**Issue №:** 4(95)

**Published:**2019.12.20

**DOI:**10.37128/2306-8744-2019-4

**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.

**ON THE POSSIBILITY OF STAGNATION OF FREE OSCILLATIONS OF A NONLINEARLY ELASTIC OSCILLATOR WITH A LINEAR VISCOUS RESISTANCE**

**DOI:**10.37128/2306-8744-2019-4-5

Olshanskiy Vasyl Pavlovych – Doctor of Physical and Mathematical Sciences, Full Professor, Petro Vasilenkj Kharkiv National Technical University of Agriculture, Tel. (066) 010-09-55, е-mail: OlshanskiyVP@gmail.com

Olshanskiy Stanislav Vasilevich – Phd in Physical and Mathematical Sciences, Petro Vasilenkj Kharkiv National Technical University of Agriculture, Tel. (057) 343-29-41, е-mail: stasolsh77@gmail.com

The free oscillations of an oscillator with a power-nonlinear elasticity characteristic under the action of linear viscous resistance are considered. Using the energy balance method, which is widespread in mechanics, the calculation of the amplitudes of free damped oscillations is reduced to calculating the roots of an algebraic equation, which has an exact analytical solution only with linear elasticity of the oscillator. In the case of an arbitrary positive indicator of nonlinear elasticity, a numerical solution of the equation is required. For this, the Newton's iterative method was used in the work, which has fast convergence of iterations at an arbitrary initial approximation. According to the results of the analysis of the coefficients of the equation established, that in the case of a rigid characteristic of elasticity, when the nonlinearity is greater than unity, the oscillations are reduced to a finite number of decaying ranges, that is, they are limited in time, and in the case of a soft characteristic of elasticity, when the nonlinearity is less than unity, they continue to infinity, as linear dissipative oscillator. The research is given by the method of energy balance and numerical integration of the differential equation of oscillations on a computer. The work of the force of viscous resistance is calculated approximately using periodic Ateb functions that accurately describe free undamped oscillations in the absence of resistance. As a result, approximate iterative dependences are obtained for calculating the amplitudes of the ranges that decay during movement. The numerical results obtained using approximate formulas and numerical computer integration of the nonlinear Cauchy problem are compared. Their satisfactory agreement was noted. A satisfactory agreement was noted between the results for both hard and soft elastic characteristics, which confirmed the adequacy of approximate analytical solutions to the dynamics problem. The main advantage of the described approximate calculation method is that there is no need to build an analytical solution to the nonlinear differential equation of motion of the oscillator, which is a rather complicated mathematical problem. Furthermore, it made it possible to establish conditions under which the oscillator with a viscous and dry friction resistance have similar oscillation properties.

**Keywords:**free damped oscillations, power-nonlinear elasticity characteristic, viscous linear resistance, energy balance method, periodic Ateb-functions.

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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.