Physics-informed remaining useful life prediction of rolling bearings under variable speed using vibration envelope features and adaptive maintenance thresholds

Suleiman Ibrahim Mohammad
Research Fellow, INTI International University, Nilai 71800, Malaysia
Asokan Vasudevan
Faculty of Business and Communications, INTI International University, Nilai 71800, Malaysia
Seif Al Bustanji
Faculty of Technical Education, Hourani Center for Applied Scientific Research, Al-Ahliyya Amman University, Amman 19328, Jordan
Qian Chen
Faculty of Liberal Arts, Shinawatra University, Pathum Thani 12160, Thailand
Jin Zhang
Faculty of Liberal Arts, Shinawatra University, Pathum Thani 12160, Thailand
Ziyu Cai
Faculty of Liberal Arts, Shinawatra University, Pathum Thani 12160, Thailand

Article ID: 4356

Keywords: rolling bearings, remaining useful life (RUL), vibration analysis, envelope features, physics-informed modelling, variable speed, accelerated life testing, health index

Abstract

The reliable estimation of remaining useful life (RUL) of rolling bearings plays a critical role in maintaining the reliability of modern industrial equipment and minimizing machine downtime. However, the conventional vibration-based prognostic methods tend to experience challenges in predicting the remaining useful life of rolling bearings in variable-speed operating environments due to issues with nonstationary signals and the lack of incorporation of physical degradation processes. This paper proposes a physics-informed approach for estimating the remaining useful life of rolling bearings using vibration envelope characteristics and accelerated life testing. The approach starts with the use of order tracking combined with envelope analysis to extract vibration envelope characteristics under variable speed conditions. A health index is constructed to represent the degradation process. The nonlinear degradation process is modelled using a physics-informed exponential degradation model. An ensemble prediction model is proposed for predicting RUL. The results demonstrate that the developed model was significantly more accurate in its predictions, with a maximum of 49% improvement in the RMSE compared to traditional models and consistent results under varied operational conditions. The use of physics-based modelling and envelope analysis increased the clarity and robustness of the model, and the acceleration of the life testing process contributed to better generalizability of the model. Moreover, the introduction of adaptive threshold values improved maintenance time prediction by over 50%, and uncertainty assessment confirmed the validity of the model.

Published

2026-07-21

How to Cite

Ibrahim Mohammad , S., Vasudevan, A., Bustanji, S. A., Chen , Q., Zhang , J., & Cai, Z. (2026). Physics-informed remaining useful life prediction of rolling bearings under variable speed using vibration envelope features and adaptive maintenance thresholds. Sound & Vibration, 60(4). https://doi.org/10.59400/sv4356

Download Citation

Issue

Vol. 60 No. 4 (2026)

Section

Article

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

[1]Chen D, Wang Y, Chai Y, et al. A polynomial speed normalized health indicator for both incipient fault detection and prognosis of variable-speed wind turbine bearings. Advanced Engineering Informatics. 2025; 66: 103455. doi: 10.1016/j.aei.2025.103455

[2]Huang X, Chen W, Qu D, et al. Remaining Useful Life Prediction Method Based on Multisensor Fusion Under Time-Varying Operating Conditions. IEEE Transactions on Instrumentation and Measurement. 2024; 73: 1–13. doi: 10.1109/TIM.2024.3378308

[3]Niu G, Liu E, Wang X, et al. A Hybrid Bearing Prognostic Method With Fault Diagnosis and Model Fusion. IEEE Transactions on Industrial Informatics. 2024; 20(1): 864–872. doi: 10.1109/TII.2023.3265532

[4]Xu T, Jiang D, Xu Z. Stage-adaptive remaining life forecast of rolling bearings via small-graph convolution. Journal of the Brazilian Society of Mechanical Sciences and Engineering. 2025; 47(11): 543. doi: 10.1007/s40430-025-05875-5

[5]Shi M, Ding C, Que H, et al. Multilayer-graph-embedded extreme learning machine for performance degradation prognosis of bearing. Measurement. 2023; 207: 112299. doi: 10.1016/j.measurement.2022.112299

[6]Kim JH, Kim BS, Park JH, et al. Optimization of accelerated life test design process for gears and bearings using equivalent damage method. Scientific Reports. 2025; 15(1): 36424. doi: 10.1038/s41598-025-20451-6

[7]Risca D, Lourenço A, Marreiros G. Continual learning for rotating machinery fault diagnosis with cross-domain environmental and operational variations. 2025; doi: 10.48550/arXiv.2504.10151

[8]Abboud D, Antoni J, Sieg-Zieba S, et al. Envelope analysis of rotating machine vibrations in variable speed conditions: A comprehensive treatment. Mechanical Systems and Signal Processing. 2017; 84: 200–226. doi: 10.1016/j.ymssp.2016.06.033

[9]Kim S, Park HJ, Seo YH, et al. A Robust Health Indicator for Rotating Machinery Under Time-Varying Operating Conditions. IEEE Access. 2022; 10: 4993–5001. doi: 10.1109/ACCESS.2022.3140755

[10]Althubaiti A, Elasha F, Teixeira JA. Fault diagnosis and health management of bearings in rotating equipment based on vibration analysis – a review. Journal of Vibroengineering. 2022; 24(1): 46–74. doi: 10.21595/jve.2021.22100

[11]Xiong J, Zhou J, Ma Y, et al. Adaptive deep learning-based remaining useful life prediction framework for systems with multiple failure patterns. Reliability Engineering & System Safety. 2023; 235: 109244. doi: 10.1016/j.ress.2023.109244

[12]He Y, Su H, Zio E, et al. A systematic method of remaining useful life estimation based on physics-informed graph neural networks with multisensor data. Reliability Engineering & System Safety. 2023; 237: 109333. doi: 10.1016/j.ress.2023.109333

[13]Li Y, Xie J, Liu T, et al. A hybrid framework driven by the integration of degradation knowledge and data for remaining useful life prediction of rolling bearings based on Transformer. Measurement Science and Technology. 2025; 36(9): 096138. doi: 10.1088/1361-6501/ae0810

[14]Li C, Li S, Feng Y, et al. Small data challenges for intelligent prognostics and health management: a review. Artificial Intelligence Review. 2024; 57(8): 214. doi: 10.1007/s10462-024-10820-4

[15]Bougacha O, Varnier C. Enhancing Decisions in Prognostics and Health Management Framework. International Journal of Prognostics and Health Management. 2021; 11(1). doi: 10.36001/ijphm.2020.v11i1.2607

[16]Javed K, Gouriveau R, Zerhouni N. Novel failure prognostics approach with dynamic thresholds for machine degradation. In: Proceedings of the 39th Annual Conference of the IEEE Industrial Electronics Society, IECON'13; 10–13 November 2013; Vienna, Austria. pp. 4402–4407. Available online: https://hal.science/hal-00907618

[17]Rezamand M, Kordestani M, Orchard ME, et al. Improved Remaining Useful Life Estimation of Wind Turbine Drivetrain Bearings Under Varying Operating Conditions. IEEE Transactions on Industrial Informatics. 2021; 17(3): 1742–1752. doi: 10.1109/TII.2020.2993074

[18]Basora L, Viens A, Chao MA, et al. A benchmark on uncertainty quantification for deep learning prognostics. Reliability Engineering & System Safety. 2025; 253: 110513. doi: 10.1016/j.ress.2024.110513

[19]Wang H, Liao H, Ma X. Remaining Useful Life Prediction Considering Joint Dependency of Degradation Rate and Variation on Time-Varying Operating Conditions. IEEE Transactions on Reliability. 2021; 70(2): 761–774. doi: 10.1109/TR.2020.3002262

[20]Alrweili H, Fawzy H. Forecasting Crude Oil Prices Using an ARIMA-ANN Hybrid Model. Journal of Statistics Applications & Probability. 2022; 11(3): 845–855. doi: 10.18576/jsap/110308

[21]Ding Y, Ding P, Zhao X, et al. Transfer Learning for Remaining Useful Life Prediction Across Operating Conditions Based on Multisource Domain Adaptation. IEEE/ASME Transactions on Mechatronics. 2022; 27(5): 4143–4152. doi: 10.1109/TMECH.2022.3147534

[22]Mohammad AAS, Mohammad SI, Al-Daoud KI, et al. A Boosted Ensemble Framework with Bio-Inspired Optimization for Multivariate Classification of Structured Data. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems. 2026; 34(04): 493–519. doi: 10.1142/S0218488526500200

[23]Liu B, Yan C, Lv M, et al. SKDAN: A Signal Knowledge-enhanced Domain Adaptation Network for remaining useful life prediction and uncertainty quantification of rolling bearings. Computers in Industry. 2026; 176: 104447. doi: 10.1016/j.compind.2026.104447

[24]Liu Y. Extended Bayesian Framework for Multicategory Support Vector Machine. Journal of Statistics Applications & Probability. 2020; 9(1): 1–11. doi: 10.18576/jsap/090101

[25]Wen H, Zhang L, Sinha JK. Early Prediction of Remaining Useful Life for Rolling Bearings Based on Envelope Spectral Indicator and Bayesian Filter. Applied Sciences. 2024; 14(1): 436. doi: 10.3390/app14010436

[26]Mohammad AAS, Mohammad SI, Vasudevan A, et al. Machine-learning-based prediction of energy requirements for biomass pyrolysis via compositional and operational parameters. Waste Disposal & Sustainable Energy. 2026. doi: 10.1007/s42768-026-00296-z

[27]Li X, Guo S, Sun D, et al. A Rolling Bearing Fault Diagnosis Method Based on Extreme Learning Machine Optimized by Improved Whale Optimization Algorithm. Facta Universitatis, Series: Mechanical Engineering. 2025; 23(4): 881. doi: 10.22190/FUME241130008L

[28]El Kabouss A, Bouhamed A. Regional Fractional Optimal Control Problem of a Bilinear Reaction Diffusion Equation Using Distributed Bounded Controls. Progress in Fractional Differentiation and Applications. 2024; 10(4): 537–544. doi: 10.18576/pfda/100402

[29]Wu P, Sheng Z, He C. Fractal Multi-Scale Physics-Informed Neural Networks for Fractal-Modified Nonlinear Oscillators. Fractals. 2026; 34(03): 2650006. doi: 10.1142/S0218348X26500064

[30]Mohammad AAS, Mohammad SI, Vasudevan A, et al. Data-driven modeling required energy of fuel production process through biomass hydrothermal liquefaction. Results in Engineering. 2026; 30: 110738. doi: 10.1016/j.rineng.2026.110738

Editor-in-Chief

Prof. Jun Yang

Institute of Acoustics, Chinese Academy of Sciences, China

ISSN

1541-0161 (Print)

2693-1443 (Online)

Publication Frequency

Bi-monthly

Indexing

Web of Science Coverage

Emerging Sources Citation Index (IF 4.2, Q1)

Elsevier Solutions

Scopus (2025 CiteScore 2.0);

Portico, etc.

About the Publisher

Academic Publishing insists on taking academic exchange and publication as the main line, carrying out comprehensive management based on science and technology, and fully exploring excellent international publishing resources. Within 5 years, it will form a strategic framework and scale with science (S), technology (T), medicine (M), education (E), and humanities and arts (H) as the main publishing fields. Academic Publishing is headquartered in Singapore and based in Malaysia, with the United States and China providing the main scientific and academic resources. At the same time, it has established long-term good cooperative relations with other publishing companies, scientific research communities, and academic organizations in more than a dozen countries and regions. Academic Publishing uses English and Chinese as its main publishing languages, mainly publishing books, journals, and conference papers in print and online. The vast majority of publications follow the international open access policy, providing stable and long-term quality and professional publications. With the joint efforts of the expert team and our professional editorial team, our publications will gradually be indexed by international databases in stages to provide convenient and professional retrieval for various scholars. At the same time, manuscripts we accept will be subject to the peer review principle, and cutting-edge and innovative research articles will be preferentially accepted for peer reference and discussion. All kinds of our publications are welcome for peer to contribute, access, and download.

more

Member of ASC

Volume Arrangement

Featured Articles

New scaling of critical damping and reduced frequency for mechanically excited systems

This paper introduces a universal framework for understanding the vibration responses of systems subjected to harmonic excitation. By examining a simplified cylinder-spring-damper model, the study refurbishes traditional scaling methods for the excitation frequency ratio and critical damping ratio. The findings indicate that in damped systems, the maximum amplitude of vibration does not align with the natural frequency. This observation leads to the introduction of a new scaling method for reduced frequency. This new approach aligns resonance peaks at the new reduced velocity of 1.0 across different damping ratios, providing a consistent characterization of vibration behavior. A new critical damping ratio of 0.707 is identified for an excited system as opposed to the traditional damping ratio of 1.0 for an unexcited system. Key properties such as maximum amplitude, phase lag, bandwidth, and quality factor are analyzed, demonstrating that the proposed reduced frequency and critical damping ratio effectively capture the dynamics of both damped and undamped excited systems. The findings offer significant insights for practical applications in engineering and various scientific fields.

Ultrasonic wave velocity as a universal metric for defect detection in timber structures: A case study on Japanese cedar wood (Cryptomeria japonica)

This study makes significant contributions to the field of ultrasonic testing (UT) by offering a novel approach to the identification of artificially introduced defects within Japanese cedar wood (Cryptomeria japonica). The findings are of particular relevance for the heritage conservation and construction sectors, where non-invasive defect detection is paramount. The study establishes a robust framework for assessing the structural integrity of timber by correlating ultrasonic wave velocity reductions with defect size and distribution. Big-sized defects led to more substantial decreases in wave velocity. The study establishes a robust framework for assessing the structural integrity of historical timber by correlating ultrasonic wave velocity reductions with defect size and distribution. This framework has the potential to be applicable to diverse wood species and defect types.

Vehicle structural road noise prediction based on an improved Long Short-Term Memory method

The control of vehicle interior noise has become a critical metric for assessing noise, vibration, and harshness (NVH) in vehicles. During the initial phases of vehicle development, accurately predicting the impact of road noise on interior noise is essential for reducing noise levels and expediting the product development cycle. In recent years, data-driven methods based on machine learning have gained significant attention due to their robust capability in navigating complex data mapping relationships. Notably, surrogate models have demonstrated exceptional performance in this domain. Numerous researchers have integrated diverse intelligent algorithms into the study of vehicle noise, leveraging advantages such as the elimination of precise modeling requirements, extensive solution space exploration, continuous learning from data, and robust algorithmic versatility. However, in NVH engineering applications, data-driven models face inherent limitations, particularly in interpretability and stability. To address these issues, this paper introduces an improved Long Short-Term Memory (LSTM) network that combines knowledge and data. Inspired by the physical information neural network concept, this approach incorporates values calculated through empirical formulas into the neural network as constraints. Comparative assessments with traditional LSTM networks highlight the advantages of this deep learning model. By integrating empirical formulas constraints, the model not only enhances interpretability but also achieves robust generalization with fewer data samples. The proposed method is validated on a specific vehicle model, showing significant improvements in prediction accuracy and efficiency.