# Numerical Simulation and Experimental Verification of the Stiffness and Stability of Thrust Pad Aerostatic Bearings

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## Abstract

Many researchers concentrate on improving the stiffness and stability of aerostatic bearings, however the contradiction between stiffness and stability is still existed. Therefore, orifice, multiple, and porous restrictors are designed to illustrate the influence of restrictor characteristics on the stability and stiffness of the aerostatic circular pad bearings. Because both the stiffness and stability of aerostatic bearings are determined by the internal pressure distribution, the full Navier-Stokes (N-S) equations are applied to solve internal pressure distribution in bearing film by using computational fluid dynamics (CFD) method. Simulation results present that the stiffness and stability of aerostatic circular pad bearings are influenced significantly by geometrical and material parameters, such as film thickness, orifice diameters, and viscous resistance coefficient. Verified by the experimental data, the micro vibration of orifice restrictor is almost the same as multiple restrictors with amplitude of 0.02 m/s^{2}, but it is much stronger than the porous restrictors with acceleration of 0.006 m/s^{2}. The optimal stiffness of multiple restrictors increased by 46%, compared to only 30.2 N/μm of orifice restrictor, and the porous restrictors had obvious advantage in the small film thickness less than 6 μm where the optimal stiffness increased to 38.3 N/μm. The numerical and experimental results provide guidance for improving the stiffness and stability of aerostatic bearings.

## Keywords

Aerostatic bearings CFD Restrictor characteristics Stability and stiffness## 1 Introduction

Aerostatic bearings, which are nearly frictionless, require low driving power and possess high accuracy of movement, have been widely used in ultra-precision machine tools. With regard to the static and dynamic characteristics of aerostatic bearings, stiffness and stability play a key role in achieving nanometer-level motion precision. Huo et al. [1] pointed out that ultra-precision micro milling machine required a high stiffness aerostatic bearings to maintain high accuracy in the presence of large cutting forces. Gao et al. [2] indicated that the stiffness and vibration of high-speed aerostatic spindles were essential for ultra-precision micro-milling machine tools. Yuan et al. [3] presented that the manufacturing precision of ultra-precision machine tool relied on the stiffness and stability of the aerostatic bearings. Wang et al. [4] demonstrated bearing vibration can have a significant influence on the machining precision of the high-precision optical components in ultra-precision diamond turning. An et al. [5] studied the influence of stability of aerostatic bearing spindles on the machining precision of ultra-precision fly-cutting machines used for processing large diameter optical components.

Among the various restrictor types found in aerostatic bearings, the orifice-type restrictor has attracted considerable attention due to its simple design and low manufacturing cost. Gao et al. [6] conducted the influence the chamber shape on performance characteristics of aerostatic thrust bearings, and found that the pressure depression, gas vortices, and the turbulent intensity which were all weakened with decreasing air film thickness. Li et al. [7] described the study on the performance of aerostatic thrust bearing with pocketed orifice-type restrictor, and presented that ignoring the influences of orifice length on the bearing’s performance resulted in large errors. Renn et al. [8] focused on the mass flow-rate characteristic through an orifice-type restrictor in aerostatic bearings, and showed that the mass flow-rate characteristic through an orifice was different from that through a nozzle.

Over the last few years, published research focuses largely on the stiffness characteristics of aerostatic bearings with orifice restrictors. Cheng et al. [9] gave a selection strategy for the design of externally pressurized journal bearings, and pointed out that the stiffness of aerostatic bearings increased obviously with the growth of supply pressure. Chen et al. [10] investigated the influences of operational conditions and geometric parameters on the stiffness of aerostatic journal bearings, and found that gas-bearing geometries had a significant effect on stiffness. Neves et al. [11] conducted theoretical investigation of the discharge coefficient influence on the stiffness performance of aerostatic journal bearings. Du et al. [12] researched the effects of pressure equalizing groove on the load capacity and stiffness of externally pressurized gas journal bearings.

Various structural designs are used to increase the stiffness of aerostatic bearings. One of these includes rectangular recesses, compared to designs with no recesses, significantly improves stiffness and is superior to spherical recesses [13]. However, increasing the stiffness of aerostatic bearings with orifice restrictors also increases micro vibration. Bhat et al. [14] studied the static and dynamic characteristics of inherently compensated orifice based flat pad air bearing system, and presented that pneumatic hammer instability tended to occur at high pressure. Nishio et al. [15] analyzed the static and dynamic characteristics of aerostatic thrust bearings with small feed holes, and confirmed that aerostatic thrust bearings with small feedholes had larger stiffness and damping coefficient than bearings with compound restrictors. Wang et al. [16] focused on the interaction of shock waves with boundary layer, which caused the instability of aerostatic bearing with orifice restrictor. Xiong et al. [17] studied the mechanism of hydrostatic spindle rotational error motion, and found that increasing of stiffness resulted in the reduction of stability.

Many investigators studies on the mechanism behind micro vibration in aerostatic bearings. Chen et al. [18, 19] demonstrated that micro-vibration was closely related to air vortices that appeared near the air inlet of the aerostatic bearings and that the strength of these air vortices could be represented by a pressure drop from the edge of the vortex to its center. Mohamed and Yoshimoto indicated that a shockwave occurred during the transition from subsonic to supersonic and that the airflow transition between laminar and turbulent flow led to a pressure drop and recovery in the bearing clearance [20, 21]. To date, there is no suitable method to address the contradictory relationship between stiffness and vibration in aerostatic bearings with orifice restrictors.

To overcome the drawbacks of orifice restrictors, the use of porous restrictors and multiple restrictors are reported. Panzera and Otsu showed that the use of a porous material as the bearing restrictor had the benefit of more uniform pressure distribution over the entire bearing surface, which created higher load capacity, stiffness, damp, and wider stability [22, 23]. Charki et al. [24] found that the stiffness of aerostatic bearings with multiple restrictors could be improved because of the smaller pressure drop near the gas inlet. Published literatures reveal that there are not comprehensive studies considering the influences of different restrictor characteristics on the stiffness and stability of aerostatic bearings. Compared to orifice restrictor, porous restrictors and multiple restrictors were designed to improve the stability and stiffness of aerostatic circular pad bearings in this paper. Taking into account that both the stiffness and the stability of aerostatic bearings were determined by the internal pressure distribution characteristics, the pressure distribution in the bearing clearance of three different restrictors was studied using CFD. The influence of film thickness, orifice diameter, and material parameters on the stiffness and stability of the aerostatic circular pad bearings was also investigated. In addition, the accuracy of the numerical simulation results was verified by the experimental data.

## 2 Numerical Modeling

### 2.1 Structures of Restrictors

### 2.2 Governing Equations

*is the velocity vector,*

**u***x*,

*y*, and

*z*are the three components of the Cartesian coordinates,

*u*,

*v*, and

*w*are the time-averaged Cartesian velocity components in the three coordinate directions,

*ρ*is the density, and

*t*is time.

*x*,

*y*and

*z*directions are derived as follows:

*p*is the pressure in bearing film, and

*F*is the force on the body. For a Newtonian fluid, the viscous stress

*τ*is proportional to the deformation rate of the fluid, given as follows:

*μ*is the dynamic viscosity, and

*λ*= − 2/3.

*k*is the fluid heat transfer coefficient,

*c*

_{ p }is the specific heat capacity,

*T*is the temperature, grad

*T*is the gradient of the temperature, and

*S*is the viscous dissipation energy.

*D*

_{ ij }is the viscous resistance coefficient matrix, and

*C*

_{ ij }is the inertial resistance coefficient matrix. The laws of conservation of mass and conservation of momentum can be rewritten as

*γ*is the porosity. The pressure distribution of porous aerostatic bearings can be obtained by solving Eqs. (5)–(8).

### 2.3 Method of Calculation

The main methods utilized for calculating the flow field characteristics of aerostatic bearings include the finite element method (FEM) [25], the finite difference method (FDM) and CFD method [26]. The Reynolds equation can be solved efficiently by the FEM and FDM. However, because of the large number of assumptions and simplifications required for the Reynolds equation, even if it is supplemented by three-region theory, it is unable to explain the phenomenon of pressure drop and recovery near the gas inlet [27]. For steady flow and a compressible fluid, the microscopic flow characteristics can be effectively captured by the full 3D N-S equation. In this study, FLUENT software, based on the CFD method, was used to solve the N-S equation, and the *k* − *ɛ* model was chosen to calculate the turbulence. The effect of swirl on turbulence is included in the RNG model, which can improve the accuracy for swirling flow near the inlet. Thus, the RNG model was chosen within the *k* − *ɛ* model, and *C*_{1} and *C*_{2} were set to 1.42 and 1.68, respectively.

### 2.4 Calculation Grids and Boundary Conditions

Computational conditions of three different restrictors

Design variable | Orifice restrictor | Multiple restrictors | Porous restrictors |
---|---|---|---|

Inlet pressure | 5 | 5 × 10 | 5 × 10 |

Outlet pressure | 0 | 0 | 0 |

Operating pressure | 1 × 10 | 1 × 10 | 1 × 10 |

Gas density | 1.189 | 1.189 | 1.189 |

Dynamic viscosity | 1.8 × 10 | 1.8 × 10 | 1.8 × 10 |

Bearing diameter | 50 | 50 | 50 |

Orifice diameter | 0.05−0.15 | 0.05−0.15 | − |

Groove width | − | 0.20 | − |

Groove depth | − | 0.10 | − |

Groove length | − | 1.50 | − |

Viscous resistance coefficient | − | − | 8 × 10 |

Inertia resistance coefficient | − | − | 2.5 × 10 |

Porosity | − | − | 0.18 |

## 3 Experimental Setup

Detailed structural and material parameters of the three experimental samples

Design variable | Orifice restrictor | Multiple restrictors | Porous restrictors |
---|---|---|---|

Bearing diameter | 50 | 50 | 50 |

Orifice diameter | 0.0665 | 0.7 | − |

Groove width | − | 0.20 | − |

Groove depth | − | 0.10 | − |

Groove length | − | 1.50 | − |

Viscous resistance coefficient | − | − | 1.258 × 10 |

Inertia resistance coefficient | − | − | 2.498 × 10 |

Porosity | − | − | 0.18 |

The stability of each aerostatic bearing was measured at the same three positions using acceleration sensors (AD10T), which possessed a sensitivity of 10 mV/g. The value of the acceleration reflects the stability of the aerostatic bearing. During the stability test, the shaft of the aerostatic bearing remained stationary.

## 4 Results and Discussion

### 4.1 Numerical Analysis

*h*= 6 μm, orifice restrictor diameter of

*d*= 0.1 mm, an operating pressure of 100 kPa, an inlet gas pressure of 500 kPa, and an outlet gas pressure of 0 kPa, a 3D view of the fluid domain pressure distribution was obtained using a post-processing function, as shown in Figure 5. According to the computed pressure distribution of the orifice restrictor, the pressure dropped sharply near the inlet of the restrictor, then decreased smoothly from the inlet to the outlet. The pressure contours of the multiple restrictors showed that the presence of multiple chambers had a significant impact on the ability to maintain a high pressure. The area of the gas film surface at high pressure clearly increased. Figure 5(c) shows the pressure contours of the porous restrictors. In this figure, Section 1 shows the pressure on the upper surface of the porous material, Section 2 represents the two-dimensional (2D) pressure distribution along the center, and Section 3 shows the pressure distribution in the bearing clearance. Based on the pressure distribution of the porous restrictors, it is obvious that the pressure distribution in the bearing clearance is more uniform than that in the orifice restrictor because the surface of the porous material contains countless small holes.

*P*is defined as the strength of the pressure drop and recovery phenomenon. It is obvious that ∆

*P*increases from 0 to almost 85 kPa, which means that the aerostatic bearings become unstable with an increase in film thickness. Under the same conditions (film thickness

*h*= 6 μm), the influence of the restrictor diameter (0.05 mm, 0.10 mm, 0.15 mm) on the pressure distribution characteristics is shown in Figure 7. ∆

*P*increases from 0.18 to 70 kPa as the orifice restrictor diameter decreases, leading to instability of the aerostatic bearings. Many investigators have demonstrated that the instability of aerostatic bearings is closely related to the air vortices that appear near the air inlet of the bearings and that the strength of the air vortices can be represented by the pressure drop between the edge of the vortex and its center. The strength of the air vortices increases as the supply pressure increases. The problem of the stability of the aerostatic bearings cannot be solved effectively by changing the diameter, length or recess shape of the orifice restrictor.

*h*= 6 μm and a supply pressure of 500 kPa, the influence of the viscous resistance coefficient (0.8 × 10

^{14}m

^{−2}, 1.2 × 10

^{14}m

^{−2}, 1.6 × 10

^{14}m

^{−2}) on the pressure distribution characteristics is also shown in Figure 9. As the viscous resistance coefficient decreased, the pressure distribution in the bearing clearance improved significantly.

*W*is the load capacity,

*P*is the gas film pressure,

*R*is the radius,

*R*

_{out}is the outer radius of the bearing clearance surface, and

*Θ*is the angle in radians.

*K*is the stiffness and

*h*is the film thickness.

### 4.2 Experimental Verification

^{2}, was weaker than that of the orifice restrictor, which had an acceleration of 0.02 m/s

^{2}, but was much stronger than that of the porous restrictors, which had an acceleration of 0.006 m/s

^{2}.

## 5 Conclusions

- (1)
The stiffness and stability of thrust pad aerostatic bearings are significantly influenced by the geometric and material parameters. For orifice restrictor and multiple restrictors, the stiffness gradually increases to a maximum and then decreases with an increase in film thickness. However, for porous restrictors, it consistently decreases.

- (2)
The optimal stiffness of multiple restrictors is significantly better than that of the orifice restrictor, by 30.34 N/μm. The porous restrictors have a clear advantage at film thicknesses less than 6 μm. Compared to the orifice restrictor, the optimal stiffness of the porous restrictors can be improved from 15.71 N/μm to 40.46 N/μm.

- (3)
The acceleration of the multiple restrictors, 0.015 m/s

^{2}, is weaker than that of the orifice restrictor, 0.020 m/s^{2}, but is much stronger than that of the porous restrictors, which has a value of 0.006 m/s^{2}. This result demonstrates that the porous restrictors have better stability than the multiple restrictors and orifice restrictor.

## Notes

### Authors’ Contributions

H-LC carried out the numerical simulation and manuscript writing. YW carried out the design and manufacture of restrictor. B-RW carried out the design and establishment of experiment platform. HY carried out the design and manufacture of thrust pad aerostatic bearings. HX carried out the analysis of experimental results and the revise of manuscript. All authors read and approved the final manuscript.

### Authors’ Information

Hai-Long Cui, born in 1989, is currently a PhD candidate and assistant professor at *Institute of Mechinery Manufacturing Technology*, *China Academy of Engineering Physics*. He received his master degree from *University of Electronic Science and technology**, China,* in 2013. His research interests include design and manufacture of aerostatic bearing. Tel: +86-0816-2480487; E-mail: cuihailong61@foxmial.com.

Yang Wang, born in 1963, is currently a professor a PhD candidate supervisor at *China Academy of Engineering Physics*. He received his PhD degree from *Northwestern Polytechnical University, China,* in 1999. His research interests include advanced optical manufacturing equipment technology. E-mail: wangyang@caep.com.

Bao-Rui Wang, born in 1961, is currently a professor and a PhD candidate supervisor at *Institute of Mechinery Manufacturing Technology, China Academy of Engineering Physics*. He received his master degree from *Sichuan University, China,* in 1999. His research interests include advanced manufacturing technology. E-mail: wangbaorui@caep.com.

Hong Yang, born in 1985, is currently an assistant professor at *Institute of Mechinery Manufacturing Technology, China Academy of Engineering Physics*. He received his PhD degree from *Sichuan University, China,* in 2012. His research interests include diamond cutting technology. E-mail: yanghong@caep.com.

Huan Xia, born in 1981, is currently an assistant professor at *Institute of Mechinery Manufacturing Technology, China Academy of Engineering Physics*. He received his master degree from *China Academy of Engineering Physics, China,* in 2009. His research interests include precision and ultra-precision machining technology. E-mail: xiahuan@caep.com.

### Competing Interests

The authors declare that they have no competing interests

### Ethics Approval and Consent to Participate

Not applicable.

### Funding

Supported by National Natural Science Foundation of China (Grant No. 51375325), NSAF (Grant Nos. U1530130), Shanxi coal based low carbon joint fund (U1610118), National Key Instrument Project (Grant No. 2016YFF0102003-02), Science Challenging Program of CAEP (Grant No. JCKY2016212A506-0106).

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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