Stability of submerged four-lobe oil journal bearings under dynamic load

95

Wear, 154 (1992) 95-108

Stability of submerged under dynamic load

four-lobe oil journal bearings

R. Pai and B. C. Majumdar Depar@nent of Mechanical Engineering Indian institute of Technology, Kharagpur (India) (Received

April 25, 1991; revised and accepted

August 28, 1991)

Abstract A non-linear

transient method was used to predict the journal centre trajectory for a submerged four-lobe oil journal bearing under (1) unidirectional instant load, (2) unidirectional periodic load, and (3) variable rotating load. The Reynolds equation was integrated using the Jakobsson-~oberg~lsson cavitation zone model, and oil film prehistory effects were taken into account. The journal eentre trajectories obtained were compared with those obtained for submerged plain cyiindrical bearings under similar loading conditions. It was observed that the excursions of the joumaf centre were subdued, unlike for the plain cylindrical bearings, where the journal centre had a large excursion before it became stable or ended in a limit cycle. Interesting trajectories were observed for the periodic load. A three-dimensional plot of the pressure distribution in the bearing clearance was obtained using computer graphics.

1. Introduction Submerged bearings were first investigated in 1957 by Jakobsson and Floberg [l]. Their work dealt with stationary cavitation. Steady state performance characteristics such as load capacity, attitude angle, oil flow for different eccentricity ratios and subambient pressures were compared with experimental results. Olsson [Z] made a theoretical and experimental investigation of non-stationary cavitation in dynamically loaded bearings. Kicinski [3] performed a linear stability analysis with the bearing under periodic external loads. A theoretical model for cavitation was proposed on the lines of the Jakobsson-Floberg-Olsson (JFO) theory and an experimental investigation was carried out. Brewe [4] made a theoretical investigation of the evolution of a vapour bubble for submerged journal bearings under dynamically loaded conditions. A comparative study was performed to determine some of the consequences of applying a non-conservative theory to a dynamic problem. More recently, the authors [5] have performed a stabiIity analysis of submerged plain journal bearings under dynamic load using a non-linear transient method. Stability studies of fluid film rotor bearing systems are impo~ant in industrial machinery. Multilobe bearings have been widely used in many industrial app~cations, because they are quite economical to manufacture and offer better performance in comparison with the full journal bearings. A linear stability analysis reduces the complexity of the problem and may be desirable from a practical standpoint. A linear analysis, however, does not reveal post-whirl orbit detail and is not adequate in situations where large vibration amplitudes are encountered. Over a period of time such vibrations

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96

may create problems such as seal failure, loss of babbit on bearing surfaces and other effects. A non-linear transient analysis may be useful in such situations. Multilobe bearings have been studied by a number of authors. Pinkus 161 made a theoretical analysis of three-lobe bearings and predicted the equilibrium position, load capacity and pressure profiles. Using a finite difference approach, Lund and Thomsen [73 analysed three-lobe bearings for fluid film stiffness and damping coefficients. Flack et

et al. [9] tested a pair of preloaded four-lobe bearings with a simple flexible rotor and determined the unbalance response and stability threshold e~eriment~ly. Li et al. [lo] examined the linear stability threshold of four multilobe bearing configurations: two-lobe, three-lobe, four-lobe and offset configurations. Non-linear transient analysis of a rigid rotor in each of these bearings was also carried out above and below the stability threshold, presenting journal orbits, bearing force and frequency content of non-linear orbit. Li et al. [ll] in another paper made an analysis of multilobe bearings where the rotor was subject to unbalance both above and below the stability threshold. Akok and Ettles [12] found stability thresholds for four basic journal bearing types. They found that increasing the groove size upto 90” and increasing the aspect ratio had a destabilizing effect on the rotor, whereas the stability was improved with a preload. In most studies on four-lobe bearings the Reynolds equation is integrated using the half-Sommerfeld or Reynolds boundary, and the lubricant is supplied by a conventional pressure feed system. The oil feed hole is usually located at the maximum film thickness zone but this does not alleviate the problem under dynamic conditions. A submerged bearing does not use an oil hole and the oil supply is maintained by continuity of flow from the edges. The submerged four-lobe bearing has received little attention and in this work an attempt is made to ascertain the stability of submerged four-lobe bearings under three different conditions of loading, using a non-linear transient method. 2. Theory

Under dynamic conditions the basic differential in the bearing clearance is (see Fig. 1)

,

,Centeted

equation

for pressure

journal

Fig. 1. A typical four-lobe bearing, offset factor $+=&‘a, preload factor &,-S/C.

distribution

97

When

non-dimensionalized

tI=xlR,

i=2zlL,

the equation

using

fi=hlC,

transforms

p=pC2/q&,

to

~(h)~)+(on)‘(bl~)=6(1-2~)~ The film thickness

T=wt

+12g

(2)

is given by (see Fig. 2)

h=(-b~@=ii?)/2a

(3)

where b = (r-x;

cos e-y;

c’=(xi2+yi2-2riR

sin e) cos O-2y;R

sin 8-C2-2RC)

a=1 and Xi =x1 -e sin20 yi =y, + OSe sin 28 Equation (3) gives the film thickness at any point on the circumference of the bearing other than those on the grooves. The following is assumed. (1) The film thickness throughout the groove region is constant and has the value of unity. (2) The pressure in the groove region is assumed to be constant and equal to atmospheric pressure, since the grooves are assumed not to carry any load. (3) The film thickness distribution around the circumference does not change in the axial direction.

L Bearing Fig. 2. Calculation of film thickness.

pad

98

3. Boundary

conditions

The boundary conditions for determining (1) Pressures at the ends of the bearing F(B, Zkl)=O

the pressure distribution are zero (a~osphe~c)

are as follows.

for OG0927~

(2) The pressure

distribution

is symmetrical

about the mid-plane

of the bearing

for 0<0<27r

~(e,o)/aY=o

(3) The boundaries of the cavitation zone are determined continuity at the boundaries (see Fig. 3). Flow on the pressure side qp=U~Uz/2-

by satisfying

the flow

(4)

W2W%2ll2~)3

is balanced by the flow in the cavitation zone and an additional flow because of boundary movement with time. (The s&ix l/2 indicate that the quantities determined concern both the cavitation and reformation boundaries.) Flow in the cavitation zone is given by (5)

q~~=~~~~~2~2

Flow due to boundary

movement

is given by

qt=L(l-P)hl/$W

(6)

qp = licav+ 4t

(7)

Substituting Wn -

~1,2lW

eqns. (4)-(6) -PI

=

(8)

(~:n(ap/~~li2)/121

Non-dimensionalizing (i/2-w1,2/aq1

in eqn. (7) we obtain

gives

- p) = (Rn(+vae)1/2)m

(9)

The filling coefficient values for a given time step are determined from the flow continuity equation in the cavitation zone for that time step. The non-dimensional form of the continuity equation can be obtained from ref. 3: a(~)/a~+

za(pi;)raT=

0

LCAV

= zlCAV/L

Fig. 3. Theoretical model of the cavitation zone 131.

99

The solution of eqn. (10) indicates that the flow M in the cavitation zone for a given time step depends on the flow &-, for the preceding time step. A numerical solution of eqn. (2) using a finite difference method with a successive over-relaxation scheme satisfying the boundary conditions gives the pressure distribution. The hydrodynamic forces can then be computed. Initially k and 4 are set equal to zero to obtain the steady state load capacity. The forces are computed from

1 81 TV==

ss

PCOS ededz-

(11)

0 h

These forces along with the steady state load are used in solving the equations of motion which give us the E, 2, 4 and 4 values for the next time step. The equations of motion are MC(d*e/di* -

l(d+/dt)*)

MC(ed*+/dt*

+ 2d+/& d&t)

The dimensionless fi~-h?Ey-WJ~o-cos

= W, + W cos 4

(13)

= W,-

(14)

W sin 4

forms of the equations

are

4=0

&fe~+2i&r&-IVJlFo+sin

(15)

4=0

(16)

where h?=MCa?JW,

and Wo= W,C2/7pR3L The second-order differential equations (15) and (16) in c and 4 are solved using a fourth-order Runge-Kutta method for constant values of h?, l%‘r and wo.

4. Unidirectional

constant load

Input data for the program are given in Table 1. Hydrodynamic forces are computed for the initial time step when g and 4 are put equal to zero. Solutions of the equations of motion using these forces gives E, 4, t and 4 values for the next time step. The new e, 6 and 4 values are used in eqns. (2) and (3) to compute the hydrodynamic forces and the equations of motion are solved again. This procedure gives us the l and 4 values at each time step and a trajectory of the journal centre can be obtained. The trajectory helps to ascertain the system stability. 5. Unidirectional The applied

periodic load load was assumed

IV= Wo[1 + sin(T/2)]

to be of the form (17)

100 TABLE

1

Input data for various load conditions Condition”

L/D

co

1

0.5

0.8

2 3

0.5 0.562

0.8 0.77

“1, unidirectional

constant

&f

5.0 5.0 5.0

o (rad s-l)

D (m)

cm

568.0 568.0 62.84

0.076 0.076 0.2

0.0039 0.0039 0.008

load; 2, unidirectional

h4 0%)

;Ipa s)

periodic

0.00689 0.~689 0.015

load; 3, variable

2286 2286 117863 rotating

233 233 2017576 load.

Where k!$ is the steady state load corresponding to the eccentricity ratio eo=0.8. At each time step, the external load is computed using the above equation. Starting from %=0.8, when g and d, are put equal to zero, the process of computing the hydrodynamic forces and solution of the equations of motion is repeated until we obtain a sufficient number of E and C#J values to plot a trajectory of the journal centre. Input data for the program are given in Table 1.

6. Variable rotating load The Ruston Homsby SVEB-X MK III engine ~nnectiRg rod bearing load data were used to simulate the variable rotating load. Figure 4 shows the polar load diagram relative to the cylinder axis and appendix 4.1 of ref. 13 gives the magnitude of load at 10” crank angle intervals. The resultant load was computed and non-dimensionalized using w= WC2/qoR3L. Appendix 4.1 also gives other related data which are presented in suitable units in Table 1. These were used to compute the hydrodynamic forces at time zero when 6 and 4 are put equal to zero. The equations of motion were then solved using these forces and the external bearing load computed. The procedure was repeated until sufficient values of E and C$were obtained so that a trajectory could be drawn. The time step for computation was chosen to match the applied load.

7. Results and discussion In the analysis, the JFO cavitation model is used to compute the cavitation and reformation boundaries. This is very important in enclosed cavities like those of submerged bearings. The analysis also includes the prehistory of the oil film, which means that the pressure distribution and the cavitation zone boundaries depend not only on the parameters connected with journal motion but also on the pressure and cavitation zone boundaries of the previous time step. Although the JFO boundaries are complicated and difficult to implement and the iterative process used is expensive by way of computer time, the cavitation model together with oil film history gives a reliable description of the dynamic state of the bearing, in particular for submerged bearings. To know for certain that we are in the right direction, we used the input data of ref. 10. The trajectory that we obtained (Fig. 5) showed the same trend as that

101

INERTIA FORCE LOOP

44 RELATIVE

360. TO CVLINOER

AXIS

Fig. 4. Polar load diagram of engine bearing.

102

X EPSILON X lb->

i

CLERRANCE SPRCE

Fig. 5. Trajectory for journal centre with unidirectional L/D=O.5, n-= -0.1.

-0.6-04

0

-0.2 x

02

oc

constant load: q,=O.OS, +=O.O, h?= 6.386,

06

(dim)

Fig. 6. Stable journal orbit with unidirectional constant load from ref. 10: L/D = 0.5, A?= 6.386, s,=os,

s,=o.s.

in ref. 10 (Fig. 6), but the stability point was shifted closer to the clearance circle. Also the amplitude was larger, indicating that larger forces were encountered. This may be attributed to the boundary conditions and prehistory of the oil film. The trajectories are plotted with the journal displacements in dimensionless units. The circular arcs formed by the lobe surfaces indicate that the bearing is preloaded

16->

Fig. 7. Trajectory I&=5.0, 7r= -0.1.

of the journal

centre

with unidirectional

constant

load: ~‘0.8,

L/D= 1.0,

Fig. 8. Trajectory of the journal centre with unidirectional &=S.O, L/D=O.S, ~-IT= -0.1.

constant load (plain bearing):

Q= 0.8,

Fig. 9. Trajectory of the journal centre with unidirectional M-6.386, LID=O.S, VT- -0.1.

constant load @lain bearing):

Q = 0.5,

104

CLEARANCE SPIKE

Fig. 10. Trajectory of the journal centre with unidirectional periodic load: Q = 0.8, += 0, fi= 5.0, LID=l.O,

T= -0.1.

(a preload factor of 0.46 was used in the simulations). The plots were scaled to contain the free clearance space within which the journal is able to move. The end of each journal rotation is indicated by an asterisk. We then used the program to make a comparative study of the stability plots obtained for four-lobe and plain cylindrical bearings. The input data and loading conditions were similar for both bearing types. Under unidirectional constant load the journal centre of a four-lobe bearing had subdued motion and became stable quickly (Fig. 7), whereas the journal centre of a plain bearing (Fig. 8, ref. 5) had a large excursion before it became stable. When we used the input data (Fig. 6) of ref. 10 for a plain bearing we found that the trajectory ended in a limit cycle (Fig. 9). Under periodic external loads the journal centre trajectory ended in a limit cycle for a plain bearing 151, whereas in a four-lobe bearing the trajectory was interesting in the sense that it did not end in a limit cycle nor did it reach a stable point (Fig. 10). This may be because of the geometry of the bearing. For the variable rotating load, the trajectory of the journal centre had a complex path similar to that of the plain bearing [S], but again the journal centre motion was

105

x :PSILON X lo-->

\

CLERRANCE SPACE

Fig. 11. Trajectory of the journal centre with variable rotating load: e,,=O.‘77,+=O, L/D =0.5625, h2=5.0, -lr= -0.1.

subdued. Instead of ending in the first quadrant, the locus ended up in the fourth quadrant (Fig. 11). A three-dimensional plot of the pressure distribution in the bearing clearance is shown in Fig. 12.

8. Conclusions The stability of submerged four-lobe and plain cylindrical bearings was studied under three different conditions of loading. The boundary conditions used and consideration of the oil film history gives a reliable description of the dynamic state of the bearings. (1) At low eccentricity ratios, four-lobe bearings were stable, while plain bearings became unstable. (2) Journal centre motion was subdued for four-lobe bearings, while for plain cylindrical bearings the journal centre had large excursions. In other words, the motion was damped out for four-lobe bearings, and it took a long time for the trajectory to

Fig. 12. A typical pressure distribution: 60-0.77, L/D;-0.5 1@=5.0, V= -0.1. reach a stable point or end in a limit cycle for the plain journal bearing. (3) Interesting trajectories were observed under unidirectional periodic load for the four-lobe bearing. For the plain bearing the trajectory ended up in a limit cycle, whereas for the four-lobe bearing the trajectory did not end in a limit cycle nor did it reach a stable point. (4) Under variable rotating load, the trajectory was complex, as for the plain bearing. Under all three loading types, the trajectory ended up in the fourth quadrant, while for the plain bearing it ended up in the first quadrant. Acknowledgment The authors thank the referees for their effort in going through the manuscript and making useful suggestions. These suggestions have been included to the best of our knowledge.

B. Jakobsson and L. Floberg, The finite journal bearing considering vapourization, Trans. Chalmers Universi@of Technobgy, Report 3, 1957 (Institute of Machine Elements, Chalmers University of Technology, Gothenburg, Sweden). K. 0. Olsson, Cavitation in ~~~1~ loaded bearings, Report 26; 1965 (Institute of Machine Elements, Chalmers University of Technology, Gotbenburg, Sweden). J. Kicinski, Influence of the flow pre-history in the cavitation wne on the dynamic characteristics of slide bearings, Wear, 211 (1986) 289-311. D. E. Brewe, Theoretical modelling of the vapour cavitation in dynamically loaded journal bearings, ASME Tmns. i. Ttil, 108 (1986) 628.

107 5 R. Pai and B. C. Majumdar, Stabihty of submerged oil journal bearings under dynamic load, Wear, 146 (1991) 125-135. 6 0. Pinkus, Analysis and characteristics of the three-lobe bearings, J. Basic Eng., 81 (1959) 49-55. 7 J. W. Lund and K. K. Thornsen, A calculation method and data for the dynamic coefficients of oil lubricated journal bearings, Topics in Fluid Film Bearing and Rotor Bearing System Design and Optimization, ASME Design Engineering Con&, Chicago, ASME, New York, 1978, pp. 245-272. 8 R. D. Flack, M. E. Leader and D. E. Lewis, An experimental determination of the instability of a Sexible rotor in four-lobe bearings, Wear, 58 (1980) 35-47. 9 R. D. Flack, M. E. Leader and P. E. Allaire, An experimental and theoretical investigation of pressure in four-lobe journal bearings, Wear, 61 (1980) 233-262. 10 D. F. Li, K. C Choy and P. E. Allaire, Stability and transient characteristics of four multilobe journal bearing aviations, TM~s. ASME, J. Lubr. Technol., 102 (1980) 291-299. 11 D. F. Li, K C. Choy and P. E. Allaire, Transient unbalance response of four multi-lobe journal bearings, Trans. ASME, J. Lube. TechnoL, 102 (1980) 300-307. 12 M. Akok and M. McC. Ettles, The effect of grooving and bore shape on the stability of journal bearings, ASLE Trmts., 23-24 (1980) 431-441. 13 J. Campbell, P. P. Love, F. A. Martin and S. C. Refique, Bearings for reciprocating machinery: a review of the present theoretical experimental and service knowledge, Proc. inst. Me&. Eng., 182 (3a) (1967-1968) 51-74. 14 B. C. Majumdar and D. E. Brewe, Stability of a rigid rotor supported on oil film journal bearings under dynamic load, NASA TM 202309, Lewis Research Centre, Cleveland, 1987. Additional references R. Holmes, Non linear performance of turbine bearings, J. Mech. Eng. Sci., 12 (6) (1970) 377. E. J. Hahn, Stability and unbalance response of centrally preloaded rotors mounted in joumaI and squeeze fitm bearings, Trans. ASME, J. Lubr. Technot., 101 (1979) 120-128. B. Humes and R. Holmes, The role of subatmospheric tilm pressures in the vibration performance of squeeze film bearings, J. Mech. Eng. SC:., 20 (5) (1978) 283-289.

C D e e45

h h L M &? P P PV R t T U W w T* ye W, We

radial clearance (m) journal diameter (m) eccentricity (m) maximum allowed ‘e‘ for journal in four-lobe bearing film thickness (m) dimensionless film thickness, &= fi /C bearing width (m) effective rotor mass (kg) dimensionless mass parameter, i%?=MCw21w0 film pressure (N m-“) dimensionless film pressure, 3 =pC2/q”R2 vapour pressure (N m-3 journal radius (m) time (s) dimensionless time, T= wt journal peripheral speed, U=wR (m s-l) load capacity (N) dimensionless load capacity, PI= WC2Jq&L hydrodynamic forces (N) dimensionless hydropic forces, r?lr = W=~~~~~3L,

(m)

tie = We~z~~~3L

108

wo % x, .z

steady state load capacity (N) dimensionless steady state load capacity, w. = WoC2/~wR3L, coordinates, x = RB, %= 2rjL

Greek symbols oil gap filling coefficient offset distance between pad and bearing centres (m) preload factor offset factor eccentricity ratio, f~=ele~~ absolute viscosity (Pa s) dimensionless coordinate, angle measured from line of centres around bearing centre angle measured from line of centres around journal centre angular coordinates at which the fib cavitates and reforms respectively dimensionless vapour pressure, ~=p~~2f~~2 attitude angle angular velocity of journal (rad s-‘)