AD-A247 566iIIl II ll ItB l li ll IiNASA AVSCOMTechnical Memorandum 105226 Technical Report 91- C- 040

Frequency Effects on the Stabilityof a Journal Bearing forPeriodic Loading

D. VijayaraghavanLewis Research CenterCleveland, Ohio

and

D.E. ByewePropulsion Directorate F LEC 3"E

U.S. Army Aviation Systems Command MAR 18 199Z,Lewis Research CenterCleveland, Ohio

Prepared for theSTLE-ASME Joint Tribology ConferenceSt. Louis, Missouri, October 13- 16, 1991

------.--- 1-ARO

AJASASYSTEMS COMMAND

92-06831

FREQUENCY EFFECTS ON THE STABILITY OFA JOURNAL BEARING FOR PERIODIC LOADING

Acoesslon For

D. Vijayaraghavan*National Aeronautics and Space Administration DT C Tf B []

Lewis Research Center Uv', :... []Cleveland, Ohio 44135

D.E. Brewe

Propulsion Directorate r L.:t ,; 4U.S. Army Aviation Systems Command "

Lewis Research Center A alihi' ilt ,Cleveland, Ohio 44135 a and/or

Diat iS~aaa

Summary required. However, the analsisof dynamically loadedbearings are usually performed for known/predicted journal

this paper, stability of aj bearing is numerically trajectories or by using linearized theory to predict the whirl.In th e an uiitijournal erii xterally The stability of the system under a known journal trajectory

predicted when an unidirectional periodic external load is can not be determined since the equations of motion are notapplied. The analysis is performed using a cavitation algorithm, considered. The linearized theory assumes the equilibrium

which mimics the JFO theory by accounting for the mass

balance through the complete bearing. Hence, the history of position as the starting point. By perturbing the journal centerthe ilmis tkenint conideatin. Te ladig pater is a small amount from equilibrium, the stiffness and damping

the film is taken into consideration. The loading ce is coefficients, critical mass and the whirl ratio are determined.taken to be sinusoidal and the frequency of the attyce is ml mutfo qiirim h tfns n apnldcys usisg 'The critical mass is important since it is a measure of stability.

varied. The results are compared with the predictions using The nonlinear film forces are linearized for the motion to bereyormaiono bo hsund cod itons, th fo rpurely analytically tractable. The linearized theory can predict thereformation. With such comparisons, the need for accurately threshold of stability but not postwhirl orbit details. On thepredicting the cavitation regions for complex loading patterns other hand, nonlinear transient analysis does not attempt to

is clearly demonstrated. For a particular frequency of loading, linearize the equations of journal motion and hence determines

the effects of mass, amplitude of load variation and frequency the locus of the journal for the operating conditions. The

of journal speed are also investigated. The journal trajectories, stability of the system can be determined from the journal

transient variations in fluid film forces, net surface velocity locus.

and minimum film thickness and pressure profiles are also Holmes (1960), in an analysis of the vibration of a rigidpresented, shaft for short sleeve bearings, determined that the occurance

of whirl is a function of steady state eccentricity, angularfrequency and clearance. The plot of the "whirl boundary"

Introduction was based on the short bearing approximation and the Gimbel

boundary conditions. Mitchell, Holmes and Byrne (1965)In order to effectively analyze the performance of a bearing used nonlinear theory with shortllong bearing approximations

for various industrial applications, a transient analysis is to determine the characteristics of oil whirl of a full journal*National Research Council-NASA Research Associate at Lewis Research Center.

bearing with the aid of analog and digital computation as well rupture and reformation boundaries. Brewe (1983) andas experimental data. Capriz (communications on Mitchell Paranjpe and Goenka (1989) applied this algorithm to analyzeet al; 1965) considered cavitation along with the nonlinear dynamically loaded bearings and found excellent agreementmotion and presented a stability diagram. However, the with experimental data.relative movement of the cavity to the minimum film thickness In this present work, the stability of the journal bearing iswas not determined and the cavity was assumed to rotate at analyzed using the nonlinear journal motion theory. Thethe whirl frequency. Jakobsen and Christensen (1968) analyzed cavitation algorithm is used to predict the full film and cavitatednonlinear transient vibrations under a constant external load regions and to determine the fluid force components. Theand presented the journal loci for various journal masses and predicted journal trajectories for the periodi- loading patternsstarting positions. Akers, Michaelson and Cameron (1971) using the cavitation algorithm are compared with theextended the work of Holmes (1960) and Capriz (1965) for predictions using Reynolds boundary conditions for both filmfinite bearings and determined the journal trajectories for rupture and reformation. The main objective of this study isboth cavitated and uncavitated bearings and for various L/D not only to analyze the system response for the dynamicratios. The out of balance load and friction force were also loading pattern; but also to highlight the need for consideringconsidered in the analysis. Kirk and Gunter (1970) used a the film history during such situations. The effect of mass,short bearing approximation in a transient analysis of journal amplitude of load variation and journal speed are alsobearings. Badgley and Booker (1969) analyzed the effect of investigated for a particular frequency of loading.initial transients on turborotor stability. Short bearing (Ocvirk),long bearing (Sommerfeld) and finite bearing (Warner)approximations were used to model the film and the film Nomenclaturewas assumed to extend 1800 between variable limits. Therest of the bearing was taken to be cavitating at zero pressure.Allaire (1979) presented the stability analysis of finite journal ap amplitude of periodic loadingbearings using linearized theory. Majumdar and Brewe (1987) c radial clearanceanalyzed the stability of a rigid rotor supported on a journal e eccentricitybearing, under unidirectional constant, periodic and variable Fr radial component of fluid film forceload patterns. The analysis included nonlinear motion but conly Reynolds boundary conditions were used for film rupture F0 circumferential component of fluid film forceand reformation. It has been clearly brought out in the above g switch functionmentioned papers that while linearized theory can predict the h film thicknessthreshold of stability due to small perturbations about the j nondimensional film thickness [b/c]equilibrium point, the details of postwhirl orbit and occurance

of stable limit cycle can be determined only by using a L bearing lengthtransient nonlinear analysis. LD length to diameter ratio of the bearing

The determination of hydrodynamic force components M mass of the systemdepends on the film model used. Most of the above analyses M nondimensional mass parameter [Mcto 2 /Wbiwere based on the restrictive assumptions of short or longbearings. The Giimbel and Swift-Stieber (Reynolds) boundary P fluid pressure

conditions treat the cavitation in a superficial manner. In Pa ambient pressuredynamically loaded bearings, the cavitation boundary will PC cavitation pressurevary in size, shape and location, and therefore, the usual zero R residualpressure and pressure gradient boundary conditions will beinadequate to handle the situation. The instantaneous pressure r bearing radiusdistribution and the location of film boundaries depend not t timeonly on the instantaneous position of the journal due to its U net surface velocitymotion but also on the history of the location of the cavitation Vf volume occupied by fluid filmboundaries preceeding the instant under consideration. V t volume o fluid lmJakobsson-Floberg (1957) and Olsson (1965) derived boundary Vt total volume of fluid elementconditions (JFO theory) for film rupture and reformation by W applied load at any instantimposing conservation of mass within the cavitated region Wb base load of periodic loadingand at the boundaries. The theory assumes that the lubricant co-ordinate axis in circumferential directionis transported through the cavitated region in the form ofstriations extending between the boundaries. Elrod (1981) z co-ordinate axis in axial directionintroduced a cavitation algorithm which conserves mass 13 bulk modulus of fluidthrough the entire bearing and automatically predicts the film Ax grid spacing in x direction

2

8 central difference operator cavitated regions and also across the boundaries. The two

E eccentricity ratio dimensional form of the universal equation can be written as

!t fluid Viscosity

p fluid density +_(h) drhU0 f3h 3 de d - --h3 gdo

PC fluid density at cavitation pressure dt A 2 12 gdx dz -12p gdz 00 attitude angle (3)

XV angular coordinate relative to maximum film wheretwchneeroctinthe independent vaibe0 is dindasthickness location

Q frequency ratio [op / w](o angular velocity of the journal in full film region (0_>1)

top frequency of periodic loading 0 = PC

uL in cavitated region (0<1)Subscript V

i nodes in circumferential direction g, the switch function introduced to remove the pressurej nodes in axial direction gradient term in the cavitated region is defined as0 initial value

1 in full film region (8 > 1)Superscript g = 0 in cavitated region (0 < I)

n1 iteration number* current value and the bulk modulus 3 which relates the pressure and den-

sity of the liquid is defined as

Reynolds Equation

The pressure profile of the fluid film within the bearing =-dp = 0-!k

clearance is governed by Reynolds equation, which is writtenin two dimensional, unsteady form, as Since, the mass conservation through the entire bearing is

taken into account via the universal equation, the boundarydh (U h d ( h3 dp )_ d ( h3°P) conditions for film rupture and reformation need not be

+t -ht2 -x 0 (1) explicitly specified. Within the full film region (g = 1),S / 21equation (3) results in a compressible form of the Reynolds

The above differential equation is applicable only within the equation.Assuming that the bearing is stationary, in the case of

full film region. The Reynolds boundary conditions to statically loaded bearings the surface velocity is only due todetermine the extent of film region can be expressed as the rotation of the journal. In dynamically loaded bearings,

the journal center also moves (whirls). In such cases,p(xI,z) p(x2 ,z) = 0 superimposing the motion of journal on the rotation, the net

surface velocity will be-x 1 z= x 2 ,Z)=0O2dx " x

U=r wo-2-) (4)

where x, and x2 are the circumferential coordinates

corresponding to film rupture and reformation respectively.Since the journal motion is superimposed on the rotation, the

Universal Equation angular frame of reference remains constant (say maximumfilm thickness) and the film thickness is determined from

Elrod and Adams (1974) developed an "universal equation"which conserves mass and is applicable for both fil film and h = 1 + £ cos (5)

3

Equations of Journal M otion ( 0 ) = - I [ g , ,2 h ,'

For ajournal at a position other than its equilibrium position, 2 3x 2 2 gxL

the plane motion, due to the unbalanced fluid forces, is +(2-g i+ .2 -g/ 114 )(Oh) -(2-gi_1 /2)(Oh),_described by the following scalar nonlinear equilibriumequations, along and normal to the line of centers (fig. 1), (11)respectively,

g1 + g1~1

d2 ( 2] =1 where 2Mcj~e t2 jj=-r+ co 6

Ld dt The above formulation is based on the implicit assumptionthat the surface velocity U is positive. However, if the

McFE d (d"02 ( de)/=F1 -Wsin (7) velocity is negative, then this flow term must remain onec dt dt )I dt = sided upwind differenced in the cavitated region (thus, forward

differenced relative to original grid scheme), to properly

The fluid film forces Fr and FO are determined from integrat- describe the physics of the problem. In such cases, the

ing the pressure distribution, which is obtained by solving following formulation shall be used

equations (1) or (3) and using the following integrals U ( 0h) _U XI (2-gi+, /(0h),,,-Fr = - j2 cos V di dz (8) i

Fr -L/2 JO -(2 - gi1 , 2 - g- 1/ 2 )(0h)t - gi-i/ 2 (0h),_1 (12)rL/2 2l r

F0 JL/2 JoP sin y d ' z (9) When the surface velocity reverses during the course of the

computation, equations (11) and (12) can be combined andPeriodic Load written in a more compact and efficient form as

The type of periodic dynamic loading envisaged is of U6i "sintsoidal variation, according to the following relationship .--- (Oh) J- 22 [(0)i_-(Oh) 1 2

W = Wbsl+ ap sin (twP t)] = Wb [I+ ap sin (rn)] (10) x[(gi+/ 2 -l)(Oh)++(2-gi+1 /2 -gi-1 /2 )(Oh).+(gi+1/ 2-1)(Oh)i-jI

(13)where Wb is the base load, ap is the amplitude coefficient,to is the frequency of periodic loading and Q is the ratio of Since the formulation of equation (3) is based on theload cycle frequency to journal frequency. assumption of a compressible film, the resultiiig algorithm

also has 0 in the full film region. For s.ufliciently largevalues of 0, the 0 values will be very close to unity and

Numerical Formulation hence this assumption does not introd, .. appreciable error.Alternately, while writing the code, tie effect of 0 can be

Cavitation Algorithm neglected in the full film region (gi-1 = gi = gi+.1 = 1) resultingin a formulation equivalent to ahIJx. Also, in order to ensure

A modified cavitation algorithm, an offshoot of the Elrod one sided differencing near the reformation boundary, i.e.,

algorithm, using type differencing procedure was developed when gi-1 = gi = 0, gi+1 = I, U > 0; gi+ 112 is set to zero.

by Vijayaraghavan and Keith (1989). The resulting formulationis simply stated below as the details of the development can Pressure duce.wTe psbe otaied romthi paer.in the x direction s differenced ashe obtained from this paper.

Convection Flow. -Th convection flow term exists both in 6 3 60--.--[h g1 (6 1 -)the full film and cavitated regions. This term is to be one sided 3x 6x x 2 t+

upwind differenced in the cavitated region and is centrally /,2 + h3 , (O -1)+ h_ g3 (, -differenced in the full film region. j+1/2

(14)

4

The expression for the flow in the z direction can be written where 0* is the currently available value of 0. At every time

in similar fashion. level, iteration is carried out by updating 0* until AO(0-0*) approaches zero.

Equation (3) applied to equation (18) producesNumerical Procedure

Reynolds Equation [- + -- al0 + -(a,,)"+- +-(az) AO + f(8)0=0* =0

Equation (1) is discretized in finite difference forms in (19)

terms of the unknown variable p and numerically solved using where a30 = a22 = 12-ha successive relaxation procedure either in explicit or implicit 2 12yform. When the finite difference formulation is cast in

successive line over relaxation (SLOR) form, the resulting Equation (19) can written in the following operator formalgebraic equations can be written as

NAO+R=O (20)

An npl + Bi jp n+l c _n+p1 D ,jpn .. +FI.JrIJ ' ijpi+lj * 1 d ~Jij+I "j where N is an operator, A0=(0 n +I - 0.) is the correction and

(15) R is the residual at 0*, or, a measure of how well equa-where the coefficients Aid, Bi and C. and right hand tion (3) is satisfied at 0*. The operator N is defined as a

n n Ch a t . product of two operators, which are approximate factors of fside of equation (15) are kown and written asrepresent grid points in the circumferential and axialdirections respectively and n denotes the iteration number.The system of equations for every j are simultaneously N = BNxN z (21)

solved. After one sweep is completed, the resulting negative where

pressures are set to zero. The procedure is repeated until thepressure values have converged according to the following Nx = I+ -- a0 +-- a,,)-

convergence criteria Bdx Bdx dxNz~ll" a 2 2B z )-"

pp. B dz dzly Pi.1 , p j I _ 0 - 3 (16) h

I;', At

Cavitation Algorithm The solution is obtained in two steps as follows

In order to obtain time-accurate transient solutions, equa- Rtion (3) is solved using equations (13) and (14) by combining Step 1: NxAO -B

an approximate factorization method with a Newton iteration Step 2: NZAO AOprocedure. The details of the procedure are discussed in detailby Vijayaraghavan and Keith (1990). Hence, only the overallidea is presented here. In the first step, AO is obtained by sweeping in the x direction

Equation (3) is a function of 0 only and can be represented (along a constant axial co-ordinate line). The final value ofby correction AO is obtained in the second step by sweeping in

the z direction (along a constant circumferential co-ordinatef(O) = 0 (17) line). The approximate factorization technique has been found

to be robust and efficient.

where f is an operator for equation (3). The Newton iterationfor a solution to equation (17) is.

Equations of Motion

+( °f / (0- 0)= 0 (18) Equations (6) and (7) are two scalar, coupled, nonlineardo* + =* ordinary differential equations. These equations are integrated

5

simultaneously using a fourth order Runge-Kutta method for Results and Discussionknown values of M, c, W, Fr and F0 and initial values of,e, de/dt and do/dt. The fourth order Runge-Kutta method To determine the effect of frequency of periodic loading,is considered to be the stansard method for solving nonlinear for the same load cycle, the frequency ratio 9 is varied fromdifferential equations, paticularly with non-smooth profiles. 0.25 to 3.0. The analyses are performed twice, using the

The integrations to determine the force components, Fr and cavitation algorithm and Reynolds equation. The solutionsF0 (eqs. (8) and (9)) are performed using Simpson's Rule. from the cavitation algorithm are compared with the solutions

of the Reynolds equation with Reynolds boundary conditions.When Q = 0.25, one load cycle occurs at every four journal

Solution Procedure revolutions. Figure 2 compares the journal trajectory, fluidfilm forces, net surface velocity and the minimum film

A submerged journal bearing without any feed groove thickness predicted by both methods for this case. Althougharrangement is considered in this analysis. The axial ends of the journal trajectory predicted by both methods is similar, thethe bearing are maintained at ambient pressure and the cavitation algorithm predicts larger eccentricity values. Thiscavitation pressure is taken to be absolute zero. Due to axial can also be observed in the differences in the minimum film

symmetry, only one half of the bearing is analyzed. Initially thickness values and the transient forces developed by thethe bearing is assumed to be filled with lubricant at ambient fluid film. At times, the radial component of the force variespressure and released at an arbitrary location other than its by as much as 30 percent. The cavitation algorithm predictsequilibrium position (table I). With Reynolds equation, at larger forces and smaller minimum film thickness values.every time step, a converged pressure distribution is The differences in the variations of net surface velocity betweendetermined. Hence, the pressure distribution is only a function the methods are not significant. However, it is interesting toof the instantaneous eccentricity ratio. However, the pressure note that the velocity becomes negative for a brief period ofdistribution of the previous time step is considered as the first time. During such occasions, it is as though the journaliteration value, for the purpose of verifying the convergence rotates in the opposite direction.according to equation (16). Whereas, with the cavitation Figure 3 represents the comparison of the journal trajectory,algorithm no such convergence is sought. The pressure fluid film forces, velocity and minimum film thickness whendistribution is obtained by marching in time. Equation (3) is Q = 0.50. While both methods predict that the journal willsolved to determine the current 0 distribution based on the 0 attain a limit cycle at high eccentricity values, the cavitationdistribution at the previous time step and the current value of algorithm predicts sharper changes in the trajectory. For thisthe eccentricity ratio. At every time step, the switch function frequency of loading, very large fluid film forces are predicteddistribution is updated. Thus, oil film history in the cavitated by the cavitation algorithm. The radial component of theregion is taken into account at every time step. The Newton force attains values as much as three times the values predictediterations are performed, if necessary, to reduce the residuals when Q = 0.25 (fig. 2). Reynolds boundary conditions do notto a low level (O(10-5)). predict such high excursions in forces, although the peaks are

From the pressure profile the force components F. and F0 about doubled compared to the case for 0 = 0.25. Largeare computed. The applied load values at every time step are excursions in minimum film thickness are predicted by thealso determined according to equation (10). Then the equations cavitation algorithm, although the minimum-minimum valuesof motion, i.e.. equations (6) and (7), are solved for known are smaller. The net surface velocity variations are similar,values of M, c, W, Fr and F0 and initial values of E, 0, de/dt except during the initial journal orbit, at which time the netand do/dt, to determine the new values of E and 4. To begin surface velocities are of opposite sign. If the motion of thethe process, dc/dt and do/dt are assumed to be zero. The journal center encloses the bearing center, the angular velocityprocedure is repeated and the trajectory of the journal, forces of the journal and the angular velocity of the journal centerdeveloped by the fluid film, net surface velocity, minimum have the same sign. Thus, according to equation (4), when thefilm thickness and the pressure distribution are recorded at angular velocity of the journal center exceeds the angularevery specified time interval. Analysis with both methods are velocity of the journal, the net velocity becomes negative. Inperformed for the same axial boundary conditions, same grid this case, the cavitation algorithm predicts an initial journalarrangement and same time steps. The computational grid center orbit that did not enclose the bearing center. Thus thehas 49 nodes in circumferential and 15 nodes in axial directions. downward motion of the journal center was of opposite signThe codes are vectorized and the CRAY XMP super computer to the rotational direction resulting in a positive net surfaceis used. The time step of 1(-5 sec is used and a typical cpu velocity. In this case also, the net surface velocity alternatestime on the CRAY was 120 sec per journal revolution. Each between positive and negative values. With the cavitationcase is run for 10 journal revolutions. The time required for algorithm, this situation causes a cavitated region to be formedone load cycle depends on the frequency ratio, Q. at both sides of the pressure hump. Instantaneous pressure

6

profiles at four events are presented in figures 4(a) and (b). mass conservation and consideration of cavitation regions areFigure 4(a) is the pressure profiles predicted by the cavitation very important during such situations. With periodic loading,algorithm during the third load cycle when (1) the the pressure the journal generally attains a limit cycle, the limit cycledeveloped is minimum, (2) the pressure developed is being smaller when the frequency of the periodic loading ismaximum, (3) eccentricity is maximum and (4) the applied larger.load is zero. Figure 4(b) also indicates the same events Having analyzed the effect of periodic loading on thepredicted using Reynolds boundary conditions, but at different performance of the journal bearing, it is desired to investigatetimes. The instantaneous journal position and its cumulative the effects of other parameters, viz., mass of the system,trajectory are also indicated in the figures. Generally, the amplitude of load variation and journal speed, on the stabilitycavitation algorithm predicts larger pressure values and larger of the bearing. The case of Q = 0.5 is considered for thiscavitated regions. Reynolds boundary conditions do not predict study due to the occurance of large excursions in the forcesany cavitated region at minimum pressure, nor does it predict during this loading frequency. Only the cavitation algorithmtwo cavitated regions split by a high pressure fluid film region is utilized for this study.for conditions (2) and (3) (figs. 4(a) and (b)). During this The mass of the system is varied from one tenth to fiveloading pattern, the pressure profile and the cavitated regions times the value used for the frequency effect study. Theundergo dramatic variations in shape, size and locations, journal trajectories are shown in figure 8. When V = 0.5, the

As shown in figure 5, when Q = 1.0, the journal trajectory journal trajectory almost immediately settles into a limit cycle;predicted by both methods are quite different. The cavitation but only on one side of the clearance circle. When M - 2.5,algorithm predicts a limit cycle around the bearing center line the journal oscillations grow with time and approach a limitwhile the predictions using Reynolds boundary conditions cycle around the bearing center. When the mass parameter isindicate the movement of the journal limited to one side of the larger, the journal trajectory attains the limit cycle faster andbearing, the amplitude of the motion becoming progressively at very high eccentricity ratios. Figure 9 compares the transientsmaller as it converges to a smaller eccentricity ratio. With variations in minimum film thickness for various massthis frequency of loading, the force levels are much smaller parameter values. With larger mass, the fluctuations areand the excursions are considerably reduced as compared to diminished but the minimum-minimum film thickness valuesthe condition with Q) = 0.5. The minimum film thickness are very small, nearly touching the bearing surface.predicted by the cavitation algorithm is also much smaller. The periodic load amplitude was varied from unity to zero.

Figure 6 is the representation of the conditions when Q = The journal trajectories are indicated in figure 10. With2.0. When the cavitation algorithm is used, the journal barely smaller amplitude, the journal movements are only on onepasses through the bearing center and the excursions are slightly side of the clearance circle. The smaller the amplitude thereduced with time. The predictions using Reynolds boundary smaller the journal excursion and the region of limit cycle.conditions indicate the journal movement was limited to one This zero amplitude case essentially represents a unidirectionalside of the clearance circle and the journal excursions reduce constant load. The journal oscillations are damped out and itwith time. The minimum film thickness predicted by the will eventually reach its equilibrium position. The variationscavitation algorithm also has wide fluctuations, with smaller in minimum film thickness are indicated in figure 11. Forminimum values. smaller amplitudes, the minimum film thickness variations

The transient conditions for Q = 3.0 are shown in fig- are smaller and the mean values are higher as compared to

ure 7. The journal trajectory predicted using the cavitation those at larger amplitudes.

algorithm has much smaller excursions as compared to the The effect of variations in journal speed is presented in

trajectory predicted using Reynolds boundary conditions. In figure 12. The frequency ratio is, however, maintained

both cases, the journal movement is limited to one side of the at 0.. The lower ralispeed, hetery inonecleaanc cicle wit th amlitde f moiondimnising at 0.5. With the lower journal speed, the trajectory is on one

clearance circle, with the amplitude of motion diminishing side of the bearing; however, growing in the amplitude of

with time. With Reynolds boundary conditions, the journal oscillation. When the journal speed is increased, the journal

trajectories for both Q = 2.0 and Q = 3.0 are similar. Further os arond the bearng ceer iinga liit cyclincrase intheloadng reqenc prduce nerlysimlar moves around the bearing center, attaining a limit cycle.

increases in the loading frequency produces nearly similar However, from the data it was observed that the lowest

journal trajectories using both methods. Due to the high minimum film thickness is obtained when the journal speed

frequency of periodic loading, the variations in load values is at 600 rpm.

occur before the fluid film can react to such variations. Hence,

the journal trajectories almost simulate the condition of aconstant load. Conclusions

In all the above cases, the cavitation algorithm predictssmaller minimum-minimum film thickness values and large A nonlinear stability analysis is performed on a systemfluid film forces. When the dynamic loading pattern undergoes supported by a plain journal bearing, under periodic dynamicsuch large variations, the memory effect of fluid film becomes loading. A cavitation algorithm was used which conservesan important criteria. Modeling the problem accounting for mass and takes the oil film history into account. These

7

solutions were compared with the solutions of Reynolds Allaire, P. E.: Design of Journal Bearings for High Speed Rotating Machinary.

equation using Reynolds boundary conditions for various Fundamentals of the Design of Fluid Film Bearings, S.M. Rohde, et a.,

loading frequencies, amplitude of periodic loading, mass e ASME, pp. 45-83.Badgley, R. H.; and Booker, J. F.: Turborotor Instability: Effects of Initial

parameter and journal speed. Comparisons on the journal Transients on Plane Motion. J. Lubr. Technol., vol. 91, no. 4, 1965,

trajectories, fluid film forces, net surface velocity, minimum pp. 625-633.film thickness and pressure profiles are presented. Booker, J. F.: Dynamically Loaded Journal Bearings - Mobility Method of

(1) It was established that the oil film history is important Solution. J. Basic Eng., vol. 87, no. 3, 1965, pp. 537-546.for these complex loading patterns. Brewe, D. E.: Theoretical Modeling of the Vapor Cavitation in Dynamically

Loaded Journal Bearings. J. Tribol., vol. 108, no. 4, 1986, pp. 628-638.(2) With periodic loading, the journal almost always attains Capriz, G.: 1965 see communications on Mitchell et al., 1965.

a stable limit cycle. When the periodic loading frequency is Elrod, H.G., Jr.; and Adams, M.L.: A Computer Program for Cavitation andhalf of the journal frequency, very large fluid forces are Starvation Problems. Cavitation and Related Phenomena in Lubrication,

developed and the journal whirls at large eccentricity ratios. Mechanical Engineering Publications, New York, 1974, pp 37-41.

With further increase in frequency of loading, the film forces Elrod, H.G., Jr.: A Cavitation Algorithm. J. Lubr. Technol., vol. 103, no. 3,1981, pp. 350-354.

are diminished, the journal oscillations are reduced and its Holmes, R.: The Vibration of a Rigid Shaft on Short Sleeve Bearings.movement is restricted to only one side of the clearance circle. J. Mech. Eng. Sci., vol. 2, no. 4, 1960, pp. 337-341.

(3) The mass plays a major part in the system response to JakobseA K.; and Christensen, H.: Non-Linear Transient Vibrations in

the loading pattern. With smaller mass, the generation of large Journal Bearings. Tribology Convention 1969, Proc. Inst. Mech. Eng.,

fluid film forces are avoided and the journal attains a stable, vol. 183, pt. 3P, 1968-69, pp. 50-56.Jakobsson, B.; and Floberg, L.: The Finite Journal Bearing, Considering

smaller limit cycle. Vaporization. Transactions of Chalmers University ef Technology,(4) The amplitude of load variation also has a pronounced No. 190, Guthenberg, Sweden, 1957.

effect on the stability of the system. The smaller the periodic Kirk, R.; and Gunter, E. J.: Transient Journal Bearing Analysis. NASA

load amplitude, the smaller the journal excursions and film CR-1549, 1970.

force levels. Majumdar, B. C.; and Brewe, D. E.: Stability of a Rigid Rotor Supported onOil Film Journal Bearings Under Dynamic Load. NASA TM- 102309

Stability analysis of dynamic loading patterns is very (AVSCOM TR 87-C-26). 1987.important for designers in order for them to understand system Mitchell, J. R.; Holmes, R; and Byrne, J.: Oil Whirl of a Rigid Rotor in 360'response. Nonlinear transient analysis and consideration of Journal Bearings : Further Characteristics. Proc. Inst. Mech. Eng.,cavitation effects along with oil film history are imperative vol. 180, pt. 1, no. 25, 1965-66, pp. 593-610.

for such studies to model the system as close to the practical Olsson, K.O.: Cavitation in Dynamically Loaded Bearing, Transactions ofChalmers University of Technology, No. 308, Guthenberg, Sweden, 1965.

situation as possible. Paranjpe, R.S.; and Goenka, P.K.: Analysis of Crankshaft Bearings Using

Mass Conservation Algorithm. Tribology Transactions, vol. 33, Num-ber 3, 1990, pp. 333-344.

VijayaraghavanD.; and Keith, Jr., T.G.: Development and Evaluation of aReferences Cavitation Algorithm. Tribol.Trans., vol. 32, no. 2, 19S9, pp. 225-233.

Vijayaraghavan, D.; and Keith, Jr. T.G.: An Efficient, Robust, and Time

Akers, A., Michaelson, S.; and Cameron, A.: Stability Contours for a Accurate Numerical Procedure Applied to a Cavitation Algorithm. J.Whirling Finite Journal Bearing. J. Lubr. Technol., vol. 93, no. 1, 1971, Tribol., vol. 112, no. 1, 1990, pp. 44-5i.

pp. 177-190.

8

TABLE I.-BEARING DATA

LID --- 0.562rm .1

C m 8.0x 10-5

Pa N/rn2 1.0133X10 5

P /2 .0vM -- 5.0

ap ----- 1.0Wb N 40X 103

N/rn 2 I.72 x10 8

I Pa.s .015rad/s 62.84 (600 rpm)

Q --- .25 to 3.0EO -----. 65

00 -----. 0

Full film -. -Cavitated

region

Figure 1.-Schematic diagram of journal bearing.

9

CAVITATION ALGOR'ITHM

PEYNOLC B.C.

(a) Journal trajectory.

Figure 2.-Periodic loading (0= 0.25).

- - CIRCUM - r I RCUN

-- A --- PAD IAL -A--- RADIAL

210 - APPLIED LOAD 210 F G AFPLIED LOAD

0- 180 180 -

z zo 1 500

120 z

i~ 90 vi 90

z

60 z 60

:3

30 30L.>

wi U0, T0~ 0 .2 -0 o , 6- 0. ,

" -30 TIMV S -30 TIME (SEC)

-60 -60

Cavitation algorithm Reynolds B.C

(b) Fluid film forces.

20B----- CAV ALG 0.6 - CAV ALG

PEYNOLDS PEYNrLDS16-.-- EYOD

00

-- 12 -I' ,

UJI

u 02 tlt o. Z,.0.o. . ~ . .

Ui It

-rrU-

LL

0 0j2 . 0.6 Q8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

z-~ of TEE~S TIME (SEC)

(c) Net surface velocity (d) Minimum film thickness

Figure 2 -Concluded.

CAVI TAT ION ALGOR ITHM

REYNOLDS B.C.

(a) Journal trajectory.

Figure 3.-Periodic loading (Q 0.50).

12

- CIRCUM

S RAD I AL

APPLIED LOAD

500 501-.{ -- C IICUM

4SO '+50 A RA0AL

400 400 e APPLIEO LOAO

350 o 3502--

300 300=" z

- 250 250

200 - 200uJ LU

0o 150 1 50

dX:

° 0d '-100 100

o so U 50Q C0

U- 0 00 0.2 0. . 0. 0. 1.00 0 O50TIME (SE

TM

-100 -100Cavitation algorithm Reynolds B.C.

(b) Fluid film forces.

20l - CAV ALG

-1-6-- REYNOLDS

- CAV ALG

S8 - REYNOLDS

,- q.0.

0 0. 0D 0 .0.6

LU TI[ME (SEU.. I

z ,2

0.2-16

-20 0.0 I I I I

0.0 0.2 0.4 0.6 0.3 1.0

TIME (SEC)

(c) Net surface velocity. (d) Minimum film thickness.

Figure 3--Concluded.

13

- 632010 ms t - 644.010 ms0 0789 t 0.931

in. Mtn nmln. fIlM

'.ma. 61lM mxflf

(i) Minimum pressure. (ii) Maximum pressure.

C -692 010 ms 9 -750 2l0msS0.936 0. 0807

+. +

(iii) Maximum eccentricity. (iv) Zero applied load.

Figure 4a.-Pressure profiles - cavitation algorithm (Q = 0.50).

14

9 - 85.790 ins t -646 190 ins-0,907 =0. 904

(i) Minimum pressure. (ii) Maximum pressure.

t-681 IQOins e -760.190 is-0.927 4c- 0,880

(iii) Maximum eccentricity. (iv) Zero applied load.

Figure 4b.-Pressure profiles - Reynolds B.C. (0 = 0.50).

CAVITATION ALGORITHM REYNOLDS B.C. CAVITATION ALGORITHM REYNOLDS B.C.

(a) Journal trajectory. (a) Journal trajectory.

1.0 1-06-- CAV ALG8s- CAV ALG REYNOLDS6 a- REYNOLDS

0.8 I0.8

0.6 x 0.6k

0. 0.

0.0 0.2 0.-.5. .000 0204 . . .TIME (SC IM SC

(b iiu imthcns.()Mnmu imtikes

Fiue5 - eidclaig(l=10) iue6 -eidclig( 2.0)

K 16

CAVITATION ALGOR[THMI REYNOLDS B.C.

(a) Journal trajectory.

-a--B CAV ALO-A-- REYNOLDS

0.8

CAY. ALG. (A-0.5) CAV. ALG. (M-2.5)0.6

K

0.2

0.0 1 1 1 1 1 1 1 1 10.0 0.2 QOI4 0.6 0.8 1 .0

TIME (SEC)

(b) Minimumnfilm thickness. CAy. ALG. (R-10.) CAV. ALG. (A-25.)

Figure 7.-Periodic loading (L2 = 3.0). Figure 8 -Effect of mass on journal trajectory (Q = 0.5).

17

0.8

10.6 I

0.2

0.0 %0.0 0.2 0.4 0.6 0.8 1.0

TIME (SEC)

=10

-- M 250.8

CAYV. A LG. ap 0~.75 CAYV. ALG. (p 0=.5 1)

S0.6

0.2

0.00.0 0.? 0.41 0.6 0.8 1 .0

T IME (SEC) CAY. ALG. OP~ 62 CAY. ALG. 'k C

Figure 9.-Effect of mass on minimum Figure I0.-Effect of amplitude of load variation on journalIfilm thickness (Q = 0.5). trajectory (Q = 0-5).

18

I.0

AP = 0.0---AP - 0.25

0.8

0.6Zz

z

.00.2 0.1+ 0.6 0.8 1.0TIME (SEC) CAy. ALG. (300 PPM) CAJ. ALG. (1200 PPM)

-0- AP - 0.60---- AP =0.75 .Z

1.0 -0- AP - 1.0 1. A- 300 Rpm

-4--1200 Rpm

0.80.8 A1 '0.6

0.4 . 0.+.

0.2 0.2

0.0 111111 1111 0.0 1 1 1 1 10.%.0 0.2 0.4 046 0.8 1.0 0.0 0.2 0.+ 0.6 0.8 1.0

TIME (SEC) TIME (SEC)

Figure I1 I-Effect of amplitude of load Figure 1 2-Efect of journal speed (Q 0+5).variation on minimum film thickness

Form ApprovedREPORT DOCUMENTATION PAGE _TOB No. 070-088

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

1991 Technical Memorandum4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Frequency Effects on the Stability of a Journal Bearing for Periodic Loading

WU-505-63-5A6. AUTHOR(S) IL161102AH45

D. Vijayaraghavan and DE. Brewe

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATIONNASA Lewis Research Center REPORT NUMBER

Cleveland, Ohio 44135- 3191

and E-6242Propulsion DirectorateU.S. Army Aviation Systenis CommandCleveland, Ohio 44135 - 3191

9. SPONSORING/MONITORING AGENCY NAMES(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING

National Aeronautics and Space Administration AGENCY REPORT NUMBER

Washington, D.C 20546- 0001 NASA TM- 105226and

U S. Army Aviation Systems Command AVSCOM TR-91-C-040

St Louis, Mo. 63120- 1798

11. SUPPLEMENTARY NOTES

Prepared for the STLE-ASME Joint Tribology Conference, St. Louis, Missouri, October 13-16, 1991.D. Vijayaraghavan, National Research Council-NASA Research Associate at Lewis Research Center; D.E. Brewe,Propulsion Directorate, U.S. Army Aviation Systems Command. Responsible person, D.E. Brewe, (216) 433-6067.

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Unclassified - UnlimitedSubject Category 34

13. ABSTRACT (Maximum 200 words)

In this paper, stability of a journal bearing is numerically predicted when unidirectional periodic external load isapplied The analysis is performed using a cavitation algorithm, which mimics the JFO theory by accounting for themass balance through the complete bearing. Hence, the history of the film is taken into consideration. The loadingpattern is taken to be sinusoidal and the frequency of the load cycle is varied. The results are compared with thepredictions using Reynolds boundary conditions for both film rupture and reformation. With such comparisons, theneed for accurately predicting the cavitation regions for complex loading patterns is clearly demonstrated. For aparticular frequency of loading, the effects of mass, amplitude of load variation and frequency of journal speed arealso investigated. The journal trajectories, transient variations in fluid film forces, net surface velocity and minimumfilm thickness and pressure profiles are also presented.

14. SUBJECT TERMS 15. NUMBER OF PAGES

Journal bearing; Dynamic loading; Stability; Frequency effects; Periodic loading; 20Nonlinear motion, Cavitation 16. PRICE CODE

A0317. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT

OF REPORT OF THIS PAGE OF ABSTRACTUnclassified Unclassified Unclassified

NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)Prescribed by ANSI Std. Z39-18298-102