Numerical investigation, modelling and vibrational dynamics of an unbalanced rigid rotor with two offset discs levitated by air foil bearings
Siddharth SUMAN, Prabhat KUMAR
Abstract. High speed rotating machines are very useful in several industries and production plants or factories. The rotating components in these machines are usually prone to multiple types of faults. These faults can cause large and heavy amount of vibrations in the assembled rotor systems during its operation. So, there is a need for analyzing the rotor behavior in the presence of faults and their identification. This paper discusses the dynamic behavior of a rigid rotor with two discs at offset positions and mounted on air foil bearings at the ends. In this system, it is assumed that the discrete unbalance fault is present at discs only. The stiffness and damping coefficients of both foil bearings are considered to be different and anisotropic in nature. Considering the forces due to foil bearings, inertia force, discs unbalance force, inertia moment, gyroscopic couple effect, the equations of motion of the rotor system have been derived in the two-dimensional transverse directions. Further, these equations are solved using a model developed in the Simulink platform. The obtained solutions of the equations are the time domain rotor displacement in the vertical and horizontal directions. It would be very interesting to investigate and study the unbalance fault effect and displacement responses with variation in the fault parameters, rotor spin speed, and stiffness and damping parameters of foil bearings.
Keywords
Foil Bearing, Gyroscopic Couple, Moment Equilibrium Method, Rotor, Unbalance
Published online 3/1/2025, 10 pages
Copyright © 2025 by the author(s)
Published under license by Materials Research Forum LLC., Millersville PA, USA
Citation: Siddharth SUMAN, Prabhat KUMAR, Numerical investigation, modelling and vibrational dynamics of an unbalanced rigid rotor with two offset discs levitated by air foil bearings, Materials Research Proceedings, Vol. 49, pp 286-295, 2025
DOI: https://doi.org/10.21741/9781644903438-29
The article was published as article 29 of the book Mechanical Engineering for Sustainable Development
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
References
[1] R. Tiwari, Rotor Systems: Analysis and Identification. Boca Raton: CRC Press, 2017. https://doi.org/10.1201/9781315230962
[2] R. Tiwari and P. Kumar, “An innovative virtual trial misalignment approach for identification of unbalance, sensor and active magnetic bearing misalignment along with its stiffness parameters in a magnetically levitated flexible rotor system,” Mechanical Systems and Signal Processing, vol. 167, p. 108540, 2022. https://doi.org/10.1016/j.ymssp.2021.108540
[3] P. Kumar and R. Tiwari, “A review: multiplicative faults and model-based condition monitoring strategies for fault diagnosis in rotary machines,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 45, p. 282, 2023. https://doi.org/10.1007/s40430-023-04203-z
[4] S. Harsha, “Nonlinear dynamic response of a balanced rotor supported by rolling element bearings due to radial internal clearance effect,” Mechanism and Machine Theory, vol. 41, pp. 688-706, 2006. https://doi.org/10.1016/j.mechmachtheory.2005.09.003
[5] R. Tomovic, V. Miltenovic, M. Banic, and A. Miltenovic, “Vibration response of rigid rotor in unloaded rolling element bearing,” International Journal of Mechanical Sciences, vol. 52, pp. 1176-1185, 2010. https://doi.org/10.1016/j.ijmecsci.2010.05.003
[6] P. Kumar, “Condition Monitoring of Mechanical and Electrical Faults in Stationary and Rotating Equipments: A Review,” in Recent Advances in Materials: Select Proceedings of ICSTE 2023, ed: Springer, 2023, pp. 297-306. https://doi.org/10.1007/978-981-99-3844-5_31
[7] E. Swanson and H. Heshmat, “Oil-Free Foil Bearings as a Reliable, High Performance Backup Bearing for Active Magnetic Bearings,” in ASME Turbo Expo 2002: Power for Land, Sea, and Air, 2002, pp. 589-598. https://doi.org/10.1115/GT2002-30291
[8] L. San Andrés and T. H. Kim, “Gas Foil Bearings: Limits for High-Speed Operation,” in World Tribology Congress III, 2005, pp. 71-72. https://doi.org/10.1115/1.2197851
[9] P. Kumar, “Dynamic analysis and identification in a cracked and unbalanced rigid rotor with two offset discs and one middle disc mounted on foil bearings,” International Journal of Dynamics and Control, vol. 12, pp. 2648-2673, 2024. https://doi.org/10.1007/s40435-024-01411-w
[10] P. Kumar, C. Arambam, L. D. Singh, and N. D. Singh, “Analysing the Dynamic Interaction Between Unbalance and Crack Responses in a Jeffcott Rotor Supported by Foil Bearings: A Numerical Study,” in Emerging Trends in Mechanical and Industrial Engineering: Select Proceedings of ICETMIE 2022, ed: Springer, 2023, pp. 971-980. https://doi.org/10.1007/978-981-19-6945-4_73
[11] P. Kumar, “Modelling, Analysis, and Identification in a Cracked and Unbalanced Jeffcott Rotor Supported on Foil Bearings,” Archive of Mechanical Engineering, vol. 71, pp. 295-322, 2024. https://doi.org/10.24425/ame.2024.150566
[12] J. S. Larsen, I. F. Santos, and S. von Osmanski, “Stability of rigid rotors supported by air foil bearings: Comparison of two fundamental approaches,” Journal of Sound and Vibration, vol. 381, pp. 179-191, 2016. https://doi.org/10.1016/j.jsv.2016.06.022
[13] A. Martowicz, J. Roemer, S. Kantor, P. Zdziebko, G. Żywica, and P. Bagiński, “Gas foil bearing technology enhanced with smart materials,” Applied Sciences, vol. 11, p. 2757, 2021. https://doi.org/10.3390/app11062757
[14] D. S. Khamari, J. Kumar, and S. K. Behera, “A Review on Modeling and Stability Aspects of Gas Foil Bearing Supported Rotors,” Tribology in Industry, vol. 44, p. 12, 2023. http://dx.doi.org/10.24874/ti.1381.09.22.01
[15] P. Kumar and R. Tiwari, “A Numerical Study on the Effect of Unbalance and Misalignment Fault Parameters in a Rigid Rotor Levitated by Active Magnetic Bearings,” in ASME 2019 Gas Turbine India Conference, 2019. https://doi.org/10.1115/GTINDIA2019-2384
[16] P. Kumar, V. Kumar, K. Kumar, and L. S. Meena, “Unbalance and Dynamic Parameters Estimation in a Rigid Rotor Mounted on Active Magnetic Bearings,” in Advances in Applied Mechanical Engineering, ed: Springer, 2020, pp. 363-371. https://doi.org/10.1007/978-981-15-1201-8_41
[17] P. Kumar and R. Tiwari, “Finite element modelling, analysis and identification using novel trial misalignment approach in an unbalanced and misaligned flexible rotor system levitated by active magnetic bearings,” Mechanical Systems and Signal Processing, vol. 152, p. 107454, 2021. https://doi.org/10.1016/j.ymssp.2020.107454
[18] P. Kumar, M. Sanket, S. Srivastav, and T. D. Madav, “Vibrational Nature of an Unbalanced Rigid Rotor System with Three Discs Secured by Two Active Magnetic Bearings,” in Recent Advances in Materials: Select Proceedings of ICSTE 2023, ed: Springer, 2023, pp. 335-345. https://doi.org/10.1007/978-981-99-3844-5_35
[19] P. Nauclér and T. Söderström, “Unbalance estimation using linear and nonlinear regression,” Automatica, vol. 46, pp. 1752-1761, 2010. https://doi.org/10.1016/j.automatica.2010.06.053
[20] I. Chatzisavvas and F. Dohnal, “Unbalance identification using the least angle regression technique,” Mechanical Systems and Signal Processing, vol. 50, pp. 706-717, 2015. https://doi.org/10.1016/j.ymssp.2014.05.002
[21] J. Yao, L. Liu, F. Yang, F. Scarpa, and J. Gao, “Identification and optimization of unbalance parameters in rotor-bearing systems,” Journal of Sound and Vibration, vol. 431, pp. 54-69, 2018. https://doi.org/10.1016/j.jsv.2018.05.050
[22] A. Shrivastava and A. R. Mohanty, “Identification of unbalance in a rotor system using a joint input-state estimation technique,” Journal of Sound and Vibration, vol. 442, pp. 414-427, 2019. https://doi.org/10.1016/j.jsv.2018.11.019
[23] P. Kumar and R. Tiwari, “Dynamic Response Analysis of an Unbalanced and Misaligned Rotor Supported on Active Magnetic Bearings and Touchdown Bearings,” in Proceedings of the 6th National Symposium on Rotor Dynamics, 2020, pp. 407-418. https://doi.org/10.1007/978-981-15-5701-9_33
[24] P. Kumar and R. Tiwari, “Effects of unbalance and AMB misalignment in a rigid rotor with an offset disc levitated by active magnetic bearings: A numerical investigation,” in 12th International Conference on Vibrations in Rotating Machinery, 2020, pp. 151-168. https://doi.org/10.1201/9781003132639
[25] A. Shrivastava and A. R. Mohanty, “Identification of unbalance in a rotor-bearing system using Kalman filter–based input estimation technique,” Journal of Vibration and Control, vol. 26, pp. 1081-1091, 2020. https://doi.org/10.1177/107754631989164
[26] P. Kumar, “Model Based Analysis and Identification of Unbalance and Misalignment in Rotor Systems Levitated by Active Magnetic Bearings,” PhD, Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, 2021.
[27] P. Kumar and R. Tiwari, “Additive Fault Diagnosis Techniques in Rotor Systems: A State-of-the-Art Review,” Sadhana, vol. 49, pp. 1-49, 2024. https://doi.org/10.1007/s12046-024-02543-7
[28] J. Baker, “Methods of rotor-unbalance determination,” Journal of Applied Mechanics, vol. 6, pp. 1-6, 1939. https://doi.org/10.1115/1.4008884
[29] K. Gupta, K. Gupta, and K. Athre, “Unbalance response of a dual rotor system: theory and experiment,” Journal of Vibration and Acoustics, vol. 115, pp. 427-435, 1993. https://doi.org/10.1115/1.2930368
[30] Y.-P. Shih and A.-C. Lee, “Identification of the unbalance distribution in flexible rotors,” International Journal of Mechanical Sciences, vol. 39, pp. 841-857, 1997. https://doi.org/10.1016/S0020-7403(96)00078-1
[31] S. Zhou and J. Shi, “Active balancing and vibration control of rotating machinery: a survey,” Shock and Vibration Digest, vol. 33, pp. 361-371, 2001. https://doi.org/10.1177/058310240103300501
[32] R. Tiwari, “Conditioning of regression matrices for simultaneous estimation of the residual unbalance and bearing dynamic parameters,” Mechanical Systems and Signal Processing, vol. 19, pp. 1082-1095, 2005. https://doi.org/10.1016/j.ymssp.2004.09.005
[33] H. F. De Castro, K. L. Cavalca, L. W. F. De Camargo, and N. Bachschmid, “Identification of unbalance forces by metaheuristic search algorithms,” Mechanical Systems and Signal Processing, vol. 24, pp. 1785-1798, 2010. https://doi.org/10.1016/j.ymssp.2009.11.012
[34] Y. Menshikov, “Identification of Rotor Unbalance as Inverse Problem of Measurement,” Advances in Pure Mathematics, vol. 2013, 2013. https://doi.org/10.4236/apm.2013.39A1004
[35] P. Kumar and R. Tiwari, “Development of a Novel Approach for Quantitative Estimation of Rotor Unbalance and Misalignment in a Rotor System Levitated by Active Magnetic Bearings,” Iranian Journal of Science and Technology-Transactions of Mechanical Engineering, vol. 45, pp. 769-786, 2021. https://doi.org/10.1007/s40997-020-00364-7
[36] P. Kumar and R. Tiwari, “Dynamic analysis and identification of unbalance and misalignment in a rigid rotor with two offset discs levitated by active magnetic bearings: a novel trial misalignment approach,” Propulsion and Power Research, vol. 10, pp. 58-82, 2021. https://doi.org/10.1016/j.jppr.2020.06.003
[37] P. Kumar and R. Tiwari, ” Identification in a Magnetically Levitated Rigid Rotor System Integrated with Misaligned Sensors and Active Magnetic Bearings,” Journal of Vibration Engineering & Technologies, pp. 1-22, 2024. https://doi.org/10.1007/s42417-024-01576-w
[38] Z. Guo, K. Feng, T. Liu, P. Lyu, and T. Zhang, “Nonlinear dynamic analysis of rigid rotor supported by gas foil bearings: Effects of gas film and foil structure on subsynchronous vibrations,” Mechanical Systems and Signal Processing, vol. 107, pp. 549-566, 2018. https://doi.org/10.1016/j.ymssp.2018.02.005