.- 

NASA 

Technical Memorandum 102309 

AVSCOM 

Technical Report 87-C-26 

Ii 

3 

Stability of a Rigid Rotor Supported 

on Oil-Film Journal Bearings 

Under Dynamic Load 

B. C . Majumdar 

Lewis Research Center 

Cleveland, Ohio 

and 

D.E. Brewe 

Propulsion Directorate 

U. S. Army Aviation Research and Technology Activity-A VSCOM 

Lewis Research Center 

Cleveland, Ohio 

. 

Prepared for the 

National Seminar on Bearings 

Madras, India, September 17- 18, 1987 

US ARM 

AVlATlO 

SYSTEMS COMMAND 

AVIATION RLT ACTIVITY 

~ 

{NASA-TN-102309) STABILITY OF A RIGID ROTOR N89-27114 

SUPPORTED ON OIL-FILM JOURNAL BEARINGS UNOEB 

DYIAl¶IIC LOAD (HAS&, Lewis Research Center) 

CSCL 20D Unclas 

G3/34 0224116 

13 P 
STABILITY OF A RIGID ROTOR SUPPORTED ON OIL-FILM 

JOURNAL BEARINGS UNDER DYNAMIC LOAD 

B.C. Majumdar* 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135 

and 

D.E. Brewe 

Propulslon Directorate 

Lewis Research Center 

Cleveland, Ohio 44135 

U.S. Army Aviation Research and Technology Activity - AVSCOM 

SUMMARY 

Most published work relating to dynamically loaded journal bearings are 

directed to determining the minlmum film thickness from the predlcted journal 

trajectories. These do not give any information about the subsynchronous whirl 

stability of journal bearing systems since they do not consider the equations 

of motion. It is, however, necessary to know whether the bearing system operation is stable or not under such an Operating condition. 

h 

c\r 

h 

M 

I w 

The purpose of the present paper Is to analyze the stability characteristics of the system. A linearized perturbation theory about the equilibrium 

point can predict the threshold of stability; however it does not indicate 

postwhirl orbit detail. The linearized method may indicate that a bearing is 

unstable for a given operating condition whereas the nonlinear analysis may 

indicate that It forms a stable limit cycle. For this reason, a nonlinear 

transient analysis of a rigid rotor supported on oil journal bearings under 

(1) a unidirectional constant load, (2) a unidirectional periodic load, and 

(3) variable rotating load are performed. 

In this paper, the hydrodynamic forces are calculated after solving the 

time-dependent Reynolds equation by a f i n 1 t e difference method wi th a successive overrelaxation scheme. Using these forces, equations of motion are solved 

by the fourth-order Runge-Kutta method to predict the transient behavior of the 

rotor. With the aid of a high-speed digital computer and graphics, the journal 

trajectories are obtained for several different operating conditions. 

INTRODUCTION 

There are two principal approaches by means of which one can analyze the 

whirl instability of a rotor supported on fluid film bearings. 

(1) linearized (perturbation) method and (2) nonlinear transient analysis. In 

These are: 

*National Research Council - NASA Research Assoclate; present address: 

Indian Institute of Technology, Dept. of Mechanical Engineering, Kharagpur721302, India. 
the linearized method a small perturbation of the journal center about its 

equilibrium point is given. 

mined after solving the basic differential equation. These coefficients are 

then used in the equations of motion to find the critical mass parameter and 

whirl ratio. The mass parameter, a function of rotor speed, is a measure of 

stability. The nonlinear transient analysis, on the other hand, gives the 

journal locus and from this one can know about the system stability. 

The stiffness and damping coefficients are deterStability analysis of finite journal bearings by the linearized method has 

been given by Allaire (ref. 11, whereas Akers, Mlchaelson, and Cameron (ref. 2) 

studied the same bearing configurations using a nonlinear transient approach. 

In reference 2 it was shown that under certain operating conditions the journal 

motion was bounded and could form a limit cycle. 

The aim of the present paper is to study theoretically the stability characteristics of finite oil journal bearings under dynamic load using a nonlinear 

transient method. A few papers (refs. 3 to 5) deal with the dynamically loaded 

bearings to predict the journal locus. As these do not consider the equations 

of motion for the prediction of the position of journal center, they cannot 

indicate whether the bearing system is stable or not. However, these are useful for estimating minimum film thickness of dynamically loaded bearings. The 

dynamical equations of motion are solved by fourth-order Runge-Kutta method to 

find eccentricity ratio and attitude angle and their derivatives for the next 

time step. These values are then introduced in the two-dimensional timedependent Reynolds equation to find the hydrodynamic forces. 

above approach a nonlinear transient analysis of a rigid rotor on oil journal 

bearings under (1) a unidirectional constant load, (2) a unidirectional periodic load, and (3) variable rotating load is performed. A number of trajectories have been obtained with the aid of a high-speed digital computer and 

graphics . 

Following the 

C 

D 

NOM E NC LATUR E 

radial clearance 

journal diameter 

eccentricity 

hydrodynamic forces 

Fr = FrC’/,,R’L, Fe = FeC2/q,R3L (dimensionless) 

film thickness, 6 = h/C (dimensionless) 

mass parameter, M = MCo2/Wo dimensionless) 

film pressure, = pC2/,,R3L 

journal radius 

dimensionless time, T = wpt 

- 

- 

2 
t time 

x, z, e, 2 

w, ii 

'09 '0 

E eccentricity ratio, E = e/C (dimensionless) 

0 absolute viscosity of oil 

coordinates, 8 = x/R, 2 = z/(L/2) (dimensionless) 

load, W = WC2/qwR3L (dimensionless) 

steady-state load, flo = WC2/qwR3L (dimensionless) 

- 

'0 atti tude angle 

angular coordinates at which film commences and cavitates, respectively, from minimum film thickness 

91 9 92 

819 82 81 = w + 91, 82 = w + 92 

a 

w angular velocity of journal 

whirl ratio, Q = wP/w (dimensionless) 

angular velocity of whirl 

THEORY 

The basic differential equation for pressure distribution in the bearing 

clearance under dynamic conditions can be written as (See fig. 1.) 

Equation I1 1 when nondlmensionalized with the following substitutions: 

8 = x/R, z = z/(L/2), h = h/C, p = pC2/qwR2, T = wpt, and R = w~/w, will read 

as 

The film thickness (dimensionless) is 

- 

h = 1 + E COS 8 (3) 

with the use of equation (31, equation (2) can be written as 

where (b = aT/aT and E = aE/aT. In this study we have used the Reynolds 

boundary conditions, which are given by: 

3 
and 

where 81 and 02 are the angular coordinates at which the film commences and 

reforms, respectively. 

Equation (4) is solved numerically for pressure by a finite difference 

method with a successive overrelaxation scheme satisfying the above boundary 

conditions. Initially, E and (b aye set equal to zero to obtain the steadystate hydrodynamic forces Fr and Fe. 

The forces are computed from 

and (6) 

where 

- F rc2 - F8C2 

3 and Fe = - 

OUR L 

Fr = - 3 OUR L 

Releasing the journal from the steady-state position, one can compute 

'p, E, and (r, for the next time step by solving the following equations of 

mot 1 on : 

E, 

MC[$-- ~($9~1 = Fr t W COS 'p 

The dimensionless form of equations (7) and (8) are: 

cos 'p = 0 - 2 .2 'r 

MQ2E - RQ ET - - 

wO 

(7) 

(9) 

4 
- 2.. 2- F0 

wO 

MR E + 2flR EQ - F+ sin Q = 0 (10) 

where 

- woc2 

- MCu2 and Wo = - 3 M=- 

wO qwR L 

The steady-state load io is obtained by letting E and (b equal to zero. 

'p. These are yolyed by using a fourth-order Runge-Kutta method for constant 

values of R, M, Fr, and Fe. 

Equations (9) and (10) are second-order differential equations in E and 

In the following section three types of load are considered. 

A Unidirectional Constant Load 

Assuming the initial conditions given In table I, the hydrodynamic forces 

under steady-state condition are found. 

obtain E, &, Q, and (b for the subsequent time step. Now the new value of 

E, E, and (9 are introduced in equations (3) and (4) to determine the hydrodynamic forces. These forces along with the steady-state load, mass parameter 

and whirl ratio are utilized for the solution of equations (9) and (10). The 

process is repeated until we get a trajectory that describes the status of the 

system. 

Equations of motion are solved to 

A Unidirectional Periodic Load 

The applied load was assumed to be 

-- 

W = W, [l + sin(: T)] (11) 

where Wo is the applied steady-state load cgnsistent with the eccentricity, 

= 0.8. At each time stee a new value of W was calculated from equa- 

- tion (11). For values of M and R given in table I and using W, F, and 

Fe, equations (9) and (10) were solved for 

step. The rest of the procedure is repeated as described. 

E, E, Q, and (9 for the next time 

Variable Rotating Load 

This type of loadlng is interesting since It pertains to an engine bearing. The data used for the analysis refer to the Ruston and Hornsby 6 VEB-X 

MKIII connecting rod bearing (ref. 5) and is listed in table I. The polar load 

diagram is represented by figure 2. 

positions of crank angle the reader may refer to appendix 4.1 of the paper 

(ref. 5). The time step AT = n/18 was taken in this case to match with the 

For the magnitude of load at different 

5 
given applied lodd at loo crank angle interval. 

nondimensionalized using the expression W = WC2/,aR3L. 

duced in the equations of motion to determine the position of the journal center (E, E, 9, and 4). The hydrodynamic forces Fr and Fe were, however, 

computed after solving equation (4). 

The resultant applied load was 

This load was introRESULTS AND DISCUSSION 

To assure ourselves that our formulation is consistent with reference 2, 

our results are shown in figure 3(a) and compared to those of reference 2 shown 

in figure 3(b). The comparison indicates very good agreement for the conditions stated in the figure. In figure 4(a) a typical three-dimensional pressure distribution for a particular time step is shown. From this figure one 

can see the cavitated region and the variation of pressure throughout the complete clearance space of the bearing. Figure 5 gives the journal locus when 

the bearing is stable under the action of constant unidirectional load. In 

figure 3, the journal trajectory within the clearance circle is shown for an 

unstable bearing. The journal locus reaches a limit cycle. This type of 

observation has been made by many workers (refs. 6 to 8) while dealing with 

short and infinitely long bearings. 

Having compared the result of the present solution with that of Akers 

et al. (ref. 21, an attempt was made to study the effect of periodic and variable load. Figure 6 shows the journal locus for a periodic load superimposed on 

the constant load given the same as those of a unidirectional constant load. 

From these two figures it may be seen that a bearing which is stable under the 

action of a unidirectional constant load can be made unstable when a periodic 

load is applied. In the latter case the journal locus reaches a limit cycle. 

The journal center trajectory of the connecting rod bearing (variable 

rotating load) is shown in figure 7. The trajectory is very complex: it neither tends to go to an equilibrium point nor to reach a limit cycle. 

mentioned that the value of the minimum film thickness for this case exceeds 

that obtained by others using the mobility method of solution. The mobility 

method (ref. 3), however, does not consider the stability of the system. 

The present solution does not take account of the transport of fluid 

through the cavitated region and consequently does not take full account of the 

oil film history. A subsequent analysis has been undertaken that will include 

the oil film history effects for these dynamic conditions and will be compared 

at a later date. A proper accounting of the mass flow in time is believed to 

be important in dynamically loaded bearings and should influence the trajectory 

of the rotor in a more realistic way. 

It may be 

CONCLUSIONS 

From the nonlinear transient analysis of an oil-film journal bearing under 

different dynamic loads with Reynolds type boundary conditions, the following 

conclusions are evident. 

6 
1. Although the analysis Is costly In terms of computer time, it gives the 

orbital trajectory within the clearance circle which is not obtainable using a 

simplified (linearized) theory. 

bearing theory, the journal locus for a finite bearing ends in a limit cycle 

2. As it is shown by others while dealing with short and infinitely long 

I for an unstable system. 

I- 3. A stable journal with a constant unidlrection load can be made unstable 

by superimposing a periodic load on the system. 

4. A more correct (cavitated) boundary condition using the prehistory of 

film may show some interesting phenomena which are not revealed in the present 

method of solutlon. 

REFERENCES 

1. Allaire, P.E., Design of Journal Bearings for High-speed Rotating 

Machinery, fundamentals of the Design of Fluid Film Bearings, ASME 

Publication, New York, 1979, pp. 45-83. 

2. Akers, A., Michaelson, S., and Cameron, A., Stability Contours for a 

Whirling Finite Journal Bearing, J. Lub. Tech., Trans. ASME, Series F, 

vol. 93, no. 1, 1971, pp. 177-190. 

3. Booker, J.F., Dynamically Loaded Journal Bearings - Mobility Method of 

Solution, J. Basic Engr. Trans. ASME, Series D, vol. 187, no. 3, 1965, 

p. 537. 

4. Horsnell, R. and McCallion, H., Prediction of Some Journal Bearing 

Characteristics Under Static and Dynamic Loading, Proc. Lubrication and 

Wear Convention, Inst. Mech. Engrs., London, 1963, pp. 126-138. 

5. Campbell, J. Love, P.P., Martin, F.A., and Refique, S.C., Bearings for 

Reciprocating Machinery: A review of the Present State of Theoretical 

Experiment and Service Knowledge, Conf. Lubrication and Wear, Proc. Inst. 

Mech. Engrs., vol. 182, (Part 3A), 1967-68, pp. 51-74. 

6. Kirk, R. and Guenter, E.J., Translent Journal Bearing Analysis, NASA 

CR-1549, June 1970. 

7. Holmes., R., The Vibration of a Rigid Shaft on Short Sleeve Bearings, J. 

Mech. Engr. Sci., vol. 2, no. 4, 1960, p. 337. 

8. Jakobsen, K. and Chrlstensen, H., Nonlinear Transient Vibrations in 

Journal Bearings, Proc. Inst. Mech. Engr. vol. 183, Part 3P, 1968-69, 

pp. 50-56. 

7 
Condition L/D I Eo 11.1 I Q o(RAD/S) D(M) C/R q(Pa-s) 

Vari ab1 e 1 0.562 I 0.724 1 5. I 1.0 I 62.84 1 0.2 I 0.0008 I 0.15 I 

rotating 

Unidirectional 

constant 

peri odi c 

Unidirectional 

FIGURE 1. - A SCHEMATIC DIAGRAM OF JOURNAL BEARING. 

1 .O 0.8 5. 0.5 ----- --- ------ ---- 

1 .O 0.8 5. 0.5 ----- --- ------ ---- 

8 
RELATIVE TO CYLINDER AXIS 

FIGURE 2. - POLAR LOAD DIAGRAn OF ENGINE BEARING. 

(A) OBTAINED FROM THE PRESENT KTHOD SOLUTION. 

(B) OBTAINED BY AKERS ET AL. [21. 

FIGURE 3. - JOURNAL CENTER TRAJECTORY. (L/D = 1.0. E, = 0.5, 3 = 30°, 

M = 10.24. n= 0.5). 

9 
FIGURE 4. - A TYPICAL PRESSURE DISTRIBUTION (L/D = 1.0, Eo = 0.8, - 

M = 5. R = 0.5. T = 0). 

FIGURE 5, - JOURNAL CENTER TRAJECTORY FOR A UNIDIRECTIONAL CONSTANT 

LOAD (L/D = 1.0.~~ = 0.8. R = 5. n = 0.5). 

FIGURE 6. - JOURNAL CENTER TRAJECTORY FOR A UNIDIRECTIONAL PERIODIC FIGURE 7. - JOURNAL CENTER TRAJECTORY FOR VARIABLE ROTATING LOAD. 

LOAD (L/D = 1.0, Eo = 0.8, K= 5, R= 0.5). 

10 
National Aeronautics and 

Space AdInlnlStrdllOn 

7. Key Words (Suggested by Author@)) 

Journal bearings; Stability; Dynamic loads; 

Bearings; Dynamics 

Report Documentation Page 

18. Distribution Statement 

Unclassified - Unlimited 

Subject Category 34 

- 2. Government Accession No. NASA TM-102309 1. Report No. 

9. Security Classif. (of this report) 

Unclassified 

AVSCOM TR 87-C-26 .- 

4. Title and Subtitle 

Stability of a Rigid Rotor Supported on Oil-Film Journal 

Bearings Under Dynamic Load 

20. Security Classif. (of this page) 1 21. NO of ;yes 22. Price' 

Unclassified 1 A03 

--~_- 

7. Author@) 

B.C Majumdar and D.E. Brewe 

9. Performing Organization Name and Address 

NASA Lewis Research Center 

Cleveland, Ohio 44135-3191 

and 

Propulsion Directorate 

U.S. Army Aviation Research and Technology Activity-AVSCOM 

Cleveland, Ohio 44135-3127 

2. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, D.C. 20546-0001 

and 

U.S. Army Aviation Systems Command 

St. Louis, Mo. 63120-1798 

3. Recipient's Catalog No. 

5. Report Date 

6. Performing Organization Code 

8. Performing Organization Report No. 

E-3727 

10. Work Unit No. 

505-63-81 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

Technical Memorandum 

14. Sponsoring Agency Code 

5. Supplementary Notes 

Prepared for the National Seminar on Bearings Madras, India, September 17- 18, 1987. B.C. Majumdar, National Research CouncilNASA Research Associate; present address: Indian Institute of Technology, Department of Mechanical Engineering, Kharagpur-721302, 

India. D.E. Brewe, Propulsion Directorate, U.S. Amy Aviation Research and Technology Activity-AVSCOM. 

6. Abstract 

Most published work relating to dynamically loaded journal bearings are directed to determining the minimum film 

thickness from the predicted journal trajectories. These do not give any information about the subsynchronous 

whirl stability of journal bearing systems since they do not consider the equations of motion. It is, however, 

necessary to know whether the bearing system operation is stable or not under such an operating condition. The 

purpose of the present paper is to analyze the stability characteristics of the system. A linearized perturbation 

theory about the equilibrium point can predict the threshold of stability; however it does not indicate postwhirl 

orbit detail. The linearized method may indicate that a bearing is unstable for a given operating condition whereas 

the nonlinear analysis may indicate that it forms a stable limit cycle. For this reason, a nonlinear transient 

analysis of a rigid rotor supported on oil journal bearings under (I) a unidirectional constant load, (2) a unidirectional periodic load, and (3) variable rotating load are performed. Ln,this paper, the hydrodynamic forces are 

calculated after solving the time-dependent Reynolds equation by a finite difference method with a successive 

overrelaxation scheme. Using these forces, equations of motion are solved by the fourth-order Runge-Kutta 

method to predict the transient behavior of the rotor. With the aid of a high-speed digital computer and graphics, 

the journal trajectories are obtained for several different operating conditions. 

'4 

'For sale by the National Technical Information Service, Springfield, Virginia 221 61 NASA FORM 1626 OCT 86 

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