Vibrations in engineering and technology

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; alexspo@ukr.net, (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: tokarchyk@vsau.vіn.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].