CFD-Based Analysis of Loading Performance and Hydrodynamic Effects in a Partial-Arc Aerostatic Radial Bearing

Abstract

With the widespread use of high-speed motorized spindles in precision machining, conventional contact loading methods are no longer adequate for stiffness loading tests under high-speed operating conditions. Non-contact loading technology based on a partial-arc aerostatic radial bearing offers an effective alternative. In this study, a CFD-based hydrodynamic model was developed for the gas-film flow field in a partial-arc aerostatic radial bearing. The effects of bearing geometric parameters, such as chamber configuration, supply-orifice structure, and eccentricity, on loading characteristics were investigated. The influence of hydrodynamic effects under high-speed rotation on the loading force stability and stiffness-testing accuracy was analyzed, and an asymmetric shallow–deep composite chamber design was proposed to mitigate these effects. The results indicate that the partial-arc aerostatic radial bearing, designed based on both static characteristics and rotational performance analysis, can effectively suppress hydrodynamic effects and improve loading force stability and stiffness-testing accuracy.

1. Introduction

With the continuous advancement of advanced manufacturing technologies, high-speed motorized spindles, as core components of precision machining equipment, play a vital role in aerospace, precision mold manufacturing, and microelectronic machining [1,2]. The operational stiffness of a motorized spindle, defined as the stiffness characteristics exhibited under rotational operating conditions, directly affects the machining accuracy, operational stability, and machining quality of the machine–tool system. In particular, under high-speed operating conditions, the accurate and reliable evaluation of operational stiffness remains a pressing issue for both academia and industry. Because conventional contact-based loading methods are subject to disturbances such as frictional heating and contact-induced vibration [3,4], they can no longer satisfy the requirements of stiffness testing under high-speed conditions.

Non-contact loading techniques have gradually become an effective means to address this problem, including electromagnetic loading and gas film loading. A major advantage of non-contact loading systems is that they enable operational stiffness testing under long-duration, high-speed operation, thereby avoiding the thermal effects and mechanical disturbances associated with conventional contact loading at high rotational speeds. Hebbale [5] investigated the influence of electromagnets on solid ferromagnetic shafts and established one of the earliest mathematical models. Autila et al. [6] employed a nonlinear two-dimensional finite element method to predict the load-carrying performance of radial bearings and conducted experimental verification using relevant apparatus; however, their analysis was limited to static conditions. Yamazaki et al. [7] proposed a non-contact measurement method based on an electromagnetic loading device and a circumferential-groove virtual cutter, enabling stiffness measurement of a spindle in the rotating state. Wang et al. [8] proposed a force-sensor-based method to accurately measure steady-state and transient loads under electromagnetic loading. However, under high-speed rotation, eddy current losses induced by electromagnetic loading can cause severe heating, thereby interfering with the measurement accuracy of displacement sensors. Meanwhile, the opposing magnetic field generated by induced eddy currents offsets the effective attractive force, leading to a pronounced attenuation of the loading force as the rotational speed increases. Song et al. [9] improved a non-contact electromagnetic loading device by designing a slotted electromagnetic loading rod, which suppresses eddy current effects to some extent; nevertheless, the influence of eddy currents remains non-negligible.

Compared with electromagnetic loading, gas hydrostatic bearings have become an important non-contact loading approach for stiffness testing of high-speed motorized spindles due to their frictionless operation and high stiffness. Gas bearings can realize high-precision non-contact loading, effectively avoiding the temperature rise and mechanical interference brought by traditional contact loading, thereby ensuring the accuracy of test results under high-speed rotation. Feng [10,11] proposed a five-degree-of-freedom non-contact gas film loading device and preliminarily established the framework structure of gas film loading stiffness testing. This device provided a theoretical basis and practical guidance for the application of gas film loading technology. However, the core component, the partial-pad gas radial bearing loading pad, was not further improved and analyzed.

The partial-arc aerostatic radial bearing is an innovative design derived from the conventional aerostatic radial bearing, and research on aerostatic radial bearings provides important theoretical references for structural improvement of the partial-arc aerostatic radial bearing. Wu et al. [12] proposed a simplified calculation method and, combined with CFD, validated the static performance of an aerostatic bearing with a multi-orifice restrictor; the results indicated that the load capacity is closely related to the supply pressure and the eccentricity ratio. Gu et al. [13] investigated the static characteristics of porous bearings under compressible gas effects using the finite difference method (FDM) and found that the eccentricity ratio and supply pressure significantly influence the load capacity and stiffness. Gao et al. [14] analyzed the enhancement in load capacity and stiffness provided by a micro-orifice restrictor using the finite element method, demonstrating that micro-orifice designs offer significant advantages over conventional orifice restrictors. Ma et al. [15] studied externally pressurized gas bearings and found that the load capacity and stiffness increase markedly with increasing supply pressure. Dal and Karaçay [16] examined the influence of asymmetric gas supply on aerostatic bearing systems and have reported that supply asymmetry not only increases load capacity but also effectively reduces the occurrence frequency of pneumatic hammer, thereby enhancing the bearing stability. Wang et al. [17] calculated the static performance of gas bearings via the finite element method, showing that factors such as rotational speed, supply pressure, and eccentricity ratio significantly affect the load capacity and stiffness. Kajiwara et al. [18] conducted numerical analyses of the load-carrying capacity of grooved gas bearings using CFD and CAE simulations and investigated the effect of asymmetric gas supply on bearing performance; they found that supply asymmetry can effectively suppress the pneumatic hammer and improve the load capacity. Renn et al. [19] investigated the mass flow characteristics of orifice-type restrictors in aerostatic bearings through experiments and CFD simulations, showing that the orifice flow exhibits evident compressible flow behavior and that its critical pressure ratio and flow characteristics differ from those of an ideal nozzle. Yang et al. [20] studied the effects of the number and distribution of throttling orifices on the static performance of aerostatic radial bearings and found that an appropriate orifice arrangement can significantly improve load capacity and enhance bearing stability and static stiffness. Lu et al. [21] proposed an aerostatic bearing with an integrated porous restrictor and found that increasing the number of throttling orifices significantly improves load capacity; specifically, a nine-orifice design achieved a load capacity 1.9 times that of a single-orifice configuration. Yi et al. [22] experimentally compared the effects of different lubricating fluids on bearing performance and found that increasing the supply pressure improves load capacity, whereas the influence of lubricant type on flow rate and stiffness cannot be neglected.

Under high-speed rotation, purely hydrostatic support theory is no longer sufficient to accurately describe the dynamic behavior of the gas film, and hydrodynamic effects have become a primary source of loading errors in the system. Zhang et al. [23] analyzed the hydrodynamic effects of a gas micro-bearing at high rotational speeds via CFD simulation and examined how the gas film pressure distribution at different speeds influences the load capacity, stiffness, and damping coefficients. Song et al. [24] investigated the flow field characteristics of aerostatic bearings at high speed using the SST k-ω turbulence model and found that an annular groove design can effectively decouple hydrostatic and hydrodynamic effects, thereby improving the load capacity. Zhang et al. [25] solved the steady-state and dynamic Reynolds equations to explore the effects of geometric parameters, rotational speed, and supply pressure on aerostatic bearing performance. In addition, An et al. [26] performed a fluid–structure interaction analysis to evaluate the influence of hydrodynamic effects on the stability and load capacity of aerostatic bearings under high-speed operation and reported that optimization of structural parameters is beneficial for performance enhancement. Yan et al. [27] proposed a simplified load capacity prediction model based on hydraulic and hydrodynamic analyses, which enables rapid prediction of the load capacity of externally pressurized gas bearings. Their results showed that hydrodynamic effects play a dominant role in bearing performance at high rotational speeds and can significantly increase load capacity. Al-Bender [28] pointed out that hydrodynamic effects may induce a pneumatic hammer, and that this instability can be effectively mitigated by optimizing the gas film design and supply pressure. Jia et al. [29] found via dynamic modeling that hydrodynamic effects can aggravate vibration under low-speed heavy-load conditions, whereas under high-speed light-load conditions, system stability can be improved by adjusting the gas film thickness. Additionally, Gao et al. [30] developed a FEM-based two-way fluid–structure interaction fast model and showed that structural deformation alters the gas film thickness, significantly affecting load capacity and stiffness, and thus, causing load prediction errors when using rigid body aerostatic theory.

To eliminate the interference caused by contact loading methods in the operational stiffness testing of high-speed motorized spindles, this study focuses on a partial-arc aerostatic radial bearing. A mathematical model of the gas film flow field was established based on the conservation of mass and momentum, and CFD simulations were performed to analyze the steady-state flow field and the dynamic pressure effect. Numerical analyses were conducted to determine the gas film pressure distribution and maximum load capacity under different geometric parameters, rotational speeds, and supply pressures. The effects of throttling orifice structural parameters and gas film chamber configurations on bearing performance were further examined, and the mechanism by which hydrodynamic effects influence the loading force stability was investigated, providing a basis for geometric optimization of the partial-arc bearing. Experiments were carried out using a partial-arc radial loading apparatus at different rotational speeds to characterize the loading behavior of the gas bearing.

2. Methods

2.1. Structure and Operating Principle of the Partial-Arc Aerostatic Radial Bearing

For the designed loading pad of the partial-arc aerostatic radial bearing, the primary objective is to ensure a sufficiently large loading force. After comparing the characteristics of various restrictor types, an orifice-type annular shallow-chamber restrictor configuration was adopted, as shown in Figure 1. This configuration not only provides a relatively large loading force but also reduces the manufacturing complexity.

The specific structural system of the partial-arc aerostatic radial bearing used in the simulations is shown in Figure 2. A bearing pair was formed between the test spindle and the partial-arc bearing, in which the aerostatic gas film provided the load required for radial loading of the spindle. Compressed gas entered through the supply passage and then flowed into the gas chamber through the throttling orifice, where it formed a gas film with the test spindle surface. By adjusting the clearance between the loading spindle and the test spindle, the gas film thickness was varied to simulate the loading process. The preliminary parameters of the partial-arc aerostatic radial bearing are listed in

Table 1. Preliminary parameters of the partial-arc aerostatic radial bearing.

Parameter	Value
Diameter of the partial-arc aerostatic radial bearing dk (mm)	25
Number of throttling orifices N	4
Throttling orifice diameter df (mm)	0.8
Orifice spacing Lm (mm)	3
Shallow-chamber width Ld (mm)	18
Wrap angle of the shallow chamber β (°)	40
Shallow-chamber depth hq (μm)	30
Gas film thickness hk (μm)	20
Supply pressure Ps (MPa)	0.7

, and air was used as the working medium.

2.2. Theoretical Model for Bearing Performance Calculation

Because of the nonlinear nature of the governing equations, the static characteristics of a partial-arc aerostatic radial bearing are generally difficult to obtain analytically. Numerical methods have therefore been widely adopted in bearing simulations as efficient and reliable alternatives. For the loading pad structure designed in this study, which features a stepped gas-film clearance, the finite volume method (FVM) provides high accuracy in handling geometrically discontinuous boundaries and enables efficiently solving complex gas-film flow fields. Accordingly, the FVM was employed to numerically evaluate the static characteristics of the loading system.

Before conducting the numerical simulations, it was necessary to establish a mathematical model of the gas film flow field. Consider an arbitrary control volume within the gas film. According to the law of mass conservation, the sum of the mass flow rates entering the control volume is equal to the rate of mass increase within it. Therefore, the continuity equation can be written as follows:

$${\displaystyle \frac{\partial (\rho )}{\partial t}}+{\displaystyle \frac{\partial (\rho v_x)}{\partial x}}+{\displaystyle \frac{\partial (\rho v_y)}{\partial y}}+{\displaystyle \frac{\partial (\rho v_z)}{\partial z}}=0$$

(1)

where ρ is the gas density; t is time; and v_x, v_y and v_z are the velocity components in the x-, y-, and z-directions, respectively.

According to the law of momentum conservation, the resultant external force acting on an infinitesimal element of the gas film, together with the net momentum flux into the corresponding control volume per unit time, is equal to the rate of change of momentum within that element. Accordingly, the governing momentum equation for the gas film flow field can be written as follows:

$$\left\{\begin{array}{l}\rho {\displaystyle \frac{d{v}_x}{dt}}=\rho X+{\displaystyle \frac{\partial {\sigma }_x}{\partial x}}+{\displaystyle \frac{\partial {\tau }_{yx}}{\partial y}}+{\displaystyle \frac{\partial {\tau }_{zx}}{\partial z}}\\ \rho {\displaystyle \frac{d{v}_y}{dt}}=\rho Y+{\displaystyle \frac{\partial {\tau }_{xy}}{\partial x}}+{\displaystyle \frac{\partial {\sigma }_y}{\partial y}}+{\displaystyle \frac{\partial {\tau }_{zy}}{\partial z}}\\ \rho {\displaystyle \frac{d{v}_z}{dt}}=\rho Z+{\displaystyle \frac{\partial {\tau }_{xz}}{\partial x}}+{\displaystyle \frac{\partial {\tau }_{yz}}{\partial y}}+{\displaystyle \frac{\partial {\sigma }_z}{\partial z}}\end{array}\right.$$

(2)

where σ_i (i = x, y, z) is the normal stress component; τ_ij (i, j = x, y, z) denotes the shear stress component; and X, Y, and Z are the body-force components in the x-, y-, and z-directions, respectively.

During the static analysis stage of the present study, the gas entering the bearing clearance was assumed to behave as an ideal gas, and the thermodynamic process was assumed to be isothermal. These assumptions simplify the problem and render the analysis of the gas lubrication process more straightforward and tractable. Therefore, the equation of state for the gas can be expressed as

$${\displaystyle \frac{p}{\rho }}={\displaystyle \frac{p_a}{{\rho }_a}}$$

(3)

where p is the gas film pressure; p_a is the ambient pressure; and ρ_a is the ambient gas density.

During the static analysis stage, air was treated as the lubricating medium. Because no rotational effect was involved and air exhibits typical Newtonian-fluid behavior, the fluid constitutive relation was introduced to establish the quantitative relationship between the stress and the deformation rate. To simplify the computation and reduce the number of unknowns, the following assumptions were adopted: no-slip velocity continuity at the fluid–solid interface; laminar flow in the lubricating film, with turbulence and vortex effects neglected; a Newtonian lubricant; negligible body forces; and negligible pressure variation across the film thickness. In addition, the fluid inertial forces were assumed to be sufficiently small to be neglected in comparison with the viscous shear stresses.

Based on these assumptions, letting the independent variables in the polar coordinate system be r and θ, the Navier–Stokes equations in polar coordinates applicable to the partial-arc aerostatic radial bearing can be written as

$${\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{h}^3{\displaystyle \frac{\partial p^2}{\partial r}}\right)+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial \varphi }}\left(h^3{\displaystyle \frac{\partial p^2}{\partial \varphi }}\right)=12\mu {\displaystyle \frac{\partial }{\partial r}}\left[rhp(v_{r1}+v_{r2})\right]+{\displaystyle \frac{12\mu }{r}}{\displaystyle \frac{\partial }{\partial \varphi }}\left[hp(v_{\varphi 1}+v_{\varphi 2})\right]$$

(4)

where r is the radial coordinate; ϕ is the circumferential coordinate, h is the gas film thickness, μ is the dynamic viscosity of the gas, v_r is the velocity component, and v_ϕ is the circumferential velocity component.

Under high-speed rotational conditions, the flow inside the gas chamber may be influenced by turbulence and related effects. Viscous shear at the rotor surface drives the gas film in the direction of rotation, leading to pronounced pressure accumulation in the downstream region and consequently causing load direction deflection and loading error. Under these conditions, the flow mechanism is no longer governed solely by the static gas supply, but is jointly affected by high-speed rotational shear, local pressure gradients, and inertial effects near the step and within the gas chamber. The transport equation for the turbulent kinetic energy in the turbulence model can be written as follows:

$${\displaystyle \frac{\partial (\rho k)}{\partial t}}+{\displaystyle \frac{\partial (\rho u_{j}k)}{\partial x_j}}=P_k-{\beta }^{*}\rho k\omega +{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\sigma }_k{\mu }_t\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]$$

(5)

The transport equation for the specific dissipation rate can be expressed as

$${\displaystyle \frac{\partial (\rho \omega )}{\partial t}}+{\displaystyle \frac{\partial (\rho u_j\omega )}{\partial x_j}}={\alpha }_1{\displaystyle \frac{\omega }{k}}{P}_k-{\beta }_1\rho {\omega }^2+{\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\sigma }_{\omega }{\mu }_t\right){\displaystyle \frac{\partial \omega }{\partial x_j}}\right]+D_{\omega }$$

(6)

where k is the turbulent kinetic energy, u_j is the velocity component, μ_t is the turbulent eddy viscosity, P_k is the turbulent kinetic energy production term, ω is the specific dissipation rate, and D_ω is the cross-diffusion term, while the remaining parameters are model constants.

After the gas film pressure is obtained, the forces exerted by the partial-arc aerostatic radial bearing in the x- and y-directions can be expressed as

$$\left\{\begin{array}{l}{F}_x={\displaystyle {\int }_0^L{\displaystyle {\int }_{-\pi /2}^{\pi /2}p}}R\cos \varphi d\varphi dz\\ F_y={\displaystyle {\int }_0^L{\displaystyle {\int }_{-\pi /2}^{\pi /2}p}}R\sin \varphi d\varphi dz\end{array}\right.$$

(7)

where F_x and F_y are the loading force components in the x- and y-directions, respectively; L is the bearing length; and R is the bearing radius.

The total load capacity is given by

$$F=\sqrt{F_x^2+F_y^2}$$

(8)

The attitude angle is given by

$$\theta =\arctan ({\displaystyle \frac{F_y}{F_x}})$$

(9)

The partial-arc aerostatic radial bearing satisfies the continuity equation; that is, the inflow rate into the bearing equals the outflow rate. For the adopted annular shallow-chamber configuration of the partial-arc aerostatic radial bearing, the mass flow rate can be expressed as

$$Q={\displaystyle \frac{\Omega R^2}{R_{g}T}}{\displaystyle {\int }_0^{2\pi }\left(\frac{p{h}^3}{2\Lambda }\frac{\partial p}{\partial z}\right)}d\varphi$$

(10)

where Λ is the bearing number, Ω is the angular velocity, T is the absolute temperature, and R_g is the gas constant.

Under high-speed rotational conditions, viscous shear traction generated at the spindle surface drives the gas film to rotate, causing the gas flow to accumulate downstream of the shallow chamber and resulting in a pronounced friction force. This phenomenon is closely related to hydrodynamic effects, and the formation mechanisms of the gas film pressure distribution and friction force can be further analyzed using lubrication theory. By integrating the surface shear stress over the effective load-carrying region of the pad, the friction force generated by the partial-arc aerostatic radial bearing can be expressed as follows:

$$F_f=R^2{\displaystyle {\int }_{-\frac{L}{2}}^{\frac{L}{2}}{\displaystyle {\int }_0^{2\pi }\left(\frac{h}{2}\frac{\partial p}{\partial \varphi }+\frac{\Lambda }{6}\frac{1}{h}\right)d\varphi dz}}$$

(11)

2.3. Computational Fluid Dynamics Model

A parametric three-dimensional CFD model of the partial-arc aerostatic radial bearing was established. The model mainly consists of three components: the gas film, the gas chamber, and the throttling orifices. The partial-arc aerostatic radial bearing adopts a symmetric single-chamber configuration with four throttling orifices. This design not only simplifies the overall manufacturing process but also provides sufficient loading force. The bearing model is shown in Figure 3.

To accurately describe the inflow and outflow behavior of the fluid and its interactions with the walls, the boundary conditions were properly defined before the model was imported into Fluent. This procedure provided the constraints required by the governing equations and ensured a unique solution. In DesignModeler, the supply orifice inlet was defined as the inlet boundary, the gas discharge surfaces surrounding the gas film were defined as outlet boundaries, the interface between the gas film and the loaded shaft was defined as the outwall, and the interface between the gas film and the inner surface of the gas bearing was defined as the inwall. Moreover, to achieve refined meshing in specific regions for the CFD simulations, a block-wise mesh number allocation scheme, as illustrated in Figure 4, was adopted. By adjusting the mesh density in each region, a finer and more suitable discretization was achieved in critical areas.

In the present study, numerical computations for the partial-arc aerostatic radial bearing were performed using the SIMPLEC algorithm for pressure–velocity coupling. Based on the classical SIMPLE scheme, SIMPLEC reduces the imbalance introduced by pressure correction into the velocity correction by optimizing the weighting factor in the pressure correction formulation. Compared with the PISO and Coupled schemes, SIMPLEC generally provides higher computational efficiency and more robust convergence for steady flow fields with relatively regular geometries and well-resolved meshes.

In the static parameter analysis stage, this study mainly investigated the effects of supply pressure and structural parameters on the loading performance of the partial-arc aerostatic radial bearing, without considering the effect of spindle rotation. Under these conditions, the gas film flow was primarily driven by the supply pressure difference. Based on the estimated characteristic parameters in the main gas-film region under the initial operating conditions, the Reynolds number was approximately 640, indicating that the flow remained within the laminar regime. Therefore, the laminar model was adopted in the static analysis to describe the gas film flow field.

To balance numerical accuracy and computational cost, a mesh independence study was conducted to ensure that the simulation results were independent of the mesh resolution. The radial loading force and tangential force were selected as the key monitoring metrics. A stepwise mesh-refinement strategy was applied to the partial-arc radial bearing by varying the mesh densities of the gas film and gas chamber, and the variation in load capacity with the total number of mesh elements was obtained. The results are summarized in

Table 2. Mesh quality summary for the partial-arc aerostatic radial bearing.

Number of Mesh Elements	Radial Loading Force (N)	Tangential Force (N)	Computational Time (h)
29,743	153.3247	12.12987	0.3
67,012	157.3247	12.15987	0.4
108,163	164.6193	12.25323	0.5
168,229	166.1791	12.51858	0.7
287,123	166.3637	12.74007	1

.

The results indicate that as the number of mesh elements increases from 29,743 to 287,123, both the radial loading force and the tangential force initially increase and then gradually approach stable values. Taking the results obtained with 287,123 elements as the high-accuracy reference, when the mesh number reaches 168,229, the relative error of the radial loading force decreases to 0.1%, while that of the tangential force is 1.73%.

Considering the trade-off between computational efficiency and accuracy, although the 287,123-element scheme provides the highest accuracy, its computational time (1.0 h) is relatively long. By contrast, the 168,229-element scheme maintains very low relative errors while reducing the computational time by approximately 30%. Therefore, 168,229 elements were ultimately selected as the baseline discretization for subsequent high-speed simulations so as to capture the flow field details under ultra-high-speed conditions while achieving an appropriate balance between computational cost and accuracy.

3. Results and Discussion

3.1. Static Performance Results and Parameter Selection

3.1.1. Shallow-Chamber Parameter Effects and Selection

To evaluate the radial support characteristics of the non-contact loader, this section focuses on the effects of the shallow-chamber width, shallow-chamber wrap angle, and shallow-chamber depth on the gas film performance. In the numerical simulations, the dimensionless eccentricity ratio was adjusted to characterize the spatial offset of the spindle rotor during loading, thereby reproducing the force state under practical operating conditions.

Figure 5 illustrates the influence of variations in shallow-chamber width on the static loading performance of the radial bearing. As shown in Figure 5a,b, at a fixed eccentricity ratio, both the radial loading force and the tangential force increase monotonically with increasing shallow-chamber width. This behavior can be attributed to the expansion of the effective high-pressure region within the gas film as the chamber width increases, which strengthens the fluid wedge effect induced by the converging clearance. The simulation results indicate that when the shallow-chamber width is 20 mm and the eccentricity ratio is 0.7, the radial loading force reaches a peak value of 147 N, while the tangential force is 0.38 N. In addition, the eccentricity ratio exerts a pronounced positive influence on the load-carrying performance: at higher eccentricity ratios, the gas film clearance is reduced more significantly, generating a larger pressure gradient and thereby providing sufficient loading capacity.

Further examination of Figure 5c,d shows that the attitude angle and mass flow rate increase simultaneously as the shallow-chamber width increases. Notably, the eccentricity ratio has a particularly strong regulating effect on the flow rate, which increases as the eccentricity ratio decreases. This is because, at lower eccentricity ratios, the gas film clearance becomes more uniform, reducing the local flow resistance and enhancing the flow field connectivity and mobility. To maintain high load capacity while effectively suppressing system power consumption and attitude angle deviation caused by excessive flow, a comprehensive evaluation was conducted, and the shallow-chamber width was ultimately set to 14 mm. This value maintains a stable gas-flow condition while satisfying the loading force requirement.

With the other structural parameters held constant, Figure 6 illustrates the influence of the shallow-chamber wrap angle on the loading characteristics of the bearing at different eccentricity ratios. As shown in Figure 6a, the radial loading force varies nonlinearly with increasing wrap angle, first increasing and then decreasing. When the wrap angle is below 70°, increasing the wrap angle effectively enlarges the region in which the gas film pressure accumulates, thereby improving the radial load capacity. However, when the wrap angle exceeds 70°, excessive circumferential spreading of the high-pressure region weakens the radial resultant component, and the loading force correspondingly decreases. As indicated in Figure 6b,c, the tangential force and attitude angle exhibit the same increase-then-decrease trend with respect to the wrap angle. When the wrap angle is 50°, the tangential force reaches its maximum, suggesting that the asymmetric pressure distribution in the gas film is most pronounced at this angle and leads to an evident circumferential offset of the spindle rotor.

As shown in Figure 6d, the mass flow rate increases monotonically with increasing shallow-chamber wrap angle, and the flow loss becomes more pronounced at higher eccentricity ratios. This is because a larger wrap angle directly increases the effective flow area of the high-pressure region, reduces local flow resistance, and thus, increases the leakage rate. Overall, when the shallow-chamber wrap angle is 70°, the bearing achieves the highest radial load capacity while maintaining relatively small tangential disturbance and attitude angle, thereby providing the most favorable overall support performance.

With the shallow-chamber wrap angle and width determined, Figure 7 illustrates the influence of shallow-chamber depth on the bearing loading performance at different eccentricity ratios. As shown in Figure 7a,b, both the radial loading force and the tangential force increase with increasing shallow-chamber depth. When the shallow-chamber depth is 70 μm and the eccentricity ratio is 0.1, the radial loading force reaches a maximum value of 151 N. This indicates that an increase in shallow-chamber depth is beneficial for improving the pressure distribution within the gas film, thereby enhancing the hydrostatic support effect.

As shown in Figure 7c,d, both the attitude angle and the flow rate increase as the shallow-chamber depth increases. This is because increasing the depth reduces the local flow resistance in the shallow-chamber region, thereby accelerating the leakage flow. Although a larger depth can significantly enhance the loading capacity, excessive gas consumption increases the burden on the supply system and may weaken the overall dynamic stability of the system. Considering the trade-off between the load capacity and the gas consumption, a shallow-chamber depth of 40 μm was selected as the final structural parameter in order to reduce the flow loss and improve the overall system efficiency while ensuring a sufficient loading force.

3.1.2. Throttling Orifice Parameter Effects and Selection

Building on the optimization of the shallow-chamber structural parameters, this section further investigates the effects of throttling orifice parameters on the loading characteristics of the partial-arc aerostatic radial bearing. As the key structural factor governing gas flow distribution and the gas film pressure field, the geometric size and spatial distribution of the throttling orifices directly determine the bearing load capacity, stiffness characteristics, and static gas consumption. This subsection focuses on the influences of throttling orifice diameter, orifice spacing, and supply pressure on system performance, aiming to provide theoretical support for optimizing the throttling configuration.

The influence of the throttling orifice diameter on the bearing loading performance at different eccentricity ratios was evaluated, and the results are shown in Figure 8. As shown in Figure 8a, at a fixed eccentricity ratio, the radial loading force increases with increasing throttling orifice diameter. When the throttling orifice diameter is 1.6 mm, the radial loading force reaches a maximum of 182.1 N. This indicates that a moderate increase in orifice diameter can effectively reduce the local throttling losses as the gas passes through the orifice and improve the pressure recovery within the gas film, thereby enhancing the static support capacity of the bearing. As shown in Figure 8b,c, the tangential force increases with increasing throttling orifice diameter, whereas the attitude angle reaches its minimum when the throttling orifice diameter is 0.6 mm. This suggests that at this diameter, the circumferential pressure distribution in the gas film is closest to being symmetric, thereby effectively suppressing the circumferential offset of the spindle rotor. As shown in Figure 8d, the mass flow rate increases with increasing throttling orifice diameter. Notably, when the throttling orifice diameter exceeds 0.8 mm, the slope of the flow rate increase becomes significantly steeper. This is because the effective flow area scales with the square of the diameter, causing the throughput of high-pressure gas through the throttling orifice to increase rapidly at larger diameters, which substantially increases the gas consumption.

A larger orifice diameter increases the gas film pressure by admitting more gas, but also markedly raises the gas consumption. A smaller diameter can suppress the tangential disturbance, but at the expense of the load capacity, and it also substantially increases the difficulty of micro-orifice machining. Considering the loading performance, flow characteristics, and manufacturing feasibility in combination, a throttling orifice diameter of 1.0 mm was selected to achieve an appropriate balance between the overall bearing performance and engineering practicality.

Figure 9 shows the influence of throttling orifice spacing on bearing loading performance. As shown in Figure 9a, the radial loading force increases with increasing orifice spacing, although the increment becomes less pronounced when the spacing exceeds 5 mm. At an eccentricity ratio of 0.7, the variation in radial loading force between different spacings is approximately 10 N. As shown in Figure 9b,c, both the tangential force and the attitude angle increase simultaneously with increasing spacing, suggesting that a larger orifice spacing aggravates the circumferential non-uniformity of the gas film pressure distribution. As shown in Figure 9d, the mass flow rate gradually increases as the orifice spacing increases and is only weakly affected by variations in the eccentricity ratio. This is because increasing the orifice spacing expands the coverage of the high-pressure gas and reduces the interference resistance in the inter-orifice flow field, thereby increasing the gas loss. Considering that excessive spacing increases the gas consumption and tangential disturbance, an orifice spacing of 5 mm was selected to maintain sufficient loading force while improving the pneumatic efficiency.

Based on the investigation of throttling orifice parameters, the influence of supply pressure on the bearing loading performance was further analyzed, as shown in Figure 10. As shown in Figure 10a–c, both the radial loading force and the tangential force increase approximately linearly with increasing supply pressure, and the radial loading force reaches a maximum of 187.3 N at a supply pressure of 0.8 MPa. This is because a higher supply pressure directly increases the absolute pressure level within the gas film and the hydrostatic stiffness, thereby significantly enhancing the static support capacity of the bearing. Meanwhile, as shown in Figure 10d, the mass flow rate increases steadily with increasing supply pressure, which can be attributed to the increased pressure drop across the throttling orifice, leading to simultaneous increases in the velocity and density of the gas entering the clearance. However, an excessively high supply pressure results in excessive gas consumption and induces a larger tangential force. Considering loading efficiency, operating economy, and the capacity constraints of the gas supply equipment, a supply pressure of 0.7 MPa was ultimately selected as the standard value to ensure good experimental reliability while meeting the high-load requirement.

3.1.3. Gas Film Thickness Effects and Parameter Selection

The influence of gas film thickness on the loading performance of the partial-arc aerostatic radial bearing is shown in Figure 11. As shown in Figure 11a, the radial loading force decreases markedly with increasing gas film thickness and reaches its maximum when the gas film thickness is 10 μm. A thinner film clearance increases the flow resistance of the gas and strengthens the hydrostatic pressure-accumulation effect, thereby producing a higher supporting pressure. As shown in Figure 11b,c, the tangential force and the attitude angle exhibit a fluctuating trend, first increasing and then decreasing, with increasing gas film thickness. However, because the overall variation amplitude is small, the influence of gas film thickness on the tangential characteristics can be regarded as negligible. In addition, Figure 11d indicates that the mass flow rate increases with increasing gas film thickness because the enlarged film clearance increases the cross-sectional area of the flow passage and reduces flow resistance. To ensure adequate radial support capacity and loading efficiency, a gas film thickness of 10 μm was ultimately selected as the design value.

3.2. Hydrodynamic Effects on Bearing Characteristics

3.2.1. Hydrodynamic Effects on Partial-Arc Bearing Loading

Based on the preceding static performance analysis, the dynamic influence of motorized spindle rotational speed on the loading characteristics of the partial-arc aerostatic radial bearing was further investigated. Under relatively high-rotational-speed conditions, not only the main gas-film region but also the local flow states in the gas chamber and the step-adjacent regions change significantly. According to the estimated local Reynolds numbers, although the Reynolds number in the main gas-film region remains within the laminar regime, the local Reynolds numbers in the gas chamber and step-adjacent regions increase significantly, and turbulence may occur under high-speed conditions. To more reasonably describe these local flow features and their influence on pressure distribution and loading performance, the SST k-ω turbulence model was adopted in the subsequent numerical calculations of the hydrodynamic effect.

In addition, under high-speed rotational conditions, viscous shear traction at the spindle surface drives the gas film to rotate, causing the gas flow to accumulate downstream of the shallow chamber and generating a pronounced friction torque. This torque may induce a slight tilt of the rear aerostatic guideway, thereby introducing a deviation in the direction of the applied load. Therefore, it is necessary to further examine the evolution of the pressure field induced by rotational speed.

As shown in Figure 12a–c, rotational speed has a significant influence on the gas film pressure distribution. As the rotational speed increases, the gas film pressure in the upstream shallow-chamber region gradually decreases, whereas the pressure in the downstream region increases significantly. This behavior is mainly attributed to the intensified hydrodynamic effects under ultra-high-speed operating conditions, which break the symmetry of the gas-film pressure distribution and drive high-pressure gas to accumulate in the direction of rotation.

To accurately evaluate the influence of rotational speed on the system stability, the gas film loading force was resolved into vector components, with particular emphasis placed on the evolution of tangential force and attitude angle with rotational speed. Figure 13 presents the quantitative relationships between the loading characteristics and rotational speed at a gas film thickness of 10 μm under different eccentricity ratios. As shown in Figure 13a, the radial loading force exhibits a clear negative correlation with the rotational speed. At a fixed eccentricity ratio, the radial loading force decreases as the rotational speed increases, with a maximum reduction of approximately 7.5%. Meanwhile, Figure 13b,c show the evolution of tangential force and attitude angle: both increase nonlinearly with increasing rotational speed, and this increase becomes more pronounced under high-eccentricity conditions. When the system operates at an ultra-high speed of 200 krpm with an eccentricity ratio of 0.7, the tangential disturbance force increases to 50 N, resulting in an attitude angle deviation of 15.57°. In addition, as shown in Figure 13d, the bearing friction torque increases concurrently with rotational speed and reaches 0.004 N·m under the highest operating condition.

These observations indicate that the asymmetric reconstruction of the pressure field induced by high-speed rotation leads to an imbalanced internal load distribution, causing the line of action of the bearing loading-force vector to deviate from the intended horizontal axis. Such circumferential deflection of the force vector introduces geometric misalignment between the loading direction and the measurement axis of the horizontal displacement sensor, thereby causing a substantial measurement error at the principal level and significantly reducing the accuracy of operational stiffness identification for the motorized spindle.

3.2.2. Hydrodynamic Suppression in Partial-Arc Bearings

In the preceding hydrostatic analysis, a symmetric structural design was adopted and was shown to satisfy the loading requirements under low-speed conditions. However, under high-speed operating conditions, as the spindle rotational speed increases, hydrodynamic effects in the gas film significantly influence the stability of the loading force, thereby introducing errors into the measurements of loading force and stiffness. Specifically, with increasing rotational speed, the pressure distribution in the gas film gradually changes from a symmetric pattern to an asymmetric one. The centrifugal and shear effects associated with high-speed rotation intensify the accumulation of hydrodynamic effects within the gas film, causing pressure to concentrate non-uniformly in the downstream region of the bearing. This, in turn, leads to a shift in the loading force and an increase in the testing error. These observations indicate that relying solely on a conventional symmetric shallow-chamber design is insufficient to suppress the adverse influence of hydrodynamic effects on the gas film pressure distribution.

To address this issue, although adjusting the shallow-chamber depth and throttling orifice configuration can improve the hydrostatic pressure distribution to some extent, the pressure gradient within the gas film still increases markedly under high-speed conditions, and the conventional symmetric design remains unable to eliminate the pressure non-uniformity. Therefore, a more flexible and effective asymmetric design in terms of the chamber depth and wrap angle is required to cope with the imbalance in gas film pressure distribution at high rotational speeds.

Accordingly, an asymmetric composite shallow-deep chamber configuration was proposed in this study. By adjusting the wrap angle and depth of the upstream shallow chamber, together with those of the downstream shallow chamber, the symmetry of the gas film flow field under the conventional design was intentionally broken to achieve a more uniform pressure distribution. Specifically, modifying the upstream shallow-chamber wrap angle and depth can effectively mitigate the centrifugal influence induced by high-speed rotation and prevent asymmetric pressure accumulation, while adjusting the downstream shallow-chamber depth further balances the pressure distribution within the gas film and suppresses the loading force offset caused by pressure non-uniformity. With this structural design, the gas film pressure distribution becomes more balanced, the stability of the loading force is improved, and the stiffness-testing accuracy of the system is significantly enhanced.

Because hydrodynamic effects in this study lead to pressure accumulation in the downstream shallow chamber of the bearing, the proposed structural design increases the depth of the downstream shallow chamber to accommodate the additional pressure while maintaining a shallower upstream chamber depth. This configuration helps balance the pressure distribution on both sides and reduce the loading force offset induced by hydrodynamic effects. As shown in Figure 14a, increasing the upstream shallow-chamber depth leads to a gradual increase in the radial loading force by approximately 15.24%, while the tangential force decreases by about 3.27%. As shown in Figure 14b, increasing the downstream shallow-chamber depth causes the radial loading force to decrease by approximately 5.91%, while the tangential force decreases by about 24.89%. Therefore, increasing the downstream chamber depth is beneficial for reducing the tangential force; however, the associated decline in radial loading force must be simultaneously suppressed.

To further reduce the influence of hydrodynamic effects, an asymmetric chamber configuration was introduced on the basis of the shallow-deep chamber design to achieve an integrated bearing design. A comparative study of the pressure contours for the partial-arc bearing with an asymmetric chamber configuration is presented in Figure 15a,b. The results show that the asymmetric chamber design significantly improves the uniformity of the pressure distribution within the gas film, alleviates the adverse impact of hydrodynamic effects on bearing performance, and thereby enhances the stability of the loading force and the accuracy of stiffness testing.

By combining the shallow-deep chamber concept with an asymmetric configuration, an asymmetric composite shallow-deep chamber structure was obtained, as schematically shown in Figure 16. As shown in Figure 17a,b, both the radial loading force and the tangential disturbance force decrease monotonically with increasing downstream shallow-chamber depth. When the chamber depth exceeds 100 μm, the flow field evolution exhibits a clear convergence trend, and the radial loading force and tangential force stabilize at 220.97 N and 12.37 N, respectively. Under the condition of an upstream shallow-chamber depth of 60 μm and a downstream shallow-chamber depth of 100 μm, compared with the shallow-deep chamber design, the radial loading force decreases by 15.13%, whereas the tangential force is reduced by 84.3%, and the attitude angle decreases to 3.21°, effectively suppressing the loading force offset induced by hydrodynamic effects.

Through CFD simulations and structural optimization, the preceding sections systematically analyzed the performance of the partial-arc aerostatic radial bearing under different operating conditions, with emphasis placed on how key design parameters—namely, shallow-chamber width, wrap angle, chamber depth, and throttling orifice diameter—affect the loading performance and stiffness-testing accuracy of the bearing. By appropriately optimizing these parameters, the loading performance and stability of the bearing were significantly improved. On this basis, a recommended set of structural parameters oriented toward loading stability and testing accuracy was obtained. With this parameter set, the bearing exhibits clear improvements in gas film pressure distribution, loading force stability, and testing accuracy, and the corresponding structural parameters are listed in

Table 3. Structural parameters of the partial-arc aerostatic radial bearing after structural adjustment.

Parameter	Value
Diameter of the partial-arc aerostatic radial bearing dk (mm)	25
Number of throttling orifices (N)	4
Throttling orifice diameter df (mm)	1
Orifice spacing Lm (mm)	5
Shallow-chamber width Ld (mm)	14
Upstream shallow-chamber wrap angle β1 (°)	20
Downstream shallow-chamber wrap angle β2 (°)	50
Upstream shallow-chamber depth hq1 (μm)	60
Downstream shallow-chamber depth hq2 (μm)	100
Gas film thickness hk (μm)	10
Supply pressure Ps (MPa)	0.7

.

A further comparison between the improved design and the baseline design is provided in

Table 4. Performance comparison of the radial bearing before and after structural adjustment.

Bearing Loading Characteristics	Before Optimization	After Optimization
Radial loading force Fx (N)	180.79	220.97
Tangential force Fy (N)	50.36	12.37
Attitude angle θ (°)	15.57	3.20
Friction torque Mq (N·m)	0.004	0.011

. The radial loading force increases by 22.2%, while the tangential force and attitude angle decrease by 75.44% and 79.41%, respectively, indicating a substantial enhancement in the overall bearing performance. A comparison of the pressure contours is shown in Figure 18a,b. In the baseline design, the pressure distribution is biased toward the downstream shallow-chamber region. After the structural parameter adjustment, the peak pressure region shifts back toward the center, and the overall pressure distribution becomes more uniform. It should be noted that the present study mainly focuses on the loading performance and directional stability of the proposed device, while the transient stability issues that may be associated with the deep-chamber structure have not been further investigated.

3.3. Experiments and Analysis

An experimental platform for loading force and stiffness measurements was established using the improved radial bearing and a five-degree-of-freedom loading device, as shown in Figure 19a,b. The platform consists of a motorized spindle system, the loading device, displacement sensors, pressure sensors, and related instrumentation. By measuring spindle stiffness and attitude angle at different rotational speeds, the feasibility of the optimized design of the partial-arc aerostatic radial bearing was validated. Owing to safety constraints of the test rig and limitations of the available experimental conditions, the rotational speed during the experimental stage was restricted to 0–10 krpm. This is because higher rotational speeds impose more stringent requirements on spindle dynamic balancing, lubrication and thermal management, rig protection, and system vibration resistance. Therefore, experimental validation of the loading capacity and directional stability of the improved structure was conducted only within the safe speed range of 0–10 krpm. However, for the hydrodynamic effects under ultra-high-speed conditions and their corresponding suppression, the related conclusions are still based mainly on numerical simulations, and direct experimental validation has not yet been achieved.

The experimental procedure was as follows. An available laboratory motorized spindle was used in the tests, with all applied loads limited to below 50 N because of its limited rigidity. To ensure stable operation of the motorized spindle at high rotational speed, the oil–air lubrication system was activated, and the spindle was operated under no-load conditions for 20 min to ensure adequate lubrication of the internal bearings. Meanwhile, the displacement sensors were adjusted such that the horizontal displacement sensor was approximately aligned with the loading handwheel. The vertical displacement sensor was then adjusted to be perpendicular to the horizontal sensor. By checking the sensor position information on the CPL-190, it was confirmed that the sensor was located at the mid-stroke position, after which the locking screws were tightened to complete fixation. The pressure sensor was mounted on the guideway and connected to the host computer, and the test software was configured to ensure that all data channels were displayed properly.

After commissioning, the loading handwheel was slowly rotated to bring the partial-arc aerostatic radial bearing close to the spindle’s dummy test shaft. The supply valve was then opened, and a 30 μm feeler gauge was inserted into the bearing clearance. By adjusting the height fine-tuning knob, a uniform gas-film distribution between the bearing and the dummy shaft was established. The motorized spindle was subsequently started, and the loading handwheel was rotated to gradually increase the loading force. During this process, the loading force and displacement data were recorded using the software. After processing the recorded data, the test results were obtained, as shown in Figure 20 and Figure 21.

As shown in Figure 20, under operating conditions of 0, 3, 8, and 10 krpm, the displacement of the motorized spindle maintains an approximately linear relationship with the radial loading force, and the fitted lines at different rotational speeds are generally consistent. No evident slope drift, curve bending, or abrupt increase in scatter is observed as the rotational speed increases. According to the preceding CFD results, hydrodynamic effects are strongly speed-dependent: high-speed shear drives the gas-film pressure field from an approximately symmetric distribution toward downstream accumulation, inducing increases in tangential force and attitude angle, thereby causing deflection of the load vector and measurement bias in its radial component. When hydrodynamic effects become stronger, the relationship typically exhibits more pronounced speed dependence, such as discernible changes in slope or intercept with rotational speed. By contrast, after adoption of the asymmetric composite shallow-deep chamber configuration, Figure 20 remains approximately linear even at 10 krpm, indicating that the sensitivity of the loading output to rotational speed is reduced and that the influence of load direction deflection on the measured radial component remains relatively small.

Furthermore, Figure 21 presents the variation in the attitude angle with rotational speed from three repeated experiments. The attitude angle remains at approximately 4.5° within the range of 0–5 krpm and increases to approximately 6.6–7.0° when the speed rises to 8–10 krpm. The three experimental curves show a high degree of agreement, indicating good repeatability and controllability of the attitude angle variation. These results are consistent with the trends predicted by the simulations and confirm that the improved structure can suppress, to a certain extent, the influence of hydrodynamic effects on loading stability and directional consistency. This enables more stable and repeatable non-contact loading at higher rotational speeds and provides more reliable loading conditions for subsequent operational stiffness identification.

4. Conclusions

This study investigates a non-contact follower-type loading method based on partial-arc aerostatic radial bearings to overcome the limitations of conventional contact loading, such as frictional heating and nonlinear disturbances, in high-speed motorized spindle stiffness testing. The research focuses on structural optimization for suppressing hydrodynamic effects under ultra-high-speed operating conditions.

(1)

Parametric modeling and performance evaluation: A parametric CFD model of the partial-arc aerostatic radial bearing was established to analyze key performance metrics, including radial loading force, tangential force, attitude angle, and gas flow rate. The model revealed the fundamental mechanism by which hydrodynamic effects induce loading force vector deflection and provided the basis for subsequent structural optimization.

(2)

Static parameter optimization and sensitivity analysis: Static parametric analysis shows that gas film thickness, throttling orifice diameter, supply pressure, and chamber geometric parameters all significantly affect the loading performance of the partial-arc aerostatic radial bearing, among which gas film thickness is the dominant factor. By comprehensively considering load capacity, gas consumption, and manufacturing feasibility, an optimal combination of structural parameters was determined, providing a basis for subsequent performance analysis and structural improvement under high-speed operating conditions.

(3)

Hydrodynamic suppression via asymmetric design: Numerical simulation results under high-speed operating conditions show that as the rotational speed increases, the pressure distribution inside the partial-arc aerostatic radial bearing becomes increasingly asymmetric, the loading force direction deviates, and the wedge effect is intensified, all of which adversely affect loading direction stability. To address these issues, an asymmetric composite shallow–deep chamber structure was proposed. This structure improves the gas film pressure distribution under high-speed conditions, reduces the deviation of the loading force, and suppresses the hydrodynamic effects induced by high-speed rotation, thereby providing a basis for further structural improvement.

(4)

Experimental validation and stability assessment: Experiments conducted in the range of 0–10 krpm show that under the optimized loading structure, the radial loading force maintains a good linear relationship with displacement, while the attitude angle varies within a relatively small range and exhibits good repeatability. The experimental results indicate that the proposed asymmetric composite shallow–deep chamber structure can effectively maintain loading direction stability within the achievable speed range and provide relatively stable non-contact loading conditions for the operational stiffness evaluation of motorized spindles. It should be noted that due to experimental limitations, direct validation in the ultra-high-speed range was not carried out in this study, and the corresponding simulation-based conclusions still require further experimental verification.