Dynamic Relationship Between Oil Temperature and BGCI in Bell 407 Helicopter A Cointegration and Autoregressive Distributed Lag Approach
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Published
May 10, 2025
lotfi Saidi
Eric Bechhofer Mohamed Benbouzid
https://orcid.org/0000-0002-3419-8583
Eric Bechhofer Mohamed Benbouzid
Abstract
The primary objective of this study was to investigate the dynamic relationship between oil temperature and the Bearing Gearbox Condition Indicator (BGCI) values of the Bell 407 helicopter. The study aims to simplify the fault diagnosis process by proposing a method that utilizes only one vibration sensor and one temperature sensor per bearing. To achieve this goal, we employ robust econometric tools, such as unit root tests, cointegration tests, and Autoregressive Distributed Lag (ARDL) models, for both long-run and short-run estimates. Our findings indicate that the variable temperature tends to converge to its long-run equilibrium path in response to changes in other variables. The results of the ARDL analysis confirmed that spectral kurtosis, inner race, cage, and ball energy significantly contributed to the increase in temperature. Furthermore, we utilized the Impulse Response Function (IRF) to trace the dynamic response paths of the shocks to the system. The identification of a cointegrating relationship between oil temperature and BGCI values suggests a practical and significant connection that can potentially be used to predict hazardous changes in oil temperature using BGCI values, which is an important implication for enhancing the safety and reliability of helicopter operations.
The study presents a promising direction for condition monitoring (CM) in rotating machinery, emphasizing the potential of integrating temperature data to simplify the diagnostic process while still achieving reliable results.
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Keywords
Autoregressive Distributed Lag (ARDL), Bearing gearbox, Cointegration, Condition monitoring (CM), , Condition indicator (CI), Oil temperature, Vibration analysis
References
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Zhao, J., Hou, L., Li, Z., Zhang, H., & Zhu, R. (2022). Prediction of tribological and dynamical behaviors of spur gear pair considering tooth root crack. Engineering Failure Analysis, 135, 106145.
Rosenkranz, L., Richter, S., Jacobs, G., Mikitisin, A., Mayer, J., Stratmann, A., & König, F. (2021). Influence of temperature on wear performance of greases in rolling bearings. Industrial Lubrication and Tribology, 73(6), 862-871.
Saidi, L., Ali, J. B., Bechhoefer, E., & Benbouzid, M. (2017, October). Particle filter-based prognostic approach for high-speed shaft bearing wind turbine progressive degradations. In IECON 2017-43rd Annual Conference of the IEEE Industrial Electronics Society (pp. 8099-8104). IEEE.
Zhang, G., Tong, B., & Yin, Y. (2020). Temperature distribution and heat generating/transfer mechanism of the circular bilayer porous bearing for thermo-hydrodynamic problem. International Journal of Heat and Mass Transfer, 149, 119134.
Ismail, A., Saidi, L., Sayadi, M., & Benbouzid, M. (2020). Remaining useful lifetime prediction of thermally aged power insulated gate bipolar transistor based on Gaussian process regression. Transactions of the Institute of Measurement and Control, 42(13), 2507-2518.
Tabrizi, A. A., Al-Bugharbee, H., Trendafilova, I., & Garibaldi, L. (2017). A cointegration-based monitoring method for rolling bearings working in time-varying operational conditions. Meccanica, 52, 1201-1217.
Zhang, Y. G., Jiang, X. F., Wu, X. W., & Ying, Z. (2012). Diagnosis of High Oil Temperature of the Bearings Based on Oil Analytical Techniques. Applied Mechanics and Materials, 224, 119-122.
Bouhadra, K., & Forest, F. (2024). Knowledge-based and Expert Systems in Prognostics and Health Management: a Survey. International Journal of Prognostics and Health Management, 15(2).
Li, H., Li, Y., & Yu, H. (2019). A novel health indicator based on cointegration for rolling bearings’ run-to-failure process. Sensors, 19(9), 2151.
Koizumi, T., & Kogiso, N. (2024). An Effectiveness Evaluation Method Using System of Systems Architecture Description of Aircraft Health Management in Aircraft Maintenance Program. International Journal of Prognostics and Health Management, 15(3).
Dass, T. D., Gunakala, S. R., Comissiong, D., Azamathulla, H. M., Martin, H., & Ramachandran, S. (2024). Investigating journal bearing characteristics incorporating variable viscosity, couple-stress lubricant, slip-velocity, magnetic fluid, and sinusoidal surface-texturing. Results in Engineering, 102338.
Babay, O., Saidi, L., Bechhofer, E., & Benbouzid, M. (2024, November). Investigation of the Dynamic Relationship between Oil Temperature and Bearing Gearbox Condition Indicator Values for the Bell 407 Helicopter Based on Cointegration Analysis. In Annual Conference of the PHM Society (Vol. 16, No. 1).
Bechhoefer, E., Schlanbusch, R., & Waag, T. I. (2016). Techniques for large, slow bearing fault detection. International Journal of Prognostics and Health Management, 7(1).
Saidi, L., & Benbouzid, M. (2021). Prognostics and health management of renewable energy systems: state of the art review, challenges, and trends. Electronics, 10(22), 2732.
Wu, K. T., Kobayashi, M., Sun, Z., Jen, C. K., Sammut, P., Bird, J., & Mrad, N. (2011). Engine oil condition monitoring using high temperature integrated ultrasonic transducers. International Journal of Prognostics and Health Management, 2(2).
Zhu, J., Yoon, J. M., He, D., Qu, Y., & Bechhoefer, E. (2013). Lubrication oil condition monitoring and remaining useful life prediction with particle filtering. International Journal of Prognostics and Health Management, 4, 124-138.
Xu, J., Li, X., Chen, R., Wang, L., Yang, Z., & Yang, H. (2021). Dynamic characteristics analysis of gear-bearing system considering dynamic wear with flash temperature. Mathematics, 9(21), 2739.
Shafique, M., Azam, A., Rafiq, M., & Luo, X. (2021). Investigating the nexus among transport, economic growth and environmental degradation: Evidence from panel ARDL approach. Transport Policy, 109, 61-71.
Ouni, M., & Ben Abdallah, K. (2024). Environmental sustainability and green logistics: Evidence from BRICS and Gulf countries by cross‐sectionally augmented autoregressive distributed lag (CS‐ARDL) approach. Sustainable Development.
Dumitrescu, E. I., & Hurlin, C. (2012). Testing for Granger non-causality in heterogeneous panels. Economic modelling, 29(4), 1450-1460.
Abboud, D., Antoni, J., Sieg-Zieba, S., & Eltabach, M. (2017). Envelope analysis of rotating machine vibrations in variable speed conditions: A comprehensive treatment. Mechanical Systems and Signal Processing, 84, 200-226.
Dickey, D. A., & Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American statistical association, 74(366a), 427-431.
Phillips, P. (1988). Testing for unit roots in time series regression. Biometrika, 71, 599-607.
Johansen, S., & Juselius, K. (1990). Maximum likelihood estimation and inference on cointegration—with appucations to the demand for money. Oxford Bulletin of Economics and statistics, 52(2), 169-210.
Pesaran, M. H., Shin, Y., & Smith, R. J. (2001). Bounds testing approaches to the analysis of level relationships. Journal of applied econometrics, 16(3), 289-326.
Narayan, P. K. (2005). The saving and investment nexus for China: evidence from cointegration tests. Applied economics, 37(17), 1979-1990.
Granger, C. W. J. (2001). Investigating causal relations by econometric models and cross-spectral methods. In Essays in econometrics: collected papers of Clive WJ Granger (pp. 31-47).
Gomez-Gonzalez, J. E., Hirs-Garzon, J., & Uribe Gil, J. M. (2020). Global effects of US uncertainty: real and financial shocks on real and financial markets. IREA–Working Papers, 2020, IR20/15.
Koop, G., Pesaran, M. H., & Potter, S. M. (1996). Impulse response analysis in nonlinear multivariate models. Journal of econometrics, 74(1), 119-147.
Soomro, A. A., Muhammad, M. B., Mokhtar, A. A., Saad, M. H. M., Lashari, N., Hussain, M., & Palli, A. S. (2024). Insights into modern machine learning approaches for bearing fault classification: A systematic literature review. Results in Engineering, 102700.J.
Zhao, J., Hou, L., Li, Z., Zhang, H., & Zhu, R. (2022). Prediction of tribological and dynamical behaviors of spur gear pair considering tooth root crack. Engineering Failure Analysis, 135, 106145.
Rosenkranz, L., Richter, S., Jacobs, G., Mikitisin, A., Mayer, J., Stratmann, A., & König, F. (2021). Influence of temperature on wear performance of greases in rolling bearings. Industrial Lubrication and Tribology, 73(6), 862-871.
Saidi, L., Ali, J. B., Bechhoefer, E., & Benbouzid, M. (2017, October). Particle filter-based prognostic approach for high-speed shaft bearing wind turbine progressive degradations. In IECON 2017-43rd Annual Conference of the IEEE Industrial Electronics Society (pp. 8099-8104). IEEE.
Zhang, G., Tong, B., & Yin, Y. (2020). Temperature distribution and heat generating/transfer mechanism of the circular bilayer porous bearing for thermo-hydrodynamic problem. International Journal of Heat and Mass Transfer, 149, 119134.
Ismail, A., Saidi, L., Sayadi, M., & Benbouzid, M. (2020). Remaining useful lifetime prediction of thermally aged power insulated gate bipolar transistor based on Gaussian process regression. Transactions of the Institute of Measurement and Control, 42(13), 2507-2518.
Tabrizi, A. A., Al-Bugharbee, H., Trendafilova, I., & Garibaldi, L. (2017). A cointegration-based monitoring method for rolling bearings working in time-varying operational conditions. Meccanica, 52, 1201-1217.
Zhang, Y. G., Jiang, X. F., Wu, X. W., & Ying, Z. (2012). Diagnosis of High Oil Temperature of the Bearings Based on Oil Analytical Techniques. Applied Mechanics and Materials, 224, 119-122.
Bouhadra, K., & Forest, F. (2024). Knowledge-based and Expert Systems in Prognostics and Health Management: a Survey. International Journal of Prognostics and Health Management, 15(2).
Li, H., Li, Y., & Yu, H. (2019). A novel health indicator based on cointegration for rolling bearings’ run-to-failure process. Sensors, 19(9), 2151.
Koizumi, T., & Kogiso, N. (2024). An Effectiveness Evaluation Method Using System of Systems Architecture Description of Aircraft Health Management in Aircraft Maintenance Program. International Journal of Prognostics and Health Management, 15(3).
Dass, T. D., Gunakala, S. R., Comissiong, D., Azamathulla, H. M., Martin, H., & Ramachandran, S. (2024). Investigating journal bearing characteristics incorporating variable viscosity, couple-stress lubricant, slip-velocity, magnetic fluid, and sinusoidal surface-texturing. Results in Engineering, 102338.
Babay, O., Saidi, L., Bechhofer, E., & Benbouzid, M. (2024, November). Investigation of the Dynamic Relationship between Oil Temperature and Bearing Gearbox Condition Indicator Values for the Bell 407 Helicopter Based on Cointegration Analysis. In Annual Conference of the PHM Society (Vol. 16, No. 1).
Bechhoefer, E., Schlanbusch, R., & Waag, T. I. (2016). Techniques for large, slow bearing fault detection. International Journal of Prognostics and Health Management, 7(1).
Saidi, L., & Benbouzid, M. (2021). Prognostics and health management of renewable energy systems: state of the art review, challenges, and trends. Electronics, 10(22), 2732.
Wu, K. T., Kobayashi, M., Sun, Z., Jen, C. K., Sammut, P., Bird, J., & Mrad, N. (2011). Engine oil condition monitoring using high temperature integrated ultrasonic transducers. International Journal of Prognostics and Health Management, 2(2).
Zhu, J., Yoon, J. M., He, D., Qu, Y., & Bechhoefer, E. (2013). Lubrication oil condition monitoring and remaining useful life prediction with particle filtering. International Journal of Prognostics and Health Management, 4, 124-138.
Xu, J., Li, X., Chen, R., Wang, L., Yang, Z., & Yang, H. (2021). Dynamic characteristics analysis of gear-bearing system considering dynamic wear with flash temperature. Mathematics, 9(21), 2739.
Shafique, M., Azam, A., Rafiq, M., & Luo, X. (2021). Investigating the nexus among transport, economic growth and environmental degradation: Evidence from panel ARDL approach. Transport Policy, 109, 61-71.
Ouni, M., & Ben Abdallah, K. (2024). Environmental sustainability and green logistics: Evidence from BRICS and Gulf countries by cross‐sectionally augmented autoregressive distributed lag (CS‐ARDL) approach. Sustainable Development.
Dumitrescu, E. I., & Hurlin, C. (2012). Testing for Granger non-causality in heterogeneous panels. Economic modelling, 29(4), 1450-1460.
Abboud, D., Antoni, J., Sieg-Zieba, S., & Eltabach, M. (2017). Envelope analysis of rotating machine vibrations in variable speed conditions: A comprehensive treatment. Mechanical Systems and Signal Processing, 84, 200-226.
Dickey, D. A., & Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American statistical association, 74(366a), 427-431.
Phillips, P. (1988). Testing for unit roots in time series regression. Biometrika, 71, 599-607.
Johansen, S., & Juselius, K. (1990). Maximum likelihood estimation and inference on cointegration—with appucations to the demand for money. Oxford Bulletin of Economics and statistics, 52(2), 169-210.
Pesaran, M. H., Shin, Y., & Smith, R. J. (2001). Bounds testing approaches to the analysis of level relationships. Journal of applied econometrics, 16(3), 289-326.
Narayan, P. K. (2005). The saving and investment nexus for China: evidence from cointegration tests. Applied economics, 37(17), 1979-1990.
Granger, C. W. J. (2001). Investigating causal relations by econometric models and cross-spectral methods. In Essays in econometrics: collected papers of Clive WJ Granger (pp. 31-47).
Gomez-Gonzalez, J. E., Hirs-Garzon, J., & Uribe Gil, J. M. (2020). Global effects of US uncertainty: real and financial shocks on real and financial markets. IREA–Working Papers, 2020, IR20/15.
Koop, G., Pesaran, M. H., & Potter, S. M. (1996). Impulse response analysis in nonlinear multivariate models. Journal of econometrics, 74(1), 119-147.
Section
Technical Papers