Investigation into the Dynamic Parameter Characterization of Water-Lubricated Bearings Under Vibration Coupling
Abstract
1. Introduction
2. Mathematical Model
2.1. Rotor System Dynamic Model
2.2. Transient Hydrodynamic Model
2.3. Calculation Procedure
2.4. Model Validation
2.4.1. Dynamic Model Validation
2.4.2. Validation of the Dynamic Characteristic Coefficient Differential Solution Model
3. Results and Discussion
3.1. Effects of External Loads
3.2. Effect of Rotational Speed
3.3. Effect of Eccentric Excitation
3.4. Effect of the Length-to-Diameter Ratio
3.5. Effect of Radial Clearance-to-Radius Ratio
3.6. Equation Fitting and Verification
4. Test Validation
4.1. Design of Test Rig
4.2. Journal Orbit Acquisition Method
4.3. Analysis of Test Results
5. Conclusions
- (1)
- The dynamic characteristic coefficients display approximately sinusoidal periodic changes under the action of eccentric excitation. The fluctuation amplitude of the journal orbit range and the dynamic characteristic coefficients increase with the increase in the eccentric excitation. Notably, the damping coefficients are significantly influenced by the eccentric excitation.
- (2)
- The main stiffness and main damping coefficients of the bearing in both the horizontal and vertical directions are proportional to the external load and inversely proportional to the rotational speed; the external load significantly influences the degree of fluctuation of the main stiffness and main damping coefficients in the vertical direction, and as the rotational speed rises, the peak-to-peak value of the main stiffness coefficient of the bearing increases, while the peak-to-peak value of the main damping coefficient decreases.
- (3)
- With the increase in the length-to-diameter ratio or the decrease in the radial clearance-to-radius ratio, the main stiffness and main damping in the horizontal direction increase, while the main stiffness in the vertical direction decreases. The reasonable design of the structure of the water-lubricated stern bearings is crucial to the influence of their dynamic behavior.
- (4)
- Based on the simulation data, the formulae for calculating the main stiffness and main damping coefficients of the dimensionless bearing were fitted, and the predictive ability of the fitted formulae was verified; the results showed that the maximum relative error of the dynamic characteristic coefficients was less than 10%.
- (5)
- In this paper, a detailed parametric study of the dynamic behavior of water-lubricated bearings under eccentric excitation was conducted to provide new insights into the transient transverse vibration modeling of marine propulsion shaft systems. Despite these contributions, this study acknowledges certain limitations. Future research should focus on improving the lining deformation model to achieve more realistic dynamic simulations of water-lubricated bearings. Additionally, further exploration of the mixed lubrication dynamics behavior of water-lubricated bearings under low-speed and heavy-load conditions is necessary to optimize the structural parameters of the bearings and improve their service capability.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
x | Horizontal direction |
y | Vertical direction |
OB | Bearing center |
OJ | Journal geometric center |
Om | Journal center of mass |
Qe | Eccentric excitation distance, mm |
e | Journal eccentricity, mm |
θ | Circumferential angle, rad |
φ | Attitude angle, rad |
Fh | Hydrodynamic force, N |
Ff | Shearing friction force, N |
Kxx | Horizontal stiffness coefficient, N/m |
Kyy | Vertical stiffness coefficient, N/m |
Cxx | Horizontal damping coefficient N·s/m |
Cyy | Vertical damping coefficient, N·s/m |
Horizontal acceleration, m/s2 | |
Vertical acceleration, m/s2 | |
mj | Journal mass, kg |
ω | Angular velocity, rad/s |
W | External load, N |
λ | Unbalanced eccentricity ratio |
c | Radial clearance, mm |
x′ | Horizontal velocity |
y′ | Vertical velocity |
kij (i,j = 1~4) | Iteration coefficients for different time steps |
Δt | Iteration timestep, s |
η | Lubricant viscosity, Pa·s |
ρ | Lubricant density, kg/m3 |
ph | Hydrodynamic pressure, Pa |
h | Water film thickness, mm |
U | Horizontal relative velocity component, m/s |
V | Vertical relative velocity component, m/s |
RB | Bearing radius, mm |
z | Axial direction |
ε | Eccentricity ratio |
δ | Liner deformation matrix |
T | Lining thickness, mm |
Eb | Bearing elastic modulus, MPa |
vb | Bearing Poisson’s ratio |
ΔFhx | Horizontal hydrodynamic force increments, N |
ΔFhy | Vertical hydrodynamic force increments, N |
Δx | Horizontal displacement disturbances, mm |
Δy | Vertical displacement disturbances, mm |
Horizontal velocity disturbances, m/s | |
Vertical velocity disturbances, m/s | |
L | Bearing length, mm |
D | Bearing inner diameter, mm |
n | Rotational speed, r/min |
Ts | Computation time, s |
F0 | Dimensionless units of force |
Dimensionless external load | |
r | Journal radius, mm |
Dimensionless eccentric excitation distance | |
L/D | Length-to-diameter ratio |
c/r | Radial clearance-to-radius ratio |
kxx | Horizontal dimensionless stiffness coefficient |
kyy | Vertical dimensionless stiffness coefficient |
cxx | Horizontal dimensionless damping coefficient |
cyy | Vertical dimensionless damping coefficient |
k0 | Dimensionless units of stiffness coefficients |
c0 | Dimensionless units of the damping coefficient |
K | Comprehensive stiffness of the bearing, N/m |
Kw | Water film stiffness, N/m |
Kl | Liner material stiffness, N/m |
σt | Tensile stress of the bearing liner material, MPa |
σf | Flexural stress of the bearing liner material, MPa |
σc | Compressive stress of the bearing liner material, MPa |
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Basic Parameters | Value |
---|---|
Bearing inner diameter, D | 50 mm |
Bearing length, L | 50 mm |
Radial clearance, c | 0.0508 mm |
Thickness of liner, T | 2 mm |
Lubricating oil viscosity, η | 0.006895 Pa·s |
Lubricating oil density, ρ | 900 kg/m3 |
Rotational Speed r/min | Eccentric Excitation Distance/μm |
---|---|
500 | 1800 |
1000 | 450 |
2000 | 110 |
Basic Parameters | Value | Basic Parameters | Value |
---|---|---|---|
Bearing inner diameter, D | 150.6 mm | Viscosity of water, η | 0.001 Pa·s |
Bearing length, L | 150 mm | Density of water, ρ | 1000 kg/m3 |
Radial clearance, c | 0.3 mm | Rotational speed, n | 500 r/min |
Lining thickness, T | 10 mm | External load, W | 1000 N |
Bearing elastic modulus, Eb | 300 MPa | Eccentric excitation distance, Qe | 0.45 mm |
Bearing Poisson’s ratio, vb | 0.48 | Iteration timestep, Δt | 0.5 ms |
Basic Parameters | Value |
---|---|
Dimensionless external load, | 28.70 |
Dimensionless eccentric excitation distance, | 0.70% |
Angular velocity, ω | 52.36 rad/s |
Length-to-diameter ratio, L/D | 1.30 |
Radial clearance-to-radius ratio, c/r | 0.45% |
Basic Parameters | Value | Basic Parameters | Value |
---|---|---|---|
Bearing inner diameter, D | 100.4 mm | Bearing elastic modulus, Eb | 3.0 GPa |
Bearing length, L | 100 mm | Bearing Poisson’s ratio, vb | 0.38 |
Radial clearance, c | 0.2 mm | Density of bearing liner, ρl | 1450 kg/m3 |
Lining thickness, T | 10 mm | Tensile stress (23 °C), σt | 150 MPa |
Viscosity of water, η | 0.001 Pa·s | Flexural stress (23 °C), σf | 230 MPa |
Density of water, ρ | 1000 kg/m3 | Compressive stress (23 °C), σc | 170 MPa |
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Zhu, H.; Jin, Y.; Liu, Q.; Ouyang, W.; He, T. Investigation into the Dynamic Parameter Characterization of Water-Lubricated Bearings Under Vibration Coupling. Lubricants 2025, 13, 123. https://doi.org/10.3390/lubricants13030123
Zhu H, Jin Y, Liu Q, Ouyang W, He T. Investigation into the Dynamic Parameter Characterization of Water-Lubricated Bearings Under Vibration Coupling. Lubricants. 2025; 13(3):123. https://doi.org/10.3390/lubricants13030123
Chicago/Turabian StyleZhu, Hongtao, Yong Jin, Qilin Liu, Wu Ouyang, and Tao He. 2025. "Investigation into the Dynamic Parameter Characterization of Water-Lubricated Bearings Under Vibration Coupling" Lubricants 13, no. 3: 123. https://doi.org/10.3390/lubricants13030123
APA StyleZhu, H., Jin, Y., Liu, Q., Ouyang, W., & He, T. (2025). Investigation into the Dynamic Parameter Characterization of Water-Lubricated Bearings Under Vibration Coupling. Lubricants, 13(3), 123. https://doi.org/10.3390/lubricants13030123