Research on the Support Performance of Internal Feedback Hydrostatic Thrust and Journal Bearing Considering Load Effect

Abstract

This study aims to analyze the impact of uniform and eccentric load conditions on the performance of internal feedback hydrostatic thrust and journal bearing. Two distinct models are established: a three-degrees-of-freedom uniform load model and a five-degrees-of-freedom eccentric load model. The support stiffness, overturning stiffness, and flow rate for both thrust and journal bearings are calculated. Additionally, numerical analysis is conducted to examine the influence of oil film thickness, inlet pressure, and restrictor size on the operational characteristics of the bearings, revealing the interplay between an eccentric load and journal bearing speed. The validity of the theoretical algorithm is verified through finite element simulation. The research outcomes hold significant guiding implications for the design and application of internal feedback hydrostatic bearings.

1. Introduction

Machine tools are evolving towards higher precision, thus requiring components with higher precision [1]. Hydrostatic bearings find extensive application in the turntables of high-precision machine tools [2,3,4]; within this domain, internal feedback hydrostatic bearings employ a constant pressure oil supply method, encompassing axial thrust bearings and journal bearings. These bearings exhibit notable advantages, characterized by a more compact structure, commendable stability, and heightened stiffness [5]. During actual machining operations, the turntable frequently operates under eccentric load conditions, owing to factors, such as workpiece shape, installation position, and cutting force. This variability results in fluctuations in the load capacity of the oil pad across different positions. Consequently, the assessment of the support performance of the hydrostatic turntable under diverse working conditions assumes paramount importance.

The assessment of the load capacity performance of hydrostatic bearings has consistently been a focal point of research [6]. Wang et al. [7] adopted a new method to determine the oil chamber pressure of sliding bearings. Wei et al. [8] proposed a method to improve the efficiency of solving the Reynolds equation for hydrostatic bearings. Gohara et al. [9] investigated the static characteristics of hydrostatic thrust bearings lubricated with water, employing a membrane restrictor at elevated speeds to enhance stiffness. Liang et al. [10] proposed a new analytical method for calculating the load capacity of hydrostatic sliding bearings, which has better convergence and higher calculation accuracy. Branagan et al. [11] studied the influence of jacking pocket structural parameters on the load-bearing performance. Wang et al. [12] proposed a calculation method for the maximum load capacity of a journal bearing and analyzed the impact of diverse structural parameters on work performance. Yu et al. [13] investigated the impact of the structural parameters of oil pads on the comprehensive tribological performance of multi-pad hydrostatic thrust bearings, establishing the correlation between optimal comprehensive tribological performance and these structural parameters. Zhang et al. [14] developed a mathematical model for the oil film between double rectangular oil pads and deduced the load capacity equation for the hydrostatic cavity. Tripkwitz et al. [15,16] carried out both theoretical and experimental investigations into the dynamic stiffness of groove hydrostatic thrust bearings. Yi et al. [17] performed comprehensive experiments on the load capacity of hydrostatic bearings, revealing that the choice of lubricant, along with its temperature and oil pressure, significantly influences bearing performance. Du et al. [18] calculated and compared the performance of hydrostatic bearings under four lubrication conditions and proposed different improved models for different operating conditions. Khakse et al. [19] conducted performance analysis on a non-concave hydrostatic/hybrid tapered sliding bearing with capillary flow holes.

The introduction of a load adds complexity to the oil film support characteristics of the static pressure turntable. Wang et al. [20] derived analytical formulas for calculating the oil film bearing capacity, bending moment, and stiffness of a constant flow static pressure turntable under eccentric load conditions. Hu et al. [21,22] applied a fluid solid thermal coupling model to study the influence of working parameters on the performance characteristics of a static pressure turntable. Liu et al. [23,24] computed the load capacity of the oil film following the tilt of the turntable and developed a method for ascertaining the maximum rotational speed of the tilted turntable. By employing a combination of simulation and experimentation, Yu et al. [25,26] investigated the impact of multi-factor coupling on the lubrication performance of the oil film and established a correlation between the speed and load. Zhang et al. [27] examined the tribological characteristics of the lubricating film on a hydrostatic turntable under a constant cutting speed, revealing a notable influence of the load on the performance of the oil film. Yu et al. [28] scrutinized the anti-eccentric load characteristics of double rectangular cavity hydrostatic bearings, revealing that the overall lubrication performance under eccentric load was inferior to that under uniform load. Wang et al. [29] studied the nonlinear support performance of a hydrostatic ram using a rectangular oil pad under cutting force and optimized its stiffness. Zhang et al. [30,31] accounted for the tilt of the turntable and formulated a five-degrees-of-freedom approximate accuracy prediction model. Zha et al. [32,33] studied the rotational and angular errors of a turntable under various load conditions and substantiated the efficacy of this approach through experimental testing.

The subject of this study is the internal feedback hydrostatic bearing, characterized by the interconnection of its upside and downside pads. A three-degrees-of-freedom uniform load model and a five-degrees-of-freedom eccentric load model are established, and the effects of different parameters on the performance of uniform load and eccentric load are discussed.

2. Theory

2.1. Mechanical Structure and Model

The configuration of the internal feedback hydrostatic turntable is depicted in Figure 1. Both upside and downside thrust plates of the thrust bearing feature a combined total of 12 pads. Each pair of opposing pads constitutes a set for a feedback oil supply. The journal bearing comprises six pads, organized into three groups, each group containing two pads. The internal feedback pad utilizes a constant-pressure oil supply, and the liquid resistance of the throttle varies in accordance with the oil film gap. Consequently, the internal feedback pad exhibits a robust capacity for load-bearing adjustments.

2.2. Governing Equations

The pad of the thrust bearing is illustrated in Figure 1a, and the corresponding parameter definitions are presented in

Table 1. Structural parameters of thrust pad.

Symbol	Name	Symbol	Name
R1	Inner diameter of pad	R2	Inner diameter of recess
R3	Out diameter of recess	R4	Out diameter of pad
φ2	Half the angle of the pad	φ1	Half the angle of the recess
bc	Width of the oil return edge	lc	Length of the oil inlet groove
tc	Width of oil inlet / return groove	Tc	Length of restrictor
Bc	Width of internal flow edge	Lc	Length of oil return groove

. The pressure values within the pads of the thrust bearing are denoted as p_ti, where i represents the pad number. Consequently, the oil film pressure on the surfaces of the land and restrictor can be expressed as follows [5]:

$$\frac{\partial }{\partial r}\left(r\cdot \frac{h_{ti}^3}{\eta }\frac{\partial p_{ti}}{\partial r}\right)+\frac{\partial }{\partial \phi }\left(\frac{h_{ti}^3}{\eta }\frac{\partial p_{ti}}{r\partial \phi }\right)=6\frac{\partial }{\partial r}\left(r\cdot U_{ti}\cdot h_{ti}\right)+6\frac{\partial }{\partial \phi }\left(V_{ti}\cdot h_{ti}\right)+\frac{\partial }{\partial r}\left(\frac{\rho V_{ti}^2{h}_{ti}^3}{\eta }\right)$$

(1)

where h_ti is film thickness, p_ti is the recess pressure, η is viscosity, ρ is oil density, U_ti is radial velocity, and V_ti is circumferential velocity.

The pad of the journal bearing is depicted in Figure 1b, with corresponding parameter definitions provided in

Table 2. Structural parameters of journal bearing.

Symbol	Name	Symbol	Name
Xd	Length of pad	Yd	Width of pad
xd	Length of recess	yd	Width of recess
Ld	Length of restrictor	ld	Width of oil inlet / return groove
bd	Width of the return oil edge	td	Width of the inlet oil groove
Bd	Width of internal flow edge	Rd	Radius of journal bearing

. The pressure value within the pad, both prior to and after the journal bearing, is denoted as p_ji and can be expressed as follows [10]:

$$\frac{\partial }{\partial x}\left(\frac{h_{ji}^3}{\eta }\frac{\partial p_{ji}}{\partial x}\right)+\frac{\partial }{\partial y}\left(\frac{h_{ji}^3}{\eta }\frac{\partial p_{ji}}{\partial y}\right)=6\frac{\partial }{\partial x}\left(U_{ji}\cdot h_{ji}\right)+6\frac{\partial }{\partial y}\left(V_{ji}\cdot h_{ji}\right)$$

(2)

where h_ji is film thickness, p_ji is the recess pressure, U_ji is x-direction velocity, and V_ji is y-direction velocity.

The shaft of the turntable rotates along with the upside and downside thrust plates at a speed of ω, and the boundary conditions for the speed are as follows:

$$\left\{\begin{array}{l}{U}_{ti}=0,V_{ti}=\omega r_t\\ U_{ji}=\omega \left(R_d-h_{ji}\right),V_{ji}=0\end{array}\right.$$

(3)

Consequently, r_t represents the radius of any point on the fan-shaped oil pad.

The rectangular gap restrictor comprises both an inlet and a return port, with pressure variations in distinct areas. The pressure boundary conditions for the thrust and journal pads are as follows:

$$p_{ti},p_{ji}=\left\{\begin{array}{l}{p}_{1i},{\Omega }_1\\ p_{2i},{\Omega }_2\\ p_s,{\Omega }_3\\ 0,{\Omega }_4\end{array}\right.$$

(4)

where p_1i and p_2i denote the target recess pressure and the opposing recess pressure, respectively. Additionally, Ω₁ stands for the recess area, Ω₂ for the oil return groove area, Ω₃ for the oil inlet area, and Ω₄ designates the external land.

A portion of the oil diverts to the opposing recess through the return groove, while the remainder enters the recess on the same side through a restrictor. Focusing on two pads as the subjects of investigation, R_c1 and R_c2 denote the return oil liquid resistance, R_t1 and R_t2 represent the internal flow liquid resistance, and R_o1 and R_o2 signify the liquid resistance of the pad. p₁ and p₂ denote the pressure in the recess of the two opposing pads; p_s is the inlet pressure. Considering R_t1 and R_t2 as two parallel hydraulic resistances, the equivalent oil circuit is illustrated in Figure 2a. The expression for the equivalent hydraulic resistance (R_a) is given by:

$$R_a=\frac{R_{t1}{R}_{t2}}{R_{t1}+R_{t2}}$$

(5)

We simplify the triangular oil circuit into a star-shaped oil circuit, as illustrated in Figure 2b. The expressions for each liquid resistance (R_k1, R_k2, R_k3) and the total liquid resistance (R_z) are as follows:

$$\left\{\begin{array}{l}{R}_z=\frac{\left(R_{k1}+R_{k2}\right)\left(R_{k3}+R_{k4}\right)}{R_{k1}+R_{k2}+R_{k3}+R_{k4}}+R_{k3}\\ R_{k1}=\frac{R_a{R}_{o2}}{R_a+R_{o1}+R_{o2}},R_{k2}=\frac{R_a{R}_{o1}}{R_a+R_{o1}+R_{o2}},R_{k3}=\frac{R_{o1}{R}_{o2}}{R_a+R_{o1}+R_{o2}}\end{array}\right.$$

(6)

According to the oil circuit relationship, the pressure in the opposite oil recess is:

$$\left\{\begin{array}{l}{p}_1=p_s-\frac{p_s}{R_z}\frac{R_{c1}+R_{k1}}{R_{c1}+R_{k1}+R_{c2}+R_{k2}}{R}_{c2}\\ p_2=p_s-\frac{p_s}{R_z}\frac{R_{c2}+R_{k2}}{R_{c1}+R_{k1}+R_{c2}+R_{k2}}{R}_{c1}\end{array}\right.$$

(7)

The expressions for each liquid resistance are as follows:

$$\left\{\begin{array}{l}{R}_{c1}=\frac{12{\eta }_0{b}_c}{\left(l_c+L_c\right){\left(h_0-\Delta h\right)}^3},R_{c2}=\frac{12{\eta }_0{b}_c}{\left(l_c+L_c\right){\left(h_0+\Delta h\right)}^3}\\ R_{t1}=\frac{12{\eta }_0{B}_c}{\left(L_c+T_c\right){\left(h_0-\Delta h\right)}^3},R_{t2}=\frac{12{\eta }_0{B}_c}{\left(L_c+T_c\right){\left(h_0+\Delta h\right)}^3}\\ R_{o1}=\frac{6{\eta }_0}{\left(h_0-\Delta h\right)\left(\frac{B-b}{l}+\frac{L-l}{b}\right)},R_{o2}=\frac{6{\eta }_0}{\left(h_0+\Delta h\right)\left(\frac{B-b}{l}+\frac{L-l}{b}\right)}\end{array}\right.$$

(8)

where $L=(R_4+R_1)/2\cdot {\phi }_1,B=R_4-R_1,l=(R_3+R_2)/2\cdot {\phi }_2,b=R_3-R_2$.

Given the curved surface of the journal bearing, the load capacity is in its normal direction. Consequently, the load capacities F_ti and F_ji of a single thrust and journal pad, respectively, are expressed as:

$$\left\{\begin{array}{l}{F}_{ti}={\displaystyle \underset{Pad}{\iint }{p}_{ti}rdrd\phi }\\ F_{ji}={\displaystyle \underset{Pad}{\iint }{p}_{ji}\cos \left(x/R_d\right)dxdy}\end{array}\right.$$

(9)

The flow rates Q_ti and Q_ji for a single thrust and journal pad, respectively, are expressed as:

$$\left\{\begin{array}{c}{Q}_{ti}={\displaystyle \underset{S_1}{\int }\left(\frac{h_{ti}}{12\eta }\frac{\partial p_{ti}}{\partial \phi }-\frac{U_{ti}{h}_{ti}}{2}\right)}{d}_r+{\displaystyle \underset{S_2}{\int }\frac{h_{ti}}{12\eta }}\frac{\partial p_{ti}}{\partial r}{d}_{\phi }+{\displaystyle \underset{S_3}{\int }\left(-\frac{h_{ti}}{12\eta }\frac{\partial p_{ti}}{\partial \phi }+\frac{U_{ti}{h}_{ti}}{2}\right)}{d}_r\\ \hspace{1em}\hspace{1em}+{\displaystyle \underset{S_4}{\int }-\frac{h_{ti}}{12\eta }}\frac{\partial p_{ti}}{\partial r}{d}_{\phi }+\frac{p_1}{R_{t1}+R_{o1}}\\ Q_{ji}={\displaystyle \underset{S_1}{\int }\left(\frac{h_{ji}}{12\eta }\frac{\partial p_{ji}}{\partial x}-\frac{U_{ji}{h}_{ji}}{2}\right)}{d}_r+{\displaystyle \underset{S_2}{\int }\frac{h_{ji}}{12\eta }}\frac{\partial p_{ji}}{\partial y}{d}_c+{\displaystyle \underset{S_3}{\int }\left(-\frac{h_{ji}}{12\eta }\frac{\partial p_{ji}}{\partial x}+\frac{U_{ji}{h}_i}{2}\right)}{d}_r\\ \hspace{1em}\hspace{1em}+{\displaystyle \underset{S_4}{\int }-\frac{h_{ji}}{12\eta }}\frac{\partial p_i}{\partial y}{d}_x+\frac{p_2}{R_{t2}+R_{o2}}\end{array}\right.$$

(10)

2.3. Analysis Model for Uniform Load

The load on the hydrostatic turntable arises from two components: axial load and radial cutting force. As depicted in Figure 3a, under uniform load conditions, the upside and downside thrust plates remain horizontally aligned. Upon loading the turntable, both the upside and downside thrust plates decrease simultaneously, and the film thickness of each pad changes concurrently, indicating uniform load capacity across all oil pads on the same side.

When the total weight of the thrust plate is denoted as W_t, the thrust bearing is in a complete force balance relationship.

$$W_t=6{\displaystyle \underset{Pad}{\iint }\left(p_{t1}-p_{t2}\right)rdrd\phi }$$

(11)

The oil film load capacity is directly proportional to the change in film thickness. As the change in oil film thickness approaches h_t0, the maximum load capacity, the expression for the maximum load, is as follows:

$${\left(W_t\right)}_{\max }=\underset{\Delta h\to h_{t0}}{\lim }6{\displaystyle \underset{Pad}{\iint }\left(p_{t1}-p_{t2}\right)rdrd\phi }$$

(12)

In numerical calculations, the relationship below can be established to approximate the solution for the maximum load. The following assumptions can be made regarding the variation in the maximum oil film thickness:

$${\left(\Delta h\right)}_{\max }=h_{t0}-0.01\times {10}^{-2}\text{mm}$$

(13)

For the model depicted in Figure 3, this study introduces a film thickness interval dichotomy method to quantify the change in film thickness under a specified load, as illustrated in Figure 4. Here, a and b denote the boundaries of the film thickness interval, with a = 0 and b = h_t0, respectively. The load capacity of the pad is determined by solving the Reynolds equation [5], and the judgmental equation g_k − f_k represents the relationship between the load capacity of pads and the external load, with a convergence threshold value of ε₁ = 1. For the thrust bearing, g_k and f_k are defined as:

$$\left\{\begin{array}{l}{g}_k=W_t\\ f_k=6{\displaystyle \underset{Pad}{\iint }\left(p_{t1}-p_{t2}\right)rdrd\phi }\end{array}\right.$$

(14)

The support stiffness (k_t) of the thrust bearing is the parallel stiffness of six pairs of pads and is expressed as:

$$k_t=\frac{W_t}{\Delta h}$$

(15)

In the absence of a load, when the shaft exhibits no eccentricity, the film thickness in each pad of the journal bearing remains constant, denoted as h_j0. However, when subjected to a radial external load W_j, the center of the shaft undergoes displacement, leading to a non-coincident center in the journal bearing, as depicted in Figure 3b. The oil film thickness at each position is no longer fixed but is contingent on the position, varying with the position angle. The angle of deviation from the load, denoted as θ, is considered positive in the clockwise direction and negative in the counterclockwise direction. Following the principle of symmetry, analysis of the change in one cycle is sufficient, θ ∈ [0, π/6]. The expression for the oil film thickness at any position is given by:

$$h=h_{j0}+e\cos \phi$$

(16)

where e is the eccentricity, equivalent to Δh, and φ is the position angle; the load application point is connected to the axial line and rotates counterclockwise.

In accordance with the correspondence of oil pads, pairs 1-4, 2-5, and 3-6 constitute the internal feedback pads. To streamline calculations, a simplified method utilizing the equivalent oil film thickness is introduced. The equivalent oil film thickness is defined as the volume of the oil film for a given pad divided by the corresponding area of that pad, and its expression is as follows:

$$h_{ei}=\frac{{\displaystyle \underset{S}{\int }hdxdy}}{S}$$

(17)

where S is the area of the pad.

Initially, compute the component forces along the x and y degrees of freedom based on the angle θ of the force application point. Subsequently, shift the force to positions where θ is 90° and 0°, calculate the oil film thickness at these locations, and ultimately determine the support forces along the x and y directions using Equation (9).

$$\left\{\begin{array}{l}{F}_{jy}=F_{j4}-F_{j1}+\left|F_{j5}-F_{j2}+F_{j3}-F_{j6}\right|\cos \frac{\pi }{3}\\ F_{jx}=\left|F_{j5}-F_{j2}-F_{j3}+F_{j6}\right|\sin \frac{\pi }{3}\end{array}\right.$$

(18)

For θ in the range [0, π], where the y-direction force is predominant, ascertain the maximum film thickness in the y direction using Equation (13). Subsequently, compute the maximum load capacity in the y direction and determine the maximum load by solving the subsequent equation.

$${\left(W_j\right)}_{\max }={\left(F_{jy}\right)}_{\max }/\cos \theta$$

(19)

The variation in film thickness for journal bearing under a load can be computed using the process illustrated in Figure 4. Specifically, a is set to 0, and b is set to h_j0. The force equations g_k and f_k for journal bearing in the x direction are expressed as:

$$\left\{\begin{array}{l}{g}_k=W_j\sin \theta \\ f_k=\left|F_{j5}-F_{j2}-F_{j3}+F_{j6}\right|\sin \frac{\pi }{3}\end{array}\right.$$

(20)

The force equations g_k and f_k in the y direction are expressed as:

$$\left\{\begin{array}{l}{g}_k=W_j\cos \theta \\ f_k=F_{j4}-F_{j1}+\left|F_{j5}-F_{j2}+F_{j3}-F_{j6}\right|\cos \frac{\pi }{3}\end{array}\right.$$

(21)

The changes in film thickness, denoted as Δh_x and Δh_y, are obtained by solving in the x and y directions, respectively. The expression for the journal bearing support stiffness k_j under uniform load is then given by:

$$\left\{\begin{array}{l}{k}_j=\sqrt{k_{jx}^2+k_{jy}^2}\\ k_{jx}=F_{jx}/\Delta h_x\\ k_{jy}=F_{jy}/\Delta h_y\end{array}\right.$$

(22)

2.4. Analysis Model for Eccentric Load

Under unbalanced loading, the turntable may incline due to an uneven weight distribution. Assumptions are made to simplify the calculations: (1) the influence of the inclined shaft is disregarded; (2) deformation of the upside and downside thrust plates under oil pressure is ignored, focusing solely on deflection caused by uneven support force due to hydrostatic pressure support; (3) to model extreme working conditions, small-load situations are excluded. Given the small tilt angle of the turntable, it consistently meets the criterion that the film thickness of the upside pad is less than the corresponding downside pad film thickness after a change in film thickness.

Given the aforementioned assumptions, the calculation method for the load capacity of journal bearings under an eccentric load remains unchanged. Nevertheless, owing to the impact of rotational speed, the centrifugal force induced by the eccentric load could significantly surpass the cutting force. Furthermore, the upside and downside thrust plates not only induce axial displacement but also exhibit tilting and swinging. Consequently, the analysis model for the turntable with five degrees of freedom under an eccentric load, as depicted in Figure 5, is formulated.

The total load weight on the thrust plate is denoted as W_t, wherein the weight of the eccentric load is W_z. Applying the force translation theorem, the force on the turntable can be resolved into an axial force and an overturning moment, where the overturning moment is W_z × d. Here, d represents the eccentricity distance, and δ is the angle between the eccentric load and the y-axis. In Figure 5, N₁–N₆ corresponds to the upside pad numbers, N₇–N₁₂ to the downside pad numbers, and the oil film thickness h_ti at any point of the pad on the thrust bearing is expressed as:

$$h_{ti}=h_0-r_t\cos \alpha \tan \gamma ,i=1,2,......,6$$

(23)

where r_t denotes the radius of any point on the plate. γ stands for the inclination angle, and α is the angle between the radius of any point on the plate and the x-axis.

The equivalent oil film thickness method is employed to compute the film thickness of the upside pad. A correlated relationship exists between the film thickness of the upside and downside oil pads after the tilt of the turntable, as outlined below.

$$h_{t1}=h_{t7},h_{t2}=h_{t8},h_{t3}=h_{t9},h_{t4}=h_{t10},h_{t5}=h_{t11},h_{t6}=h_{t12}$$

(24)

Combining the aforementioned equation yields the thickness of the oil film on the downside oil pad.

When inclined, the torque decomposes, and the torque balance in the rotational directions of the x- and y-axes for the thrust bearing is described by the following equation:

$$\left\{\begin{array}{l}{W}_{z}d\cos \delta =\left[{\displaystyle \sum _{i=1,5}\left(F_i-F_{i+6}\right)}-{\displaystyle \sum _{i=2,4}\left(F_i-F_{i+6}\right)}\right]\cos \frac{\pi }{3}\frac{R_1+R_4}{2}+F_e\\ W_{z}d\sin \delta =\left[{\displaystyle \sum _{i=1,5}\left(F_i-F_{i+6}\right)}-{\displaystyle \sum _{i=2,4}\left(F_i-F_{i+6}\right)}\right]\sin \frac{\pi }{3}\frac{R_1+R_4}{2}\\ W_t={\displaystyle \sum _{i=1}^6\left(F_i-F_{i+6}\right)}\end{array}\right.$$

(25)

where F_e = F₆ − F₁₂ + F₉ − F₃, F_i refers to the load capacity of a single oil pad, which can be derived from Equation (9).

The calculation process for the support performance of the hydrostatic turntable under an eccentric load is illustrated in Figure 6. Here, a and b denote the Δh interval of film thickness change, and c and d represent the interval of the inclination angle change Δγ. The limiting condition for the change in the inclination angle is to avoid interference between the plate surface and the pad. Hence, the boundary conditions for oil film thickness and inclination angle are as follows:

$$\left\{\begin{array}{l}a=0\\ b=\min \left(h_{ti}\right),i=1,2,...,12\\ c=0\\ d=\arctan \frac{h_{t0}}{R_4}\end{array}\right.$$

(26)

In the Figure 6, |g_t − f_t| < ε₂ and |g_a − f_a| < ε₃ are the judgment formula for force balance and torque balance, coefficient ε₂ = ε₃ = 1, and we construct the judgment equation as follows

$$\begin{array}{l}{g}_t=W_t\\ f_t={\displaystyle \sum _{i=1}^6\left(F_i-F_{i+6}\right)}\\ g_a=W_{z}d\cos \delta +W_{z}d\sin \delta \\ f_a=\left[{\displaystyle \sum _{i=1,5}\left(F_i-F_{i+6}\right)}-{\displaystyle \sum _{i=2,4}\left(F_i-F_{i+6}\right)}\right]\cos \frac{\pi }{3}\frac{R_1+R_4}{2}+F_e\\ \hspace{1em}+\left[{\displaystyle \sum _{i=1,5}\left(F_i-F_{i+6}\right)}-{\displaystyle \sum _{i=2,4}\left(F_i-F_{i+6}\right)}\right]\sin \frac{\pi }{3}\frac{R_1+R_4}{2}\end{array}$$

(27)

Iteratively adjust the oil film thickness and inclination angle. If the inclination angle undergoes changes, it is imperative to compute the minimum pad film thickness and revise the range of film thickness alterations until convergence conditions are satisfied. In adherence to the principle of structural symmetry, the eccentricity load angle should only vary within the range [0, π/3].

The overturning stiffness k_u is

$$k_u=\frac{W_{z}d}{\gamma }$$

(28)

To ensure that the maximum capacity of the journal bearing under eccentric load exceeds the centrifugal force, it is imperative to adjust the speed in accordance with the magnitude of the centrifugal force for optimal adaptation to working conditions. The formula for calculating centrifugal force is

$$F_L=m{\omega }^2{R}_d$$

(29)

where m is the eccentric load mass.

3. Simulation Verification

Considering the intricate nature of its oil path, the establishment of a unified model for fluid simulation poses challenges due to the substantial computational demand and inherent difficulties in achieving convergence. Consequently, a pair of pad fluid simulation models, illustrated in Figure 7, are developed using SolidWorks and subsequently imported into the Fluent module in ANSYS for simulation analysis.

To address the notable differences in structural dimensions between the oil circuit and oil film thickness, automatic grid division is implemented for both the oil circuit and oil pad. The oil film thickness is segmented into swept grids, and an interface is introduced at the grid transition point for interpolation calculations. The structural parameters of the pad are detailed in

Table 3. Calculation parameters of thrust and journal bearing.

Symbol	Input Value	Symbol	Input Value
R1/mm	239	R2/mm	249
R3/mm	298	R4/mm	308
φ2/°	28.9	φ1/°	26.9
bc/mm	2	lc/mm	63.26
tc/mm	3	Tc/mm	85
Bc/mm	6	Lc/mm	70
Xd/mm	164.37	Yd/mm	110
xd/mm	144.51	yd/mm	90
Ld/mm	60	ld/mm	60
bd/mm	2	td/mm	6
Bd/mm	4	Rd/mm	180
ω/r·min−1	60	η0/Pa·s	0.008
ps/Pa	1.5 × 106	ρ/kg·m3	897.1

. Displacement of the thrust plate or shaft under uniform load conditions is altered, and the inclination angle of the oil film under eccentric load conditions is adjusted to validate the accuracy of the equivalent film thickness method. The combined external force of two pads is utilized as a comparative verification indicator for both theoretical predictions and simulation results.

The pressure field distribution for the thrust and journal bearing pads, each with a 6 μm offset under uniform load, is illustrated in Figure 8 and Figure 9. To evaluate the pressure along this path, 100 points are strategically sampled along the marked dotted line in the figures. It is evident from the visual representation that the pressure within the return groove of the restrictor equals the pressure within the opposing oil recess, confirming the characteristic wherein the internal feedback pad pressure is regulated by the opposing restrictor. Figure 10 presents the simulation and theoretical outcomes under varying conditions of uniform load and inclined film thickness. In both uniform and eccentric load scenarios, the load capacity and flow rate errors for both thrust and journal pads remain below 10%, thereby affirming the validity of the proposed methodology in this study.

4. Results and Discussion

4.1. Analysis of Work Performance under Uniform Load

4.1.1. Influence of Oil Film Thickness

In Figure 11a,c, the support stiffness of both the thrust and journal bearings consistently decreases with an increasing load. The figures demonstrate that smaller film thickness corresponds to greater stiffness. Despite varying film thicknesses, the maximum load capacity remains constant. The thrust bearing is approximately 64,833 N, while the journal bearing is around 17,510 N. Figure 11b,d reveal that as the load increases, the flow rate also rises, with a larger film thickness causing a more pronounced change in flow rate.

4.1.2. Influence of Inlet Pressure

Figure 12 reveals notable differences in the trends of support stiffness, maximum load capacity, and flow rate under varying inlet pressures. As the pressure proportionally rises from 1.2 MPa to 1.5 MPa, the thrust and journal bearings exhibit approximately a 25% increase in stiffness, maximum load capacity, and maximum flow rate. The figure illustrates that, with a proportional increase in pressure, the distance between distinct curves also proportionally expands, signifying pressure’s proportional amplification role in supporting performance.

4.1.3. Influence of Eccentric Load Angle

In Figure 13a, the support stiffness is significantly smaller at an offset angle of 0° compared to other angles. This discrepancy arises because, at a 0° offset angle, only the y direction generates support force, and the combined force in the x direction is 0, indicating an absence of the stiffness effect in the x direction. At other angles, the change in support stiffness for the journal bearing is negligible. Figure 13b demonstrates that when the radial load is below 12,000 N, the impact of varying deflection angles on the flow rate is insignificant. Although an increase in the deflection angle raises the maximum radial load, the difference in the maximum flow rate is not substantial.

4.1.4. The Influence of Restrictor Size

From Figure 14a, it is evident that a greater oil inlet groove length corresponds to increased support stiffness. For thrust bearing, an increase in l_c from 33.26 mm to 63.26 mm resulted in a modest maximum load increase of only 7.4%, coupled with a 6.3% rise in flow rate (as indicated in Figure 14b). Notably, for journal bearing, l_c = 40.285 mm exhibits significantly better stiffness than other dimensions (see in Figure 14c). Moreover, in Figure 14d, an elevation in l_c from 20.285 mm to 50.285 mm leads to a 75.6% surge in the journal bearing’s maximum flow rate, indicating a relatively more pronounced flow rate change.

With an increase in the width of the oil return edge, there is a simultaneous decrease in the support stiffness, maximum load, and flow rate. In Figure 15a, widening the width from 2 mm to 5 mm results in a 27.5% decrease in the thrust bearing’s maximum load, along with a corresponding 27.5% reduction in minimum stiffness. Figure 15b indicates a substantial 53.1% decrease in the maximum flow rate. Figure 15c demonstrates that increasing the width from 2 mm to 5 mm leads to a 35.5% decrease in the journal bearing’s maximum load, a 41.4% decrease in minimum stiffness, and a 39.4% decrease in the maximum flow rate (Figure 15d).

In Figure 16a,b, an extension in restrictor length results in decreased stiffness and maximum load, accompanied by an increased flow rate. This phenomenon is attributed to the extended restrictor length, which reduces the internal flow resistance, improves oil flow to the same side, and mitigates the external force. Figure 16c,d reveal that the journal bearing undergoes more pronounced changes in the stiffness and flow rate, but the trend aligns with that of the thrust bearing. As L_d increases from 40 mm to 70 mm, the journal bearing experiences a 19.5% decrease in maximum load, a 19.2% increase in maximum flow rate, and an approximately 19.1% decrease in minimum stiffness within the calculation range.

In Figure 17a, it is evident that when B_c extends from 6 mm to 12 mm, both the maximum load and the minimum support stiffness of the thrust bearing increase by 19.7%. Notably, a substantial change in the stiffness of the journal bearing occurs as B_d increases from 4 mm to 8 mm, as illustrated in Figure 17c. Figure 17b,d depict that as the internal flow edge width increases, the flow rate decreases. In summary, increasing the width of the internal flow edge results in a proportional increase in the maximum load and minimum stiffness of the thrust bearing.

4.2. Analysis of Work Performance under Eccentric Load

The preceding analysis reveals that the journal bearing’s stiffness is significantly downside than that of the thrust bearing. In this study, we assume the upside and downside plates, as well as the shaft, to be rigid bodies. When an eccentric load is applied, the shaft undergoes a certain deflection angle. However, it is essential to note that the inclination angle resulting from the eccentric force is minimal. Furthermore, the moment generated by the journal bearings is substantially smaller in comparison to that of the thrust bearings. This observation provides confirmation that the assumptions made in this study are justified.

In addressing the eccentric load condition, it is imperative not to set the load excessively high. This precaution prevents surpassing the film thickness load limit due to the tilting of the thrust plates, leading to an inability for the equation to converge. Consequently, in this study, when examining eccentric loads, the total axial load is restricted to 2.5 t. The complexity of eccentric load conditions with multiple variables, determining the limit eccentric load, remains challenging. However, utilizing the method outlined in this paper enables the calculation and evaluation of the given total load and eccentric load size. To analyze the impact of various parameters under eccentric load, this study confines the selected parameters within a reasonable range, all falling under equations with solutions.

4.2.1. Influence of Oil Film Thickness

In Figure 18a, it is evident that the overturning stiffness decreases with an increase in eccentric load, and the smaller the film thickness, the greater the overturning stiffness. Figure 18b illustrates that as the film thickness decreases, the flow rate also decreases, aligning with the pattern identified in the earlier analysis. In Figure 18a, at an eccentric load of 8000 N, the overturning stiffness for a 20 μm film thickness is 1.5-times that for 50 μm. Figure 18b indicates that the flow rate variation within the calculated range is not significant.

4.2.2. Influence of Inlet Pressure

From Figure 19a, it is evident that the overturning stiffness gradually decreases with an increase in eccentric load and higher oil supply pressure. As the eccentric load rises from 1000 N to 8000 N, the overturning stiffness decreases by 16.7% at an oil inlet pressure of 1.2 MPa. In contrast, at 2.1 MPa, the overturning stiffness only decreases by 2.9%. This observation indicates that greater oil supply pressure enhances the stability of the thrust bearing against eccentric loads. In Figure 19b, a higher oil inlet pressure significantly improves the flow rate. Within a specific range, an increase in the eccentric load has an insignificant effect on flow rate variation, suggesting that the internal feedback hydrostatic bearing contributes to stabilizing the flow rate.

4.2.3. The Influence of Restrictor Size

As shown in Figure 20a,b, both the overturning stiffness and flow rate increase simultaneously with the widening of the oil inlet groove. Within the calculated range, the overturning stiffness experiences an approximately 10% increase, and the flow rate sees an approximately 11% increase as the length of the oil inlet groove extends from 43.26 mm to 73.26 mm.

From Figure 21a, it is evident that the overturning stiffness decreases more noticeably with an increase in the return channel width. Specifically, when b_c = 5 mm, the overturning stiffness experiences a significant decrease of 26.5% within the calculated range. This observation suggests that a greater width of the oil return edge reduces the resistance of capsizing of the rotary table. Examining Figure 21b, it is apparent that at b_c = 5 mm, the flow rate experiences a more pronounced increase, improving by approximately 13.2%.

From Figure 22a,b, it is evident that with an increase in the length of the restrictor, overturning stiffness decreases, and, simultaneously, the flow rate increases. For instance, under a maximum load of 8000 N, the overturning stiffness experiences a decrease of approximately 16.7% as L_c increases from 75 mm to 105 mm. Meanwhile, the flow rate increases by a modest 4.1%. Therefore, the increase in the length of the restrictor does not contribute favorably to improving the overturning resistance.

As shown in Figure 23a, the overturning stiffness decreases with an increase in the internal flow edge width. As illustrated in Figure 23b, the flow rate experiences a decrease with an increase in the internal flow edge width. Overall, with an eccentric load of 8000 N, for instance, with an increase in B_c from 6 mm to 8 mm, the overturning stiffness shows an improvement of 14.2%, and the flow rate experiences a decrease of 5.6%. Similarly, with an increase in B_c from 10 mm to 12 mm, the overturning stiffness shows a modest improvement of 3.7%, and the flow rate experiences a decrease of 2.9%.

4.2.4. Speed Adjustment of Journal Bearing

In deriving the load capacity discussed earlier, the selected oil demonstrated minimal viscosity, thereby making the impact of rotational speed on load capacity inconsequential, without accounting for the temperature viscosity effect. Figure 24 illustrates that the relationship between rotational speed and eccentric load weight is limited by the maximum load capacity of the journal bearing. When faced with substantial eccentric loads, a reduction in the rotational speed is enacted to counteract centrifugal forces. Conversely, with smaller eccentric loads, an increase in the rotational speed can be judiciously implemented. The application of this strategy in practical scenarios holds promise for improving the machining efficiency.

5. Conclusions

In this study, a computational model of the even and partial load characteristics of the internal feedback hydrostatic turntable is established to analyze and explore the stiffness and flow rate characteristics of the thrust and journal bearings under varying loading conditions. The key findings are as follows:

(1)

Both the support and overturning stiffness of the thrust bearings are enhanced with an increase in inlet pressure, oil inlet groove length, and internal flow edge width and a decrease in oil film thickness, oil return edge width, and restrictor length. In comparison to other parameters, the effects of restrictor length and oil inlet groove length on stiffness and flow rate are relatively small.

(2)

The eccentric load angle significantly influences the stiffness of the journal bearing, with stiffness noticeably smaller at a bias load angle of 0° compared to other angles. Moreover, adjusting the centrifugal force under an eccentric load can be achieved by changing the rotational speed, making it more adaptable to various working conditions.

(3)

For both thrust and journal bearings, the relationship between the flow rate and load capacity can be roughly categorized into two stages: stabilization and rapid growth. Moreover, increased flow rate results in enhanced power, demonstrating that the internal feedback hydrostatic bearing plays a role in stabilizing the power within a specific load range.