Solving a nonlinear problem for a one-sided dynamically loaded sliding thrust bearing

P. E. Fedotov ; N. V. Sokolov

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 32 (2024) no. 2, pp. 180-196.

Résumé

The purpose of this study is to propose an efficient numerical method for solving the inverse nonlinear problem of the movement of the compressor rotor collar in a fluid film thrust bearing. Methods. A periodic thermoelastohydrodynamic (PTEHD) mathematical model of hydrodynamic and thermal processes in a bearing is constructed under the condition of the rotor collar motion. Within the framework of the model, an inverse nonlinear problem of determining the position of the collar under a given external load is formulated. An iterative solution method is proposed, which utilizes the solution of the direct problem. To reduce computational costs, a modified Dekker-Brent method is employed in conjunction with a modified Newton's method. Results. Numerical experiments have been conducted, demonstrating the effectiveness of the proposed approaches. The suggested methods significantly reduce the required computational resources by minimizing the number of calls to the target function in the optimization problem. A software suite has been developed that allows for the calculation of the nonlinear system of rotor motion under various physical and geometric parameters. Conclusion. An efficient set of numerical methods for solving the inverse nonlinear problem of the motion of the rotor collar in the compressor fluid film thrust bearing is proposed. The method's effectiveness lies in substantial savings of computational resources. The method's efficiency has been demonstrated in numerical experiments.

Détail

Keywords: fluid film thrust bearing, differential equations, inverse nonlinear problem, zeroin, Dekker-Brent method, Newton's method

Bibliographie

[1] Khadiev M. B., Khamidullin I. V., Kompressory v tekhnologicheskikh protsessakh. Raschet podshipnikov skolzheniya tsentrobezhnykh i vintovykh kompressorov, KNITU, Kazan, 2021, 260 pp.

[2] Maksimov V. A., Batkis G. S., Tribologiya podshipnikov i uplotnenii zhidkostnogo treniya vysokoskorostnykh turbomashin, FEN, Kazan, 1998, 429 pp.

[3] Kostyuk A. G., Dinamika i prochnost turbomashin : ucheb. posobie dlya studentov vuzov, 3, pererabot. i dop., MEI, Moskva, 2007, 476 pp.

[4] Nekrasov A. L., Raschetnyi analiz nelineinykh kolebanii rotorov turbomashin v podshipnikakh skolzheniya, dis. \ldots kand. tekhn. nauk: 05.04.12, MEI, Moskva, 1998, 125 pp.

[5] Khisameev I. G., Maksimov V. A., Batkis G. S., Guzelbaev Ya. Z., Proektirovanie i ekspluatatsiya promyshlennykh tsentrobezhnykh kompressorov, 2, ispr. i dop., FEN, Kazan, 2012, 671 pp.

[6] Sokolov N. V., Khadiev M. B., Khavkin A. L., Khusnutdinov I. F., “Kharakter osevykh kolebanii rotora pri peremennykh rezhimakh raboty tsentrobezhnoi kompressornoi ustanovki”, Kompressornaya tekhnika i pnevmatika, 4 (2018), 29–32

[7] Lund J. W., “Review of the concept of dynamic coefficients for fluid film journal bearings”, ASME Journal of Tribology, 109:1 (1987), 37-41 | DOI

[8] Zhu Q., Zhang W. J., “A Preliminary nonlinear analysis of the axial transient responseof the sector-shaped hydrodynamic thrust bearing-rotor system”, ASME Journal of Tribology, 125:4 (2003), 854–858 | DOI

[9] Sokolov N. V., Khadiev M. B., Maksimov T. V., Futin V. A., Odnostupenchataya tsentrobezhnaya kompressornaya ustanovka: praktikum, KNITU, Kazan, 2019, 152 pp.

[10] Khadiev M. B., Zinnatullin N. Kh., Nafikov I. M., “Mekhanizm pompazha v tsentrobezhnykh kompressorakh”, Vestnik Kazanskogo tekhnologicheskogo universiteta, 17:8 (2014), 262–266

[11] Heshmat H., Pinkus O., “Mixing inlet temperatures in hydrodynamic bearings”, ASME Journal of tribology, 108:2 (1886), 231–244 | DOI

[12] Uskov M. K., Maksimov V. A., Gidrodinamicheskaya teoriya smazki: etapy razvitiya, sovremennoe sostoyanie, perspektivy, Nauka, Moskva, 1985, 143 pp.

[13] Sokolov N. V., Khadiev M. B., Maksimov T. V., Fedotov E. M., Fedotov P. E., “Mathematical modeling of dynamic processes of lubricating layers thrust bearing turbochargers”, Journal of Physics: Conference Series, 1158:04219 (2019), 138–151 | DOI

[14] Sokolov N. V., Khadiev M. B., Fedotov P. E, Fedotov E. M., “Trekhmernoe periodicheskoe termouprugogidrodinamicheskoe modelirovanie gidrodinamicheskikh protsessov upornogo podshipnika skolzheniya”, Vestnik Samarskogo universiteta. Aerokosmicheskaya tekhnika, tekhnologii i mashinostroenie, 20:3 (2021), 138–151 | DOI

[15] Sokolov N. V., Khadiev M. B., Fedotov P. E., Fedotov E. M., “Mathematical model of a dynamically loaded thrust bearing of a compressor and some results of its calculation”, Mesh methods for boundary-value problems and applications. Lecture notes in computational science and engineering, 141 (2022), 461–473 | DOI

[16] Maksimov V. A., Khadiev M. B., Fedotov E. M., “Opredelenie gidrodinamicheskikh i teplovykh kharakteristik upornykh podshipnikov matematicheskim modelirovaniem”, Vestnik mashinostroeniya, 2004, no. 6, 39–45

[17] Sokolov N. V., Khadiev M. B., Fedotov P. E., Fedotov E. M., “Chislennoe issledovanie vliyaniya klassa vyazkosti smazki na rabotu upornogo podshipnika skolzheniya”, Matematicheskoe modelirovanie i chislennye metody, 2023, no. 1, 92–111 | DOI

[18] Golubev A. I., Tortsovye uplotneniya vraschayuschikhsya valov, Mashinostroenie, M., 1974, 214 pp.

[19] Fedotov P. E., Fedotov E. M., Sokolov N. V., Khadiev M. B., $Sm2Px3Tx\tau$ — Dinamicheski nagruzhennyi upornyi podshipnik skolzheniya pri postanovke pryamoi zadachi, Svid-vo o gosud. registratsii programmy dlya EVM # 2020615227, 2020

[20] Sokolov N. V., Khadiev M. B., Fedotov P. E., Fedotov E. M., “Vliyanie temperatury podachi smazochnogo materiala na rabotu upornogo podshipnika skolzheniya”, Vestnik mashinostroeniya, 2023, no. 1, 47–55 | DOI

[21] Sokolov N. V., Khadiev M. B., Fedotov P. E., Fedotov E. M., “Sravnenie kvazitrekhmernoi i polnoi trekhmernoi postanovok raboty upornogo podshipnika skolzheniya”, Vestnik Samarskogo universiteta. Aerokosmicheskaya tekhnika, tekhnologii i mashinostroenie, 22:3 (2023), 143–159 | DOI

[22] Fedotov P. E., “Numerical solution of the one-sided compressor thrust bearing dynamics equation”, CEUR Workshop Proceedings, 2837 (2021), 54–75

[23] Savin L. A., Solomin O. V., Ustinov D. E., “Metod prostranstvennogo dvizheniya zhestkogo rotora na oporakh zhidkostnogo treniya”, Vestnik Samarskogo gosudarstvennogo aerokosmicheskogo universiteta im. akademika S. P. Koroleva (natsionalnogo issledovatelskogo universiteta), 2006, no. 2-1, 328–332

[24] Korneev A. Yu., “Analiz dinamiki zhestkogo rotora na konicheskikh gidrodinamicheskikh podshipnikakh skolzheniya metodom traektorii”, Vestnik mashinostroeniya, 2013, no. 12, 24–28

[25] Forsythe G. E., Malcolm M. A., Moler C. B., “Computer Methods for Mathematical Computations”, Prentice-Hall series in computational mathematics, Prentice-Hall, Englewood Cliffs, NJ, 1977, 259 pp. | DOI | MR | Zbl