DOI : 10.17577/IJERTV6IS040355
- Open Access
- Total Downloads : 246
- Authors : Manoj Kundu, Dr. Swapan Kumar Mazumder
- Paper ID : IJERTV6IS040355
- Volume & Issue : Volume 06, Issue 04 (April 2017)
- DOI : http://dx.doi.org/10.17577/IJERTV6IS040355
- Published (First Online): 17-04-2017
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Nonlinear Stability Analysis of Flexibly Supported Finite ISO VISCOUS Oil Journal Bearings Including Bearing Surface Deformation
Manoj Kundu1 S. K Mazumder2
Assistant professor, Department of Mechanical Engineering, Professor, Department of Mechanical Engineering, DR. B.C Roy Engineering College, Durgapur. DR. B.C Roy Engineering College, Durgapur.
West Bengal, India West Bengal, India
Abstract: This paper analyses the stability characteristics of a rigid rotor mounted in flexibly supported hydrodynamic oil journal bearings considering pressure depended viscosity including the effect of elastic distortion on the surface of bearing liner. This theoretical analysis is intended to show how the effect of elastic distortion along with the flexible support on the journal bearing performance considering pressure depended viscosity can be calculated for three-dimensional bearing geometries. The deformation equations for bearing surface will be solved simultaneously with hydrodynamic equations considering constant viscosity. A Non-linear time transient method is used to simulate the journal and bearing centre trajectory and thereby to estimate the stability parameter. In this analysis forth order Runge-Kutta method is used to determine the locus of the journal and the bearing centre for the various operating conditions. The stability of the system is determined from the combined stability effect in journal and bearing centre. It has been found that stability decreases with increase of of the elasticity parameter of the bearing.
Keyward – Journal bearing, surface deformation, variable viscosity, eccentricity ratio, Reynolds equation
-
INTRODUCTION
Journal bearings are widely used in rotating machineries to support large loads at mean speed of rotation. Regardless of significant advancement in lubrication technology, these bearings fail due to metal to metal contact when they operate below certain minimum speed especially during starting and stopping operations. In order to save cost of replacing the bearing, these bearings are provided with flexible liner. But the deformation of liner, affects the performance characteristics of the bearing particularly at high values of eccentricity ratio.
Many investigators notably ODonoghue et al. [10], Brighton et al. [11] and Majumder et al. [9], Jain et al. [ 13 ], Chandrawat and Sinhasan [ 5 ], Oh and Huebner [19] solved the journal bearing problem considering the effect of elastic distortion of the bearing liner. ODonoghue et al. [10 ] dealt
distortion in the bearing liner on bearing performance. Majumder [9] had done the stability analysis also by linearised method. The displacement equations thus derived were compared with those of ODonoghue et al. [10] for two dimensional elasticity problem with axial displacements reduced to zero. The displacement equations and form of film pressure tallied completely with Mazumder et al. [9] and stability performance analysis is done considering liner deformation through parametric study of the various variables like eccentricity ratio, slenderness ratio, Poisson ratio, liner thickness to radius ratio with variation in pressure depended viscosity.
Fig. 1: shows the schematic diagram of the flexibly supported oil journal bearing with flexible liner:
-
BASIC THEORY
Using the normal assumptions in the theory of hydrodynamic lubrication, modified Reynolds equation for dynamic conditions with fluid in rotating coordinate systems derived from Navier-Stokes equations and continuity equation in the bearing clearance of an oil-lubricated bearing as follows:
the analysis with the infinitely long bearing approximation.
p p p p
2 h h
Brighton et al.[11] described the methods of solution for
x x z
z 6 R 1 . t x 12 t
finite journal bearing considering the effect of elastic
(1)
distortions. Majumder et al. [9] used the numerical methods to determine the effects of elastic
Where h is the oil film thickness, p is the oil film pressure,
is the oil viscosity, is the angular velocity of journal and
R is the journal radius.
L
2 2
2 m z
The arrangement of journal bearing system with bearing
liner is shown in a schematic diagram (figure 1) above.
p cos sin n dz d
L
tan 1 0 0
(7)
m,n L
Equation (1) when non-dimensionalised with the following substitutions,
2 2
p cos
0 0
2 m z L
cos n dz d
x
-
h
z
-
p c2
, p t , p L
, h , z
R c
L 2 , p R2
2 2 2
(8)
The following modified Reynolds equation considering
p0,0 L p d dz
0 0
variable viscosity is obtained in non-dimensional form:
These displacements are substituted in the stress-strain
p
(2)
relationships using Lames constants. The six components of
h 3 p
h 3 6 1 .
h 12 h
z
z
stresses are then used in the equations of equilibrium to
obtain the following three displacement equations,
Boundary conditions for equation (1) are as follows
-
The pressure at the ends of the bearing is assumed to be
The first term of the right-hand side of equation (5) is 1 p .
2 0,0
zero (ambient): p , 1 0
-
The pressure distribution is symmetrical about the mid-
Using the end condition of the bearing (i,e p 0 at z L 2 ) we can obtain p0,0 . This term does not contribute any
plane of the bearing: p , 0 0
z
-
Cavitation boundary condition is given by:
deformation at z 2. Its effect for the other values of z is
included in the total deformation. The boundary conditions of the inner radius are
p
r p ,
r
0, r z 0
(9)
2 , z 0 and
p , z 0 for 2
After non-dimensionalisation, the equation (6), (7) and (8)
becomes
Where 2 is the angular coordinate at which the film
2 1
1
2 2
cavitates.
The oil fluid film thickness, h , in the case of flexible bearing
p m,n
p cos m z cos n d z d
2 0 0
(10)
can be written as,
2 1
2
h c ecos
(3)
p cos m z sin n d z d
where is the deformation of the bearing surface and it is a
2 1
0 0
function of and z .
p cos m z sin n d z d
tan1 0 0
(11)
Therefore, h 1 cos
(4)
m,n
2 1
h
Where
h , e
and
p cos m z cos n d z d
0 0
c c c
and
1 2 1
(12)
Before finding solution to equation (1) satisfying the
p0,0
p d d z
appropriate boundary conditions, the elastic deformation 0
0 0
p c2
p c2
is obtained by a method similar tothat of Majumder et al. [9] and Brighton et al. [11]. In present calculation
where p
m,n
m,n , p
0
R2
0
R2
, z z
L 2
the three displacement components
u,v & w in
r, & z
The outer surface of the bearing is rigidly enclosed by the
directions are solved simultaneously satisfying the boundary conditions with an approximate method, as Brighton et al. [11], to evaluate the displacements.
The pressure distribution in the bearing clearance of the rigid bearing is first calculated by solving two dimensional steady state Reynolds equation. The film pressure is then expressed
in double Fourier series of the form:
housing, preventing any displacement of the outer surface. The ends of the bearing are prevented from expanding axially, but are free to move circumferentially or radially.
The displacement components in r, and z directions are found from the pressure distribution, which has been expressed in a Fourier series. It is apparent that the displacements will also be harmonic functions.
l
p
m
l
pm,n n
cos 2 m z cos n
L
m,n
(5)
These displacements were substituted in the stress-strain relationships using Lame's constants. The six components of
Where l indicates that the first term of the series is halved.
stresses were then used in the equations of equilibrium to obtain the following three displacement equations.
p and as follows,
d 2 u *
*
-
C *
d u *
*
2 u *
* n
d v*
m ,n
m,n
L
1
2 2
C 2 C n 2
d y y d y y
C 1
y d y
2 2
p cos
4 0 0
2 m z L
cos n dz d
(6)
C * 1 n
2
v*
k 2 u * C * 1k
d w*
0
(13)
pm,n L
2 L
2
p cos
0 0
2 m z L
2
sin n dz d
-
d y
d 2 v* 1
dv*
* 2 v*
2
2 *
* n
du*
-
-
-
METHOD OF ANALYSIS
2
d y y d y
1 C n k v
y
C 1
y d y
The modified film pressure p was first obtained from
-
C
* 1 n
2
y
u *
n k C
* w*
1
0
y
(14)
equation (2) in finite difference form assuming a constant film shape and using Gauss-Seidel method with successive over relaxation scheme. The convergence criterion adopted
d 2 w*
1 d w* n 2
d u *
for pressure is
_
_ 5
. w* C * k 2 w*/
k C * 1
1 pnew pold 10 .
2 2
d y y d y y
d y Then this pressure distribution was expressed as a double
k C 1
* u*
y
v*
*
n C 1 . k 0
y
(15)
Fourier series as given by equation (5). The deformation equation (22) was then calculated for a given F using distortion coefficients from equation (20). The film thickness
Where, E , C* 2 , k 2 m ri &
equation was then modified using equation (4). The process
E
2 1
1 1 2 L
was repeated until a compatible film shape and pressure distribution was determined.
_
-
Fluid film forces: At any point on the journal the film
C 2 k w
The boundary conditions are,at
y 1,
pressure is p and the film force is p R d d z , where R d d z
*
d u* 1
C
pm,n
* n v* u* *
(16)
is any small segment at an angle with the line of centres. This will have components p R d d z cos in the direction
d y
d v* n u* v*
d y y y
y y
(17)
along the line of centres and p R d d z sin in the direction normal to the line of centres.
Component Fr of the oil fluid film forces along the line of
d w*
u* k
(18)
centres is given by,
L
2 2
(23)
d y
and at
b
y ,
u* v* w* 0
(19)
Fr 2 p cos R d dz
0 1
a
The equations (13), (14) and (15) expressed first in finite difference form solving the displacement equations with the boundary conditions (16-19) the values of the distortion coefficient dm ,n were obtained and expressed as,
where 1 and 2 are angular coordinates at which the fluid film commences and cavitates respectively.
Component F of the oil fluid film forces perpendicular the line of centres is given by
L
dm,n
u*
R p
(20)
2 2
F 2 p sin
0 1
R d dz
(24)
The radial deformation of the bearing surface will be
Using
-
p c2
,
z
0
0
p
-
,
2
Fr c
, and F c2 ,
u
R2
L 2 F r
R3 L
F
0
R3 L
or,
d
R p r
cos n
cos 2 m z
the non-dimensional form is given by,
m,n i
m,n L
1 2
(25)
Considering the bearing clearance is very small in compare to the diameter of the journal, the total radial deformation
F r p cos d d z
0 1
1 2
(26)
will be
R p0,0 d0,0 d
R p cos n
cos 2 m z
(21)
F p sin d d z
0 1
2 m0 n0
m,n 0,0
m,n m,n
m,n L
-
-
-
Steady state load
From the film forces in r and directions, neglecting the
After non-dimensionalisation, the equation (21) becomes
time dependent term in Reynolds equation, the resultant film
2 1 F d
p0,0 0.0
pm,n
dm,n
cos n
m,n
cos m .
z
force which is balanced by the load applied to the shaft can
m 0 n 0
m,n 0,0
(22)
be calculated and the angle between load W 0 and the line of
centres (i,e attitude angle 0 ) are determined by
Where, and
R3 and is replaced by E
c F
2
0
E c3
2 1
W0
F r
2
F
(27)
Knowing the distortion coefficient
dm,n
and using the
(28)
tan1 F
expressions for
pm,n &m,n from equations (10), (11) and
0
F r
also for
p0,0 from equation (12) the radial deformation in the
where
and
are the dimensionless steady state
-
_ F r F
inner bearing surface at any point ( , z) was computed.
hydrodynamic forces in r and directions respectively. Since the film pressure has been obtained numerically for all the mesh points, integrations in equations (25) and (26) can
be easily performed numerically by using Simpsons 1/ 3 rd.
where,
rule to get F r
and F
. The steady state load (W 0 ) and the
.
.
2
1
attitude angle ( )
are then calculated by using equaions
C 2 . Cos
.Sin .
Fr Sin
2
-
F Cos
0
(27) and (28).
M .W0 .
.
.
-
-
Equation of Motion:
2
1
D 2 . .Sin
-
.Cos
.
Fr Cos
-
F Sin
-
-
W0
0
M .W .2
The equation of motion for a rigid rotor supported on four identical flexibly supported bearings are given by,
A1 Sin ,
_
G X b
_
H Yb
E C G
F D H
A2 Cos ,
A3 .Cos ,
A4 .Sin
-
-
Solution scheme:
For stability analysis, a non-linear time transient analysis is carried out using the equations of motion [equations (29) to
(32)] to compute a new set of
, , X b , Yb
and their
derivatives for the next time step for a given set of
L D,
Fig. 2: Coordinate system of hydrodynamic fluid film forces in circumferential & radial direction
steady state eccentricity ratio 0 , deformation factor F, mass
_
parameter M .The fourth order Runge-Kutta method is used
M r .
d 2 X
r
dt 2
d 2Y
Fr Sin F Cos
(29)
(30)
for solving the equations of motion. The hydrodynamic forces are computed every time step by solving the partial differential equation for pressure satisfying the boundary
M r . r Fr Cos F Sin W0 dt 2
d 2 X
b
dXb
(31)
conditions.
-
Stability Analysis
Mb .
b
dt2
F Cos Fr Sin B. dt
-
KXb
Mb.
2
d Y
b F Sin FrCos B. dt2
dYb KY dt b
(32)
To study the effect of bearing surface deformation on journal centre trajectory of flexibly supported bearings a set of
trajectories of journal and bearing centre has been studied
The relation between rotor & bearing motion are given
by,
and it is possible to construct the trajectories for numbers of complete revolution of the journal the plots shows the
X r X b e Sin Yr Yb eCos
(33)
(34)
stability of the journal when the trajectory of journal and bearing centre ends in a limit cycle. Critical mass parameter for a particular eccentricity ratio, slenderness ratio and
The above two equations are substituted in equations of
motion. Finally the equations of motion are expressed in non-dimensional form as follows,
deformation factor is found when the trajectories end with limit cycle (Fig. 10 & Fig. 11) or it changes its trend from stable to unstable.
dX b
(35)
X b d
-
-
RESULTS AND DISCUSSION
dYb
(36)
Fig.3:Variation of critical mass parameter with deformation factor for various eccentrycity ratio
mr=0.2,bb=0.02,kb=10,H/R=0.3,L/D=1.0,
=0.5, =0.4
=0.3
=0.5
=0.6
=0.7
=0.8
Yb d
1
_
X b _ _
.
F Cos Fr Sin .W0 .B. X b W0 .K .X b
(37)
18
16
14
12
10
8
6
4
2
0
0
m. M .W .2
(38)
_ 1 .
Yb _
F Sin Fr Cos .W0 .B.Y b W0 .K .Yb
2
m. M .W0 .
-
d
(39)
d
-
d
(40)
d
A3.F A4 .E
(41)
0 0.2 0.4 0.6 0.8 1
deformation factor
critical massparameter
A .A A .A
2 3 1 4
A2 .E A1.F A2 .A3 A1.A4
(42)
A Effect of Eccentricity ratio :
-
Effect of slenderness ratio L D:
Figure.3 shows that the critical mass parameter of journal
Effect of slenderness ratio L D on the critical mass
bearings as a function of deformation factor F for
L D 1.0, H R 0.3, 0.4 when eccentricity ratio 0 is
parameter of the bearing can be studied from figure 5. Here, dimensionless critical mass parameter of journal bearings is
Fig.6: Variation of critical mass parameter with deformation factor for different H/R
m=0.2, bb=0.02, kb=10.0, L/D=1.0, =0.6, =0.5, =0.4
7
6
considered as a parameter. From the figure it is found that when other parameters remain same as eccentricity ratio
shown as a function of deformation factor F for
0
0.6, H R 0.3, 0.4. It is found that when other factors
0
increases the critical mass parameter increases. Further, for the eccentricity ratio beyond 0.6 the family of the curves shows drooping trend which becomes more significant up to F 0.4 . For the eccentricity ratio 0.6 the characteristics are
remain unaltered, an increase in mass parameter.
L D decreases the critical
_
more or less horizontal meaning that the mass parameter M
Fig.4:Variation of critical mass parameter with deformation factor for various poisons ratio
m=0.2,bb=0.02,kb=10,L/D=1.0,=0.6
,H/R=0.3,=0.5
H/R=0.2
5
4
3
2
1
0
critical mass parameter
remains unaffected with a change in F . The stability threshold falls rapidly with F at higher eccentricity ratio.
deformation factor
1
0.6 0.8
0.2 0.4
=0.3
=0.4
0
=0.2
7
6
5
4
3
2
1
0
1
0.6 0.8
0.2 0.4
H/R=0.3 H/R=0.4
0
critical mass paramter
0
-
Effect of liner thickness to journal radius ratio H R: In figure.6 the dimensionless critical mass parameter of journal bearings is shown as a function of deformation factor
deformation factor
F
for L D 1.0, 0.4,
0.6, liner thickness to journal
radius ratio H R
is considered as a parameter. It is
.B Effect of Poissons ratio :
observed from the figure that as H R
increases the
Fig.7:Variation of critical mass parameter with
deformation factor for various damping co-efficient m=0.2,kb=10,=0.6,H/R=0.3,L/D=1.0 =0.5,=0.4
Figure.4 is the plot of dimensionless critical mass parameter of journal bearing as a function of deformation factor F for
dimensionless critical mass parameter decreases.
bb=0.02
7
6
5
4
3
2
1
0
critical mass parameter
LD 1.0, R 0.3, 0
0.6
when Poissons ratio is
considered as a parameter. A scrutiny of the figure reveals that as Poissons ratio increases, the dimensionless critical mass parameter increases. Further, the family of the curves shows declining trend i.e., critical mass parameter decreases with increase in deformation factor. The decreasing trend of the curve is very slow.
Fig.5:Variation of critical mass parameter with deformation factor for various slenderness ratio m=0.2,bb=0.02,kb=10,=0.6,H/R=0.3,
=0.5,=0.4
7
6
5
4
3
2
1
0
L/D=0.5
1
0.5
deformation factor
bb=0.03 bb=0.09
0
-
Effect of support damping co-efficient bb:
Effect of damping co-efficient (bb) on dimensionless critical mass parameter of the bearing can be studied from figure 7. Here, dimensionless critical mass parameter of journal bearings is shown as a function of deformation factor F for
1
0.5
deformation factor
L/D=1.0 L/D=1.5 L/D=2.0
0
critical mass parameter
0
0.6, H R 0.3, 0.4 . It is found that whe other factors
remain unaltered, an increase in bb increases the critical mass parameter.
Fig.8:Variation of critical mass parameter with
deformation factor for various mass ratio bb=0.02,kb=10,L/D=1.0,H/R=0.3,=0.6,=0.5,=0.4
deformation factor
0.8
0.6
0.4
0.2
m=0.2 m=0.3
0
m=0.1
7
6.5
6
5.5
5
4.5
4
3.5
3
critical mass parameter
-
Effect of mass ratio m :
critical mass parameter
Figure.8 is the plot of dimensionless critical mass parameter of journal bearing as a function of deformation factor F for L D 1.0, H R 0.3, 0 0.6 when mass ratio is considered as a parameter. A scrutiny of the figure reveals that as mass ratio increases, the dimensionless critical mass parameter decreases.
7
6.5
6
5.5
5
4.5
4
3.5
3
Fig.9:Variation of critical mass parameter with deformation factor for various stiffness co-efficient m=0.2,bb=0.02,=0.6,H/R=0.3,L/D=1.0,=0.4,
=0.5
kb=10 kb=5 kb=1
kb=0.5
0 0.2 0.4 0.6 0.8
deformation factor
-
Effect of support stiffness co-efficient kb:
In figure.9 the dimensionless critical mass parameter of journal bearings is shown as a function of deformation factor
-
-
CONCLUTION
Numerical methods are used to determine the effects of elastic distortions in the bearing liner on bearing stability of finite journal bearing:
-
The stability decreases as the bearing liner is made more flexible for high eccentricity ratios (i.e., 0 > 0.8). For 0 < 0.5, the flexibility of the bearing liner had little or no effect on stability.
-
Bearing is highly stable when L D is small but drops as L D increases from 0.5 to 2.0. This stability drops as deformation factor increases.
-
The hydrodynamic pressure and hence the stability
is reduced as the bearing liner becomes more flexible, especially at eccentricities greater than 0.8.
-
As the Poisson ratio increases the stability increases but drop sharply when bearing liner is made more flexible.
-
As the liner thickness to radius ratio increases the stability decreases but drop when bearing liner is made more flexible
NOMENCLATURE
F
for
LD 1.0, 0.4, 0
0.6, stiffness co-efficient is
a r
Inner radius of the bearing liner [ m ]
considered as a parameter. It is observed from the figure that as kb increases the dimensionless critical mass parameter
increases.
i
b r0
Outer radius of the bearing liner [ m ]
c Radial clearance [ m ]
R Journal radius [ m ]
-
Journal diameter [ m ]
dm ,n
m, n
Distortion coefficient of m ,n harmonic Axial and circumferential harmonics
e Eccentricity [ m ]
e
Steady state eccentricity [ m ]
0
-
Youngs modulus [ N / m2 ]
-
Elasticity parameter or deformation factor,
0 R
3
c 3 E
FS Shear force on journal surface [ N ]
-
Nondimensional fluid film force along the line of
F r centers
-
Nondimensional fluid film force perpendicular the M b
M
F line of centres M
Mass of bearing Mass ratio
-
Non-dimensional steady state fluid film forces
m b
Critical Mass Parameter
F r0 , F0 r
h Oil film thickness [ m ]
h
Steady state oil film thickness [ m ]
0
-
Non-dimensional oil film thickness
h
H Thickness of bearing liner [ m ]
L Length of bearing [ m ]
p Oil film pressure [ Pa ]
_ M .c. 2
r
M
WO
0
R2
B
c2
_
K kb
_
B bb
Viscosity Parameter
Bearing support stiffness coefficient Bearing support damping coefficient
REFERENCES
p0 Steady state film pressure [ Pa ]
_
p Dimensionless oil pressure
Q End flow of oil [ m3 / s ]
Q Nondimensional End flow
-
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B.Tower, 1st report on friction experiments, Proc. Inst. Mech. Engrs., 1883, 632-664, 1884, 29-35; Part II: 1885, 58-70; Part III: 1888, 173-205; Part IV:1891, 111-140.
-
O.Reynolds, On the theory of lubrication and its application to Mr. Beauchamp Towers experiments, including an experimental determination of the viscosity of Olive oil, Phil. Trans. Royal Soc., London, 177, 1886, 157-234.
u ,v , w
Components of fluid velocity in the x, y, and z direction, respectively. [ m / s ]
-
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U Shaft peripheral speed [ m / s ]
-
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W0
W 0
x, y, z
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-
, y, z
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_
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Poissons ratio
Eccentricity ratio
0 Steady state eccentricity ratio
Attitude angle [ rad ]
0 Steady state attitude angle [ rad ]
1 Angular coordinates at which the fluid film commences [ rad ]
2 Angular coordinates at which the fluid film cavitates [ rad ]
Angular velocity of journal [ rad / s]
Whirl ratio. [ p ]
Deformation of bearing surface. [ m ]
Steady state deformation of bearing surface. [ m ]
0
-
-
c Non-dimensional deformation of bearing surface
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-
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-
D.K.Brighton, C.J.K.Hooke and J.O'Donoghue , A theoretical and experimental investigation on the effect of elastic distortions on the performance of Journal bearing, Tribology convention 1968, Proc. of Institute of Mechanical Engineers, Vol 182,Part 3N, 1967 – 1968, pp 192 – 200.
-
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-
S.C.Jain, R.Sinhasan and D.V.Singh, Elastohydrodynamic analysis of a cylindrical Journal bearing with a flexible bearing shell, Wear, March 1981, pp 325 – 335.
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D. Benasciutti, M. Gallina, M. Gh. Munteanu, F. Flumian,2012
,
X b
Yb
X r
Yr
M r
Lames constants
Coordinate of bearing centre in x-direction
Coordinate of bearing centre in y-direction Coordinate of rotor centre in x-direction
Coordinate of rotor centre in y-direction Mass of rotor or journal
A Numerical Approach for Static and Dynamic Analysis of Deformable Journal Bearings World Academy of Science, Engineering and Technology 67 , 2012,pp-778-783
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Jose´ A. Va´zquez, Lloyd E. Barrett, Ronald D. Flack A Flexible Rotor on Flexible Bearing Supports: Stability and Unbalance Response Transactions of the ASME, Journal of Vibration and Acoustics, APRIL 2001, Vol. 123 pp 137-144