An Analysis of Hertzian Contact and Inner Race Crack-Induced Vibrations in a Coupled Shaft-Bearing System
Received: 28 June 2025 | Revised: 23 July 2025 and 2 August 2025 | Accepted: 20 August 2025 | Online: 8 December 2025
Corresponding author: Gilbert Emmanuel Ophel
Abstract
A dynamic model of a coupled shaft-bearing system was developed to examine the vibrational behavior caused by Hertzian contact mechanics and localized defects on the inner race. The model integrates mechanical energy balance, nonlinear Hertzian contact stiffness, and the Lagrangian formalism, considering the radial loads, system damping, and geometric characteristics. Numerical simulations executed via the fourth-order Runge–Kutta method reveal distinct vibrational signatures between the healthy and cracked bearing conditions. Under healthy operating conditions, broadband spectral energy is observed without significant peaks, indicating a uniform vibration behavior. The introduction of an inner race crack generates amplitude modulation, increased vibration levels, and pronounced spectral peaks between 600 Hz and 800 Hz. These peaks are consistent with defect-passing frequencies and nonlinear impact phenomena. This study reveals that the local defects significantly impact the dynamic response by producing complex, nonstationary vibration patterns that can be detected through time-domain analysis and power spectral density estimation using the power spectrum of the Fast Fourier Transform (FFT) technique.
Keywords:
shaft–bearing system, Hertzian contact, inner race crack, vibration analysis, power spectrum analysisDownloads
References
S. H. Gawande, K. Balashowry, K. A. Raykar, and K. H. Munde, "Estimation of contact stresses in EN31 rolling contact bearings for screw compressor using Gauss quadrature & statistical analysis," Australian Journal of Mechanical Engineering, vol. 22, no. 5, pp. 894–910, 2023. DOI: https://doi.org/10.1080/14484846.2023.2195098
D. Javanmardi and M. A. Rezvani, "Rail vehicle axlebox roller bearing life and failure analysis based on the Hertz contact theory, finite element modeling, and field observations," World Journal of Engineering, vol. 21, no. 6, pp. 1183–1192, Nov. 2023. DOI: https://doi.org/10.1108/WJE-01-2023-0010
J. Shi et al., "Three-Dimensional Dynamic Modeling and Vibration Analysis of Deep Groove Ball Bearings Under Compound Fault Excitation," Social Science Research Network, Rochester, NY, Jan. 18, 2025. DOI: https://doi.org/10.2139/ssrn.5102377
J. Hu, J. Shi, Y. Yu, W. Huang, C. Shen, and Z. Zhu, "Three-Dimensional Dynamic Analytical Modeling of Deep Groove Ball Bearings with Fault Excitation," in 2023 International Conference on Sensing, Measurement & Data Analytics in the era of Artificial Intelligence (ICSMD), Xi’an, China, Aug. 2023, pp. 1–6. DOI: https://doi.org/10.1109/ICSMD60522.2023.10490904
J. Li, Y. Qi, P. Wu, J. Ding, and H. Ding, "Research on wear and grinding models of high-speed turnouts based on semi-Hertz contact," Scientific Reports, vol. 15, no. 1, Jan. 2025, Art. no. 531. DOI: https://doi.org/10.1038/s41598-024-85016-5
Y. Bella, A. Oulmane, and M. Mostefai, "Industrial Bearing Fault Detection Using Time-Frequency Analysis," Engineering, Technology & Applied Science Research, vol. 8, no. 4, pp. 3294–3299, Aug. 2018. DOI: https://doi.org/10.48084/etasr.2135
H. Jia et al., "Short-time variational mode decomposition," Signal Processing, vol. 238, Jan. 2026, Art. no. 110203. DOI: https://doi.org/10.1016/j.sigpro.2025.110203
M. Al-Sa’d, T. Jalonen, S. Kiranyaz, and M. Gabbouj, "Quadratic Time-Frequency Analysis of Vibration Signals for Diagnosing Bearing Faults." arXiv, Feb. 08, 2024.
C. Zeng, "Frequency-domain defect imaging in anisotropic media based on coordinate transformation," Physical Review Applied, vol. 23, no. 1, 2025, Art. no. 014034. DOI: https://doi.org/10.1103/PhysRevApplied.23.014034
R. Suwondo, I. Hidayat, M. Suangga, M. Keintjem, and J. Walewangko, "Dynamic Assessment of a Railway Bridge using Operational Modal Analysis and Fast Fourier Transform: A Comparative Study with Finite Element Analysis," Engineering, Technology & Applied Science Research, vol. 15, no. 1, pp. 19200–19206, Feb. 2025. DOI: https://doi.org/10.48084/etasr.9202
A. F. Khalil and S. Rostam, "Machine Learning-based Predictive Maintenance for Fault Detection in Rotating Machinery: A Case Study," Engineering, Technology & Applied Science Research, vol. 14, no. 2, pp. 13181–13189, Apr. 2024. DOI: https://doi.org/10.48084/etasr.6813
B. X. Tchomeni Kouejou, D. F. Sozinando, and A. Anyika Alugongo, "Modeling and Analysis of Drill String–Casing Collision under the Influence of Inviscid Fluid Forces," Applied Sciences, vol. 13, no. 6, Jan. 2023, Art. no. 3557. DOI: https://doi.org/10.3390/app13063557
Z. Shi, G. Zhang, J. Liu, X. Li, Y. Xu, and C. Yan, "Influences of inclined crack defects on vibration characteristics of cylindrical roller bearings," Mechanical Systems and Signal Processing, vol. 207, Jan. 2024, Art. no. 110945. DOI: https://doi.org/10.1016/j.ymssp.2023.110945
S. Wang, J. Du, C. Li, S. Xia, S. Jiang, and J. Sun, "Crack propagation analysis of slewing bearings in wind turbines applying a modified sub-model technology," Engineering Failure Analysis, vol. 153, Nov. 2023, Art. no. 107556. DOI: https://doi.org/10.1016/j.engfailanal.2023.107556
C. Zhou, Y. Liu, W. Teng, H. Zhang, H. He, and C. Zhou, "Dynamic Modeling and Stability Analysis of a Heavy-Duty Flywheel Rotor-Bearing System with Two Cracks," International Journal of Structural Stability and Dynamics, vol. 22, no. 09, July 2022, Art. no. 2250103. DOI: https://doi.org/10.1142/S0219455422501036
Q. Ding, Z. Feng, Y. Zhang, and W. Sun, "Dynamic analysis of slant cracked rotor system considering nonlinear oil film force," Plos One, vol. 19, no. 1, 2024, Art. no. e0294293. DOI: https://doi.org/10.1371/journal.pone.0294293
Z. Wang, G. Li, X. Zhou, H. Zhang, Z. Lin, and S. Jia, "Dynamic analysis of deep groove ball bearing with localized defects and misalignment," Journal of Sound and Vibration, vol. 568, Jan. 2024, Art. no. 118071. DOI: https://doi.org/10.1016/j.jsv.2023.118071
L.-H. Yang, Y. Sun, Z.-B. Yang, Z. Mao, and X.-F. Chen, "Coupling vibration mechanism of rotating shaft–disc–blade system with blade crack—A systematical investigation on the effect of crack, condition, and structure parameters," Thin-Walled Structures, vol. 205, Dec. 2024, Art. no. 112398. DOI: https://doi.org/10.1016/j.tws.2024.112398
H. Wen, L. Zhang, and J. K. Sinha, "From Envelope Spectra to Bearing Remaining Useful Life: An Intelligent Vibration-Based Prediction Model with Quantified Uncertainty," Sensors, vol. 24, no. 22, Jan. 2024, Art. no. 7257. DOI: https://doi.org/10.3390/s24227257
D. Ruan, C. Gühmann, and J. Yan, "A new solving method for ball bearing dynamics model based on optimization with nonlinear constraints: Contact angles as iterative variables," Journal of Manufacturing Processes, vol. 106, pp. 338–346, Nov. 2023. DOI: https://doi.org/10.1016/j.jmapro.2023.09.076
Y. Yang, X. Qi, Y. Wang, M. Wang, B. Wen, and J. Zhai, "Dynamic Characteristics of a Cylindrical Roller Bearing with Cage Cracks," Lubricants, vol. 13, no. 1, Jan. 2025, Art. no. 25. DOI: https://doi.org/10.3390/lubricants13010025
P. Dong, Z. Gao, D. Xu, and S. Nyassi, "Rolling stiffness of contact surfaces during the rolling process of elastic particles," Powder Technology, vol. 445, Sept. 2024, Art. no. 120077. DOI: https://doi.org/10.1016/j.powtec.2024.120077
P. H. Jain, S. P. Bhosle, A. J. Keche, and R. G. Desavale, "A Dynamic Model of Outer Race Defective Bearing Considering the Unbalanced Shaft-Bearing System With Experimental Simulation," Journal of Tribology, vol. 146, no. 1, Oct. 2023, Art. no. 014301. DOI: https://doi.org/10.1115/1.4062689
M. Wang, X. Fu, X. Yan, and L. Teng, "A New Chaos-Based Image Encryption Algorithm Based on Discrete Fourier Transform and Improved Joseph Traversal," Mathematics, vol. 12, no. 5, Jan. 2024, Art. no. 638. DOI: https://doi.org/10.3390/math12050638
. M. Iqbal, A. K. Madan, and N. Ahmad, "Vibration and acoustic signal-based bearing fault diagnosis in CNC machine using an improved deep learning," Iran Journal of Computer Science, vol. 7, no. 4, pp. 723–733, Dec. 2024. DOI: https://doi.org/10.1007/s42044-024-00205-9
R. Wang, S. Zhang, S. Liu, W. Liu, and A. Ding, "A bearing fault diagnosis method for high-noise and unbalanced dataset," Smart and Resilient Transportation, vol. 5, no. 1, pp. 28–45, Dec. 2022. DOI: https://doi.org/10.1108/SRT-04-2022-0005
Downloads
How to Cite
License
Copyright (c) 2025 Gilbert Emmanuel Ophel, Bernard Xavier Tchomeni, Alfayo Anyika Alugongo, Desejo Filipeson Sozinando
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain the copyright and grant the journal the right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) after its publication in ETASR with an acknowledgement of its initial publication in this journal.
