ADA AFWAL-TR SELF EXCITED OSCILLATIONS IN JOURNAL BEARINGS. Wilbur L. Hankey. Aerodynamics and Airframe Branch Aeromechanics Division

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1 ADA AFWAL-TR SELF EXCITED OSCILLATIONS IN JOURNAL BEARINGS Wilbur L. Hankey Aerodynamics and Airframe Branch Aeromechanics Division March 1983 Final Report for Period June September 1981 Approved for public release; distribution unlimited FLTGHT nyviam4cs LfABCRATODv AF wqln"t AFOONAUICrAL LA0i-Arn;I.1cS WVIfG4vrPAVTrP 3N ftfp, OHI AIRFO~ Y~~''~rft4la)8 3 07~

2 NOTICE When Government drawings, specifications, or other data are used for any purpose other than in connection with a definitely related Government procurement operation, the United States Government thereby incurs no responsibility nor any obligation whatsoever; and the fact that the government may have formulated, furnished, or in any way supplied the said drawings, specifications, or other data, is not to be regarded by implication, or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture use, or sell any patented invention that may in any way be related thereto. This report has been reviewed by the Office of Public Affairs (ASD/PA) and is releasable to the National Technical Information Service (NTIS). At NTIS, it will be available to the general public, including foreign nations. This technical report has been reviewed and is approved for publication. WILBUR L. HANKEY LO C. KEEL, ' Project Engineer Chief, Aerodynamics & Airframe Branch FOR THE COMMANDER CJEYALIER, Col, USAF Jr, erodechanics Division "If your address has changed, if you wish to be removed from our mailing list, or if the addressee is no longer employed by your organization please notify AFWAL/FIHM, W-PAFB, OH to help us maintain a current mailing list". I Copies of this report should not be returned unless return is required by security considerations, contractual obligations, or notice on a specific document.

3 UNCLASS I FIED SECURITY CLASSIFICATION OF THIS PAGE (When Veto Entered) READ INSTRUCTIONS REPORT DOCUMAENTATION PAGE BEFORE COMPLETING FORM 1REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER 4. TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED Self Excited Oscillations in Journal Bearings June 1981-Sep 1981 S. PERFORMING O~tG. REPORT NUMBER 7. AU THOR(&) S.I CONTRACT OR GRANT NUMBER(&) Wilbur L. Hankey 9 PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK Aerodynamics & Airframe Branch AREA & WORK UNIT NUMBERS Aeromechanics Division 2307N603 I. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Flight Dynamics Laboratory (AFWAL/FIM) 12 March 1983 AF Wright Aeronautical Laboratories, AFSC 13. NUMBER OF PAGES Wright-Patterson AFB, OR MONITORING AGENCY NAME & ADDRESS(iI different from Controlling Offi'ce) IS. SECURITY CLASS. (of this report) 16. DISTRIBUTION STATEMENT (of this Report)I Anproved for public release- distribution unlimited. UNCLASSIFIED 15s. DECLASSIFICATION DOWNGRADING SCHEDULE 17. DISTRIBUTION STATEMENT (of the abstract ent ered in Block 20, if different from Report) IS. SUPPLEMENTARY NOTES III KEY WORDS (Continue on reverse side if necessary and identify by block number) Self-Excited Oscillations Lubrication Theory Bearing Technology Fluid Mechanics 20. ABSTI(ACT (Continue on reverse side If necessary and Identify by block number) UXnder certain rotation rates, a rotor, supported by an oil-lubricated journal bearing, can experience excessive shaft vibration. Two types of oscillations can exist called oil whirl and oil whip, Oil whirl produces a circular orbit in a direction of the same sense as the shaft rotation but at a value of about 45% of the shaft rotation speed. Oil whip is a more violent oscillation which occurs at the resonance frequency of the shaft system. Both of these oscillation are predicted in this regor b tanin2 an analytic DD I JA, EDTONO NOV S5 IS OBSOLETE UNCLASSIFIED I SECURITY CLASSIFICATION OF THIS PAGE 'Whein Data Entered)

4 I1NCLAqSIVIED SECURITY CLASSIFICATION OF THIS PAGE(Iften Data Enterod) -description of the fluid forces and inserting these into the dynamic equations for the rotor system. Oil whl is shown to be an auto rotation, fl, of the journal centroid at a rate of one half of the shaft rotational speed. Oil whip 4q shown to occur when the frequency of autorotation exceeds the natural freq~uency of the shaft system. i.e. Q2 > 1k/. UNCLASSIFIED SECURITY CLASSIFICAIOW4 OF T~PAGE'Wen Date Enterod)

5 FOREWORD This report is the result of research performed in the Computational Aerodynamics Group, Aerodynamics and Airframe Branch, Aeromechanics Division, Flight Dynamics Laboratory from June 1981 to September This report was prepared under Work Unit 2307N603 "Computational Fluid Dynamics". Dr Wilbur L. Hankey is the Task Manager. The author is indebted to Dr Agnieszka Muszynska for suggesting this research topic and for fruitful discussions concerning the physical phennmpua. The material presented herein is related to several other efforts ot self-excited oscillations being accomplished in the Computational Aerodynamics Group. ns r " or#

6 TABLE OF CONTENTS Section Page 1. Introduction Ii. Governing Dynamic Equations III. Governing Fluid Mechanic Equations IV. Dynamic Equations in Polar Form V. Analysis of Results VI. Conclusions References

7 LIST OF ILLUSTRATIONS Figure 1 Rotor System 2 Oil Whirl and Oil Whip 3 Journal Bearing Coordinates vi

8 LIST OF SYMBOLS a b c Cq e Fr F g h k L m n p radius of inner cylinder difference in radii between inner and outer cylinders damping coefficient rotary derivative of force coefficient eccentricity of cylinder polar coordinate components of fluid force acceleration of gravity gap thickness = b(l-e cos 8') bending spring constant of shaft width of journal bearing mass of rotor system mass unbalance moment arm pressure q rotational rate perturbation = 6-Q Q r,o t u v x,y X,Y z volume flow rate between cylinders polar coordinates time fluid velocity in 0 direction fluid velocity in r direction Cartesian coordinates Cartesian components of fluid force gap coordinate - r-a constant = -- ab(a - 2 coefficient = 2 (l-a-e 2 ) 6 radial perturbation of orbit j vil

9 LIST OF SYMBOLS (Cont'd) T fluid dynamic viscosity a paametr =(2+e-) parameter(1-27) e 6e 2 shear stress w shaft forced rotation rate autorotation rate eigenvalue of characteristic equation Viii

10 SECTION I INTRODUCTION Under certain rotation rates, a rotor, supported by an oil-lubricated journal bearing, can experience excessive shaft vibration (Fig. 1). Two types of oscillations can exist called oil whirl and oil whip (Ref 1-6). Oil whirl produces a circular orbit in a direction of the same sense as the shaft rotation but at a value of about 45% of the shaft rotational speed (Fig. 2). Oil whip is a more violent oscillation which occurs at the resonance frequency of the shaft system. Although this phenomenon has been investigated for some time and identified as a self-excited oscillation due to viscous effects, no satisfactory analytic prediction method has been developed. It is the purpose of this note to provide such an analysis. k, A.

11 SECTION II GOVERNING DYNAMIC EQUATIONS Consider an unbalanced rotor with mass, m, shaft elasticity, k, rotating at speed, w, under the influence of fluid forces, X and Y (Ref. 1, 2). Hence 2 m x + kx = m n w 2 m y + ky = m n w cos w t + X sin w t - mg + Y The classical method (Ref. 2) for describing the damping forces is x = -ck Y -c The two dynamic equations are uncoupled and can be readily solved as follows (for small damping, in which (2m2 << k where 2m x = r e cos (W t cos Wt 0o (1 -W 0 ect 2 2m nw_ y = r a e sin (W t + 4) + n22 2 sin ina t - 2 W) -W W o o 2 = k-- r and 0 are ascertained from initial conditions. 0 m 0 nm These results, however, give little insight into the cause of oil whirl or whip. The reason for the failure is believed to be the inability to adequately describe the fluid forces. Therefore another mathematical model for the prediction of these forces was developed.

12 i SECTION III GOVERNING FLUID MECHANIC EQUATIONS In this section the fluid motion between two non-concentric rotating cylinders will be analyzed to obtain the fluid dynamic force on the inner cylinder representative of a journal bearing (Fig. 3). The radius of the inner cylinder is 'a' and the radius of the outer cylinder is 'a+b', where i - 1. To analyze this problem two coordinate systems must be a defined, i.e., an inertial system which is fixed at the center of the outer cylinder (x,y) and a rotating coordinate system with a moving origin referenced to the center of the inner cylinder (x', y'). The set of polar coordinates associated with these two systems are (r,e) and (r',e') respectively. The inner cylinder is considered to be rotating at a constant value of 0 about the center of the outer cylinder with an orbit radius of eb. The incompressible Navier-Stokes equations in polar coordinates for this rotating system are as follows: au' + (r'v) = 0 where Du + uv _ 2 _u 2av = Dt r' D-- pr'we pr'e'r, - + V(V u 2 r'2ae, 2 a - 20v 2 Dv u = - p + v(v2v v 2 Dt r' par' r,2, D a a a D t + v ar' + u r'36' 2 a + a + a 2 a r,2 r ar' r,2 ae,2 Consistent with lubrication theory these equations may be greatly simplified for small gap thicknesses. 3

13 Let r' = a+z where << 1 a and Y <' 1 U where z has limits between zero and h, the gap thickness. Using the law of cosines a relationship for h may be obtained. (a+b) 2 = (be) 2 + (a+h) 2-2be (a+h) cos (-O') For the case at hand, - << 1, hence a h(b')=b (1-e cos 6') is a good approximation. An order of magnitude analysis of the Navier-Stokes equations produces the classical lubrication equations. h Q = udz = flow rate 0 dpe 3 2 ade' = - pressure gradient with boundary conditions u(e', z=o) = (w-6) a u(o', z=h) = - 6a The pressure must satisfy a periodic boundary condition. p(o') = p(e'+2tr) Integrating the pressure gradient equation twice with respect to z produces a relationship for u. d 2 U=aud' 2 + Clz + C 2 Applying the two boundary conditions for u on the cylinder walls eliminates the two integration constants. u+6a d- ---z(z-h) (z-h) 2a d '- - ( -

14 Note that from this expression the shear stress on the wall of the inner cylinder can be obtained as follows: u h dp a d' d2a h T=iz(z=O) Integrating u with respect to z produces the flow rate. Q udz ( ah - h3 d 0 or rearranging this equation. b 2dp/de' 1-a-e cos6' 12a 6) (l-e cos 0') where ab( - This equation can be integrated with respect to 0' to determine the pressure distribution around the cylinder. Ap 6' S 0 do' kd6 do' The following integral is required: (Ref. 7) 0 C (1-a-c cos 0') d -e a sin 0' + e(b-a) sin 0' (1ecs0)3 =- 2(- 2 o 2(1-2 )2(1ecsV 2 2ta +(0-e (-e 2-5/2 a) tn- tol-ei V1-c a 2- where B = 2 (l-a-e ) The periodic boundary condition on pressure, i.e., p(o') - p(o' + 27), requires that the coefficient of the last term must vanish. Therefore, O-e 2 a=0, provides a relationship between the flow rate (Q) and rotational speed (. '.

15 2(1-e 2 ) = 2 + e 22 ah( -8) Using this condition to eliminate a and 8 produces the final relationship for the pressure distribution over the cylinder (called the Harrison Equation, Ref. 8-12) )e sin 8' e -12 (- (2-e cos Ap2 b 2(2+e 2 ) ) (l-e cos 6')2 One additional integration will produce the desired relationship for the forces on the inner cylinder. 2t 27 Fr = -al S Ap cos 0' de' -al 5 T sin 6' do' = 0 o 0 21T 21T Fe = -al S Ap sin e' d6' -al T cos 6' de' = 0 where a(e) (2+e 2 ) (1 - /l-e 2 ) 6e 2 For ii << 1 a 24wia3Le! w + b - 2 b 2(2+e2) 2 a ) +a b2+ (2 )ja2 a Fe 24fpa 3 Le b b 2 (2+e2) V- 2 2 (1 - a- )] b The small term, C- in the above expression for F is the only remaining a e contribution of the shear stress integral indicating that the pressure term dominates. 6

16 Having obtained a description of the fluid forces on non-concentric rotating cylinders, we can now return to the dynamic equations for the rotor system. I.

17 SECTION IV DYNAMIC EQUATIONS IN POLAR FORM Since we found it convenient to express the fluid forces in polar co-ordinates, we choose to similarly convert the dynamic equations. Utilize the following transformation. Let x = r cos 0 ; F = X cos 6 + Y sin e r y = r sin 0; F e = Y cos e + X sin 0 The governing dynamic equations in polar form become m(r-r e ) + kr = mnu cos (0-wt) - mg sin 0 + F r 2 m(2 r + r )= -mn sin (e-ut) -mg cos e + F e These equations are non-linear and must be linearized to continue further. Consider only small perturbations in both r and r - be + 6(t) - Q + q(t) From the analysis of the fluid forces we determined the following F -0 r F0 =m be Cq (0-Q) where Q = q b0 ( a ba u 2 a 2 and C = -24ipa L q ~ 3 2 vl q (2+e2) e 2 b Note, the effect of the shear stress term (-) in the expression a for 0 reduces the value of Q to slightly below 50% of w. This result is in qualitative agreement with the experimental data (Ref. 2). Using these relationships and neglecting higher order terms produces 8

18 the following set of linear equations. k _ ~2 2 k 2 S + ( - ) 6-2Qbeq = nw cos (U-w)t-g sin 1 t+f /m-(- - f2 )be m r in 2 2Q6 + beq - (1be C )q = - nw sin (Q-w)t-g cos Q t q The right hand sides of these equations contain the forcing functions and lead to the particular solution. The homogeneous solution contains the source of the self-excited oscillation and can be obtained as follows: Let and be = Ae~t Be Setting the right hand sides to zero we obtain B X +( _S2) -2x A A-S2C q. This produces a cubic characteristic equation. 3_ 2 k( Q2 C k - Q 2, =0 X Q~ C X + (I - q m q m Routh's discriminant states that no positive real root exists provided all coefficients and determinants of this equation are positive. Hence, stable motion occurs provided 1. C q < 0 2. _k 2 >0 m I.

19 SECTION V ANALYSIS OF RESULTS In the previous section two conditions were derived for stable motion of the journal bearing. If the two conditions are satisfied then a steady autorotation will persists at 6=Q=0.5 w and r-be, producing a circular orbit. This motion is descriptive of oil whirl for which experimental evidence (Ref. 2) indicates a stable autorotation at about 6: 0.45 w. Our predictions indicate that C is always negative. q 3 24njia L C q= < 0 mib (2+e 2 ) e 2 Hence, the first condition is always satisfied. Stable rotation is a consequence of the fact that df 0 /do - C is negative, forcing the rotation rate q to return to the autorotation value (P = ) if perturbations occur on either side of this value. Also equilibrium is attained at zero torque or F, = 0 which occurs when 0 =. Also, experiments show that oil whirl occurs only if Q < Fk (Fig. 2) which verifies the second condition. m If the second condition is violated i.e., Q > A, structural diverm gence of the shaft occurs in which the centrifugal force of the whirl becomes greater than the elastic restoring force of the shaft. ihe eigenvalue, X, becomes positive indicating a departure solution. The journal deflection will become excessive and strike the bearing wall during rotation creating unusual journal position patterns. This motion at the divergence state is descriptive of oil whip. 10

20 SECTION VI CONCLUSIONS Self-excited oscillations in journal bearings called oil whirl and oil whip have been predicted by obtaining an analytic description of the fluid forces and inserting these into the dynamic equations for the rotor system. Oil whirl is shown to be an autorotation, Q, of the journal centroid at a rate of one half the shaft rotational speed. This value is in good agreement with experiment. Stable rotation is a consequence of the fact that df /de is negative forcing the rotation rate to return to 0 the autorotation value () if perturbations occur on either side of this value. Oil whip is shown to occur when the frequency if autorotation exceeds the natural frequency of the shaft system, i.e. 0 > A m At this value structural divergence occurs and the centrifugal force exceeds the elastic restoring force. Positive eigenvalues indicate departure solutions which are in agreement with experiment. Although simplifying approximations to the Navier-Stokes equations produced satisfactory results, it is recommended that additional research be accomplished to assess the consequences of this mathematical model. II.

21 REFERENCES 1. Muszynska, A., "On Rotor Dynamics, (Survey)". Nonlinear Vibration Problems, Zag adnienia Drgan Nieliniowcyh, Muszynska, A., "Dynamic Analysis in Turbomachinery Design", University of Wisconsin Report, Jan Robertson, D., "Whirling of a Journal", Phil Mag 15.1, p. 113, Cameron, A., "Oil Whirl in Bearings". Engineering, 179, p. 237, London, Shawki, G.S.A., "Whirling of a Journal Bearing", Engineering 179, p. 243, Partitsky, H., "Contribution to the Theory of Oil Whip", Trans ASME 75, p. 1153, Bois, P., "Tables of Indefinite Integrals", Dover Publications, Inc., Harrison, W.J., Trans Camb Phil Soc, Vol 23, p. 373, Freeman, P., "Lubrication and Friction', Pitman & Sons, Ltd, Smith, D.M., "Journal Bearings in Turbomachinery", Chapman and Hall, Ltd., Slaymaker, R.R., "Bearing Lubrication Analysis", Wiley & Sons, Inc., Tipei, N., 'Theory of Lubrication", Stanford University Press,

22 I-HA~0V oournal BEARING Figure 1. ROTOR SYSTEM 13

23 (DYNAMIC / I WI MOTION, rpm) F- WO (ROTOR pm) Figure 2. OIL WHIRL AND OIL WHIP 14

24 y OIL. BEARING Figure 3. JOURNAL BEARING COORDINATES 11