Design of the Hydrostatic Thrust Spherical Bearing with Restrictors (Fitted Type)

The hydrostatic bearings are broadly observed in industrial applications owing to their potential characteristics of low running friction, high load-carrying capacity, high stiffness and small viscous dissipation. Spherical bearings have advantages over other configurations of bearings because of their self-aligning property and also their capacity to accept both radial and thrust loads. Operation of the spherical bearings is unaffected by angular miss-alignment. They are mainly used in machine tools, precision measuring instruments, hydraulic piston pumps and motors, gyroscope gimbals, telescopes, radar tracking units, air craft engines and rotary drum of centrifugal separator for enriched uranium. The main objective of this part of study is to handle the fitted bearing design taking into consideration the combined effects of rotational inertia, surface roughness and the fluid film viscosity variability. The mathematical expressions derived in part 2 for this type of bearings with its different configurations (with and without recess; hemispherical and partial hemispherical seats) have been used in the detailed bearing design. Improving the previous design through developing some mathematical expressions, a new conception is adopted to reveal the bearing consistency showing how the fluid flow rate pulsation or intermittent could be prevented. The detailed design procedures show the advantages of this method to get the central pressure ratio mathematically through maximizing the flow rate with minimizing the supply pressure which means minimum pump power to get maximum discharge. This design showed very high mean static stiffness of all the bearing configurations and an acceptable temperature rise for a heavy duty bearing without the need to a use a cooling system.


Introduction
Nowadays humanity is much more dependent upon the mechanical machines for their daily needs. To take care of smooth operations of the relatively moving surfaces of the machine without wear and tear, many different bearing types and configurations (dynamic and hydrostatic) have been designed. Hydrostatic bearing design requires adjustment of a number of parameters including the bearing geometry, restrictor size and type, supply pressure, central pressure, surface roughness, rotational speed, lubricating fluid properties and bearing eccentricity to optimize performance. The flow restrictor is necessary to provide a pressure drop between the supply and the inlet pressures, where design of flow restrictors influences bearing stiffness, pumping power, supply pressure, and lubricant flow. While the load carrying capacity is dependent on the bearing geometry, eccentricity, rotational speed, surface roughness, fluid viscosity and the central pressure it is very difficult to design a robust, consistent and highly stiffened thrust spherical bearing.
Dowson and Taylor [1] studied the hydrostatic spherical bearing with adiabatic lubricant flow considering the inertia effect on the bearing performance concluding that the inertia affects positively the fitted bearing performance and the adiabatic temperature rise decreases with the increase in eccentricity pointing to neither the central pressure ratio nor the bearing static stiffness.
Row [2][3] studied and designed the externally pressurized bearing concluding that the multi-recess bearing is usually preferred than that of the single recess and the design pressure ratio should be 0.5 for the spherical bearing considering lubricant constant viscosity without pointing to the bearing stiffness.
Shijiang and Decheng [4] studied the spherical bearing with what is called a pressure tracing system which is suitable for heavy duty forging presses; it is concluded that in practical designs throttling ratio (pressure ratio inverse) should be greater than 1.5, i.e. the inlet pressure should be less than (2/3) the supply pressure and this pressure tracing system results in more stiffened bearing. Toshiharu Kazama [5] offered an optimal bearing design concluded that for the fitted type of thrust spherical bearings the optimum design pressure ratios are (2/3 and 0.6907) for capillary tube and orifice restrictors respectively without pointing to the bearing stiffness.
Satish C. Sharma et al [6 -7] described theoretically the performance of circular and journal hydrostatic bearings having recesses of different geometric shapes. The computed results indicated that: to get an improved performance a proper selection of the geometric shape of the recess in conjunction with the type of restrictor and the value of the restrictor design parameter is essential.
Lin and Chiang [8] studied and designed the hydrostatic circular thrust bearing showing that the central pressure ratio could be between 0.5 and 0.7 and the stiffness depends on the restrictor parameter, radius ratio and the film height.
Ahmad W. Yacout et al [9] studied the fitted hydrostatic thrust spherical bearing considering the combined effects of the inertia and the surface roughness on the bearing performance showing that the central pressure ratio is (2/3) regardless the restrictor type and offering a bearing design based on minimum losses and optimal restrictor dimensions.
Lu Lihua et al [10] proposed a fluid-structure interaction model to calculate the static performance of hydrostatic bearing where the temperature rise, the pressure distribution and static stiffness were calculated by computational fluid dynamics (CFD). When comparing the calculated results with those got empirically, the static stiffness showed 78% drop. Elescandarany [11][12] studied the effect of the fluid viscosity variation on the fitted hydrostatic thrust spherical bearing performance with and without recess taking into consideration the effects of the inertia and the surface roughness showing that the central pressure ratio is (2/3) and offering a bearing design with temperature rise less than 14 degrees centigrade, in addition to a comparison between the two bearings designed in Yacout et al [9] and in Elescandarany [11][12].
Due to the lack of theoretical researches handling the detailed design of such type of bearings taking into consideration the effects of the surface roughness, the inertia, lubricant variable viscosity and the bearing geometry which could not only be difficult but also tedious, the present study offers detailed procedures and calculations to design a robust, consistent and highly stiffened fitted hydrostatic thrust spherical bearing with its different configurations using a capillary tube and orifice restrictors revealing the mathematical relations between the seat dimensions and the eccentricity and determining the maximum central pressure ratio which gives the bearing its consistency, the localized static stiffness and its mean value, the geometry factor and the temperature rise and distribution 'theoretically and numerically' where all these items were absent in the author previous studies [9,[11][12]. All bearing parameters were calculated for the different bearing configurations using both restrictors to enable the selector choosing the most appropriate configuration serving his purposes.

Design Considerations
There are five groups of factors should be considered when this type of bearings is designed:

Submitted Data
The bearing designer could be provided with: (L, N and R)

Selected Data
Selecting the geometrical group to be: Selecting the manufacturing group: (ʌ = 5→ 100 or ξ,= 0.05→ 1) Selecting the fluid group to be for instance:

Calculated Data
Operational group could be calculated.

Design Mathematical Expressions
The mathematical expressions governing the bearing design and performance could be found in the Appendix.

Design Procedures
It is necessary to have a comprehensive idea about the sequence of the design steps depending on the data provided, data selected and data calculated.

Central Pressure Ratio Determination (ß)
One of the design difficulties is determining the central pressure ratio mathematically, putting aside the industrial experience which allows the designer to select its value to be between (0.5and 0.7), it will be derived mathematically when using a capillary and orifice restrictors showing the drawbacks of selecting it by experience.

Capillary Tube Restrictor
Following Yacout [9], the derivation could be found in the Appendix.

Orifice Restrictor
Following Yacout [9], the derivation could be found in the Appendix.

Speed Parameter Determination (S)
Once the supply pressure (p s ) is determined, the speed parameter (S) could be calculated as from Yacout [9] as: S = 3 ρ Ω 2 R 2 / 40 p i Where: Ω = 2 π N &p i = ßp s In turn, the dimensionless load carrying capacity (W), flow rate (Q) and friction torque (M) could also be calculated at any given eccentricity (e)

Eccentricity Determination (e)
Calculating (M) and (Q) at any (e) allows calculating the pump power (P p ), frictional power and the total power loss (P t ) at different values of (e) as from [9] and from the appendix equations (8-10): Then eccentricity (e) that gives the minimum total power losses could be graphically determined.

The Pressure Distribution (P)
Once determining the eccentricity (e) and the speed parameter(S), the pressure distribution could be calculated by equitation (2).

Load Carrying Capacity Determination (w)
Getting the eccentricity, speed parameter and the central pressure ratio (ß) the dimensionless load carrying capacity (W) could be calculated using equation (5), in turn the load (w) as: w =π R 2 ß P s W

Temperature Rise
Temperature rise could be determined theoretically as: or numerically as in equation (7).

Design Example
It is required to design a bearing with a radius of (50 mm), maximum rotor speed (N =300 r. p. s) and lifting a load of (L = 40 KN).

Design Example Calculations
For this example; a bearing with its four different configurations has been designed to find out which one is the best for this case. Table 1.
It is preferable to choose one bearing configuration to reveal and perform the design procedures and calculations.
A recessed hemispherical configuration is suggested.

Bearing Configuration and Lubricant Fluid Data
Geometrical group: Manufacturing group: (ξ = 0.05) Restrictor data as put in the theoretical analysis (C d = 0.6 for orifice, n c = 20 for capillary and Re= 2000) Fluid group:

The Bearing Geometry Factor Determination (J f )
The dimensionless load carrying capacity (W) in the stationary state is less than the unity depending on the bearing configuration. Hence the geometry factor is (1/W).

Determination of the Primary Supply Pressure and
Speed Factor (p l &S l ) 3

The Design Supply Pressure Evaluation (ps)
The design supply pressure which could lift the load (L) at the stationary state (N=0) is: 6 1.0076 7.6394 (10) Then: 6 7.69785(10) / s p N m =

The Design Eccentricity Determination (e)
Determining the design supply pressure, the design eccentricity could be determined graphically as before from figure (2-c) to be: (e = 1.4 mm).

Load Carrying Capacity (W)
As  Hence, this ultimate load should not be exceeded.

Temperature Rise and Stiffness Determination
The temperature rise could be numerically and theoretically calculated whereas the theoretical stiffness calculation is quite difficult.

Temperature Rise
The temperature rise could be numerically calculated through applying equation (7) at the rotational state to be (20.26. C) and theoretically as:

Stiffness Determination (λ)
The stiffness is calculated numerically through an infinitesimal change in the eccentricity to get the bearing main stiffness at the rotational state as (203, 404 / m µ ) with capillary and orifice restrictors respectively, figure (4-c).

Results
This design technique produces a consistent and highly stiffened bearing in addition to a good temperature rise that needn't to any cooling system where the central pressure ratio is (2/3),mean stiffness is in range of (100 -400 N/m) and a temperature rise (20 -28 C) for all the bearing configurations.

Discussion
The bearing is designed based on minimum power losses, minimum flow rate and optimal restrictor dimensions taking into consideration the rotational inertia, the surface roughness and the lubricant viscosity variation [9, 11 & 12].

Central Pressure Ratio
This central pressure ratio has been derived based on previous work of Yacout [9] avoiding the solution of the 3 rd order equations (18, 22) where their solution haven't given a conception about the bearing consistency. The present study offered a new idea about these equations, through their differentiation, where equating their 1 st derivative, relating to the central pressure ratio, and equating to zero have given maximum values of the quantities (a1, a2) to be (4/27) and a unique value(2/3) for the central pressure ratio. Rowe and stout [2] show that the pressure ratio is (0.4-0.6), Shijiang [4] concluded that it should be (≤ 2/3), Kasama [5] showed this ratio as (0.66) and Lin [8] showed that the critical pressure ratio is (0.4902), and the operational value (0.7). Figure (1) shows the quantities (a1, a2) versus the central pressure ratio (β) through drawing equations (18, 22). It is clear that if the quantities (a1, a2) have any values less than (4/27) will result in two values of (β) which will make the flow rate severely or slightly changes depending on the difference between these two values causing a pulsating or intermittent fluid flow and could lead to the bearing deterioration due to the change of the load carrying capacity. This can't support Lin [8] in the critical pressure ratio where at (β = 0.4902) the quantity (a1 or a2)will be about (0.1224)which gives another (β) value of about (0.815); the difference between these two ratios is quite big.

Temperature Rise
The temperature rise calculated numerically or theoretically showed approximately the same results which give the priority to the theoretical solution for its simplicity, but the numerical one allows the designer to get an idea about the distribution of this temperature rise figure 3. Elescandarany [11][12] showed that the temperature rise is about 14 degrees centigrade for a light load capacity (about one tenth of that of the present bearing) and didn't pay attention to the theoretical calculation.

Bearing Stiffness
One of these design technique difficulties is calculating the bearing stiffness where it is a function of the film thickness and the thickness is variable. Hence, the present study offered a new conception about calculating the stiffness numerically where the stiffness is calculated as the mean stiffness of a sample of localized stiffness calculated at a (100) different thicknesses along the bearing (4)shows the local stiffness and its mean one. Rowe [2] designed a hydrostatic spherical bearing with (53, 20 N/m) average radial and axial stiffness respectively while the present design technique produces a bearing of a very high mean stiffness (200-300 N/m) and (400-600 N/m) for capillary and orifice restrictors respectively according to the bearing configuration.

Other Bearing Parameters
The other different calculations of the chosen bearing configuration could be found in Table 1. Table 1 shows the details of each configuration to enable the designer selecting the best suitable one for the client required purpose. For instance, a partial hemi-spherical with or without recess, orifice restrictor, has the highest mean stiffness, acceptable flow rate and temperature rise but with more power losses. If the least power losses is needed a hemi-spherical with or without recess, orifice restrictor, could be selected and so on.

Conclusion
From the previous results and discussion it could be said that the present design technique is reliable and dependable to design a highly consistent and stiffened bearing recommending and preferring the orifice restrictor.
The four bearing configurations have approximately the same quality.

Future Work
Part 4 will be the last part of this series and will offer a restrictor-less bearing design with a comprehensive analysis of the bearing stiffness.

A2.1 Simplified Governing Equation
The simple governing equation of this type of bearings considering the centripetal inertia by Dowson [1] is: which was firstly modified by Yacout [9,11] to express the surface roughness and finally to express the viscosity variation respectively.

A2.2 Modified Governing Equation
The modified governing equation for this type of bearings and its solution from Yacout [9][10][11] is: