stability performance comparison of rotor-bearing systems

International Journal of Rotating Machinery 2005:1, 16–22c© 2005 Hindawi Publishing Corporation

Rotor-Bearing System Stability Performance ComparingHybrid versus Conventional Bearings

Julio Gomez-MancillaLaboratorio de Vibraciones & Rotodinamica ESIME, Instituto Politecnio Nacional, Edificio 5, 3er Piso, Colonia Zacatenco,07300 Mexico, Distrito Federal, MexicoEmail: [email protected]

Valeri NosovLaboratorio de Vibraciones & Rotodinamica ESIME, Instituto Politecnio Nacional, Edificio 5, 3er Piso, Colonia Zacatenco,07300 Mexico, Distrito Federal, MexicoEmail: [email protected]

Gerardo Silva-NavarroSeccion de Mecatronica, Centro de Investigacion y de Estudios Avanzados del IPN, A.P. 14-740, 07360 Mexico,Distrito Federal, MexicoEmail: [email protected]

Received 1 May 2004

New closed-form expressions for calculating the linear stability thresholds for rigid and flexible Jeffcott systems and the imbalanceresponse for a rotor supported on a hybrid bearing are presented. For typical bearings characteristics, expressions yield stabil-ity thresholds practically equal to those reported by Lund (1966). The hybrid bearing design has a single injection port whoselocation is so chosen to stabilize the bearing performance and to reduce the steady equilibrium attitude angle. Rotordynamics co-efficients graphs for conventional and pressurized bearings, as functions of bearing equilibrium eccentricity and/or Sommerfeldnumber, are presented. Using the rotordynamics coefficients into the expressions for the corresponding velocity thresholds andthe imbalance response, the system stability and vibration performances are estimated and analyzed. When comparing the Jeffcottflexible shaft supported on two journal bearings of the conventional type with the hybrid type, the results show a clear superiorityof the pressurized design as far as stability behavior is concerned. Specifically for cases of flexible shafts with similar character-istics to those used in industry, the analysis shows that this design yields velocity thresholds 25%–40% higher compared to theconventional circular ones. Also this bearing displays nonlinear feeding pressure behavior, and it is capable of reducing the syn-chronous vibration amplitude in most speed ranges, except around the critical speed; moreover, for certain Jeffcott configurationsthe amplitude reduction can be substantial.

Keywords and phrases: Jeffcott rotor, hybrid bearing, imbalance response, pressurized rotordynamics coefficients, stabilitythreshold.

1. INTRODUCTION

By hybrid we mean a circular journal bearing that workswith both, a hydrodynamic wedge plus a lubricant injectedat substantial pressure. Bently et al. [1, 2] proposed a hy-brid bearing design capable of removing the oilwhirl bear-ing instabilities. As presented by Kucherenko and Gomez-Mancilla [3], turbomachinery seals located near the midspan

This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

exert a great dynamic influence in spite of their small values.In [3] the equations of motion using dimensionless parame-ters are developed, allowing us to notice how small changeson the stiffness largely affect the stability properties. Oil-whirl is a milder instability problem in comparison to oil-whip and steamwhirl, all occurring at subsynchronous vi-bration frequencies. In general a moderate nonlinear bearingforce is enough to control growth of oilwhirl vibration am-plitudes, but this is not so for the other two instabilities. Onthe other hand, acting at a susceptible location by producinga relatively small change in midspan effective stiffness Bentlyet al. [2] has put up a rotor-bearing-seal system capable

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Stability Comparison of Rotors on Hybrid-Conventional Bearings 17

of postponing the oilwhip instability. Therefore, location ofthe active control element is of fundamental importance toachieve high dissipating efficiency, that is, the squeeze filmdampers are conveniently located at strategic points. Al-though bearing locations near the rotor ends are not optimal,here shaft ends are bearing locations used to estimate systemstability performance.

Recently some interesting bearing design capable ofchanging effectively the stability and behavior of supportedrotors have been proposed by Bently et al. [1, 2], Gomez-Mancilla [4], Dimofte and Hendricks [5], just to mentionthree works. The rotor-bearing stability is considerably in-fluenced by the fluid film, modeled by the rotordynamics co-efficients, which in turn are functions of machine operationspeed, lubricant viscosity temperature, and so forth. In theLaboratorio de Vibraciones & Rotodinamica ESIME, ESIME-IPN, it has been noted that rotor-bearing instabilities can becontrolled and the operational stable range and the vibrationperformance can be substantially improved by incrementingthe lubricant pressure supply in a bearing. Therefore, ade-quate changes of the rotordynamics coefficients, caused bythe lubricant pressure supply increment, will favorably post-pone the stability threshold of a machine, sometimes in aconsiderable manner (Ordonez-Pantoja [6]). Briefly it canbe mentioned that there exist other bearing designs such asthe tilting pad, the multilobe, and elliptic bearings, all ofwhich are intrinsically more stable when compared to thecircular one studied here. See, for instance, [7] for stabil-ity performances of circular, four-lobe, and tilting pad bear-ings. Nevertheless this study does not only analyze stabil-ity issues but also compares rotor-bearing synchronous vi-bration responses, where this hybrid design performs quitewell.

In this paper, a formula for the calculus of the stabilitythreshold, which is obtained through the Lienard-Chipartcriterion is used (Antonio-Garcia et al. [8] ; Afanasiev et al.[9]; Demidovich [10]). The geometric configuration of thebearing shown in Figure 1a provides the same benefits as theconventional lubricant pressure supply required to achievethe classical supply lubrication. For the infinitely rigid shaft,the hybrid bearing leads to a very high increase in the speedthreshold and a significant but moderate improvement as theshaft flexibility increases. Hence the advantage of modifyingthe bearing can be seen in improving the stability thresh-old.

2. ROTORDYNAMICS COEFFICIENTS IN CASEOF LUBRICANT INJECTION

The physical configuration is shown in Figure 1a, wherethe lubricant injection port is located on the upper top ofthe bearing. The purpose of this design is to exert a pres-sure downward force especially useful at high Sommerfeldnumbers, when the attitude angles are particularly large.There is no danger in reducing the minimum oil film since acircular bearing is generally more unstable at small eccentric-ities and large attitude angles, a situation which is correctedby the injection pressure force.

45◦ 45◦

Fr

X

Y

(a)

Peripherial coordinate X

Axi

alY

Maximum pressure = 610 451.6

(b)

Figure 1: The analyzed bearing: (a) physical description and geo-metrical configuration and (b) typical pressure field used to com-pute coefficients; notice the feed port pressurized area at Ps = 10.

The rotordynamics coefficients (ki j , ci j) for Figure 1aconfiguration are computed using the CHUMA software de-veloped by Gomez-Mancilla and Dimarogonas [11] for fixedmultipad journal bearings with or without preload. CHUMAsolves numerically the Reynolds equation, estimating theequilibrium point and corresponding pressure field usingthe Stieber-Swift boundary conditions, then computes bear-ing reaction forces and expanding them into Taylor seriesobtains linear and nonlinear bearing stiffness and dampingcoefficients. An incompressible lubricant and a groove depthsignificant enough to maintain a constant pressure zone inthe port area is assumed as illustrated in Figure 1b.

The dimensionless increase of bearing pressure aboveclassical supply is expressed by

Ps = psP

, (1)

18 International Journal of Rotating Machinery

where ps denotes the pressure supply due to additional lubri-cant injection in the bearings and P = F0/LD is the bearingstatic pressure load, where F0 is the bearing static load. Typ-ical pressure supply values Ps are 0 (classical supply withoutadditional pressurization), 3, 6, and 10. The value ps = 0Prefers to the case of maintaining a sufficient lubricant sup-ply to keep a constant fluid film between the journal and thebearing.

Program outputs include the stiffness ki j and damping ci jcoefficients in the dimensionless form, which are defined asfollows:

ki j = Cr

F0ki j , ci j = ω Cr

F0ci j , (2)

where Cr denotes the bearing radial clearance and ω the op-eration angular speed.

Figures 2 and 3 illustrate different behavior for the rotor-dynamics coefficients presented as functions of the Sommer-feld number and using the coordinates (x, y) in Figure 1a.Due to the existence of a negative portion in the kxy curve(printed in dashed line), absolute values for this coefficientare plotted. In Figure 3a it is important to notice that the kxyaxis crossing, from negative to positive values, occurs at sig-nificantly larger Sommerfeld numbers, a situation which canbe interpreted as beneficial for bearing stability.

3. STABILITY THRESHOLD FOR RIGID ROTORS

The Jeffcott model equations describe the motion of a sym-metrical rotor-bearing system with a 2m disc mass located inthe middle of the shaft and supported on two hydrodynamicbearings on each rotor end. The shaft can be rigid or flexible.The Jeffcott rotor equations can be presented in dimension-less form as follows:

p2{X ′′

Y ′′}

+[Ci j]{X ′

Y ′}

+[Ki j]{X

Y

}= p2a

Cr

{cos τsin τ

}, (3)

where p2 = Crmω2/F0, X = X/Cr , Y = Y/Cr , ω is the angularspeed, Cr is the bearing radial clearance, and a is the discresidual imbalance. The variables with quoted marks denotederivatives with respect to the dimensionless time τ = ωt.The corresponding characteristic equation for the system (3)is given by the fourth-order polynomial

P4(λ) = λ4 +Σ(c)p2

λ3 +p2Σ(k) + ∆(c)

p4λ2 +

∆(c, k)p4

λ +∆(k)p4

,

(4)

where

Σ(c) = cxx + cyy , Σ(k) = kxx + kyy ,

∆(c) = cxxcyy − cyxcxy ,

∆(k) = kxxkyy − kyxkxy ,

∆(c, k) = cxxkyy + cyy kxx − cxykyx − cyxkxy.

(5)

10110010−110−2

Sommerfeld

10−3

10−2

10−1

100

101

102

ki jci j

kyx

cyxcxy

kxykxy

L/D = 1Y

X

(a)

10110010−110−2

Sommerfeld

10−1

100

101

102

kiicii

kxx

kyy

cyy

cxx

L/D = 1Y

X

(b)

Figure 2: Stiffness and damping rotordynamics coefficients ofcylindrical bearing for Ps = 0, equivalent to conventional circularjournal bearings: (a) cross-coupled coefficients and (b) direct coef-ficients.

The characteristic polynomial (4) can be rewritten as

P4(λ) = λ4 + b1λ3 + b2λ

2 + b3λ + b4, (6)

where

b1 = Σ(c)p2

, b2 = p2Σ(k) + ∆(c)p4

,

b3 = ∆(c, k)p4

, b4 = ∆(k)p4

.

(7)

The necessary and sufficient conditions for stability areobtained from the Lienard-Chipart criterion (Afanasiev etal. [9]; Demidovich [10]). In particular, for the characteristic


10110010−110−2

Sommerfeld

10−2

10−1

100

101

102

103

ki jci j

kyx

kyx

kxy

kxy

cyxcxy

L/D = 1Y

X

(a)

10110010−110−2

Sommerfeld

100

101

102

103

kiicii kxx

kyy

kxx

kyy

cyy

cxx

cyy

cxx

L/D = 1Y

X

(b)

Figure 3: Stiffness and damping rotordynamics coefficients of hy-brid bearing design for Ps = 10: (a) cross-coupled coefficients and(b) direct coefficients.

polynomial P4(λ) in (6) the necessary and sufficient condi-tions for stability are that all its coefficients bi > 0, i = 1, . . . , 4and

∆1 =∣∣b1∣∣ > 0, ∆3 =

∣∣∣∣∣∣∣b1 1 0b3 b2 b1

0 b4 b3

∣∣∣∣∣∣∣ > 0, (8)

where | · | denotes the determinant function. The square ofthe stability threshold p2

RS,thr for a rigid shaft configuration,obtained through the Lienard-Chipart criterion (8), is ob-tained as

p2RS,thr =

∆(c)∆(c, k)Σ(c)∆2(c, k) + ∆(k)Σ2(c)− Σ(k)∆(c, k)Σ(c)

. (9)

0.90.80.70.60.50.40.30.20.1

Eccentricity

100

101

102

p2 RS,

thr

Stable

Unstable

Ps = 10

Ps = 6

Ps = 3

Ps = 1

(a)

0◦30◦

60◦

90◦

1

0.8

0.6

0.4

0.2

Eccentricity

Attitude angle

0P, S = 0.14160P, S = 0.28333P, S = 0.14163P, S = 0.2833

6P, S = 0.14166P, S = 0.283310P, S = 0.141610P, S = 0.2833

(b)

Figure 4: Rigid shaft stability behavior for conventional and hybriddesigns: (a) p2

RS,thr for different lubricant pressure supplies and (b)equilibrium locus for S = 0.1416, 0.2833 where eccentricities andattitude angles for four injection pressures are indicated.

Figure 4a illustrates a very significant improvement onthe threshold as the feed pressure is increased from Ps = 0to Ps = 10, nearly one order of magnitude for p2

RS,thr. In ad-dition, Figure 4b clearly indicates that increasing of injectionpressure effectively reduces the equilibrium attitude angle,estimated at two different Sommerfeld numbers, a featurecharacterizing stability.


4. STABILITY THRESHOLD FOR FLEXIBLEROTORS

The Jeffcott model with shaft flexibility and mass imbalanceat the disc is obtained as

mx + k(x − x0

) = maxω2 cosωt −mayω

2 sinωt,

my + k(y − y0

) = maxω2 sinωt + mayω

2 cosωt,(10)

kxxx0 + kxy y0 + cxxx0 + cxy y0 = k(x − x0

),

kyxx0 + kyy y0 + cyxx0 + cyy y0 = k(y − y0

),

(11)

where the coordinates (x, y) and (x0, y0) are the disc cen-ter mass and the bearing center mass displacements, respec-tively; k = K/2, with K describing the shaft stiffness; ax, ayare components of the disc mass imbalance, and ω is shaftangular speed. Equations (10)-(11) in dimensionless coordi-nates are rewritten as

p2x′′ +1γ

(x − x0

) = p2[(

axcr

)cos τ −

(aycr

)sin τ

],

p2 y′′ +1γ

(y − y0

) = p2[(

axcr

)sin τ +

(aycr

)cos τ

],

(12)

kxxx0 + kxy y0 + cxxx′0 + cxy y

′0 =

1γ

(x − x0

),

kyxx0 + kyy y0 + cyxx′0 + cyy y

′0 =

1γ

(y − y0

),

(13)

where x = x/Cr , y = y/Cr , x0 = x0/Cr , and y0 = y0/Cr

are dimensionless displacements, γ = δ/Cr is the dimension-less static deflection with respect to a shaft midspan static de-flection δ = F0/k. The characteristic equation of the system(12)-(13) is the sixth-order polynomial

λ6γ2∆(c) + λ5[γΣ(c) + γ2∆(c, k)]

+ λ4[

1 + γ2∆(k) + γΣ(k) + 2γ∆(c)p2

]

+ λ3[Σ(c)p2

+ 2γ∆(c, k)p2

]

+ λ2[Σ(k)p2

+∆(c)p4

+ 2γ∆(k)p2

]

+ λ∆(c, k)p4

+∆(k)p4

= 0,

(14)

where notations (5) are used. This polynomial can be ex-pressed as

P6(λ) = b0λ6 + b1λ

5 + b2λ4 + b3λ

3 + b4λ2 + b5λ + b6,

(15)

where

b0 = γ2∆(c), b1 = γΣ(c) + γ2∆(c, k),

b2 = 1 + γ2∆(k) + γΣ(k) + 2γ∆(c)p2

,

b3 = Σ(c)p2

+ 2γ∆(c, k)p2

, b4 = Σ(k)p2

+∆(c)p4

+ 2γ∆(k)p2

,

b5 = ∆(c, k)p4

, b6 = ∆(k)p4

.

(16)

Application of the Lienard-Chipart criterion to the char-acteristic polynomial (15) leads to the necessary and suffi-cient conditions for stability of the system, that is, all its co-efficients bi > 0, i = 0, . . . , 6 and

∆1 =∣∣b1∣∣ > 0, ∆3 =

∣∣∣∣∣∣∣b1 b0 0b3 b2 b1

b5 b4 b3

∣∣∣∣∣∣∣ > 0,

∆5 =

∣∣∣∣∣∣∣∣∣∣∣

b1 b0 0 0 0b3 b2 b1 b0 0b5 b4 b3 b2 b1

0 b6 b5 b4 b3

0 0 0 b6 b5

∣∣∣∣∣∣∣∣∣∣∣> 0.

(17)

Therefore the square of the stability threshold p2FS,thr for the

rotor-bearing system that considers the shaft flexibility isgiven by (Bently et al. [2]; Antonio-Garcia et al. [8])

p2FS,thr =

∆(c)∆(c, k)Σ2(c)A + ∆2(c, k)Σ(c) + ∆(k)Σ3(c)− Σ(k)∆(c, k)Σ2(c)

,

(18)

where

A = γ[∆3(c, k) + ∆(c, k)∆(k)Σ2(c)− ∆2(c, k)Σ(c)Σ(k)

].

(19)

It is important to note that, when the shaft flexibility is set tozero (γ = 0), then p2

FS,thr becomes p2RS,thr, which corresponds

to the stability threshold for rigid shafts.Figure 5 shows the stability results for both cases. In

Figure 5a the conventional circular bearing rotordynamicscoefficients for the long bearing are used, and in Figure 5bthe hybrid case with pressure supply Ps = 10. Here, squareof stability threshold values are plotted against the bearingequilibrium eccentricity for different shaft flexibility values(γ), considering the previous rotordynamics coefficients. No-tice that for γ = 0.1 the improvement is dramatic, whilefor γ = 4 the efficiency of the hybrid bearing is much di-minished; yet the benefit that pressurized lubricant supplyachieves can be highly appreciated for stabilization and con-trol purposes.

Finally, the steady state imbalance responses of the Jef-fcott rotor supported on conventional and hybrid bearingsfor different pressure supplies are shown in Figure 6. Both


0.80.70.60.50.40.30.20.10

Eccentricity

10−1

100

101

102

p2 FS,t

hr (Rigid rotor)

0P

γ = 0

γ = 0.1γ = 1

γ = 2

γ = 4

γ = 6

(a)

0.90.80.70.60.50.40.30.20.10

Eccentricity

10−3

10−2

10−1

100

101

102

103

p2 FS,t

hr

(Rigid rotor)

10P

γ = 0γ = 0.1

γ = 1γ = 2γ = 4γ = 6

(b)

Figure 5: Stability threshold curves p2FS,thr for different shaft flexibil-

ities γ: (a) conventional circular bearing (0P, infinitely long bearingcoefficients) and (b) hybrid, for pressure supply Ps = 10.

system configurations are identical except in the disc masswhere Figure 6b has twice the mass as compared to Figure 6a.Observe that in both figures injecting lubricant while tres-passing the critical speed is contra-productive but after thisregion, and only in case of Figure 6a, the pressurized in-jection can achieve reductions in the absolute disc ampli-tude.

5. CONCLUSIONS

Stability characteristics of conventional circular and hybridbearing designs based on the Lienard-Chipart stability cri-terion, as far as equilibrium eccentricities and attitude an-gle are analyzed and compared. Rotordynamics coefficients

76543210

Sommerfeld

0

0.5

1

1.5

2

2.5

R/a

10P

6P3P

0P

(a)

76543210

Sommerfeld

0

0.5

1

1.5

2

2.5

3

R/a

10P6P

3P0P

(b)

Figure 6: Steady state imbalance response of a Jeffcott rotor sup-ported on hybrid bearing trespassing the critical speed: (a) semi-flexible shaft and (b) flexible shaft with γ equal to half of the case(a).

for the hybrid bearing design are numerically computed anddata are curve-fitted and input into expressions to comparestability thresholds for both, conventional circular journaland hybrid bearing designs under diverse pressure supplies.The study comprises analysis of rigid rotors and flexible Jeff-cott rotors supported on typical and hybrid bearings; thresh-olds curves for the conventional circular bearing type aremuch similar to those obtained by Lund [12]. Concerningthe bearing equilibrium locus, the eccentricity increases andthe attitude angle is reduced as the injection pressure in-creases.

Also study on the imbalance response of a simplifiedJeffcott rotor supported on conventional and this hybridbearing is performed; it is shown that the mass imbalance


vibration can be effectively reduced (Ordonez-Pantoja [6]).For this design, nonactive control and steady state vibrationin Figure 6 leads to amplitude reduction capacities (at certainspeed regions), where nonlinear behavior with respect to lu-bricant feed pressure is observed. Using the rotordynamicscoefficients fitted curves into the stability expressions herederived, as well as the imbalance response performance ofother type of bearings can be studied and evaluated.

Based on the previous expressions, in the analysis of thepresent design, and probably of other hybrid bearings (e.g.,the Servofluid), proves that a proper pressurization designlinked to an adequate vibration control system can be ap-plied to improve rotating machine performances supportedon these components.

ACKNOWLEDGMENTS

This work was partially sponsored by Consejo Nacional deCiencia y Tecnologia (CONACyT), Mexico, through researchproject 38711-U. Also thanks to the SNI and the EDI schol-arships, granted by CONACyT and Instituto Politecnico Na-cional de Mexico, respectively.

REFERENCES

[1] D. Bently, J. Grant, and P. Hanifan, “Active-control fluid bear-ings for a new generation of machines,” Orbit, vol. 20, no. 2,1999.

[2] D. Bently, T. Elridge, and J. Jensen, “Externally pressurizedbearings allow rotor dynamic optimization,” in Proc. Interna-tional Symposium on Stability and Control of Rotating Machin-ery (ISCORMA ’01), p. paper 3018, Lake Tahoe, Calif, USA,August 2001.

[3] V. V. Kucherenko and J. C. Gomez-Mancilla, “Bifurcations ofan exactly solvable model of rotordynamics,” Internat. J. Bifur.Chaos Appl. Sci. Engrg., vol. 10, no. 12, pp. 2689–2699, 2000.

[4] J. Gomez-Mancilla, “Technological developments in rotordy-namics,” Project and final report, Feria Internacional El Sal-vador, Mexico, DF, Mexico, 1997.

[5] F. Dimofte and R. Hendricks, “Active controlled fluid filmbased on wave bearing technology,” in Proc. InternationalSymposium on Stability and Control of Rotating Machinery (IS-CORMA ’01), p. paper 3014, Lake Tahoe, Calif, USA, August2001.

[6] A. Ordonez-Pantoja, “Design and analysis of a controllablehybrid bearing,” M.S. thesis, Escuela Superior de IngenierıaMecanica y Electrica (ESIME), Instituto Politecnico Nacional,Mexico, DF, Mexico, 2003.

[7] G. Meng and R. Gash, “Stability and stability degree of acracked flexible rotor supported on journal bearings,” ASMETrans. Journal of Vibration and Acoustics, vol. 122, no. 2, pp.116–125, 2000.

[8] A. Antonio-Garcia, J. Gomez-Mancilla, and V. Nosov, “Sta-bility threshold speed calculation for rigid and flexible jeffcottrotors,” in Proc. 3er Congreso Nacional de Ingenieria Electrome-canica y de Sistemas (ICESE ’02), IPN, Mexico, DF, Mexico,November 2002.

[9] V. N. Afanasiev, V. B. Kolmanovskii, and V. R. Nosov, Math-ematical Theory of Control System Design, Kluwer AcademicPublishers, Boston, Mass, USA, 1996.

[10] B. P. Demidovich, Lectures on the Mathematical Theory of Sta-bility, Nauka Press, Moscow, Russian, 1967.

[11] J. Gomez-Mancilla and A. D. Dimarogonas, “MAQUI/CHUMA/FFT,” Clave SEP-86441, CopyRight, Mexico, Df,Mexico, 1996.

[12] J. W. Lund, Self-Excited, Stationary Whirl Orbits of a Journalin a Sleeve Bearing, Ph.d. dissertation, Rensselaer PolytechnicInstitute, Troy, NY, USA, 1966.

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