Theoretical Study on Dynamics Quality of Aerostatic Thrust Bearing with External Combined Throttling

The article considers the design, mathematical modelling and theoretical study of dynamics quality on aerostatic thrust bearing with external combined air double throttling and flow cavity. The use of the finitedifference method for solving boundary value problems for linearised and Laplace-transformed Reynolds equation allowed to obtain its solution with high accuracy. The quality indicators of bearing dynamics were studied depending on the damping and throttling resistances, as well as on the volume of the cavity. It is established that the presence of a cavity contributes to a significant improvement in the quality indicators of the bearing dynamics. Based on the analysis of the data obtained, conclusions are drawn confirming the hypothesis that with a targeted choice of the values of the external throttling system parameters, a design can be obtained that, in comparison with the known aerostatic bearings, will have much better dynamic characteristics.


INTRODUCTION
Aerostatic bearings are widely used in various fields of mechanical engineering, providing high accuracy, low friction and low heat [1 -6].
Passive type aerostatic bearings with throttle control of air discharge have received the main practical application. Structures having input throttling resistances in the form of nozzles with simple diaphragms and shallow pockets at their outlet have the highest load characteristics [7,8]. To reduce the number and increase the diameter of the nozzles on the working surface of the bearing sleeve along the nozzle location line, micro grooves are applied [9,10].
Aerostatic bearings with inlet throttling resistances in the form of nozzles with annular diaphragms have one and a half times less compliance and worse load characteristics [11].
Aerostatic bearings have an increased tendency to unstable operating modes (such as "pneumatic hammer") due to the high compressibility of the air. Traditional methods for increasing the stability of such bearings are usually reduced to the ultimate reduction in the volume of cavities (grooves, pockets) enclosed between external chokes and the carrier gas layer, which complicates the construction of the bearing and complicates its manufacture [12,13].
The insufficiently studied method of increasing the stability of aerostatic bearings based on the use of closed resonant cavities connected by a throttling channel to the supply pocket of the bearing is quite noteworthy [14][15][16]. Such bearings may well have better dynamic characteristics and, therefore, external chokes can be placed in them, which would provide the least flexibility without compromising the quality of the dynamics of the bearings.
The purpose of this theoretical study is to examine the dependence of the dynamic characteristics of the bearing on the ratio of the resistances of the damping and throttling nozzles, as well as to verify the assumption that, with the presence of an additional flow resonant cavity, the cavity should improve the dynamic characteristics of the bearing. Fig. 1 shows the design diagram of the considered aerostatic bearing, which has a fixed thrust bearing 1 and a rotating heel 2 of radius r b . The mating working surfaces of the heel and the thrust bearing are separated by a thin carrier gas gap 6, which is created by external injection of compressed air and non-contact balances the effect of the external load f. Heel 2 has a simple diaphragm 3, through which compressed air is pumped into the inter-throttle resonant cavity 5. From the cavity compressed air enters the carrier gas gap of the bearing through a system of damping annular diaphragms 4, evenly spaced around the circle of radius r c . The fundamental difference between the considered bearing and conventional bearings is that cavity 5 is made flowing and is located in the discharge path between throttle 3 and damping annular diaphragms 4.

DESIGN SCHEME AND MATHEMATICAL MODEL OF THE BEARING
The following notation is used in the work: μ is the coefficient of dynamic viscosity of air; T is the absolute air temperature; ℜ is the gas constant; γ = 1.41 is the adiabatic exponent [14]; r b , r c are the radii of the bearing; r is the current coordinate of the radius; t is the current time; c k , c p are the coefficients of outflow from the throttling and damping nozzles; m is the mass of the heel 2; p (r, t) is the pressure function in the carrier gas gap; q p (t), q k (t), q h (t) are the mass air flow rates through the throttles 3, 4 and the carrier gap 5, respectively; f is the external load.
A theoretical study of the bearing dynamics quality is performed using dimensionless quantities.
The following are taken for their scale: r b for radii; p a for pressures; h 0 for the current gap thickness; t 0 for the current time; 3  for forces, where h 0 is the gap thickness in the so-called "design point" mode. The "design mode" corresponds to dimensionless static pressures P k = P k0 , P p = P p0 and air film thickness H = 1, which can be calculated through the throttling and damping coefficients of the lubricant supply system into the bearing gap At ς = 0, the developed mathematical model corresponds to an aerostatic bearing with a single external throttling with simple diaphragms, and at ς = 1 -an aerostatic bearing with a single external throttling with annular diaphragms. Intermediate values ς correspond to a bearing with a combined external double throttling and an inter-throttle resonant cavity.
Using (1), we determined the dimensionless static analogues of boundary pressures The dynamics of the bearing under consideration is described by a system of dimensionless equations of continuity of air flow through cavity 5, carrier gap 6, and the equilibrium equation of forces acting in a conservative system (d'Alembert principle) [17] 0, 0, where the flow rate through resistance 3 [18].
flow rate through resistance 4 [18] ( , ), flow rate taking into account the compressibility of the lubricant in the cavity 5 [7] , τ flow rate in the lubricant gap with thickness H [7] Prandtl function for flow rate from diaphragms [18,19] ( ) as well as dimensionless parameters: τ is the dimensionless time, is gas constant.
The pressure function P(R,τ) is a solution of the nonstationary Reynolds equation [5] ( ) with initial condition 0 ( ,0) ( ) P R P R = (12) and boundary conditions is the solution of a stationary problem Using (2), we write the system of equations of the stationary equilibrium of the bearing in the "design point" mode As follows from (5), (12) At known pressures of the "design point" mode and supply pressure, similarity criteria for the throttles can be found

METHOD FOR CALCULATING THE STATIC AND DYNAMIC CHARACTERISTICS OF THE BEARING
When calculating the static characteristics of the bearing, P s , R c , χ, ς were used as input parameters. Then, at H = 1, the pressures of the "design point" mode P k0 , R p0 were determined by formulas (2) and the similarity coefficients A h , A k , A p by formulas (16), (17).
To construct the load characteristic H(F) by varying the pressure P k ∈ [1, P s ], we calculated the static bearing capacity and determined the current thickness H of the bearing carrier gap by solving the system of nonlinear equations arising from (3) -(5), (7), (14) ( ) System (19) can be reduced to a nonlinear equation 2 3 / 2 The equation (20) was solved by the numerical bisection method [20], then gap H was determined using the formula In studying the bearing and finding a solution to the problem (11) − (13), it was assumed that the deviation of dynamic functions from their stationary values is insignificant. Moreover, dynamic functions were represented as where D 0 , ∆D(τ) are the static part and the small dynamic component of the function.
Linearised and Laplace-transformed equations (2) have the form where, in the general case, the Laplace transform of the deviation of the dynamic function ΔD(τ), s is the variable of the Laplace transform. The Laplace transforms of the mass flow deviations included in (21) can be written as For bearing capacity and inertia, the corresponding Laplace transforms can be written as The functions W Δ , h Q Δ are determined by the distributed pressure transform P Δ , which, as follows from (6) − (8), is a solution to the boundary value problem ( ) ( ) 3 σs , , where H, P is the static gap and the function of the static pressure distribution.
A finite-difference grid method was applied to problem (28), which allows one to obtain its solution with high accuracy. The variable s in this case acts as a complex parameter.
For this, segments [0, R c ] and [R c , 1] were divided into n equal parts and, having performed the transition from the differential form to the algebraic form in (28), we obtained a system of algebraic equations for each segment ( ) where g is the grid step, i = 1, 2,..., n.
The boundary conditions for the first segment are the equations 2 1 0 for the second one 0 , 0.
where U is the desired function, For R = 0, the boundary condition is the dependence which approximates the first derivative of the function U(R) at the endpoint R = 0 with an accuracy of the order of O (g 2 ) [21]. Problem (30), (33) was solved by the sweep method [22], accepting 1 .
Here, α, β, s, P are the input quantities, A w A q are the desired coefficients for the functions or for the transforms of the bearing capacity and air flow rate at the entrance to the carrier gap of the bearing, where β = 0 corresponds to the segment [0, R c ], β = 1 to segment [R c , 1].
When calculating the criteria of the bearing dynamics for each value of the complex variable s, numerically solving the problems Prb(0, 0, s, P, A w00 A q00 ), Prb(1, 0, s, P, A w10 A q10 ), Prb(1, 0, s, P, A w10 A q10 ), Prb(1, 1, s, P, A w11 A q11 ), we can get the output parameters and find the quantities  [23], which gives a solution to the system (38) in the form of a relation of determinants of matrices whose elements, as mentioned, are the values of polynomials of the variable s, and therefore this relation itself will also be the value of a rational function of this variable. This circumstance is the basis and is used below nor the development of an algorithm for determining the coefficients of the transfer function of the bearing, as a dynamic system.
Coefficients A wk , A wh were calculated using the Simpson method of calculating a certain integral [24]. Coefficients A qk , A qh were calculated using terminal finite-difference formulas providing quadratic accuracy O(g 2 ) for calculating the derivatives of the functions U(R) in expressions (19) at the ends of the integration intervals [25].
The Laplace analogue of the dynamic compliance of the bearing is the function where according to Cramer's rule [23] ( ) ( )

m k p k h w k k k h s w k h s D A A A A A A AA A A
Static compliance of the bearing in the "design point" mode As mentioned above, in the general case, the matrix components of the system (39) are rational functions of the Laplace variable. Therefore, function K(s) will also be a rational function of this variable and can be represented in the form  where m < n, m > 0, n > 0; n, m are natural numbers, a i , b i are real numbers. Number m is constant for the transfer function K(s) and is determined by its smallest natural value, for which the infinite limit is ( ) 0. Numerical experiments showed that for a given transfer function m = 2. This completely corresponds to the difference in the polynomial orders of the transfer functions of compliance of gas-static bearings, the models of which take into account the influence of the heel mass on their dynamics [26 -28].
To determine n, the value of which depends on the accuracy of ε calculations, an iterative algorithm is used, the stopping criterion of which is the fulfillment of the condition Having set a sufficiently small ε and n = 2, we solved the system of equations (43), then checked condition (44). If it was not fulfilled, then at a new iteration, n was increased by one and the system (43) was again solved until condition (44) is satisfied.
Calculations showed that at ε = 10 -4 the order of the characteristic polynomial in most cases is in the range n = 3-6.
To assess the quality of the dynamics of the bearing, the root criteria were used [30]: The degree of stability η characterises the speed of the system, i.e. the rate of attenuation of its free oscillations.
The criterion "attenuation of oscillations over a period" ξ can be applied to the assessment of the stability margin of the bearing. The smaller ξ, the greater the oscillation will have a transient response, and the system − a smaller margin of stability. It is believed that a dynamical system is well damped if ξ ≥ 90% [30].
The oscillation index M = |K(jω r )/K 0 |, representing the relative magnitude of the maximum amplitudefrequency modulus at the resonance frequency ω r reduced to static compliance K 0 , characterises the vibrational properties of the bearing. It is known that the larger M and the higher the resonance peak of the frequency response, the higher the tendency of the dynamic system to fluctuations in frequency response and, conversely, the smaller M, the greater its stability margin is.

THEORETICAL ANALYSIS, RESULTS AND DISCUSSION
The dimensionless input parameters were used in the calculations: the shaft mass M p = 1, the boost pressure P s , the radius R c , the coefficients χ and ς, the number n of division of the segments of the finite difference grid, as well as the "compression number" σ and volume V p . The latter two parameters affect only the dynamic characteristics of the bearing.  Fig. 2 shows the curves of the static load characteristics of the bearing, which are obtained by solving a system of nonlinear equations (19). The curve ς = 0 corresponds to the bearing with simple diaphragms, ς = 1 corresponds to the bearing with annular diaphragms.
As mentioned above, at the "design point" mode H = 1, the compliance of the latter is 1.5 times higher than the compliance of the former. It follows that, from the point of view of ensuring the minimum conformity of the bearing, values close to ς = 0 are desirable. Other curves on Fig. 2 correspond to the bearing with combined throttling.
A more detailed idea of the static compliance of the bearing is given by presented by the curves in Fig. 3, which are obtained by calculating the function K(s) at s = 0 in the "design point" mode of the load curve H(F) of Fig. 2 for which H = 1.
The curves confirm the conclusions made on the basis of the analysis of load characteristics that less flexibility corresponds to a conventional bearing with simple diaphragms (ς = 0), and the greatest compliance is supported by the bearing with annular diaphragms (ς = 1).
Other curves relate to the bearing with combined external throttling.
The data presented in the graphs also show that if the dependences K 0 (ς) are monotonic functions, then the dependences K 0 (χ), on the contrary, are unimodal extremal functions. For the given curves, a minimum of compliance occurs at χ = 0.42. Fig. 4 shows the curves of the dependence η (ς), which allow to evaluate the influence of the tuning coefficient ς on the dynamics of the bearing with external combined throttling in the vicinity of the "design mode" mode.
The main conclusion is that, given the optimal selection of the values of the dynamic parameters σ and V p , this dependence is extreme, and the greatest response speed η ≈ 0.9 takes place at ς = 0.1 -0.2, i. e. near the setting factor of the best from the point of view of minimum compliance flexibility ς = 0. As the curves show, the speed of the bearing with combined throttling is at least 4 times higher than that for the bearing with annular diaphragms (ς = 1), which of the single throttle designs is It is the best in terms of dynamic indicators. The data presented in Fig. 5 show that the criterion for the speed η of the bearing extremely depends on the parameters V p and σ. Fig. 6 demonstrates the idea of the vibrational properties of the characteristics, as the curves show the dependences of the oscillation damping index for the period ξ (ς). It can be seen that the extreme modes ς = 0 and ς = 1, corresponding to the single throttle bearings, are characterised by unacceptably high oscillation, or, as follows from Fig. 3, the instability that can occur in bearings with simple diaphragms.
In the range 0.1 < ς <0.8 at V p > 20 and σ > 10, the quality of the dynamics of the bearing corresponds to ξ = 100%, i. e., aperiodic transition characteristics, providing the bearing with a guaranteed margin of stability.
The best values of the analysed parameters ς, σ, V p from the point of view of minimum oscillation of the system are given in Fig. 7, which shows the dependences of the oscillation index M. The unsteady movement of the bearing not only at the "design mode", but also in the range of permissible loads on the structure.  Fig. 8 shows the curves that prove these conclusions. As follows from the data above, the bearing tends to the boundary of its stability (η = 0) at the maximum load, which corresponds to the clearance H = 0. The bearing can lose stability at extremely small loads, when the gap H is 2 to 3 times higher than the calculated operating value H = 1. It is also seen that by increasing the value of the parameter σ this boundary can be substantially shifted to the area of large gaps. It was found that the increase in the volume V p also contributes to the expansion of the range of stable operation of the bearing at extremely small loads.

AN EXAMPLE OF DIMENSIONAL CALCULATION OF THE BEARING
Let us consider a bearing with an outer radius r b = 4·10 -2 m. Let us take environmental pressure p a = 0,1MPa, air viscosity μ = 18·10 -6 Pa·s, heel mass m p = 3 kg. The dimensionless quantities are P s = 5, R c = 0.5, χ = 0.42, ς = 0.2, σ = 10, V p = 30, η = 0.75. Using the expressions for the dimensionless mass at M p = 1 and the "compression number" σ, we found the gap h 0 of the "design point" mode, the time scale t 0 , the decay time of the transition characteristic t h [30], and the volume of cavity 7 In this case, the bearing capacity of the bearing in the "design point" mode is w = 720 [N], the static compliance k 0 = 1.9·10 -8 [m/N].

CONCLUSION
A mathematical model of the dynamics of the axial aerostatic bearing has been formulated, numerical methods for determining its static and dynamic characteristics have been developed.
The application of the approach based on the finitedifference method for solving boundary value problems for the linearised and Laplace-transformed Laplace-Reynolds equations to the study of bearing dynamics made it possible to obtain its solution with high accuracy and thereby eliminate the issue of the accuracy of calculating bearing stability criteria that arise when using approximate analytical methods [2,7,9].
Theoretical analysis showed that the optimal choice of parameters of the throttles and the volume of the resonant cavity can significantly improve the dynamic characteristics of the considered bearing. The considered bearing has the highest dynamic characteristics for the volume of the inter-throttle resonant cavity by an order of magnitude and more than the volume of the carrier gas gap, which allows the creation of aerostatic bearings with actively controlled throttling resistances (elastic diaphragms, piezoelectric regulators, etc.), as well as the use of double combined external throttling with inter-throttle resonant chamber for closed axial, as well as radial and radial-axial aerostatic bearings.
The dynamic characteristics of such bearings is many times better, which makes it possible to significantly improve the load characteristics at the same dynamic performance with aerostatic bearings having single external throttling with simple or annular diaphragms due to increased pressure of the injected air.
Thus, an external throttling system is a universal means of improving the quality of the dynamics of gasstatic supports when using devices that help to maintain their low compliance.