rub orbit chaos

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Full Rubs, Bouncing and Quasi Chaotic Orbitsin Rotating Equipmentby F. K. CHOY J. PADOVAN and J. C. YUDepartment of Mechanical Engineering, University of Akron, Akron, OH 44313,U.S.A.

ABSTRACT : In order to improve performance, closer tolerances are usually required in highperformance turbomachinery. This often results in an increase in rotor casingJrea1 rubsensitivities. The objective of this paper is to investigate the eflects ofmass,support sti@essand blade sttjrness during a rotor-casing rub event. Special emphasis will be given to de ningthe tuningldetuning effects of the system during such variations in mass and sttj%ess. Theoverall model will incorporate the influence of: (i) casing and rotor inertia, (ii) casing androtor support stt@tess, (iii) contact friction induced during rub interaction, (iv) single andmultiple blade contact, as well as v> lateral and radial blade st@ness eflects. The main thrustwill be to investigate the dtyerent regimes of rubbing, i.e. the development of full rubs, rigidbouncing and essentially chaotic behavior due to the changes in rotor-to-casing mass andsttjrness ratios, as welI as blade sttjrness andfriction efSects.

NomenclatureCCCEe,(f;,) cG)c(.L) RF)nP;;,(F,)c(F,.),(FJR(F,)RIKCKIm,m,(C)c(t)c

casing damping coefficientshaft equivalent damping coefficientblade Youngs modulusrotor eccentricityrub forces exerted in the casing radial directionrub forces exerted in the casing transverse directionrub forces exerted in the rotor radial directionrub forces exerted in the rotor transverse directionnormal rub forcetangential friction force between the tip and the casingcasing external load in X-directioncasing external load in Y-directionrotor external load in X-directionrotor external load in Y-directionmoment of inertiastiffness of casing supportshaft equivalent stiffnessblade radial lengthcasing massrotor massunit vector for casing radial directionunit vector for casing transverse direction

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F. K. Choy et al.unit vector for rotor radial directionunit vector for rotor transverse directionrotor center angular precession accelerationphase angle between timing mark and rotor mass centerblade radial deflectioncasing motion vectorrotor motion vectormass center phase anglefrictional coefficient between blade tip and casingcasing precession angleangle between casing motion vector and relative motion vectorrotor center precession angleangle between rotor motion vector and relative motion vectorrotational speed of rotor

I. IntroductionIn recent years increased attention has been given to analysing the dynamics of

rub interactions in high performance rotating equipment, i.e. turbines, pumps etc.To improve performance, closer tolerance is typically required. This has led toincreased rub sensitivities. In studies to date prototypically the rub interactionmodels have either considered the casing as rigid (l-lo), or as purely an elasticallysupported structure (6,11,12). Such assumptions do not account for the potentialtuning/detuning effects of casing inertia.

In the context of the foregoing, this paper will consider the effects of casinginertia on the overall rotor-bladecasing interaction problem. Special emphasis willbe given to defining the tuning/detuning effects. Overall, the model will incorporatethe influence of

(i) casing and rotor inertia,(ii) casing and rotor support stiffness,

(iii) contact frictions induced during rub,(iv) single and multiple blade contact, as well as(v) lateral and radial blade stiffness effects.

To gain a thorough understanding of the overall influence of casing mass, awider ranging parametric study is presented. This considers system sensitivity tovariations in

(i) rotor/casing mass ratio,(ii) rotor/casing stiffness and

(iii) blade stiffness and friction effects.Special emphasis will be given to ascertain parameter ranges wherever full rubs,rigid bouncing and essentially chaotic behavior are noted.

In the sections which follow, detailed discussions are given on(0 model development,(ii) solution procedure and

(iii) the system response to wide-ranging parameter variations.

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Chaotic Behavior in Rotating EquipmentI I . Equations of Motion

For the rotor/casing system shown in Fig. 1, the equations of motion can bedeveloped for both rotor and casing motion (4,5, 9) independently when no rubinteraction occurs.The rotor equation can be written in the radial and transverse coordinates shownin Fig. 2 as:(ir), direction :rn,(Jr -@S,) + C,B, + K,6, = m,eu { o(cos 9 cos Q, + sin 8 sin ar)

- a(cos 0 sin O, -sin 8 cos Qr)> + (F.JR cos @,+ (F,)R sin QD, (1)and(i,J R direction :m,(6,6, + 2&J,) + C,&,S, = m,e,(m2(sin 0 cos @, - cos 0 sin B,,)

-cz(sin6sin@,,+cos8cos@~)}-(F,),sin


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F. K. Choy et al.

FIG. 2. Rotor-bladeecasing coordinate system

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Chaotic Behavior in Rotating Equipment(iO)L.direction :

m,(6


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F. K hoy et al.I I I . Solution Procedure

In order to simulate a realistic rub event, the rotor system is assumed to beoperating a steady state orbit with some initial mass imbalance without any rubinteraction between the rotor and the casing. With a suddenly induced mass-imbalance, the rotor vibration orbit is increased and eventually rub interactionsoccur, For the purpose of capturing the transient dynamics of the system, (9))(12)are rewritten in the form (4, 5)Jr = (l/m,){m,e,[02(cos OcosQ,+sin 0 sin O,,)

- a(cos 0 sin @, sin 0 cos a,.)] + F,)R cos D,+ Fy)R sin @,+F,,cos@,,-pF,,sin@,,-C&-K,&} -t-@S,, (13)

6, = (1 /m,6,) {m,e,,[w(sin 19 os Q, - cos 0 sin 0,)- a(sin 8 sin O, + cos 0 cos


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Chaotic Behavior in Rotating Equipment

(9(ii)

(iii)(iv)(v)

(vi)

Radial Blade Deflection (nils)FIG. 3. Nonlinear blade stiffness due to single/multiple blade rub interaction.

The rotor-bearing assembly is assumed to be a Jeffcott rotor simulation,see Fig. 1,linearized effective damping and stiffness characteristics are assumed at therotor and casing geometric center,the turbine/impeller blades are assumed to be fixed at the rotor disk mount-ing (cantilevered),the effect of nonlinear blade stiffnesses are generalized to handle multipleblade participation rub interaction (Fig. 3)the casing structure is assumed to be supported by linearized stiffness anddamping characteristics which can vibrate independently of the rotormotion,the rotor assembly is assumed to operate initially at steady-state motionunder a small imbalance wherein the casing is initially at rest. An additionalmass imbalance is suddenly induced into the rotor system to excite thetransient vibration of the system.

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F. K. Clzo_vt al.With the above assumptions, the equations of motion for the analytical model arepresented in the following section.

V. Di scussionAs noted earlier, the main thrust of this paper is to establish :

(i) The tuningdetuning effects of the relative rotor/casing stiffness and massratios,

(ii) the nonlinear effects of blade stiffness, i.e. multi-blade interactions and(iii) the system parametric characteristics leading to full rubs.

For instance, Figs 46 illustrate the effects of casing stiffness. In particular, therotor, casing and relative orbits are displayed. As can be seen, as the casingis gradually stiffened, its orbit progressively shrinks. In contrast, for softeningcharacteristics, increasingly bouncier/loopier type trajectories are excited. Theseare an outgrowth of the potential excitation of subharmonics, i.e. the naturalfrequencies of some system subcomponents. For example, Fig. 7 illustrates thedevelopment of l/3 harmonic loops. These involve excitation of the casing wherethe frequency is l/3 of the rotating imbalance. Similar responses may be excited atl/2, I /4, l/S etc. . . of the rotating speed.

Continuing, the tuning and detuning effects of mass variations are depicted inFigs S-10. Three basic response types are excited, i.e.

(i) full rubs for very light casings,(ii) rigid bouncing for very heavy casings and

(iii) essentially chaotic behavior for intermediate values.Three-dimensional views of the stiffness and mass effects on the steady-state rub

force, blade stress and blade deflection are depicted in Fig. 11. Note that for lowcasing to rotor mass ratios, significantly lower steady-state response levels areattained. As the mass ratio is increased, significantly higher steady-state values areexcited. Similar trends are also depicted for the transient -three-dimensional plotsgiven in Fig. 12. Here the various isolated peaks are a result of the tuning anddetuning of the various system component frequencies.

Next, we consider the generation of continuous and intermittent rubs. These areillustrated in Figs 13-l 5. Figures I3 and 14 depict the effects of blade stiffness onthe generation of rubs. As can be seen, blade stiffness is reduced when the inter-mittent rub is replaced by a full rub. Similar trends are noted as the casing to rotormass ratio is raised. Here the response varies between full rub to highly intermittenttrajectories, see Fig. 15.

Lastly, we shall consider the nonlinearity effects induced by multiple blade rubs.Note Fig. 13, it follows that transitions from single to multiple blade rubs aremarked by discontinuous load deflection behavior. In particular, the response ishighly stiff at transitions and gradually softens until the next transition occurs.Such behavior leads to a saturation of peak loads. This is clearly seen in Figs 16and 17 where a distinct saturation is noted in the various responses.

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0.004 1 Kc = 2000 lb/in. 0.004 K = 5000 lb/in.

-0.004 - -0.004

T-0.004 0.004 -0.004 0.004

Kc - 200000 lb/in.

I- -0.004 0.004 -0.004 0.004ROTOR ORSILT IN INCHES

FIG 4. Rotor motion orbit trajectories with change in casing stiffness.

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F. K. Choy et al.

0.003 - Kc = 2000 lb/in. 0.003.

0.003

-0.003 -0.003

Kc = 5 lb/in. 0.003 Kc = 200000 lb/in.

-0.003 0.003

P

-0.003 0.003

CASING ORBIT IN INCHES

FIG. 5. Casing motion orbit trajectories with change in casing stiffness.

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0.004

-0.004

0.004

-0.004

Kc = 2000 lb/in. 0.004. Kc = 5000 lb/in.

-0.004 1F

-0.004 0 004 -0.d04 0.004Kc = 50000 lb/in. 0.004 K = 200000 lb/in.

-0.004

-0.006 0.006 +-0.004 0.004

RELATIVE ORBIT IN INCHES

FIG. 6. Rotor-casing relative motion orbit trajectories with change in casing stiffness.

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F. K. Choy et al.

0.004

-0.004

0.004

-0.004

Kc = 5000 lb/in

c-4d

-0.004 0.004 0

Kc = 2000 lb/in.

-0.004 0.004


Ilil.06MAXIMUM BLADE STRESS

FIG. 7. Relative orbit and blade stress with change in casing stiffness.

L0.06

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0.004

-0.004

0.004

-0.004

WC = 64 lb 0.004 WC = 16 lb

-0.004

-0.004 0.004

WC = 8 lb O.QO4

-0.004

-0.004 0.004

-0.004 0.004

-0.004 0.004

ROTOR ORBIT IN INCHES

FIG 8 Rotor motion orbit trajectories with change in casing mass.

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F. K. Choy et al.

0.003

-0.003

4-0.003 - -0.003

T F-0.003 0.003 -0.003 0.003

0.003 - WC 64 b 0.003 WC = 16 lb

W -8lb 0

-0.003 0.003

.003

-0.003 0.063

CASING ORBIT IN INCHES

FIG. 9. Casing motion orbit trajectories with change in casing mass.

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0.004

-0.004

0.004

-0.004

WC 64 lb 0.004 WC 16 lb

-0.004

-0.004 0.604

WC = 8 lb 0.004 wc 2 lb

-0.004

-0.004 0.004

-0.004 0.004

-0.004 0.004


FIG. 10. Rotor-casing relative orbit trajectories with change in casing mass.

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Sae

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T R

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F. K. Choy et al.

.4

0II

E = 500000000.4

0.2E = 5000000

Time n econds I

E = 3

E = 1

Txme in Seconds 0.07FIG. 13. Normalized blade stress with change in blade stiffness.

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E - 50000000 E 30000000.04 0.04

2.a 0 0.07 0 0:07

E - 5 0.2

FIG. 14. Radial blade displacement with change in blade stiffness.

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F. K. Choy et al.

0.004

WC = 0.5 lb

, _,0_.004L0.07

0.006

Time in Seconds

-0.004

0.0041C = 8 lbI0

-0.004 0.004

-0.bo4 0.004

-0.004 0.004Relative Orbit i n Inches

FIG. 15. Radial blade displacement and rotor relative orbit with change in casing mass.

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0Time n Seconds

FIG. 16. Nonlinearity between blade radial displacement, radial rub force and maximumblade stress.

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E = 1000000

E = 1000000

Time in Seconds

FIG. 17. Nonlinearity between blade radial displacement and maximum blade stress duringa full rub event.

VI . ConclusionsOverall, the foregoing case studies lead to the following conclusions concerningthe influence of rotor/casing/blades and stiffness and associated mass effects

namely :(1)(2)3)

4)5)

Large nonlinearity due to rub interactions is induced by the discontinuities dueto single/multiple blade participation rub interactions.The decreasing in casing stiffness will result in larger and bouncier/loopierrotor orbit trajectories. (The increase of subharmonic orbit components.)Rotor orbit trajectories can shift from rigid bouncing to chaotic behavior thento full rub phenomena with decrease of casing stiffness for very lightweight-casing system.Level of rotor steady response will decrease with the decrease of casing mass.The system can shift from intermittent behavior to full rub under similaroperating conditions with either a decrease in blade stiffness or a decrease incasing mass.

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Chaotic Behavior in Rotating EquipmentReferences

1) R. F. Beatty, Differentiating rotor response due to radial rubbing, Trans. ASME,J. Vibration Acoust. Stress Reliability Des., Vol. 107, p. 151, Apr. 1985.

(2) D. E. Bently, Forced Subrotative Speed Dynamic Action of Rotating Machinery,ASME paper 74-DET-16, Petroleum, Mech. Engng Conf., Dallas, TX, 1979.(3) D. W. Childs, Rub induced parametric excitation in rotors, ASME. Mech. Des.,

Vol. 10, p. 640, 1979.(4) F. K. Choy and J. Padovan, Investigation of rub effects on rotor-bearing-casing

system response, Proc. 40th Mech. Failure Prevention Group Symp. Nat1 BureauStandards, Gaithersburg, MA, Apr. 1985.

(5) F. K. Choy and J. Padovan, Nonlinear transient analysis of rotor-casing rub events,J. Sound Vibration, Vol. 113, No. 2, 1987.

(6) F. K. Choy, J. Padovan and W. Li, Rub in high performance turbomachinery :modelling; solution methodology; and signature analysis, J. Mech. Syst. SignalProcess., Vol. 2, No. 2, p. 113, 1988.

(7) E. F. Ehrich, The dynamic stability of rotor/stator radial rubs in rotating machinery,ASME J. Engng Ind., p. 1025, Nov. 1969.

(8) A. Muszynska, Partial Lateral Rotor to Stator Rubs, ZMech. E. Conf. Publicationon Third Int. Conf. on Rotating Machinery, Sept. 1984.

(9) J. Padovan and F. K. Choy, Nonlinear dynamics of rotor/blade/casing rub inter-actions, ASME J. Turbomachinery, Vol. 198, No. 4, p. 527, Oct. 1987.

(10) A. Tondl, Note on the identification of subharmonic resonances of rotors. J. SoundVibration, Vol. 3 1, p. 119, 1973.

11) Y. Choi and S. T. Noah, Nonlinear steady state response of a rotor support system,ASME J. Vibration Acoust. Stress Reliability Des., Vol. 109, p. 255, July 1987.

(12) H. D. Nelson, W. L. Meacham, D. P. Fleming and A. F. Kascak, Nonlinear analysisof rotor-bearing systems using component mode synthesis, ASME J. Engng Power,Vol. 105, p. 606, July 1983.

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rub orbit chaos

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