Engineering Tribology Episode 2 Part 1 doc

Start Acquire parameters Film thickness Sliding speed Pad width Viscosity Viscosity-temperature exponent Specific heat Thermal conductivity Density Inlet temperature Adiabatic or isother

FIGURE 5.12 Coded mesh of the control volume used in thermohydrodynamics [8].

The parameter ‘E’ is used to enforce flow directionality or convection into the controlling finite difference equation (5.52) There are two values of ‘E’ and the selection is based on the difference between lubricant flow ‘in’ and ‘out’ of the control volume.

E = E 1 if E 1 > 0 and E = 0 if E 1 ≤ 0 (5.59)

E 1 = |a E | + |a W | + |a N | + |a S | − |a P | (5.60)The inequality in equation (5.59) allows for reversal of flow which otherwise causesnumerical instability

The subscripts for all the mass flow terms, e.g (ρw) n, are lower case denoting that an average

between velocities at the central and peripheral node is taken; for example: w n = 0.5(w N + w P ).

‘S p ’ and ‘S c’ are terms representing viscous heating where allowance is made for the stronginfluence of temperature on viscosity These two quantities are derived from the basicexpression for viscous heating which is caused by the shearing of the lubricant:

S is the intensity of viscous heating [W/m3];

The controlling equation for ‘S p ’ and ‘S c’ is based on the assumption of a linear dependence ofthe heat source term on temperature:

ηp is the predicted dynamic viscosity of the lubricant [Pas];

η0 is the dynamic viscosity of the lubricant at some reference temperature ‘T 0’ [Pas];

γ is an exponent of viscosity-temperature dependence (typically γ = 0.05) [K-1]

Treatment of Boundary Conditions in Thermohydrodynamic Lubrication

As mentioned already, the boundary conditions necessary when viscous heating is modelledare considerably more complicated than in the isoviscous case

The finite difference equations presented are arranged so that they allow solution by methodsappropriate to elliptic differential equations This means that if iteration is used, the direction

or order in which nodes are iterated does not affect the solution If the equations were of aparabolic type then it would be necessary to iterate in the down-stream direction, i.e apply amarching procedure but when reverse flow occurs this method generally fails A marchingprocedure is a process of establishing nodal values in a specified sequence

The temperature boundary conditions at the interfaces of the hydrodynamic film varyaccording to the heat transfer mode of the bearing It is thus necessary to modify the finitedifference mesh to provide a means of solving the thermohydrodynamic equations for thespecified boundary conditions An example of the modified mesh is shown in Figure 5.13.The mesh can be applied to solve both the isothermal and adiabatic cases

If the isothermal bearing is studied, then the boundary conditions at the sliding surfacessimplify to a fixed temperature for the boundary nodes Iteration is then confined to interiornodes without any need for extra arrays of imaginary nodes apart from at the outlet and inlet

On the other hand, if an adiabatic bearing is to be analyzed, then the boundary condition atthe pad surface changes to an unknown pad temperature but with a zero temperaturegradient normal to the plane of the lubricant film In this case it is necessary to invoke animaginary array of temperature nodes above the pad surface with values of temperaturemaintained equal to the adjacent pad surface temperature Iteration then includestemperature nodes at the interface between the pad and hydrodynamic film Even for anadiabatic pad, the temperature on the pad at the bearing inlet which involves just one node,

remains the same as the lubricant inlet temperature.

Extra row of temperature nodes

for adiabatic pad

Extra row of nodes in velocity and temperature for outlet boundary condition

Extra row of

nodes for possibility

RUNNER

F B B

F F

V V V V

V

V V

V V

B Unknown value of temperature and velocity if backflow occurs

There will only be a negligible variation in ‘T’ with respect to ‘x’ compared to changes within

the bearing where heat generation occurs This condition can be accommodated by supplying

an extra column of nodes with temperatures equal to the adjacent node's outlet temperature.The iteration procedure then includes the extra nodes at the bearing outlet Whereverreverse flow at the inlet of the oil film occurs, temperatures are iterated on the boundarywith the assumption that ∂u/∂x and ∂T/∂x remain constant across the boundary Thiscondition is met by another column of nodes up-stream of the bearing inlet which maintainthe values of temperature and velocity calculated by linear extrapolation from nodetemperatures inside the bearing

Computer Program for the Analysis of an Infinitely Long Pad Bearing in the Case of Thermohydrodynamic Lubrication

A computer program ‘THERMAL’ for the analysis of both the isothermal and adiabatic

infinitely long pad bearings is listed and described in the Appendix For bearings which areneither isothermal nor adiabatic an estimation of the effects of bearing heat transfer can bededuced from a comparison of data from the adiabatic and isothermal conditions which

represents lower and upper limits of load capacity respectively The program is based on atwo-level iteration in temperature and pressure and its flow chart is shown in Figure 5.14.

An initial constant temperature field equal to the oil inlet temperature is assumed and apressure solution calculated from the resulting viscosity field A new temperature field isthen derived from the viscous shearing terms created by the pressure field This newtemperature field is then used to produce a second viscosity field which completes the firstcycle of iteration This iteration cycle is repeated until adequate convergence in the pressurefield between successive iterations of temperature is reached On completion of the iteration,pressure is integrated with respect to distance to obtain film force per unit length and the data

is then printed to complete the program

Example of the Analysis of an Infinitely Long Pad Bearing in the Case of Thermohydrodynamic Lubrication

The computer program ‘THERMAL’ described in the previous section provides a means of

calculating the reduction in load capacity of a bearing due to heating effects To demonstratethis effect the load capacity of a typical industrial bearing operating under conditions similar

to those studied by Ettles [9] was analyzed by this program The bearing parameters werechosen as typical of an industrial bearing

The selected values of controlling parameters were as follows: bearing width (i.e length in

the direction of sliding) 0.1 [m], maximum film thickness 10 -4 [m], minimum film thickness

5×10 -5 [m], lubricant viscosity temperature coefficient 0.05, lubricant specific heat 2000 [J/kgK], lubricant density 900 [kg/m3] and lubricant thermal conductivity 0.15 [W/mK] Two values of viscosity were considered, 0.05 [Pas] and 0.5 [Pas] at the bearing inlet temperature of 50°C The performance of the bearing was studied over a range of sliding speeds from 1 to 100 [m/s] for the lower viscosity and 0.3 to 20 [m/s] for the higher viscosity Sliding speed values used for computation were 0.3, 1, 3, 10, 20, 30 and 100 [m/s] For higher speeds the computing time

required to obtain convergence was unfortunately far too long for practical use

The calculated temperature distributions within an isothermal and adiabatic bearing areillustrated in Figure 5.15 The temperature fields were obtained for a lubricant inlet viscosity

of 0.5 [Pas] and bearing sliding speed of 10 [m/s] It can be seen from Figure 5.15 that the

maximum temperature occurs at the outlet of the bearing

Start

Acquire parameters

Film thickness Sliding speed Pad width Viscosity Viscosity-temperature exponent Specific heat

Thermal conductivity Density

Inlet temperature Adiabatic or isothermal pad

Special settings of iteration parameters?

No

Yes Acquire iteration

parameters

Initialize temperature, viscosity and pressure fields

A Use preset values

End

A

Yes No

Compute velocity field in direction of

Calculate reduced values of coefficients:

AE/E, AW/E etc.

Print out pressure, viscosity, temperature, U and W

as fractions of maximum values; print out load

Calculate M and N integrals

Solve 1-D Reynolds equation for pad using current values of viscosity Iteration

Compute velocity field normal to

direction of sliding W(I,K)

Compute coefficients of temperature iterations:

AE, AW, AT, AB, S, SP, SC, B, AP etc.

Allow for flow reversal in calculation of E

Calculate T(I,K) residual Convergence of T(I,K)

residual less than termination value?

Number of sweeps over the limit?

Update viscosity field VISC(I,K) with new

temperature values using relaxation factor Calculate cumulative residual

from pressure terms P(I)

No Convergence of P(I)residual less than termination value?

Number of sweeps over the limit?

Find maximum values

A strong effect of pad heat transfer on the temperatures inside the lubricant film is clear The

maximum temperature in the isothermal bearing is 71°C, compared to 116°C for the adiabatic

bearing The location of the maximum temperature is also different for these bearings Forthe isothermal bearing the peak occurs close to the middle of the bearing A small decline intemperature beyond this maximum is due to improved thermal conduction with reducedfilm thickness The location of the peak temperature in the adiabatic bearing is at the down-stream end of the pad at the interface with the lubricant The lubricant is progressivelyheated to higher temperatures as it passes down the bearing and the pad surface becomesvery hot as it is remote from any source of cooling

FIGURE 5.15 Computed temperature field in isothermal and adiabatic pad bearing at high

sliding speed

The dependence between bearing load (defined as load per unit length divided by the product

of sliding speed and viscosity) and sliding speed for both the adiabatic and isothermal bearing

at two different viscosity levels is shown in Figure 5.16 Defining the bearing load as load perunit length divided by the product of sliding speed and viscosity allowed for comparison ofthe heating effects on lubricants of different viscosity and at various sliding speeds

At low sliding speeds close to 1 [m/s], the load parameter converges to a common value of about 640,000 [dimensionless] This indicates that load under these conditions is proportional

to the product of sliding speed and viscosity which agrees well with isoviscous theory ofhydrodynamic lubrication As the sliding speed is increased, the load parameter declines andheating effects are gradually becoming evident It can be seen from Figure 5.16 that thethreshold sliding speed at which decreases in load capacity from the isoviscous level becomesignificant is lowered by the higher lubricant viscosity At high sliding speeds, the rise inlubricant viscosity may not provide as large an increase in load capacity as might be expected

It can also be seen that an isothermal bearing has a higher load capacity than an adiabatic

bearing Improvements in cooling of a real bearing can therefore bring an improvement inload capacity.

1050.1

0.05 Pas

0.5 Pas

FIGURE 5.16 Computed effect of lubricant heating on relative load capacity of a pad bearing

5.6.2 ELASTIC DEFORMATIONS IN A PAD BEARING

Almost all plain bearings operate with very small clearances and a requirement of nearly flatsliding surface All bearings are made of material with a finite elastic modulus, if theydeform or bend there may be a significant deviation from the optimum surface geometryconsiderably affecting the bearing performance Pad bearings are particularly vulnerable to

this phenomenon which is known in the literature as ‘crowning’ In a Michell bearing, the

pad bends about the pivot point to form a curved or crowned shape which has a much lowerload capacity than a rigid pad This effect can become extreme at small film thicknesses,where even very limited deflections due to bending may severely distort the film geometry

To illustrate this problem a one-dimensional pad has been selected as an example since therelevant elastic deflections can be found from simple bending theory The two-dimensionalcase would require the analysis of deflections in a plate which is far more complex [9] Elasticdeflections combined with thermohydrodynamic effects have also been analysed and a stronginteraction between these effects has been found [9,11]

An example of a computer program ‘DEFLECTION’ for analysis of an elastically deforming

one-dimensional pivoted Michell pad bearing is listed and described in the Appendix.Thermal effects, although significant, have been omitted in the program because oflimitations of computing speed The controlling equations of the bearing are the isoviscousReynolds equation and the elastic deformation equation:

M' is the local bending moment [Nm];

E is the elastic modulus of the pad material [Pa];

I is the second moment of area of the pad [m4]

The pad is modelled as an infinitely long plate of uniform thickness so that ‘I’ (in terms of

second moment of area per unit length) is a constant The bearing load is assumed to besupported at the pivot The pivot is located at the calculated centroid of hydrodynamicpressure A two level iteration procedure is used in this analysis The isoviscoushydrodynamic pressure field is first computed by iteration and then the bending momentsare found and the resulting pad deflection calculated The hydrodynamic pressure field isthen re-iterated and a new series of pad deflections is found The process is repeated until thepad deflections converge to sufficient accuracy

Hydrodynamic pressure is found from a finite difference equivalent of the one-dimensionalisoviscous Reynolds equation The one-dimensional isoviscous Reynolds equation (4.25) can

) ( i

The bending deflection equation is applied with the following boundary conditions:

· the bending moment ‘M'’ and shear force normal to the pad ‘S’ are equal to zero at

both ends of the pad bearing,

· the pad is balanced at the pivot point and there are no other forms of support tothe pad,

· pad deflection and deflection slope dz'/dx are zero at the pivot.

With these conditions, for x < x c where ‘x c’ is the position of the pressure centroid, the

expressions for ‘S’ and ‘M'’ are:

For x > xc the expressions for ‘S’ and ‘M'’ can be written as:

where ‘B’ is the width of the pad [m].

The deflection of the pad for all ‘x’ is found by integrating of (5.69) twice with respect to ‘z’

and is given by:

one-of a deformable pad bearing

Effect of Elastic Deformation of the Pad on Load Capacity and Film Thickness

The computer program ‘DEFLECTION’ described above can provide useful information for

the mechanical design of hydrodynamic bearings For instance, the effect of pad thickness andelastic modulus of pad material on the load capacity can be assessed with the aid of thisprogram The effect of pad thickness on load capacity is demonstrated as an example ofpossible applications of this program

It is of practical importance to know how thick the bearing pad should be to providesufficient rigidity for a particular size of bearing and nominal film thickness A reduction inthe hydrodynamic film thickness can increase load capacity but at the same time it alsoincreases bearing sensitivity to elastic distortion Optimization of bearing characteristics is

therefore essential to the design process The computed load capacity of a bearing of 1 [m] pad width, lubricated by a lubricant of 1 [Pas] viscosity versus pad thickness is shown in Figure 5.18 The Young's modulus of the pad's material is 207 [GPa] The hydrodynamic film thickness is 2 [mm] at the inlet and 1 [mm] at the outlet of the pad.

Start

Acquire parameters

Maximum film thickness Minimum film thickness Sliding speed

Bearing width Lubricant viscosity Pad thickness Pad elastic modulus Special settings of

iteration parameters?

No

Yes Acquire mesh

Integrate pressure to find shear force

Print out pressures, deflections and load capacity

Calculate DHDX from film thickness

Solve 1-D Reynolds equation for

Double integration of bending moment

to find deflection

No Is residual deformation

small enough?

Use preset values

Calculate initial film thickness;

set pressure and deflections to zero

Integrate shear force to find bending moments

Calculate deflection with relaxation factor between current and previous iteration Calculate new film thickness = undeformed film thickness + new deflection

Yes

FIGURE 5.17 Flow chart of program to compute load capacity of an elastically deforming

one-dimensional pivoted pad bearing

It can be seen from Figure 5.18 that as the pad thickness is reduced from 200 [mm] to 30 [mm] the load capacity declines by 70% When the thickness of the pad is 200 [mm], then the load capacity is identical to that of the rigid pad At the pad thickness of 100 [mm], load capacity is only reduced by about 10% as compared to a rigid pad It can thus be concluded that 100 [mm]

is close to the optimum pad thickness for this particular bearing The relationships between

the film thickness and pressure for pads of 100 [mm] and 30 [mm] thickness are shown in

Figures 5.19 and 5.20 respectively

Straight reference line

FIGURE 5.19 Effect of elastic deflection on the hydrodynamic film thickness and pressure

profile for a 100 [mm] thick pad.

It can be seen from Figure 5.19 that with a pad thickness of 100 [mm], elastic deflection is

small and the pressure field is essentially the same as for a rigid bearing When the pad

thickness is reduced to 30 [mm], however, the geometry of the bearing is distorted from a

tapered wedge to a converging-diverging film profile For this pad thickness, the divergencebeyond the minimum film thickness is sufficiently small so that cavitation does not occur

The pressure profile does, however, shift forward along with the pivot point as shown inFigure 5.20 With further reduction in either the pad thickness or the film thickness,cavitation occurs and this causes a severe reduction in load capacity Cavitation causes theeffective load bearing area to shrink and the load capacity declines even if specifichydrodynamic pressures remain high.

0 100

FIGURE 5.20 Effect of elastic deflection on the hydrodynamic film thickness and pressure

profile for a 30 [mm] thick pad.

From the example presented it is clear that the performance of pad bearings depends on theuse of high modulus materials and thick bearing sections Low modulus materials such aspolymers, although attractive as bearing materials, would require a metal backing for allbearings with the exception of very small pad sizes

5.6.3 CAVITATION AND FILM REFORMATION IN GROOVED JOURNAL BEARINGS

Cavitation occurs in liquid lubricated journal bearings to suppress any negative pressuresthat would otherwise occur In the numerical analysis of complete journal bearings, asopposed to partial arc bearings, cavitation and reformation must be included in thenumerical model For partial arc bearings, it can be assumed that the inlet side of the bearing

is fully flooded and cavitation is usually limited to a small area down-stream of the loadvector Full 360° journal bearings are lubricated through the lubricant supply holes orgrooves and the cavitation and reformation fronts that form around the grooves or holescontrol the load capacity of the bearing For the standard configuration of two groovespositioned perpendicular to the load line, a cavitation front forms down-stream of eachgroove and a reformation front is located up-stream of each groove This is illustrated inFigure 5.21 which shows the cavitation and reformation fronts on an ‘unwrapped’ lubricantfilm

A method of predicting the location of the cavitation and reformation fronts is required fornumerical analysis of the grooved bearing The cavitation front can be determined byapplying the Reynolds condition that all negative pressures generated during computationsare set to zero It is found that if this rule is applied then not only are all the negative