Research on the influence of a micro-groove-orifice structure and its layout form on the static characteristics of aerostatic journal bearings under a high gas supply pressure

Abstract

To solve the problems of a low gas supply pressure and low load capacity of traditional orifice-compensated aerostatic journal bearings (OCAJBs), this paper presents the design of a double-row gas supply micro-groove-orifice aerostatic journal bearing (MGOAJB). By establishing new governing equations and introducing the SST k-ω turbulence model, the influence of the micro-groove-orifice (MGO) structure and its layout form on the static characteristics of aerostatic journal bearings under a high gas supply pressure are studied. It is found that under high gas supply pressure, the MGO structure can significantly improve the load capacity and stiffness and reduce the gas consumption of the MGOAJB compared with the OCAJB. With additional MGO structures in the eccentric direction of the spindle, the load capacity and stiffness of the bearing increase substantially, and the gas flow also increases. The layout direction of the MGO has a prominent influence on the static characteristics of the MGOAJB. Axial MGOs have a higher load capacity and stiffness and lower gas flow. The accuracy of the simulation method is verified by experiments. This study provides a theoretical basis and method for the design of aerostatic journal bearings under a high gas supply pressure.

Keywords

High gas supply pressure aerostatic journal bearing micro-groove-orifice layout form static characteristics

Introduction

Aerostatic bearings have the advantages of low friction, high speed, low heating, and high precision.¹ The gas supply pressure of aerostatic bearings is generally low, usually between 0.4 and 0.6 MPa,² and their load capacity and stiffness are also low. They are generally used in light load and precision situations, which limits their application fields.³ However, with the development of science and technology and the continuous improvement in production requirements, the demand for high-loading and high-precision aerostatic bearings is increasingly urgent. Since an aerostatic bearing is a kind of gas bearing with an external pressure supply, increasing the gas supply pressure is an effective method to improve the load capacity. However, in practice, taking the widely used orifice-compensated aerostatic bearing as an example, when the gas supply pressure increases, the flow velocity of high-pressure gas accelerates after entering the gas film, reaching supersonic status, resulting in a sharp drop in gas pressure inside the film and even a negative pressure phenomenon, thus reducing the load capacity and stiffness.^4–7 From a theoretical point of view, when the gas supply pressure is high, the gas flow velocity is accelerated, and the influence of inertia force cannot be ignored.⁸ Studies have proven that the traditional Reynolds equation used to solve the gas film pressure distribution is no longer applicable at this time,⁹ and the governing equation needs to be re-established to solve. In Hasegawa and Izuchi,⁵ the inertial effect of gas is considered, and the flow is assumed to be a stable two-dimensional flow. Under the condition of an aerostatic bearing with a parallel surface, the approximate solution of the N−S equation of the positive pressure relation of compressible viscous fluid is solved. Classic fluid lubrication theory assumes that the flow state is laminar. Jean et al.¹⁰ puts forward the theory of nonlaminar lubrication considering the influence of inertia force in turbulent flow and gives the working characteristics of bearings in a turbulent flow state. In recent years, through the combination of the turbulence model and computer simulation methods, researchers have accurately solved the gas film flow field of aerostatic thrust bearings and ball bearings under a high gas supply pressure and a supersonic state. In Sun et al.,^11,12 the shear stress transport (SST) k-ω turbulence model is used to calculate the changes in the gas film flow pressure, Mach number and Reynolds number of a thrust bearing under a high gas supply pressure (0.9–1.5 MPa), and the results are compared with the Reynolds lubrication equation and experimental test results. It is concluded that the SST k-ω model can effectively simulate and predict the distribution of pressure and velocity in the gas film of aerostatic thrust bearings under the condition of high gas supply pressure. In Wang and Bao,¹³ a numerical solution method of the full N−S equation is adopted, the inertia term is retained, the k-ε turbulence model is introduced into the simulation calculation, and simulation calculations of the aerostatic ball bearing of the large gas film clearance are carried out. The results show that the simulation calculation results are in agreement with the actual measurement results when the turbulence model is used. In Li et al.,¹⁴ the CFD method is used to study the flow field in the gas film of an aerostatic thrust bearing with a single gas supply hole. The formation process of supersonic phenomena is studied, and the formation mechanism of eddy currents is revealed.

Orifice-compensated aerostatic bearings are widely used in engineering, and their load capacity and stiffness can be improved by increasing the number and row of orifices.¹⁵ However, the consequent problem is that the gas consumption of bearings increases and energy consumption intensifies; moreover, when the row and number of orifices increase to a certain extent, the improvement of bearing performance is not very significant. For this reason, researchers put forward a structural optimization method of setting pressure-equalizing grooves in aerostatic bearings.^16–18 By analyzing the influence of different shapes and geometric parameters of pressure-equalizing grooves on load capacity, it is found that the pressure-equalizing groove structure can suppress the attenuation of gas film pressure far away from the orifices and alleviate the phenomenon of high-pressure gas congestion and redundancy in the chamber, which can significantly improve the load capacity of aerostatic bearings. According to Du et al.,^19,20 pressure-equalizing grooves are set up in the inner wall of aerostatic journal bearings, which are connected with gas supply holes. The influence of length, depth, quantity and location of pressure-equalizing grooves on load capacity and stiffness is studied. It is found that the circumferential pressure-equalizing groove has a great influence on the bearing performance of the aerostatic journal bearing, but the axial pressure-equalizing groove is more conducive to improving the load capacity. Even if only one or two axial grooves are set on the inner wall of the bearing, the effect is still prominent. In Zhao et al.,^21,22 a deforming elastic pressure-equalizing groove structure is adopted to improve the orifice-compensated aerostatic thrust bearing and floating slide channel, which proves the feasibility and superiority of this structure. However, this deforming elastic pressure-equalizing groove technology has a high material requirement, which limits its application. In Yan and Zhang,²³ a variety of pressure-equalizing groove schemes are set up on the aerostatic guide. By optimizing the distribution form of the pressure-equalizing groove, the influence of the pressure-equalizing groove on the aerostatic guide is revealed. It is found that the load capacity and static stiffness of the aerostatic guide can be greatly improved by using the pressure-equalizing groove, and the service accuracy and life of the bearing can be improved.

To solve the problems of low gas supply pressure and low load capacity of traditional orifice-compensated aerostatic journal bearings (OCAJBs), this paper takes advantage of the advantages of pressure-equalizing groove structures, connects them with orifices to form a micro-groove-orifice (MGO) structure, and designs a double-row gas supply micro-groove-orifice aerostatic journal bearing (MGOAJB). By establishing new governing equations and introducing the SST k-ω turbulence model, the static characteristics (load capacity, stiffness, and gas flow) of the MGOAJB under a high gas supply pressure are calculated and analyzed. Compared with the OCAJB, the gas supply pressure, load capacity, and stiffness of the MGOAJB are effectively improved, and the gas consumption is reduced. The influence of the layout form (number, position, and layout direction) of the MGO structure on the static characteristics of the MGOAJB under a high gas supply pressure is further studied, which provides a theoretical basis and method for the design of aerostatic journal bearings under a high gas supply pressure.

Structural design of the MGOAJB

Figure 1(a) shows the structure diagram of the MGOAJB. The bearing adopts a double-row gas supply, and each row has eight orifices with equal spacing along the circumference direction. $α$ is the position angle of the axis of each orifice relative to the center of the bearing. An axial pressure-equalizing groove is set at the outlet of the orifice on the inner surface of the bearing, and the orifices at the corresponding positions of the front and rear rows are connected along the axial direction to form a MGO structure. The MGOAJB in Figure 1(a) has $z$ =8 MGO structures, where $z$ is the number of pressure-equalizing grooves, namely, the number of MGO structures.

Figure 1.

Schematic diagram of the structure dimensions of the MGOAJB and OCAJB: (a) structure diagram of the MGOAJB and (b) structure diagram of the OCAJB.

where $e$ is the eccentricity of the spindle (μm), $D$ is the bearing inner diameter (mm), $B$ is the bearing width (mm), $d$ is the diameter of the orifice (mm), $h_{g}$ is the groove depth of the MGO (μm), $b$ is the groove width of the MGO (mm), and $l$ is the groove length of the MGO (mm). To avoid the adverse phenomenon caused by the increase in the gas volume in the MGO,^19,20 the size of the MGO should be as small as possible on the premise of improving the load capacity and stiffness. The depth $h_{g}$ and width $b$ of the groove are determined by $h_{m}$ and $d$ , usually $h_{g} = (5 ~ 10) h_{m}$ and $b = (2 ~ 3) d$ , respectively. In addition, $h_{m}$ is the average thickness of the gas film when eccentricity $e = 0$ (μm).

To reflect the influence of the MGO structure on the static characteristics of aerostatic journal bearings under a high gas supply pressure, this paper takes the orifice-compensated aerostatic journal bearing (OCAJB) as a comparison reference, and its structure is shown in Figure 1(b). To ensure that the bearing always has the characteristics of orifice compensation, without cylindrical surface compensation (the load capacity and stiffness of cylindrical surface compensation are lower than orifice compensation),⁸ a cylindrical stingy cavity is arranged at the outlet of the orifice to ensure that the restrictive section is always the orifice cross-section, where $d_{1}$ and $h_{1}$ are the diameter and depth of the cylindrical cavity, respectively. According to Liu et al.,⁸ $d_{1} > d^{2} / 4 h_{m}$ and $h_{1} > d / 4 - h_{m}$ .

The structural dimension parameters of the MGOAJB and OCAJB are shown in Table 1.

Table 1.

Structural dimension parameters of the MGOAJB and OCAJB.

Structural dimension parameters	Value
Bearing inner diameter $D$ (mm)	80
Bearing width $B$ (mm)	80
Orifice diameter $d$ (mm)	0.2
Average thickness of the gas film $h_{m}$ (μm)	15
Groove depth $h_{g}$ (μm)	75
Groove width $b$ (mm)	0.5
Groove length $l$ (mm)	50
Cylindrical cavity diameter $d_{1}$ /(mm)	0.8
Cylindrical cavity depth $h_{1}$ (μm)	75

Theoretical model and simulation of the static characteristics of the MGOAJB under a high gas supply pressure

Theoretical calculation

Gas flow

The gas flow through a single orifice⁸ is

q_{orifice} = C_{D} A_{0} ρ \sqrt{2 RT} {(\frac{κ}{κ - 1})}^{1 / 2} {(β^{1 / 2} - β^{\frac{κ + 1}{κ}})}^{1 / 2}

(1)

where $C_{D}$ is the fluid coefficient, $A_{0} = π d^{2} / 4$ is the orifice cross section (mm²), $d$ is the diameter of the orifice, $ρ$ is the gas density (kg/m³), $R$ is the gas constant, $T$ is the gas temperature (K), $κ$ is the gas isentropic exponent, and $β = p_{d} / p_{0}$ is the pressure coefficient, where, $p_{0}$ is the gas supply pressure (MPa) and $p_{d}$ is the pressure behind the orifice (MPa).

When the gas passes through a single pressure-equalizing groove, the gas flow² is

q_{groove} = \frac{ϕ}{24 μ RT} [\bar{B} {(h + h_{g})}^{3} + (1 - \bar{B}) h_{m}^{3}] (p'_{0}^{2} - p'_{d}^{2})

(2)

where $p'_{0}$ is the pressure of the gas into the pressure-equalizing groove (MPa), $p'_{d}$ is the outlet pressure of the pressure-equalizing groove (MPa), $ϕ = π D / l$ , and $\bar{B}$ is the ratio of the groove area to the gas film area.

According to the lubrication theory proposed in Wang,² the existence of a pressure-equalizing groove is equivalent to the secondary compensation of an orifice, so the pressure behind the orifice is equal to the pressure when the gas enters the pressure-equalizing groove, that is,

p_{d} = p'_{0}

(3)

The gas flow entering the gas film through the orifice or pressure-equalizing groove² is

\begin{matrix} q_{film 1} = \frac{(p_{d}^{2} - p_{a}^{2}) π {Dh}_{m}^{3}}{24 μ RTB n} (orifice), \\ q_{film 2} = \frac{(p'_{d}^{2} - p_{a}^{2}) π {Dh}_{m}^{3}}{24 μ RTB z} (groove) \end{matrix}

(4)

where $n$ is the number of orifices and $p_{a}$ is the environmental pressure (MPa)

According to the principle of conservation of mass, the following relations are obtained:

q_{orifice} = q_{film 1}, q_{groove} = q_{film 2}

(5)

Substituting equations (1), (2), and (4) into equation (5), and using equation (3), the outlet pressure of the pressure-equalizing groove $p'_{d}$ and the gas flow of the MGOAJB can be obtained.

Turbulence governing equations

When calculating the static characteristics of aerostatic bearings, the Reynolds equation is usually used as the theoretical model, the derivation process of the Reynolds equation is carried out under the condition that the Reynolds number is not too large (laminar flow), and the momentum equation in the governing equation can be simplified.⁸ However, when the gas supply pressure increases, the Reynolds number at the gas film inlet increases significantly, making the gas flow more complex, and the inertia force term in the momentum equation cannot be ignored. Therefore, the Reynolds equation of classic lubrication theory is no longer applicable,⁹ and the governing equation needs to be re-established for theoretical modeling under a high gas supply pressure.

To investigate the phenomenon that the Reynolds number in the gas film increases when the gas supply pressure increases, the Reynolds average N−S equation (RANS) method in the turbulence model is used to derive the theoretical model of the pressure distribution in the gas film flow field. The tensor forms of its governing equations are as follows^11,24:

Continuity equation:

\frac{\partial ρ}{\partial t} + div (ρ u) = 0

(6)

Momentum equation:

\frac{\partial}{\partial t} (ρ u_{i}) + \frac{\partial}{\partial x_{j}} (ρ u_{i} u_{j}) = - \frac{\partial p}{\partial x_{i}} + \frac{\partial}{\partial x_{j}} (μ \frac{\partial u_{i}}{\partial x_{j}} - ρ \bar{{u'}_{i} {u'}_{j}}) + S_{i}

(7)

Energy equation:

\frac{\partial}{\partial t} (ρ T) + \frac{\partial}{\partial x_{j}} (ρ u_{i} T) = \frac{\partial}{\partial x_{j}} (\frac{λ}{c_{p}} \frac{\partial T}{\partial x_{j}} - ρ \bar{T' {u'}_{j}}) + S_{T}

(8)

Gas state equation:

p = p (ρ, T)

(9)

Ideal gas equation:

p = ρ RT

(10)

where $u$ is the velocity vector of the flow field; $ρ$ is the gas density; $u_{i}$ and $u_{j}$ are the velocity tensor corresponding to each coordinate axis; $λ$ is the heat transfer coefficient of the gas; $c_{p}$ is the specific heat capacity; the superscript “—” represents the average with respect to time; $u'_{i}$ , $u'_{j}$ , and $T'$ indicate the pulsation value of the corresponding physical quantity; $S_{i}$ is the generalized source term on each coordinate axis; and $S_{T}$ is the viscous dissipation term.

The term related to the turbulent pulsation value $- ρ \bar{{u'}_{i} {u'}_{j}}$ in the governing equation is the Reynolds stress term. Since it is a new unknown quantity, a new expression should be established for the Reynolds stress term (i.e. the turbulence model equation is introduced) to relate the turbulent pulsation value to the time-mean value to solve the equation.

- ρ \bar{{u'}_{i} {u'}_{j}} = μ_{t} (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}}) - \frac{2}{3} (ρ k + μ_{t} \frac{\partial u_{i}}{\partial x_{i}}) δ_{ij}

(11)

where $μ_{t}$ is the turbulent viscosity, $u_{i}$ is the time-mean velocity, $k$ is the turbulent kinetic energy, and $δ_{ij}$ is the Kronecker delta.

k = \frac{1}{2} \bar{{u'}_{i} {u'}_{j}} = \frac{1}{2} (\bar{{u'}^{2}} + \bar{{v'}^{2}} + \bar{{w'}^{2}})

(12)

Researchers have found that the turbulent viscosity in the SST k-ω turbulence model solves the transport of turbulent shear stress in the inverse pressure gradient boundary layer and can effectively simulate the high gas supply pressure near the gas film inlet under complex flow conditions.^11,24 Therefore, this paper adopts this turbulence model to calculate the pressure distribution in the gas film flow field.

The turbulent viscosity $μ_{t}$ is defined as

μ_{t} = \frac{ρ a_{1} k}{max (a_{1} ω, Ω F_{2})}

(13)

where $Ω$ is the absolute value of the eddy, $a_{1}$ =0.31, $F_{2}$ is the mixing function, and $ω = ε / k$ is the turbulence specific dissipation rate, where $ε$ is the turbulence dissipation rate.

F_{2} = \tanh {[max (2 \frac{\sqrt{k}}{0.99 ω y}, \frac{500 μ}{ρ y^{2} ω})]}^{2}

(14)

The transport equation of the turbulent kinetic energy $k$ and the turbulence specific dissipation rate $ω$ ²⁴ of the SST k-ω turbulence model are as follows:

Transport equation of turbulent kinetic energy $k$ :

\frac{\partial}{\partial t} (ρ k) + \frac{\partial}{\partial x_{i}} (ρ k u_{i}) = \frac{\partial}{\partial x_{j}} [(μ + σ_{k} μ_{t}) \frac{\partial k}{\partial x_{j}}] + G_{k} - Y_{k}

(15)

Turbulence specific dissipation rate $ω$ equation:

\frac{\partial}{\partial t} (ρ ω) + \frac{\partial}{\partial x_{i}} (ρ ω u_{i}) = \frac{\partial}{\partial x_{j}} [(μ + σ_{ω} μ_{t}) \frac{\partial ω}{\partial x_{j}}] + G_{ω} - Y_{ω} + D_{ω}

(16)

where $G_{k}$ is the generation term of turbulent kinetic energy $k$ , $G_{ω}$ is the generation term of the turbulence specific dissipation rate $ω$ , $Y_{k}$ , and $Y_{ω}$ are the divergence terms of $k$ and $ω$ , respectively, $D_{ω}$ is the cross-diffusion term, and $σ_{k}$ and $σ_{ω}$ are the Prandtl numbers of $k$ and $ω$ , respectively.

Load capacity and stiffness

The source of the load capacity of an aerostatic journal bearing is the ability to support an external load caused by the difference in pressure distribution on the upper and lower surfaces of the bearing.⁸ The load capacity is calculated as^19,20:

W = \frac{p_{a} D^{2}}{4} \int_{0}^{B} dx \int_{0}^{2 π} p \cos α d α

(17)

The stiffness of the bearing is the ratio of the change in load capacity and the increment of eccentricity, that is,

K = \frac{Δ W}{Δ e}

(18)

where $K$ is the bearing stiffness, $Δ W$ is the variation in load capacity with eccentricity, and $Δ e$ is the increment of eccentricity.

Simulation calculation

Boundary conditions and meshing

According to the method and steps of CFD simulation calculation, as shown in Figure 2, the gas flow field model of the MGOAJB is established by taking the gas in the MGO (including the orifice and pressure-equalizing groove) and the gas in the gas film (the gap between the inner surface of the bearing and the spindle) as the research object. Before the high-pressure gas enters the orifice through the gas supply channel, the flow needs to be stabilized in a pressure-stabilizing chamber. Therefore, the inlet of the pressure-stabilizing chamber is usually set as the pressure inlet in the simulation calculation.²³ The outlet at both ends of the gas film is set as the pressure outlet and connected with the external atmosphere. Since the double-row gas supply structure adopted in this paper has symmetry in the axial direction, to simplify the calculation, the midpoint of the bearing axis is taken as the symmetry face, and half of the structure is taken for calculation. In this paper, SolidWorks software is used for modeling, and the x_t format is saved for output, which can completely retain the thickness characteristics of the gas film for subsequent mesh division.

Figure 2.

Gas film flow field model and computational boundary.

This paper uses the mesh tool of the Workbench module in ANSYS 18.2 software to mesh the model of the gas film flow field. The gas film is very thin (μm as unit), which is quite different from other structure sizes (mm) in the model. In order to improve the accuracy and reflect the thickness characteristics of the gas film, this paper adopts the method of partition mesh division. First, the model shown in Figure 2 is divided into two parts: MGOs and gas film. Then, the MGO part is divided by the hex dominant method. The mesh size should not only ensure the accuracy of reflecting the shape characteristics of the MGO structure but also avoid unnecessary computational pressure due to the mesh density. Through trial and error, the body size of the MGO part is chosen to be 0.05 mm. The grid cell size of the gas film is of the same order of magnitude as that of the MGO as much as possible. In this paper, the gas film is divided into 750 and 2100 equal parts in the axial and circumferential directions by the sweep method and edge sizing commands, respectively. In the direction of the gas film thickness, to reduce the pressure on the computer hardware as much as possible under the condition of ensuring the calculation accuracy, the thickness is divided into five layers by the face meshing command, as shown in Figure 3 after the meshing is completed.

Figure 3.

Computing grid of the gas film flow field.

Simulation calculation

Usually, the gas supply pressure of the aerostatic bearing is 0.4–0.6 MPa. Since this paper studies the static characteristics of the aerostatic journal bearing under a high gas supply pressure, referring to previous research^11,12 and experimental equipment, this paper adopts 0.9, 1.1, 1.3, and 1.5 MPa as the gas supply pressure, which is connected with the pressure inlet of the gas film flow field model, and the pressure outlet is connected with the outside atmosphere. In this paper, the Fluent module in ANSYS is used to simulate the flow field. The SST k-ω turbulence model is widely used in industrial applications due to its high computational convergence and accuracy in complex shear flows, swirling flows and separated flows. Therefore, the SST k-ω turbulence model is adopted in this study to simulate the gas film flow field under a high gas supply pressure. In this paper, the SIMPLEC algorithm is used for iterative calculation, and the relaxation factor is adjusted to improve the convergence of the calculation. The simulation calculation process is shown in Figure 4.

Figure 4.

Flowchart of the simulation calculation.

Results and analysis

Influence of the MGO structure on the static characteristics

As shown in Figure 5, under a high gas supply pressure ( $p_{0}$ = 0.9, 1.1, 1.3, 1.5 MPa), compared with the OCAJB, the load capacity of the MGOAJB is increased by 108.97%–193.05%, 94.64%–170.95%, 85.70%–158.49%, 76.32%–136.40%, respectively. This shows that under a high gas supply pressure, the MGO structure can play the role of pressure equalization so that the load capacity of the aerostatic journal bearing can be significantly improved. From the above data, it is found that with increasing gas supply pressure, the load capacity of the MGOAJB increases continuously, but the increase amplitude shows a trend of gradual decrease. For example, when the gas supply pressure is $p_{0}$ = 0.9 MPa, the load capacity increases by 108.97%–193.05%, while when it is $p_{0}$ = 1.5 MPa, the load capacity increases by 76.32%–136.40%, and the increase amplitude decreases. The reasons for this phenomenon may be that the gas supply pressure is increased to a certain degree (e.g. this article $p_{0}$ = 1.5 MPa), the gas flow increases, and the speed increases. In the case of the bearing size parameters being unchanged, its structure has a certain inhibition on the gas flow, and the flow condition becomes more complicated, resulting in an increase in the load capacity under a certain influence. Therefore, although the MGO structure can effectively improve the load capacity of aerostatic bearings under a high gas supply pressure, the gas supply pressure needs to match the bearing structure size. For the bearing size and MGO parameters described in this paper, the bearing performance can be significantly improved when the gas supply pressure is $p_{0}$ =0.9–1.5 MPa.

Figure 5.

Load capacity and stiffness of the MGOAJB under a high gas supply pressure: (a) $p_{0}$ = 0.9 MPa, (b) $p_{0}$ = 1.1 MPa, (c) $p_{0}$ = 1.3 MPa, and (d) $p_{0}$ = 1.5 MPa.

As shown in Figure 5, under a high gas supply pressure ( $p_{0}$ = 0.9, 1.1, 1.3, 1.5 MPa), the stiffness of the MGOAJB is also significantly improved compared with that of the OCAJB. The stiffness of the bearing is increased by 109.11%–173.85%, 95.32%–156.45%, 90.65%–141.09%, and 79.38%–129.97%, respectively. Affected by the load capacity with the change in gas supply pressure, although bearing stiffness increases with the increase in gas supply pressure, the increase amplitude also shows a trend of gradual decrease. The figure also shows that the load capacity increases first and then decreases with an increase of eccentricity $e$ , while the stiffness gradually decreases with an increase of eccentricity $e$ . That is because when the eccentricity $e$ increases from zero, the pressure difference between the upper and lower surfaces of the bearing increases, making the load capacity increases.⁸ However, when the eccentricity is high, the gas film thickness in the eccentric direction becomes extremely thin and the flow channel becomes very narrow, so that high-pressure gas into the gas film is hindered. As a result, the increase of load capacity gradually becomes gentle, and even the load capacity decreases. According to equation (18), the stiffness is the ratio between the increment of load capacity $Δ W$ and the increment of eccentricity $Δ e$ . With an increasing eccentricity, the increment of the load capacity gradually decreases. Thus, the stiffness in the figure decreases with an increasing eccentricity.

As shown in Figure 6, to study the principle of the MGO structure to improve the load capacity of the aerostatic bearing, taking the gas supply pressure $p_{0}$ = 1.5 MPa as an example, a 3D cloud map and contour profile of the gas film pressure distribution of the MGOAJB and OCAJB are shown. When the MGO structure is adopted, high-pressure gas enters the pressure-equalizing groove after passing through the orifice. Under the action of pressure equalization, the high-pressure area (red part in Figure 6) in the gas film is distributed along the pressure-equalizing groove of the MGO structure (Figure 6(a)), which expands the high-pressure area of the aerostatic bearing film and flattens the pressure gradient. It can inhibit the attenuation of the gas film pressure far from the orifice and improve the load capacity and stiffness of the aerostatic journal bearing.

Figure 6.

3D cloud map and contour profile of the gas film pressure distribution of the aerostatic journal bearing ( $p_{0}$ = 1.5 MPa, $e$ = 10 μm): (a) MGOAJB and (b) OCAJB.

As shown in Figure 7, with increasing gas supply pressure, the gas flow into the gas film of the aerostatic journal bearing shows an increasing trend, indicating that the gas flow increases with increasing gas supply pressure. Under the same gas supply pressure, the gas flow of the MGOAJB is lower than that of the OCAJB (3.94%–5.26%). Combined with the above conclusions, it can be concluded that the load capacity and stiffness of the MGOAJB are significantly higher than those of the OCAJB, and the gas flow of the MGOAJB is lower. Therefore, when the aerostatic journal bearing works for a long time, the MGO structure can save the high-pressure gas in the gas source device and offers better economy.

Figure 7.

Gas flow of the MGOAJB under high supply pressure.

Influence of the number and position of MGO structures on static characteristics

As shown in Figure 8, $z$ = 2, 3, and 4 axial MGO structures are set on the inner surface of the bearing, among which the number of axial MGOs located in the eccentric direction of the spindle is 1, 2, and 3, respectively, and the number of axial MGOs far away from the eccentric direction is 1 and the position is the same. The structure with $z$ = 8 axial MGOs in Figure 1(a) is taken as a comparison. The influence of the number and position of MGOs on the static characteristics of aerostatic journal bearings is studied.

Figure 8.

Schematic diagram of the different numbers and locations of MGO structures: (a) $z$ = 2, (b) $z$ = 3, (c) $z$ = 4, and (d) $z$ = 8.

As shown in Figure 9, under a high gas supply pressure ( $p_{0}$ = 0.9–1.5 MPa), the number and position of MGOs have a significant impact on the load capacity and stiffness of aerostatic journal bearings. Under the condition that the number and position of MGOs away from the eccentric direction of the spindle are unchanged, the load capacity and stiffness of the MGOAJB gradually increase with the increase in the number of MGOs in the eccentric direction, and the load capacity and stiffness increase by 13.26%–54.30% when $z$ = 3 compared with $z$ = 2. When $z$ = 4, the load capacity and stiffness increase by 8.35%–38.72% compared with $z$ = 3. However, if the number of MGOs in the direction away from the spindle eccentricity increases, the load capacity and stiffness do not continue to increase but decrease. For example, when $z$ = 8, pressure-equalizing grooves are set at the outlet of each orifice so that the number of MGO structures reaches the highest value, but compared with the $z$ =4 case, the load capacity and stiffness are reduced by 3.56%–16.23%. The reason for this phenomenon can be analyzed from the formation principle of the load capacity of aerostatic journal bearings. The load capacity is derived from the pressure difference between the upper and lower surfaces of aerostatic journal bearings, and the compensating effect of orifices and the gas film thickness difference caused by spindle eccentricity are important reasons for the formation of pressure difference between the upper and lower surfaces of bearings.⁸ When $z$ = 8, because the MGO structure spans the upper and lower surfaces of the bearing, the gas film pressure increases not only on the lower surface of the bearing (where the film is thinning) but also on the upper surface (where the film is thickened), thus reducing the pressure difference between the upper and lower surfaces of the bearing and reducing the load capacity. Therefore, in the design of the MGOAJB, it is necessary to reasonably arrange the position and number of MGO structures according to the force of the spindle and the eccentric direction.

Figure 9.

Load capacity and stiffness of aerostatic bearings with different numbers and positions of MGO structures under a high gas supply pressure: (a) $p_{0}$ = 0.9 MPa, (b) $p_{0}$ = 1.1 MPa, (c) $p_{0}$ = 1.3 MPa, and (d) $p_{0}$ = 1.5 MPa.

As shown in Figure 10, when only one axial MGO structure is set in the eccentric direction of the spindle ( $z$ = 2), the high-pressure area of the gas film is distributed and extended along the axial direction, which improves the load capacity and stiffness of the aerostatic journal bearing. With additional axial MGOs in the eccentric direction of the spindle, the range of high pressure areas in the gas film continues to expand ( $z$ = 3). However, due to the long distance between the MGOs, the high-pressure areas fail to be effectively connected in the circumferential direction, and the improvement in the load capacity and stiffness is still insufficient. When the number of MGOs in the eccentric direction continues to increase, the high-pressure area further expands and gradually begins to connect in the circumferential direction ( $z$ = 4), and the load capacity and stiffness of the aerostatic journal bearing are effectively improved. However, when the number of MGOs away from the eccentric direction increases ( $z$ = 8), the pressure difference between the upper and lower surfaces of the bearing is reduced, and the load capacity and stiffness are reduced. Figure 10 also shows that the distribution of the gas film pressure of the MGOAJB mainly depends on the layout of the pressure-equalizing groove in the MGO structure. Although the gas film pressure on both sides of the pressure-equalizing groove (in the circular direction in Figure 10) has been increased to a certain extent, the high-pressure gas entering the gas film is still prominently distributed along the axial direction under the pressure-equalizing effect of the axial MGO.

Figure 10.

Contour profile of the gas film pressure distribution of the MGOAJB with different numbers and positions of MGO structures ( $p_{0}$ = 1.5 MPa, $e$ = 10 μm): (a) $z$ = 2, (b) $z$ = 3, (c) $z$ = 4, and (d) $z$ = 8.

As shown in Figure 11, under a high gas supply pressure, different numbers of MGOs have a significant effect on the gas flow. For the axial MGO structure, the gas flow increases with additional axial pressure-equalizing grooves. When $z$ = 2, the bearing gas consumption is the smallest (13.26%–54.30% lower than that when $z$ = 3), and the energy saving effect is the best, but the improvement of load capacity is limited. The difference in gas flow between $z$ = 3 and $z$ = 4 is very small (1.62%–1.72%), which is significantly lower than that of the OCAJB (23.73%–24.92%), and the load capacity increases substantially. Considering the improvement of load capacity and energy consumption, a more ideal structure scheme is attained. As the number of MGOs continues to increase $z$ = 8, the gas flow increases significantly, although it is lower than that of the OCAJB (3.94%–5.26%), the load capacity and stiffness decreases (3.56%–16.23%) compared with the $z$ = 4 case. Therefore, when designing an MGOAJB, the position and number of MGO structures should be reasonably configured according to the working conditions, in conjunction with the load direction and gas consumption.

Figure 11.

Gas flow of the MGOAJBs with different numbers and positions of MGO structures under a high gas supply pressure.

Influence of the MGO layout direction on the static characteristics

To study the influence of the layout direction of the MGO on the static characteristics of the MGOAJB under a high gas supply pressure, four kinds of the MGOAJB are designed: an axial MGO, through-type circumferential MGO, segmented type circumferential MGO and oblique MGO. For convenience of expression, they are correspondingly expressed as Layout-I, Layout-II, Layout-III and Layout-IV, as shown in Figure 12. The four kinds of bearings are the same: double-row gas supply, bearing size, groove depth, groove width and groove section shape. The specific parameters are shown in Table 1. To avoid the influence of the number and position of MGOs on the results, for the axial MGO and oblique MGO, the number of MGO structures is $z$ = 8, and the gas supply pressure is $p_{0}$ = 0.9, 1.1, 1.3, and 1.5 MPa.

Figure 12.

Structural schematic diagram of aerostatic journal bearings with different MGO layout directions: (a) layout-I (axial MGO), (b) layout-II (through-type circumferential MGO), (c) layout-III (segmented type circumferential MGO), and (d) layout-IV (oblique MGO).

As shown in Figure 13 and referring to the data in Figure 5, under a high gas supply pressure ( $p_{0}$ = 0.9–1.5 MPa), compared with the OCAJB, the load capacity of Layout-II (through-type circumferential MGO) is increased by 41.10%–66.06%, 42.51%–58.36%, 34.92%–50.62%, 17.51%–33.88%, and this layout can also provide pressure equalization under a high gas supply pressure, which can significantly improve the load capacity of aerostatic journal bearings. However, under the same gas supply pressure, compared with the Layout-I case (axial MGO), the load capacity is reduced by 32.98%–51.85%, 26.75%–44.75%, 27.03%–46.51%, and 31.96%–66.29%, a significant decrease. The load capacity of Layout-III (the segmented type circumferential MGO) is significantly higher than that of Layout-II but still significantly lower than that of Layout-I (15.21%–39.09%, 12.10%–33.70%, 9.70%–33.81%, 9.86%–48.98%). The load capacity of Layout-IV (oblique MGO) is high but slightly lower than that of Layout-I (2.57%–9.41%, 1.81%–11.39%, 4.15%–13.14%, 2.88%–17.86%).

Figure 13.

Load capacity and stiffness of MGOAJBs with different MGO layout directions under a high gas supply pressure: (a) $p_{0}$ = 0.9 MPa, (b) $p_{0}$ = 1.1 MPa, (c) $p_{0}$ = 1.3 MPa, and (d) $p_{0}$ = 1.5 MPa.

Under a high gas supply pressure, although the stiffness of Layout-II is increased by 38.71%–66.06%, 42.51%–58.36%, 34.91%–50.35%, and 18.13%–43.82% compared with the OCAJB case, it is significantly lower than that of the other three MGO layout structures. Compared with the Layout-I case, the Layout-III results are reduced by 15.21%–32.47%, 13.02%–30.50%, 9.70%–26.47%, and 9.86%–28.53%, respectively. The stiffness of Layout-IV is close to that of Layout-I but slightly lower (2.63%–14.09%, 3.41%–13.61%, 4.15%–11.62%, 4.31%–14.47%).

As shown in Figure 14, according to the principle of the pressure-equalizing effect of the MGO structure, Layout-I has a significant axial pressure equalization effect. According to the analysis in the previous section, by increasing the number of axial MGOs, the high-pressure area in the gas film can be expanded and connected along the circumferential direction, which makes up for the deficiency of a single axial MGO structure in increasing the gas film pressure, expands the range of high-pressure area in the gas film, and improves the load capacity of aerostatic journal bearings (Figure 14(a)). Layout-II connects the high-pressure gas in the eccentric direction of the spindle to the low-pressure area far from the eccentric direction through the pressure-equalizing groove and equalizes the gas film pressure in the whole circumferential direction so that the pressure difference between the upper and lower surfaces of the bearing becomes significantly smaller, reducing the load capacity and stiffness (Figure 14(b)). Layout-III is an improvement of Layout-II. The original through-type circumferential MGO is divided into two sections according to the different eccentric directions so that the upper and lower surfaces of the bearing are pressure-equalizing, and the loss of pressure difference is smaller, so it has higher load capacity and stiffness compared with Layout-II. However, for the double-row gas supply journal bearing in this study, Layout-III is unable to expand and connect the high-pressure area in the gas film along the axial direction effectively, and its effect on axial pressure equalization is limited (Figure 14(c)). As shown in Figure 14(d), although Layout-IV offers both axial and circumferential pressure equalization, the high-pressure area is asymmetric, resulting in uneven force on the upper and lower surfaces of the bearing and reduced load capacity.

Figure 14.

Contour profile of the gas film pressure distribution of MGOAJBs with different MGO layout directions under a high gas supply pressure ( $p_{0}$ = 1.5 MPa, $e$ = 10 μm): (a) layout-I (axial MGO), (b) layout-II (through-type circumferential MGO), (c) layout-III (segmented type circumferential MGO), and (d) layout-IV (oblique MGO).

As shown in Figure 15, the layout direction of the MGO has a significant impact on the gas flow of the MGOAJB. The gas flow of Layout-II is the highest, 6.87–9.12% higher than that of the OCAJB. Although the load capacity and stiffness of Layout-IV are higher, the gas flow is close to that of Layout-II (1.81%–2.71%) and higher than that of the OCAJB (5.25%–6.11%), so the gas consumption of the gas source device is higher. The gas flows of Layout-I and Layout-III are similar (4.24%–4.84%) and lower than that of the OCAJB (3.94%–9.52%). Therefore, considering the influence of the four layout directions of the MGO on the load capacity, stiffness and gas flow of the MGOAJB, Layout-I (axial MGO) is more suitable for aerostatic journal bearings with a double-row gas supply, as it can not only improve the load capacity and stiffness of the MGOAJB but also reduce the gas consumption and has better economy.

Figure 15.

Gas flow in different MGO layout directions under a high gas supply pressure.

Experimental verification

As shown in Figure 16, the aerostatic journal bearing test bench is mainly composed of an air compressor, gas cylinders, air filter, testing bearings, bearing supports, spindle, loading device, drive motor, displacement sensor, data collection device and computer. The testing bearing is distributed at both ends of the spindle, which is fixed on the test bench by the bearing supports and supports the spindle rotation. The drive motor is connected to the spindle through a flexible coupling to drive the spindle to rotate. The displacement sensor is installed on the support, the test data are transmitted to the data collection device, and the data are processed by the data processing software on the computer.

Figure 16.

Aerostatic bearing test equipment: (a) schematic diagram of test bench and (b) test bench.

Under a high gas supply pressure, the radial load and displacement of the spindle are measured by the loading device and displacement sensor and drawn into a curve, namely, the load capacity curve of the aerostatic journal bearing, as shown in Figure 17. With the continuous increase in the gas supply pressure, the load capacity continues to increase. Under a high gas supply pressure, the load capacity of the MGOAJB is significantly higher than that of the OCAJB (33.7%–102.6%), and the larger the spindle eccentricity is, the more significant the difference. Comparison of the simulation calculations indicates that they are consistent with the variation trend of the experimental data. Although errors and assembly accuracy are unavoidable in the processing of testing bearings, resulting in certain differences between the experimental data and the simulation calculation, the results of the simulation calculation and experimental data are close to each other (5.8%–9.5%), indicating accurate calculation the load capacity of aerostatic journal bearings under high supply pressure; thus, the simulation method has high accuracy and reference value.

Figure 17.

Comparison of simulation and experimental data under a high gas supply pressure: (a) $p_{0}$ = 0.9 MPa, (b) $p_{0}$ = 1.1 MPa, (c) $p_{0}$ = 1.3 MPa, and (d) $p_{0}$ = 1.5 MPa.

Conclusion

To solve the problems of a low gas supply pressure and low load capacity of the traditional OCAJB, this paper designs a double-row gas supply MGOAJB. By establishing new governing equations and introducing the SST k-ω turbulence model, the static characteristics (load capacity, stiffness and gas flow) of the MGOAJB under a high gas supply pressure are studied, and the influence of the layout form (number, position and layout direction) of the MGO structure on the static characteristics of the MGOAJB under a high gas supply pressure is further studied. The main conclusions are as follows:

(1) Under a high gas supply pressure, the MGO structure can improve the pressure distribution of the gas film flow field and expand the range of the high-pressure area of the gas film, which makes the load capacity and stiffness of the MGOAJB significantly higher than those of the OCAJB. The MGO structure can reduce the gas consumption so that the MGOAJB has better economy.

(2) Under a high gas supply pressure, the position and number of MGO structures have a notable influence on the static characteristics of the MGOAJB. With additional MGO structures in the eccentric direction of the spindle, the load capacity and stiffness of the bearing increase substantially, but when the number of MGO structures away from the eccentric direction of the spindle increases, the load capacity and stiffness slightly decrease. The gas flow increases with additional MGO structures.

(3) Under a high gas supply pressure, the layout direction of MGO structures has a prominent influence on the static characteristics of the MGOAJB. Axial MGOs have a higher load capacity and stiffness and lower gas flow. Through-type circumferential MGOs have a lower load capacity and stiffness but higher gas flow. As an improvement, the segmented type circumferential MGO improves the load capacity and stiffness and reduces the gas flow to a certain extent. The load capacity and stiffness of the oblique MGO are close to those of the axial MGO, but the gas flow is higher than that of the axial MGO, and the gas consumption is higher.

Footnotes

Appendix

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The full name of the funding project is: supported by “the Fundamental Research Funds for the Central Universities”, Grant No. 2572016AB70.

ORCID iDs

Wei Cui

Shusen Li

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