International Journal of Scientific & Engineering Research Volume 2, Issue 8, Auguest-2011 1
ISSN 2229-5518
Effect of turbulence on the performance of an inverted three-lobe Pressure dam bearing
N. K. Batra, Gian Bhushan and N. P. Mehta
Key Words: Inverted three- lobe pressure dam bearing, finite element method, multi-lobes
—————————— ——————————
HE analysis of multi-lobe bearings was first published
The Reynolds Equation for the laminar flow is:
Tby Pinkus[1]. This was followed by Lund and
h
p h
p 6R h 12sin 12e cos 1
3 3
Thomson[2] and Malik, et al [3], who gave some
x x
z z x
design data which included both static and dynamic characteristics for laminar as well as turbulent flow
regimes.
The above equation is non-dimensionalized by making the following substitutions:
.
. .
The analytical dynamic analysis Nicholas and Allaire[4,7 ]
x 2x 2x, e ,
and
and Mehta, et al[9,11] and Mehta and Singh[10] has shown
R D c
2
that cylindrical pressure dam bearings are found to be very
x , z , h ,
2p c
,
x z h
p
stable. Also an experimental stability analysis of such types
of bearings [5] showed that the analytical stability analysis
D L 2c
R
(2)
reflects the general trends in the experimental data.
The non- dimensionalized equation thus obtained is:
3 2
3
The effect of turbulence on the stability was first
h
p D
h
p h
investigated by Constantinescu [6] who concluded that the
x k x x
L
z k z z
2 x
stable zone decreases with increase in Reynolds number for plain journal bearing. Nicholas and Allaire[7] found that turbulence increases the stability of finite pressure dam
a sin 2x cos 2x
(3)
bearings. Soni, et al [8] found that turbulence improves the stability of non circular bearings. Mehta [9-13] found that the threshold speed increases where zone of stability decreases with increase in turbulence in pressure dam bearings. The present study is undertaken to investigate the effect of turbulence on the performance of inverted three- lobe pressure dam bearing supporting either rigid or flexible rotor.
Corresponding author: N.K.Batra- Research scholar of NIT, Kurukshetra, Haryana(India)
E mail: nkbatraeng@rediffmail.com
Gian Bhushan: Assoc. prof, MED, NIT, Kurukshetra, Haryana(India) E mail: aroragian@ yahoo.com
N.P Mehta: Director, M.M. University, Mullana(Ambala)
Email: drnpmehta7@gmail.com
Where k x = k z =12 for laminar flow (4)
k x =12 + 0.0136 Re 0.9
k z =12+0.0043 Re 0.96 for turbulent flow
(5)
These correction factors for the turbulent flow were suggested by Constantinescu and Galetuse [14]. The various assumption made in the derivation of Reynolds equation are that the fluid is Newtonian, no slip occurs at bearing surface, inertia forces are neglected , oil viscosity is constant and curvature is negligible as the height of the fluid film is very small as compared to span and length. The Reynolds equation is analysed for a pressure profile using finite element method [15]. The value of this Reynolds number based on major radial clearance is
IJSER © 2011
International Journal of Scientific & Engineering Research Volume 2, Issue 8, Auguest-2011 2
ISSN 2229-5518
defined as
Vc
Rec =
(6)
C0 CxxC yy CxyC yx
C1 K xxC yy K yyCxx K xyC yx K yxCxy
Local Reynolds number at any nodal point Re is
C2 Cxx C yy
C3 K xxK yy K xy K yx
Re = Vh =2 Re
h (7)
C4 K xx
K yy
(8)
k x and k z
are calculated by calculating Re from equation
For a rigid rotor, the value of F (dimensionless flexibility) is
taken as zero. The system is considered as stable if the real
( 7) for each point in the finite element mesh and turbulence
correction factors are applied at the nodal point level.
Fig.1 An inverted three–lobe pressure dam bearing.
part of all roots is negative. For a particular bearing geometry and eccentricity ratio, the values of dimensionless speed are increased until the system becomes unstable. The maximum value of speed for which the bearing is stable is then adopted as the dimensionless threshold speed. The stability threshold curves divide any figure into two major zones. The zone above this curve is unstable, whereas the zone below is stable. The minimum value of this curve is termed as the minimum threshold speed. Mostly, the curve has a vertical line, towards the left side of which the bearing is stable at all speed. This portion is called the zone of infinite stability.
The present analysis has been done for the bearing with the following parameters
Figure 1 shows the geometry of an inverted three- lobe
L/D=1.0,
=1.5
L = 0.8,
S
pressure dam bearing. Each lobe of the bearing is analysed d d
separately. Since, the pressure profile has to be symmetrical
L t =0.25,
s =850
=100
about the bearing centre line, only half of the lobe is taken
for analysis. The film area of the hydrodynamic oil film to
be analysed by finite element method is divided in to mesh
of finite elements. Elements involving linear variation of pressure are simplest for dynamic analysis of two dimensional hydrodynamic problems and the same has been considered for the present analysis.
An analysis based on variational principle is used to
derive a set of simultaneous algebraic equations with nodal film pressure unknown. Fluid pressure at nodal points is taken by applying the Reynolds boundary conditions. The resulting matrix is stored in the banded form. Stiffness and damping coefficients are determined separately for each lobe and then added [16]. The values of these stiffness and damping coefficients, shaft flexibility, and dimensionless speed are then used to evaluate the coefficients of the characteristic equation [17], which is a polynomial of the
6th order for flexible rotors as given below(for a rigid rotor, F=0):
This characteristic equation has been given by
The value of ellipticity ratio (δ) = 0.5 is selected for present study. The effect of turbulence on attitude angle, eccentricity ratio, oil film coefficient has been also observed by varying the Reynolds numbers in steps (0, 1500, 3000, and 6000). The effect of flexibility of rotor on the stability of bearing is considered by using non zero values of F (0.5 and 2.0) in the characteristics equation.
The effect of turbulence on the performance of inverted three lobe pressure dam bearing is shown in figures 2 to 9. The static characteristics are shown in figure 2 to 6 where’s dynamic characteristics are shown from figure 7 to 9. The various relations to calculate minimum oil film thickness, oil flow and friction coefficients are given by Mehta [9].
s6 F 2v4C s5v4 F 2C FC s4v2 v2 F 2C 2FC
v2 v2 FC
0 1 2
3 0 4
s3v2 2FC C s2 2Fv3C v2C C sC C 0
1 2
Where
3 4 0 1 3
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International Journal of Scientific & Engineering Research Volume 2, Issue 8, Auguest-2011 3
ISSN 2229-5518
Figure 2 Effect of turbulence on eccentricity ratio
Figure 3 Effect of turbulence on attitude angle
Figure 4 Effect of turbulence on minimum film thickness
Figure 5 Effect of turbulence on oil flow coefficient
Figure 6 Effect of turbulence on friction coefficient
Figure 7 Effect of turbulence on the stability of inverted three lobe bearing supporting a rigid rotor (F=0)
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International Journal of Scientific & Engineering Research Volume 2, Issue 8, Auguest-2011 4
ISSN 2229-5518
Figure 8 Effect of turbulence on the stability of inverted three lobe bearing supporting a flexible rotor (F=0.5)
Figure 9 Effect of turbulence on the stability of inverted three lobe bearing supporting a flexible rotor (F=2.0)
The effect of turbuleance on the eccentricity ratio of the bearing is shown in the figure 2. It is observed from the
Figure 5 and 6 shows the effect of turbulence on the oil – flow and friction coefficients, respectively. Both are observed to increase with increase in the turbulence level for a particular value of sommerfeld number.
Figure 7 shows the effect of turbulence on the stability of
inverted three lobe pressure dam bearing supporting a rigid rotor. The zone of infinite stability as well as the value of minimum threshold speed is observe to be increase with the turbulence level ,thus indicating better stability of the bearing at higher turbulence levels. The values of zone of stability of Reynolds number 0, 1500, 3000and 6000 are up to S=0.684, 1.59, 1.786, and 1.094 while the values of minimum threshold speed are15.6, 18.3, 19.05, and 46.8 respectively.
The effect of turbulence on the stability of inverted three- lobe bearing supporting a flexible rotor is shown in figures
9 and 10. The same effects, as for the case of a rigid rotor,
are observed for the bearing supporting a flexible rotor.
1. The values of eccentricity ratio, oil flow coefficients and friction coefficient increases with increase in turbulence.
2. The attitude angle increases with increase in turbulence
value of the sommerfeld number less than 1.9 while it decreases for value of sommerfeld number greater than 1.9
3. Turbulence increases both the minimum threshold speed
and zone of infinite stability of inverted three lobe pressure dam bearing supporting either rigid or flexible rotor, thus increasing its stability.
c : Radial clearance
cm
plots that the eccentricity ratio increases with increase in
the value of turbulence, for a particular value of
: Minimum film thickness
for a centered shaft
sommerfeld number. As the bearings become more stable
Cxx, Cxy , Cyx, Cyy
: Oil - film damping coefficients
when operating at higher eccentricity ratio, the effect of
turbulence indicates a better operation of the bearing. The variation of attitude angle vs sommerfeld number is shown in figure 3 with different turbulence levels. The figure
C xx, C xy , C yx, C yy
: Dimensionless oil-film
coefficients C xx Cxx c / W
depicts that with increase in Reynolds number the attitude
angle the increases with low value of sommerfeld number and decreases with increase in the value of sommerfeld number. However for a particular value of Reynolds number, the attitude angle increases with increase in sommerfeld number, the increase being more at lower values of Reynolds number. The variation of minimum film thickness with sommerfeld number with different turbulence level is shown in figure 4. The minimum film
thickness is observed to reduce with the increase in
C0 , C1, C2 , C3, C4 : Coefficients of the characteristic equation
D : Diameter
e : Eccentricity
F : Dimensionless shaft flexibility, W / ck
h : Oil-film thickness, c1 cos
h : Dimensionless oil-film thickness,
h / 2c
turbulence level for a particular value of the sommerfeld 2k
: Shaft stiffness
number.
K xx, K xy , K yx, K yy
: Oil-film stiffness coefficients
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International Journal of Scientific & Engineering Research Volume 2, Issue 8, Auguest-2011 5
ISSN 2229-5518
K xx, K xy , K yx, K yy : Dimensionless oil- film stiffness
Dam Bearings with Rotor Flexibility Effects, ASLE Trans., vol.29, no. 4, pp. 548-557, 1986.
[10] N. P. Mehta, and A. Singh, Stability of Finite Orthogonally-Displaced
coefficients, K xx
L : bearing length
K xx c / W
Pressure Dam Bearings, ASME Journal of Tribology, vol. 109, no. 4, pp.
718-720, 1987.
[11] N. P. Mehta, S. S. Rattan and G. Bhushan, Static and Dynamic
N : Journal rotational Speed
Oi : Lobe center of lobe i (i =1, 2, 3, 4)
p : Oil-film pressure
R : Journal radius
NLD R 2
Characteristics of Four-Lobe Pressure Dam Bearings, Tribology Letters,
vol. 15, no. 4, pp. 415-420, 2003.
[12]. N. P. Mehta, S. S. Rattan and Rajiv verma , Stability analysis of circular
pressure dam hydrodynamic journal bearings with couple stess lubricant, vol. 5 , no. 10, 2010.
[13] S S Rattan and N P Mehta. Effect of turbulence on the performance of a
Three lobe pressure dam bearing, .Proceedings of NACOMM, IIT Kharagpur, 1993, p 1
S : Sommerfeld no
W c
V : Peripheral speed of journal
W : bearing external load
[14] V N Constantinescu and S Galetuse. Pressure Drop due to Inertia forces in
Step Bearings,. ASME journal of lubrication Technology, 1976, p 167.
[15] G Bhushan, S S Rattan and N P Mehta Effect of Pressure Dams and Relief –
tracks on the performance of Four lobe Bearings, Journal of The institute
x, z
: coordinates for bearing surface (x- peripheral, z-along shaft axis)
of Engineers (India), vol. 85, 2005, p 194.
[16] N. P. Mehta, A. Singh, and B. K. Gupta, Stability of Finite Elliptical
Pressure Dam Bearings with Rotor Flexibility,. ASLE preprint no 84-LC-
: Attitude angle
. . .
: whirl rate ratio, /
5C-1, presented at the ASME/ASLE Lubrication Conference, San Diego,
USA, October 22-24, 1984
[17] E .J. Hahn, The Excitability of Flexibility Rotors in Short Sleeve Bearings,.
ASME Journal of Lubrication Technology, 1975, p 105.
. . .
: squeeze rate ratio,
/
: Eccentricity ratio, e / c
: Ellipticity ratio, (1 cm / c)
: Angle measured from the line of centers in the direction of rotation
g : Oil-groove angle
: Fluid density
: Average fluid viscosity
: Rotational speed
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