Evaluation of Seal Effects on the of Fluid Machinery

Internationl Journal of Rotating Machinery1995, Vol. 2, No. 2, pp. 85-92Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1995 OPA (Overseas Publishers Association)Amsterdam B.V. Published in The Netherlands

by Harwood Academic Publishers GmbHPrinted in Singapore

Evaluation of Seal Effects on the Stability of RotatingFluid Machinery

T. IWATSUBOFaculty of Engineering, Kobe University, 1-1 Rokkodai-cho Nada-ku, Kobe, Japan

B. C. SHENGFaculty of Engineering, Kobe University, 1-1 Rokkodai-cho Nada-ku, Kobe, Japan

KUBOTA Corporation, 1-2-27 Shiromi Chuo-ku Osaka City, Japan

The stability of typical rotating fluid machinery such as single and multi-stage pumps is evaluated by using the finite elementmethod. The individual contribution of the impellers, bearings and seals to the stability and the dynamic interactions of thesefluid elements are examined. Various types of bearings and seals, such as annular smooth, parallel grooved and damper seals,are compared for better rotor stability. The effect of the operating conditions on the stability is also investigated. The resultsshow that rotor stability can be easily improved by replacing the unstable fluid elements.

Key Words: Rotating fluid Machinery; Bearing; Impeller; Non-contacting seal; Rotor stability; Logarithmic decrement

INTRODUCTION

n recent years the operating conditions of rotatingfluid machinery tend towards higher speed and higher

pressure along with the rapid progress made in thetechnology in industry and space development. In orderto raise the efficiency and prevent leakage flow at highpressure, the clearance of fluid elements such as bear-ings, non-contacting seals and impellers has to be de-signed possibly smaller. However, with higher pressure,higher speed and smaller clearance, greater fluid forcesoccur and sometimes these forces cause unstable vibra-tion.To prevent such unstable vibration, firstly the dynamic

characteristics of these fluid elements and their indi-vidual contribution to rotor stability must be made clear.Work is being done, and many results have been obtainedby various researchers. Secondly, the combined effect ofthese fluid elements on rotor stability and their interac-tion must be investigated. Yang et al. [12] (1985)investigated the effect of annular smooth and taper sealson the stability of the single-stage pump rotor system.

Diewald et al. [3] (1987) showed a procedure to inves-tigate the effect of annular smooth and grooved seals andimpellers on the stability of the Jeffcott rotor and themulti-stage pump rotor system. These researches showthat the rotor stability is strongly affected by the fluidelements, and the contribution of these elements to rotor

stability varies according to the operating conditions.In this paper, continuing the research of Yang et al.

[12] (1985), the stability of typical rotating fluid machin-ery such as the single-stage and multi-stage pumps whichconsist of impellers, bearings and non-contacting seals isevaluated by using the finite element method. Theindividual contribution of the impellers, bearings as wellas seals to the stability, and the dynamic interactions ofthese fluid elements are investigated. The contribution tothe rotor stability is evaluated by the logarithmic decre-ment. For the linear and non-cross-coupled inertia rotor

system, the total logarithmic decrement of the rotor

system can be represented as the sum of the individualdecrements. Therefore, the stability of the rotor systemcan be easily improved by changing the unstable fluidelements in the design stage. In the investigation, some

86 T. IWATSUBO AND B.C. SHENG

types of bearings and seals such as annular smooth,parallel grooved and damper seals are compared to seekbetter ones for the rotor stability. The effect of theoperating conditions on stability is also studied.

EQUATIONS OF MOTION ANDLOGARITHMIC DECREMENT

The analytical models of the single and multi-stagepumps are shown in Fig. and Fig. 2. For generalapplication, the non-symmetrical single-stage pump istaken. These rotor systems consist of bearings, impellersand seals. For the convenience of analysis, the impeller islooked upon as a disk which has the same mass and samemoment of inertia as the practical impeller. If the rotorrotates with a steady angular velocity to, the equation ofmotion of the disk elements in the coordinates illustratedin Fig. 2 is as follows:

(IMP] + [MJr]){/]} o[O]{4} {F} (i)

where [M and IMP] are respectively the mass matricesfor translational and bending motions; [Gd] is gyroscopicmatrix; {F} is the force vector acting on the disks.The equation of motion of the journal elements is

given by the expression

([MI +[Mr]){qe} to[Ge]{g]e} + [ge]{qe} ={fe} (2)

where { qe} { qt, qr} r {x, y, 4x, 4,} r, [M,e], [M], [ae],[Ke] are the mass matrices of translational and bendingmotions, gyroscopic matrix and stiffness matrix respec-tively; {Fe} is the fluid force vector of bearings, impel-lers and seals, and they are expressed as follows:

FIGURE 2 Model of multi-stage pump.

{F} [C]{4} + [K]{qt}

{FI} [Mi]{]t} + [Ci]{Ot} + [Ki]{qt}

{Fs } [Ms] { aJt } + [Cs] { 4t } + [Ks] { q,}

where [MI], [Ms] are the inertia coefficient matrices;[C,], [CI], [Cs] are the damping coefficient matrices;[K,], [KI], [Ks] are the stiffness coefficient matrices.Substituting Eq.(3) into Eq. (2), and combining theequations of disk elements and journal elements, theequation motion of the rotor system can be obtained.

[M]{4} + [C]{4} + [K]{q} {F} (4)

where {F} is the external force not including the fluidforces of bearings, impellers and seals. To determine theeigenvalues and eigenvectors, the characteristic equationof Eq. (4) is rewritten in the following form:

[C] [M]

or

[D]{:) + [E]{z) {0} (6)

Impeller

Seal lJ], Seal2

Bearing ng 2

1o

FIGURE Model of single-stage pump.

Letting the generalized solution of Eq. (6) have theform

then Eq. (6) yields

{} {#}e’’ (7)

{[/])+ [D]-1 + [E]}{O} {0} (s)

where X and {c}} are the eigenvalue and eigenvector,respectively. They can be obtained by solving Eq. (8).

In general, the eigenvalues are conjugate complex, andexpressed in the following form:

)k O -I" joi, h O --jo (9)

STABILITY OF FLUID MACHINERY 87

where means i-th natural mode. The logarithmic dec-rement of the rotor system is defined as

O

i-" -2’rr (10)o.)

In order to investigate the effects of the fluid elements onthe rotor stability, the logarithmic decrements of the fluidelements have to be determined. These logarithmicdecrements can be obtained by the eigenvalue andeigenvector through the following transformation [9](1982).

Substituting the i-th conjugate eigenvalues )ti, ti andi-th conjugate eigenvectors {/}, {/} into the charac-teristic equation of Eq. (4), the following equations areobtained.

x + X,[C]{,,} + I/q{,,} {0}

[M]{ i} -- i[C]{ i} -it- [K]{ i} {0} (11)

Furthermore, expressing the mass matrix, dampingmatrix and stiffness matrix as the sum of symmetric partsM*, C*, K* and unsymmetric parts AM, AC, AK, thenthe above equations become

)k/2[M* -[- AM]{,/} -t- NiEC*+ AC]{q/i} -[- [g* -- AK],{,i}{0}

[M*+ AM]{i} -+- -i[C*n- AC]{i} nt- [K*nt- mg]

{ q/i} {0} (I 2)

where [AM] -[AM]r [AC], -[AC]r, [AK]-[A/flr

Premultiplying equations of (12) by {(pi} r and {i}r,and introducing the following expressions

T{* i} [M ]{tpi} {i}r[M*]{ i} mi

T{* i} [C 1{/i} {i}T[M*]{ 1 i} ci

(h -)m + (h + -2)mmi--l-()kint--i)c; ()k+ hi)mc "nt- 2Ak 0 (14)

Because the unsymmetric mass Ami is caused by theimpellers and seals, and it is usually much smaller thanthe sysmmetric mass m, it is ignored here. SubstitutingEq. (9) into Eq. (14), the real part of the eigenvalues isobtained.

O,)iC "-t- m k(15)Oti 2toim -[- A ci

Therefore, the logarithmic decrement is expressed asfollows"

i qTtOi Ci -[- 7rAkii 27r-

to 60 m q- toiAci/2(16)

For the present rotor systems, c*i and Ak; can beexpressed as the sum of the fluid elements.

Ci {-i}T[c*]{li}

nB nsTE({IIIBi}[C*B]j{qIBi}j) -[- E({qISi}j [Cs]j{lllSi}j)

j=l j=l

-- E({ li};[Cl*]j{lIi}j)j=l

?’/B /’/S

E(CBi)j + E (Csi)j -- E (CIi)jj=l j=l j=l

mki {- i}T [Ak]j{ i}

rlB

E({m}[Ak]y {@rely) +j=l j=l

{* i} [K ]{1/i} { i}T[M*]{ I1 i} ki

{ i}T[AM]{,i} {,}T[AM]{, i} J A m

{- i}T[AC]{qli} {,i}T[Ac]{- i} j A C

{-i}T[AK]{qji} {,i}T[AK]{-i} j A k (13)

the difference of the two equations of Eq. (12) can bewritten as follows"

-- E({ li};[Akl]j{l11i}j)j=l

nB ns,(AN Bi)j nt- E (Aksi) + E(Ak i/)j=l j=l j=l

(17)

Substituting the above expressions into Eq. (16), thelogarithmic decrement expressed as the sum of theindividual elements is obtained.


//B /’/S /’/I

j=l j=l j=l

where nB, and ns, and n are the total numbers of thebearings, seals and impellers, respectively

EXAMPLE OF ANALYSIS

Stability of Single-Stage Pumps

The calculation is based on the conditions in Table 1. Inthe calculation, the bearing dynamic coefficients andimpeller dynamic coefficients of references [7] (1984)and [11] (1987) are used, while the dynamic coefficientsof seals are obtained by the calculating method given in[5] (1989) and [6] (1990).The loci of eigenvalues of Eq. (8) for single-stage

pump rotor systems with smooth, damper and parallelgrooved seals are illustrated in Fig. 3. Here, only the twoeigenvalues in or near to the unstable region (o > 0) aregiven. In these figures, coo is the first eigenfrequency ofthe rotor without seals and bearings. The numbersmarked on the loci are the ratios at rotating speed N tocoo In the present rotating region, the stability of the rotorsystem is dependent on the eigenvalue of root 1. There-fore, the stability is discussed as to this eigenvalue bymeans of the logarithmic decrement.From the point of view of energy, the rotor system is

releasing energy to the outside so that the system tends to

TABLESpecification of single-stage pump

Bearing length (mm) 60Bearing diameter (mm) 60Bearing clearance (mm) 0.06Seal length (mm) 20 ---40Seal diameter (mm) 120Seal clearance (mm) 0.2--0.6Impeller width (mm) 80Impeller diameter (mm) 300Impeller clearance (mm) 0.5Pressure difference (M Pa) 4.9Oil viscosity (mPa.s) 16.7Oil temperature (C) 60Water temperature (C) 50

2.0

1.0

0.5

With smooth seal

we 538.2 rad/s

2.4 a 2.0

a. rootl

,5.0

0.1 0,2

Real

(a) With smooth seals

With damper seal

2.0 we 538.2 rad/s

root2 2.4 , 2.0

12

’ .o rootl

-0.2 -0.1 0 0.1 0.2

a ()(b) With damper

With grooved seal2.0 Wo 538.2 rad/s

1.5 ,)t2

1.0

0.5

1.6 2.43.0

5.0.0( 1.2

’7 root1

0.4,/

-0.1 0 0.1 0.2

Real (ai/COo)(c) With parallel grooved seals

FIGURE 3 Complex eigenvalues.


-1.00.8

0.5

o

FIGURE 4 Logarithmic decrements.

stabilize with the lapse of time, if the logarithmicdecrement is positive; while, if the logarithmic decre-ment is negative, the rotor system is absorbing energyfrom the outside so that the system enlarges the ampli-tude and tends to unstabilize with the lapse of time [9](1982). Figure 4 shows the logarithmic decrements of thebearings, seals, impellers and the total rotor systems. Forthis natural mode, the contribution of the impeller to thestability is very small. The logarithmic decrements of theseals are positive, which means the stable effect on therotor stability, but the logarithmic decrements of thebearings are almost negative, which means an unstableeffect on the rotor stability.The logarithmic decrements of the rotor systems with

smooth, damper and parallel grooved seals are shown inFig. 5. The results show that the logarithmic decrementsof rotor systems with smooth and damper seals aresimilar, and the stability of these systems are better thanthat of rotor systems with parallel grooved seals. Inactual machinery, investigation of rotor stability with aspecific rotating speeds or in a specific rotating region isusually necessary. The logarithmic decrements of rotorsystems with different seals at specific rotating speedN/too are shown in Fig. 6, where the mode shape ofthe rotor system is shown, too. These results are drawn inthe form of bar graphs for the convenience of clarity.From these results the contribution of every fluid elementto rotor stability can be immediately recognized. Therotor stability can be improved by replacing the unstableelements with stable elements. This will be discussed innext section.

2.0

1.5

0.5

positive Wo 538.2 rad/snegative

With grooved seal

With damper seal

/ .,’/ With smooth seal /

/i/I/ I/

O.8 1.0Nondimentional frequency (co/COo)

FIGURE 5 Comparison of logarithmic decrement.

1.2

9O T. IWATSUBO AND B.C. SHENG

1.2

0.8

0.4

B ’Bearings ’Stable"impeller

S "Seals "Unstable

T Total

,4,-

Smooth seal Damper seal Grooved seal

FIGURE 6 Logarithmic decrement at (N/oo=l).

The influence of preswirl velocity Vt in seals on rotorstability is investigated, and the results are shown in Fig.7. This figure shows that positive preswirl velocity exertsan unstable influence on rotor stability; while negativepreswirl velocity exerts a stable influence on rotorstability. This result is in agreement with the individualresearch of seals [4]---[6].

Stability of Multi-Stage Pumps

The specification of multi-stage pumps is illustrated inTable 2. The logarithmic decrements of rotor systems

0.1

w0 538.2 rad/s With smooth seal

1.04Nondimentional frequency (co/co o)

FIGURE 7 Effect of preswirl velocity on logarithmic decrement.

TABLE 2Specification of multi-stage pump

Bearing length (mm) 80Bearing diameter (mm) 60Bearing clearance (ram) 0.06Seal length (mm) 20 ---40Seal diameter (ram) 100Seal clearance (mm) 0.2"-0.6Impeller width (mm) 40Impeller diameter (mm) 300Impeller clearance (mm) 0.5Pressure difference (MPa) 4.9Oil viscosity (mPa.s) 16.7Oil temperature (C) 60Water temperature (C) 50

and fluid elements at N oo 19671.6 rpm are studiedand shown in Fig. 8, where the mode shapes of rotor

systems are shown, too. In the investigation, some ofseals or bearings are changed in order to improve rotorstability and find the interactions of these fluid elements.

In the case of (a), five parallel grooved seals are usedin the rotor system. The total logarithmic decrementshows a negative value because of the unstable effects ofthe seals and the bearings. If the parallel grooved sealis replaced by a damper seal, the rotor system becomesstable (b). Furthermore, by replacing all the groovedseals with damper seals, the rotor system becomes morestable (c). It is found that after the replacement of theseals, not only the logarithmic decrements of the sealsbut also those of the bearings are changed. If the circlebearings are replaced with tilting pad bearings instead ofthe seals, as shown in (d), the rotor stability can also beimproved. In this case, the logarithmic decrements of theseals are also changed. It is considered that the interac-tive variation of the fluid elements is caused by thevariation of the natural mode, because the logarithmicdecrement is dependent on the natural mode, which hasbeen demonstrated in the previous section. According tothe above discussion, the interaction of other elementsmust be considered when evaluating the effect of thefluid element on rotor stability.

CONCLUSION

The present analysis supports the following conclusions:

1. In rotating fluid machinery, the investigation of theeffect of the individual fluid element on rotor stabilityis an effective method for the purpose of theevaluation of rotor stability and dynamic design.


Beadng

Im )ellerl Impeller4

s...,r-! i-i N l-is.., Bearing

1.0N 19671.6 pm

319.6 rad/$

With grooved eals

B $ I S I S I I S B To

"Stable (+) "Unstable (-)

(a) With 2 circle journal bearings and grooved seals

1.0

0.5

._.o 0.5

.E

N. 19671.6 rpm

=0" 36.5 ra/s

With grooved alsdamper seal (seal 1)

Bt S I S I S I S= I S B Total

Stable (+) ’Unstable (-)

(b) With 2 circle journal bearings,4 grooved seals and damper seal

N 19671.6 rpm487.4 rad/s

With dmaper seals

ToBI St $2 12 $3 I3 $4 14 $5 B2’Stable (+) ’Unstable (.)

(c) With 2 circle journal bearings and 5 damper seals

N 19671.6 rpm

0" 285.1 wadis

1.0 With grooved seals --Tilting pad bearings. 13.5

N__n I-IBI S] I] $2 12 $3 ]3 Sn In $5 B

m--] Stable (+) "Unstable (.)

(d) With tilting pad bearings and grooved seals

Total

FIGURE 8 Logarithmic decrement of multi-stage pump.

2. Rotor stability can be improved by replacing theunstable or other elements. Such replacementsusually cause a variation on the logarithmicdecrements of the other elements.

3. The effect of preswirl velocity in seals on the stabilityis consistent with that ofthe individual research onseals.

Nomenclature

C (N. s/m)F (N)K (N/m)M (kg)N (rpm)

V, (rex,y,z

XD

Damping coefficientForceStiffness coefficientInertia coefficientRotating speed of rotorNumber of fluid elementsPreswirl velocity in sealCoordinatesReal part of eigenvalueLogarithmic decrementEigenvalueRotationary coordinatesEigenvectorEigenfrequency or imaginary part ofeigenvalueEigenfrequency of rotor system withoutfluid

Subscript

B BearingI ImpellerS Seal

Superscript

T Transposed (matrix)

References

Black, H, F., 1969. "Effects of Hydraulic Forces in Annular PressureSeals on the Vibrations of Centrifugal Pump Rotors," Trans. ASMEJ. Mech. Eng. Sci., Vol. 11, No. 2, pp. 206-213.

Childs, D. W., 1983. "Finite-Length Solutions for Rotordynamic

Coecients of Turbulent Annular Seals," Trans. ASME. J. Lubr.Technol., Vol. 105, pp. 437-445.

Diewald, W., and Nordmann, R., 1987. "Dynamic Analysis of Centrifu-gal Pump Rotors With Fluid-Mechanical Interactions," ASME.Vibrations Conf. pp. 571-580.

Iwatsubo, T., Sheng, B. C., and Matsumoto, T., 1989. "An ExperimentalStudy on the Static and Dynamic Characteristics of Pump AnnularSeals (2nd Report, The Dynamic Characteristics for Small Concen-tric Whirling Motion)," Trans. JSME Series C, (in Japanese), Vol. 55,No. 510, pp. 317-322.

Iwatsubo, T., Sheng, B. C., 1989. "Evaluation of Dynamic Character-istics ofParallel Grooved Seals by Theory and Experiment)," Prec. of1st Asia Vibration Conference, pp. 498-504.

Iwatsubo, T., Sheng, B. C., 1990. "An Experimental Study on the Staticand Dynamic Characteristics of Damper Seals," Prec. of IFToMMConference, pp. 307-312.

JSME., 1984. "Data Collection of Static and Dynamic Characteristics

of Journal Bearings," Publishing Company of Japanese Industry (inJapanese).

Kanki, H., and Kwakami, T., 1984. "Experimental Study on theDynamic Characteristics of Pump Annular Seals," I. Mech. E.,C297/84, pp. 159-166.

Kurohashi, M., Iwatsubo, T., Kawabata, N., and Fujikawa, T., 1982.


"Evaluation of Rotor Stability by Means of Logarithmic DecrementsDerived from Energy for the Individual Elements," Trans. JSME (inJapanese), Vol. 48, No. 430, pp. 825-832.

Nordmann, R., Dietzen, E J., and Janson, W., 1986. "Rotordynamic

Coefficients and Leakage Flow ofParallel Grooved Seals and SmoothSeals," NASA cp-2443, pp. 129-153.

Ohashi, H., and Shoji, H., 1987. "Lateral Fluid Forces on WhirlingCentrifugal Impeller (2nd Report: Experiment in Vanetess Diffuser),"Trans. ASME. J. Fluids Eng., Vol. 109, pp. 100-106.

Yang, B. S., Iwatsubo, T., and Kawai, R., 1985. "Effects ofSeals on theStability ofPump Rotor System," Trans. JSME (in Japanese), Vol. 51,No. 470, pp. 2479-2486.

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Evaluation of Seal Effects on the of Fluid Machinery