Vibrations in engineering and technology

Olshanskiy Vasiliy Pavlovich – Doctor of Physical and Mathematical Sciences, Professor, Kharkiv Petro Vasylenko National Technical University of Agriculture, Kharkov.

(e-mail: OlshanskiyVP@gmail.com).

Olshanskiy Stanislav Vasilevich – PhD in Physics and Mathematics, Kharkiv Petro Vasylenko National Technical University of Agriculture, Kharkov; (e-mail: stasolsh77@gmail.com).

The generalization of the well-known classical Hertz problem of the impact of two solids, taking into account contact deformations, was achieved by introducing into consideration the internal hysteresis energy losses, accompanying the dynamic deformation of bodies upon impact. In contrast to the classical version of the theory, in the generalized problem they do not coincide, but the graphs of the dependence of the impact force on the convergence of the centers of mass of the bodies at the stages of their compression and stretching differ. Therefore, the differential equations of motion at these stages have different values of the coefficients. They also correspond to different durations in time of the mentioned stages, and the duration of compression is longer than the duration of decompression. The first integrals of the nonlinear differential equations of motion are expressed in elementary functions and the second integrals are expressed through periodic Ateb-functions. Compact formulas have been derived for calculating the maxima: the convergence of the centers of mass of the bodies, the impact force, the dimensions of the contact area, the pressure at the center of this area. A compact expression was also obtained for the coefficient of speed recovery for a direct central impact, moreover, its value is less than unity and does not depend on the speed of the collision and the masses of the bodies subjected to the impact. This is consistent with the well-known Newton hypothesis, which is traditionally used in the theory of partially elastic stereomechanical shock.

1. Pisarenko G.S., Kvitka A.L., & Umansky E.S. (2004) Opir materialiv [Materials Resistance]. – Kyiv: Vyshcha shkola.

2. Gurniak L.I., Gutsulyak Yu.V., & Yuzkiv T.B. (2005) Opir materialiv [Materials Resistance]. – Lviv: Novyi svit.

3. Goldsmith W. (1965) Udar. Teoriya i fizicheskie svojstva soudaryaemyh tel [Impact. Theory and physical properties of the colliding bodies]. – M.: Stroyizdat.

4. Kilchevsky N.A. (1976) Dinamicheskoe kontaktnoe szhatie tverdyh tel. Udar. [Dynamic contact compression of solids. Blow]. – Kiev : Naukova Dumka.

5. Panovko Y.G (1977) Vvedenie v teoriyu mekhanicheskogo udara. [Introduction to the theory of mechanical shock] – M.: Nauka.

6. Timoshenko S.P., Yang D.H., & Weaver U. (1985) Kolebaniya v inzhenernom dele [Fluctuations in engineering] – M: Mashinostroenie.

7. Naduty V.P., Yagnyukov V.F., & Yagnyukova I.V. (2014) Vzaimodejstvie kuskov materiala s futerovannym valkom vibracionnogo klassifikatora [Interaction of pieces of material with a lined roll of a vibration classifier]. Vibracii v tekhnike i tekhnologiyah, 1, 94-99 [in Russian].

8. Olshanskiу V.P., Olshanskiу S.V., & Tishchenko L.M. (2012) Kolebaniya sterzhnej i plastin pri mekhanicheskom udare [Oscillations of rods and plates during a mechanical shock] – Kharkov: Miskdruk.

9. Larin A.А. (2013) Ocherki istorii razvitiya teorii mekhanicheskih kolebanij [Essays history of development theory of mechanical oscillations] – Sevastopol: Weber.

10. Olshanskiy V.P., & Olshanskiy S.V. (2018) Ateb-sy`nus u rozvyazku zadachi Gercza pro udar [Ateb-sin in solving the Hertz problem]. Visnyk NTU «KhPI». – Kharkiv : NTU «KhPI», Seriia : Matematychne modeliuvannia v tekhnitsi ta tekhnolohiiakh. 3 (1279), 98-103 [in Ukrainian]. 

11. Gricik V.V., Nazarkevich M.A. (2007) Matematichnі modelі algoritmіv і realіzacіja Ateb-funkcіj [Mathematical models of algorithms and implementation Atte functions] Dopovidi Natsionalnoi akademii nauk Ukrainy, 12, 37-42 [in Ukrainian].