**Issue №:** 2(101)

**Published:**2021.06.29

**DOI:**10.37128/2306-8744-2021-2

**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.

**OSCILLATIONS OF A PULSE LOADED OSCILLATOR WITH A SQUARE RESISTANCE IN THE COMPOSITION OF THE DISSIPATIVE FORCE**

**DOI:**10.37128/2306-8744-2021-2-4

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, Kharkiv, 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, Kharkiv, Ukraine; 61022; Slipchenko_M@ukr.net, тел.: (066) 7120989

Tverdokhlib Igor – candidate of technical sciences, associate professor of the department of general technical disciplines and occupational safety, Vinnytsia National Agrarian University: Vinniytsia, st. Sonyachna 3, VNAU, 21008, e-mail: igor_tverdokhlib@yahoo.com.

Kupchuk Ihor – Candidate of Technical Sciences (Ph. D in Engeneering), Associate Professor of the Department of General Technical Disciplines and Labor Protection, Vinnytsia National Agrarian University (3, Sonychna St., Vinnytsia, 21008, Ukraine, +380978173992, kupchuk.igor@i.ua, http://orcid.org/0000-0002- 2973-6914).

The unsteady oscillations of a dissipative oscillator caused by an instantaneous impulse of the force are described. The case is considered when the dissipative force consists of quadratic viscous resistance and dry friction, and the theoretical results are generalized to the case of the sum of three forces. The third is the force of positional friction. Formulas for calculating the ranges of oscillations have been constructed In this case, the Lambert function of negative and positive arguments is used. It is a tabulated special function. Its value can also be calculated using its known approximations in elementary functions. It is shown that, due to the action of the dissipative force, the process of post-pulse oscillations consists of a finite number of cycles and is limited in time. This is due to the presence of dry friction among the resistance components. Examples of calculations that illustrate the possibilities of the stated theory are given. In order to check the reliability of the derived calculation formulas, numerical computer integration of the differential equation of motion was also carried out. The convergence of the numerical results obtained by two different methods is shown. Thus, it has been confirmed that with the help of analytical solutions it is possible to find the extreme displacements of the oscillator without numerically solving its nonlinear differential equation of motion. Using Lambert function and the first integral of the equation of motion made it possible to derive precise calculation formulas for determining the range of oscillations caused by the pulsed load of the oscillator. The derived formulas are suitable for calculating the value of the instantaneous impulse applied to the oscillator, which refers to the inverse problems of mechanics. Thus, by measuring the maximum displacement of the oscillator, it is possible to identify the initial velocity or instantaneous impulse applied to the oscillator. The performed numerical computer integration of the output differential equation confirmed the adequacy of the obtained analytical solutions, which concern not only direct, but also inverse problems of dynamics.

**Keywords:**nonlinear oscillator, dissipative force with quadratic resistance, instantaneous impulse, Lambert function, post-impulse oscillation swing.

1. Loytsyanskiy, L. G., & Lurye, A. I. (2006). Kurs teoreticheskoy mekhaniki [Course of Theoretical Mechanics]. (T. 2). Moskva : Drofa. [In Russian].

2. Bogolyubov, N .N., & Mitropol'skiy Yu. A. (1974). Asimptoticheskiye metody v teorii nelineynykh kolebaniy [Asymptotic methods in the theory of nonlinear oscillations]. Moskva : Nauka. [In Russian].

3. Cveticanin, L. (2009). Oscillator with strong quadratic damping force. Publications of the Mathematical Institute. 85, 99, 119-130. doi:10.2298/PIM0999119C

4. Cveticanin L. (2018). Strong nonlinear oscillstors. Analitical Solution, 2 nd edition. Springer : Cham. doi:10.1007/978-3-319-58826-1.

5. Olshanskiy, V. P., & Olshanskiy, S. V. (2014). Funktsiya Lamberta v zadache kolebaniy matematicheskogo mayatnika [Lambert function in the problem of oscillations of a mathematical pendulum]. Visnyk NTU «KhPI» Seriya: Matematychne modelyuvannya v tekhnitsi ta tekhnolohiyakh [Bulletin of NTU "KhPI". Series: Mathematical modeling in engineering and technology]. 18 (1061), 116-119. [In Russian].

6. Blekhman, I. I. (Ed.) (1979). Vibratsii v tekhnike. Spravochnik v shesti tomakh. T. 2. Kolebaniya nelineynykh mekhanicheskikh sistem [Vibrations in technology. Reference book in six volumes. T. 2. Oscillations of nonlinear mechanical systems]. Moskva : Mashinostroyeniye. [In Russian].

7. Vasylenko, M. V., & Alekseychuk, O. V. (2004). Teoriya kolyvanʹ i stiykosti rukhu [Theory of oscillations and stability of motion]. Kyyiv : Vyshcha shkola. [In Ukrainian].

8. Corless, R. M., Gonnet, G. H., Hare, D. E. G, et al. (1996). On the Lambert W function. Adv. Comput. Math. 5, 1, 329-359. doi: 10.1007/BF02124750

9. Fariel, Shafee. (2007). Lambert function and new-extensive form of entropy. IMA Journal of Applied Mathematics. 72, 785-800. doi:10.1093/imamat/hxm039

9. Olshanskiy, V. P., Slipchenko, M. V., Spolnik, O. I., & Burlaka, V. V. (2020). Nelinijni kolyvannya dysypatyvnyh oscylyatoriv [Nonlinear oscillations of dissipative oscillators]. Kharkiv : Miskdruk. [In Ukrainian].

11. Panovko, Ya. G. (1980). Vvedeniye v teoriyu mekhanicheskikh kolebaniy [Introduction to the theory of mechanical vibrations]. Moskva : Nauka. [In Russian].

12. Olshanskiy, V. P., Slipchenko, M. V., Spolnik, O. I., & Zamriy, M. A. Vilʹni kolyvannya ostsylyatora za nayavnosti kvadratychnoho vyazkoho oporu ta sukhoho tertya [Free oscillations of the oscillator in the presence of quadratic viscous resistance and dry friction]. Vibratsiyi v tekhnitsi ta tekhnolohiyakh [Vibrations in engineering and technology]. 2(97), 33-40. [In Ukrainian].

13. Olshanskiy, V. P., & Olshanskiy, S. V. (2013) Funktsiya Lamberta v zadachakh ballistiki materialnoy tochki [Lambert function in material point ballistics problems]. Kharkov : Savchuk A. O. Publ. [In Russian].

14. Olshanskiy, V. P. (2020). Pro aproksymatsiyu funktsiyi Lamberta [On the approximation of Lambert's function]. Visnyk NTU «KHPI». Seriya: Matematychne modelyuvannya v tekhnitsi ta tekhnolohiyakh [Bulletin of NTU "KhPI". Series: Mathematical modeling in engineering and technology]. 1 (1355), 64-68. [In Ukrainian].

15. Dyagel', R. V., & Lapshin, V. V. (2011). O nelineynoy vyazkouprugoy modeli kollinearnogo udara Khanta-Krossli [On the nonlinear viscoelastic model of the Hunt-Crossley collinear impact]. Izvestiya RAN. Mekhanika tverdogo tela [Bulletin of the RAS. Rigid body mechanics]. 5, 164-173. [In Russian].

16. Borovin, G. K., & Lapshin, V. V. (2018). Obobshchennaya model udara Gertsa-Khanta-Krossli [Generalized Hertz-Hunt-Crossley Impact Model]. Vesnik MGTU im. N.E. Baumana. Seriya: Yestestvennyye nauki [Bulletin of MSTU. N.E. Bauman. Series: Natural Sciences]. 6, 18-30. doi: 10.18698/1812-3368-2018-6-18-30 [In Russian].

17. Biderman, V. L. (1980). Teoriya mekhanicheskikh kolebanij [Theory of mechanical oscillations]. Moskva : Vysshaya shkola. [In Russian].

18. Suryaninov, N. G., Dashhenko, A. F., & Belous, P. A. (2000). Teoreticheskie osnovy dinamiki mashin [Theoretical foundations of machine dynamics]. Odessa : OGPU. [In Russian].

19. Yanyutin, Ye. G., Yanchevskiy, I. V., Voropay, A. V., & Sharapata, A. S. (2004). Zadachi impul'snogo deformirovaniya elementov konstruktsiy [Problems of impulse deformation of structural elements]. Kharkov: KhNADU. [In Russian].

20. Yanyutin, Ye. G., Voropay, A. V., Povalyayev, S. I., & Yanchevskiy, I. V. (2010). Identifikatsiya nagruzok pri impul'snom deformirovanii tel. Chast 2 [Identification of loads during impulse deformation of bodies. Part 2]. Kharkov: KhNADU. [In Russian].

21. Voropay, A. V. (2018). Integral'nyye uravneniya Volterra v nekorrektnykh zadachakh nestatsionarnogo deformirovaniya plastin [Volterra integral equations in ill-posed problems of unsteady deformation of plates]. Kharkov: Lider. [In Russian].

22. Yanytin, E. G., & Voropay, A. V. (2004). Controlling nonstationary vibration of a plate by means of addition loads. Int. Jour. of Solid and Struct. 41, 4919-4926. doi: 10.1016/j.ijsolstr.2004.04.022

23. Ramm, A. G. (2005). Inverse problems : Mathematical and analytical techniques with applications to engineering. Sringer. doi: 10.1007/b100958

24. Gang, Yan, & Li, Zhou. (2009). Impact load identification of composite structure using genetic algorithms. Journal of Sound and Vibration. 319, 3-5, 869-884. doi: 10.1016/j.jsv.2008.06.051

About 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

**Keywords:**машинобудування

**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

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, 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.