillations of a linearly elastic oscillator under the combined action of the forces of quadratic viscous resistance and Coulomb dry friction. The calculations involve a table of the specified special function of the negative argument. It is shown that the presence of viscous resistance reduces the duration of free oscillations to a complete stop of the oscillator. The set dynamics problem is also approximately solved by the energy balance method, and a numerical integration of the nonlinear differential equation of motion on a computer is carried out. The satisfactory convergence of the numerical results obtained in various ways confirmed the suitability of the derived closed formulas for engineering calculations. In addition to calculating the magnitude of the oscillations, the energy balance method is also used for an approximate solution of the inverse problem of dynamics, by identifying the values of the coefficient of quadratic resistance and dry friction force in the presence of an experimental vibrogram of free damped oscillations. An example of identification is given. This information on friction is needed to calculate forced oscillations, especially under resonance conditions. It is noted that from the obtained results, in some cases, well-known formulas follow, where the quadratic viscous resistance is not associated with dry friction.
1. Babakov, I. M. (2004). Teoriya kolebanij [Oscillation theory]. Moskva : Drofa. [In Russia].
2. Burlaka, V. V., Olshanskiy, V. P., & Malec, O. M. (2014). Do rozraxunku kolyvan` mexanizmiv pry spil`nij diyi syl sukhogo ta vyazkogo tertya [To the calculation of oscillations of mechanisms under the joint action of dry and viscous friction forces]. Mekhanizaciya silskogospodarskogo vyrobnycztva: Visnyk XNTUSG – Mechanization of agricultural production: Bulletin of KhNTUSG. Kharkiv, 148, 19-24. [In Ukrainian]
3. Olshanskiy, V. P., Tishhenko, L. M., & Olshanskiy, S. V. (2016). Dynamika dysypatyvnykh oscylyatoriv [Dynamics of dissipative oscillators]. Kharkiv : Miskdruk. [In Ukrainian].
4. Blekhman I. I. (Ed) (1979) Vibraczii v tekhnike. Spravochnik v shesti tomakh. Kolebaniya nelinejny`kh mekhanicheskikh sistem [Vibration in technology. Handbook in six volumes. Oscillations of nonlinear mechanical systems]. (Vols. 2). Moskva : Mashinostroenie. [In Russia].
5. Panovko Ya. G. (1980). Vvedenie v teoriyu mekhanicheskikh kolebanij [Introduction to the theory of mechanical vibrations]. Moskva : Nauka. [In Russia].
6. Vasylenko, M. V., & Aleksejchuk, O. M. (2004). Teoriya kolyvan` i stijkosti rukhu [Theory of oscillations and stability of motion]. K. : Vyshcha shkola. [In Ukrainian].
7. Olshanskiy, V. P., & Olshanskiy, S. V. (2014). Funkcy`ya Lamberta v zadache kolebanyj matematycheskogo mayatnyka [Lambert function in the problem of oscillations of a mathematical pendulum]. Visnyk NTU «KhPI». Seriya: Matematychne modelyuvannya v tekhnici ta tekhnologiyah [Bulletin of NTU "KhPI". Series: Mathematical modeling in engineering and technology]. Kharkiv, 18(1061). 116-119. [In Russia].
8. Polyanin, A. D., & Zaitsev, V. F. (2007). Handbook of exact solution for ordinary differential equation. Boca Raton : CRC press.
9. Rao S. S. (2016). Mechanical vibrations (6th ed.). Pearson.
10. Lojczyanskij, L. G., & Lur`e, A. I. (2006) Kurs teoreticheskoj mekhaniki [The course of theoretical mechanics]. (Vols. 2). Moskva : Drofa. [In Russia].
11. Olshanskiy, V. P., & Olshanskiy, S. V. (2013). Funkcziya Lamberta v zadachakh ballistiki materialnoj tochki [Lambert function in ballistic problems of a material point]. Kharkov : Izdatel Savchuk A. O. [In Russia].
12. Olshanskiy, V. P., Burlaka, V. V., & Slipchenko, M. V. (2019). Vilni zatukhayuchi kolyvannya oscylyatora z kombinovanym oporom [Free damped oscillations of the oscillator with combined resistance]. Visnyk KhNTUSG: Suchasni napryamky tekhnologiyi procesiv pererobnykh i kharchovykh vyrobnycztv [Bulletin of KhNTUSG: Modern directions of technology of processes of processing and food productions]. Kharkiv, 207, 22-33. [In Ukrainian].
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
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Type of publication : Scientific
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ISSN: 2306-8744 (printed version), (online)
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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.