Solution of the equation of force of impact of solids expressed by the Ateb-sine

: 53-60
Received: August 01, 2019
Revised: August 23, 2019
Accepted: August 30, 2019
Petro Vasilenko Kharkiv National Technical University of Agriculture
Kharkiv Petro Vasylenko National Technical University of Agriculture
Petro Vasilenko Kharkiv National Technical University of Agriculture

A nonlinear differential equation of the force of direct central quasistatic impact of elastic bodies bounded in the area of their contact by rotation surfaces is compiled. To determine the coefficients of the equation and the order of its degree nonlinear force, we used the well-known solution of the axisymmetric contact problem of elasticity theory, constructed in due time by I. Ya. Shtaerman, for the case of dense static contact of bodies, when the order of their boundary surfaces is not lower than the second. In the case of the second order, it goes into the well-known static solution of G. Hertz, whose assumption in the theory of shock is also taken here in the formulation of the dynamics problem. A closed analytic solution of the composite differential equation with respect to the force of impact as a function of time is constructed. It is expressed through Ateb-sine. This function also describes the process of motion of the centers of mass of bodies in the stages of their compression and expansion. Compact formulas are derived for calculating the maximum values of the impact force, the approach of the centers of mass of the bodies and the duration of the impact. Thanks to the use of the Ateb-sine and its approximation by elementary functions, it was possible to obtain a fairly simple scan of the fleeting process of mechanical shock in time. It is shown that well-known dependencies that describe the impact of elastic balls follow from the derived formulas. Examples of calculations are given in which the influence of various factors on the main characteristics of the impact is investigated. It is noted that the theory set forth concerns only the impact of bodies with low velocities, when plastic deformation does not occur during their dynamic compression. To extend the theory beyond the limits of elasticity, it is necessary to determine a constant for the stage of compression  in the impact force equation not by calculation, but by experimental method. Then, during compression and decompression of bodies, the impact force will be described by different analytical expressions, and the speed recovery coefficient will become less than one, which is consistent with practice.

[1] I. V. Kuzio, Ya. A. Zinko, T.-N. M. Vankovich, et al, Teoretychna mekhanika [Theoretical mechanics]. Kharkiv, Ukraine: Folio Publ., 2017. [in Ukrainian].

[2] Ya. G. Panovko, Vvedenie v teoriyu mehanicheskogo udara [Introduction to mechanical shock theory]. Moscow, Russia: Nauka Publ., 1977. [in Russian].

[3] N. A. Kilchevskiy, Dinamicheskoe kontaktnoe szhatie tverdih tel [Dynamic contact compression of solids bodys]. Kyiv, Ukraine: Naukova dumka Publ., 1977. [in Russian].

[4] V. Goldsmit, Udar. Teoriya i fizicheskie svoisnva soudaryaemih tel [Shock. Theory and physical properties of colliding bodies]. Kyiv, Ukraine: Naukova dumka Publ., 1977. [in Russian].

[5] S. P. Timoshenko, Kolebaniya v inzhenernom dele [Oscillations in engineering]. Moscow-Leningrad, Russia: Fizmatgiz Publ., 1959. [in Russian].

[6] A. P. Filippov, Kolebaniya deformiruemyh system [Oscillations of deformable systems]. Moscow, Russia: Mashinostroenie Publ., 1970. [in Russian].

[7] N. V. Smetankina, Nestaczionarnoe deformirovanie, teromouprugost` i optimizacziya mnogoslojny`kh plastin i czilindricheskikh obolochek [Unsteady deformation, thermoelasticity and optimization of multilayer plates and cylindrical shells]. Kharkiv, Ukraine: Mis’kdruk Publ., 2011. [in Russian].

[8] V. P. Olshanskiy, L. M. Tishchenko, S. V. Olshanskiy, Kolebaniya sterzhnej i plastin pri mekhanicheskom udare [Oscillations of rods and plates during mechanical shock]. Kharkiv, Ukraine: Mіs'kdruk Publ., 2012. [in Russian].

[9] P. M. Senik, “Pro Ateb-funkcii” [“About Ateb-functions”], Dopovidi AN URSR. Seriya A [Reports of Academy of Science of USSR. Series A], vol. 1, pp. 23–27, 1968. [in Ukrainian].

[10] P. M. Senik, “Obernennya nepovnoyi Beta-funktsii” [“Incomplete Beta-function inversion”], Ukrainskyi matematychnyi zhurnal [Ukrainian mathematical journal], vol. 21, issue 3, pp. 325–333, 1969. [in Ukrainian].

[11] A. M. Vozniy, “Zastosuvannia Ateb-funktsii dlia pobudovy rozviazku odnoho klasu istotno neliniinykh dyferentsialnykh rivnian” [“Application of Ateb functions to construct a solution of one class of essentially nonlinear differential equations”], Dopovidi AN URSR. Seriya A [Reports of Academy of Science of USSR. Series A], vol. 9, pp. 971–974, 1970. [in Ukrainian].

[12] H. T. Drogomyretska, “Pro intehruvannia spetsialnykh Ateb-funktsii” [“On the integration of special Ateb-function”], Visnyk Lvivskoho universytetu. Seriia mekhaniko-matematychna [Bulletin of Lviv University. The mechanical and mathematical series], vol. 46, pp. 108–110, 1977. [in Ukrainian].

[13] V. V. Grytsyk, N. A. Nazarkevich, “Matematychni modeli alhorytmiv i realizatsiia Ateb-funktsii” [“Mathematical models of algorithms and implementation of Ateb-functions”], Dopovidi Natsionalnoyi akademiyi nauk Ukrayiny [Reports of the National Academy of Sciences of Ukraine], vol. 12, pp. 37–42, 2007. [in Ukrainian].

[14] B. I. Sokil, “Pro zastosuvannya Ateb-funktsiy dlya pobudovi rozvyazkiv deyakih rivnyan, yaki opisuyut neliniyni kolivannya odnovimirnih seredovisch” [“On the application of Ateb-functions for construction of solutions of some equations describing nonlinear oscillations of one-dimensional medium”], Dopovidi Natsionalnoyi akademiyi nauk Ukrayiny [Reports of the National Academy of Sciences of Ukraine], vol. 1, pp. 55–58, 1997. [in Ukrainian].

[15] B. I. Sokil, H. I. Lishchyns’ka, “Asymptotychnyi metod i periodychni Ateb-funktsii u doslidzhenni kolyvalnykh protsesiv rukhomykh neliniino pruzhnykh odnovymirnykh system” [“Asymptotic method and periodic Ateb functions in the study of oscillatory processes of moving nonlinearly elastic one-dimensional systems”], Visnyk natsionalnoho universytetu “Lvivska politekhnika”. Dynamika, mitsnist ta proektuvannia mashyn i pryladiv [Bulletin of Lviv Polytechnic National University. Dynamics, durability and design of machines and appliances], vol. 556, pp. 57–64, 2006. [in Ukrainian].

[16] P. Ya. Pukach, Yakisni metody doslidzhennia neliniinykh kolyvalnykh system [Qualitative methods for investigating nonlinear oscillation systems]. Lviv, Ukraine: Lviv Polytechnic Publ., 2014. [In Ukrainian].

[17] P. Ya. Pukach, I. V. Kuzio, “Resonance phenomena in quazi-zero stiffness vibration isolation systems”, Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu [Scientific Bulletin of the National Mining University], issue 3, pp. 62–67, 2015.

[18] I. V. Kuzio, T.-N. M. Vankovich, Ya. A. Zinko, Teoretychna mekhanika. Dynamika. Knyha 1 [Theoretical mechanics. Dynamics. Book 1]. Lviv, Ukraine: Rastr-7 Publ., 2012. [In Ukrainian].

[19] L. A. Cveticanin, “Review on dynamics of mass variable system”, Journal of the Serbian Society for Computational Mechanics, vol. 6, issue 1, pp. 56–74, 2012.

[20] L. Cveticanin, T. Pogany, “Oscillator with a sum of noninteger-order nonlinearities”, Journal of Applied Mathematics, vol. 2012, pp. 1–20, 2012.

[21] V. P. Olshanskiy, S. V. Olshanskiy, “Ateb-synus u rozviazku zadachi Hertsa pro udar” [“Ateb-sine in solving the Hertz problem of impact”], Visnyk Natsionalnoho tekhnichnoho universytetu Kharkivskyy politekhnichnyy instytut”. Seriia: Matematychne modeliuvannia v tekhnitsi ta tekhnolohiiakh [Bulletin of the National Technical University “Kharkiv Polytechnic Institute”. Series: Mathematical Modeling in Engineering and Technology], vol. 151, pp. 324–333, 2014. [in Ukrainian].

[22] V. Ol'shanskii, O. Spol'nik, M. Slipchenko, et al., “Modeling the elastic impact of a body with a special point at its surface”, Eastern-European Journal of Enterprise Technologies, vol. 1, no. 7 (97), pp. 25–32, 2019.

[23] V. P. Olshanskiy, S. V. Olshanskiy, “Uzahalnena zadacha mekhanichnoho udaru v teorii Hertsa” [“A generalized problem of mechanical shock in Hertz theory”], Vibratsii v tekhnitsi ta tekhnolohiiakh [Vibrations in engineering and technology], vol. 4 (91), pp. 70–75, 2018. [in Ukrainian].

[24] I. Ya. Shtayerman, Kontaktnaya zadacha teorii uprugosti [Contact problem of the theory of elasticity]. Moscow-Leningrad, Russia: Gostehizdat Publ., 1949. [in Russian].

[25] I. S. Gradshtein, I. M. Ryzhik, Tablitsy integralov, sum, ryadov i proizvedeniy [Tables of integrals, sums, series and products]. Moscow, Russia: Nauka Publ., 1962. [in Russian].

V. Olshanskiy, V. Burlaka, M. Slipchenko, "Solution of the equation of force of impact of solids expressed by the Ateb-sine", Ukrainian Journal of Mechanical Engineering and Materials Science, vol. 5, no. 2, pp. 53-60, 2019.