Theoretical Study of the Dynamic Quality of an Aerostatic Thrust Bearing with a Microgroove and Simple Diaphragms

Abstract

This paper presents the results of a study of the dynamic performance of an aerostatic thrust bearing with a microgroove and simple diaphragms. The objective of this study was to determine the influence of the lubrication gap thickness and the volume of the microgroove and pockets on the structural dynamics. Unlike most studies that typically use the second-order harmonic oscillator equation as the characteristic equation, the root criteria are determined with high accuracy when the characteristic equation is of an order no lower than the fourth order. The presented formulas allow one to find the optimal calculated dimensional gap, microgroove and pocket volume in terms of the best dynamic performance. For a well-damped thrust bearing, the required response speed and sufficient stability margin can only be achieved within a narrow range of 1–2 times the bearing gap volume. Calculations have shown that to ensure satisfactory thrust bearing dynamics, the calculated gap should not exceed 10–15 µm.

1. Introduction

Aerostatic thrust bearings are widely used in various fields of mechanical engineering, including machine tool building, providing low heat generation, high precision and low friction [1,2,3,4,5,6].

In practice, simple diaphragm thrust bearings are most commonly used, as they provide the least compliance [7,8,9,10]. However, such feeders have a drawback, they are prone to pneumatic hammer-type instability [11,12,13,14]. This is because air-filled pockets are located between the feeder openings and the supporting lubricant gap. The compressibility of the air in these pockets leads to the instability of the thrust bearings or an insufficient margin of stability.

Studies of the dynamic quality of aerostatic bearings are typically conducted in dimensional form for several specific sets of values via ready-made software products [11,12,13,14,15,16]. These are essentially calculation recommendations, since they do not allow for general conclusions to be drawn about the dynamics of such structures on the basis of an analysis of the dependencies for dynamic criteria generally accepted in automatic control theory, both for the relative air volumes in the pockets and for other important quantities that have a decisive influence on the quality of the thrust bearing dynamics.

The reasons for the absence of such recommendations are primarily that the boundary value problems for the Reynolds differential equation, which governs the air pressure function in the supporting gap, are extremely complex and lack analytical quadratures [17,18,19]. Moreover, even for simplified linearized problems, there are no analytical solutions; therefore, it is impossible to obtain characteristic equations in analytical form, which can be used to obtain information on the stability, stability margin, and quality of the dynamics of aerostatic thrust pads.

This article presents the results of a theoretical study of the dynamic performance of an aerostatic thrust bearing with simple diaphragms, focusing on the effect of the lubricant volume contained in the pockets and microgrooves. The latter are designed to increase the bearing capacity and reduce the compliance of the supporting lubricant film. A mathematical model of the thrust bearing dynamics with simple diaphragms has been created, and a numerical finite-difference method has been developed that enables one to obtain, with a specified accuracy, analytical relationships for the characteristic equation and for the transfer function of the thrust bearing’s dimensionless compliance via a linearized dynamic model of the structure.

2. Calculation Scheme and Mathematical Model of the Bearing

Figure 1 shows the design diagram of a structure that has a fixed base (1), an aerostatic thrust bearing (2) hermetically connected to it, and a movable element (5) with a mass m_s and a radius r_b.

The working surfaces of the heel and the thrust bearing are separated by a thin supporting gas gap of thickness h, which is created by the external injection of compressed air under pressure p_s and noncontactively balances the action of the external load f. The thrust bearing of the simple diaphragm type has a circle of radius r_c, a microgroove (6) with holes (3) of diameter d, evenly spaced along it, at the outlet of which, under pressure p_c, there are pockets (4) filled with compressed air of volume v.

3. Mathematical Model of Bearing Dynamics

The model includes the equation of force equilibrium of the moving element and the equation of balance of the lubricant flow rate

$$\left\{\begin{array}{l}w-f_{in}=f,\\ q_s-q_v-q_c=0,\end{array}\right.$$

(1)

where

$$w=2\mathrm{\pi }{\displaystyle \underset{0}{\overset{r_b}{\int }}r(p-p_a)dr,}$$

(2)

$$f_{in}=m\ddot{h}$$

(3)

are the bearing capacity of the thrust bearing, the inertial force of the movable element and the external load.

$$q_s={\Gamma }_{0}G(p_s,p_c),$$

(4)

$$q_v={\displaystyle \frac{v}{E_p}}{\dot{p}}_c,$$

(5)

$$q_h=-{\displaystyle \frac{\mathrm{\pi }{h}^3}{12\mathrm{\mu }{R}_0{T}_0}}\left[{\left(r{\displaystyle \frac{\partial p^2}{\partial r}}\right)}_{R=R_c+0}-\hspace{0.33em}{\left(r{\displaystyle \frac{\partial p^2}{\partial r}}\right)}_{R=R_c-0}\right]$$

(6)

where μ is the coefficient of the dynamic viscosity of air; R₀ is the gas constant; T₀ is the absolute air temperature; E_p is the reduced modulus of the elasticity of air; ${\Gamma }_0={\displaystyle \frac{\mathrm{\pi }n{d}^2}{4}}\sqrt{{\displaystyle \frac{\mathrm{\gamma }}{R_0{T}_0}}{\left({\displaystyle \frac{2}{\mathrm{\gamma }+1}}\right)}^{{\displaystyle \frac{\mathrm{\gamma }+1}{\mathrm{\gamma }-1}}}};$ γ = 1.4 is the adiabatic index [16]; r is the current radius; and t is the current time.

$$G(p_1,p_2)=\left\{\begin{array}{l}{p}_1,\hspace{0.33em}\hspace{1em}{p}_2/p_1\le {\Gamma }_c,\\ {\Gamma }_a{p}_1\sqrt{{\left({\displaystyle \frac{p_2}{p_1}}\right)}^{{\displaystyle \frac{2}{\mathrm{\gamma }}}}-{\left({\displaystyle \frac{p_2}{p_1}}\right)}^{{\displaystyle \frac{\mathrm{\gamma }-1}{\mathrm{\gamma }}}}}\end{array}\right.,\hspace{0.33em}\hspace{1em}{p}_2/p_1>{\Gamma }_c$$

(7)

is an adiabatic function of the flow rate from the diaphragm;

$${\Gamma }_a=\sqrt{{\displaystyle \frac{2}{\mathrm{\gamma }-1}}{\left({\displaystyle \frac{\mathrm{\gamma }+1}{2}}\right)}^{{\displaystyle \frac{\mathrm{\gamma }+1}{\mathrm{\gamma }-1}}}}, {\Gamma }_c={\left({\displaystyle \frac{2}{\mathrm{\gamma }+1}}\right)}^{{\displaystyle \frac{\mathrm{\gamma }}{\mathrm{\gamma }-1}}}\approx 0.528.$$

The pressure function satisfies the boundary value problem for the nonstationary Reynolds equation [16].

$$\left\{\begin{array}{l}{h}^3{\displaystyle \frac{\partial }{\partial r}}\left(rp{\displaystyle \frac{\partial p}{\partial r}}\right)=12\mathrm{\mu }r{\displaystyle \frac{\partial }{\partial t}}\left(ph\right),\\ p(r,0)=p_0(r),\\ {\displaystyle \frac{\partial p(0,t)}{\partial r}}=0,\hspace{0.33em}p(r_c,t)=p_c(t),\hspace{0.33em}p(r_b,t)=p_a,\end{array}\right.$$

(8)

where p_a is the ambient pressure and p₀ (r) is the stationary function of pressure.

4. Mathematical Model in Dimensionless Form

To study the quality of the thrust bearing’s dynamics, it is convenient to reduce its mathematical model to a dimensionless form. This reduces the number of variable parameters and thereby increases the informational scope of the study. The following scales are adopted: p_a for pressures; r_b for radii; working gap h₀, to which the thrust bearing is adjusted to withstand the calculated load, for the current thickness of the lubricating gap; t₀ for current time; $h_0^3 p_a^2/\left(\mathrm{\mu }{R}_0{T}_0\right)$ for mass flow rates; and $\mathrm{\pi }{r}_b^2{p}_a$ for forces. Dimensionless quantities are denoted by capital and Greek letters.

In the design mode, the pressure is P_c = P_c0 and the gap is H = 1. For this mode, we introduce a normalized pressure adjustment coefficient at the air outlet from the throttle as follows:

$$\mathrm{\chi }={\displaystyle \frac{P_{c0}^2-1}{P_s^2-1}}\in [0,1].$$

(9)

Using (9), we define

$$P_{c0}=\sqrt{1+\mathrm{\chi }\left(P_s^2-1\right)}.$$

(10)

The system of dimensionless equations corresponding to (1) and (2) takes the form

$$\left\{\begin{array}{l}W-F_{in}=F,\\ Q_s-Q_v-Q_h=0,\end{array}\right.$$

(11)

Here, the bearing capacity of the design is

$$W=2{\displaystyle \underset{0}{\overset{1}{\int }}R(P-1)dR},$$

(12)

and the inertial force of a moving element is

$$F_{in}=M_p\ddot{H},$$

(13)

where $M_p={\displaystyle \frac{m{h}_0}{\mathrm{\pi }{t}_0^2{r}_b^2{p}_a}}\hspace{0.33em}$ is the mass of the moving element.

The flow rate through the diaphragm is

$$Q_s=A_{s}G(P_s,P_c),$$

(14)

where $A_s={\displaystyle \frac{{\Gamma }_0\hspace{0.17em}}{h_0^3{p}_a}}.$

Considering the compressibility of the lubricant in the pockets and microgrooves,

$$Q_v=A_v{\dot{P}}_p,$$

(15)

where $A_v=\mathrm{\sigma }V,$ and

$$\mathrm{\sigma }={\displaystyle \frac{12\mathrm{\mu }{r}_0^2}{h_0^2{p}_a{t}_0}}\hspace{0.33em}$$

(16)

is the so-called compression number of the gas film [20],

The formula for the dimensionless gas flow rate in a lubricant gap of thickness H has the form [21]

$$Q_h=-H^3\left[{\left(R{\displaystyle \frac{\partial P^2}{\partial R}}\right)}_{R=R_c+0}-{\left(R{\displaystyle \frac{\partial P^2}{\partial R}}\right)}_{R=R_c-0}\right],$$

(17)

The corresponding (8) boundary value problem for the nonstationary Reynolds equation takes the form

$$\left\{\begin{array}{l}{H}^3{\displaystyle \frac{\partial }{\partial R}}\left(RP{\displaystyle \frac{\partial P}{\partial R}}\right)=\mathrm{\sigma }R{\displaystyle \frac{\partial }{\partial \mathrm{\tau }}}\left(PH\right),\\ P(R,0)=P_0(R),\\ {\displaystyle \frac{\partial P(0,\mathrm{\tau })}{\partial R}}=0,\hspace{0.33em}P(R_c,\mathrm{\tau })=P_c(\mathrm{\tau }),\hspace{0.33em}P(1,\mathrm{\tau })=1.\end{array}\right.$$

(18)

Comparing the costs in the design mode H = 1, we find the coefficient $A_s={\displaystyle \frac{A_h\left(P_{c0}^2-1\right)}{G(P_s,P_{c0})}}.$

The static pressure function P₀ (R) can be obtained by solving the boundary value problem (18) in the absence of oscillations of the moving element.

This occurs when the right-hand side of the partial differential equation of the boundary value problem vanishes. This condition significantly simplifies the solution of the problem.

5. Static Characteristics of the Bearing

In the absence of oscillations, the boundary value problem (18) is simplified and takes the form

$$\left\{\begin{array}{l}{\displaystyle \frac{d}{dR}}\left(R{\displaystyle \frac{d{P}_0^2}{dR}}\right)=0,\\ {\displaystyle \frac{d{P}_0(0)}{dR}}=0,\hspace{0.33em}{P}_0(R_c)=P_c,\hspace{0.33em}{P}_0(1)=1.\end{array}\right.$$

(19)

The solution to the boundary value problem (19) is a function of the static pressure in the bearing gap as

$$P_0(R)=\left\{\begin{array}{l}{P}_c,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0\le R\le R_c,\\ \sqrt{\left(P_c^2-1\right){\displaystyle \frac{\ln R}{\ln R_c}}+1},\hspace{1em}{R}_c<R\le 1,\end{array}\right.$$

(20)

where P_c is the static pressure at the outlet of the diaphragm.

Using (17) and (20), we find the steady-state air flow rate in the carrier gap to be

$$Q_h=A_h{H}^3\left(P_c^2-1\right),$$

(21)

where $A_h=-1/\ln R_c.$

The static bearing capacity of the thrust bearing according to (12) and (20) is determined by the formula

$$W=R_c^2\left(P_c-1\right)+2{\displaystyle \underset{R_c}{\overset{1}{\int }}R\left(\sqrt{\left(P_c^2-1\right)\frac{\ln R}{\ln R_c}+1}-1\right)dR}.$$

(22)

The integral included in (22) does not have an analytical quadrature, so it was calculated via Simpson’s numerical quadrature rule [22].

The pressure P_c is conveniently used as a parameter when determining the static dependencies of the gap H, lubricant flow rate Q, and thrust bearing compliance K on the external load F. According to the static flow rate balance Equations (14) and (21), the gap can be calculated via the following formula:

$$H=\sqrt[3]{{\displaystyle \frac{A_{s}G(P_s,P_c)}{A_h\left(P_c^2-1\right)}}}.$$

The compliance of the thrust bearing can be conveniently calculated using an approximate finite difference formula of the second order of accuracy O(λ²) as follows:

$$K=-{\displaystyle \frac{\partial H}{dF}}={\displaystyle \frac{H(P_c-\mathrm{\lambda })-H(P_c+\mathrm{\lambda })}{W(P_c+\mathrm{\lambda })-W(P_c-\mathrm{\lambda })}},$$

(23)

where λ is a small number (in the calculations, we took λ = 0.001), F = W.

The compliance of the thrust bearing can be conveniently calculated via an approximate finite difference formula of the second order of accuracy O(λ²) as follows:

Figure 2 shows the graphs of the dependence K(χ) for the design load mode F, for which the gap is H = 1. The pressures in Figure 2, Figure 3 and Figure 4 are calculated via the formula $P_s=2+0.5i.$

The dependences are clearly strictly unimodal functions with a single extremum minimum. The figure also shows that the minimum degree of compliance dependence K(χ) depends on supply pressure P_s.

Table 1. Parameters of minimum compliance K from supply pressure P_s.

N	Ps	Pc	χ	Pc/Ps	K
1	2.5	1.889	0.489	0.755	0.975
2	3.0	2.185	0.472	0.728	0.731
3	3.5	2.483	0.459	0.709	0.586
4	4.0	2.781	0.449	0.695	0.490
5	4.5	3.080	0.441	0.685	0.421
6	5.0	3.379	0.434	0.676	0.370

presents the data for which the curves reach the minimum compliance value. These data are important for bearing design, as its construction involves adjusting the parameters for minimum compliance.

Let us construct the dependences K(H) for the found optimal values of the parameter χ, the supply pressure P_s from Table 1 and the calculated gap H = 1. These dependences are shown in Figure 3.

Notably, the abscissa of the minimum K(H) does not coincide with the calculated gap H = 1 since there are gaps for which the compliance K is 4–6% less. The minimum K (H) occurs at H₁ ≈ 0.89. If it is necessary to adjust the bearing so that the minimum compliance K (H) occurs at the gap H = 1, then its scale must be changed while taking into account the value of H₁.

For the new scale, the dependences K(H) are shown in Figure 4. As follows from the graphs, the minimum of the function K(H) now occurs at H = 1, that is, in the mode of adjusting the bearing to the calculated load.

Table 2. Correction data for adjusting the K compliance to a minimum in the design gap mode H = 1.

N	Ps	H1	Pc1	χ	Pc1/Ps	K
1	2.5	0.895	2. 051	0.611	0.821	0.975
2	3.0	0.891	2.408	0.600	0.803	0.731
3	3.5	0.888	2.765	0.591	0.790	0.586
4	4.0	0.886	3.124	0.584	0.781	0.490
5	4.5	0.883	3.485	0.579	0.774	0.421
6	5.0	0.881	3.848	0.575	0.770	0.370

summarizes the correction data so that, at a given supply pressure P_s and radius R_c, the minimum compliance occurred in the mode of the calculated gap H = 1.

Figure 5 and Figure 6 show the dependences of the bearing capacity W and air flow rate Q in the gap H on a scale considering H₁ when the lowest compliance K occurs at H = 1.

Of particular interest is the study of the quality of the dynamics of the thrust bearing, adjusted to the working calculated gap H = 1.

It is evident that with increasing supply pressure P, the bearing capacity W increases proportionally.

The dependence W (H) to the left and right of the point H = 1 decreases more weakly; that is, the bearing compliance is higher than in the range adjacent to the mentioned gap to which the bearing is adjusted. This means that the minimum compliance occurs in the middle part of this dependence.

Similarly, with increasing P_s, the lubricant flow rate Q increases proportionally. Moreover, the larger the gap H is, the greater the flow rate Q, even though the pressure in the bearing gap decreases as the gap increases. This means that the gap between the mating surfaces of the thrust bearing lubricated by compressed air has a dominant influence on flow rate.

6. Formula for the Transform of the Bearing Dynamic Compliance

Let us turn to the nonlinear differential equations of thrust bearing dynamics (11). The greatest difficulty in calculating the dynamic quality criteria is finding a solution to the nonlinear boundary value problem for the Reynolds Equation (18). The problem can be simplified by finding a solution for small deviations of the dynamic functions from their static values P_c and H.

Let us represent the dynamic functions P_c, H and F in the form

H (τ) = H − ΔH(τ), P_c (τ) = P_c ΔP_c (τ),
F (τ) = F + ΔF (τ), P (R,τ) = P₀ (R) + ΔP (R,τ).

(24)

Assuming that

$$\left|\Delta H\right|<<H, \left|\Delta P_c\right|<<P_c, \left|\Delta F\right|<<F_0, \left|\Delta P\right|<<P_0,$$

(25)

Let us linearize the boundary value problem (18)

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial }{\partial R}}\left[R{\displaystyle \frac{\partial \left(P_0\Delta P\right)}{\partial R}}\right]={\displaystyle \frac{\mathrm{\sigma }R}{H^3}}{\displaystyle \frac{\partial }{\partial \mathrm{\tau }}}\left(\Delta P-P_0\Delta H\right),\\ {\displaystyle \frac{\partial \Delta P(0,\mathrm{\tau })}{\partial R}}=0,\hspace{0.33em}\Delta P(R_c,\mathrm{\tau })=\Delta P_c(\mathrm{\tau }),\hspace{0.33em}\Delta P(1,\mathrm{\tau })=0.\end{array}\right.$$

(26)

Applying the Laplace integral transform (24) to (26) with respect to the current time τ, we obtain a boundary value problem for a linear ordinary differential equation

$$\left\{\begin{array}{l}R{\displaystyle \frac{d^2\left(P_0\overline{\Delta P}\right)}{d{R}^2}}+{\displaystyle \frac{d\left(P_0\overline{\Delta P}\right)}{dR}}={\displaystyle \frac{\mathrm{\sigma }sR}{H^3}}\left(\overline{\Delta P}-P_0\overline{\Delta H}\right),\\ {\displaystyle \frac{d\overline{\Delta P}(0,s)}{dR}}=0,\hspace{0.33em}\overline{\Delta P}(R_c,s)={\overline{\Delta P}}_c(s),\hspace{0.33em}\overline{\Delta P}(1,s)=0,\end{array}\right.$$

(27)

where s is the Laplace transform variable, which plays the role of a complex parameter, and where $\overline{\Delta P}(R,s),\overline{\Delta H}(s),{\overline{\Delta P}}_c(s)$ are the Laplace transforms of the deviations of the corresponding dynamic functions.

Since problem (27) does not have an analytical solution, we apply the finite-difference method of sweeping the second order of accuracy with respect to its step [23]. To do this, we divide each integration region [0, R_c] and [R_c, 1] by an even number n of equal parts and replace the differential form of the boundary value problem (27) with an algebraic one as follows:

$$\begin{array}{l}{R}_i{\displaystyle \frac{P_{0,i+1}{\overline{\Delta P}}_{i+1}-2P_{0,i}{\overline{\Delta P}}_i+P_{0,i-1}{\overline{\Delta P}}_{i-1}}{g^2}}+\\ +{\displaystyle \frac{P_{0,i+1}{\overline{\Delta P}}_{i+1}-P_{0,i-1}{\overline{\Delta P}}_{i-1}}{2g}}={\displaystyle \frac{\mathrm{\sigma }\mathrm{s}{R}_i}{H^3}}\left(H{\overline{\Delta P}}_i-P_{0,i}\overline{\Delta H}\right),\\ i=1,2,\dots ,n,\end{array}$$

(28)

where $g=\left\{\begin{array}{l}{R}_c/n,\hspace{1em}R\in [0,R_c],\\ (1-R_c)/n,\hspace{1em}R\in [R_c,1]\end{array}\right.$ is the grid step.

The boundary conditions for the first segment are as follows:

$${\displaystyle \frac{-{\overline{\Delta P}}_2+4{\overline{\Delta P}}_1-3{\overline{\Delta P}}_0}{2g}}=0,\hspace{1em}{\overline{\Delta P}}_n={\overline{\Delta P}}_c.$$

(29)

The first boundary condition is the equality to zero of the right first derivative of the sought function of the second order of accuracy [24], which is an algebraic approximation of this boundary condition (27).

For the second condition, we have

$${\overline{\Delta P}}_0={\overline{\Delta P}}_c,\hspace{1em}{\overline{\Delta P}}_n=0.$$

(30)

Using the superposition method, we represent the desired pressure transform in the form

$$\overline{\Delta P}(R)=U_c(R){\overline{\Delta P}}_c+U_h(R)\overline{\Delta H}.$$

(31)

Taking into account (31), the system of linear Equation (28) can be represented in general form

$$c_{1,i}{U}_{i+1}-c_{2,i}{U}_i+c_{3,i}{U}_{i-1}+c_{4,i}=0,$$

(32)

where

$$c_{1,i}=\left(2R_i+g\right)P_{0,i+1},\hspace{0.33em}{c}_{2,i}=2R_i\left(2P_{0,i}+\mathrm{\alpha }\right),\hspace{0.33em}{c}_{3,i}=\left(2R_i-g\right)P_{0,i-1},\hspace{0.33em}{c}_{4,i}=\mathrm{\beta }{R}_i{P}_{0,i},\hspace{0.33em}$$

U is the desired function U_c or U_h, $\mathrm{\alpha }={\displaystyle \frac{g^2\mathrm{\sigma }s}{H^2}},\mathrm{\beta }={\displaystyle \frac{2g^2\mathrm{\sigma }as}{H^3}},$ a = 0 corresponds to the function U_c, and α = 1 corresponds to the function U_h.

On the basis of (29) for R = 0, the boundary condition will be

$$-3U_0+4U_1-U_2=0.$$

(33)

Problems (30), (32) and (33) were solved via a recurrence formula as follows:

$$\begin{array}{l}{U}_{i-1}=X_i{U}_i+Y_{i.},\\ i=1,2,3\dots ,n-2,n-1,n\end{array}$$

(34)

To find the initial values of the sweep coefficients, consider Equation (32) for i = 1:

$$c_{1,1}{U}_2+c_{3,1}{U}_0-c_{2,1}{U}_1+c_{4,1}=0.$$

(35)

As follows from (33)

$$U_2=-3U_0+4U_1.$$

(36)

By substituting (36) into (35) after simple manipulations followed by comparison with (34) at R = 0, we find formulas for the initial sweep coefficients in the region $R\in [0,R_c]$

$$X_1={\displaystyle \frac{4c_1^1-c_2^1}{3c_1^1-c_3^1}},\hspace{1em}{Y}_1={\displaystyle \frac{c_4^1}{3c_1^1-c_3^1}}.$$

(37)

For the other end of the segment of this region R = R_c, U_n = 1 − a.

For the region $R\in [R_c,1]$, the corresponding boundary conditions give

X₁ = 0, Y₁ = 1 − a, and U_n = 0.

Substituting (34) into (32), we find recurrence formulas for the direct run

$$X_{i+1}={\displaystyle \frac{c_1^i}{c_2^i-c_3^i{X}_i}},Y_{i+1}={\displaystyle \frac{c_3^i{Y}_i+c_1^i}{c_2^i-c_3^i{X}_i}}.$$

The backsweep was performed via Formula (34) for i = n, n − 1, …, 2, 1.

The function U(R) obtained as a result of the run is generally a numerical array of the form

$$U=(U_0,U_1,U_2,\dots ,,U_{n-2},U_{n-1},U_n).$$

(38)

The number n of partitions of the integration segments is determined by the accuracy of the calculation of the dynamic quality criteria (when analyzing the calculated data, the values of n that ensure the specified accuracy are given below).

Having performed the run for a given value of the Laplace transform variable s, we find the coefficients of the transform of the deviation of the bearing capacity as follows:

$$\overline{\Delta W}=A_{wc}{\overline{\Delta P}}_c+A_{wh}\overline{\Delta H}.$$

(39)

Coefficient $A_{wc}=A_{wc1}+A_{wc2},\hspace{0.33em}\hspace{0.33em}{A}_{wh}=A_{wh1}+A_{wh2}$ formulas (36) were found via Simpson’s quadrature formula [20] as follows:

$$A_w=2{\displaystyle \sum _{i=0}^n{k}_i}{R}_i{U}_i,\hspace{1em}k=1,4,2,4,2,\dots ,2,4,1.$$

Four U arrays were obtained by sweeping as follows. Setting a = 0 (region R ∈ [0, Rc]), with b = 0, we obtain the array U_c1 and with a = 1, the array U_h1. Then, via Formula (39), we find the coefficients A_wc1 and A_wh1. Similarly, setting b = 1 (region R ∈ [0, Rc]), for a = 0, we obtain the array U_c2, and for a = 1, the array U_h2; then, using (37), we find the coefficients A_wc2 and A_wh2.

Figure 7, Figure 8, Figure 9 and Figure 10 show graphs of the functions U(R) for all four combinations of parameter values (a, b) for s = 0, 2, 4, and 6.

It is clear that the functions are smooth. Therefore, Simpson’s numerical formulas for calculating the coefficients of the bearing capacity transform and the formula for the air flow rate in the bearing gap should yield highly accurate results with a small number of integration domain partitions.

Taking into account (13), the transform of the deviation of the inertial force takes the form

$${\overline{\Delta F}}_{in}=-M_p{s}^2\overline{\Delta H},$$

(40)

To find the time scale, we take M_p = 1. Then,

$$\hspace{0.33em}{t}_0={\displaystyle \frac{1}{r_b}}\sqrt{{\displaystyle \frac{m{h}_0}{\mathrm{\pi }{p}_a}}.}$$

(41)

Substituting (41) into (16), we obtain the formula for the “compression number” [17]

$$\mathrm{\sigma }={\displaystyle \frac{12\mathrm{\mu }{r}_b^3}{h_0^2}}\sqrt{{\displaystyle \frac{\mathrm{\pi }}{m_p{p}_a}}.}\hspace{0.33em}$$

(42)

Next, via (17), we find a formula for determining the transform of the deviation of the air flow rate in the carrier gap.

$${\overline{\Delta Q}}_h=A_{ch}{\overline{\Delta P}}_c+A_{qh}\overline{\Delta H}.$$

(43)

To do this, we first perform linearization (17) and then perform the Laplace transform. As a result, we found

$$\overline{\Delta Q_h}=-3A_h{H}^2\left(P_c^2-1\right)\overline{\Delta H}+2H^3\left\{{\left[R{\displaystyle \frac{d\left(P_0\overline{\Delta P}\right)}{dR}}\right]}_{R=R_c+0}-{\left[R{\displaystyle \frac{d\left(P_0\overline{\Delta P}\right)}{dR}}\right]}_{R=R_c-0}\right\}.$$

(44)

To approximate the right and left derivatives for R = R_c with the use of finite difference formulas of the second order of accuracy [20] we found that

$$\begin{array}{l}{\left[R{\displaystyle \frac{d\left(P_{0}U\right)}{dR}}\right]}_{R=R_c+0}={\displaystyle \frac{R_0}{2g}}\left(-P_{0,2}{U}_2+4P_{0,1}{U}_1-3P_{0,0}{U}_0\right),\\ {\left[R{\displaystyle \frac{d\left(P_{0}U\right)}{dR}}\right]}_{R=R_c-0}={\displaystyle \frac{R_n}{2g}}\left(3P_{0,n}{U}_n-4P_{0,n-1}{U}_{n-1}|+P_{0,n-2}{U}_{n-2}\right).\end{array}$$

(45)

Using the arrays U found in a fourfold run and previously used to find the transform of the deviation of the bearing capacity (38) according to Formulas (42) and (43), we find the coefficients $A_{ch}=A_{ch2}-A_{ch1},\hspace{0.33em}{A}_{qh}=A_{qh2}-A_{qh1}.$

The transformer of flow rate deviation through throttles

$${\overline{\Delta Q}}_s=A_{sd}{\overline{\Delta P}}_c,$$

(46)

where

$$A_{sd}=A_s{\displaystyle \frac{\partial G(P_s,P_c)}{\partial P_c}}=\left\{\begin{array}{l}0,\hspace{0.33em}\hspace{1em}{P}_c/P_s\le {\Gamma }_c,\\ {\Gamma }_a{\displaystyle \frac{2{\left(\frac{P_c}{P_s}\right)}^{\frac{2-\mathrm{\gamma }}{\mathrm{\gamma }}}-(\mathrm{\gamma }+1){\left(\frac{P_c}{P_s}\right)}^{\frac{1}{\mathrm{\gamma }}}}{4\sqrt{{\left(\frac{P_c}{P_s}\right)}^{\frac{2}{\mathrm{\gamma }}}-{\left(\frac{P_c}{P_s}\right)}^{\frac{\mathrm{\gamma }-1}{\mathrm{\gamma }}}}}}{p}_1\end{array}\right.,\hspace{0.33em}\hspace{1em}{P}_c/P_s>{\Gamma }_c.$$

(47)

Transformer flow rate deviation due to air compressibility in pockets and microgrooves

$${\overline{\Delta Q}}_v=\mathrm{\sigma }Vs\hspace{0.17em}{\overline{\Delta P}}_c.$$

(48)

The analog of (11) is the system of equations

$$\left\{\begin{array}{l}\overline{\Delta W}-{\overline{\Delta F}}_{in}=\overline{\Delta F},\\ {\overline{\Delta Q}}_s-{\overline{\Delta Q}}_v-{\overline{\Delta Q}}_h=0.\end{array}\right.$$

(49)

Substituting (38), (40), (44), (46) and (48) into (49), we obtain a system of linear equations in transforms

$$\left\{\begin{array}{l}{A}_{wc}{\overline{\Delta P}}_c+\left(A_{wh}+s^2\right)\overline{\Delta H}=\overline{\Delta F},\\ \left(A_c-\mathrm{\sigma }Vs\right){\overline{\Delta P}}_c-A_{qh}\overline{\Delta H}=0,\end{array}\right.$$

(50)

where $A_c=A_{qc}-A_{sd}.$

Having solved (50), we obtain the Laplace image of the dynamic compliance of the thrust bearing as follows:

$$K(s)=-{\displaystyle \frac{\overline{\Delta H}}{\overline{\Delta F}}}={\displaystyle \frac{A_c-\mathrm{\sigma }Vs}{A_{wc}{A}_{qh}+\left(A_c-\mathrm{\sigma }Vs\right)\left(A_{wh}+s^2\right)}}.$$

(51)

This formula enables one to calculate one value of the function K(s) using a given value of the variable s.

7. Rational Interpolation of the Compliance Transfer Function

To calculate the quality criteria of the thrust bearing dynamics, Formula (49) alone is not sufficient. It is also necessary to find a rational transfer function of the dynamic system in the form of a ratio of polynomials of the variable s, which has the following general form [25]:

$$K(s)={\displaystyle \frac{b_0+b_{1}s+b_2{s}^2+\dots +b_m{s}^m}{1+a_{1}s+a_2{s}^2+\dots +a_n{s}^n}}.$$

(52)

where m < n; a_i, b_i are real numbers.

The difference in powers n − m is equal to the smallest natural number for which

$$\underset{s\to \infty }{\lim }\left[s^{n-m}K(s)\right]={\displaystyle \frac{b_m}{a_n}}\ne 0.$$

Numerical experiments have shown that for a given transfer function, n − m = 2. As mentioned, the number n is determined by the accuracy of the calculation of the quality criteria of the dynamics—the larger n is, the higher the accuracy of the calculations.

The coefficient $b_0=K(0)$ represents the static compliance of the thrust bearing. In addition to (24), this is another formula for calculating it.

To find the coefficients of the function K(s), it is convenient to use Formula (52) in the form

$$b_1+b_{2}s+b_3{s}^2+\dots +b_m{s}^{m-1}-K(s)\left(a_1+a_{2}s+\dots +a_n{s}^{n-1}\right)=K_b(s),$$

(53)

where $K_b(s)=\left[K(s)-b_0\right]/s.$

Equation (53) has k = n + m unknown coefficients. We calculate $e=exp\left(-{\displaystyle \frac{2\mathrm{\pi }i}{k}}\right)$, where i is the imaginary unit. We set $s_1=1$ and find $s_{j+1}=e\hspace{0.17em}{s}_j,\hspace{0.33em}j=1,2,\dots ,k-1.$ Let us denote $K_j=K\left(s_j\right),$ $K_{b,j}=K_b\left(s_j\right),$ and $j=1,2,\dots ,k.$

Note that $e_i{e}_j=e_{\left(i+j\right)\hspace{0.33em}\mathrm{mod}\hspace{0.33em}k}.$ Taking this into account, we obtain a system of linear equations as follows:

$$M x=y,$$

(54)

where

$$M=\left[\begin{array}{cccccc}{s}_1& s_1& s_1& \dots & K_1{s}_1& \dots \\ s_1& s_2& s_3& \dots & K_2{s}_1& \dots \\ s_1& s_3& s_5& \dots & K_3{s}_1& \dots \\ \dots & \dots & \dots & \dots & \dots & \dots \\ s_1& s_k& s_1& \dots & K_k{s}_1& \dots \end{array}\right], x=\left[\begin{array}{c}{b}_1\\ b_2\\ \dots \\ a_{n-1}\\ a_n\end{array}\right], y=\left[\begin{array}{c}{K}_{b,1}\\ K_{b,2}\\ \dots \\ K_{b,k-1}\\ K_{b,k}\end{array}\right].$$

Multiplying Equation (54) by the inverse Fourier transform matrix [21] yields

$${\Phi }_{i,j}^{*}={\displaystyle \frac{1}{k}}{s}_{[(i-1)(j-1)]\mathrm{mod}k+1}^{*};\hspace{1em}i,j=1,2,3,\dots ,k,$$

where the complex conjugate number is marked with an asterisk; we obtain the following system of equations:

$${\Phi }^{*}Mx={\Phi }^{*}y.$$

By performing multiplication, we obtain a matrix equation

$$Ax=z,\hspace{1em}z={\Phi }^{*}y.$$

Matrix A has the cellular structure

$$A=\left[\begin{array}{cc}E& C\\ 0& D\end{array}\right],$$

(55)

where E and 0 are the identity and zero matrices of sizes $m\times m$ and $n\times m$, respectively; C and D are real Toeplitz matrices of sizes $m\times n$ and $n\times n$, respectively [20]; and z is a real vector.

From (55), it follows that the coefficients of denominator (52) can be found by solving a simpler system of equations containing only n equations [26]:

Da = d,

(56)

where a = (a₁, a₂, a₃, …, a_n), and d is a vector composed of the last n elements z as in

$$d_i=z_{m+i},\hspace{1em}i=1,2,\dots ,n.$$

After the solution (56) is obtained, one can find the coefficients of the numerator (52) via the formulas

$$b_i=z_i-{\displaystyle \sum _{j=1}^n{A}_{i,n+j}\hspace{0.17em}{a}_j},\hspace{1em}i=1,2,\dots ,m.$$

(57)

If it is necessary to calculate only the dynamic stability criteria of the thrust bearing, then only the denominator coefficients (52) are needed. In this case, calculations via Formula (57) are not necessary.

8. Analysis of the Bearing Dynamic Quality

To evaluate the quality of the thrust bearing dynamics, the following root criteria were used: the degree of stability η, the normalized degree of stability η₀, and the damping of oscillations over a period ξ [22]. The first and second criteria represent the largest real parts of the roots of the characteristic and normalized characteristic equations, respectively, which are taken with opposite signs. The third criterion allows one to judge the oscillation of the transient process and the stability margin of the structure during oscillation of the moving element caused by the disturbance of an external force. At ξ = 100%, the transient process has an aperiodic nature, which indicates a high stability margin of the dynamic system. For well-damped dynamic systems, oscillation of at least ξ = 90% is permissible [22].

The root criteria are determined with high accuracy when the characteristic equation is of an order no lower than the fourth order (n = 4 was assumed in the calculations). This demonstrates that the second-order harmonic oscillator equation used in most studies on aerostatic bearing dynamics is insufficient for accurately calculating the thrust bearing dynamic performance criteria.

Of greatest interest is the influence of the parameters V and σ on the quality of the dynamics in the calculated load mode on the thrust bearing at stationary values of the gap H = 1 and the pressure in the microgrooves P_c = P_c0. Note that these parameters V and σ affect only the quality of the dynamics of the structure.

Figure 11 shows the dependences of the degree of stability η on the compression number σ for different volumes V.

It is evident that for V > 0 and σ < 35, the thrust bearing is always unstable (η < 0). Taking into account (16), σ is inversely proportional to the square of the gap scale h₀². This means that when adjusting the thrust bearing to support the design load with large gaps, it will always be unstable. For σ > 35, the η (σ) dependence has the characteristic of a unimodal function containing an extremum maximum.

With the same steady-state parameters, the highest response speed of the thrust bearing was recorded at σ = 61.4 and V = 1.51. This means that there is a calculated gap h₀ at which the thrust bearing will have the best dynamic quality. In this case, the criteria for the quality of the thrust bearing dynamics were η_opt = 0.502, η_0,opt = 0.567, and ξ_opt = 98%. These are very high indicators. These values are extremely close to the ideal thrust bearing dynamics, for which η₀ = 1.0 and ξ = 100% [22].

As seen from the graphs in Figure 11, the duration of the transient processes of an excessively damped thrust bearing will be 4 or more times slower than that of a thrust bearing with the optimal σ = σ_opt, which is undesirable when this design is used in metal-cutting machines, since it will contribute to a decrease in the accuracy of metalworking.

The effect of volume V on the quality of the thrust bearing dynamics for different σ values is clearly shown in Figures 13 and 14. For small volumes V < 1, the thrust bearing, although stable (η > 0), clearly has a small stability margin (ξ < 80%) for any σ.

Figure 12 shows graphs of the influence of the parameters σ and V on the stability margin of the thrust bearing. In the region where σ < σ, the transient characteristics clearly oscillate in nature (ξ < 100%). Consequently, in this region, a stable thrust bearing may have an insufficient stability margin, indicating proximity to the stability limit, when the transient characteristics are oscillatory in nature.

Notably, for small V < 1, the criterion ξ < 80% is also small, that is, too small a volume, commensurate with the volume of the bearing gap and even less (V < 1), indicating too small a margin of stability of the thrust bearing. Only at V > 1 does ξ = 100% occur. A high margin of stability can also be achieved at σ/σ_opt > 3. However, the degree of stability will be much less than its optimum. This means that a thrust bearing with such characteristics will have such high damping that the duration of the transient response will be far from optimal. The duration of the transient process is inversely proportional to the degree of stability [25].

This is a remarkable conclusion, as there is a persistent belief that the smaller the volume of pockets and microgrooves, the better the thrust bearing’s dynamic performance. This belief turned out to be unfounded.

As shown in the graphs in Figure 13 and Figure 14, the thrust bearing has satisfactory dynamics at σ > 56. In this case, the stability margin ξ > 90% occurs at 1.2 < V < 3.5. This means that the thrust bearing will have the best dynamic quality when the ratio of the volume of the pockets and microgrooves to the volume of the bearing gap in the calculated load mode is within the specified limits.

Until now, we have discussed the dynamics of the thrust bearing under the calculated static conditions for a single point of the load characteristic W (H) at H = 1. However, during operation, the thrust bearing will have a load deviation from the calculated value when the gap may deviate from the calculated value but not by more than 50%.

Figure 15 and Figure 16 show the dependences of the dynamic criteria on the gap in the vicinity of H = 1 for V = 1.5. The larger σ is, that is, the smaller the calculated dimensional gap h₀, the wider the stability region for the gap H. The same conclusion can be drawn regarding the stability margin ξ.

The bearing has better dynamics at H < 1. This implies that thrust bearing loading is a factor in improving the thrust bearing’s dynamic performance. For H > 1, the stability margin ξ rapidly decreases, reaching critical values. However, with proper selection of the parameters σ and V, acceptable dynamic performance can be achieved.

In addition to root criteria, frequency criteria for assessing the quality of dynamic systems are used.

For this purpose, the amplitude–frequency characteristics A(ω) are constructed, with the help of which the relative oscillation index M corresponding to the greatest amplitude of the characteristic is found as follows:

M = max A(ω)/A(0).

The M-index is a performance criterion for automatic control systems, characterizing their susceptibility to oscillation. The smaller the system’s stability margin is, the greater its susceptibility to oscillation and the higher the resonant peak M. The oscillation index is used to assess the system’s performance during transient processes and to determine the stability margin.

Figure 17 shows the amplitude–frequency characteristics A(ω) for different values of volume V. Here, ω is the dimensionless oscillation frequency. At σ = 70, the peaks of A(ω) increase, indicating that the stability margin decreases with increasing volume V.

Figure 18 shows the dependence of the oscillation index M on the volume V for different σ values. With increasing σ (in other words, with decreasing calculated gap h₀), the stability margin clearly increases.

The largest stability margin is at the minimum point of the function M (V). The smallest values of M (V) occur in the range of 1 < V < 2.

9. An Example of a Dimensional Calculation of the Bearing

Let us consider a thrust bearing with an outer radius r_b = 4·10⁻² m. Let us take the ambient pressure p_a = 0.1 MPa, the air viscosity μ = 1.82·10⁻⁶ Pa·s, and the mass of the moving element m_p = 3 kg. We also accept the dimensionless quantities σ = 70, V = 1.5, and η = 0.4. Using the expressions for the dimensionless mass at M_p = 1 and the compression number σ, we find the calculated gap h₀, the time scale t₀, the attenuation time of the transient response th [22] and the volume of the pockets and microgrooves v as follows:

$$\begin{array}{l}{h}_0=r_b\sqrt[5]{{\displaystyle \frac{144\mathrm{\pi }{r}_b{\mathrm{\mu }}^2}{m_p{p}_a{\mathrm{\sigma }}^2}}} = 13.3·{10}^{-6} [\mathrm{m}],\\ t_0={\displaystyle \frac{1}{r_b}}\sqrt{{\displaystyle \frac{m{h}_0}{\mathrm{\pi }{p}_a}}} = 0.28·{10}^{-3} [\mathrm{s}],\\ t_h\approx {\displaystyle \frac{3t_0}{\mathrm{\eta }}} = 2.1·{10}^{-3} [\mathrm{s}],\\ v=\mathrm{\pi }{r}_b^2{h}_{0}V= 100·{10}^{-9} [\mathrm{m}].\end{array}$$

In this case, the bearing capacity of the support in calculation mode is w = 850 [N], and the static compliance k₀ = 1.2·10⁻⁸ [m/N].

10. Conclusions

This paper presents the results of a study on the dynamic performance of an aero-static thrust bearing with a microgroove and simple diaphragms. The data obtained from the analysis compensate for the extremely uninformative results obtained by the researchers in dimensional form.

It has been shown that in the parameter space of the “compression number” σ and the dimensionless volume V, which affect only the dynamic performance of the thrust bearing, for each set of values of the dimensionless parameters P_s, R_c, χ, and H, there exists exactly one set of σ and V values for which the thrust bearing will exhibit the best dynamic performance. Using these values, it is possible to find the corresponding optimal calculated dimensional gap h₀ and the dimensional volume v of the microgrooves and pockets in terms of the best dynamic performance.

However, ensuring the optimal response and stability margin for gap variations in the range of 0.5 < H < 1.5 is difficult. For a well-damped thrust bearing, the required response and sufficient stability margin can be achieved for gap variations in the aforementioned vicinity within a rather narrow range of dimensionless volume 1 < V < 2. Satisfactory bearing dynamics can be achieved for volume values of 0.5 < V < 4.