chapter 6 bearing and seal systems - iitg.ac.in · chapter 6 bearing and seal ... load type both...

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

CHAPTER 6

BEARING AND SEAL SYSTEMS

All rotating machineries are supported by bearings and fitted with seals. The bearings clearly

constitute a vital component in any turbo-machine and a good understanding of their dynamic

behaviour is a pre-requisite to the prediction of the machine’s properties. The influence of bearings on

the performance of rotor-bearing systems has been recognised for many years. One of the earliest

attempts to model a journal bearing was reported by Stodola (1925) and Hummel (1926). They

represented the fluid-film as a simple spring support, but their model was incapable of accounting for

the observed finite amplitude of oscillation of a shaft operating at a critical speed. Concurrently,

Newkirk (1924) and Newkirk and Taylor (1925) described the phenomenon of bearing induced

instability, which he called oil whip, and it soon occurred to several investigators that the problem of

rotor stability could be related to the properties of the rotor dynamic parameters i.e. RDPs (the rotor

dynamic (or dynamic) parameters are also known as: bearings/seals force (or moment) coefficients;

added-mass, damping and stiffness coefficients; linearized rotor dynamic parameters; dynamic

impedances).

Seals are mainly used to reduce the leakage of working and lubricating fluids through the interface

between machine parts. Some leakage is inevitable, and it results in axial fluid velocities though the

seal in the direction of the pressure drop. The present day requirement of critical sealing applications

have a diverse range of operating condition requirements such as (i) cryogenic temperature, (ii) hard

vacuum, (iii) ultra-clean systems, (iv) leakage control to 10-12 cc/sec, (v) pressures over 100 bar, (vi)

temperatures exceeding 800°C, (vii) hard-to-handle liquids and gases, (viii) high pressure pulsations

and (ix) rotor speeds as high as 105 rpm. These extreme conditions of seals are challenging tasks on

the space age aviation and aerospace industries. The importance of calculations of RDPs of seals

arose in the late 1970s in regard to vibration problems related to high-pressure oxygen turbopump of

the space shuttle main engine. Compressors used in many industries also had the instability problems

within the operating speed range. Seals in high-speed operations of turbo-machines leads to

instability. The main factor, which governs the instability, is RDPs of seals. RDPs of seals are greatly

dependent on many physical and mechanical parameters such as lubricant and working fluid

temperatures, pressure drop, seal clearances, surface roughness and patterns, rotor speeds, eccentricity

and misalignments.

Designers should know the types of bearing and seal that could be used, and performance

characteristics associated with each of them. The main aim will be towards the operation of bearings


272

and seals under the action of dynamic loads, in particular the RDPs of different bearings and seals

types are discussed. These characteristics have a major influence on the overall system dynamics.

Bearing types are (i) rolling element bearings (ii) oil film hydrodynamic bearings (iii) gas bearings

(iv) squeeze film bearings and (v) hydrostatic bearings. Dynamic seals can be classified as (i) plain

seals (ii) roughened seals (iii) contact seals and (iv) brush seals. Table 6.1 one compares the criterion

of selection between the rolling and journal bearings.

Table 6.1 Criteria for selection of the bearing type

S.N. Parameter Rolling bearing Journal bearing

1. Starting friction Low High

2. Space Less axial space Less radial space

3. Load type Both radial and axial Radial or axial

4. Failure time Gradual Sudden

5. Clearance Less More

6. Deflection of shaft Less More

7. Assembly Simple Tedious

8. Maintenance Less More

9. Replacement Cheaper Costly

10. Load carrying capacity Moderate High

11. Electrical insulation No Yes

12. Noise High Low

13. High overload For short periods For moderate periods

6.1 Rolling Element Bearings

Rolling element bearings or simply rolling bearings are the most common type of bearings, it requires

less boundary dimensions and can transmit heavy and variable loads of various forms and can easily

installed and serviced. Figure 6.1 shows a cut section of the ball and roller bearings. Figures 6.2 and

6.3 show, respectively, the nomenclature and various basic geometries of a most simple ball deep

groove bearing. Various types of rolling element bearings (refer Figure 6.4) are: deep groove ball

bearings, angular contact ball bearings, self-aligning ball bearings, cylindrical roller bearings,

spherical roller bearings, tapered roller bearings, needle roller bearings, thrust ball/roller bearings and

linear re-circulating ball bearings. Bearings are selected on the basis of the magnitude and direction of

the loading and speeds. Our aim is to study bearing elastic deformation and the evaluation of the

linear and non-linear bearing stiffness. The bearing stiffness plays a particularly important role in the

determination of the overall rotor bearing system dynamic characteristics e.g. critical speeds and

stability analysis. These are not usually documented in manufacturer’s selection catalogues. Damping


273

in rolling element bearings is very small, however, it may be significant in case lubricant is trapped

between the outer ring and bearing housing, due to squeeze-film action.

(a) Ball Bearing (b) Roller Bearing

Figure 6.1 A cut section of rolling bearings

Figure 6.2 Nomenclature of a ball bearing

Figure 6.3 A radial ball bearing


274

Figure 6.4 Types of rolling bearings and rolling elements

Bearing Systems

Table 2.2 summarizes various bearing types available, together with their relative merits. Very high

speeds are those speeds having dN number more than 3 millions, where d is the bore diameter in mm

and N is the bearing operating speed in rpm. High-speed bearings find applications in aerospace and

space technologies. Low speed bearings (i) for gyroscopes have very long life of the order of 15 years

and they are very high precision bearings. Lubrication for such bearings are quite challenging since it

requires about one to two drops per year. Micro and nano-pumps may have very useful applications

for such lubrication flow rate. (ii) In rotary kiln in cement factories have 2 to 3 rpm, however, the bore

diameter of the size of 4 m.

Estimation of the elastic parameters of bearings involves establishing a relationship between the

incident load on the bearing and its resultant deformation. The classical solution for the local stress

and deformation of two elastic bodies apparently contacting at a single point was established by Hertz

in (1896). Hertz's analysis is applied to surface stresses caused by a concentrated force, applied

perpendicular to the surface. In the determination of contact deformation versus load, the concentrated

load applied normal to the surface alone, is considered, for most rolling element bearing applications.

It is possible to determine how the bearing load is distributed among the balls or rollers, after having

determined how each ball or roller in a bearing carries load. To do this it is necessary to develop load-


275

deflection relationships for rolling elements contacting raceways. Most rolling bearing applications

involve steady-state rotation of either the inner or outer raceways or both. Rolling element centrifugal

forces, gyroscopic moments and frictional forces and moments do not significantly influence this load

distribution in most applications. Theoretical models (Palmgren, 1959; Ragulskis et al., 1974; Harris,

1984; Eschamann et al. 1985; Stolarski, 1990) are available for estimation of bearing stiffnesses under

static loading conditions. The stiffness characteristics of the rolling element bearing are studied under

combined loading i.e. under the action of radial and axial force and the moment, hence there will be

radial, axial and angular displacement. All the displacement are coupled i.e. they will depend upon the

combined effect of all the forces and moments acting. The stiffness matrix obtained is of size 5×5. To

obtain displacement a set of five equilibrium equations are solved for a given loading (Lim and

Singh).

Table 2.2 Type of rolling bearings

Bearing type Suitable for load/speed type Other remarks

Deep groove

ball bearings

Radial and/ or light axial. Very

high speed.

Available with shields and seals. Available in

double row form to accommodate higher radial

loads.

Angular

contact

bearings

Larger axial loads than deep

groove ball bearing. Normally

axial load at least as large as

radial load. Very high speed.

Available in ‘matched pairs’ to provide accurate

axial positioning of shaft.

Four-point

(duplex

bearing)

High load-carrying capacity. To

be used when axial load greater

than radial load. Moderate

speeds.

Available in double row form

Self-aligning bearings

Lower radial and axial load-carrying capacity than deep

groove ball bearing. Moderate

speeds

Can accommodate large amounts of misalignment

Cylindrical

roller bearings

Large radial loads and very

light axial loads. High speed.

Separate inner and outer rings. Available in ‘match

assembly’ form for tight control over internal clearance.

Spherical roller

bearings

Very high radial loads and light

axial loads. Moderate speeds

Non-separable. Operate at lower speed than

cylindrical roller. Can accommodate misalignment

Tapered roller

bearings

Can support very large radial

and axial loads. Moderate

speeds

Lower operating speed than angular contact

bearings. Provides a very rigid shaft mounting

Needle roller

bearings

Can support large radial loads

at speeds similar to those of cylindrical roller bearings.

Very small bearing outer diameter. Can be used

without one or both rings to save space provided the seatings are surface treated.

Thrust ball

bearings

Axial loads only (can be in

either direction).

Axial load must always be present. Can function

with a small amount of misalignment when used

with a spherical seating washer. Available in

cylindrical and needle roller form for very large

loads


276

6.1.1 Bearing Elastic Deformation

The amount by which a shaft, supported by bearing, can be displaced from its concentric position by a

given load is of our interest. This displacement depends upon the elastic deformation of the bearing

raceways and of the rolling elements themselves. The elastic deformation of these components

depends upon the geometry of the bearing and on the bearing internal clearance or preload (i.e. the

negative clearance). The bearing internal clearance is an important factor because it determines the

size of the stressed area of the rings (i.e. the load zone) as shown in Figure 6.5. Lesser the clearance

then more and more rolling elements will share the external applied load from the shaft. However,

preload may reduce the life of bearings high fatigue stresses in rolling elements.

Figure 6.5 Load zone in rolling bearings

For a given load, the size of the contact area determines the magnitude of stresses in the bearing

components, and so determines the amount of elastic deformation. In case of a ball bearing the contact

area between the rolling element and the raceway is zero when no load is applied; ‘point contact’ is

said to occur. For two bodies with point contact, made of same material and subjected to a

compressive load F, from Hertzian contact theory it can be written that the total deformation δ is

given by

23

2s

b b

Fc

D D

δ =

(6.1)

where Db is the diameter of the ball (in mm) and δc is a deformation constant which depends upon

material properties and the geometry of contacting surfaces (Harris, 2001). In case of rolling element

bearings the elastic deformation can take place at both contact between the inner raceway and the

cr = 0, Ψl = 900

(b) No clearance

Ψl

Ψl Ψl

cr < 0, Ψl > 900

(c) Negative internal

clearance (preload)

cr > 0, Ψl > 900

(a) Positive clearance

Load

direction

Inner

ring


277

rolling element, and also at contact between the outer raceway and the rolling element. The total

deformation at the location of one rolling element is therefore given by

( )2/3

2i o

b

Fc c

d Dδ δ

δ = +

(6.2)

which can be rewritten as

32

nF K δ= (6.3)

with

12

32

i o

bn

DK

c cδ δ

= +

(6.4)

where nK ( in N mm-3/2 ) depends mainly upon bearings curvature ratio f as defined in equation (6.5)

and other terms i.e. Db, Dm and α, where Dm is the pitch diameter of the bearing and α is the contact

angle (refer Figures 6.2 and 6.4). The subscripts: o represents outer and i represent inner. We have

( )2 b

b

r Df

D

−= (6.5)

where r is the radius of curvature of the raceway (as shown in Figure 6.6) in mm.

Figure 6.6 The contact angle in the ball bearing


278

For detailed accurate calculation of nK readers are referred to Harris (2001). If dependence of nK on

the bearing size (D) and contact angle α is neglected then values of nK may be obtained directly

from and for 0.1inner outerf f≈ < (which is usually the case for deep groove ball bearings), as

(Palmgren, 1959)

12

0.35

34300n bK D

f= (Approximate value) (6.6)

For roller-bearings the corresponding deformation constant ln

K is dependent only on the effective

length of rollers themselves, le and approximately given as

0.5226200ln eK l= (6.7)

The effective length of rollers is that which is actually in contact with the raceway, usually this is the

actual roller length minus the roller corner radii as shown in Figure 6.7. The relationship between the

deformation constant, the applied compressive load and the elastic deformation at a single roller is

given by

1.08

lnF K δ= (6.8)

Figure 6.7 A crowned roller showing the crown radius and the effective length

In roller bearings there is “line contact” between the rolling element and bearing raceways. Equations

(6.3) to (6.8) indicate relationships of elastic deformation at one rolling element under the action of a


279

single compressive force applied at the element. In real bearings where more than one element is in

compression, the effect at each element should be incorporated as shown in Figure 6.8(a).

Figure 6.8 (a) Displacement of the inner ring with respect to the outer ring (b) Displacement of a

rolling element located at an angle ψ.

Let the bearing inner ring is displaced from the concentric position by a distance xm, part of which

consists of the radial internal clearance cr. The elastic deformation in the direction of applied load will

be

m rx x c= − (6.9)

and the elastic deformation at any rolling element position, ψ, as Figure 6.8(b), is given by

cosm rx x cψ= − (6.10)

Setting deformation to zero in equation (6.10) the load zone can be obtained by

1cos rl

m

cx

ψ − =

(6.11)

For a given bearing, with known angular positions of rolling elements, a displacement of the inner

ring xm may be assumed and thus the resulting elastic deformation at each rolling element calculated

from equation (6.10). Contact forces at each element may then be evaluated from equations (6.3) and

(6.6). These are added as vector to give the net radial forces Fr applied to the bearing in order to

produce the assumed displacement xm. In case there is a difference between the estimated net radial

force and applied radial force, a new displacement is then chosen and above procedure is repeated

until it converges up to the desired decimal of accuracy. However, if we know the displacement then

xm xm

Center of

outer ring Center of

inner ring

External load

direction

lψψ

ψcosmx

ψ


280

the load required to produce the displacement can be obtained by straightforward method i.e.

rm FFxx →→→ )()( ψψ .

If the fraction of the net radial load applied that is transmitted through the rolling element directly in

line with the applied load in known than the resulting inner ring displacement xm may be calculated

directly, from equation (6.3) for the ball bearing, as

23

mm r

n

Fx c

K

= +

(6.12)

where Fm is the force on the rolling element directly in line with the applied radial load and cr is the

internal clearance. Similarly, for roller bearing, we have

11.08

l

mm r

n

Fx c

K

= +

(6.13)

Approximate relationships (neglecting effect of bearing clearance and geometry) given by Palmgren

(1959) are

1

2 384.36 10 m

m

b

Fx

D

− = ×

for ball-bearings (6.14)

and

0.910

0.83.06 10 m

m

e

Fx

l

− = ×

for roller bearings (6.15)

where Fm is in N and Db and le are in m. Before equation (6.12) can be applied, the maximum

compression force, Fm, on a single element must first be determined. The relationship between Fm and

net radial force applied, Fr, is approximately related as

14m r

cF F

Z= (6.16)


281

here Z is the number of rolling element and c1 is a number which depends upon the applied load, the

number of rolling element, the deformation constant nK and the bearing clearance. For approximate

calculations, we have 1c = 5 for both ball & roller bearing. The variation of elastic deformation of the

most stressed rolling element, (xm – cr), with the applied load is shown in Figure 6.9.

Figure 6.9 Variation of stiffness of bearings with the clearance/preload

Once the relationship between the applied radial force and the bearing elastic deformation has been

established, the bearing stiffness can be calculated. The stiffness of rolling bearings is highly non-

linear with displacement of the inner ring with respect to the outer ring. On substituting F = Fm and

rm cx −=δ into equations (6.3) and (6.8), we get

32[ ]m n m rF K x c= − (for ball bearing) (6.17)

and

1.08[ ]lm n m rF K x c= − (for roller bearing) (6.18)

Equating equations (6.17) and (6.18) gives the load deformation relationship for a single ball which is

sharing maximum load or which is just below the radial external load. The bearing stiffness k can be

obtained by differentiating with respect to displacement xm to give

( ) 0.5

1 1

1.5 ( )mrm n m r

m m

dFdF Z Zk x K x c

dx c dx c= = = − (for ball bearing) (6.19)

and

(xm-cr)

µm

(xm-cr)

µm

cr=0

cr=20

cr=60

cr= -10

cr= -20

cr=0

cr=60

cr= -5

cr= -10

(a) Ball-bearing (b) Roller-bearing

0.0005 0.001 0.0015 0.0025 0.005 0.0075 5.1/ mmnCFr δ

08.1/ mmnCFlr δ

10

20

30

40

0 0


282

( ) 0.08

1 1

1.08 ( )l

mrm n m r

m m

dFdF Z Zk x K x c

dx c dx c= = = − (for roller bearing) (6.20)

where Z is the number of rolling element, c1 is a number depends upon a material property and

geometry of bearings. The value of xm that should be used in these expressions should be that which is

caused to occur as a consequence of the steady load, which is applied, to the bearing. It should be

noted that effectively we are linearising the bearing stiffness at static equilibrium by the assumption of

small oscillations about this point, which may cause due to the unbalance. If large amplitudes of

vibration are expected to occur at the bearing, such that the bearing stiffness non-linearity becomes a

significant factor in machine response calculations, then an effective stiffness can be defined as

force amplitude

displacement amplitudek = (6.21)

When the unbalance load is greater than the steady load, then

( )unbalance

m r

Fk

x c=

− (6.22)

The discussion above has made no reference to effects of movement of rolling elements around

raceways on the bearing force-displacement relationship and the stiffness. In fact, although the

number and location of rolling elements in the load zone changes as the bearing is rotated, the

displacement and the stiffness of the bearing are little affected. Hence rotation of the bearing need not

be allowed for in stiffness calculations. For more detail in this regard readers are referred to Ragulskis

et al., 1974.

Example 6.1 Obtain the load zone angle of a rolling bearing for the following cases (i) cr = xm (ii) cr =

- xm (iii) cr = 0.5 xm (iv) cr = - 0.5 xm; where xm is the displacement of inner ring from the concentric

position and cr is the radial clearance. Draw load zones for each case.

Answer:

Case I: r mc x=

Lϕ is the load zone angle ( ) 1 0cos cos (1) 0r mc x −= = =

None of the bearing element is loaded.

Bearing inner

ring

Case (i)


283

Case II

r mc x= −

1 0/ cos ( 1) 180L r mc xϕ −= = − =

All the rolling elements taking part in load sharing.

Case III

0.5r mc x=

( )1 1 0cos / cos (0.5) 60L r mc xϕ − −= = =

Very less number of rolling elements will take part in load sharing

Case IV

0.5r mc x= −

( )1 1 0cos / cos ( 0.5) 120L r mc xϕ − −= = − =

Sufficient number of rolling elements will take part in the load sharing.

Example 6.2 The specification of a SKF 7218 angular contact ball bearing are as follows: Inner and

outer grove curvature radius = 11.64 mm, Inner raceway diameter = 102.92 mm outer raceway

diameter 147.9 mm, bore diameter = 90.0 mm, outer diameter = 160.0 mm, ball diameter = 22.25 mm,

number of balls = 16 and preload = 0.6 µm. Plot bearing radial stiffness versus elastic radial

deformation. For plotting purpose take six points i.e. take elastic deformations: 0.002 mm, 0.006 mm,

0.01 mm, 0.014 mm, 0.018 mm and 0.022 mm.

Answer: From equation (6.19), for the stiffness of the ball bearing, we have

( ) 0.5

1

1.5 ( - )m n m r

Zk x K x c

c= with

1 5 (Approx)c =

where n is the number of balls = 16 ; 3(- ) preload 0.6 µm 0.6 10 mmrc

−= = = × . From equation

(6.6), we have

1 2

0.35

34300n bK D

f=

with

Case (ii)

0180=Lϕ

060=Lϕ Case (iii)

0120=Lϕ Case (iv)


284

(2 ) (2 11.64 22.25)

0.04322.25

b

b

r Df

D

− × −= = =

where raceway curvature 11.64 mm and diameter of ball 22.25 mmbr D= = = = . Hence,

1 2 1.5n 0.35

34300(22.25) 474276.5 N/mm

(0.043)K = =

Hence the stiffness can be obtained as

( ) ( )1 2161.5 474276.5 0.0006

5mk x x= × × × +

Figure 6.1 shows the variation of the stiffness with displacement due to static loads. The stiffness

values for various bearing static displacements are also given in Table 6.3.

Table 6.3 Bearing stiffness

( )5

5

6

6

6

6

(mm) (N/mm)

0.002 3.08 10

0.006 11.87 10

0.010 2.07 10

0.014 2.94 10

0.018 3.8 10

0.022 4.71 10

m mx k x

×

×

×

×

×

×

Figure 6.? Bearing stiffness

Exercise 6.1 List down parameters on which the stiffness of rolling element bearings depends.

Alternately, more accurate determination of the stiffness of rolling contact bearings can be done as

follows. From Hertz’s theory of elasticity for the point loading, we have the following load-

deformation relation

2 /3~Qδ or 3/ 2

pQ K δ= (6.23)

In general it can be written as

nQ Kδ= (6.24)


285

where δ is the deflection, Q is the load and K is the load-deflection factor, n is equal to 3/2 for the

ball bearing (point contact) and 10/9 for the roller bearing (line contact).

Total elastic forces at the points of contact of the ith ball with the inner and outer rings as shown in

Figure 6.10 can be expressed as

nQ Kδ= (6.25)

with

1/ 1/

1

(1/ ) (1/ )

n

n ni o

KK K

=

+ (6.26)

where n=3/2, i and o is represent the inner and outer contact points, respectively. From theory of

elasticity for contact problems (Timoshenko and Goodier, 1950), we have

2 / 3

2 2(1 ) (1 )3

2 2

I II

I II

Q

E E

ρν νδ δ

ρ∗ − −

= +

∑∑

(6.27)

with

1/ 3

*

2

2

2

πδ

π κℜ = ℑ

;

1/ 22/ 2

2

2

0

11 1 sin d

π

φ φκ

− ℑ = − −

∫ ;

1/ 22/ 2

2

2

0

11 1 sin d

π

φ φκ

ℜ = − −

∫

(6.28)

where subscripts I and II refer to the first and second contacting bodies; Q refers to normal force

between contacting bodies; E refers to Young’s modulus; ν refers to Poisson’s ratio, the elliptical

eccentricity parameter κ = a / b, a and b are the semi-major and semi-minor axis of the elliptical

contact area for the point contact, ρ∑ is curvature sum, δ∗ is the dimensionless contact

deformation. Equation (6.27) can be rearranged as

Figure 6.10 Ball and raceways


286

( )

12 25 / 2

3 / 2 3 / 2

1/ 2

(1 ) (1 )2

3

I II

I II

QE E

ν νδ δ

ρ

−−∗

− − = + ∑ (6.29)

On comparing equations (6.23) and (6.29), we have

( ) ( )15 / 2 2 2 3 / 2

1/ 2(1 ) (1 )2

3

I IIp

I II

KE E

ν νρ δ

− −− ∗

− − = +

∑ (6.30)

where p refers to the point contact either i or o. For the steel ball and the steel raceway contact, we

have

( ) ( )3/ 2

1/ 25 1.52.15 10 N/mmpK ρ δ−

− ∗= × ∑ (6.31)

which can be written for the inner and outer ring contact as

( ) ( )3 / 2

1/ 25 * 1.52.15 10 N/mmi i iK ρ δ−

−= × ∑ and ( ) ( )

3 / 21/ 25 * 1.52.15 10 N/mmo o oK ρ δ

−−

= × ∑

(6.32)

with

1 1 24

1i

iD f

γρ

γ

= − + −

∑ ; 1 1 24

1o

oD f

γρ

γ

= − − −

∑ (6.33)

cos0

m

D

dγ =

�

; ii

rf

D= ; o

o

rf

D= and

1 1

2 2( ) ( )m i oD d d D d= + ≈ + (6.34)

where D is the ball diameter, md is pitch diameter and α is contact angle, id and od are inner and

outer ring raceway contact diameter respectively and ir and or are inner and outer groove radius

respectively. We have

1 2

1( )

1 24

1

ii

i

fF

f

γγ

ργγ

+−

=− +

−

and

1 2

1( )

1 24

1

oo

o

fF

f

γγ

ργγ

−+

=− −

+

(6.35)

where ( )F ρ is curvature difference. The plot of *δ as function of ( )F ρ is shown in Figure 6.11.


287

To find K from equation (6.32), we need: (i) type of bearing (ii) inner raceway diameter id (iii) outer

raceway diameter od (iv) ball diameter Db (v) number of ball Z (vi) inner groove radius ir (vii) outer

groove radius or (viii) pitch diameter Dm (viii) material modulus of elasticity EI and EII, and (ix)

material Poisson’s ratio, and I IIν ν of two contact bodies.

Figure 6.11 *δ as function of ( )F ρ

chapter 6 bearing and seal systems - iitg.ac.in · chapter 6 bearing and seal ... load type both...

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