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51 H Hashimoto, S Wada and M Sumitomo, The Effects of Fluid Inertia Forces on the Dynamic Behaviour of

Short Journal Bearings in Superlaminar Flow Regime, Transactions ASME, Journal of Tribology, Vol 110,

1988, pp 539-547.

52 J.A Tichy, Measurements of Squeeze-Film Bearing Forces and Pressures, Including the Effect of Fluid Inertia,

ASLE Transactions, Vol 28, 1985, pp 520-526.

53 L.A San Andres and J.M Vance, Effect of Fluid Inertia on Squeeze-Film Damper Forces for Small-Amplitude

Circular-Centered Motions, ASLE Transactions, Vol 30, 1987, pp 63-68.

54 W.J Harrison, The Hydrodynamical Theory of Lubrication with Special Reference to Air as a Lubricant,

Trans Cambridge Phil Soc., Vol 22, 1913, pp 39-54.

55 W.A Gross, Fluid Film Lubrication, John Wiley, New York, 1980.

56 A Burgdorfer, The Influence of the Molecular Mean Free Path on the Performance of Hydrodynamic Gas

Lubricated Bearings, Transactions ASME, Journal of Basic Engineering, Vol 81, 1959, pp 94-100.

57 B Bhushan and K Tonder, Roughness-Induced Shear- and Squeeze-Film Effects in Magnetic Recording, Parts

1 and 2, Transactions ASME, Journal of Tribology, Vol 111, 1989, Part 1, pp 220-227, Part 2, pp 228-237.

58 Y Mitsuya, Stokes Roughness Effects on Hydrodynamic Lubrication, Part 2 - Effects under Slip Flow Boundary

Conditions, Transactions ASME, Journal of Tribology, Vol 108, 1986, pp 159-166.

59 S Haber and I Etsion, Analysis of an Oscillatory Oil Squeeze Film Containing a Central Gas Bubble, ASLE

Transactions, Vol 28, 1985, pp 253-260.

60 D.W Parkins and W.T Stanley, Characteristics of an Oil Squeeze Film, Transactions ASME, Journal of

Lubrication Technology, Vol 104, 1982, pp 497-503.

61 V.T Morgan and A Cameron, Mechanism of Lubrication in Porous Metal Bearings, Proceedings Conf on Lubrication and Wear, Inst Mech Engrs., London, 1957, pp 151-157.

62 F.E Cardullo, Some Practical Deductions from the Theory of the Lubrication of Cylindrical Bearings,

Transactions ASME, Vol 52, 1930, pp 143-153.

63 Cz M Rodkiewicz, K.W Kim and J.S Kennedy, On the Significance of the Inlet Pressure Build-up in the

Design of Tilting-Pad Bearings, Transactions ASME, Journal of Tribology, Vol 112, 1990, pp 17-22.

64 J.A Cole, Experimental Investigation of Power Loss in High Speed Plain Thrust Bearings, Proceedings Conf.

on Lubrication and Wear, Inst Mech Engrs., London, 1957, pp 158-163.

65 D Dowson, Non-Steady State Effects in EHL, New Directions in Lubrication, Materials, Wear and Surface Interactions, Tribology in the 80's, edited by W.R Loomis, Noyes Publications, Park Ridge, New Jersey, USA, 1985.

66 J.A Cole and C.J Hughes, Visual Study of Film Extent in Dynamically Loaded Complete Journal Bearings,

Lubrication and Wear Conf., Proc Inst Mech Engrs., 1957, pp 147-149.

67 E.R Booser, CRC Handbook of Lubrication, Volume II, CRC Press, Boca Raton, Florida, 1984.

5 H Y D R O D Y N A M I C S

5.1 INTRODUCTION

The differential equations which arose from the theories of Reynolds and later workersrapidly exceeded the capacity of analytical solution For many years some heroic attemptswere made to solve these equations using specialized and obscure mathematical functionsbut this process was tedious and the range of solutions was limited A gap or discrepancyalways existed between what was required in the engineering solutions to hydrodynamicproblems and the solutions available Before numerical methods were developed, analoguemethods, such as electrically conductive paper, were experimented with as a means ofdetermining hydrodynamic pressure fields These methods became largely obsolete with theadvancement of numerical methods to solve differential equations This change radicallyaffected the general understanding and approach to hydrodynamic lubrication and othersubjects, e.g heat transfer It is now possible to incorporate in the numerical analysis of thebearing common features such as heat transfer from a bearing to its housing The application

of traditional, analytical methods would require to assume that the bearing is eitherisothermal or adiabatic Numerical solutions to hydrodynamic lubrication problems can nowsatisfy most engineering requirements for prediction of bearing characteristics andimprovements in the quality of prediction continue to be found In engineering practiceproblems like: what is the maximum size of the groove to reduce friction before lubricantleakage becomes excessive, or how does bending of the pad affect the load capacity of abearing?, need to be solved

In this chapter the application of numerical analysis to problems encountered in

hydrodynamic lubrication is described A popular numerical technique, the ‘finite difference

method’ is introduced and its application to the analysis of hydrodynamic lubrication is

demonstrated The steps necessary to obtain solutions for different bearing geometries andoperating conditions are discussed Based on the example of the finite journal bearing it isshown how fundamental characteristics of the bearing, e.g the rigidity of the bearing, theintensity of frictional heat dissipation and its lubrication regime, control its load capacity

5.2 NON-DIMENSIONALIZATION OF THE REYNOLDS EQUATION

Non-dimensionalization is the substitution of all real variables in an equation, e.g pressure,film thickness, etc., by dimensionless fractions of two or more real parameters This process

extends the generality of a numerical solution A basic disadvantage of a numerical solution

is that data is only provided for specific values of controlling variables, e.g one value offriction force for a particular combination of sliding speed, lubricant viscosity, film thicknessand bearing dimensions Analytical expressions, on the other hand, are not limited to anyspecific values and are suited for providing data for general use, for example, they can beincorporated in an optimization process to determine the optimum lubricant viscosity Acomputer program would have to be executed for literally thousands of cases to provide acomprehensive coverage of all the controlling parameters The benefit of non-dimensionalization is that the number of controlling parameters is reduced and a relativelylimited data set provides the required information on any bearing

The Reynolds equation (4.24) is expressed in terms of film thickness ‘h’, pressure ‘p’, entraining velocity ‘U’ and dynamic viscosity ‘η’ Non-dimensional forms of the equation's

variables are following:

h is the hydrodynamic film thickness [m];

c is the bearing radial clearance [m];

R is the bearing radius [m];

L is the bearing axial length [m];

p is the pressure [Pa];

U is the bearing entraining velocity [m/s], i.e U = (U 1 + U 2 )/2;

η is the dynamic viscosity of the bearing [Pas];

x, y are hydrodynamic film co-ordinates [m]

The Reynolds equation in its non-dimensional form is:

Although any other scheme of non-dimensionalization can be used this particular scheme is

the most popular and convenient For planar pads, ‘R’ is substituted by the pad width ‘B’ in

the direction of sliding

5.3 THE VOGELPOHL PARAMETER

The Vogelpohl parameter was developed to improve the accuracy of numerical solutions ofthe Reynolds equation and was introduced by Vogelpohl [1] in the 1930's The Vogelpohl

parameter ‘M v’ is defined as follows:

in the final solution, i.e d n M v /dx * n where n > 2, unlike the dimensionless pressure ‘p*’ This

is because, where there is a sharp increase in ‘p*’ close to the minimum of hydrodynamic film thickness ‘h*’, ‘M v ’ remains at moderate values Large values of higher derivatives cause significant truncation error in numerical analysis The characteristics of ‘M v ’ and ‘p*’ for a

journal bearing at an eccentricity of 0.95 are shown in Figure 5.1

Degrees around bearing

Vogelpohl parameter

M v

Dimensionless pressurep*

p*, M v

L /D = 1

ε = 0.95

360° bearing

FIGURE 5.1 Variation of dimensionless pressure and the Vogelpohl parameter along the

centre plane of a journal bearing [4]

It can be seen from Figure 5.1 that the introduction of the Vogelpohl parameter does not

complicate the boundary conditions in the Reynolds equation, since wherever p* = 0, also M v

= 0 (zero values of ‘h’, i.e solid to solid contact, are not included in the analysis) As discussed later in this chapter, wherever cavitation occurs, the gradient of ‘M v’ adjacent and normal to

the cavitation front is zero like that of ‘p*’.

Numerical solutions of the Reynolds equation are obtained in terms of ‘M v ’ and values of

‘p*’ found from the definition M v /h * 1.5 = p*.

5.4 FINITE DIFFERENCE EQUIVALENT OF THE REYNOLDS EQUATION

Journal and pad bearing problems are usually solved by ‘finite difference’ methods although

‘finite element’ methods have also been employed [2].The finite difference method is based

on approximating a differential quantity by the difference between function values at two ormore adjacent nodes For example, the finite difference approximation to ∂Mv / ∂x* is given

FIGURE 5.2 Illustration of the principle for the derivation of the finite difference

approximation of the second derivative of a function

The finite difference equivalent of (∂ 2 M v / ∂x * 2 + ∂2 M v / ∂y * 2 ) is found by considering the nodal variation of ‘M v ’ in two axes, i.e the ‘x’ and ‘y’ axes A second nodal position variable is introduced along the ‘y’ axis, the ‘j’ parameter The expressions for ∂M v / ∂y * and ∂ 2 M v / ∂y * 2 are exactly the same as the expressions for the ‘x’ axis but with ‘i’ substituted by ‘j’ The coefficients of ‘M v ’ at the ‘i’-th node and adjacent nodes required by the Reynolds equation

which form a ‘finite difference operator’ are usually conveniently illustrated as a ‘computingmolecule’ as shown in Figure 5.3

The finite difference operator is convenient for computation and does not create anydifficulties with boundary conditions When the finite difference operator is located at theboundary of a solution domain, special arrangements may be required with imaginary nodesoutside of the boundary The solution domain is the range over which a solution isapplicable, i.e the dimensions of a bearing There are more complex finite differenceoperators available based on longer strings of nodes but these are difficult to apply because ofthe requirement for nodes outside of the solution domain and are rarely used despite their

greater accuracy The terms ‘F’ and ‘G’ can be included with the finite difference operator to

form a complete equivalent of the Reynolds equation The equation can then be rearranged

to provide an expression for ‘M v,i,j’ i.e.:

+( (R L

C 1 = 1

δx* 2

C 2 = δy* 1 2

This expression forms the basis of the finite difference method for the solution of the

Reynolds equation Its solution gives the required nodal values of ‘M v’

−2 δx* 2

−2 δy* 2

1

δx* 2

1 δy* 2

1 δy* 2

δy*

δx*

To boundary

of solution domain

j-1

j j+1

FIGURE 5.3 Finite difference operator and nodal scheme for numerical analysis of the

Reynolds equation

Definition of Solution Domain and Boundary Conditions

After establishing the controlling equation, the next step in numerical analysis is to definethe boundary conditions and range of values to be computed For the journal or pad bearing,

the boundary conditions require that ‘p*’ or ‘M v’ are zero at the edges of the bearing and alsothat cavitation can occur to prevent negative pressures occurring within the bearing The

range of ‘x*’ is between 0 - 2π (360° angle) for a complete bearing or some smaller angle for a partial arc bearing The range of ‘y*’ is from -0.5 to +0.5 if the mid-line of the bearing is

selected as a datum A domain of the journal bearing where symmetry can be exploited to

cover either half of the bearing area, i.e from y* = 0 to y* = 0.5, or the whole bearing area is

shown in Figure 5.4 Nodes on the edges of the bearing remain at a pre-determined zerovalue while all other nodes require solution by the finite difference method Whensymmetry is exploited to solve for only a half domain, it should be noted that nodes on themid-line of the bearing are also variable and the finite difference operator requires an extracolumn of nodes outside the solution domain, as zero values along the edge of the solutiondomain cannot be assumed This extra column is generated by adopting node values fromthe column one step from the mid-line on the opposite side In analytical terms this isachieved by setting:

M v,i,jnode +1 = M v,i,jnode −1 (5.10)

where ‘jnode’ is the number of nodes in the ‘j’ or ‘y*’ direction.

A split domain reduces the number of nodes but when analyzing a non-symmetric ormisaligned bearing then a domain covering the complete bearing area is necessary The

domain in this case is the complete bearing with limits of ‘y*’ from -0.5 to +0.5 and the

mid-line boundary condition vanishes

Sliding direction Load

Extra row for half bearing

Extra row of nodes for overlap with x* = 0

FIGURE 5.4 Nodal pressure or Vogelpohl parameter domains for finite difference analysis of

hydrodynamic bearings

Calculation of Pressure Field

It is possible to apply the direct solution method to calculate the pressure field but this isquite complex In practice the pressure field in a bearing is calculated by iteration procedureand this will be discussed in the next section

Calculation of Dimensionless Friction Force and Friction Coefficient

Once the pressure field has been found, it is possible to calculate the friction force and frictioncoefficient from the film thickness and pressure gradient data As discussed already inChapter 4 the frictional force operating across the hydrodynamic film is calculated byintegrating the shear stress ‘τ’ over the bearing area, i.e.:

τ is the shear stress [Pa];

η is the dynamic viscosity of the lubricant [Pas];

U is the entraining velocity [m/s];

h is the hydrodynamic film thickness [m];

p is the hydrodynamic pressure [Pa];

F is the friction force [N];

x is the distance in the direction of sliding [m];

y is the distance normal to the direction of sliding [m]

In a manner similar to the computation of pressure, the equation for friction force can be

expressed in terms of non-dimensional quantities From (5.1) h = h*c, x = x*R and

p = p*(6U ηR)/c 2 and substituting into (5.12) yields:

dp*

dx* = Uη

c

1 h* + 3h*

µ is the coefficient of friction;

W is the bearing load [N]

Load on a journal bearing is often expressed as:

h* cav

h*

where:

h cav * is the dimensionless film thickness at the cavitation front;

h* is the dimensionless film thickness at a specified position down-stream of the

cavitation front

The average or ‘effective’ coefficient of friction is proportional to the lubricant filled fraction

of the clearance space and within the cavitated region ‘p*’ and ‘dp*/dx*’ are equal to zero.

The symbol for ‘effective’ dimensionless shear stress is ‘τe *’ Assuming a simple

proportionality between fluid filled volume and total shear force, an average value of ‘τ*’ e

that allows for zero shear stress between streamers of lubricant, is given by:

τ* e = h* cav

This value of dimensionless shear stress is included in the integral for dimensionless frictionforce (eq 5.17) with no further modification

Values of h*, ∂h*/∂x*, ∂h*/∂y* and ∂ 2 h*/ ∂x * 2 are also required in computation and the

expressions for these are:

where:

ε is the eccentricity ratio;

t is the misalignment factor;

β is the attitude angle

Note that the variation in ‘h*’ due to misalignment is dependent on ‘x*’ whereas the variation in ‘h*’ due to eccentricity is also controlled by the attitude angle.

The derivatives of ‘h*’ are found by direct differentiation of (5.24), i.e.:

Numerical Solution Technique for Vogelpohl Equation

The nodal values of ‘M v ’ are conveniently arranged in a matrix with ‘i’ and ‘j’ as the column

and row ordinates The coefficients in equation (5.9) can also be organized into a ‘sparse’matrix with all coefficients lying close to the main diagonal It is therefore possible to solveequation (5.9) by matrix inversion but this requires elaborate computation Programming isgreatly simplified when iterative solution methods are applied The Gauss-Seidel iterativemethod is used in this chapter All node values are assigned an initial zero value and thefinite difference equation (5.9) is repeatedly applied until convergence is obtained

5.5 NUMERICAL ANALYSIS OF HYDRODYNAMIC LUBRICATION IN IDEALIZED JOURNAL AND PARTIAL ARC BEARINGS

A numerical solution to the Reynolds equation for the full and partial arc journal bearings isnecessary for the calculation of pressure distribution, load capacity, lubricant flow rate andfriction coefficient when the bearings are neither ‘infinitely long’ nor ‘infinitely narrow’

This condition is valid for bearings with L/D ratio in the range 1/3 < L/D < 3, where ‘L’ is the bearing length and ‘D’ is the bearing diameter Equation (5.9) is solved numerically in order

to find the dimensionless pressure field corresponding to equation (5.2) and the other

important bearing parameters An example of the flow chart of the computer program

‘PARTIAL’ for the analysis of a partial arc or full 360°, isothermal, rigid and non-vibrating

journal bearing is shown in Figure 5.5 while the full listing of the program with description

is provided in the Appendix The program provides a solution for aligned and misalignedjournal bearings Misalignment has a pronounced effect on bearing characteristics but cannot

be modelled by either the infinitely long or narrow bearing theories Numerical methodshelp to overcome this problem

Example of Data from Numerical Analysis, the Effect of Shaft Misalignment

The computed solution to the classical Reynolds equation as applied to journal bearings hasbeen comprehensively used to obtain basic information for bearing design An example ofthis data was shown in Chapter 4 for the 360° journal bearing (Figure 4.32) Tables of data for

load and attitude angle as a function of eccentricity, L/D ratio and partial arc angle can be found in [e.g 4,5,6] A computer program ‘PARTIAL’ for the analysis of a partial arc bearing is

listed in the Appendix The program calculates the dimensionless load, attitude angle, Petroffmultiplier and dimensionless friction coefficient for a specified angle of partial arc bearing,

L/D, eccentricity and misalignment ratios The solution is based on an isoviscous model of

hydrodynamic lubrication with no elastic deflection of the bearing

Start

Acquire bearing parameters

Eccentricityε

L /D ratio Arc angleαMisalignment parametert

Special settings

of finite difference mesh?

No

Yes Acquire number

of I and J nodes

of I and J nodes

Specify iteration limits?

No

Yes Acquire limits

Number of iteration cycles Relaxation factor

Residual termination size

Set initial zero values of

M(I,J) and P(I,J)

Set initial values of arc position

as bisecting minimum film thickness

A Use preset number

Use preset values