Water-lubricated stern tube bearings of ship propulsion systems develop severe friction-induced vibration and noise at low speed and heavy load, threatening reliability and acoustic stealth. Lin et al. (2019) proposed a three-degree-of-freedom mode-coupling model in which stochastic rough-surface perturbations, synthesized by inverse FFT from a prescribed power spectral density, act as an excitation source, and identified a mode-proximity amplification effect — vibration grows markedly as two modal frequencies approach, even in a linearly stable system — validated only on steel plates sliding in air. The model rests on simplifying assumptions (constant Coulomb friction, prescribed PSD, three degrees of freedom, linear supports) and has not been translated to actual water-lubricated bearings. Later work (2019–2026) developed realistic friction laws, richer shafting models, and physics-based mixed-lubrication excitation only in separation; no study has integrated stochastic rough-surface excitation with realistic friction, adequate modal density, and water-lubricated validation. The integrated question remains open as of February 2026.

Background

Marine stern tube bearings support the propeller shaft where it exits the hull; water-lubricated versions replace mineral oil for environmental reasons, but water's low viscosity gives a fragile lubricant film, so under low-speed heavy-load operation the bearing operates in mixed lubrication — a regime in which the shaft surface and the polymer/rubber bearing liner partially touch through asperity (surface-peak) contact. The resulting friction can excite self-sustained vibration, audibly as low-frequency 'chatter' and high-frequency 'squeal'. Classical mechanisms include stick–slip from a negative friction–velocity slope (the Stribeck effect, i.e., friction coefficient rising as sliding speed falls toward zero), mode-coupling instability in which frictional cross-coupling of two structural modes with nearly equal frequencies destabilizes the system, sprag–slip (geometric locking), and lock-in of structural modes. Stability is commonly studied by complex eigenvalue analysis: the linearized equations of motion MX¨+CX˙+KX=F\mathbf{M}\ddot{\mathbf{X}}+\mathbf{C}\dot{\mathbf{X}}+\mathbf{K}\mathbf{X}=\mathbf{F} are solved for eigenvalues whose real parts turn positive (instability) as friction coefficient or contact stiffness varies; time histories are obtained by direct integration (e.g., the Newmark method). Because the true rough-surface geometry is stochastic and unknown in detail, it is naturally described by its power spectral density (PSD, the distribution of height fluctuation energy over spatial frequency); a random surface realization consistent with that PSD can be synthesized by inverse fast Fourier transform (IFFT) and, when the shaft slides at speed vv, mapped into a temporal displacement excitation q(t)q(t) at the contact through ω=vΩ\omega=v\Omega. Lin et al. (2019, Tribology International 131:661–671) built exactly this kind of three-degree-of-freedom (3-DOF) shaft–bearing model with an IFFT-generated stochastic roughness perturbation entering the linear contact force N=−kc(y1−y2−q(t))N=-k_c(y_1-y_2-q(t)), studied stability by complex eigenvalues, and simulated response with the Newmark scheme. They found that peak frequencies of friction-induced vibration and noise follow the system's natural frequencies, that the stochastic roughness acts as an excitation source, and — the paper's new result — that forced-response peaks are strongly amplified when two mode frequencies approach each other even while the system remains linearly stable. Validation was qualitative: hand-slid steel plates in air whose modes were identified by finite-element analysis (FEM) and knock experiments (hammer-impact modal tests). The model's idealizations are explicit in its formulation: linear springs and dampers, Coulomb friction with constant coefficient μ\mu, a prescribed analytic PSD, and only three degrees of freedom, while real shaft–bearing systems are effectively infinite-DOF assemblies with stick–slip, velocity-dependent friction, and time delays. The paper positions itself as a fundamental study of the mechanism, and the translation to actual water-lubricated stern tube bearings — with fluid interaction, elastomeric bearing materials, and operational environment effects — remains open.

Problem Statement

Establish a predictive, experimentally validated account of friction-induced vibration and noise of the water-lubricated marine stern tube bearing–shaft system that removes the acknowledged limitations of the 3-DOF stochastic-rough-surface model of Lin et al. (2019). Specifically: (1) build a coupled model that simultaneously represents (i) realistic friction laws (velocity-dependent/Stribeck and stick–slip friction rather than constant-μ Coulomb), (ii) the rough-interface excitation derived from measured surface topography or physics-based mixed-lubrication asperity contact rather than a prescribed analytic PSD, and (iii) adequate modal density, i.e., a continuous-shaft/multi-mode shafting description rather than a 3-DOF truncation; (2) within this integrated model, determine how the stability boundaries, the mode-proximity amplification phenomenon (growth of forced-response peaks as two modal frequencies approach in a linearly stable system), and the predicted noise spectra are modified relative to the 2019 predictions; and (3) validate the mechanism and predictions on water-lubricated stern tube bearing experiments with fluid interaction, elastomeric bearing materials, and representative operating conditions, replacing the steel-plate-in-air qualitative analogy.

References

  1. Lin, C.-G.; Zou, M.-S.; Sima, C.; Liu, S.-X.; Jiang, L.-W. (2019). Friction-induced vibration and noise of marine stern tube bearings considering perturbations of the stochastic rough surface. Tribology International 131:661–671. DOI: 10.1016/j.triboint.2018.11.026. https://doi.org/10.1016/j.triboint.2018.11.026
  2. Guo, X.; Zhou, X.; Huang, J.; Yan, Z.; Yang, Y.; Liu, R.; Ouyang, W.; Jin, Y. (2026). Friction-Induced Vibration in Ship Water-Lubricated Bearings: A Review. Lubricants 14(3):107. DOI: 10.3390/lubricants14030107. https://doi.org/10.3390/lubricants14030107
  3. Lin, C.-G.; Zou, M.-S.; Zhang, H.-C.; Qi, L.-B.; Liu, S.-X. (2021). Influence of different parameters on nonlinear friction-induced vibration characteristics of water lubricated stern bearings. International Journal of Naval Architecture and Ocean Engineering 13:746–757. DOI: 10.1016/j.ijnaoe.2021.10.003. https://doi.org/10.1016/j.ijnaoe.2021.10.003
  4. Zhang, Z.; Duan, N.; Lin, C.; Hua, H. (2020). Coupled dynamic analysis of a heavily-loaded propulsion shafting system with continuous bearing-shaft friction. International Journal of Mechanical Sciences 172:105431. DOI: 10.1016/j.ijmecsci.2020.105431. https://doi.org/10.1016/j.ijmecsci.2020.105431
  5. Lin, C.-G.; Yang, Y.-N.; Chu, J.-L.; Sima, C.; Liu, P.; Qi, L.-B.; Zou, M.-S. (2023). Study on nonlinear dynamic characteristics of propulsion shafting under friction contact of stern bearings. Tribology International 183:108391. DOI: 10.1016/j.triboint.2023.108391. https://doi.org/10.1016/j.triboint.2023.108391
  6. Lin, C.-G.; Zou, M.-S.; Qi, L.-B.; Sima, C.; Chu, J.-L. (2023). Propeller-shaft-hull coupling acoustic radiation characteristics induced by friction self-excitation of the stern bearing. Ocean Engineering 287:115769. DOI: 10.1016/j.oceaneng.2023.115769. https://doi.org/10.1016/j.oceaneng.2023.115769
  7. Lin, C.-G.; Wang, H.; Sima, C.; Han, Y.-F.; Zhang, W.; Zou, M.-S. (2025). Research on the calculation methods for time-domain friction excitation force in mixed lubrication contact between ship shaft and bearings. Ocean Engineering 337:121811. DOI: 10.1016/j.oceaneng.2025.121811. https://doi.org/10.1016/j.oceaneng.2025.121811
  8. Kuang, F.; Mu, H.; Huang, J.; Wan, G.; Yuan, C.; Zhou, X. (2024). Theoretical and In Situ High Speed Measurement Study on Friction-Induced Vibration in Water Lubricated Rubber Bearing-Shaft System. Tribology Transactions 67(1):47–61. DOI: 10.1080/10402004.2023.2289526. https://doi.org/10.1080/10402004.2023.2289526
  9. Han, H.S.; Lee, K.H. (2019). Experimental verification of the mechanism on stick-slip nonlinear friction induced vibration and its evaluation method in water-lubricated stern tube bearing. Ocean Engineering 182:147–161. DOI: 10.1016/j.oceaneng.2019.04.078. https://doi.org/10.1016/j.oceaneng.2019.04.078
  10. Wu, C.; Shang, D.; Xiao, Y.; Zhang, C.; Liu, Y. (2024). Analytical expressions of friction-induced self-excited vibration amplitudes of the marine rubber bearing-shaft system. Journal of Low Frequency Noise, Vibration and Active Control 43(1):20–40. DOI: 10.1177/14613484231190261. https://doi.org/10.1177/14613484231190261
  11. Gao, K.; Huang, Q.; Xie, Z. (2025). Stability analysis of friction self-excited vibration of ship propulsion shaft-bearing system based on modal-coupling mechanism. Journal of Mechanical Science and Technology 39(5):2563–2574. DOI: 10.1007/s12206-025-0414-z. https://doi.org/10.1007/s12206-025-0414-z