analysis vibration characteristics rotor-bearing system...

International Journal of Rotating Machinery1995, Vol. 1, No. 3-4, pp. 277-284Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1995 OPA (Overseas Publishers Association)Amsterdam B.V. Published under license byGordon and Breach Science Publishers SA

Printed in Singapore

Analysis of Vibration Characteristics for aRotor-Bearing System Using Distributed Spring and

Damper Model

D. C. HAN, S. H. CHOI and Y. H. LEEDepartment of Mechanical Design and Production Engineering, College of Engineering, Seoul National University, San 56-1,

Shinlim-Dong, Kwanak-Gu, Seoul, 151-742, Korea

In this paper, we modelled the journal beatings as distributed springs and dampers to investigate the influence of beatingwidth on the vibration characteristics of a rotor-beating system. A quadratic function is used as a shape function forhydrodynamic journal bearings. And a trapezoidally or a quadratically distributed model is used as a shape function forhydrostatic journal bearings. A finite element method is applied for analyzing the vibration characteristics of a rotor-bearingsystem.

Dimensionless governing equations are derived for a uniform rotor supported by two journal bearings, which aremodelled as point supported or distributed springs and dampers. For these models, natural frequencies for various beatinglength ratios and stiffness ratios between the bearing and the shaft are calculated and compared with each other. Then thestability limits of a rotor supported by two cylindrical journal beatings are calculated for the point supported or distributedsprings and dampers.

Key Words: Natural Frequency; Vibration; Bearing Width; Journal Bearing; Length Ratio

INTRODUCTION

present, the development of turbomachinery tendsto increase the rotating speed for the purpose of

decreasing the weight to power ratio of machines. So, thevibration is one of the most important problems to besolved. In analyzing the dynamic behavior of rotor-bearing systems, various mathematical models have beendeveloped and used over the past few decades. A finiteelement approach is one of the most widely usedmethods for this purpose.

In the early finite element investigations, many effec-tive models considering gyroscopic effect, axial force,torque, internal viscous damping, internal hystereticdamping and shear effects for uniform or tapered shaftswere developed by Nelson [1980], 0zguven and Ozkan[1984], Rouch and Kao [1979], and Gmtir and Rodrigues[1991]. The bearings were simplified as point supportedor uniformly distributed springs and dampers by Moure-latos and Parsons [1987]. It is a reasonable assumptionwhen using ball bearings or short width journal beatings.

However, in the case of a long width journal bearing thepressure created in journal is different along the axialdirection as well as radial direction. So, when the bearingwidth increases for the sake of large bearing loadcapacity, the bearing force is no more a point force. Thepressure distribution in axial direction of a hydrody-namic journal bearing is quadratic function or trapezoi-dal function for a hydrostatic journal bearing.

In the present study, journal bearings are modelled asquadratically and trapezoidally distributed springs anddampers along the shaft. For these models natural fre-quencies for various design parameters are calculatedand compared to each other. And the stability limits ofthe rotating speed are calculated for an example rotorusing two types of support model.

MODELING AND GOVERNINGEQUATIONS

The finite element equation for a typical rotor-bearingsystem can be written as

278 D.C. HAN ET AL.

[M]{q} + ([C] f[G]){q) + ([Ks] + [K]){q) {f)(I)

From these approximated shape functions we canrewrite equation (2) as

where subscript, B, denotes the bearing element.The stiffness and damping matrices of a bearing,

shown in FIGURE 1, can be written as [ ]Kyyfi(s) Kyzfi(s)[K] .J [c-ITKzyfi(S) Kzzfi(s)

[K] J []T[kyy(S) kyz(S)]kzy(S kzz(S) [ ]Cyyfi(s) Cyzfi(s)[C3] ) []TCzyfi(s) Czzfi(s)

[]ds (4)

[C] 3[]T[cyy(S) Cyz(S)]Czy(S Czz(S)[]ds (2)

where s denotes an axial distance along element and [y]the matrix of the translational displacement shape func-tion of a finite shaft element defined by equation (3),

{V W}T [xI/’] {q, q2 q3 q4 q5 q6 q7 q8}T (3)

Shape functions, kij and cij, were assumed as Dirac deltafunction(assumed a bearing as point supported one) orconstant value(assumed as uniformly distributed one).However, a typical bearing analysis shows that thedistribution shapes of springs and dampers are differentfrom those assumptions.FIGURE 2 and FIGURE 3 show the shapes of

dynamic coefficients of typical hydrodynamic and two-line feed hydrostatic journal bearings, respectively. Weapproximate these shape functions as quadratic o,r trap-ezoidal functions, which are shown in FIGURE 4. Theseapproximated shapes are shown by dashed lines inFIGURE 2 and FIGURE 3.

Bearing

where 0,1, and 2 represent point supported, uni-formly and quadratically distributed models, respec-tively.

For a two-line feed hydrodynamic bearing we canrewrite equation (2) as follows.

Kyyf3(s) Kyzfa(s) ][K] .J [t]TKzyf2(s Kzzf3(s)

[]ds

Cyyf2(s)Cyzf2(s) 1[C3 ) [1[I]TCzyf2(s Czzf2(s)

[@]ds (5)

The shape functions, fi, are given in APPENDIX.

NONDIMENSIONALIZATION

In this study, we consider the uniform shaft supported bytwo journal bearings as in FIGURE 5 to estimate theeffects of bearing width on vibration characteristics. Thegoverning equations of this system are given by equation(1).The equations of this model shown in FIGURE 5 can

be rewritten by using the following dimensionless quan-tities

Distributed Springs _._,,, " Y / L, Z z / L, tOot (6and dampers .-_ZzZ,_7 where o is the first natural frequency of a simply

,y .d/.. supposed Euler beam as

7 o --4 (7)

Then, using the parameters in equation (6) the nondi-mensionalized equations are derived as

[M]@ + [C] [G] {} + ([] + {]){} {f}.(8)

The generalized eigenvalue problem can be caed out,after reaanging equation (8) into a system of first order

FIGURE Finite shaft element in a bearing, differential equations of the fo

ANALYSIS OF VIBRATION 279

Axial distance Axial distance Axial distance Axial distance

calculated approximated

FIGURE 2 Distribution shapes of spring and damping coefficients for hydrodynamic bearing.

where

[A]=

[( ][#B]f {F} {-}o,--; +

(9)

(10)The damped natural frequencies of the system are then

obtained by finding the eigenvalues of the dynamicmatrix [D] which is given by

[D] [A]- [B] (11)

NUMERICAL EXAMPLES AND RESULTS

Example 1

In order to investigate the various effects of the rotor-bearing system on the natural frequencies, we use fourdifferent models. They are models supported by point, by

uniform pressure, quadratically and trapezoidally distrib-uted pressure over the length of the bearing, respectively.From the earlier analyses of bearings, we can approxi-mate the magnetic bearings by uniformly distributedmodel, the hydrodynamic journal bearings by quadrati-cally distributed model and the two-line feed hydrostaticjournal bearings by hybrid model, that is, uncoupledsprings are approximated by trapezoidally .distributedand coupled springs and all dampers are approximatedby quadratically distributed model. We use 9 shaftelements in point supported model and 8 elements inother models. All damping coefficients and the coupledterms of spring coefficients are set to zero and theuncoupled terms of spring coefficients are set by

Cyy Czz Cy Czy Ky Kzy 0 (12)

Kyy Kzz KB (13)

Other data for these cases are shown in TABLE 1.We calculate the first two natural frequencies of the

system varying bearing length ratio(B/L) for two differ-ent values of B/L. These results are shown in FIGUREs6 and 7. Because the deflection shape of the shaftbetween the nodes is approximated by cubic function infinite element modeling, the stiffness of the distributed

Axial distance Axial distance Axial distance Axial distance

calculated approximated

FIGURE 3 Distribution shapes of springs and dampers for two-line feed hydrostatic bearing.

280 D.C. HAN ET AL.

Bearing Shaft

FIGURE 4 Approximated shape functions of rotor dynamic coefficients.

models is less than the point supported model. Theuniformly distributed model is stiffer against rotationthan other distributed models. So the natural frequenciesof point supported and uniformly distributed models areoverestimated from these results. Point supported anduniformly distributed models give similar results, andquadratically or trapezoidally distributed model givessimilar results. (Some results are plotted as if they havesame values.) So we can use alternative model for thevibration analysis. FIGURE 7 shows the difference ratiobetween point supported model and other models definedby

to 100 1, 2, 3 (14)

where 60kp is the k-th natural frequency of the pointsupported model and 6Oki is the k-th natural frequency ofother models. From these results it can be observed thatthe difference is considerable as the bearing length ratiois increased.

Similar calculations are carried out with varying thestiffness ratio of bearings for the beating length ratio of0.1. From FIGUREs 8 and 9 it can be seen that differ-ences between other models are increased as the bearingstiffness ratio is increased, but the difference ratio isalmost independent of the beating stiffness ratio exceptfor the 1st natural frequency of uniformly distributedmodel.

Example 2

We consider the rotor model which has three discs andsupported by two cylindrical journal bearings as shownin FIGURE 10. The stability limit of the rotating speed iscalculated for two types of support models, i.e., pointsupported model and quadratically distributed one. Theresults are shown in FIGURE 11. (The results of Glinickeet al. [1980] are used as the rotor-dynamic coefficients ofthe cylindrical journal bearings for some B/D ratios. Thedata of the discs are shown in TABLE 2.)From these results it is found that the stability limit of

distributed model is lower than that of point supportedone in the case of small B/D. The reason is that thelogarithmic decrements of distributed model are smallerthan those of point supported one.

FIGURE 5 Simple rotor-bearing model.


0.8

0.5-

0.4

2’nd lat’urai Fr’eq.

1st Natural Freq. (C):

--O--- Point Supported Model--< Constantly distributed Model

Hydrodynamic Model/- Hydrostatic Model

0.04 0.08 0.12 0.16

8 0.4-

0.3-

0.00 0.04

2’nd Natural Freq.-

1st Natural Freq.. -_

----O--- Point Supported Model---x Constantly distributed Model

[ Hydrodynamic Model/ Hydrostatic Model

0.08 0.12 0.16

B/L B/L

(a) Case (b) Case 2

FIGURE 6 First two natural frequencies for some bearing length ratio.

100

10-4_0 0.04 0.08 0.

B/L

, 1.00-

0.01

10-4 .........12 0.16 0 0.04 0.08 0.12 0.16

B/L

(a) Case (b) Case 2

o 1st Natural Freq. of Constantly distributed model --o- 2nd Natural Freq. of Constantly distributed model

---x 1st Natural Freq. of Hydrodynamic model --x--- 2nd Natural Freq. of Hydrodynamic model

a- 1st Natural Freq. of Hydrostatic model 2nd Natural Freq. of Hydrostatic model

FIGURE 7 Differenceratio of two models for some bearing length ratio.

282 D.C. HAN ET AL.

0.5-

0.4-

0.3

0.0-0

2nd Natural Freq.

j

1st Natural Freq.

Point Suported Model--< Constantly distributed Model

/ Hydrodynamic Model

+-- Hydrostatic" Model

0.1 0.2 0.3 0.4 0.5 0.6

KB/K L

0.5-

0.4-

; 0.:3-

0.2

0.1

,"2nd Natural Freq.

/ /r / lstNaturalFreq.1,I J ----O---- Point Suported Model

---X Constantly distributed Model

Hydrodynamic Model

+-- Hydrostatic Model

0.1 0.2 0.3 0.4 0.5 0.6

(a) Case

FIGURE 8 First two natural frequencies for some bearing stiffness ratio.

KB/K L(b) Case 2

0.01

10.0

10-3

10-40 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6

KB/K L KB/K L(a) Case (b) Case 2

o st Natural Freq. of Constantly distributed model --o- 2nd Natural Freq. of Constantly distributed model--x 1st Natural Freq. of Hydrodynamic model x 2nd Natural Freq. of Hydrodynamic model

a- 1st Natural Freq. of Hydrostatic model 2nd,Natural Freq. of Hydrostatic modelFIGURE 9 Difference ratio for some bearing stiffness ratio.


Disk Disk2 Disk3

I5 I_ 600

FIGURE 10 Rotor model for stability analysis.

85O

800

650

6O0

Point Supported ModelConstantly distributed Model[

----Hydrodynamic:Model 1"!

unstable

..:..:::.:.’.,.-........stable " ’

0.5 0.75 1.25

Bearing width ratio [B/D}

FIGURE 11 Stability limit of a example rotor.

TABLEData of the two example cases

Case Case 20.03 0.052.81 2.810.4 0.40.02 0.02

TABLE 2Data of the discs

Disc (= Disc 3) Disc 2200 25040 4030 50

D/LE/kG

KB/KLD/to

Outer DiameterInner DiameterWidth

CONCLUSIONS

The finite element approach is used in analyzing theinfluence of bearing width on the vibration characteris-tics of rotor-bearing systems where various models ofbearing supports are used. It is suggested that hydrody-namic and hydrostatic journal bearings should be ap-proximated by quadratically and trapezoidally distrib-uted springs and dampers. Then, the first two naturalfrequencies of the uniform shaft, supported by twobearings, are calculated and compared for the pointsupported model, the uniformly distributed model andhydrostatically or hydrodynamically distributed model.And we calculate the stability limit of the simple rotormodel supported by two cylindrical journal bearings.From this analysis, it is observed that the point supportedor uniformly distributed bearing models overestimate thenatural frequencies and the logarithmic decrements. Sowe should consider the effects of bearing width as thebeating length ratio, B/L, is increased.

However, in the present studies the bearing data arecalculated on the assumption that the portion of the shaftin the bearing is rigid, so iterative method betweenflexible rotor analysis and bearing analysis will givemore realistic results for a real rotor bearing system. (Itis in progress.)

Acknowlegements

This research was carried out at the Turbo and Power MachineryResearch Center and was funded by the Korea Science and TechnologyEngineering Foundation. Their supports are gracefully acknowledged.

Nomenclature

[M], [C], [G], [K] mass, damping, gyroscopic and stiffnessmatrices

[lql], [], [J], [K] nondimensional forms of mass,damping, gyroscopic and stiffnessmatrices

[D] dynamic matrix

{q} displacement vector of the system.(V,W) centerline displacement in (Y, Z)

directions

f} force vector of the system

fi shape functions of bearing coefficients;i=0, 1,2,3

fI spin speed of shaft[KeB], [CeB] stiffness and damping matrices of

bearing part of finite shaft element

y, z coordinates of horizontal and verticaldirections

kij, cij shape function of bearing stiffness anddamping to the direction i,

284 D.C. HAN ET AL.

Kij, Cij the stiffness and the damping coefficientof a bearing to the direction i,length of finite shaft element

B bearing widthL length of a shaftEI bending stiffness per unit curvatureG shear modulusk shear factorKB uncoupled stiffness of bearingKe 48EI/L stiffness of a simply suported beamm mass per unit length shaft-(bar) nondimensional property

References

Glienicke, J., D. C. Han. and M. Leonhard, 1980. Practical determina-tion and use of bearing dynamic coefficients, Tribology international,Dec., pp. 297-309.

Gmtir, T. C. and Rodrigues, J. D., 1991. Shaft Finite Elements for RotorDynamics Analysis, Transactions ofASME, Vol. !3, pp. 482-493.

Mourelatos, J. E and Parsons, M. G., 1987. A Finite Element Analysisof Beams on Elastic Foundation including Shear and Axial Effects,Computers & Structures, Vol. 27, No. 3, pp. 323-331.

Nelson, H. D., 1980. A Finite Rotating Shaft Element Using Timosh-enko Beam Theory, Journal of Mechanical Design, Vol. 106, No. 4,pp. 793-803.

Ozguven, H. N. and Ozkan, Z. L., 1984. Whirl Speeds and UnbalanceResponse of Multibearing Rotors Using Finite Elements, Journal ofVibration, Acoustics, Stress and Reliability in Design, Vol. 106, No.1, pp. 72-79.

Rouch, K. E. and Kao, J. S., 1979. A Tapered Beam Finite Element ForRotor Dynamics Analysis, Journal of Sound and Vibration, Vol. 66,No. 1, pp. 119-140.

APPENDIX

The shape function of the bearing coefficient.

fo(s) 8

f(s) 6s(1 s) /L

fl(s) 1/L

2f3(s)

11 + 212 + 13

(--S + 12 + 13)/13[S- 11 12] lwhere [s] 1, s->0

Is] 0, s<0

The shape function of the shaft.

[111 0 0 12 113 0 0 1/4][I,l/]--OtI/1 2003 1/40

+ [oi + 13]

[s]+ 1- Is- 11]+

o 3vz + 2v 13 v

o2- l(v-2v2+v3) [32=(v-v2)Ot 31)2- 21,3 I V

a, (-v -t- v3) 134 (-v -I- v2)

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