Rtiwari Rd Book 06c

8/10/2019 Rtiwari Rd Book 06c

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Dr R Tiwari, Associate Professor, Dept. of Mechanical Engg., IIT Guwahati, ([email protected])

303

6.3 Dynamic Seals

6.3.1 Classification of Seals

Seals are broadly classified as liquid and gas seals according to the working fluid used in the system.

The most common working fluids are water, air, nitrogen, Triflurobromomethane (CBrF 3 ), liquid

oxygen, liquid hydrogen etc. In addition, they can be categorized as static and dynamic seals. Static

seals are used where the two surfaces do not move relative to one another. Gasket-type seals are static

seals (Fig. 1). Dynamic seals are used where sealing takes place between two surfaces having relative

movement viz. rotary, reciprocating, and oscillating. The main focus of the present paper is on rotary

seals. It has wide variety of applications in high-speed, high-pressure and cryogenic temperature

conditions of aviation and space industries such as in turbine stages, turbo-pumps, compressors, gear

boxes, etc. Rotary seals can be subdivided into two main categories as clearance seals and contact

seals. Clearance seals are circumferential non-contacting seals (Fig. 2a). In contact seals, the contact is

formed by positive pressure, while in the case of clearance seals; they operate with positive clearance(no rubbing contact). The most commonly used material for dynamic seals (especially for rotary seals)

are stainless steel, bronze, aluminium, nickel-based alloys, Polytetrafluroethane etc. Fig. 2(a) shows a

typical rotary seal with the clearance exaggerated. Rotary seals based on geometry can be classified as

(i) Ungrooved plain seals (or Smooth annular seals): (a) Straight (Fig. 2b), (b) Tapered (Fig. 2c) and

(c) Stepped (Fig. 2d). In geometry they are similar to journal bearings but the clearance/radius

ratio is as low as two times and as high as ten times (or more) large to avoid rotor/stator contact.

(ii) Grooved/Roughened surface seals: (a) Porous surface seals (b) Labyrinth seals (Figs. 3(a-d)), (c)

Helically grooved / Screw seals (d) Circular hole or triangular patterns seals and (e) Honeycomb

patterns seals (Fig. 4). These seals are used in centrifugal and axial compressors and pumps and in

turbines. Different internal surface patterns of seals are shown in Fig. 5.

(iii) Contact seals: (a) Brush seals (Fig. 6a) (b) Face seals and (c) Lip seals (Fig. 6b)) Because of

rubbing, these seals are used commonly in low speed pumps, or where the working fluid can act as

a coolant. Contact seals provide much lower leakage rates than either of non-contact seals (Adams,

1987), however, the latter can operate at very high speed and pressure conditions.

(iv) Floating-ring oil seals: The ring whirls or vibrates with the rotor in the lubricating oil, but does

not spin. They are used in high-pressure multi-stage centrifugal compressors.


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304

High pressure fluid

Hydraulic end thrust

GasketCompressive load

Fig. 1. Static seal (gasket)

Highpressure

Lowpressure

Flow

SealRotor

Fig. 2(a). Rotor-seal assembly

Flow

Seal

Rotor

Fig. 2(b). Straight annular seal

Flow

Seal

Rotor

Fig. 2(c). Tapered annular seal (converging)

Flow

Seal

Rotor

Fig. 2(d). Stepped annular seal

Expandingcavity

Chalk vaneGroovedepth

Fig. 3(a). Labyrinth seal (teeth-on-stator)

Stator

Rotor

Labyrinthseal

Fig. 3(b) Labyrinth seal (teeth-on-rotor)

Rotor

Stator

LabyrinthFlow

Fig. 3(c) Labyrinth seal (teeth-on-stator and teeth-on-rotor) axial flow type


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305

Labyrinthseal

Stator

Impellor

(Rotor)

Leakage

Fig. 3(d) Labyrinth seal radial flow type

Cell depth Cell size

Honeycombhousing

Shaft

Fig. 4. Honeycomb seal

Unwrap(a) Plain seal

(c) Labyrinth seal

(d) Helically grooved seal

Unwrap

(e) Honeycomb seal

(f) Hole pattern roughness seal

(g) Triangular pattern roughness seal

(b) Plain seal with porous material

Fig. 5. Different internal surface patterns on seals


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306

Brush

Leak flow

Fig. 6(a). Brush seal

Metal stiffner

Rubber lipFluid tobe sealed

Garter spring

Fig. 6(b). Lip seal

6.3.2 Theoretical Estimation of Dynamic Coefficients of Seals

In this chapter, basic governing equations to obtain dynamic coefficients of smooth annular turbulent

seals (smooth seals) are presented. Dynamic coefficients are calculated from the approximate solution

of the bulk flow theory for the configuration of the test rig. Effects of rotor speeds, seal dimensions

and operation conditions on these dynamic coefficients are also presented and discussed in detail.

Basic governing equations and solution

In an annular seal, flows are usually turbulent because of high Reynolds numbers at which they

operate. Black and his co-workers (Black 1969, Black and Jensen 1970) were the first to attempt to

identify and model the rotor dynamics effects of turbulent annular seals using bulk flow models

(similar to those of Reynolds lubrication equations). Bulk flow models employ velocity components,

( , )zu z and ( , )u z , that are averaged over the clearance, where zu and u are the velocities in

the directions and zand are the coordinates as shown Figure (4.1). Black and Jensen used several

heuristic assumptions in their model, such as the assumption that / 2u R = , whereRis the radius

of the seal and is the rotor speed. Moreover, their governing equations do not reduce to

recognizable turbulent lubrication equations. These issues caused Childs (1983b) to publish a revised

version of the bulk flow model and the present section will focus on Childs' model.

The geometry of the seal annulus which is filled with fluid is sketched in Figure 4.1, and is described

by coordinates of the meridian of the gap as given byZ(s) andR(s), 0 < s < L, where the coordinate,

s, is measured along that meridian and tis the time. The clearance is denoted byH(s, , t) where theunperturbed value of H is (s). Equations governing the bulk flow are averaged over the clearance.

This leads to a continuity equation of the form (4.1)

1( ) ( ) 0s s

H H dRHu Hu u

t s R R ds

+ + + =

(4.1)


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307

where su and u are velocities averaged over the local clearance.

Figure 4.1. Fluid filled annulus between a rotor and a stator for turbulent lubrication analysis

The axial and circumferential momentum equations are as follows

21 ss sr s s ss

u u u u uP dRu

s H H R ds t R

= + + + +

(4.2)

1 s r ss

u u u u u uP Ru

R H H t R s R s

= + + + +

(4.3)

The approach used by Hirs (1973) is employed to determine the turbulent shear stresses, ssand s,

applied to the stator by the fluid in thesand directions respectively, which takes the following form

12 21 ( / ) ( )

2

s

s

mmss s s s

s s

s

A uu u Re

u u

+

= = + (4.4)

and stresses, srand r, applied to the rotor by the fluid in the sand directions respectively and are

obtained as1

2 21 {( ) / } ( )( ) 2

mmsr r r s

s s

s

A uu R u Re

u u R

+

= = + (4.5)

where the local meridional Reynolds number is given as

Rotor

Stator

Co-ordinateand

velocityu - Normal

to sketch

Z(s)R(s)

H(s,,t)

sss

rs

su


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308

/s sRe Hu = (4.6)

and constantsAs,Ar, msand mare chosen to fit the available data on turbulent shear stresses. Childs

(1983a) uses typical values of these constant.

As =As= 0.066; ms= m= -0.25 (4.7)

In the following subsection, the solution for the governing equation are presented and discussed in

details.

Approximate dynamic coefficients of seals

In the present subsection, the theoretical and computational analysis performed by various researchers

has been compiled. Lomakin (1958) was the first to propose a theoretical model of a plain seal, which

predicted that the axial pressure drop across the seal caused a radial stiffness, independent of shaft

rotation. The Lomakin radial direct stiffness (kd) is given by

2

0.25/4.7 with 0.079 /1.5 2 /

d e

P L Ck R R

L C

= =

+ (1)

where P is the pressure drop andR,Land Care the radius, axial length and radial clearance of the

seal, respectively. If the direct stiffness were the only effect of the plain seal, then its effect on critical

speeds would be easily and accurately predictable. Blacks work (1969, 1971) provided the major

initial impetus for the extensive research and the state of the art design information developed on this

topic over the last 35 years. Black developed the classical theory for turbulent annular seals,

considering the axial fluid flow caused by a pressure drop along the seal, the rotational fluid flow as a

consequence of the shaft rotation and a relative motion of the seal between the rotor and housing.

Black (1969, 1971) and Childs (1983a, b) formulated and extended Lomakins theory in terms

applicable to the rotor dynamic analysis of centrifugal pumps. Black, Childs and others have shown,

however, that kd increases with shaft speed (at constant P) and that the seal also produces cross-

coupled stiffness (kc), direct and cross-coupled damping (cd and cc), and direct inertia coefficients.

Moreover, the pressure drop will vary with the speed in most turbomachineries and the rotor dynamiceffects are quite complex.

Clearances, pressures and velocities are divided into mean components (subscript 0) that would

pertain in the absence of whirl, and small linear perturbations (subscript 1) due to the eccentricity, ,

rotating at the whirl frequency, :


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309

0 1 0 1

0 1 0 1

;

;s s s

H H H P P P

u u u u u u

= + = +

= + = + (4.8)

These expressions are substituted into governing equations (4.1-4.3) to yield a set of equations for

mean flow quantities and a second set of equations for perturbation quantities; terms which are of

quadratic or higher order in are neglected. Resulting zeroth-order equations define the leakage and

the circumferential velocity development and are solved by numerical methods. From the first order

equations, the time and dependency is eliminated to obtain the pressure distribution solution which

is then integrated and along and around the seal clearance to yield reaction force components. From

rotor dynamic force components, following rotor dynamic coefficients and constants are obtained

(Childs, 1983).

*2

20)(

4

1kTaak

d

= ;

= kTakc

)(2

11 ;

= cac d 1 ;= cTac

c)(

2; = mamd 2 (4.9)

with

C

RLPk = ; Tkc = ; 2Tkm = (4.10)

V

LT= (4.11)

EAa 5.20 = ;

++=

61

221 E

BEAa

;

+=612 E

Aa

(4.12)

21 ++=A ;

2

2

41

71

b

bB

+

+= ;

)1(2

1

BE

++

+= (4.13)

/L C = ; ca RRb /= ; VC

Ra = and

CRR

c

= (4.14)

where k, m and c are the stiffness, mass and damping coefficients, k , c and m are reference

values of corresponding quantities, oa 1a and 2a are dimensionless coefficients, is the speed of

the rotor, Tis the transit time as given in equation 4.11, Lis the length of the seal, Vis the average

axial stream velocity, is the entrance loss coefficient, is the fluid density, is the friction

coefficient, Ris the radius of the seal, Cis the clearance of the seal and Pis the difference between

pressures at the inlet and the exit of the seal. Subscripts dand crepresent the direct and cross-coupled


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terms, respectively.Rais the Reynolds number for the axial flow and Rc is the Reynolds number for

the circumferential flow for smooth annulus seals. Dimensional coefficients are thus functions of ,

and b. To determine coefficients 0a , 1a and 2a coefficients and b are required for the

frequently occurring value of =0.5. From Childs (1983a), we have

[ ] 375.024/1 )2/1(1066.0 bRa += (4.15)

To calculate the average velocity V is inserted into equation (4.14). The expression for Vcan be

obtained from the fundamental relationship for the pressure difference,

2

2)21( VP

++= (4.16)

So, the average axial stream velocity can be expressed as

)21(

2

++

=

PV (4.17)

Since the desired value of is also function of V and thereby , it is best obtained iteratively. From

the , the dynamic coefficients can be obtained for different speed . Figure 4.2 shows an algorithm

for the solution of dynamic coefficients of seals.


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Figure 4.2. Flow chart for the theoretical estimation of dynamic coefficients of seals

Start

Input:, , P,, C, L, RSetN = 0 rad/s

= + 0.001Calculate V, b,Ra

Calculate LC/1 = ,

[ ]375.0

24/1 )2/1(1066.0 bRa += = 1e

CalculateA,B,E, a0, a1, a2Calculate k , c , m

Calculate kd, k

c, c

d, c

c,m

d

SetN =N+1

End

IfN> 5301

Yes

Yes

No

If e 10-

= 0 and e= 1

No


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Numerical simulation results and discussion

In this subsection, numerical results of dynamic coefficients of seals are presented for the rotor speed

up to 50,000 rpm. The input data are taken as mentioned in Table 4.1.

Table 4.1. Input data for numerical simulation of dynamic coefficients of seals

Length of the seal 11, 22, 33 and 44 mm

Radius of the seal 22 mm

Clearance of the seal 0.2 and 0.4 mm

Dynamic viscosity of water at 32 oC 0.810-6m2/s

Entrance loss coefficient 0.5

Inlet pressure 3, 6, 16, 41, 81 bar

Seal exit pressure 1 bar

Speed of the rotor 1 to 50,000 rpm

Seals dynamic coefficients are dependent on speeds, seal dimensions and pressure differences. The

stiffness (kdand kc), damping (cdand cc) and mass (md) coefficients are presented for various speeds

(), pressure differences (P) and ratiosL/D.

Figures 4.3 to 4.15 show the variation of the direct and cross-coupled stiffness and damping and direct

inertia coefficients with respect to the speed up to 50000 rpm, for different values of clearances (0.2

and 0.4 mm),L/Dratios (0.25, 0.50, 0.75 and 1.00) and pressure differences (2, 5, 15, 40 and 80 bar).

The effects of these variables on seal dynamic coefficients are discussed in detail in followingsections.

Effect of rotational speeds and pressure differences

Direct stiffness coefficients increase with increase in the pressure difference (Figure 4.3). At low-

pressure differences (2 and 5 bars), the direct stiffness coefficient becomes negative as shown in

Figure 4.3. The direct stiffness coefficient reaches maximum nearly at 5000 rpm and then slowly

declines as shown in Figure 4.3. The cross-coupled stiffness linearly increases with the rotor speed

and also increases with the pressure difference (Figure 4.4). The direct damping coefficient increase

slightly to the speed, however, it increases with the pressure difference (Figure 4.5). The cross-

coupled damping increases linearly with the speed but, insensitive to the pressure difference (Figure

4.6). The direct inertia coefficient increases sharply with the rotor speed and it is almost insensitive to

the pressure difference (Figure 4.7).

Effect of L/D ratios


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L/D ratio has significant effect on rotor dynamic coefficients of seals. The direct stiffness increases

with the increase inL/Dratio. ForL/D= 1.00, after reaching a maximum value nearly to 8000 rpm it

starts declining and becomes negative with increase in the rotor speed. At L/D=0.25, the direct

stiffness coefficient always has positive values (Figure 4.8). The cross-coupled stiffness and the direct

and cross-coupled damping coefficients increase with the increase in L/D ratio as shown in Figures

4.9-4.10.

Effect of seal clearances

Doubling the clearance show a huge drop in the direct stiffness and damping coefficients, while

increasing speeds up to 50,000 rpm. The drop in the cross-coupled stiffness and damping and direct

inertia coefficients with increase in clearance is also significant (Figures 4.13-4.15).

Figure 4.3. Direct stiffness coefficients for C=0.2 mm,L/D=0.25 and P=2 to 80 bar.

Figure 4.4. Cross-coupled stiffness coefficients for C=0.2 mm,L/D=0.25, P=2 to 80 bar.


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Figure 4.5. Direct damping coefficients for C=0.2 mm,L/D=0.25, P=2 to 80 bar.

Figure 4.6. Cross-coupled damping coefficients for C=0.2 mm,L/D=0.25, P=2 to 80 bar.

Figure 4.7. Direct inertia coefficients for C=0.2 mm,L/D=0.25, P=2 to 80 bar.


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Figure 4.8. Direct stiffness coefficients for C=0.2 mm, P=40 bar,L/D=0.25-1.00.

Figure 4.9. Cross-coupled stiffness coefficients for C=0.2 mm, P=40 bar,L/D=0.25-1.00.

Figure 4.10. Direct damping coefficients for C=0.2 mm, P=40 bar,L/D=0.25-1.00


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Figure 4.11. Cross-coupled damping coefficients for C=0.2 mm, P=40 bar,L/D=0.25-1.00

Figure 4.12. Direct inertia coefficients for C=0.2 mm, P=40 bar,L/D=0.25-1.00

Figure 4.13. Direct and cross-coupled stiffness coefficients for P=40 bar,L/D=0.25, C=0.2 & 0.4mm.


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Figure 4.14. Direct and cross-coupled damping coefficients for P=40 bar,L/D=0.25, C=0.2 & 0.4

mm.

Figure 4.15. Direct inertia coefficients for P=40 bar,L/D=0.25, C=0.2 and 0.4 mm.

Basic governing equations to obtain dynamic coefficients of the smooth-annular turbulent seals (i.e.

smooth seals) are explained briefly. Dynamic coefficients are calculated from the bulk flow theory for

a seal dimension and effects of rotor speeds, seal dimensions and operation conditions on dynamic

coefficients of seals are presented and discussed in detail.

6.3.3 Fluid-Film Dynamic Force Equations

A model of a typical annual (or clearance) seal is shown in Fig. 2(a). The geometrical shape of a

clearance seal is similar to that of a hydrodynamic bearing; however, they are different in the

following aspects. To avoid contact between a rotor and a stator, the ratio of the clearance to the shaft

radius in seals is made few times (2 to 10 times) larger than that of hydrodynamic bearings. The flow

in seals is turbulent and in hydrodynamic bearings it is laminar. Therefore, unlike hydrodynamic

bearing, one cannot use the Reynolds equation for analysis of seals. When a rotor vibrates, a reaction


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force of the fluid in the seal acts on the rotor. In case of a small vibration around the equilibrium

position, the fluid force can be linearized on the assumption that deflections andx y are small.

The general governing equations of fluid-film forces on seals, which has small oscillations relative to

the rotor, is given by the following linearized force-displacement model (Childs et al., 1986)

xy xy xyx xx xx xx

y yx yx yxyy yy yy

k c mf k c mx x x

f k c mk y c y m y

= + +

(2)

where fx and fy are fluid-film reaction forces on seals in x and y directions. k, c, m represent the

stiffness, damping and added-mass coefficients, subscripts:xxand, yyrepresent the direct andxyand

yx represent the cross-coupled terms, respectively. These coefficients vary depending on the

equilibrium position of the rotor (i.e. magnitude of the eccentricity), rotational speed, pressure drop,

temperature conditions etc. The off-diagonal coefficients in equation (2) arise due to fluid rotation

within the seal and unstable vibrations may appear due to these coefficients. Equation (2) is applicableto liquid annular seals. But for the gas annular seals, the added-mass terms are negligible. For small

motion about a centered position (or with very small eccentricity) the cross-coupled terms are equal

and opposite (e.g., kxy = -kyx = kcand cxy = -cyx = cc) and the diagonal terms are same (e.g., kxx = kyy = kd

and cxx = cyy = cd) (Childs et al., 1986). Considering these relationships and neglecting the cross-

coupled added-mass terms, equation (2) takes the following form

0

0

x d c d c d

y dc d c d

f k k c c mx x

f mk k y c c y y

= + +

(3)

where subscripts: dand crepresent direct and cross-coupled, respectively. The RDPs largely affect the

performance of the turbomachineries as they lead to serious synchronous and sub-synchronous

vibration problems. Whirl frequency ratio, f= kc /(cd) is a useful non-dimensional parameter for

comparing the stability properties of seals. For circular synchronous orbits, it provides a ratio between

the destabilizing force component due to kc and the stabilizing force component due to cd. In

experimental estimation of RDPs of seals, these coefficients (of equation (2) and (3)) are determined

with the help of measured vibrations data from a seal test rig.

The more recent textbooks on rotor dynamics include information on rotor dynamic characteristics of

rotary seals. Vance (1988), Childs (1993), Krmer (1993), Rao (2000), Adams (2001) and Tiwari et

al. (2005) provide a good introductory treatments of seal dynamics.

References:


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Admas, M.L. Jr, 2001, Rotating Machinery Vibration, Marcel Dekker, Inc., New York.Changsen, W., 1991, Analysis of Rolling Element Bearings, Mechanical Engineering Publications

Ltd., London.Childs, D.W., 1993, Turbomachinery Rotordynamics: Phenomena, Modeling and Analysis, John

Wiley & Sons, Inc., New York.El-Sayed H. R., 1980, Wear, 63, 89-94.Stiffness of deep-groove ball bearing.Eschmann, P., Hasbargen, I. and Weigand, K., (1985),Ball and Roller Bearings, Theory, Design and

Application. John Wiley and Sons:New York.Gargiulo E.P., 1980,Machine Design, 52, 107-110. A simple way to estimate bearing stiffness.Harris, T.A., 2001,Rolling Bearing Analysis, Wiley, New York.Hertz, H., (1896), Miscellaneous Papers, Macmillan, London, 163-183. On the contact of rigid elastic

solids and on hardness.Hummel, C., 1926, Kristische Drehzahlen als Folge der Nachgiebigkeit des Schmiermittels im

Lager, VDI-Forschungsheft, 287.Johnson. T.L., 1991, Contact Mechanics, 2ndedition, McGraw-Hills, New York.Jones, A. B., 1946,Analysis of Stresses and Deflections, New Departure Engineering Data, Briston.Jones A.B., 1960, Transactions of ASME, Journal of Basic Engineering, 309-320, A general theory

for elastically constrained ball and radial roller bearings under arbitrary load and speedconditions.

Kr

mer E., 1993, Dynamics of Rotors and Foundations, Springer-Verlag, New York.R. Kashyap and R. Tiwari, 2006, Prediction of Heat Generations and Temperature Distributions atCritical Contact Zones of High-Speed Rolling Bearings,Proceedings of 18th National & 7thISHMT-ASME Heat and Mass Transfer Conference, January 4-6, 2006, IIT Guwahati.

Lim, T. C. and Singh, R., (1990a), Journal of Sound and Vibration 139 (2), 179-199. VibrationTransmission Through Rolling Element Bearings, Part I: Bearing Stiffness Formulation.

Lim, T. C. and Singh, R., (1990b), Journal of Sound and Vibration 139 (2), 201-225. VibrationTransmission Through Rolling Element Bearings, Part II: System Studies.

Lim, T. C. and Singh, R., (1991), Journal of Sound and Vibration 151 (1), 31-54. VibrationTransmission Through Rolling Element Bearings, Part III: Geared Rotor System Studies.

Lim, T. C. and Singh, R., (1992), Journal of Sound and Vibration 153 (1), 37-50. VibrationTransmission Through Rolling Element Bearings, Part IV: Statistical Energy Analysis.

Lim, T. C. and Singh, R., (1994), Journal of Sound and Vibration 169 (4), 547-553. Vibration

Transmission Through Rolling Element Bearings, Part V: Effect of Distributed Contact Loadon Roller Bearing Stiffness Matrix.Newkirk, B.L., 1924, Shaft Whipping, General Electric Review, pp. 169.Newkirk, B. L. and Taylor, H.D., 1925, Shaft Whipping due to Oil Action in Journal Bearing,"

General Electric Review, 559-568.Palmgren, A., (1959),Ball and Roller Bearing Engineering, 3rd ed., Burbank.Ragulskis, K. M., Jurkauskas, A. Yu., Atstupenas, V. V., Vitkute, A. Yu., and Kulvec, A. P., (1974),

Vibration of Bearings. Vilnyus: Mintis Publishers.Rao, J. S., 2000, Vibratory Condition Monitoring of Machines, Narosa Publishing House, New

Delhi.Schweitzer, G., Bleuler, H. and Traxler, A., 1994, Active Magnetic Bearing: Basics, Properties and

Application of Active Magnetic Bearings. Vdf Hochschulverlag AG an der ETH, Zrich.Smith, D.M., 1969,Journal Bearings in Turbomachinery, Chapman and Hall, London.

Stodola, A., 1925, Kritische Wellenstrung infolge der Nachgiebigkeit des lpolslers im Lager(Critical shaft perturbations as a result of the elasticity of the oil cushion in the bearings),Schweizerische Bauzeitung, Vol. 85,No. 21, May.

Stolarski, T. A., (1990), Tribology in Machine Design. Oxford: Heinemann Newnes.Timoshenko, S. and Goodier, J., 1951, Theory of Elasticity, 2ndedition, McGraw-Hills, New York.Vance, J.M., 1998, Rotordynamics of Turbomachinery,John Wiley & Sons Inc, New York.


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Rtiwari Rd Book 06c