Issue №: 2 (93)
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.
SOLUTION OF THE EQUATION OF IMPACT THE ELASTIC BODY, ONE OF WHICH CONE
Ольшанський Василь Павлович – доктор фізико-математичних наук, професор, Харківський національний технічний університет сільського господарства ім. Петра Василенка, тел. (066) 010-09-55, e-mail: OlshanskiyVP@gmail.com
Ольшанський Василий Павлович – доктор физико-математических наук, профессор, Харьковский национальный технический университет сельского хозяйства им. Петра Василенка, тел. (066) 010-09-55; e-mail: OlshanskiyVP@gmail.com
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
Ольшанський Станіслав Васильович – кандидат фізико-математичних наук, Харківський національний технічний університет сільського господарства ім. Петра Василенка, тел. (057) 343-29-41, e-mail: stasolsh77@gmail.com
Ольшанський Станислав Васильевич – кандидат физико-математических наук, Харьковский национальный технический университет сельского хозяйства им. Петра Василенка, тел. (057) 343-29-41; e-mail: stasolsh77@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 impact interaction of elastic bodies with a small initial velocity is considered, when one of them is limited in the contact zone with another conical surface of revolution. Using the well-known solution of the static contact problem of the theory of elasticity, which found I. Shtaermann, and the assumptions of G. Hertz, which he made when creating his own theory of quasistatic impact of solids, compiled a nonlinear differential equation of impact force as a function of time. His closed analytical solution, which describes the process of dynamic interaction of bodies in time, is expressed through the periodic Ateb-sine. To simplify the use of the obtained analytical solution in the calculations, a separate table of the specified special function has been compiled and its approximation with elementary functions has been proposed, the relative error of which is less than one percent. In order to confirm the reliability of the constructed solutions, the integration of the equation of impact force on the computer was carried out in parallel. A good agreement is established between the results obtained by the constructed analytical solution and the numerical integration of the nonlinear Cauchy problem on a computer for a second-order differential equation. Compact formulas for the maxima of the impact force and the magnitude of the compression of bodies, as well as the formula for the duration of the impact process are derived. It is noted that the obtained results can be used in determining the dynamic loads acting on the rubber-lined rolls of the vibration classifier when pieces of solid raw materials fall on them. Examples of calculations are given and a comparative analysis of the results is carried out.
1.Pysarenko H.S., Kvitka O.L. & Umanskyi E.S. (2004) Opir materialiv [Materials Resistance]. – Kyiv : Vyshcha shkola. 2.Shkelov L.T., Stankevych A.M. & Poshivach D. V. (2011) Opir materialiv [Materials Resistance]. – Kyiv : Vipol.
3.Olshanskiу V.P., Olshanskiу S.V. (2012) Sravnenie priborov konsolnoy balki pri udare, vyichislennyih po teoriyam Koksa i Sen-Venana [Comparison of cantilever beam devices at impact, calculated according to the theories of Cox and Saint-Venant]. Vibratsii v tekhnitsi ta tekhnolohiiakh : Vseukrainskyi naukovo-tekhnichnyi zhurnal. – Vinnytsia, 4 (68), 94-97 [in Ukrainian].
4.Olshanskiу V.P., Olshanskiу S.V. (2013) Vyichislenie dinamicheskih peremescheniy balki pri neuprugom udare po teoriyam Koksa i Sen-Venana [Calculation of the dynamic displacements of a beam with an inelastic impact on the theories of Cox and Saint-Venant]. Strength problems, 3, 148-156 [in Ukrainian].
5.Нerts H. ber die Berrhrung Fester Elastischer Krper. Jornal Reine und Andewandte Mathematik, 1882. B 92, 156-171.
6.Goldsmit V. (1965) Udar. Teoriya i fizicheskie svoystva soudaryaemyih tel [Impact. Theory and physical properties of the colliding bodies]. – M.: Stroyizdat, 447.
7.Kilchevsky N. A. (1976) Dinamicheskoe kontaktnoe szhatie tverdyih tel. Udar [Dynamic contact compression of solids. Blow]. – Kiev: Naukova dumka, 319.
8.Filippov A.P. (1970) Kolebaniya deformiruemyih sistem [Oscillations of deformable systems]. – Moscow: Mechanical Engineering, 734.
9.Olshanskiу V.P., Tishchenko L.N. & Olshanskiу S.V. (2012) Kolebaniya sterzhney i plastin pri mehanicheskom udare [Swings of rods and plates with mechanical impact]. – Kharkiv : Miskdruk, 320.
10.Shtaerman I.Ya. (1949) Kontaktnaya zadacha teorii uprugosti [Contact problem of the theory of elasticity]. – M.-L. : Gostekhizdat, 272.
11.Gricik V.V., Nazarkevich M.A. Matematychni modeli alhorytmiv i realizatsiia Ateb-funktsii [Mathematical models of algorithms and realizations Ateb-functions] Dopovidi Natsionalnoi akademii nauk Ukrainy, 12, 37-42 [in Ukrainian].
12.Pukach P. Ya. (2014) Yakisni metody doslidzhennia neliniinykh kolyvalnykh system [Qualitative methods for investigating nonlinear oscillatory systems]. – Lviv: Lvivska politekhnika, 288.
13.Olshanskiу V.P., Olshanskiу S.V. (2018) Ateb-synus u rozviazku zadachi Hertsa pro udar [Ateb-sinus in the solution of the Hertz problem]. Visnyk NTU“KhPI”. Matematychne modeliuvannia v tekhnitsi ta tekhnolohiiakh. № 3, 98-103 [in Ukrainian].
14.Abramovits M., Stigan I. (1979) Spravochnik po spetsialnyim funktsiyam (s formulami, grafikami i matematicheskimi tablitsami ) [Handbook of special functions (with formulas, graphs and mathematical tables) ]. – M.: Nauka, 832.
15.Dyakonov V.P. (2003) Maple 8 v matematike, fizike i obrazovanii [Maple 8 in Mathematics, Physics and Education]. – M.: Solon-Press, 656.
16.Nadutyiy V. P. (2014) Vzaimodeystvie kuskov materiala s futerovannyim valkom vibratsionnogo klassifikatora [The interaction of pieces of material with a lined roll vibration classifier] Vibratsii v tekhnitsi ta tekhnolohiiakh : Vseukrainskyi naukovo-tekhnichnyi zhurnal. – Vinnytsia, 1 (73), 94-99.
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
The journal "Vibrations in Engineering and Technologies" is included in the list of technical scientific publications of Ukraine
(Category "B", Order of the Ministry of Education and Science of Ukraine dated July 2, 2020 No. 886)
Old version of the site: http://vibrojournal.vsau.edu.ua/
The journal "Vibrations in Engineering and Technology" is indexed by the following databases and catalogs:
The certificateof massmedia State 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.
