Mathematical modeling of the rotor dynamics of a turbomachine on gas foil bearings subjected to vibration

Abstract

BACKGROUND: The use of gas foil bearings is a promising development in the field of turbomachinery due to their economy, autonomous operation capability, and durability. However, gas foil bearings have lower load capacities than other types of bearings. However, turbomachines are complicated, dynamic systems that must meet high standards of safety, sustainability, and durability against external mechanical factors like vibration, shock, etc.

AIM: Development of a mathematical model of rotor dynamics to predict the displacement of the rotor in foil bearings for maintaining separation between the rotor and the housing while being subjected to vibration.

METHODS: A mathematical model of the dynamics of a stiff rotor on gas foil bearings was built and analyzed, taking into account the flexibility of the bearing bushing supports and the housing of the turbomachine. Stationary and transient modes of operation, including the transient modes combined with random vibration, are simulated. The system of ordinary derivatives equations describing the mathematical model was solved by the Rado IIA method. Random vibration was modeled using digital Fourier transformation. The modeling results were analyzed by discrete Fourier transformation and short-time Fourier transformation.

RESULTS AND CONCLUSIONS: Rotor movement trajectories were obtained and the results were compared with author’s previous experimental data. Upper bound of maximal displacements was obtained. The maximum values of rotor displacement can be used to set the optimal values of blade tip gaps.

Keywords

foil bearings / gas bearings / air cycle machine / turboexpander / radial inflow machine / transport environmental control system / turbomachinery / vibration / random vibration

References

[1] Sukhomlinov IYa, Golovin MV Hermetic centrifugal refrigeration compressor on the gas-dynamic bearings. Compressor technology and pneumatics. 2014;6:6–10. (in Russ).

[2] Сухомлинов И.Я., Головин М.В. Герметичный холодильный центробежный компрессор на газодинамических подшипниках // Компрессорная техника и пневматика. 2014. № 6. С. 6–10.

[3] Polikarpov AV, Vikulov AP, Zotov SN, et al. Oilfree centrifugal electric compressor with foil gasdynamic bearings. Refrigeration Technology. 2020;109(2):36–44. doi: 10.17816/RF104085 (in Russ).

[4] Поликарпов А.В., Викулов А.П., Зотов С.Н., и др. Безмасляный центробежный электрокомпрессор на лепестковых газодинамических подшипниках // Холодильная техника. 2020. Т. 109. № 2. C. 36–44. doi: 10.17816/RF104085

[5] Shchedukhin SI, Polikarpov AV, Vikulov AP, et al. Bezmaslyanyy turbodetander prirodnogo gaza na lepestkovykh gazodinamicheskikh podshipnikakh. Refrigeration Technology. 2017;106(6):46–51. doi: 10.17816/RF99254 (in Russ).

[6] Щедухин С.И., Поликарпов А.В., Викулов А.П., и др. Безмасляный турбодетандер природного газа на лепестковых газодинамических подшипниках // Холодильная техника. 2017. Т. 106 № 6. С. 46–50. doi: 10.17816/RF99254

[7] Zvonarev PN. Razrabotka metoda rascheta radial’nyh uprugogazdinamicheskih podshipnikov s predvaritel’no naprjazhennymi lepestkami dlja malyh turbomashin nizkotemperaturnyh ustanovok [dissertation] Moscow; 2005. (in Russ).

[8] Звонарев П.Н. Разработка метода расчета радиальных упругогаздинамических подшипников с предварительно напряженными лепестками для малых турбомашин низкотемпературных установок: дис. ... канд. тех. наук. М., 2005.

[9] Sytin AV. Reshenie kompleksnoj zadachi rascheta harakteristik radial’nyh lepestkovyh gazodinamicheskih podshipnikov [dissertation]. Orel; 2008. (in Russ).

[10] Сытин А.В. Решение комплексной задачи расчета характеристик радиальных лепестковых газодинамических подшипников: дис. ... канд. тех. наук. Орел, 2008.

[11] Bonello P, Pham H. The efficient computation of the nonlinear dynamic response of a foil air bearing rotor system. J. Sound Vibr. 2014;333:3459–3478. doi: 10.1016/j.jsv.2014.03.001

[12] Bonello P., Pham H. The efficient computation of the nonlinear dynamic response of a foil air bearing rotor system // J. Sound Vib. 2014. Vol. 333. P. 3459–3478. doi: 10.1016/j.jsv.2014.03.001

[13] Andrés LS, Rubio D, Kim TH. Rotordynamic performance of a rotor supported on bump type foil gas bearings: experiments and predictions. ASME J. Eng. Gas Turbines Power. 2007;129(3):850–857. doi: 10.1115/1.2718233

[14] Andrés L.S., Rubio D., Kim T.H. Rotordynamic performance of a rotor supported on bump type foil gas bearings: experiments and predictions // ASME J. Eng. Gas Turbines Power. 2007. Vol. 129, N. 3. P. 850–857. doi: 10.1115/1.2718233

[15] Powell JW, Tempest MC A Study of High Speed Machines with rubber Stabilized Air Bearings. ASME J. Lubric. tech. 1968;90(4):701–707. doi: 10.1115/1.3601702

[16] Powell J.W., Tempest M.C. A Study of High Speed Machines with rubber Stabilized Air Bearings // ASME J. Lubric. tech. 1968. Vol. 90, N. 4. P. 701–707. doi: 10.1115/1.3601702

[17] Waumans T, Peirs J, Al-Bender F, et al. Aerodynamic journal bearing with a flexible, damped support operating at 7.2 million DN. J. Micromech. Microeng. 2011;21:104014. doi: 10.1088/0960-1317/21/10/104014

[18] Waumans T., Peirs J., Al-Bender F., et al. Aerodynamic journal bearing with a flexible, damped support operating at 7.2 million DN // J. Micromech. Microeng. 2011. Vol. 21. P. 104014. doi: 10.1088/0960-1317/21/10/104014

[19] Gu Y, Ma Y, Ren G. Stability and vibration characteristics of a rotor-gas foil bearings system with high-static-low-dynamic-stiffness supports. J. Sound Vibr. 2017;397:152–170. doi: 10.1016/j.jsv.2017.02.047

[20] Gu Y., Ma Y., Ren G. Stability and vibration characteristics of a rotor-gas foil bearings system with high-static-low-dynamic-stiffness supports // J. Sound. Vib. 2017. Vol. 397. P. 152–170. doi: 10.1016/j.jsv.2017.02.047

[21] Peshti YV. Gas lubricant. Moscow: Bauman Moscow State Technical University; 1993. (in Russ).

[22] Пешти Ю. В. Газовая смазка. Учебник для вузов. М. : МГТУ им. Баумана, 1993.

[23] Kim D. Parametric studies on static and dynamic performance of air foil bearings with different top foil geometries and bump stiffness distributions. J. Trib. 2007;129(2):354–364 doi: 10.1115/1.2540065

[24] Kim D. Parametric studies on static and dynamic performance of air foil bearings with different top foil geometries and bump stiffness distributions // J. Tribol. 2007. Vol. 129, N. 2. P. 354–364 doi: 10.1115/1.2540065

[25] Wanner G, Hairer E. Solving ordinary differential equations II. Stiff and differential algebraic problems. Moscow: Mir; 1999.

[26] Хайрер Э., Ваннер Г. Решение обыкновенных дифференциальных уравнений. Жесткие и дифференциально-алгебраические задачи. М.: Мир, 1999.

[27] Tishchenko IV, Nikolaev VS, Merkulov VI. Experimental Study of a Dynamics Rotor of an Aircraft Air Cycle Machine with Foil Gas Bearings. In: Refrigeration and cryogenic equipment, air conditioning and life support systems: Third international scientific and practical conference. November 19–20, 2019; Moscow, Russia. Bauman Moscow State Technical University.

[28] Тищенко И.В., Николаев В.С., Меркулов В.И. Экспериментальное исследование динамики ротора авиационного турбохолодильника на газодинамических подшипниках. // Холодильная и криогенная техника, системы кондиционирования и жизнеобеспечения: Третья международная научно-практическая конференция: материалы конференции, Москва, 19 ноября 2020 года. МГТУ им. Н.Э. Баумана.

[29] Nikolaev VS, Abalakin SA, Tishchenko IV. Comparison of efficiency losses due to leaks for turbine units of aviation air conditioning systems with petal-type gas-dynamic bearings and ball bearings. Refrigeration Technology. 2022;111(1):13–20. doi: 10.17816/RF96964

[30] Николаев В.С., Абалакин С.А., Тищенко И.В. Сравнение потерь эффективности из-за перетечек для турбоагрегатов авиационных систем кондиционирования на лепестковых газодинамических подшипниках и шарикоподшипниках // Холодильная техника. 2022. Т. 111, № 1. С. 13–20. doi: 10.17816/RF96964

RIGHTS & PERMISSIONS

Nikolaev V.S., Tishchenko I.V.