Dynamic Quality of an Aerostatic Thrust Bearing with a Microgroove and Support Center on Elastic Suspension

Abstract

The disadvantage of aerostatic bearings is their low dynamic quality. The negative impact on the dynamic characteristics of the bearing is exerted by the volume of air contained in the bearing gap, pockets, and microgrooves located at the outlet of the feeding diaphragms. Reducing the volume of air in the flow path is a resource for increasing the dynamic quality of the aerostatic bearing. This article presents an improved design of an axial aerostatic bearing with simple diaphragms, an annular microgroove, and an elastic suspension of the movable center of the supporting disk. A mathematical model is presented and a methodology for calculating the static characteristics of a bearing and dynamic quality indicators is described. The calculations were carried out using dimensionless quantities, which made it possible to reduce the number of variable parameters. A new method for solving linearized and Laplace-transformed boundary value problems for transformants of air pressure dynamic functions in the bearing layer was applied, which made it possible to obtain a numerical solution of problems sufficient for practice accuracy. The optimization of the criteria for the dynamic quality of the bearing was carried out. It is shown that the use of an elastic suspension of the support center improves its dynamic characteristics by reducing the volume of compressed air in the bearing layer and choosing the optimal volume of the microgroove.

1. Introduction

To produce the load capacity in aerostatic bearings, flow rate limiters are usually used, which automatically regulate the pressure in the bearing air layer [1,2,3,4,5,6]. In practice, diaphragms are used as flow limiters, in which the limitation of the flow is created due to the air pressure, and capillaries, in which the pressure is regulated by the forces of viscous friction in the air lubricant. Diaphragms are used more often, among which nozzles with annular and simple diaphragms are distinguished [7,8].

The vulnerability of aerostatic bearings to instability is known, and oftentimes, this circumstance is decisive when choosing flow limiters. Bearings with annular diaphragms are always stable; however, compared to similar bearings with simple diaphragms, they have 1.5 times higher compliance [9]. The load capacity of bearings with annular diaphragms is also lower due to the discreteness of the feeders, while the use of simple diaphragms united by microgrooves helps to eliminate the noted drawback. At the same time, the presence of air pockets at the outlet of simple diaphragms and microgrooves is the reason for the loss of stability of the bearings and the extremely low quality of their dynamics [9,10]. The volume of air contained in the bearing layer also has a negative impact on the dynamics of the unit. Thus, a decrease in the volume of air in its flow path is a resource for increasing the dynamic quality of the aerostatic bearing.

Figure 1 shows a diagram of an improved circular aerostatic bearing with the shaft, 1 and base, 2, which is connected to a supporting disk, 3, of radius r₀. The bearing is powered from a source of compressed air through a hole, 8, of diameter d, from which air enters the microgroove, 7, made on the disk, 3, along the circumference of the radius r₁. The bearing center, 5, is supported by an elastic suspension, 4, in the form of a thin ring of thickness δ. This ring, under the action of the pressure difference p_s − p_k > 0 in the hollow layer, 6, and the bearing layer, provides the necessary deformation h_d of the material of the disk, 3, and the displacement of the central disk, 5, relative to the shaft, 1, by the value h_s, where p_s is the injection pressure and p_k is the pressure at the outlet of the diaphragms, 8.

Compared to a conventional support (h_d = 0), the volume of air contained in the bearing layer will be less due to deformation of h_d, where the gap h_s on the central circle will always be less than the gap h on the outer ring. In addition, due to deformation of the ring, 4, the vibration pattern of the bearing layer will change, which can also contribute to an increase in the dynamic quality of the bearing.

This paper considers a mathematical model of the stationary state of the bearing and calculates its optimal modes. On this basis, non-stationary mathematical modeling of the effectiveness of the proposed method for improving the dynamic characteristics and the calculation and study of the root stability criteria of the bearing were carried out. An example of the design calculation of the bearing is given.

2. Static Model of Bearing Movement

The static model includes the air mass flow balance equation:

$$q_k-q_h+q_{hs}=0,$$

(1)

as well as two equations of power balance of the movable elements of the support:

$$h_d=k_e\left(w_p-w_{hs}\right),$$

(2)

$$w=f,$$

(3)

$$w=w_h+w_{hs}.$$

(4)

where q_h, q_hs, and q_k are the mass flow rates of air through the gaps h, h_s, and the hole, 8, respectively, w_h, w_hs, and w_p are the power reactions of compressed air in these gaps and the cavity, 6, and w is the load capacity of the support [11].

The study of static characteristics was carried out in a dimensionless form. The scales of values are taken as: p_a—for air pressures, r₀—for radial dimensions, h₀—for the thickness of throttling slotted gaps, $\frac{\mathsf{\pi }{h}_0^3{p}_a^2}{12\mathsf{\mu }\Re T}$—for mass air flow rates, and $\mathsf{\pi }{r}_0^2{p}_a$—for axial forces, where h₀ corresponds to the gap h in the bearing that perceives the design load f₀, p_a is the ambient pressure, μ is the air viscosity, $\Re$ is the universal gas constant, and T is the absolute gas temperature. Furthermore, dimensionless quantities are designated by capital Latin letters.

Dimensionless flow through the hole, 8, is determined by the expression [12]:

$$Q_k=A_k\mathsf{\Pi }(P_s,P_k),$$

(5)

where Π is the Prandtl outflow function, and the aperture similarity criterion is:

$$A_k=\frac{3\mathsf{\mu }\mathsf{\Gamma }{d}^2\sqrt{\Re T}}{p_a{h}_0^3},$$

(6)

where $\mathsf{\Gamma }=2\sqrt{\mathsf{\gamma }}{\left(\frac{2}{\mathsf{\gamma }}\right)}^{\frac{\mathsf{\gamma }+1}{2(\mathsf{\gamma }-1)}}$ and γ is an adiabatic air expansion index [9].

The pressure distribution function in the bearing layer [2] is:

$$P(R)=\{\begin{array}{l}{P}_k,R\le R_1,\\ \sqrt{\left(P_k^2-1\right)\frac{LnR}{Ln{R}_1}+1},R_1\le R\le 1\end{array}.$$

(7)

Taking into account Equation (7), the formula for the flow rate in the bearing layer can be written in the form:

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

(8)

where $A_h=-\frac{1}{Ln{R}_1}.$

Dimensionless force reactions were found by the following formulas [2,12]:

$$W_{hs}=2{\displaystyle \underset{0}{\overset{R_1}{\int }}R\left(P-1\right)dR=R_1^2(P_k-1),}$$

(9)

$$W_h=2{\displaystyle \underset{R_1}{\overset{1}{\int }}R\left(P-1\right)dR=2J+R_1^2-1,}$$

(10)

$$J={\displaystyle \underset{R_1}{\overset{1}{\int }}R\sqrt{\left(P_k^2-1\right)\frac{\ln R}{\ln R_1}+1}\hspace{0.17em}dR,}$$

(11)

$$W_p=2{\displaystyle \underset{0}{\overset{R_1}{\int }}R\left(P_s-1\right)dR=R_1^2\left(P_s-1\right).}$$

(12)

In a dimensionless form, the model will include equations similar to (1)–(4):

$$Q_k-Q_h+Q_{hs}=0,$$

(13)

$$H_d=K_e\left(W_p-W_{hs}\right),$$

(14)

$$H_s=H-H_d,$$

(15)

$$W=F,$$

(16)

as well as, obviously, $H_s=H-H_d.$

When calculating the static characteristics of the bearing, dimensionless quantities were used as input parameters: radius R₁, injection pressure P_s, deformation H_d0 of the ring, 4, at the design point H = 1, and normalized coefficient χ of the relative resistance of the diaphragms, 8:

$$\mathsf{\chi }=\frac{P_k^2-1}{P_s^2-1}\in [0,1],$$

(17)

which was used to calculate the pressures and the diaphragm similarity criterion.

Using Equation (17), we obtain the formula for calculating the pressure in the microgroove, 7:

$$P_k=\sqrt{1+\mathsf{\chi }\left(P_s^2-1\right)}.$$

According to (5)–(17), it is possible to find dimensionless force reactions, deformation H_d, coefficient A_h, flow rate Q_h (flow rate Q_hs = 0), and a similarity criterion for the diaphragms:

$$A_k=\frac{Q_h}{\mathsf{\Pi }\left(P_s,P_k\right)}.$$

(18)

By sequentially setting pressure values $P_k\in \left[1,P_s\right]$ in small steps, the corresponding force reactions, deformation H_d, gap $H=\sqrt[3]{\frac{Q_k}{A_h\left(P_k^2-1\right)}}$, and gap H_s can be calculated. In the calculations, integral (12), which has no analytical quadrature, is calculated by the Simpson method [13].

Having written down the equations of the model in small Δ increments, we found the dimensionless bearing compliance to be:

$$K=-\frac{\Delta H}{\Delta F}=\frac{A_4-A_5}{A_3\left(A_1+A_2\right)},$$

(19)

where:

$$A_1=\frac{2P_k}{\ln R_1}{\displaystyle \underset{R_1}{\overset{1}{\int }}\frac{R\ln RdR}{\sqrt{\left(P_k^2-1\right)\frac{\ln R}{\ln R_1}+1}}},$$

$$A_2=R_1^2,A_3=3A_h{H}^2\left(P_k^2-1\right),A_4=2A_h{H}^3{P}_k,$$

$$A_5=A_k\frac{\partial \mathsf{\Pi }\left(P_s,P_k\right)}{\partial P_k}=A_k\{\begin{array}{l}0,\hspace{0.17em}{P}_k/P_s\le 0.5,\\ \frac{0.5P_s-P_k}{\sqrt{\left(P_s-P_k\right)P_k}}\end{array},P_k/P_s>0.5.$$

3. Static Characteristics of the Bearing

Static compliance K of the bearing does not depend on the elasticity coefficient of the ring, 4, deformation H_d, and gap H_s. The graphs in Figure 2 and Figure 3 show the influence of the parameters R₁, P_s, and χ on this characteristic at the design point H = 1.

As can be seen from Figure 2, the curves of the dependence of the bearing compliance K on the adjustment factor χ have an extreme character. The optimal value of the parameter of the relative resistance of the diaphragms is in the range χ = 0.45–0.46.

Of interest is the nature of the change in the deformation H_d and the gap H_s from the load F. Figure 3 shows the dependences for four functions H(F), K(F), H_d(F), and H_s(F). The first of them is a load characteristic, the second represents the dependence of the compliance K on the load F, and the last two show the nature of the dependence of the deformation H_d and the gap H_s. The curves correspond to the calculated deformation H_d0 = 0.5. Formula (27) makes it possible to determine the coefficient of elasticity K_e of the ring, 4, at which the deformation H_d0 is ensured. At R₁ = 0.75 and P_s = 4, the coefficient K_e = 0.731.

As can be seen from Figure 3, at low loads of F when the pressure drop across the moving center is the greatest, the deformation is highest, as a result of which gap H_s turns out to be much less than gap H. In comparison with the usual support due to the deformation of the suspension, 4, a significantly smaller volume of the bearing layer is provided, which should have a beneficial effect on the dynamic quality of the structure.

Deformation of H_d in the region of low and moderate loads contributes to an effective decrease in the volume of the bearing layer. Consequently, the effect created by deformation can affect almost the entire range of loads.

The dependence of the volume of the bearing layer V on the load F at different values of deformation H_d0 is clearly shown in Figure 4.

Curve H_d0 = 0 corresponds to the dependence V(F) of a conventional bearing, and the remaining curves for which H_d0 > 0 correspond to a bearing with a deformable suspension, 4. It is seen that deformation contributes to a significant decrease in the volume V of the bearing layer. Thus, at load F = 1 and deformation H_d0 = 0.4 at the design point, the volume decreases by almost one and a half times, and at H_d0 = 0.6, it decreases more than twice.

The conclusion about the effectiveness of the proposed improvement can be made based on the study of non-stationary characteristics of the bearing.

4. Dynamic Model of Bearing Movement

In the unsteady mode, pressure p_k(t), gaps h(t) and h_s(t), deformation h_d(t), flow rates q_h(t) and q_k(t), and forces w_h(t), w_hs(t), w_p(t), and w(t) become functions of the current time t. Moreover, new non-stationary functions appear, such as flow:

$$q_v=\frac{v_k}{\Re T}\frac{d{p}_k}{dt},$$

(20)

due to the compressibility of air in the microgroove 7, and the force of inertia:

$$w_i=m_s\frac{d^{2}h}{d{t}^2},$$

(21)

of the shaft, 1, mass m_s.

Taking this into account, the model of the bearing dynamics will take the form:

$$q_k-q_{hs}+q_h-q_v=0,$$

(22)

$$h_d=k_e\left(w_p-w_h\right),$$

(23)

$$w=w_h+w_{hs},$$

(24)

$$w-w_i=f.$$

(25)

The dynamic processes caused by small perturbations ΔF(τ) relative to the equilibrium state of the external dimensionless static load F, where τ is the dimensionless current time, are investigated. Linearization is applied to model (22)–(25), which is based on the assumption that in the investigated dynamic process, the variables change so that their deviations from steady–state values remain small at all times. The integral Laplace transform with respect to the current time was applied to the linearized model, as a result of which transformants of the deviations of the input load function $\overline{\Delta F}(s)$ and output functions were obtained, where s is the variable of the Laplace transform [14]. The transformed linearized dimensionless mathematical model of the dynamics of small vibrations of the bearing is described by the equations:

$${\overline{\Delta Q}}_k-{\overline{\Delta Q}}_h+{\overline{\Delta Q}}_{hs}-{\overline{\Delta Q}}_v=0,$$

(26)

$${\overline{\Delta H}}_d=K_e\left({\overline{\Delta W}}_p-{\overline{\Delta W}}_{hs}\right),$$

(27)

$$\overline{\Delta H}-{\overline{\Delta H}}_s-{\overline{\Delta H}}_d=0,$$

(28)

$$\overline{\Delta W}={\overline{\Delta W}}_h+{\overline{\Delta W}}_{hs},$$

(29)

$$\overline{\Delta W}-{\overline{\Delta W}}_i=\overline{\Delta F}.$$

(30)

The function of dimensionless pressure in the bearing layer in the general case satisfies the boundary value problem for the Reynolds equation [2]:

$$\{\begin{array}{l}\frac{\partial }{\partial R}\left(R{H}^3\frac{\partial P^2}{\partial R}\right)=2\mathsf{\sigma }R\frac{\partial (PH)}{\partial \mathsf{\tau }},\\ \frac{\partial P}{\partial R}(0,\mathsf{\tau })=0,P(R_1,\mathsf{\tau })=P_k,P(1,\mathsf{\tau })=1,\end{array}$$

(31)

where $\mathsf{\sigma }=\frac{12\mathsf{\mu }{r}_0^2}{h_0^2{p}_a{t}_0}\hspace{0.17em}$ is the number of squeezing and t₀ is the scale of the current time [9].

The transformed linearized problem corresponding to Equation (31) has the form:

$$\{\begin{array}{l}\frac{d}{dR}\left[R\frac{d\left(P\overline{\Delta P}\right)}{dR}\right]=\frac{\mathsf{\sigma }s}{H_{}^3}R\left(H\overline{\Delta P}+P\overline{\Delta H}\right),\\ \frac{d\overline{\Delta P}}{dR}(0,s)=0,\overline{\Delta P}(R_1,s)={\overline{\Delta P}}_k(s),\overline{\Delta P}(1,s)=0,\end{array}$$

(32)

where $\overline{\Delta P}(R,s),\overline{\Delta H}(s),{\overline{\Delta P}}_k(s)$ are the Laplace transformants of the corresponding function deviations and H, P(R) are static values of the gap and pressure distribution functions.

The boundary value problem (32) for the segment $R\in \left[R_a,R_b\right]$ can be represented as:

$$\{\begin{array}{l}R\frac{d^2\left(P\overline{\Delta P}\right)}{d{R}^2}+\frac{d\left(P\overline{\Delta P}\right)}{dR}=\frac{\mathsf{\sigma }s}{H^3}R\left(H\overline{\Delta P}+P\overline{\Delta H}\right),\\ {\mathsf{\alpha }}_a\overline{\Delta P}(R_{\text{a}},s)+(1-{\mathsf{\alpha }}_a)\frac{d\overline{\Delta P}}{dR}(R_{\text{a}},s)={\mathsf{\beta }}_a{\overline{\Delta P}}_a(s),\\ {\mathsf{\alpha }}_b\overline{\Delta P}(R_{\text{b}},s)+(1-{\mathsf{\alpha }}_b)\frac{d\overline{\Delta P}}{dR}(R_{\text{b}},s)={\mathsf{\beta }}_b{\overline{\Delta P}}_b(s),\end{array}$$

(33)

where $\overline{\Delta P}(R,s),\overline{\Delta H}(s),{\overline{\Delta P}}_a(s),{\overline{\Delta P}}_b(s)$ are Laplace deviations of the corresponding functions and H and P(R) are the static gap and pressure function, respectively.

Using the superposition method, we represent the required function in the form:

$$\overline{\Delta P}(R,s)=U_a(R,s){\overline{\Delta P}}_a+U_b(R,s){\overline{\Delta P}}_b+U_h(R,s)\overline{\Delta H}.$$

(34)

Substituting Equation (34) into Equation (33) and performing the separation of variables, we obtain problems for determining functions $U_a,U_b,U_h,$ which can be written in the following general form:

$$\{\begin{array}{l}R\frac{d^2\left(PU\right)}{d{R}^2}+\frac{d\left(PU\right)}{dR}=\frac{\mathsf{\sigma }s}{P{H}^3}R\left(HU+\mathsf{\lambda }P\right),\\ {\mathsf{\alpha }}_{a}U(R_{\mathrm{a}},s)+(1-{\mathsf{\alpha }}_a)\frac{dU}{dR}(R_{\mathrm{a}},s)={\mathsf{\beta }}_a,\\ {\mathsf{\alpha }}_{b}U(R_{\mathrm{b}},s)+(1-{\mathsf{\alpha }}_b)\frac{dU}{dR}(R_{\mathrm{b}},s)={\mathsf{\beta }}_b.\end{array}$$

(35)

For λ = 1, α_a = 1, α_b = 1, β_a = 0, and β_b = 0, we obtain a boundary value problem for the function U_h for any model, and for λ = 0 we obtain problems for the functions U_a or U_b.

The algebraic finite difference method [15] was applied to the solution of problem (35). For this, the segment $\left[R_{\mathrm{a}},R_b\right]$ was divided into an even number n of segments, and a system of algebraic equations is written for the internal nodes of the grid:

$$R_i\frac{P_{i+1}{U}_{i+1}-2P_i{U}_i+P_{i-1}{U}_{i-1}}{g^2}+\frac{P_{i+1}{U}_{i+1}-P_{i-1}{U}_{i-1}}{2g}=\frac{\mathsf{\sigma }\mathrm{s}{R}_i}{H^3}\left(H{U}_i+\mathsf{\lambda }{P}_i\right),$$

(36)

where $g=\left(R_b-R_a\right)/n$ is the grid step, U_i are the values of the function at the grid nodes, and i = 1, 2, …, n−1.

System (36) is supplemented with the boundary conditions:

$$\{\begin{array}{l}{\mathsf{\alpha }}_a{U}_0+\frac{(1-{\mathsf{\alpha }}_a)}{2g}(-3U_0+4U_1-U_2)=(1-{\mathsf{\lambda })\mathsf{\beta }}_a,\\ {\mathsf{\alpha }}_b{U}_n+\frac{(1-{\mathsf{\alpha }}_b)}{2g}(3U_n-4U_{n-1}+U_{n-2})=(1-{\mathsf{\lambda })\mathsf{\beta }}_b.\end{array}$$

(37)

When deriving Equation (37), derivatives of the second order of accuracy O(g²) at the end of segment [16] were used.

Systems (36) and (37) were solved by the sweep method [17]. For this, the recurrent formula is applied:

$$U_{i-1}=x_i{U}_i+y_i.$$

(38)

The first two equations from Equations (36) and (37) have the form:

$$\{\begin{array}{l}{a}_1{U}_0+a_2{U}_1-a_3{U}_2=a_0,\\ b_1{U}_0-b_2{U}_1+b_3{U}_2=b_0,\end{array}$$

(39)

where:

$$a_1={\mathsf{\alpha }}_a-\frac{3(1-{\mathsf{\alpha }}_a)}{2g},a_2=\frac{2(1-{\mathsf{\alpha }}_a)}{g},a_3=\frac{1-{\mathsf{\alpha }}_a}{2g},a_0=(1-\mathsf{\lambda }){\mathsf{\beta }}_a,$$

$$b_1=\frac{P_0}{g}\left(\frac{R_1}{g}-\frac{1}{2}\right),b_2=R_1\left(\frac{2P_1}{g^2}+\frac{\mathsf{\sigma }s}{H^2}\right),b_3=\frac{P_2}{g}\left(\frac{R_1}{g}+\frac{1}{2}\right),b_0=\frac{R_1{P}_1\mathsf{\lambda }\mathsf{\sigma }s}{H^3}.$$

Using Equations (38) and (39), we found the initial sweep coefficients:

$$x_1=-\frac{c_2}{c_1},y_1=\frac{c_0}{c_1},$$

(40)

where $c_1=b_1+\frac{b_3{a}_1}{a_3},c_2=\frac{b_3{a}_2}{a_3}-b_2,c_0=b_0-\frac{b_3{a}_0}{a_3}.$

Equation (36) is presented in the form:

$$a_i{U}_{i+1}-b_i{U}_i+c_i{U}_{i-1}=d_i,$$

(41)

where:

$$a_i=\frac{P_{i+1}}{g}\left(\frac{R_i}{g}+\frac{1}{2}\right),b_i=R_i\left(\frac{2P_i}{g^2}+\frac{\mathsf{\sigma }s}{H^2}\right),c_i=\frac{P_{i-1}}{g}\left(\frac{R_i}{g}-\frac{1}{2}\right),d_i=\frac{\mathsf{\lambda }\mathsf{\sigma }{R}_i{P}_{i}s}{H^3}.$$

Substituting Equation (41) into Equation (38), we found recursive formulas for the sweep coefficients:

$$x_{i+1}=\frac{a_i}{b_i-c_i{x}_i},y_{i+1}=\frac{c_i{y}_i-d_i}{b_i-c_i{x}_i}.$$

(42)

The value of U_n required for the backward sweep is obtained from the second equation in Equation (37) and the last equation in Equation (36):

$$U_n=\frac{e_6-e_5{y}_n}{e_4+e_5{x}_n},$$

(43)

where:

$$e_3={\mathsf{\alpha }}_b+\frac{3(1-{\mathsf{\alpha }}_b)}{2g},e_2=\frac{2(1-{\mathsf{\alpha }}_b)}{g},e_1=\frac{1-{\mathsf{\alpha }}_b}{2g},e_0=(1-\mathsf{\lambda }){\mathsf{\beta }}_b,$$

$$e_4=a_{n-1}-\frac{c_{n-1}{e}_3}{e_1},e_5=\frac{c_{n-1}{e}_2}{e_1}-b_{n-1},e_6=d_i-\frac{c_{n-1}{e}_0}{e_1}.$$

The load capacity factors are expressed by the integral:

$$A_w=2{\displaystyle \underset{R_1}{\overset{R_2}{\int }}RUdR}.$$

(44)

In the general case, the formula for the flow rate transformant in the bearing layer has the form:

$$\begin{array}{l}{\overline{\Delta Q}}_h=-3H^{2}R\frac{d{P}^2}{dR}\overline{\Delta H}-2H^3\frac{d\left(P\overline{\Delta P}\right)}{dR}=A_{qh}\overline{\Delta H}+A_{qa}{\overline{\Delta P}}_a+A_{qb}{\overline{\Delta P}}_b,\\ \end{array}$$

where:

$$\begin{array}{l}{A}_{qh}=A_{q0}-2H^{3}R\frac{d\left(P{U}_h\right)}{dR},A_{qa}=-2H^{3}R\frac{d\left(P{U}_a\right)}{dR},\\ A_{qb}=-2H^{3}R\frac{d\left(P{U}_b\right)}{dR},A_{q0}=\frac{3\mathsf{\lambda }{H}^2\left(P_b^2-P_a^2\right)}{\ln \left(R_a/R_b\right)}.\end{array}$$

At the edges of the segment $\left[R_a,R_b\right]$:

$$\begin{array}{l}{B}_a=-2H^{3}R{\frac{d\left(PU\right)}{dR}}_{R=R_a}=\frac{R_a{H}^3}{g}\left(P_2{U}_2-4P_1{U}_1+3P_0{U}_0\right),A_{qa}=\left(A_{q0}+B_a\right),\\ B_b=-2H^{3}R{\frac{d\left(PU\right)}{dR}}_{R=R_b}=-\frac{R_b{H}^3}{g}\left(3P_n{U}_n-4P_{n-1}{U}_{n-1}+P_{n-2}{U}_{n-2}\right),A_{qb}=\left(A_{q0}+B_b\right).\end{array}$$

The transformant of the force reaction of the bearing layer thickness H_s is determined by the Simpson formula [13]:

$${\overline{\Delta W}}_{hs}=2{\displaystyle \underset{0}{\overset{R_1}{\int }}R\overline{\Delta P}dR}=A_{whs}{\overline{\Delta H}}_s+A_{wks}{\overline{\Delta P}}_k,$$

(45)

where $A_{whs}=2{\displaystyle \underset{0}{\overset{R_1}{\int }}R{U}_{h}dR},A_{wks}=2{\displaystyle \underset{0}{\overset{R_1}{\int }}R{U}_{k}dR}.$

The transformed function of the flow rate in the bearing layer is:

$${\overline{\Delta Q}}_{hs}=A_{qhs}{\overline{\Delta H}}_s+A_{qks}{\overline{\Delta P}}_k.$$

(46)

The force reaction of the bearing layer in the gap of thickness H is:

$${\overline{\Delta W}}_h=2{\displaystyle \underset{R_1}{\overset{1}{\int }}R\overline{\Delta P}dR}=A_{wh}\overline{\Delta H}+A_{wk}{\overline{\Delta P}}_k,$$

(47)

where $A_{wh}=2{\displaystyle \underset{R_1}{\overset{1}{\int }}R{U}_{h}dR},A_{wk}=2{\displaystyle \underset{R_1}{\overset{1}{\int }}R{U}_{k}dR}.$

The flow rate transformant at the inlet to the gap of thickness H is:

$${\overline{\Delta Q}}_h=A_{qh}\overline{\Delta H}+A_{qk}{\overline{\Delta P}}_k.$$

(48)

The coefficients of Equations (44)–(48) were found as a result of a fourfold solution of problem (35) with the values of the parameters $\mathsf{\lambda },{\mathsf{\alpha }}_a,{\mathsf{\beta }}_a,{\mathsf{\alpha }}_b,{\mathsf{\beta }}_b$ given in

Table 1. Coefficients of transformants for special cases of the boundary value problem (Equation (35)).

Gap	λ	αa	βa	αb	βb	Coefficients
Hs	1	1	0	1	0	$A_{qhs}$,$A_{qks}$
	0	0	0	1	1	$A_{whs}$,$A_{wks}$
H	1	1	0	1	0	$A_{qh}$,$A_{qk}$
	0	1	1	1	0	$A_{wh}$,$A_{wk}$

.

The transformation formula for the flow rate due to the air compressibility in the microgroove, 7, has the form:

$${\overline{\Delta Q}}_v=A_{v}s{\overline{\Delta P}}_k,$$

(49)

where $A_v=\mathsf{\sigma }{V}_k,V_k=\frac{v_k}{\mathsf{\pi }{r}_0^2{h}_0}$ is the dimensionless volume of the microgroove.

The flow function through the diaphragm, 8, is determined by the formula:

$${\overline{\Delta Q}}_k=A_k{\overline{\Delta P}}_k.$$

(50)

The formula for the dimensionless transformant of the inertia force has the form:

$${\overline{\Delta W}}_i=M_s{s}^2\overline{\Delta H},$$

(51)

where $M_s=\frac{m_s{h}_0}{\mathsf{\pi }{r}_0^2{t}_0^2{p}_a}$ is the dimensionless mass of the shaft, 5.

Substituting Equations (44)–(51) into Equations (26)–(41), we obtained a system of linear equations with input $\overline{\Delta F}$ and output $\overline{\Delta H},{\overline{\Delta H}}_s,{\overline{\Delta H}}_d,{\overline{\Delta P}}_k$ functions:

$$A\left[\overline{\Delta V}\right]=\left[E\right]\overline{\Delta F},$$

(52)

where A is a complex matrix:

$$\left[V\right]=\left[\begin{array}{c}\overline{\Delta H}\\ {\overline{\Delta H}}_s\\ {\overline{\Delta H}}_d\\ {\overline{\Delta P}}_k\end{array}\right],\left[E\right]=\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right].$$

The dynamic compliance of the bearing is determined by the transfer function:

$$K(s)=-\frac{\overline{\Delta H}}{\overline{\Delta F}}=-\frac{\left|A_H\right|}{\left|A\right|},$$

(53)

where $\left|A\right|$ is the determinant of the matrix A of system (52) and A_H is the matrix formed from matrix A by replacing the first column with [E] in accordance with Cramer’s rule [16].

5. Characteristic Polynomial and Criteria for the Dynamic Quality of the Bearing

The bearing dynamics model is a non-linear system with distributed parameters. After linearization, it becomes linear, but it still remains a system with distributed parameters, since transfer function (53) can be obtained by numerical methods based on the solution of the above–mentioned differential equations.

To assess the dynamic quality of the bearing, transfer function (53) was approximately represented in the form:

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

(54)

where m < n, m > 0, and n > 0, n and m are natural numbers, and a_i and b_i are real numbers.

The number m is constant for the transfer function K(s) and is determined by its smallest natural value m, for which:

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

Numerical experiments have shown that for a given transfer function, m = 2. This corresponds to the difference in the orders of the polynomials of the transfer functions of the aerostatic and hydrostatic bearings, the models of which take into account the effect of the shaft mass inertia on their dynamics [11,18]. The unknown transfer function coefficients were found using the iterative method described in [19].

To assess the dynamic quality of linear systems, the following root criteria were used: the degree of stability η and damping of oscillations over the period ξ. The degree of stability η characterizes the speed of the system, while the criterion ξ is suitable for assessing the stability margin of the system [18,19,20].

6. Dynamic Characteristics of the Bearing

The calculation of the criteria for the dynamic quality of the bearing was carried out at a unit dimensionless mass M_s = 1, varying the parameters P_s, R₁, χ, H_d0, σ, and V_k. The graphs in Figure 5, Figure 6 and Figure 7 show curves for P_s = 4, R₁ = 0.75 and χ = 0.45.

Among the parameters that do not affect the static characteristics but affect the dynamic characteristics of the bearing are the deformation parameter H_d0, the compression number σ, and the volume of the microgroove V_k. The influence of these parameters is of particular interest, since they are a resource for optimizing the dynamic characteristics of the bearing.

Figure 5 and Figure 6 show the dependences of the stability degree η and damping for the period ξ on the volume V_k at a fixed σ = 25 and different values of deformation H_d0 for the regime of the design point.

The graphs show that an ordinary support with a rigid center 5 (H_d0 = 0) is stable only for a small V_k. However, with an increase in H_d0, when the rigid center, 5, acquires the ability to perform additional displacement, the support becomes stable even at the volume of the microgroove, which is larger by one order of magnitude or more than the volume of the bearing layer. Moreover, even a small displacement has a significant effect on the stability of the support: for example, an unstable conventional support already becomes stable at H_d0 = 0.1. A further increase in H_d0 contributes to an increase in the response rate, which, depending on the values of other parameters, can fluctuate within a significant range of H_d0 = 0.25–0.5. For the graph in Figure 5, the maximum response rate falls on H_d0 = 0.32. A further increase in H_d0 leads to a decrease in the response speed, while the support remains stable.

The graph in Figure 6 shows the dependences of the criterion ξ on the transient characteristic’s oscillation. This makes it possible to assess the stability margin of the bearing.

It can be seen that at H_d0 > 0.1 near the extremum points, there is a rapid increase in the values of the criterion ξ, which indicates a decrease in the oscillation of the system and, consequently, an increase in its damping. Already at H_d0 > 0.2 for V_k > V_k,opt, where V_k,opt is the volume corresponding to the maximum value of the stability degree η, the criterion ξ > 90%, which characterizes the bearing as a well–damped dynamic system [14,20,21].

Figure 7 shows the dependence of the stability degree η on the parameter σ, which also only affects the dynamic properties of the bearing. It can be seen that at the design point, the bearing can be both stable (η > 0) and unstable (η < 0). An increase in the volume V_k of the microgroove promotes an expansion of the stability area and an increase in the performance of the bearing. The dependences η (σ) are also extreme. This means that each value of the volume V_k corresponds to the optimal value of the number σ from the point of view of speed. With an increase in V_k, the speed peak shifts towards a lower σ, which corresponds to large values of the dimensional gap h₀. In this case, the peaks themselves describe an extreme curve, which indicates that there is also an optimal value V_k in terms of speed. Together with the previously made conclusions, this indicates that for each set of values of the parameters P_s, R₁, and χ, there is an optimal set of values for the parameters H_d0, σ, and V_k, which only affect the dynamics of the bearing. However, optimization of the design point mode does not guarantee that the bearing will have optimal dynamic characteristics over the entire range of load variation F. In this range, with large volumes of microgrooves, such a bearing may even be unstable. As the analysis of the calculated data has shown, the instability region can occupy up to half of this range, which falls on small and moderate loads.

From the graph in Figure 8, which shows the curves of the dependence of the stability degree η on the load F, it can be seen that at large volumes V_k and small deformations H_d0 in the region of small and medium loads, the bearing is unstable. Stability in this range is provided only for H_d0 > 0.4.

Analysis of the graph in Figure 8 shows that the best in terms of performance is the curve corresponding to H_d0 = 0.5. For this mode, stability takes place in the entire range of loads at σ = 25, V_k = 16.

7. An Example of a Bearing Design Calculation

To conclude the study, we give an example of calculating the dimensional characteristics of the bearing, corresponding to σ = 25, V_k =16.

Let us take the bearing radius r₀ = 40.10–3 m and the ambient pressure p_a = 0.1 · 10⁶ Pa. The maximum dimensionless bearing load F_max = 2.42, and the maximum load $f_{\max }=\mathsf{\pi }{r}_0^2{p}_a{F}_{\max }=1.2\cdot {10}^3\mathrm{N}.$ The dimensionless design load at χ = 0.45 is F = 1.42, and the dimensional design load is $f=\mathsf{\pi }{r}_0^2{p}_{a}F=0.71\cdot {10}^3\mathrm{N}.$

Using the expression for the dimensionless mass of the shaft at M_s = 1, we find the formula for calculating the scale of the real time:

$$t_0=\frac{1}{r_0}\sqrt{\frac{m_s{h}_0}{\mathsf{\pi }{p}_a}}.$$

(55)

Substituting Equation (55) into the expression for the compression number σ of problem (32), we obtain the formula for calculating the gap at the design point:

$$h_0=r_0\sqrt[5]{\frac{\mathsf{\pi }{r}_0}{m_s{p}_a}{\left(\frac{12\mathsf{\mu }}{\mathsf{\sigma }}\right)}^2}.$$

(56)

Taking the mass of the shaft m_s = 5 kg and the viscosity of the air μ = 17.2 · 10⁻⁶ Pa.s, using Equation (56), we find the gap h₀ = 18 · 10⁻⁶ m. Using Equation (55), we find the scale of the real time t₀ = 0.42 · 10⁻³ s. The well–known formula [14] for determining the duration t_p of the decay of the transient response at η = 0.07 gives $t_{пп}=\frac{3t_0}{\mathsf{\eta }}=0.02\hspace{0.17em}s.$

At temperature T = 293 K, a gas constant $\Re$ = 287.14 m²/s² Kz and an adiabatic air expansion index γ = 1.4, we calculate $\mathsf{\Gamma }=2\sqrt{\mathsf{\gamma }}{\left(\frac{2}{\mathsf{\gamma }}\right)}^{\frac{\mathsf{\gamma }+1}{2(\mathsf{\gamma }-1)}}$ = 1.7. Dimensionless pressure in the microgroove $P_k=\sqrt{1+\mathsf{\chi }\left(P_s^2-1\right)}=$ 2.78. From the condition of equality of the flow rates Q_k and Q_h, we find the criterion for the similarity of the diaphragm $A_k=\frac{A_h\left(P_k^2-1\right)}{\mathsf{\Pi }\left(P_s,P_k\right)}=12.8$. Let us take the number of diaphragms n_k = 3. Using Equation (15), we calculate the diameter of simple diaphragms $d=h_0\sqrt{\frac{A_k{p}_a{h}_0}{3\mathsf{\mu }{n}_k\mathsf{\Gamma }\sqrt{\Re T}}}=0.4\cdot {10}^{-3}\mathrm{m}.$

Using Formula (14), we determine the dimensionless design lubricant flow rate Q = 23. The dimensional mass flow rate $q=\frac{\mathsf{\pi }{h}_0^3{p}_a^2}{12\mathsf{\mu }\Re T}Q=2.4\cdot {10}^{-3}\frac{\mathrm{kg}}{\mathrm{s}}.$

The structural volume at design point $q=\frac{\mathsf{\pi }{h}_0^3{p}_a^2}{12\mathsf{\mu }\Re T}Q=2.4\cdot {10}^{-3}\frac{\mathrm{kg}}{\mathrm{s}}.$

The cross–sectional area of the microgroove is $s_k=\frac{v_k}{2\mathsf{\pi }{r}_0{R}_1}=1.44\cdot {10}^{-6}{\mathrm{m}}^2,$ while its depth is $l_k=\sqrt{s_k}=$1.2 · 10⁻³ m. For dimensionless K_e = 0.731, we obtain the dimensional coefficient of elasticity $k_e=\frac{K_e{h}_0}{\mathsf{\pi }{r}_0^2{p}_a}=4.67\cdot {10}^{-6}\frac{m}{N}.$

8. Conclusions

This paper proposes an improved technical solution for an axial aerostatic bearing with an elastic suspension of the supporting disk, simple diaphragms, and an annular microgroove. The results of mathematical modeling and theoretical research of stationary and non-stationary operation modes of support are presented, and the possibility of improving its dynamic characteristics by reducing the volume of the central part of the bearing layer is shown. It has been established that the optimal choice of the compression number, the volume of the microgroove, and the coefficient of elasticity of the suspension, which ensures the displacement of the support center, provides the bearing with high performance and a high margin of stability. It is shown that the use of a moving center makes it possible to reduce the volume of the bearing layer by a factor of two or more and, due to this, to increase the speed of the bearing by 3–4 times. With optimal adjustment of the bearing parameters, the oscillation of the transient processes decreases until their transition to a qualitatively new state—they become almost aperiodic, which indicates that the bearing acquires the properties of a well-damped dynamic system. An example of the design calculation of the structure is given.