Engineering Tribology Episode 2 Part 3 pps

FIGURE 6.8 Diaphragm pressure sensor valve developed by Mohsin [5].There are two basic limitations to diaphragm bearings; it is necessary to adjust the spring force on the diaphragm to m

FIGURE 6.8 Diaphragm pressure sensor valve developed by Mohsin [5].

There are two basic limitations to diaphragm bearings; it is necessary to adjust the spring

force on the diaphragm to match the bearing stiffness and high stiffness is only maintained

up to a certain load where bearing deflection rises sharply with any further increase in load.More detailed information about pressure sensing valves can be found in [2]

6.5 AEROSTATIC BEARINGS

In some applications a gas is used to lubricate the bearings and these are known as aerostaticbearings The mechanism of film generation is the same as in liquid bearings Gas lubricatedbearings offer some advantages such as:

· gas viscosity increases with temperature thus reducing heating effects during load or abnormal operating conditions,

over-· some gases are chemically stable over a wider temperature range than hydrocarbonlubricants,

· a non-combustible gas eliminates the fire hazard associated with hydrocarbons,

· if air is selected as the hydrostatic lubricant, then it is not necessary to purchase orrecycle the lubricant,

· gases can offer greater cleanliness and non-toxicity than fluid lubricants

The viscosities of some gases used in aerostatic bearings are shown in Table 6.1

TABLE 6.1 Viscosities of various gases at 20°C and 0.1 [MPa] pressure

Aerostatic bearings are used in the precision machine tool industry, metrology, computerperipheral devices and in dental drills, where the air used in the bearing also drives the drill.They are particularly useful for high speed applications and where precision is required sincevery thin film thicknesses are possible The main disadvantage, however, is that the loadcapacity of gas lubricated bearings or aerostatic bearings is much lower than the load capacity

of fluid lubricated bearings of the same size The most commonly used gas is air, but othergases such as carbon dioxide and helium have been used in specialized systems, e.g nucleartechnology

The analysis of aerostatic bearings is very similar to liquid hydrostatic bearings The maindifference, however, is that the gas compressibility is now a distinctive feature and has to beincorporated into the analysis Since the pressures generated in these bearings are muchlower than in liquid lubricated bearings, ambient pressure cannot be neglected and is alsoincluded in the analysis For example, assuming isothermal behaviour of the gas, the mainperformance parameters of the aerostatic flat circular pad bearings can be calculated from thefollowing formulae [6]

Pressure Distribution

The pressure distribution is found from a simplified form of the compressible Reynoldsequation for radial coordinates with angular symmetry and negligible sliding velocity Thederivation of the pressure distribution can be found in [5] and the final form of the equationfor pressure at any given radial position between the recess and bearing edge is:

p s is the supply pressure [Pa];

p r is the recess pressure [Pa];

p a is the ambient pressure [Pa];

R is the outer radius of the bearing [m];

R 0 is the radius of the recess [m];

r is the radius from the centre of the circular thrust bearing [m]

In spite of the apparent complexity of the above expression, the pressure distributionapproximates to a linear decline of pressure with distance from the recess

where:

h is the film thickness [m];

η is the gas dynamic viscosity [Pas];

K* = W

where:

W is the bearing load capacity [N];

p r is the recess pressure [Pa];

R is the outer radius of the bearing [m]

Values of ‘K*’ as a function of the ratio of bearing radius to recess radius are shown in Figure

FIGURE 6.9 Values of load factor as a function of bearing geometry for a circular pad

aerostatic thrust bearing

Friction Torque

The friction torque or friction coefficient of a gas bearing is not usually considered because itsvalue is extremely small unless the bearing is operating at a very high sliding speed If it isnecessary to estimate a friction torque, a Petroff approximation can be used In other words,the shear rate should be multiplied by the gas viscosity and bearing area The viscosity ofmost gases is not greatly affected by temperature so that the ambient temperature value ofviscosity can safely be used even if the gas leaves the bearing at a high temperature At highsliding speeds, turbulent flow of gas in the bearing may occur and this should be checked bycomputing the local Reynolds number based on film thickness The friction torque for flat

circular pad bearings can be calculated from equation (6.11) provided that laminar flowprevails.

EXAMPLE

Calculate the load capacity of a 0.1 [m] radius flat circular aerostatic bearing with radiusratio of 2, and recess pressure of 1 [MPa] From Figure 6.9, K* = 0.63 The load capacity isthen:

W = π × 10 6× 0.1 2× 1 × 0.63 = 19 792 [N].

Power Loss

The power loss for most slow sliding speed gas bearings is determined by the pumping powerrequired since the friction power loss is extremely small As mentioned already the pumpingpower is the product of gas flow rate and the total pressurization required (including losses incapillaries and supply lines) If the friction power loss is significant then this should be added

to the pumping power loss to give the total power loss The friction power loss for flatcircular pad bearings can be calculated from equation (6.12)

6.6 HYBRID BEARINGS

Hybrid bearings function by the combined action of hydrostatic and hydrodynamiclubrication A bearing technology resembling a hybrid bearing was described in Chapter 4where a pressurized lubricant supply is used to prevent metallic contact during starting orstopping of a Michell pad bearing The principle of augmenting hydrodynamic lubricationwith a hydrostatic effect or vice versa has been developed further than this limitedapplication The distinction between a true ‘hybrid bearing’ and the pressurized lubricantsupply to a journal bearing is that in the latter case, the purpose of the extra supply is tosupply cool lubricant into the hottest part of the bearing

An important difference between hybrid and hydrostatic bearings is the absence of recesses inthe hybrid bearings Recesses cause reduced hydrodynamic pressures in the loaded parts ofthe bearing which are where hydrostatic gas or liquid outlets are usually positioned Asdiscussed previously, lubricant supply outlets are usually located remote from the loadedpart of the bearing for efficient hydrodynamic lubrication An example of a hybrid journalbearing is shown in Figure 6.10

Where several lubricant outlets are used it is important to avoid interconnection of thesupply lines If the supply lines are connected then recirculating flow of lubricant will occurwhich reduces the hydrodynamic pressure

The basic parameters of these bearings such as pumping power and size are designed as for ahydrostatic bearing and any hydrodynamic effect which improves bearing performance isregarded as a bonus [2]

6.7 STABILITY OF HYDROSTATIC AND AEROSTATIC BEARINGS

Hydrostatic bearings are subject to vibrational instability particularly under variable loads orwhere a gas is used as the lubricant The mechanism causing vibrational instability is thesame as discussed already in hydrodynamic lubrication, i.e a resonance dependent on thestiffness and damping coefficients of the load-carrying film and the coupled mass ‘Oil whirl’can also occur in hydrostatic and hybrid journal bearings at high speeds in a similar manner

HYDROSTATIC LUBRICATION 279

to hydrodynamic bearings Most high-speed externally pressurized bearings are aerostatic andthe analysis of the corresponding vibrational stability is highly specialized For the practicalprevention of vibrational instability it is important to minimize the depth of the recess in ahydrostatic bearing since this increases the storage capacity of energy in the bearing,particularly so if gas is used The depth of the recess should only be a small multiple of thedesign film thickness, e.g only 10 to 20 times larger but not 100 times larger A high bearingstiffness can raise the critical vibration speed in these bearings The bearing stiffness may beincreased by supplying a gas under higher pressure The method of gas or liquid supply isalso important, restrictors and capillary compensation are associated with vibrationproblems More information on bearing stability can be found in [2] and [5]

as aerostatic lubrication, can provide very low friction even at extremely high sliding speedsbecause of the low viscosity of gases Quasi-ideal characteristics of zero wear and friction areobtained with hydrostatic or aerostatic lubrication at low to medium contact stresses but amore complicated technology, e.g the application of an external high pressure pump, isrequired in comparison to other forms of lubrication Bearing stiffness in these bearings canalso be manipulated more easily than with other types of bearings to suit specific designrequirements

REFERENCES

1 L.G Girard, Nouveaux Systeme de Locomotion Sur les Chemins de Fer, Bachelier, Paris, 1852, pp 40.

2 W B Rowe, Hydrostatic and Hybrid Bearing Design, Butterworths, 1983.

3 A Cameron, Basic Lubrication Theory, Ellis Horwood Ltd, 1981.

4 J.P O'Donoghue and W.B Rowe, Hydrostatic Bearing Design, Tribology, Vol 2, 1969, pp 25-71.

5 M.E Mohsin, The Use of Controlled Restrictors for Compensating Hydrostatic Bearings, Advances in Machine Tool Design and Research, Pergamon Press, Oxford, 1962.

6 W.A Gross, Fluid Film Lubrication, John Wiley and Sons, 1980.

7 J.N Shinkle and K.G Hornung, Frictional Characteristics of Liquid Hydrostatic Journal Bearings,

Transactions ASME, Journal of Basic Engineering, Vol 87, 1965, pp 163-169.

8 H Opitz, Pressure Pad Bearings, Conf Lubrication and Wear, Fundamentals and Application to Design,

London, 1967, Proc Inst Mech Engrs., Vol 182, Pt 3A, 1967-1968, pp 100-115.

9 R Bassani and B Piccigallo, Hydrostatic Lubrication: Theory and Practice, Elsevier, Amsterdam, 1992.

10 J.K Scharrer and R.I Hibs Jr., Flow Coefficients for the Orifice of a Hydrostatic Bearing, Tribology Transactions, Vol 33, 1990, pp 543-550.

The existence of elastohydrodynamic lubrication was suspected long before it could be proved

or described using specific scientific concepts The lubrication mechanisms in conformalcontacts such as those encountered in hydrodynamic and hydrostatic bearings were welldescribed and defined and the reasons for their effectiveness were well understood.However, the mechanism of lubrication operating in highly loaded non-conformal contacts,such as those which are found in gears, rolling contact bearings, cams and tappets, althougheffective was poorly understood The wear rates of these devices were very low whichimplied the existence of films sufficiently thick to separate the opposing surfaces, yet thisconclusion was in direct contradiction to the calculated values of hydrodynamic filmthicknesses The predicted values of film thickness were so low that it was inconceivable forthe contacting surfaces to be separated by a viscous liquid film In fact, the calculated filmthicknesses suggested that the surfaces were lubricated by films only one molecule inthickness In experiments specifically designed to permit only lubrication by monomolecularfilms, much higher wear rates and friction coefficients were obtained This apparentcontradiction between the empirical observation of effective lubrication and the limits ofknown lubrication mechanisms could not be explained for a considerable period of time Theentire problem acquired an aura of mystery and many elaborate experiments and theorieswere developed as a result From the view point of an engineer, the answers to the questions

of what controls the lubrication mechanism and how it can be optimized are very important,since heavily loaded point contacts are often found, and provision for effective lubrication ofthese contacts is critical

In the 1940's a substantial amount of work was devoted to resolving elastohydrodynamicsand the first realistic model which provided an albeit approximate solution forelastohydrodynamic film thickness was proposed by Ertel and Grubin The work waspublished by Grubin in 1949 [1] It was found that the combination of three effects:

hydrodynamics, elastic deformation of the metal surfaces and the increase in the viscosity ofoil under extreme pressures are instrumental to this mechanism This lubrication regime isreferred to in the literature as elastohydrodynamic lubrication which is commonlyabbreviated to EHL or EHD At this stage, it should be realized that elastohydrodynamiclubrication is effectively limited to oils as opposed to other viscous liquids because of thepressure-viscosity dependence It was shown theoretically that under conditions of intensecontact stress a lubricating oil film can be formed The lubricated contacts in which thesethree effects take place are said to be operating elastohydrodynamically, which effectivelymeans that the contacting surfaces deform elastically under the hydrodynamic pressuregenerated in the layer of lubricating film The lubricating films are very thin, in the range of0.1 to 1 [µm], but manage to separate the interacting surfaces, resulting in a significantreduction of wear and friction Although this regime generally operates between non-conforming surfaces, it can also occur under certain circumstances in the contacts classified asconformal such as highly loaded journal and pad bearings which have a significantcomponent of contact and bending deformation However, enormous loads are required forthis to occur and very few journal or pad bearings operate under these conditions Significantprogress has been made towards a complete understanding of the mechanism ofelastohydrodynamic lubrication The pioneering work in this field was conducted by Martin(1916) [2], followed by Grubin (1949) [1] and was continued by Dowson and Higginson [3],Crook [4], Cameron and Gohar [5] and others.

In this chapter the fundamental mechanisms of film generation in elastohydrodynamiccontacts, together with the methods for calculating the minimum film thickness in rollingbearings and gears will be outlined Some particular characteristics of elastohydrodynamiccontacts such as traction and flash temperature will be discussed, along with the methods oftheir evaluation

7 2 CONTACT STRESSES

From elementary mechanics it is known that two contacting surfaces under load will deform.The deformation may be either plastic or elastic depending on the magnitude of the appliedload and the material's hardness In many engineering applications, for example, rollingcontact bearings, gears, cams, seals, etc., the contacting surfaces are non-conformal hence theresulting contact areas are very small and the resulting pressures are very high From theview point of machine design it is essential to know the values of stresses acting in suchcontacts These stresses can be determined from the analytical formulae, based on the theory

of elasticity, developed by Hertz in 1881 [6-8] Hertz developed these formulae during hisChristmas vacation in 1880 when he was 23 years old [7]

Simplifying Assumptions to Hertz's Theory

Hertz's model of contact stress is based on the following simplifying assumptions [6]:

· the materials in contact are homogeneous and the yield stress is not exceeded,

· contact stress is caused by the load which is normal to the contact tangent planewhich effectively means that there are no tangential forces acting between thesolids,

· the contact area is very small compared with the dimensions of the contactingsolids,

· the contacting solids are at rest and in equilibrium,

· the effect of surface roughness is negligible

ELASTOHYDRODYNAMIC LUBRICATION 283

Subsequent refinements of Hertz's model by later workers have removed most of theseassumptions, and Hertz's theory forms the basis of the model of elastohydrodynamiclubrication

Stress Status in Static Contact

Consider two bodies in contact under a static load and with no movement relative to eachother Since there is no movement between the bodies, shearing does not occur at theinterface and therefore the shear stress acting is equal to zero According to the principles ofsolid mechanics, the planes on which the shear stress is zero are called the principal planes.Thus the interface between two bodies in a static contact is a principal plane on which aprincipal stress ‘σ1’ is the only stress acting It is also known from solid mechanics that themaximum shear stress occurs at 45° to the principal plane, as shown in Figure 7.1

If the contact load is sufficiently high then the maximum shear stress will exceed the yieldstress of the material, i.e τmax > k, and plastic deformation takes place Material will then

deform (slip) along the line of action of maximum shear stress The maximum shear stress

‘τmax’, also referred to in the literature as ‘τ45°’ since it acts on planes inclined at 45° to theinterface (in static contacts), is given by:

the maximum shear stress occurs at approximately 0.6a, where ‘a’ is the radius of the contact

area

Stress Status in Lubricated Rolling and Sliding Contacts

In a lubricated rolling contact, the contact stresses are affected by the lubricating filmseparating the opposing surfaces and the level of rolling and sliding The hydrodynamic filmgenerated under these conditions and the relative movement of the surfaces causesignificant changes to the original static stress distribution and will be discussed later

Rolling, in general, results in an increase in contact area and a subsequent modification ofthe Hertzian stress field in both dry and lubricated conditions The most critical influence onsubsurface stress fields, however, is exerted by sliding To illustrate the effect of sliding on thestress distribution, consider two bodies in contact with some sliding occurring between them.Frictional forces are the inevitable result of sliding and cause a shear stress to act along theinterface, as shown in Figure 7.3

The frictional stress acting at the interface is balanced by rotating the planes of principalstresses through an angle ‘φ’ from their original positions when frictional forces are absent.The magnitude of the angle ‘φ’ depends on the frictional stress µq acting at the interfaceaccording to the relation:

φ = 1/2cos −1 (µq/k)

k k

45°

FIGURE 7.1 Stress status in a static contact; σ1, σ3 are the principal stresses, p is the

hydrostatic pressure, k is the shear yield stress of the material.

x b

1

2

0.173 0.236

0.267 0.283

0.290 0.295

0.300

0.251

FIGURE 7.2 Subsurface stress field for two cylinders in static contact; p max is the maximum

contact pressure, b is the half width of the contact rectangle [9].

The variation with depth below the interface of the principal shear stress ‘τmax’ for a cylinderand the plane on which it slides is shown in Figure 7.4 The contours show the principalshear stress due to the combined normal pressure and tangential stress for a coefficient of

friction µ = 0.2 [9] It can clearly be seen that as friction force increases, the maximum shear

stress moves towards the interface Thus there is a gradual increase in shear stress acting atthe interface as the friction force increases This phenomenon is very important in crackformation and the subsequent surface failure and will be discussed later

7.3 CONTACT BETWEEN TWO ELASTIC SPHERICAL OR SPHEROIDAL BODIES

Elastic bodies in contact deform and the contact geometry, load and material propertiesdetermine the contact area and stresses The contact geometry depends on whether thecontact occurs between surfaces which are both convex or a combination of flat, convex and

FIGURE 7.3 Stresses in a contact with sliding; σ1, σ3 are the principal stresses, p is the

hydrostatic pressure, k is the shear yield stress of the material, µ is the coefficient

of friction, q is the stress normal to the interface or compressive stress due to

load, φ is the angle by which the planes of principal stress are rotated from thecorresponding zero friction positions to balance the frictional stress

x b

1

0.312 0.300 0.280

0.140 0.200

0.220 0.240

0.180 0.260

0.160

FIGURE 7.4 Subsurface stress field for a cylinder sliding on a plane; p max is the maximum

contact pressure, b is the half width of the contact rectangle [9].

Geometry of Contacting Elastic Bodies

The shape of the contact area depends on the shape (curvature) of the contacting bodies Forexample, point contacts occur between two balls, line contacts occur between two parallelcylinders and elliptical contacts, which are most frequently found in many practicalengineering applications, occur when two cylinders are crossed, or a moving ball is in contactwith the inner ring of a bearing, or two gear teeth are in contact The curvature of the bodies

can be convex, flat or concave It is defined by convention that convex surfaces possess a

‘positive curvature’ and concave surfaces have a ‘negative curvature’ [7] The following general rule can be applied to distinguish between these surfaces: if the centre of curvature

lies within the solid then the curvature is positive, if it lies outside the solid then the curvature is negative This distinction is critical in defining the parameter characterizing the

contact geometry which is known as the reduced radius of curvature

· Two Elastic Bodies With Convex Surfaces in Contact

The configuration of two elastic bodies with convex surfaces in contact was originallyconsidered by Hertz in 1881 and is shown in Figure 7.5

Body B

FIGURE 7.5 Geometry of two elastic bodies with convex surfaces in contact

The reduced radius of curvature for this case is defined as:

R x is the reduced radius of curvature in the ‘x’ direction [m];

R y is the reduced radius of curvature in the ‘y’ direction [m];

R ax is the radius of curvature of body ‘A’ in the ‘x’ direction [m];