Static Characteristics of a Hydrostatic Thrust Bearing with a Membrane Displacement Compensator

The article discusses the design and also presents a mathematical model and method for calculation of static characteristics of a hydrostatic thrust bearing with a membrane-type displacement compensator. Formulae for calculation of compliance and load characteristics of the bearing are also presented, as well as deformation and compliance of the membrane at which zero compliance of the bearing is guaranteed.


INTRODUCTION
Units with non-contact hydrostatic slide bearings are widely used in metal-cutting machines for precision and high-speed machining, micro-machining, in heavy and unique machine tools. Hydrostatic bearings provide precision and cleanliness of processing, load characteristics, vibration resistance, and high durability of ultra-hard cutting tools that cannot be provided by other types of sliding and rolling bearings [1][2][3].
The main prospects for further research and development in this area of science and technology are in the area of creation and complex use of spindle assemblies and guides with adaptive hydrostatic bearings, which have regulators of active lubricant injection, in metal-cutting machines. Further improvement of technical solutions, development of theory and methods for optimal design of spindle assemblies and guides with adaptive hydrostatic bearings is an urgent scientific and technical problem of mechanical engineering, the solution of which can significantly increase the accuracy and productivity of machining with metalcutting equipment [4,5].
Modern hydrostatic bearings should have a compact and technologically advanced design with elastically built-in, elastomeric, floating and other regulators of performance characteristics, surpass their counterparts in accuracy, load-bearing capacity, compliance and other important performance characteristics.
Mechanical engineering, and particularly heavy machine-tool construction, needs such hydrostatic bearings with active displacement compensators that have the ability to provide low positive, as well as zero and negative compliance [6,7]. In comparison with bearings with active flow rate compensation, such structures are distinguished by significantly lower energy consumption and the ability to compensate for movements with amplitudesat which other hydrostatic bearings would be obviously inoperative [6]. In addition, bearings with displacement compensation have more stable load characteristics and significantly better dynamics when operating in low compliance modes [8].
The role of active displacement compensators can be performed by elastic rings or membranes, the use of which is preferable due to longer retention of stable elastic characteristics [9,10].
Shown in Figure 1, the bearing structure contains a fixed 1 and a movable 2 disks, which are connected to each other by an annular membrane 3, which together with Disk 2 acts as a movement compensator. Elements 1 -3 form the base of the bearing. The suspended part is Disk 4, which takes an external load f and is separated from the base of the bearing by a lubricating gap of thickness h. The bearing is powered by the inlet throttle 5, which is supplied with lubricant under pressure p s . At the outlet of the throttle, the lubricant enters the flow cavity 6 under pressure p k and then, having overcome the carrier layer, flows out of the bearing.
where r 0 , r 1 are the outer and inner radii of the membrane and the movable disk respectively, g(r) is the load on the membrane, δ is the membrane thickness, ν is the Poisson's ratio, E is the elastic modulus of the membrane material [11].
The bearing capacity w is the sum of integral hydrostatic forces 0 1 where w 1 is the hydrostatic force applied to the suspended Part 4 from the side of the central circle of radius r 1 ;w 2 is the hydrostatic force applied simultaneously to the annular periphery of the suspended Part 4, on the movable Disk 2, and the membrane. The study of static characteristics of the bearing is carried out in a dimensionless form. The scale for the main variables is as follows: the outer radius r 0 for linear dimensions, p s for pressures, πr 0 2 p s for forces, h 0 3 p s /12µ for volumetric flow rates of the operating fluid flow, h 0 for the lubricating gap and small displacements, where h 0 is the thickness of the gap h in the calculated point, µ is the viscosity of the lubricant.
Below, dimensionless variables are designated in capital letters.
Solution to problem (1) for dimensionless pressure function would be: where R 1 is dimensionless inner radius of Disk 2 and the membrane, P k is dimensionless pressure at the outlet of the throttle and at the entrance to the lubricating gap.
Taking into account (3) -(5), formulas were obtained for dimensionless hydrostatic forces and load capacity of the bearing Dimensionless flow rates through the bearing and the throttle are determined by the formulas 3 1 , ( 1 ) , (9) where A d is the throttle parameter. When solving boundary problem (2), it is convenient to represent the main differential equation and boundary conditions in the following form: where U(R), Ф(R) are dimensionless functions of the deflection and inclination angle of the membrane, The solution of the boundary problem (10) for functions Ф and U was obtained by numerical grid sweeping method [12]. For this, segment where R i = 1+iτ, Ф i = Ф(R i ), U i = U(R i ), τ = (1-R 1 )/n is the grid step.
The deflection of the membrane U 0 =U(0) determines the axial displacement H m of the movable Disk 2.
The design conditions of the bearing were found from condition h = h 0 , i.e., at H = 1 and thedefined value of the throttle resistance adjustment factor, which in the design mode is equal to pressure P k . Having comparedflow rates (8) and (9) in this mode, we found the throttle parameter In the calculations, the following parameters were used as input parameters: χ, R 1 , K m , P k , B. Taking into account the fact that the thickness of the bearing gap is three orders of magnitude smaller than its characteristic dimensions, it has been assumed that B = 1000.

MEMBRANE DEFLECTION CHARACTERISTICS
Calculations have shown that the deflection function U(R) is proportional to parameters P k , K m and therefore, can be represented as It has been found that if the deflection model takes into account only the edge contact of the membrane with the movable disk (R x ≈ R 1 ), then the function V(R) calculated when solving problem (11) turns out to be convex (Fig. 2, curve 1), which does not correspond to the operating conditions of the bearing, since for design reasons this curve should be monotonic. In the calculations, the convex section is eliminated by increasing radius Rx, which means the area of surface contact between the membrane and the diskto which the hydrostatic force W 2 is applied. This causes a decrease in the specific load (pressure) on the membrane from the side of the movable ring 2, which should contribute to increase in membrane deflection. The minimum radius R x of the membraneto Disk 2 contact has been found among monotonic functions V(R) corresponding to the condition dV/dR ≤ 0. In Fig.  2, this corresponds to curve 2, for which R x = 0.638. Figure 3 shows the dependences V(R) at different values of R 1 . As can be seen from the graphs, these relations significantly depend on this radius.
In the range R 1 ∈ [0.35, 0.65], the value H R = V(R 1 ) with an error of no more than 0.01 can be calculated using the approximate formula 1 1 16.19 26.85 11.2. The resulting dependence enables to obtain a simple model for calculation of static characteristics of a bearing in analytical form and obtain formulae for the characteristic modes of its compliance.

BEARING CHARACTERISTICS
Bearing compliance is determined by the formula K = -dH s /dW, where , where according to (6) P k = W/A w . Substituting χ = P k , in (14), we find the formula to determine the bearing compliance in the design loading mode Formula (15) allows us to find the value of the cylindrical compliance of the membrane at which the bearing in the design mode will have zero compliance So, at χ = 0.5 and R 1 = 0.5, the bearing has zero compliance at K m0 = 4.55. Figure 4 shows the dependences of the bearing compliance K 0 in the design loading mode on the throttle setting parameter χ. With increase in compliance K m , the bearing compliance K 0 decreases and at K m = K m0 and at a certain value of χ the bearing reaches zero compliance(K 0 = 0).With a further increase in K m and negative compliance (K 0 < 0) curvesK 0 (χ)become extremal. The minimum of this dependence occurs at 0 / χ 0.
Differentiating (15) and solving this equation, we find the optimal value of the coefficient χ = 0.5. Graphs in Fig. 4 confirm this result.  Figure 5 shows the load characteristics H s (W) of the bearing for the modes of positive, zero and negative compliance. The calculation of the characteristics was carried out in a parametric form according to formulas (13), (6), (7)with pressure (0,1) k P ∈ as parameter.
It can be seen that in active displacement compensation modes the characteristics retain a fairly wide range of stable compliance in the region of moderate and large loads up to the maximum ones.  Figure 6 give more details about the bearing compliance in the range of its loads.

Curves in
The graphs show that the minimum of the compliance function K(W) corresponds to the load W k >W 0 , where W 0 is the load corresponding to the design clearance H = 1. For these curves, W 0 = 0.271.
The minimum value of compliance and the corresponding load can be found from equation

CONCLUSION
The results obtained allow us to conclude that open hydrostatic thrust bearing with membrane displacement compensators have a number of important properties that significantly expand the area of efficient application of active hydrostatic bearings in metal-cutting machines in order to improve the processing quality. In particular, they make it possible to compensate for large deformations of the elastic system of machine tools by using the negative compliance mode, which is difficult or impossible to achieve using conventional hydrostatic bearings or bearings with compensation of the working fluid flow rate. The discovered proportional relation between the membrane deflection and the product of its cylindrical compliance and the applied pressure has enabled us to build a simple engineering model based on elastic properties of the membrane. This model is a basis to develop an analytical method for calculation of parameters and static operating characteristics of the bearing.