## Hydrostatic Lubrication

6.5.3.1 Introduction

Load relationships of a two dimensional hydrostatic bearing are given in (6.1), showing its dependence on viscosity and land clearance.

H3 P H3 load capacity of bearing viscosity (kg/m/s) flow rate (m3/s) clearance (m)

The use of sea water as the working fluid, with a viscosity much lower than that of normal lubricating oils, results in either the need for smaller land clearances or higher flow rates to achieve these high forces. It is hence necessary to develop expressions for the load capacity of the bearings, furthermore, the use of 3 dimensional analysis is required. This section details the logical process of hydrostatic bearing selection and explains simplifications and assumptions used as the basis for the final results. Although some bearings can be optimised for a general case, it is illuminating to design the bearing for a particular VHM, allowing numerical examples to compare topologies.

A VHM capable of producing 100 kW at a speed of 1 m/s is used, consisting of 3 single phase modules. Each of the 4 poles making up a module would have 12 X 12 mm wide magnets with an axial depth of 1m. One set of poles was modelled using

With the maximum rated current of 15 Amps flowing in the 240 turn coils, the model was run with the translator at various levels between the two fixed stator poles in order to calculate the resultant forces acting on it. The maximum axial force reacted was 18 kN for one pair of pole faces, implying 36 kN per module and a total of 108 kN for the 3 phases. The corresponding gap closure force between the translator and one side of the stator was 36 kN per face at the design airgap of 1mm, rising to 54 kN if the translator was allowed to deviate to within 0.05 mm of the translator.

When the translator sits in the mid point of the two stators there is no net force on it and hence no criteria with which to design a lubrication system. A small deviation from the midpoint, however, results in a large net force. The magnetic stiffness may hence be used. If the bearing system has a higher stiffness than the magnetic system, the implication is that for a given deviation from the midpoint of the airgap, the bearing forces will dominate and the translator will be centred. The stiffness at either side of the design clearance was calculated to be almost 25 MN/m, resulting in a 5kN force imbalance with a 0.2 mm deviation from the design clearance.

### 6.5.3.2 Flow assumptions

Much of the work in this section is based on laminar flow and hence relies on the Reynolds' number, Re, as given by (6.2), being sufficiently low, typically taken as less

FEA.

Where p v density (kg/m3) average velocity (m/s)

Assuming the flow and dimensions of Figure 6.12, where the fluid flow lines are parallel and there is no leakage through the corners, the velocity of a fluid can be calculated for a particular square bearing and substituted into the Reynolds' equation, giving (6.3).

Where w L

x recess width (m) recess length (m)

land width (m)

### 6.5.3.3 Regulation

For a hydrostatic pad to work, there needs to be some sort of control over the fluid flow through it. If the bearing deviates from its design clearance, for example if the gap reduced, the pressure within the bearing recess needs to be increased in order to apply a greater force to the bearing and attempt to recover the design clearance. Figure 6.13 shows three possible configurations to achieve this of increasing mechanical complexity and effectiveness. In each one, the position of the translator is deduced from the recess pressure, which varies with bearing clearance.

In Figure 6.13 A the pads are fed from a constant supply source through a fixed restrictor, typically a capillary or an orifice. The pressure drop across the restrictor allows for a variation in bearing recess pressure to compensate for deviation from the design condition. In Figure 6.13B the restrictors are variable, for example constant flow valves, which gives improved stiffness to the bearings. The ultimate system is given in Figure 6.13C, where each recess is fed from a separate constant flow pump, which,

assuming incompressible fluid, guarantees a fixed land width. Clearly the increasing effectiveness comes at the cost of increasing mechanical complexity.

6.5.3.4 Stiffness of opposed pads

The load capacity, F, of a single rectangular pad hydrostatic bearing is a combination of that provided by the recess chamber and the bearing land. It is given by (6.4).

Where PR = recess pressure (Pa)

Assuming laminar flow, the pressure drop across a flow between two parallel plates, of length Lpp, may be expressed by (6.5).

12yx H 3 L

pp J

Where

If the exit pressure, P2, is the surrounding pressure, and P1 is given as the relative pressure in the recess, from here on defined simply as PR, then the hydraulic resistance, Rb, of the rectangular pad may be defined by (6.6).

Again, this is based on the assumption that the flow is as shown in Figure 6.12 and the corners are hence ignored.

Considering Figure 6.14, the diagram of an equivalent circuit comprising of a single pad bearing and a restrictor, the recess pressure, PR, can be expressed as a function of the two hydraulic resistances and PS, the supply pressure.(6.7)

Where Rr =

restrictor resistance (Pa/m )

Restrictor r bearing

Figure 6.14: Equivalent circuit for a single pad

For maximum stiffness of a bearing operating at the intended clearance, PR should equal half PS. This is analogous to the maximum power transfer theory in electrical engineering, which states that internal and external resistances should be equal for maximum power transfer.

The resistance of the bearing, Rb, must therefore be equal to the restrictor resistance Rr at the design clearance. Defining this value at the reference pressure, Rbo, allows the value of the bearing resistance to be expressed non dimensionally, (6.8):

Equation (6.7) may now be expressed as below, (6.9).

Consider now the set of opposed pads shown in Figure 6.15, in which the translator has been offset by h.

Figure 6.15: Opposed pads given a small displacement, h, from the design clearance, H

The reference resistance of the system is now given by (6.10).

The absolute hydraulic resistance of pad 1, and its non dimensional equivalent are given in(6.11) and (6.12) respectively. Similar relationships exist for pad 2.

Combining equations (6.4) to (6.12) and considering the resultant force on the pair of opposed pads when given the small displacement h shown in Figure 6.15 gives the net downward force as:

Remembering that the stiffness is defined as the derivative of force with respect to displacement, and setting h to zero, allows the stiffness of the opposed bearings at the design clearance to be expressed, (6.14).

Where k0 = stiffness of opposed pads at design clearance (N/m)

6.5.3.5 Pumping Power of Opposed Pads

The power required to pump fluid around a hydraulic circuit may be expressed as (6.15).

Where HP = pumping power (W)

Rtot = total resistance of hydraulic circuit At the design clearance the total power for the opposed pad bearing system, represented by the hydraulic circuit shown in Figure 6.16, can be expressed as (6.16), incorporating four resistances defined by (6.10).

Figure 6.16: Equivalent circuit for opposed pad bearings

Figure 6.16: Equivalent circuit for opposed pad bearings

Using a combination of (6.14) to (6.16), it is possible to express the pumping power required for a given set of opposed pads solely as a function of its dimensions and the required stiffness.

6.5.3.6 Optimisation for stiffness

The size of the pads is made dimensionless by using the substitutions given in (6.18), thus allowing the power to be expressed in a dimensionless manner, (6.19).

2 L'2 + x'2 +L'2 x'2 +2L' x'+2L'2 x'+2L'x'2 Where HP0 is the reference power, defined as that dissipated when L' and x' are equal to unity.

The graph of Figure 6.17 uses (6.19) and shows the relative pumping power required by bearings of different aspect ratios to achieve any equal stiffness. The width, w, of the bearing recess is the dimension which dictates the size of bearing.

25 20 15

25 20 15

Figure 6.17: Dimensionless graph of relative pumping power verses relative pad size for any given stiffness

Figure 6.17: Dimensionless graph of relative pumping power verses relative pad size for any given stiffness

The total clearance, H, is assumed constant and is hence not a variable. From this it is clearly desirable to make L', the ratio of length to width of the bearing recess, and x', the ratio of land width to recess width, as large as possible in order to reduce power consumption. Physically this means that an increase in either the recess length or the land width reduces the pumping power required to obtain a specific stiffness. This simplified analysis hence implies that larger bearings are more desirable.

### 6.5.3.7 Friction Power

The frictional force of the bearing, i.e. the force required by the moving member to overcome the internal shear forces of the lubricant, acts in three distinct areas. These correspond to the two pairs of bearing lands, perpendicular and parallel to the plain of motion, and the bearing recess.

It is customary to ignore the power dissipated by shearing in the recess part of the pads when operated at low speeds. The friction force is hence approximated to that occurring over the bearing lands only, as given by (6.20).

Where

= friction force (N) = friction power (W) = velocity (m/s) = area of two parallel surfaces (m2) Equation (6.20), contrary to that found in the stiffness section, 6.5.3.6, implies that a bearing with a short length, short land width and large value of clearance is a more desirable design for low power dissipation.

### 6.5.3.8 Total Power Minimum Power

As a result of the two conflicting conditions outlined in the preceding sections, it is necessary to consider the total power dissipated by a given bearing. As the frictional power is dependent on the velocity of the bearing, whereas the pumping power is not, in order to allow a dimensionless comparison it is necessary to put some actual values into both equations. The values used are given in Table 6-1, where the stiffness is half that of the VHM because there will be one bearing on each side of the translator.

Velocity |
v |
0.5 m/s |

Dynamic viscosity |
u |
0.0015 kg/ms |

Stiffness at equilibrium position |
k0 |
12.5 x106 N/m |

Using these values, it is still possible to obtain the dimensionless value of total power, by dividing the calculated value of actual power by Ht0, the power dissipated when L' and H' are equal to unity. The actual and dimensionless total power are given in (6.21)and (6.22) respectively, with definitions given in (6.23).

0.75 x 10 3 x (L + x)2 (wL + wx + Lx)2 + 7.72 x 1015 x H6(w + L)

Inspection of the order of magnitude of the two parts of the numerator of equation (6.22) can be used to describe the behaviour of the bearing. The left hand term, which corresponds to the friction power, will be negligible unless either the clearance H' is very small, or L' is very large. At all other values, the right hand term, corresponding to he pumping power, will dominate.

The power dissipated by friction is therefore ignored in further analysis as it has been demonstrated to be negligible when compared with pumping power at realistic bearing dimensions.

### 6.5.3.9 Opposed Pads for 100 kW device

Ignoring the friction losses in the bearing allows the conclusions of the stiffness section, 6.5.3.6, to again be employed. It was stated there that the bearing should be large. The size of the bearings will be limited by the electrical machine design, in which case it is necessary to consider specific examples. In keeping with the concept of a modular VHM, i.e. one where the machine is essentially made up of independent phases the number of which dictates the overall power of the machine, the bearings may be designed such that there will be one pair of opposed pads for each pole. It will be assumed that one pad will run along the entire length of a stator pole.

Using the VHM outlined in 6.5.3.1 above, the total length of the bearing, consisting of recess and perpendicular lands, can therefore be up to 0.144 m. The total width of the bearing has no real constraint, and so is arbitrarily confined to 10% of that of the translator, i.e. 0.1 m, giving ( 6.24).

The pumping power for these bearings, calculated using (6.17), is given in Figure 6.18. The values shown must be multiplied by 12 for the entire 100 kW machine, as the machine comprises of three modules, with two pole pairs per module and a bearing on each side of the stator.

x 10

x 10

At any design clearance, H, there is a value of width, w, which gives a minimum pumping power. This value of w is constant irrespective of the value of H and equal to 0.0427 m. The pumping power becomes very large as the land width and recess width become very large, tending towards 2 MW, clearly unsuitable for use as the lubrication system for a 100 kW generator. The limit of the validity of these equations must be remembered, which assumes laminar flow. Figure 6.19 shows the Reynolds' number, from (6.3), plotted against its accepted limit for laminar flow, 2000.

At any point above the 2000 mark, the flow must be assumed to be turbulent and the equations governing its flow become invalid. In order to maintain laminar flow it is necessary to limit the value of design clearance, H, for a given value of recess width, w. The equation of the upper limit is given in (6.25), derived by equating the Reynolds' number, (6.3), to the flow through one bearing and restrictor (obtained from (6.14) and (6.16)).

Figure 6.19: Reynolds' number for flow out of bearing compared with 2000, maximum to

Figure 6.19: Reynolds' number for flow out of bearing compared with 2000, maximum to

The maximum allowable clearance for condition of minimum power dissipation is obtained by substituting the value of 0.0427 into Equation (6.25), which yields 0.23 mm. The minimum value of pumping power for clearance values up to and including this value can be satisfactorily used to design a bearing using the assumption of laminar flow.

This methodology of finding the dimensions of minimum power dissipating recess width for a given external bearing dimension and then using Equation (6.25) to find the maximum allowable clearance, can equally be applied to any external dimension.

The external bearing length, Lex in Figure 6.2, is chosen as half, equal to and double the 0.144 m VHM face width. The external width of bearing is then altered between a square and a rectangle of aspect ratio 10.

Figure 6.20 shows the pumping power and maximum clearance permitted for this range of bearings and demonstrates that by enlarging the external size of the bearings, the tolerance on the clearance can be relaxed and the overall pumping power required to achieve the desired stiffness is reduced. For a square bearing of total length 0.288 m, for example, it is possible to construct a bearing that can have a design clearance of almost 0.45 mm and requires around 4 kW. Unfortunately this bearing adds 60% onto ensure laminar flow

ensure laminar flow

the width of the machine and is twice the length of one pole. A more realistic size bearing, which requires 5 kW, is a square bearing of external width 0.0728 m and can have a clearance up to 0.15 mm with a required input pressure of 5 bar. The corresponding internal dimensions for all possible bearings are given in Figure 6.21.

Concluding remarks

The behaviour of opposed pad bearings has been investigated using simple laminar fluid flow theory. The concept of hydrostatic lubrication and the need for flow regulation was introduced and used to derive a simple expression for the stiffness of a pair of opposed pads. Dimensionless optimisation techniques were then developed to explore the behaviour of aspect ratio and bearing size. The power dissipated by friction was deemed negligible compared with the pumping power. For particular external bearing dimensions, it was found that an optimum value existed for the recess dimensions that implied minimum power dissipation. This value was used, in combination with an upper limit on the design clearance that ensured laminar flow, to design sets of bearings capable of supporting the translator of a 100 kW VHM. A 0.0728 m external dimension square 5 kW bearing was proposed, which required a clearance of 0.15 mm and supply pressure of 5 bar.

### 6.5.3.10 Self-Regulating Bearings Introduction

The hydrostatic bearings introduced above require flow regulation between the pressure source and the actual bearing. This source of hydraulic resistance is necessary to regulate the recess pressure in response to loads applied to the bearing. By its very nature, then, there must be a pressure drop across it and hence power loss within it. In self regulating bearings, the design of the bearings is such that there is no need for this added resistance and so there is a potential for increased efficiency. Principle of operation

It is crucial that the overall hydraulic resistance of a self regulating bearing remains constant regardless of the position of the land being supported, thus ensuring a constant flow rate.

Consider the rotating shaft of Figure 6.22. The fluid enters from the right and is split into two equal resistance paths. It is clear that these resistance paths will be equal regardless of the position of the shaft, due to the opposing sets of identical lands. If the input pressure remains constant, then so too does the pressure dropped across the bearing as a whole. The pressure diagram of Figure 6.23 shows the pressure distribution, and hence force distribution, for a small upwards displacement.

The upper chamber is at a greater pressure than the lower, and hence there is a net force downwards, opposing the initial force required for the displacement. As the total hydraulic resistance has not been altered, the lubricant supply source, and hence any other bearings connected to it, remains unaltered by this displacement.

The principle of self regulation is applicable to linear, as well as rotary, slideways. It has been proposed, by Bassani and Piccigallo [90], that the system shown in Figure 6.24 could be used as a hydrostatic slideway, which would be independently self regulating in two perpendicular directions.

rit.

Figure 6.24: Self regulating opposed pad hydrostatic slideway

### Two dimensional flow

For the case where axial flow can be ignored, and the flow is purely two dimensional as shown in Figure 6.24, Bassani and Piccigallo have presented a very extensive theoretical analysis linking the parameters of the bearing together. For example, (6.26) links the vertical stiffness of the arrangement to the geometry and flow rate.

Where Bi & B2 are pad sizes in orthogonal directions

In order to achieve two dimensional flow, Bassani and Piccigallo suggested the use of lateral seals. Referring to the logic behind the use of hydrostatic bearings in wave energy devices, it is clear that if lateral seals are to be used, then the advantage of their use, i.e. near infinite life, would be seriously diminished. The only way to investigate the bearing with unsealed ends would be to consider the case of an infinite, or near infinite length bearing. It is conceivable that this would leave the end effects as negligible in the overall bearing behaviour.

Experimental work by the author demonstrated that three dimensional affects were both apparent and significant in the case of a long bearing and it was evidently not acceptable to claim that end effects would be insignificant, even in scenarios where the bearing axial length was long compared with its land width, Figure 6.25.

The testrig presented here was built with the intention of verifying a two dimensional stiffness formula presented by Bassani and Piccigallo [90]. The clearances were set to the minimum achievable tolerance in the workshop. When connected to mains water pressure almost all the flow was leakage flow through the end plates, with a very small flow rate emerging through the intended upper clearance. Increasing the flow rate was technically possible, but would have clearly resulted in a very power intensive bearing.

For continued investigation of self regulating hydrostatic bearings, therefore, it is necessary to consider flow in three dimensions and investigate alternative end configurations. Concluding remarks

Self regulating bearings have the theoretical advantage of removing the power loss associated with flow restrictors required by other hydrostatic bearings. Although their use in three dimensions has been documented, to achieve this requires the use of lateral seals. Initial simple three dimensional analysis, not presented here, revealed that the true flow pattern through a self regulating bearing is too complex to be adequately accounted for using the intuitive simplifications suitable for opposed pad bearings. To design a true self regulating bearing with constant hydraulic resistance would require three dimensional fluid analysis, possibly using an FEA program such as FLUENT [91]. Verification would involve measuring a variety of pressures and flows, constituting an almost independent research project from the body of this thesis.

### 6.5.3.11 Conclusions

The concept of hydrostatic lubrication using sea water as the medium to withstand the high airgap closing forces of a 100 kW VHM has been investigated. Using an assumption of laminar flow, a relationship between pad size and required stiffness was derived for a set of opposed bearings. The most satisfactory design, which dissipated 5 kW of pumping power, required a bearing clearance of 0.15 mm, which may cause manufacturing problems. If the analysis were extended to turbulent flow, i.e. one where all the load was taken by the bearing recesses and not by the bearing lands, smaller bearings or bearings with greater land clearances would probably be possible.

Further work would be desirable into turbulent flow bearings and possibly more detailed analysis of self regulating bearings.

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