628.822 Analytical model of oscillations of the roller moving along a surface in a hydrodynamic lubrication regime

Ivanov V. A. (Политехнический институт СФУ), Yerkayev N. V. (ICM SB RAS)

LUBRICATION LAYER, HYDRODYNAMIC LUBRICATION, ROLLER OSCILLATION, ASYMPTOTIC SERIES EXPANSION


doi: 10.18698/2309-3684-2018-3-4966


This article deals with the model of normal oscillations of the roller moving along the surface with a constant velocity in a presence of a liquid lubrication layer. Pressure distribution along the lubrication layer is obtained as a result of integration of the Reynolds equation taking into account both tangential and normal velocities of the roller with respect to the surface. A damping coefficient is determined as that of proportionality between the normal velocity and corresponding variation of the carrying capacity. After special normalizations, the problem is reduced to the stiff ordinary differential equation with small parameter multiplied on the highest order derivative term. For this equation, analytical solution is derived by method of asymptotic expansion on a singular small parameter. This solution contains regular terms of series expansion, as well as boundary layer functions decreasing rapidly with time. Characteristic decreasing time for these functions is proportional to the small parameter. The obtained analytical solutions is applied for the problem of roller relaxation to the new equilibrium state after sharp increase of the external loading. A peculiarity of this process is a rapid increase of the pressure peak just after the loading jump, which afterwards is gradually relaxing to the new stationary value corresponding to the increase external loading.


[1] Kantha Shoba M., Manikandan M. Parametric optimization of cylindrical roller bearing and compare with FEA. International Journal of Innovative Research in Technology, Science & Engineering, 2016, vol. 2, no. 5.
[2] Galakhov M.A, Usov P.P. Differentsial'nye i integral'nye uravneniya matematicheskoy modeli teorii treniya [Differential and integral equations of the mathematical model of the friction theory]. Moscow. Nauka, Fizmatlit Publ., 1990. 280 p.
[3] Terent’ev V.F, Erkaev N.V. Tribonadezhnost' podshipnikovykh uzlov v prisutstvii modifitsirovannykh smazochnykh kompozitsiy [Tribo-durability of bearing units in a presence of modified lubricant compositions]. Novosibirsk, Nauka, 2003. 142 p.
[4] Kapitsa P.L. Zhurnal tekh. Fiziki – Journal of engineering physics, 1955, vol. 25, no. 4, pp. 747-762.
[5] Levandovskiy V.A., Nesterenko V.I., Gundar' V.P. Vestnik SNU im. V. Dalya – Bulletin of the SNU Dahl, 2011, vol.1, no. 4 (158), pp. 95-100.
[6] Besportochnyy A.I. Asimptoticheskie metody v kontaktnoy gidrodinamike. [Asymptotic methods in fluid mechanics contact]: dis. kand. fiz.-mat. nauk. Moscow. MFTI, 2014. 225 p.
[7] Besportochnyy A.I. Trudy MFTI – Proceedings of MIPT, 2011, vol. 3, no. 1, pp. 28-34.
[8] Ciulli E., Bassani R. Influence of vibrations and noise on experimental results of lubricated non-conformal contacts. Engineering Tribology, 2006, vol. 220, pp. 319-331.
[9] Stacke L-E., Fritzson D. Dynamic behaviour of rolling bearings: simulations and experiments. Proc Instn Mech Engrs, 2001, vol. 215, pp. 499-508.
[10] Vasil'eva A.B., Butuzov V.F. Asimptoticheskie metody v teorii singulyarnykh vozmushcheniy [Asymptotic methods in the theory of singular perturbations]. Moscow, Vysshaya Shkola Publ., 1990. 208 p.
[11] Vasil'eva A.B., Butuzov V.F. Asimptoticheskie razlozheniya resheniy singulyarno vozmushchennykh uravneniy. [Asymptotic expansions of solutions of singularly perturbed equations]. Moscow. Nauka, 1973. 272 p.
[12] Besportochnyy A.I., Galakhov M.A. Matematicheskoe modelirovanie v tribotekhnike. [Mathematical modeling in tribology] Moscow. MFTI, 1991. 88 p.
[13] Galin L.A. Kontaktnye zadachi teorii uprugosti i vyazkouprugosti. [Contact problems of the theory of elasticity and viscoelasticity]. Moscow, Fizmatlit Publ., 1980. 304 p.
[14] Galakhov M.A., Gusyatnikov P. B., Novikov A.P. Matematicheskie modeli kontaktnoy gidrodinamiki. [Mathematical models of contact hydrodynamics]. Moscow, Fizmatlit Publ., 1985. 296 p.
[15] Tikhonov A. N. Matematicheskii sbornik – Mathematical collection, , 1952, vol. 31(73), no. 3, pp. 575-586.
[16] Vasil'eva A. B. Uspekhi matematicheskikh nauk – Russian Math., 1963, 18, vol. 3, pp. 15-86.
[17] Aleksandrov А.А., Dimitrienko Yu.I. Matematicheskoe modelirovanie i chislennye metody — Mathematical Modeling and Computational Methods, 2014, no. 1 (1), pp. 3–4.
[18] Zarubin V.S., Kuvyrkin G.N. Matematicheskoe modelirovanie i chislennye metody — Mathematical Modeling and Computational Methods, 2014, no. 1, pp. 5–17.


Иванов В.А., Еркаев Н.В. Аналитическая модель колебаний ролика, движущегося вдоль твердой поверхности в режиме гидродинамической. Математическое моделирование и численные методы, 2018, № 3, с. 49–66.



Download article

Количество скачиваний: 587