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Vibrations in engineering and technology

space SCIENTIFIC JOURNALS OF VINNITSA NATIONAL AGRARIAN UNIVERSITY

Issue №: 3 (102)

Published: 2021.09.28
DOI: 10.37128/2306-8744-2021-3


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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ABOUT THE ANALYTICAL SOLUTION OF THE EQUATION OF IMPACT FORCE OF TWO ELASTIC BODIES

DOI: 10.37128/2306-8744-2021-3-4
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Olshanskiy Vasyl – Doctor of Physical and Mathematical Sciences, Full Professor, Professor of the Department of Reliability and Strength of Machines and Construction, State Biotechnological University, Alchevskikh str. 44, Kharkiv, Ukraine; 61022; OlshanskiyVP@gmail.com, тел.: (066) 0100955

Slipchenko Maksym  – Candidate of Technical Sciences (Ph. D.), Associate Professor, Head of the Department of Reliability and Strength of Machines and Construction of State Biotechnological University, Alchevskikh str. 44, Kharkiv, Ukraine; 61022; Slipchenko_M@ukr.net, тел.: (066) 7120989

Solona Olena – 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, Solnyschaya St., Vinnytsia, Ukraine,21008, e-mail: solona_o_v@ukr.net).

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).

 

Annotation

A nonlinear differential equation of the force of direct central impact of elastic bodies of revolution, which have a singular point on the boundary contact surface, where its curvature is infinite, is compiled. To determine the coefficients of the equation and the order of its power nonlinearity, the well-known solution of the axisymmetric contact problem of the theory of elasticity, constructed by I. Ya. Shtaermann, is used. In the formulation of the dynamic problem, the classical assumptions of the theory of quasi-static impact proposed by H. Hertz were also used. The constituted equation of impact force is reduced to the Bernoulli equation and its closed analytical solution is constructed, which is expressed in terms of the Ateb-sine. Analytical time dependences of the impact force and the convergence of the centers of mass of elastic bodies are obtained. Compact formulas have been derived for calculating the maxima of these quantities, as well as the durations of the process of compression and impact of bodies. Compact approximations of Ateb-sine by elementary functions are proposed. Thanks to these approximations, it was possible to obtain a fairly simple analytical sweep in time of a fast-flowing mechanical process. Traditionally, in other works such a scan was obtained by numerical solution of the corresponding integral equations that determine the force of an impact. Examples of calculations are given in which the influence of various factors on the main characteristics of a body impact with a small initial velocity is investigated. The limitation on the collision rate is due to the elastic formulation of the problem, where the possibility of plastic deformations is excluded. As a result of this formulation, the need to determine the rate of recovery rate has disappeared, for it is equal to one. Comparison of numerical results is carried out, to which the obtained analytical solutions and the numerical integration of the impact force equation on a computer lead. Small divergences of the results confirmed the accuracy of the derived calculation formulas. Numerical results relate to the impact of a steel body on a fixed rubber half-space, the analogue of which is observed in practice when falling pieces of mineral raw materials on the rolls of a vibration classifier lined with rubber.

Keywords: H. Hertz theory, equations of force of impact, special point on the surface, analytical solution, Ateb-sine and its approximation.

List of references

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2. Goldsmit V. (1965) Udar. Teoriya i fizicheskie svoystva soudaryaemyih tel [Impact. Theory and physical properties of the colliding bodies]. M.: Stroyizdat. [In Russian].

3. Kilchevsky N.A. (1976) Dinamicheskoe kontaktnoe szhatie tverdyih tel. Udar [Dynamic contact compression of solids. Blow]. Kiev: Naukova dumka. [In Russian].

4. Shtaerman I.Ya. (1949) Kontaktnaya zadacha teorii uprugosti [Contact problem of the theory of elasticity]. M.-L. : Gostekhizdat. [In Russian].

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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

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/

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.