Rtiwari Rd Book 06b

Dr R Tiwari, Associate Professor, Dept. of Mechanical Engg., IIT Guwahati, ([email protected])

288

6.2 Hydrodynamic Oil-Lubricated Journal Bearings

Journal bearings consist of a circular section length of shaft (journal) rotating inside a bearing bush

that is normally circular. The journal diameter is usually 99.8 – 99.9% of that of the bush, and the

clearance space between the two in partially filled by the lubricating fluid. At zero rotational speed,

under a steady load, the journal rests at the bottom of the clearance space. When it rotates, oil is

dragged along by the journal due to the oil viscosity and a thin film of oil is built up between the

journal and bush so that under normal operating conditions there is no surface-to-surface contact.

When designing for steady loads the problem is to ensure that there is sufficient film clearance to take

up the effects of the surface undulations and likely changes in load during the machine’s life. To

ensure that the bearing is not prone to self-excited vibrations, with these factors in mind, most journal

bearings operate at steady state eccentricity ratios (journal eccentricity/radial clearance) of about

0.6 – 0.7. Journal bearing can have a significant effect on a machine’s vibration characteristics. The

oil film behaves like a complicated arrangement of springs and dampers and so influences the

machine critical speeds and imbalance response. Moreover, bearing fluid film forces can cause rotor

instability, which results in serious levels of self-excited vibration (e.g. oil-film whirl). The most

common forms of journal bearing are given in Figure 6.12.

Plain cylindrical Circumferential Partial arc Lemon bore

3 lobe 4 lobe Offset halves Tilting pad

Figure 6.12 Various geometric shapes of hydrodynamic bearings

Plain Cylindrical bearings

This type of bearings are simplest to manufacture and the most common in use. Oil inlet port (and

outlet port when present) is frequently on the bearing horizontal center line and may be either circular

(“hole entry”) or in the form of an axial groove type of recessed pocket to aid oil distribution. The

plain bearing is able to carry very heavy loads except in the direction towards the oil feed pocket.


289

Circumferential Groove Bearings

An alternate form of the plain cylindrical bearing is the circumferential groove bearing in which the

surface is effectively divided into two half-bearing in which the surface is effectively divided into two

half-bearings by a circumferential oil distribution groove. This type of bearing is frequently used

when the direction of the load may change significantly. However the magnitude of the load, which

can be applied, is less than half of that of the plain bearing.

Partial-Arc Bearings

In another variation of the plain cylindrical bearing the lower part of the bearing bush bore is relieved.

Typically, 60 – 1200 of included arc of surface are left to carry the load whose direction is virtually

constant. The advantage obtained by relieving the bush bore is to reduce friction losses in the bearing

whilst retaining most of the bearing load-carrying capacity.

Lemon Bore Bearing

It is effectively two partial arc bearings mounted in opposition to each other, and is manufactured by

removing material from the interface between the upper and lower halves of what was originally a

circular bearing. The resulting bearing has a load-carrying capacity, which is more resistant to oil-film

whirl. The stiffness and damping in the horizontal direction are relatively low.

Three-Lobe and Four-Lobe Bearings

They provide greater resistance to oil-film whirl and relatively large oil-film stiffness and damping

values, which are dependent on the orientation of lobes. Their load-carrying capacity is only about

half of that of the partial arc bearing.

Offset Halves Bearings

Able to support loads which are slightly less than those carried by the partial arc bearing. It has high

oil-film stiffness and damping. It has very high resistance to oil-whirl but suitable for machines which

rotate in one direction only.

Tilting Pad Bearings

It has 100% resistant to oil-film whirl. It contains number of arcs, each of which is independently

mounted on a fulcrum whose axis of rotation runs parallel with that of the journal. The pads are thus

able to find their own equilibrium alignment position during bearing operation. It is most costly to

manufacture and has a load-carrying capacity which is only about half of that of the plain cylindrical

bearing. It is generally used only when oil-film whirl cause a problem.


290

6.2.1 Basic Concepts and Assumptions of Bearing Models

For a given bearing and rotational speed, lubrication theory may be used to calculate the reaction

forces on the journal from the lubricant film are functions of the displacements of the journal from

bearing center and of the instantaneous journal center velocities and accelerations. Hence, for small

amplitude motions, measured from the static equilibrium position (see Figure 1) of the journal (u0, v0),

a first order Taylor series expansion yields

ymfxmf

ym

yc

yk

ymxmycxcykxk

ymxmycxcykxk

yxx

vu

x

xy

vu

x

xy

vu

x

xy

yyyxyyyxyyyxvy

xyxxxyxxxyxxux

−=ℜ−=ℜ

∂

∂ℜ=

∂

∂ℜ=

∂

∂ℜ=

++++++ℜ=ℜ

++++++ℜ=ℜ

y

),(),(),(

where

000000

0

0

(1)

and analogously the remaining bearing dynamic coefficients may be defined. In the matrix form of

equation (1) all diagonal terms are called direct coefficients and off-diagonal terms are called cross-

coupled terms. The latter terms arise due to the fluid rotation within the bearing. ℜ is the reaction

force of fluid film on the journal, f is the external excitation force on the journal, m is the journal mass, u0 and v0 are the static equilibrium position of the journal from the bearing center, x and y are

the displacements of the journal from its static equilibrium position, yx and are the instantaneous

journal center velocities and yx and are the instantaneous journal center accelerations, in the vertical

and horizontal directions respectively. The “dot” indicates the time derivatives and kij, cij and mij (i, j =

x, y) are bearing stiffness, damping and added-mass (also termed the virtual fluid-film mass or inertia)

coefficients respectively. The indices of the stiffness, damping and added-mass coefficients have the

following significance: the first index gives the direction of loading which produces the elastic

(damping/inertia) force and the second index gives the direction of the displacement

(velocity/acceleration). Because (u0, v0) is the equilibrium position, then 0u

ℜ equals the static load, W.

The equilibrium position depends on a unique value of the dimensionless Sommerfeld number

(22 )/()/)(/( DLcRWRLS rΩ= µ ), where µ is the lubricant viscosity, Ω is the journal rotational

speed, D is the bearing bore, R is the journal radius, L is the bearing length and cr is the bearing radial

clearance. The Sommerfeld number, S, defines the operating conditions (speed, lubricant viscosity,

static load and geometry). The dynamic coefficients are evaluated for a particular static equilibrium position, which is a function of the Sommerfeld number, S, which means that for a given application,

they are functions of rotor speed. Moreover, bearing dynamic coefficients could be external excitation

frequency, ω, dependent.

It should be noted that equation (1) is a complete form of linearised fluid-film dynamic equation and it

contains twelve stiffness, damping and added-mass coefficients. Consistent with the assumptions

inherent in reducing the Navier-Strokes equations to the Reynolds equation, the conventional laminar,

thin film lubrication theory ignores the inertia forces in the fluid-film (Pinkus and Sternlicht, 1961;

Schlichting, 1960). This is theoretically justified for small values of the Reynolds number (of the

order of 1). On the other hand, the assumption of laminar flow ceases to be valid when there is

transition to either Taylor vortex flow or to turbulent flow which, for fluid-film cylindrical journal


291

bearing, occurs at a Reynolds number of approximately 1000 to 1500. Thus, there is an intermediate

range, say for values of Reynolds number of the order of 100, where added-mass effects may become

noticeable (several times the mass of the journal itself) without affecting the assumption of laminar

flow. The added-mass coefficients represent the mass of the bearing fluid-film (Reinhardt and Lund,

1975) but are significant only in exceptional cases and in most analyses the added-mass of the bearing

film are ignored. The stiffness and damping coefficients can be obtained by a finite difference

solution of the perturbed Reynolds equation (Lund and Thomsen, 1978).

From lubrication theory (ignoring inertia effects) damping coefficients are symmetric but stiffness

coefficients are not. Therefore principal directions do not exist (although this was assumed by Hagg

and Sankey (1956) and Duffin and Johnson (1966-67)), and in the experimental determination of the

coefficients, it is necessary to obtain two independent sets of amplitude-force measurements. Lund

(1987) emphasised the experimental measurement of the bearing coefficients and established more

uniform agreement with analytical calculations by considering the influence of thermal and elastic

deformations and practical problems of manufacturing and operating tolerances of bearing geometry,

clearance and lubricant viscosity. Although the load-displacement characteristics of a journal bearing

is evidently non-linear, the concept of linear dynamic coefficients is still used for modern rotor

dynamic calculations for unbalance response, damped natural frequencies and stability since

experience has demonstrated the usefulness of the coefficients. Clearly the applicability of these

techniques is dependent on small amplitudes of vibration at the journal (relative to the bearing clearance).

For squeeze-film bearings the governing equation for the fluid-film reaction force has a similar form

to equation (1) with negligible stiffness coefficients and no static load.

ycxcymxm

ycxcymxm

yyyxyyyxy

xyxxxyxxx

+++=ℑ

+++=ℑ (2)

Mittwollen et al. (1991) showed theoretically and experimentally that hydrodynamic thrust bearings,

which are often treated as an axial support, might affect the lateral vibration of a rotor-bearing system.

If no axial force is present, then the resulting reaction moments of a thrust bearing can be written in

terms of stiffness and damping coefficients (Jiang and Yu, 1999 and 2000)

ψθψθ

ψθψθ

ψθψθ

ψθψθ

yyyyy

xxxxx

cckk

cckk

+++=ℵ

+++=ℵ (3)

where kxθ, etc. represent moment dynamic coefficients of the thrust bearing, and θ and ψ are angular displacements (slopes) in x and y directions, respectively.

All of the bearing models discussed thus far are linearised models. A few researchers have considered

non-linear bearing models and these will be described in appropriate places.

6.2.1 Reynolds Equation and Approximate Solutions

The characteristic of hydrodynamic bearings can be derived theoretically by correctly modeling the

bearing fluid film. The equation which does this was derived by Reynolds (1886) and is (Cameron,

1981)


292

3 3

6 ( ) 2 ( )h p h p

U h hx x y y x t

ρ ρρ ρ

µ µ ∂ ∂ ∂ ∂ ∂ ∂ + = + ∂ ∂ ∂ ∂ ∂ ∂

(6.23)

where x locates the distance around the bearing circumference of the point under consideration

(measured from some arbitrary datum), y is the position in the axial direction, ρ is the lubricating

density,µ is the dynamic viscosity, h is the film clearance, p is the lubricant pressure, U is the

tangential velocity of the journal surface and t is the time. The film clearance is very small compared

with the journal diameter and so the effects of curvature can be ignored for the purposes of evaluation

of the lubricant pressure variation. For the purpose of mathematical modeling, the lubricant film may

be considered as through it were unwrapped from around the journal as indicated in Figure 6.8.

Figure 6.8 Hydrodynamic bearing static equilibrium film thickness

In Figure 6.8, θ is the angular position around the bearing measured from JB produced, φ is the

attitude angle, L is the bearing length, B is the bearing center, J is the journal center, R is the journal

radius and JB = e is the eccentricity. In most applications the lubricant density does not vary

substantially throughout the oil film. Its viscosity might vary; a constant effective viscosity based on

the lubricant thermal balance may be used for calculations. With above assumptions and for steady

state operation (i.e. 0h

t

∂=

∂) we get from equation (6.23)

θ

h

φ

R

J

B U

0 π 2π

Angular position, θ

0 2πR Circumferential position, x

Axial position

L

Film thickness, h


293

3 36

p p hh h U

x x y y xµ

∂ ∂ ∂ ∂ ∂ + = ∂ ∂ ∂ ∂ ∂ (6.24)

Above equation describes the variation of lubricant pressure in both the axial and circumferential

directions. An approximate solution (closed form) may be obtained by using the “short bearing”

(Ocvirk, 1952) approximation (where pressure variation in the circumferential direction is assumed to

be negligible compared with that in the axial direction and so p

x

∂∂

is set to zero). The eight linearized

stiffness and damping coefficients depend on the steady state operating conditions of the journal, and

in particular upon the angular speed. For the short bearing, the dimensionless bearing stiffness and

damping coefficients, ij ij rk k c W= , ij ij rc c c W= , and i, j = x, y, as a function of the steady

eccentricity ratio, ε, of the bearing are given as (Smith, 1969, Journal Bearings in Turbo Machinery)

( ) ( )2 2 2 2 4

2

2 16

1xy

Qk

π π π ε π ε ε

ε ε

− − −=

−;

( ) ( ) ( )( )

2 2 2 2 4

2

4 32 2 16

1yy

Qk

π π ε π ε ε

ε

+ + + −=

−

( ) ( ) ( )2 2 2 2 4

2

32 2 16

1yx

Qk


ε ε

− + + + −=

−; ( ) ( )2 2 4

4 2 16xxk Qπ π ε ε= + −

( ) ( ) ( )2 2 2 22 1 2 8

xx

Qc

π ε π π ε ε

ε

− + −= ; ( ) ( )2 2 2

8 2 8xy yxc c Qπ π ε ε= = − + −

( ) ( )

( )

2 2 2 2 4

2

2 2 24

1yy

Qc


ε ε

+ − +=

− (6.25)

with

( ) 1 22 2 2

1( )

1 16Q ε

π ε ε=

− +

To determine stiffness and bearing coefficients of a short bearing, the Sommerfeld number


294

2

r

DLN rS

W c

µ =

(6.26)

is first determined, where W is the load on the bearing, r is the bearing radius, D is the journal

diameter, L is the length of bearing, µ is the viscosity of lubricant at operating temperature,

( 2 )nπΩ = the angular speed of journal, n is the number of revolutions per seconds and cr is the

radial clearance. The eccentricity ratio of the journal center defined as, r

e cε = , where e is the

journal eccentricity. We can then determine the eccentricity ratio under steady state operating

conditions by

( )( )

222

2 2 2

1

1 16

LS

D

ε

πε π ε ε

− = − +

(6.27)

For the “long bearing” approximation (where the pressure variation in the axial direction is assumed

to be negligible compared with that in the circumferential direction and so p

y

∂∂

is set to zero). This

approximation also enable closed-form solution of equation (6.26) to be obtained, provided that

appropriate boundary conditions (Figure 6.9) are selected to enable evolution of the constants of

integration. (Cameron, 1981; Pinkus and Sternlickt, 1961 and Rao, 1993).

6.2.2 Finite Bearings

“Half Sommerfeld” condition

(set p = 0 if computed p < 0)

Reynolds condition 0dp

dθ= when p =0

Angular position θ

‘Full Summerfeld’

condition

(p=0 at θ=0 & θ=2π

Lubricant pressure, p

Figure 6.9 Boundary conditions used in journal bearing analysis


295

Real bearings are neither infinitely long nor infinitely short. Most bearings have a length to diameter

ratio in the range 0.5 to 1.5, so although the approximate solutions described above may be used in

preliminary design calculations, final design studies will often require a more realistic analysis. For a

more general solution of the Reynolds equation it is necessary to use numerical method. Most

common numerical technique used in hydrodynamic journal bearing analysis is the finite difference

method, because of its relative simplicity and because of the ease with which it can be adopted to suit

most bearing geometries. In the first instance the problem should be reduced to a two dimensioned

one by considering the oil film to be unwrapped from around the journal. The oil film is then divided

into a number of sections of finite size by describing it in terms of mesh nodes, as indicated in Figure

6.10. In between each nodes described, a number of further points are considered, for example those

surrounding the node i, j are shown in Figure 6.11. Since the Reynolds equation describes the

behavior of the lubricant at any location in the fluid film, it can be written for every specific node

contained within the finite-difference mesh.

Equation (25) may be written for the node i, j as

1 1, ,

, 1 , , , 1 1, , , 1,3 3 3 3 2 21 1 1 12 2 2 2

, , , ,2 2 2 2

6i j i j

i j i j i j i j i j i j i j i j

i j i j i j i j

h hp p p p p p p p

h h h h Ux x y y x

µ+ −+ − + −

+ − + −

−− − − −

− + − =∆ ∆ ∆ ∆ ∆

which can be simply written as

, 1 2 1, 3 1, 4 , 1 5 , 1i j i j i j i j i jp k k p k p k p k p+ − + −= + + + + (6.28)

where 1

,2

i jh

+ is the film thickness between (i, j) and (i, j + 1) nodes and k1, k2 etc. are constants whose

values are known for every nodes. Equation (6.28) may then be written for every node in the mesh,

except for those where the lubricant pressure is already known (for example at the nodes representing

y∆

x∆

Circumferential position

Axial

position

i, j+1

i+1, j i-1, j

i, j-1

i, j i, j+1/2

Node point designation

Further points

considered

Figure 6.10 Figure 6.11


296

the ends of the bearing where the lubricant pressure must be equal to the ambient pressure). These

equations can then be solved simultaneously. Alternatively, an iterative method of solution may be

employed such as successive relaxation where the value of the pressure at each node is determined

successively according to the most up-to-date values of the terms on the right-hand side of the

equation (6.28) (initially, all pressure might be set to zero). This evaluation of pressure is repeated for

all nodes several times over until the change in the value of lubricant pressure at any node is no

greater than a small fraction of 1% of the lubricant pressure. At first consideration it may appear to be

difficult to impose the correct Reynolds conditions, because of the uncertainty of where the lubricant

pressure becomes zero. It appear that only the half-sommerfeld conditions can be imposed by setting

any negative pressure to zero, after solution has been found. However, if a second iteration procedure

or solution of simultaneous equations, is carried out with these pressure set to zero then different

nodes will be found to have sub-zero pressure (negativeve pressure). This process may be repeated

until there is no change in the lubricant film edge position. In the case of iterative procedures the

process can be speeded up by assigning any sub-zero pressure to be zero as and when the arc

evaluated. When this process is completed it is found that, because the Reynolds equation is a

continuous function, the final pressure distribution corresponds to the Reynolds boundary conditions

with the constraint of 0p

θ∂

=∂

at the trailing edge of the lubricant film automatically catered for. Once

the lubricant pressured have been evaluated for a particular journal attitude angle and eccentricity (i.e.

then film thickness will be known), the corresponding forces provided by the lubricant on the journal

may be evaluated by integrating numerically all of the element forces associated with each node. For

example, each node pressure is considered to contribute to force on the journal over an area X∆ &

Y∆ surrounding it, when integrating using a trapezium rule.

Figure 6.12 A closer view of the grid and node around circumference

X∆

The shading indicates the area

over which pressure pi,j is

considered to act

Y∆ pi,j

ψ

Various

node

positions

Some node

position around

the bearing

circumference


297

00 300

600

900 0 0.1 0.2 0.4 0.6 0.8

L/D=0.5

L/D=1.0

φ , Altitude angle

Eccentricity ratio

ε =e/c L/D=1.0 0.5 0.25

10-2 10-1 1 10

Sommerfeld number

2

r

DLN RS

W c

µ =

Eccentricity ratio ε

Thus, if the node is situated at some angle ψ to the horizontal direction as shown in Figure 6.12, then

the lubricant force on the journal in the horizontal direction (from left to right) is

∑∑= =

∆∆=m

j

n

i

ijijh yxpF1 1

cosψ (6.29)

and that in the vertical direction (upwards) is

∑∑= =

∆∆=m

j

m

i

ijijv yxpF1 1

sinψ (6.30)

where i is the axial node positions (Total n such nodes) and j is the circumferential node position

(total m such nodes). The above equations for lubricant forces acting on the journal should also

include the tangential friction forces, in most cases however, the friction forces are small when

compared with the pressure forces. For bearing designed to carry vertical loads only (for example

gravity loads) the relationship between eccentricity ratio ε and journal attitude angle φ may be

determined by investigating different values of φ for a given value of ε until the value of Fn is found

to be zero. This trial and error method enables corresponding value of ε, φ and Sommerfeld number

S to be found. The steady-sate journal loci for plain cylindrical bearings and their relationship with

sommerfeld number are shown in Figure 6.13. Where L is the bearing length, D is the bearing

diameter, W is the load on the baring, R the bearing radius, C is the radial clearance and N the shaft

rotational speed in rev./sec.


298

Figure 6.13 Typical variation of (a) eccentricity ratio with Sommerfeld number and (b) eccentricity

ratio with journal altitude angle for plain cylindrical bearings for L/D = 0.5 and 1.0

6.2.3 Friction

Friction forces in fluid film bearings arise out of the energy dissipation associated with the shearing

of the fluid film. From the elementary fluid mechanics the shear stress at some point in a fluid layer is

given by

dz

duµτ = (6.31)

where z is in film thickness h direction, µ is fluid dynamics viscosity and du

dz is velocity gradient

across the fluid film. In a hydrodynamic the velocity gradient term arises partly from the variation in

pressure around the journal causing lubricant flow and partly as result of the journal rotation

‘dragging’ lubricant around the clearance space. Thus, for a point at the journal surface equation

(6.31) becomes

1

2

dp h U

dx hτ µ

µ

= +

(6.32)

The first term in the bracket is the pressure-induced term and the second term is the velocity-induced

term. The total drag force on the journal is then given by ( )x Rdθ=

22

0 02

2

Ld

d

L

dp h UF Rd Rd dy

Rd h

π π µθ θ

θ

+

−

= +

∫ ∫ ∫ (6.33)

where pressure-induced component is assumed to be present only when the lubricant pressure is

positive i.e. over the region 0 < θ < π + d. The velocity-induced component is present wherever there

is lubricant, for example all around the clearance space (with reasonable approximation). It is possible

to develop closed-form expressions for the integral in equation (6.33) in the case of the infinitely long

and infinitely short bearings (Pinkus and Sternlicht, 1961). In the case of finite bearings the

integration may be carried out numerically and expressed as


299

∑∑∑∑∑∑= == == =

∆∆+∆∆+∆∆

=n

i

m

lj ij

n

i

l

j ij

n

i

l

j ij

ij

d yxh

KRUyx

h

RUyx

dx

dphF

11 11 1 2

µ (6.34)

where l is the node corresponding to the edge of the pressurized section of the lubricant film

(circumferential node position). Equation (6.34) is a slight variation of equation (6.33) in that the last

term of (6.34) now allows for the break up of the lubricant film, where K is a constant whose value is

less than unity. The magnitude of K may be determined empirically, and is the concern of current

research. This feature of real lubricant film is not allowed for in equation (6.33).

6.2.4 Lubricant Flow Rate

In case of both the infinitely short and infinitely long bearing there is a lubricant pressure variation in

the axial direction for 0 < θ < 2π causing a flow of lubricant out of the ends of the bearing. This flow

rate, allowing for both ends of the bearing, is given by basic fluid mechanics theory as

∫=

π

θµ

0

3

122 Rd

dy

dphQ (6.35)

For the finite bearing, using a trapezium rule for integration for the finite difference mesh shown in

Figure 6.10, we have

∑=

∆

=

m

j ij

ijx

dy

dphQ

1

3

122

µ (6.36)

A typical variation of the flow rate is shown in Figure 6.14.


300

Figure 6.14 Variation of flow rate with eccentricity ratio

6.2.5 Dynamic characteristics

Hydrodynamic bearing films are flexible, i.e. when a dynamic load is applied to the bearing, as a

consequence of imbalance for example, the journal is caused to orbit about the static equilibrium

position which it would otherwise take up when carrying only a static load. The effective stiffness and

damping associated with this flexibility of the lubricating film have a most significant effect on

system critical speeds, on forced response and on system stability. For these reasons it is important to

be able to evaluate the bearing dynamic characteristics at the design stage. The approach which is

usually adopted is to consider there to be some small displacements dx and dy of the journal away

from its static equilibrium position, in the horizontal and vertical directions respectively. The journal

center is also assumed to possess velocities xd and yd in horizontal and vertical directions

respectively (whereas under the action of only a steady load its velocity would be zero). The effects of

these changes in the state of the journal on the lubricant film forces are then evaluated. When there is

no dynamic load on the journal, the lubricant steady-sate film forces in the horizontal and vertical

directions could be expressed as functions of the journal displacements, x and y, and velocities, x and

y , away from the bearing center; that is

),,,(1 yxyxfFx= and ),,,(2 yxyxfFy

= (6.37)

where x and y are both zero. When there are changes in these displacements and velocities as

described above, the new values of lubricant film forces will be x xF dF+ and y yF dF+ respectively.

0.1=D

L

5.0=D

L

25.0=D

L

Eccentricity Ratio, /e cε =

Dim

ensionless flow rate, 2Q/(UcL)


301

These forces may also be expressed using equation (6.37) as a four-variable Taylor series, neglecting

small terms, as

1( , , , ) x x x xx x

F F F FF dF f x y x y dx dy dx dy

x y x y

∂ ∂ ∂ ∂+ = + + + +

∂ ∂ ∂ ∂

(6.38)

and

y

Fyd

x

Fxd

y

Fdy

x

FdxyxyxfdFF

yyyy

yy

∂

∂+

∂

∂+

∂

∂+

∂

∂+=+ ),,,(2 (6.39)

The change in the lubricant film forces are then given by

( )

x x x xx x x x

xx xy xx xy

F F F FdF F dF F dx dy dx dy

x y x y

k dx k dy c dx c dy

∂ ∂ ∂ ∂= + − = + + +

∂ ∂ ∂ ∂

= + + +

(6.40)

and

( )

y y y y

y y y y

yx yy yx yy

F F F FdF F dF F dx dy dx dy

x y x y

k dx k dy c dx c dy

∂ ∂ ∂ ∂= + − = + + +

∂ ∂ ∂ ∂

= + + +

(6.41)

where kij and cij are known as the oil-film stiffness and damping coefficients respectively, i and j can

take values of x and y. It is these quantities, which must be known in order to calculate the overall

system response to imbalance and stability. Values of kxx and kyx may be found by calculating the

lubricant film forces 1x

F and 1y

F when a small displacement dx is imposed on the journal (from its

static equation position) with dy , xd , and yd all set to zero. The process is then repeated with a

displacement of – dx to calculate 2x

F and 2y

F . The stiffness coefficients kxx and kyx are then given by

1 2 1 2and2 2

x x y y

xx yx

F F F Fk k

dx dx

− −= = (6.42)

Similarly, values of the other stiffness and damping coefficients may be evaluated by imposing small

variations in dy , xd , and yd on the journal, and investigating their effect on the lubricant film

forces, as


302

3 4 3 4 5 6

5 6 7 8 7 8

; ; ;2 2 2

; ; 2 2 2

x x y y x x

xy yy xx

y y x x y y

yx xy yy

F F F F F Fk k c

dy dy dx

F F F F F Fc c c

dx dy dy

− − −= = =

− − −= = =

(6.43)

It should be noted that when proceeding in the manner described above care should be taken when

formulating Reynolds equation (6.24) in finite difference form, as the quantity t

h

∂∂

is no longer zero

when xd or yd are non-zero, and appropriate value of U is also dependent on xd and yd . Solutions

for several types of bearing have been carried out (Lund et al., 1965; Homer, 1960, Someya, 1988). In

addition to the ‘spring’ and ‘damping’ coefficients introduced above, some authors have suggested the

use of ‘inertia’ coefficient and ‘moment’ coefficients. Although such coefficients usually have only a

small effect on bearing dynamics. Inertia coefficients can be defined as

),,,,,( yxyxyxFx = 0 and ),,,,,( yxyxyxFy = 0 (6.44)

Hence

x xx xx yx

F FdF dx dy m dx m dy

x y

∂ ∂= + + = + +

∂ ∂ … …

(6.45)

where m is the inertia coefficients. Moment coefficients can be defined as

),,,,,,,( φθφθ yxyxFx = 0 (6.46)

Hence

………… +++=+∂

∂+

∂

∂+= φθφ

φθ

θ φθ dkdkdF

dF

dF xxxx

x .. (6.47)

and

………… +++=+∂

∂+

∂

∂+= φθφ

φθ

θ φθθθ dkdkdM

dM

dM xxx .. (6.48)

where , , xk kθθ θ are moment stiffness coefficients.

Rtiwari Rd Book 06b

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Transcript of Rtiwari Rd Book 06b