Investigations on the Frictional Hysteresis Effect of Multi-Leaf Journal Foil Bearing: Modeling, Predictions and Validations
Abstract
:1. Introduction
2. Modeling of MLJFB with Coulomb Frictions
2.1. Nonlinear Curve Beam Element of the Foil Leaf
2.2. Contact Constraints Inside MLJFB including Frictions
2.2.1. Frictional Foil–Foil Contact
- (1)
- Contact energy formulations for sticking contact nodes pair
- (2)
- Contact energy formulations for the sliding contact nodes pair
2.2.2. Foil-Bearing Sleeve Contact
2.2.3. Rotor–Foil Contact
2.3. Boundary Conditions of the Elastic Foil Leaves
2.4. Iteration Formula and Calculation Algorithms
2.4.1. Iteration Formula
2.4.2. Calculation Algorithms
3. Results and Discussions
3.1. Influences of Nonlinear Large Foil Deformation in MLJFB
3.2. Frictional Hysteresis Characteristics of MLJFB
3.2.1. Influence of the Friction Coefficient
3.2.2. Influence of Foil Leaf Boundary Conditions
3.2.3. Influence of the Foil Leaf Number
3.2.4. Influence of Bearing Radial Clearance
3.2.5. Influence of Foil Leaf Structural Parameters
- (1)
- Initial foil leaf radius
- (2)
- Foil leaf thickness
- (3)
- Foil overlapping ratio
4. Experiments
5. Conclusions
- (1)
- The consideration of the geometrical nonlinearity of the larger foil deformation leads to the consistent results of foil assembly, rotor insertion and static loading under different initial foil configurations.
- (2)
- A fixed boundary of foil leaves can cause insufficient sliding motions between foil leaves and results in less dissipated energy by friction compared with hinged boundaries.
- (3)
- Under the same radial clearance, a larger foil leaf number, foil leaf thickness, initial foil leaf radius and foil overlapping ratio tend to increase the preload effect after rotor insertional assembly and result in a more compact and stiff foil structure. With a bearing size of 35 × 35 mm2, large values of the above parameters (Nf = 12, tf = 0.2 mm, Rf = 2.1 × Rr, λ = 0.45) can lead to the performance degeneration of the Coulomb friction energy dissipation. There are optimized values of these parameters, enabling the MLJFB to possess a larger area of frictional hysteresis loops.
- (4)
- A decreased radial clearance leads to insufficient foil deformations, thus reducing the frictional energy dissipation. A larger radial clearance can also result in less dissipated energy due to the limited rotor displacement by the rigid bearing sleeve under a larger static load.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
Ba, BL, BN | total, linear and nonlinear strain matrix of the curve beam element |
Cini | bearing radial clearance (m) |
Dr | rotor diameter (m) |
result vector after the variation calculation with respect to | |
ey | rotor displacement in the y direction (μm) |
Ea | total axial strain of the curve beam |
En | geometrical nonlinear strain of the curve beam |
Fy | bearing reaction force in the y direction or static bearing load (N) |
normal contact gap between adjacent foils | |
tangent contact gap between adjacent foils | |
contact gap between the foil leaf and bearing sleeve | |
contact gap between the rotor and foil leaf | |
gap vector between adjacent foil leaves | |
GU | general contact force vector |
KU | general tangent matrix |
bearing width (m) | |
foil leaf length (m) | |
, | unit normal and tangent vectors on the deformed contact surface |
, | unit normal and tangent vectors of node A |
general displacement vector per contact nodes pair | |
Q | general displacement vector of all contact nodes pairs |
, | unit normal and tangent vectors of interpolating node F |
Nw, Nu | interpolating functions of the curve beam element at position |
Nf | number of foil leaves |
Rfa | inscribed circle radius after foil assembly (m) |
Rs | inner radius of the bearing sleeve (m) |
Rr | rotor radius (m) |
Rf | free-foil radius (m) |
s | position of the interpolating contact node on the master curve element |
Sh | frictional hysteresis loop area (N·m) |
tf | foil thickness (m) |
displacement vector of the curve beam element | |
w, v | radial and tangential displacements of the curve beam element (m) |
ws | width of the foil installation groove (m) |
flag of the sliding direction | |
μ | friction coefficient between foil leaves |
εs | elastic strain of the curve beam element |
εt | penalty factor in the tangent direction |
foil installation angle (°) | |
threshold of the foil leaf rotating angle (rad) | |
nodal rotational angle of the curve beam element (rad) | |
foil overlapping ratio, | |
Lagrange multiplier or contact force in the normal direction | |
contact energy per contact nodes pair |
Appendix A
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Parameters | Values |
---|---|
Rotor diameter Dr/mm | 35 |
Rotor radius Rr/mm | 17.5 |
Bearing width L/mm | 35 |
Foil leaf number | 5, 8, 12 |
Foil leaf radius Rb/mm | 1.2 × Rr, 1.5 × Rr, 1.8 × Rr, 2.1 × Rr |
Foil leaf thickness tf/mm | 0.1, 0.15, 0.2 |
Overlapping ratio λ | 0.35, 0.4, 0.45 |
Initial radial clearance Cini/μm | 50, 100, 150, 200 |
Friction coefficient between foil leaves μ | 0, 0.1, 0.2, 0.3 |
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Li, C.; Du, J.; Li, J.; Xu, Z. Investigations on the Frictional Hysteresis Effect of Multi-Leaf Journal Foil Bearing: Modeling, Predictions and Validations. Lubricants 2022, 10, 261. https://doi.org/10.3390/lubricants10100261
Li C, Du J, Li J, Xu Z. Investigations on the Frictional Hysteresis Effect of Multi-Leaf Journal Foil Bearing: Modeling, Predictions and Validations. Lubricants. 2022; 10(10):261. https://doi.org/10.3390/lubricants10100261
Chicago/Turabian StyleLi, Changlin, Jianjun Du, Jie Li, and Zhenni Xu. 2022. "Investigations on the Frictional Hysteresis Effect of Multi-Leaf Journal Foil Bearing: Modeling, Predictions and Validations" Lubricants 10, no. 10: 261. https://doi.org/10.3390/lubricants10100261