Stability of Flexibly supported Finite Oil Journal Bearings including Fluid Inertia and surface Roughness Effect: A Non-linear Transient Analysis

Stability of Flexibly supported Finite Oil Journal Bearings including Fluid Inertia and surface Roughness Effect: A Non-linear Transient Analysis

A. K. Bandyopadhyay1,

Assistant Professor Department of Mechanical Engineering,

DR. B.C Roy Engineering College, Durgapur, West Bengal, India

S. K. Mazumder2,

Professor,

Department of Mechanical Engineering, DR. B.C Roy Engineering College, Durgapur, West Bengal, India

M. C. Majumdar3 Professor,

Department of Mechanical Engineering, NIT,Durgapur,

West Bengal,India

Abstract: The aim of this study is to analyse the Non-linear transient stability of finite oil journal bearing including the effect of fluid inertia and bearing surface roughness. The inertia effect is usually ignored in view of its negligible contribution compared to viscous force.However,fluid inertia effect is to be taken in the analysis when modified Reynolds number is around one.This investigation deals with the stability of flexibly supported finite rough oil journal bearing with fluid film inertia effect using finite difference method.An attempt has been made to evaluate the critical mass parameter. A non-linear time transient method is used to simulate the journal centre trajectory to estimate the stability parameter,which is a function of speed.

In the present work,a modified form of Reynolds equation is developed to include the combined influence of fluid inertia and surface roughness for the analysis of finite oil journal bearing.The modified average Reynolds equation considering inertia effect with flow simulation model of rough surfaces (Patir and Cheng [1, 2] is solved by a finite difference method with a successive over-relaxation scheme (Gauss-Siedel),while the equation of motion of both the journal and bearing are solved by the fourth-order Runge-Kutta method. The stability increases with the increase of eccentricity ratio and modified Reynolds numbers.

Keyword- Modified Reynolds number, stability, critical massparameter, Surface Roughness parameter,Fluid film inertia.

1. INTRODUCTION

   Ever-increasing demand for the hydrodynamic journal bearing systems to operate under high speed and high eccentricity makes it imperative to design this class of bearings accurately.In such cases,the familiar assumptions of smooth surface can no longer be employed to accurately predict the performance of journal bearings systems as no machining surfaces are perfectly smooth. Therefore,it is imperative to include the influence of surface roughness and

   fluid inertia effects in the design and analysis of journal bearings

   In classical lubrication theory,the Reynolds equation has been used to provide an explanation for the process of hydrodynamic lubrication.TheNavier-Stokes equation is reduced to the Reynolds equation under the assumptions that the inertia forces of the lubricant is negligible and the flow is laminar.The validity of this conventional thin film theory is justified for small values of Reynolds number.Recently,owing to some practicalapplications,the need to include inertia effects has arisen because of the increasing number of lubrication problems which involve moderately large Reynolds numbers.Such application include large size bearing operating with non-conventional lubricants,bearings and seals operating with non- conventional lubricants such as liquid metals and water,the use of high speed bearings etc.In these cases,it is adequate to extend the Reynolds equation to include the inertial effects.

   The effect of fluid inertia has been studied by many researchers for turbulent flow using long and short bearing approximations.However,there are few publications which deal only with the intermediate regime for finite oil journal bearings.Reinhardt and Lund [19] studied the dynamic characteristics based on first-order perturbation solution starting from the Navier-stokes equation.Banerjee et al.[5] introduced an extended form of Reynolds equation to include the effect of fluid inertia adopting an iteration scheme.

   Kakoty and Majumdar [7- 9] carried out a first order perturbation technique in modified Reynolds number as was done by Reinhardt and Lund [19], to study the stability of an oil journal bearing.

   The hydrodynamic lubrication theory of rough surfaces has been subject of growing interest as the bearing surfaces, in

   practice are all rough.Stochastic concept introduced by Tzeng and Saibel [15] has fascinated many researchers and simulated a fair amount of work in this field.A theoretical analysis of the effect of surface roughness in a finite width bearing was done by Christensen et.al [16] based upon stochastic theory of hydrodynamic lubrication. A modified Reynolds equation considering combined effect of turbulence and surface roughness was derived by Hashimoto and Wada

   [18] to a high speed journal bearing .Majumdar and Ghosh[13] studied the stability of rigid rotors supported on finite rough oil journal bearings using perturbationmethod.Non-linear transient stability analysis has been performed by R.Turaga et.al[2,6]to study the sub- synchronous whirl stability of a rigid rotor supported on two symmetric hydrodynamic bearings with rough surfaces subjected to unidirectional constant load. Theoretical analysis to study the effect of support stiffness and damping on the transient response of flexibly supported rotor bearing systems,considering surface roughness effect was done by Ramesh,. J. et.al.[14].

   An attempt is being made here to study the effect of fluid inertia and surface roughness effect on the stability of oil film

   Fig.1 The schematic diagram of flexibly supported oil Journal Bearing

   journal bearings under unidirectional constant load.The

   2

   (1)

   governing equations are deduced starting from the Navier-

   Re

   u u u v u w D u p u

   L

   _ 2

   Stokes equation and flow continuity equations.These

   equations are identical (except for time dependent terms) to _

   y

   z y

   the ones developed by Constatinescu and Galetuse[3] which also include turbulent flow regime.In the present study the authors are particularly concerned with the laminar flow

   regime.Since closed-form solution is not possible,an attempt

   p 0

   _

   y

   (2)

   2 (3)

   R e

   w u w v w w D w D p w

   2

   is made to solve the system of nonlinear partial differential equation using Gauss-Siedel iteration method in a finite

   difference scheme.

   _ _

   y L z L z _

   y

   _

   u v D w 0

   (4)

   _

   L _

   A nonlinear time transient method is used to simulate the journal centre trajectory and thereby to estimate the stability

   y z

   • z _ y x

   parameters,which is a function of speed.

   Where, z ,

   L 2

   y ,

   c R

   , p .t , p ,

2. BASIC THEORY

_ _

u

v

u , v

_ w p

p c2

and

1. Considering fluid Inertia Effect only

   The modified average Reynolds equation for fully lubricated surfaces is derived starting from the Navier-Stokes equations and the continuity equation with few assumptions. The non- dimensional form of the momentum equations and the continuiy equation for a journal bearing may be written as (Figure.1)

   R c w R R2

   c c2

   Re Re . R

   The variation in the density with time is considered to be negligible. Since there is no variation in pressure across fluid film the second momentum equation is not used.

   The fluid film thickness can be given as

   h c e cos

   _

   h 1 cos

   (5)

   (6)

   h

   where,

   h , e ,

   c c

   After Constantinescu and Galetuse[ 3 ] the velocity components are approximated by the parabolic profiles. The velocity components may be expressed in non-dimensional form as follows:

   • y

     2

     y

     y

     (7)

     u Q 2

     h

     h

     h

     2

     y

     y

     (8)

     w Qz 2

     h

     h

     _

     Q and Qz are dimensionless flow parameter in and z

     direction respectively.

     Substituting these two into momentum equations and integrating give

     2

     Q h p R I

     (9)

     Fig. 2. Two rough surfaces in relative motion

     2 e X

     The combined roughness

     1 2

     (15)

     2

     (10)

     2 2 2

     Q h

     D p R I

     and has a variance

     1

     2

     (16)

     z 2 L e Z

     z

     The ratio of h

     is an important parameter showing the

     Where,

     1

     _

     1 _ Q 1 1

     1

     (11)

     effects of surface roughness.

     To study surfaces with directional properties the surface

     1

     Q h 1 h

     1 Q Q 2 h

     h 2

     I

     3

     3 6

     2 10

     characteristic can be used. The parameter can be viewed

     X 2 1 1

     1 Q

     1 D Q

     1 D 1

     1 Q

     as the length to width ratio of a representative asperity. There

     h Q

     h Qz

     h

     Q

     z

     3 5

     z

     2

     _

     30 L

     6 L 5

     2 z

     are mainly three sets of asperity patterns are identified purely

     1. Transverse roughness pattern <1

        1 h 1 _ Qz 1 1

        1 h

        6 Qz 6 h

        6 Qz 5 Q 2 (12)

     2. Isotropic roughness pattern =1

        Z

        I h

        2 1 1

        1 Q

        1

        Q 1 D

        Q

     3. Longitudinal roughness pattern >1

     h

     Q z

     h Qz

     h Qz z

     Considering the bearing and journal surface are rough surface

     6 5

     2 30

     15 L

     z

     having random roughness amplitudes of the two surfaces

     From continuity equation one can obtain the following form of modified Reynolds equation in rotating coordinate system

     considering fluid inertia effect.

     hT can be written as

     D 2

     hT

     h f d

     h 3

     p

     h 3

     p 12 h

     (13)

     L

     h

     (17)

     z

     z

     Where

     f is the probability density function of composite

     h

     D

     6 1.0 2.0 2 Re h I X L

     h Iz

     roughness. Where

     1 and 2 are the random roughness

     z

2. Considering surface roughness effect

   It has been reported by many researcher that the surface roughness patterns significantly influence the steady state and dynamic characterises of hydrodynamic bearings. Consider

   amplitudes of the two surfaces measured from their mean levels. 1 and 2 are the standard deviations.

   For a Gaussian distribution, the normal probability function of is

   two real surfaces with normal film gap h in the sliding motion. Local film thickness hT is defined to be of the form

   f 1 e

   2

   2

   2 2

   (18)

   hT h 1 2

   (14)

   From equation (17) and (18) we have

   Where h is the normal film thickness (compliance) defined as the distance between levels of the two surfaces. 1 and

   hT

   1

   2 h

   h e

   2

   2 2 d

   (19)

   2 are the random roughness amplitudes of the two surfaces measured from their mean levels.

   After integration we have

   _ 2

   • h

   2

   _

   We assume

   and have a Gaussian distribution of hT h 1 erf

   h e 2

   (20)

   1 2 2

   2 2

   heights with zero mean and standard deviations 1

   respectively.

   and 2

   _

   c 1

   Where,

   c

   or _ .

   Where, is called surface roughness parameter.

   _ 2

   experimentation, is found to depend on these parameters

   h h

   1 0.5* h s

   hT

   1 erf

   e

   (21)

   through the functional form:

   2

   2

   2

   V

   h ,

   V

   ( h , )

   (28)

   • s

   r1 s 1

   r 2 s 2

   Differentiating h with respect to x, z and , we get

   T

   (22)

   Where Vr1 and Vr 2 are the variance ratios given by:

   hT 1 1 erf h h 1 1 erf h sin

   2

   2

   2

   2 2

   2

   V 1

   V 2

   1 V

   (29)

   (23)

   r1 ' r 2 r1

   hT 1 1 erf h h 1 1 erf h cos

   is a positive function of

   h and the surface pattern

   2

   2 2

   2

   s

   parameter of the given surface.

   The shear flow factor is plotted as a function of h and

   hT 1 1 erf h h 0

   (24) s

   2

   2

   in [1, 2].starting with zero for purely longitudinal

   z

   z

   roughness , the shear flow factor increases with

   As h 0

   z

3. Pressure Flow Factors

decreasing ,and retains highest value for purely transverse roughness 0 .Through numerical simulation and using nonlinear least square program they are of the form:

Patir and Cheng [1] and [2] introduced pressure flow factors

A H 1 e 2 H 3 H 2

H 5

(30)

x and z in circumferential and axial direction are obtained through numerical simulation. The pressure flow simulation

s

1

Where H=h/.For extrapolation beyond H=5 the following relation should be used:

factors are given by the empirical relation of the form:

A e0.25H

H 5

(31)

1 CerH for 1 s 2

x (25)

The coefficients

A , A , , ,

are listed as functions

x

1 CH r

for 1

(26)

1 2 1 2 3

of in Table 2.

Where H h .The constants C and r are given as

A functions of in Table.1

Table 2: Coefficients o equations (26),(27) for s (range

	A1	1	2	3	A2
1/9	2.046	1.12	0.78	0.03	1.856
1/6	1.962	1.08	0.77	0.03	1.754
1/3	1.858	1.01	0.76	0.03	1. 561
1	1.899	0.98	0.92	0.05	1.126
3	1.560	0.85	1.13	0.08	0.556
6	1.290	0.62	1.09	0.08	0.388
9	1.011	0.54	1.07	0.08	0.295

H 0.5 )

z is equal to x

value corresponding to the directional

properties of the z profile. In functional form it is given as:

h ,

h , 1

(27)

z x

Table 1. Coefficients of equations (25), (26) for x

	C	r	Range
1/9	1.48	0.42	H > 1
1/6	1.38	0.42	H > 1
1/3	1.18	0.42	H > 0.75
1	0.90	0.56	H > 0.5
3	0.225	1.5	H > 0.5
6	0.520	1.5	H > 0.5
9	0.870	1.5	H > 0.5

Now introducing pressure flow factors

x and z

with

shear flow factors

s we get modified Reynoldss equations

D. Shear Flow Factors

Similar to the pressure flow factors, the shear flow factor is a

considering combined effect of fluid inertia and surface roughness in dimensionless form as:

function of the film thickness and roughness parameters only.

D 2

z

(32)

h 3 p

h 3

p 12 h

z

However, unlike x which only depends on the statistics of

X

L z

the combined roughness , and the shear flow factors

h

_ s

D

6 1.0 2.0 6 2 Re hT I X L

• hT Iz

  depends on the statistical parameter of

  1 and 2

  separately

  z

  .Therefore, s

  is a function of h , the standard deviations

  1 and 2 and the surface pattern parameters

  1 and

  2 of

  the two opposing surfaces. Through numerical

  Or,

  D 2

  1

  be good enough for the present study. Since the bearing is

  _

  h 3 p

  h 3

  p 12

  1 erf

  h

  cos

  symmetrical about its central plane ( z =0),only one half of

  X

  L

  • z

z

1

z

2

2

the bearing needs to be considered for the analysis.

6 1.0 2.0 1 erf

h sin

1. Fluid film forces

   2

   2

   The non-dimensional fluid film forces along line of centers and perpendicular to the line of centers are given by

   _

   D

   (33)

   _ F C 2

   1 2 _ _

   e

   T

   X L T z

   Fr 3 p cos d d z

   6 s 2 R h I h I

   z

   r

   R L

   0 1

   (34)

   Where, I x and I z are same as equation (11) and (12) above,

   _ F C 2

   1 2 _ _

   (35)

   Boundary conditions for equation (33) are as follows

   F 3 p sin d d z

   1. The pressure at the ends of the bearing is assumed to be zero (ambient):

      R L

      where1 and 2

      0 1

      are angular coordinates at which the fluid

      p

      , 1 0

   2. The pressure distribution is symmetrical about the mid- plane of the bearing:

   p , 0 0

   film commences and cavitates respectively.

2. Steady state load

   The steady state non-dimensional load and attitude angle are given by

   z

   W F r 2 F 2

   (36)

   3. Cavitation boundary condition is given by:

   0 0 0

   p

   _

   2 , z 0 and p , z 0 for 1 2

   tan1 F

   0

   (37)

   o _

   Fr

   The equations (9), (10), (11), (12) and (33) are first expressed in finite difference form and solved simultaneously using Gauss-Siedel method in a finite difference scheme.

   0

   Since the steady state film pressure distribution has been obtained at all the mesh points, integration of equations (34) and (35) can be easily performed numerically by using

   III METHOD OF SOLUTION

   Simpsons 1/ 3 rd. rule to get

   F r and F

   . The steady state

   To find out steady-state pressure all the time derivatives are set equal to zero in Equations. (9), (10), (11), (12) and (33).

   load W 0 and the attitude angle

   0

   are calculated using

   For 0 0.2 the pressure distribution and flow parameters

   z

   Q and Q are evaluated from inertia less ( Re* 0 ) solution, i.e., solving classical Reynolds equation. These values are then used as initial value of flow parameters to

   equations (36) and (37).

3. Equation of Motion

The equation of motion for a rigid rotor supported on four identical flexibly supported bearings are given by,

solve Eqs.(9) and (10) simultaneously for

Q and Qz Using

Guss-Siedel method in a finite difference scheme. Then

update I x & I z and then calculate Q and Qz for use to solve Eq.(33) with particular surface roughness pattern and surface roughness parameter for new pressure p with

inertia effect by using a successive over relaxation scheme.

The latest values of

Q ,

Qz and p are used iteratively to

solve the set of equations until all variables converges. The convergence criterion adopted for pressure is

_ _

5 and also same criterion for Q

1 pnew pold 10

and Qz . For higher eccentricity ratios 0 0.2the initial values for the variables are taken from the results corresponding to the previous eccentricity ratios. Very small

increment in is to be provided as

Re* increases. The

procedure converges up to a value of Re* 1.5 which should

where,

.

.

2

C 2 . Cos .Sin.

1 F Sin F Cos

r

0

M .W . 2

.

.

2

D 2 . .Sin .Cos.

1 Cos F Sin W

r 0

0

M .W . 2

d 2 X

r

Fig. 4: Hydrodynamic fluid film forces in circumferential & radial direction

(38)

A1 Sin , A2 Cos , A3 .Cos , A4 .Sin

_

G X b

M r .

r

dt 2

2

Fr Sin F Cos

_

H Yb

M . d Yr F Cos F Sin W

(39)

E C G

r dt 2 r

d 2 X

b

0

dXb

(40)

F D H

Mb .

b

dt2

d 2Y

F Cos Fr Sin B. dt

dY

• KXb

(41)

D. Solution Scheme:

Mb. b F Sin FrCos B. b KYb

For stability analysis, a non-linear time transient analysis is

dt2 dt

The relation between rotor & bearing motion are given by,

carried out using the equations of motion to compute a new

X r X b e Sin

Yr Yb eCos

(42)

(43)

set of ,, Xb ,Yb & their derivatives for the next time step

_

for a given set of. Re*, , L / D, 0 , M (Mass parameter) for a

particular roughness parameter, . The forth order Runge-

The above two equations are substituted in equations of motion. Finally the equations of motion are expressed in non- dimensional form as follows,

Kutta method is used for solving the equations of motion. The hydrodynamic forces are computed for every time step by solving the partial differential equation for pressure satisfying the boundary conditions.

X b

dX b

d

(44)

Yb

_

dYb d

F Cos F Sin

(45)

E. Stability Analysis

To study the combined effect of fluid inertia and surface roughness on journal centre trajectory of flexibly supported

1 r

(46)

bearings a set of trajectories of journal centre and bearing has

X b

.

_ _ 2

been studied and it is possible to construct the trajectories for

m. M .W0 . .W0 .B. X b W0 .K .X b

F Sin F Cos

_

numbers of complete revolution of the journal the plots

1 r

(47)

shows the stability of the journal when the trajectory of

Yb _

.

m. M .W .2

0

d

d

• d

.W0 .B.Y b W0 .K .Yb

(48)

(49)

journal and bearing centre ends in a limit cycle. Critical mass parameter for a particular eccentricity ratio, slenderness ratio, modified Reynolds number, surface roughness parameter and roughness pattern is found when the trajectories ends

with limit cycle (Fig. 9& Fig.10) or it changes its trend from

d

A3.F A4 .E A2 .A3 A1.A4

A2 .E A1.F A2 .A3 A1.A4

(51)

(50)

stable to unstable.

2.00 Stable

0.00

0.30 0.40 0.50 0.60 0.70 0.80

Ecentricity ratio

Re*=1.5 Re*=1.0 Re*=0.5

Re*=0

4.00

Re*=0

8.00

6.00

Re*=1.0

Unstable

Re*=0.5

10.00

Critical mass parameter

IV RESULTS AND DISCUSSIONS

Figure 5.Variation of Critical mass parameter with ecentricity ratio including fluid inertia and surface roughness effect and different modified Reynolds number

L/D=1,Vr1=0,=6,=6

12.00

Re*=1.5

Figure 6.Variation of Critical mass parameter with ecentricity ratio including fluid inertia and surface roughness effect for difeerent modified Reyolds number

L/D=2.0,Vr1=0

12.00

Re*=1.

10.00 Unstable

Re*=1.0

Ecentricity ratio

0.80

0.70

0.60

0.50

0.40

2.00

0.00

0.30

Stable

4.00

Re*=0

6.00

Re*=1.5 Re*=1.0 Re*=0.5

Re*=0

Re*=0.5

8.00

Critical mass parameter

A Effect of Modified Reynolds Number (Re*)with eccentricity ratio

=6

Unstable =1/6

Critical mass parameter

Figure 5 and 6 shows the variations of stability at different values of Re*(0.5,1.0 and 1.5) and L/D (1.0 and 2.0).From the figure smaller L/D ratio gives better stability in the case of inertialess solution for all eccentricity ratios.

Figure 7.Variation of Critical mass parameter with surface roughness parameter for different roughness pattern parameter

L/D=1.0,Re*=1.5,Vr1=0.5,0=0.5

7.00

6.50

6.00

=1

1

2

3

Ro4ughness pa5rameter 6

7

8

5.00

4.50

4.00

Stable

5.50

B Effect of Roughness Pattern () with Surface roughness Parameters

Figure 7 shows the variation of mass parameter with surface roughness parameter for various roughness pattern parameters.It is seen that stability is better for transversely oriented roughness pattern.

Figure 8.Variation of critical mass parameter with surface roughness parameter for different variance ratio.

L/D

Vr1=1.0

Vr1=0.5

8.00

6.00

4.00

2.00

0.00

Vr1=0

Vr1=0.5

Vr1=1.0

1 3 5 7 9

Surface roughness parameter

Vr1=0

Stable

Unstable

14.00

12.00

10.00

Critical mass parameter

C Effect of variance ratio (Vr) with Surface roughness Parameter

Figure 8 shows the variation of mass parameter with surface roughness parameter for different variance ratio.when the journal surface is rough and the bearing surface is smooth

Vr1 1.0,the stability is seen to decrease sharply for small

values of 3.On the other hand, when the journal

surface is smooth and the bearing surface is rough (i.e., Vr1 0 ),the bearing is highly stable for small roughness parameter 3.A bearing having identical roughness

structure (i.e.,Vr1 0 .5) gives intermediate values of stability.

Fig. 9. Journal centre Trajectory of flexibly supported finite oil journal bearing with fluid inertia and surface roughness effect.

Re = Reynolds number,

cR

Re = Modified Reynolds number, c R

R e

t = time in s

u, v w = velocity components in x, y, z directions in

m/s

_ _ _

u,v, w

x, y z

= dimensionless velocity components

= coordinates

_ _

,Y , Z

Fig. 10.Bearing centre Trajectory of flexibly supported finite oil journal

= Dimensionless coordinates, x , y , z

R c L 2

bearing with fluid inertia and surface roughness effect _

h d

d

V d cos d sin

V CONCLUSIONS

1. The effect of inertia on the stability is affected considerably W0 at higher L/D ratios and eccentricity ratios (Figure 5 and _ 6).The probable reason may be that higher L/D ratios and W eccentricity ratios,the circumferential component of flow will 0

   = steady-state load bearing capacity in N

   W c

   2

   = Dimensionless steady-state load 0

   R3 L

   be overtaking the axial flow.The inertia effect of the

   circumferential flow will possibly add more stiffness in the film,thereby improving the stability.It is also noted that higher Re* means higher surface speed of the shaft (when other parameter remain constant).This will further increase couette flow which is a part of circumferential flow.One can see this particular effect for L/D=2.0 and for 0.5(Figure 6).

2. When the journal surface is very rough 3 and the bearing is smooth,the stability decreases drastically.

3. When the bearing surface is very rough 3 and the

   , 0 = Eccentricity ratio e c (dimensionless),

   steady-state eccentriity ratio

   = Density of the lubricant (kg m-3)

   = Angular velocity of journal(rad s-1)

   p = Angular velocity of whirl (rad s-1)

   = Whirl ratio, p

   = Absolute viscosity of lubricating

   film (N s m-1)

   = Attitude angle

   journal is smooth,the stability improves significantly. Q

   Qz	=	Dimensionless flow parameter in z direction
   _
   Q	=	Dimensionless side leakage
   1 ,	2 =	Angular coordinates at which film commences and cavitates.
   	=	Surface pattern parameter
   	=	Roughness Parameter, c
   H	=	h
   , =		Pressure flow factors
   s	=	Shear flow factor

4. The stability can be improved by employing higher L/D ratio.

= Dimensionless flow parameter in direction

_

NOMENCLATURE

c = Radial clearance (m)

D = Diameter of Journal (m)

e = eccentricity (m)

F , F

= Hydrodynamic forces (N) ,

_ _

F r , F

2

Dimensionless Hydrodynamic film forces

2

_

F

Fr C

3

_

,

F

F C

3

R L R L x s

h = Film thickness, (m)

_

h = Dimensionless film thickness, h/c

L = Length of the bearing in m

= Composite r.m.s roughness,= 2 2

1

2

1 , 2 = Standard deviation of 1 and 2

p = Film pressure in Pa

_

p = dimensionless film pressure

p c2

1 , 2 = Random roughness amplitudes (heights) of

surfaces.

R2

= Combined roughness [m], = 1 + 2

R = radius of journal in m

Vr = Variance ratio,

2

2

Vr1 1 , Vr 2 2

X b = Coordinate of bearing centre in x-direction

Yb = Coordinate of bearing centre in y-direction

X r = Coordinate of rotor centre in x-direction

Yr = Coordinate of rotor centre in y-direction

M r = Mass of rotor or journal

M b = Mass of bearing

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