M Rades - Dynamics of Machinery 2

Mircea Rades

Dynamics

of Machinery II

2009

Preface

This textbook is based on the second part of the Dynamics of Machinery lecture course given since 1993 to students of the English Stream in the Department of Engineering Sciences (D.E.S.), now F.I.L.S., at the University Politehnica of Bucharest. It grew in time from a postgraduate course taught in Romanian between 1985 and 1990 at the Strength of Materials Chair and continued within the master course Safety and Integrity of Machinery until 2007.

Dynamics of Machinery, as a stand alone subject, was first introduced in the curricula of mechanical engineering at D.E.S. in 1993. To sustain it, we published Dynamics of Machinery in 1995, followed by Dinamica sistemelor rotor-lagăre in 1996 and Rotating Machinery in 2005.

The course aims to: a) increase the knowledge of machinery vibrations; b) further the understanding of dynamic phenomena in machines; c) provide the necessary physical basis for the development of engineering solutions to machinery problems; and d) make the students familiar with machine condition monitoring techniques and fault diagnosis.

As a course taught to non-native speakers, it has been considered useful to reproduce, as language patterns, full portions from English texts. For the students of F.I.L.S., the specific English terminology is defined and illustrated in detail.

Basic rotor dynamics phenomena, simple rotors in rigid and flexible bearings as well as examples of rotor dynamic analyses are presented in the first part. This second part is devoted to the finite element modeling of rotor-bearing systems, fluid film bearings and seals, and instability of rotors. The third part treats the analysis of rolling element bearings, gears, vibration measurement for machine condition monitoring and fault diagnosis, standards and recommendations for vibration limits, balancing of rotors as well as elements of the dynamic analysis of reciprocating machines and piping systems. No reference is made to the vibration of discs, impellers and blades.

July 2009 Mircea Rades

Prefata

Lucrarea se bazeaza pe partea a doua a cursului de Dinamica masinilor predat din 1993 studentilor Filierei Engleze a Facultatii de Inginerie in Limbi Straine (F.I.L.S.) la Universitatea Politehnica Bucuresti. Continutul cursului s-a largit in timp, pornind de la un curs postuniversitar organizat intre 1985 si 1990 in cadrul Catedrei de Rezistenta materialelor si continuat pana in 2007 la cursurile de masterat in specialitatea Siguranta si Integritatea Masinilor. Capitole din curs au fost predate din 1995 la cursurile de studii aprofundate si masterat organizate la Facultatea de Inginerie Mecanica si Mecatronica.

Dinamica masinilor a fost introdusa in planul de invatamant al F.I.L.S. in 1993. Pentru a sustine cursul, am publicat Dynamics of Machinery la U. P. B. in 1995, urmata de Dinamica sistemelor rotor-lagare in 1996 si Rotating Machinery in 2005, ultima continand materialul ilustrativ utilizat in cadrul cursului.

Cursul are un loc bine definit in planul de invatamant, urmarind: a) descrierea fenomenelor dinamice specifice masinilor; b) modelarea sistemelor rotor-lagare si analiza acestora cu metoda elementelor finite; c) inarmarea studentilor cu baza fizica necesara in rezolvarea problemelor de vibratii ale masinilor; si d) familiarizarea cu metodele de supraveghere a starii masinilor si diagnosticare a defectelor.

Fiind un curs predat unor studenti a caror limba materna nu este limba engleza, au fost reproduse expresii si fraze din lucrari scrise de vorbitori nativi ai acestei limbi. Pentru studentii F.I.L.S. s-a definit si ilustrat in detaliu terminologia specifica limbii engleze.

In prima parte se descriu fenomenele de baza din dinamica rotoarelor, raspunsul dinamic al rotoarelor simple in lagare rigide si lagare elastice, precum si principalele etape ale unei analize de dinamica rotoarelor. In aceasta a doua parte se prezinta modelarea cu elemente finite a sistemelor rotor-lagare, lagarele hidrodinamice si etansarile cu lichid si gaz, precum si instabilitatea rotoarelor. In partea a treia se trateaza lagarele cu rulmenti, echilibrarea rotoarelor, masurarea vibratiilor pentru supravegherea functionarii masinilor si diagnosticarea defectelor, standarde si recomandari privind limitele admisibile ale vibratiilor masinilor, precum si elemente de dinamica masinilor cu mecanism biela-manivela si vibratiile conductelor aferente. Nu se tratează vibratiile paletelor, discurilor paletate si ale rotilor centrifugale.

Iulie 2009 Mircea Rades

Contents

Preface i

Contents iii

5. Finite element analysis of rotor-bearing systems 1 5.1 Rotor component models 1

5.2 Kinematics of rigid body precession 3

5.2.1 Main reference frames 3

5.2.2 Rigid-body precession 3

5.2.3 Small rotations of the spin axis 4

5.3 Equations of motion for rotor components 7

5.3.1 Thin disks 7

5.3.2 Uniform shaft elements 13

5.3.3 Bearings and seals 31

5.3.4 Flexible couplings 35

5.4 System equations of motion 36 5.4.1 Second order configuration space form 37

5.4.2 First order state space form 40

5.5 Eigenvalue analysis 40

5.5.1 Right and left eigenvectors 41

5.5.2 Reduction to the standard eigenvalue problem 42

5.5.3 Campbell and stability diagrams 43

5.6 Unbalance response 47

5.6.1 Modal analysis solution 47

5.6.2 Spectral analysis solution 48

5.7 Kinematics of elliptic motion 49

5.7.1 Elliptic orbits 49

5.7.2 Decomposition into forward and backward circular motions 52

5.7.3 Variable angular speed along the ellipse 54

5.8 Model order reduction 56

DYNAMICS OF MACHINERY iv

5.8.1 Model condensation 56

5.8.2 Model substructuring 62

5.8.3 Stepwise model reduction methods 68

References 76

6. Fluid film bearings and seals 77 6.1 Fluid film bearings 77

6.2 Static characteristics of journal bearings 78

6.2.1 Geometry of a plain cylindrical bearing 79

6.2.2 Equilibrium position of journal center in bearing 80

6.3 Dynamic coefficients of journal bearings 83

6.4 Reynolds equation and its boundary conditions 84 6.4.1 General assumptions 86

6.4.2 Reynolds’ equation 87

6.4.3 Boundary conditions for the fluid film pressure field 89

6.5 Analytical solutions of Reynolds’equation 90 6.5.1 Short bearing (Ocvirk) solution 90

6.5.2 Infinitely long bearing (Sommerfeld) solution 99

6.5.3 Finite-length cavitated bearing (Moes) solution 99

6.6 Physical significance of the bearing coefficients 103

6.7 Bearing temperature 107

6.7.1 Approximate bearing temperature 108

6.7.2 Viscosity-temperature relationship 109

6.8 Common fluid film journal bearings 112

6.8.1 Plain journal bearings 112

6.8.2 Axial groove bearings 112

6.8.3 Pressure dam bearings 114

6.8.4 Offset halves bearings 116

6.8.5 Multilobe bearings 117

6.8.6 Tilting pad journal bearings 124

6.8.7 Rayleigh step journal bearings 128

6.8.8 Floating ring bearings 129

6.9 Squeeze film dampers 130 6.9.1 Basic principle 130

CONTENTS v

6.9.2 SFD design configurations 132

6.9.3 Squeeze film stiffness and damping coefficients 133

6.9.4 Design of a squeeze film damper 135

6.10 Annular liquid seals 137 6.10.1 Hydrostatic reaction. Lomakin effect 138

6.10.2 Rotordynamic coefficients 139

6.10.3 Final remarks on seals 146

6.11 Annular gas seals 147

6.12 Floating contact seals 150 6.12.1 Design characteristics 151

6.12.2 Seal ring lockup 154

6.12.3 Locked-up oil seal rotordynamic coefficients 155

References 159

7. Instability of rotors 161 7.1 Whirling of rotating shafts 161

7.2 Instability due to rotating damping 164

7.2.1 Planar rotor model 165

7.2.2 Qualitative effect of damping 166

7.2.3 Whirl speeds of rotor with rotating damping 168

7.2.4 Quantitative effects of damping 172

7.3 Whirl in hydrodynamic bearings 173

7.3.1 Oil-whirl and oil-whip phenomena 174

7.3.2 Half frequency whirl 176

7.3.3 Onset speed of instability 178

7.3.4 Crandall’s explanation of journal bearing instability 179

7.3.5 Stability of linear systems 187

7.3.6 Instability of a simple rigid rotor 189

7.3.7 Instability of a simple flexible rotor 194

7.3.8 Instability of complex flexible rotors 199

7.4 Interaction with fluid flow forces 199

7.4.1 Steam whirl 199

7.4.2 Impeller-diffuser interaction 201

DYNAMICS OF MACHINERY vi

7.5 Dry friction backward whirl 203

7.5.1 Rotor-stator rubbing 203

7.5.2 Dry friction whirl 204

7.6 Instability due to asymmetric factors 206

7.6.1 Parametric excitation 207

7.6.2 Shaft anisotropy 207

7.6.3 Asymmetric inertias 219

7.6.4 Finite element analysis of asymmetric rotors 226

References 229

Index 233

5. FINITE ELEMENT ANALYSIS

OF ROTOR-BEARING SYSTEMS

This chapter presents the background of finite element techniques used for the prediction of rotordynamics behavior. The inertia, gyroscopic and stiffness matrices are established for axi-symmetric disks and uniform shaft segments. Journal bearings are described by linearized models with generally non-symmetric stiffness and damping matrices. Seals, dampers and couplings are also modeled by corresponding matrices. The forcing vectors due to unbalance are established for constant angular velocity. Model order reduction and condensation techniques are also considered.

5.1 Rotor component models

For the analytic prediction of the rotordynamic response, the main components of a machine with rotating assemblies must be first identified and modeled.

The actual system (Fig. 5.1, a) is replaced by a physical model (Fig. 5.1, b), usually a discrete-parameter model, comprising: a) shaft segments (Bernoulli/Timoshenko, uniform/conical); b) bearings (isotropic/orthotropic, undamped/damped); c) disks (rigid/flexible, thin/thick); d) seals; e) dampers (squeeze-film); f) couplings; and g) pedestals (plus the static support structure).

A set of mathematical equations consistent with the modeling assumptions is then generated for each component of the system.

In the following, the inertia, gyroscopic and stiffness matrices are established for circular disks and axi-symmetric shaft segments, starting from expressions of the kinetic and potential energies, and using Lagrange’s equations:

iiii

QqU

qT

qT

t=

∂∂

+∂∂

−∂∂&d

d , n,..,,i 21= , (5.1, a)

DYNAMICS OF MACHINERY 2

where T – kinetic energy, U – potential energy, iq – generalized displacement, iq& – generalized velocity, and iQ – generalized force. The response analysis of general unsymmetrical rotors is beyond the aim of this introductory presentation.

a

b

Fig. 5.1 (from [5.1])

In actual situations, the generalized forces are derived by identifying physically a set of generalized coordinates and writing the virtual work in the form

∑=

=n

kkk qQW

1δδ . (5.1, b)

Rotary inertia and shear deformations are considered for axi-symmetrical shafts, while external and internal damping is neglected. Bearings are described by linearized models with generally non-symmetric stiffness and damping matrices. The forcing vectors due to unbalance are established considering constant angular velocity.

Then, the system equations of motion in fixed coordinates are set up. Results of eigenvalue analysis and unbalance response calculations are presented in a form useful for engineering analysis.

5. FEA OF ROTOR-BEARING SYSTEMS 3

5.2. Kinematics of rigid body precession

In order to write the equations of motion of a rigid disk attached to a flexible shaft, it is necessary first to determine the components of its angular velocity.

5.2.1 Main reference frames

The analysis is carried out with respect to two reference frames: X,Y,Z – an inertial frame and x,y,z – a frame rotating with the rotor (Fig. 5.2). The analysis is restricted to the case when the disk rotates with constant angular velocity Ω (rad/sec) about the spin axis Ox.

Fig. 5.2

5.2.2 Rigid-body precession

Two possible motions of the rigid disk are shown in Fig. 5.3.

a b

Fig. 5.3


The motion in which the Ox axis traces out a cone (Fig. 5.3, a), with angular velocity ω , is called a precession when point M moves along a closed orbit, and whirling when it moves along a spiralling orbit. In forward precession

0>Ωω , while in backward precession 0<Ωω . In synchronous precession Ωω = . In some papers, whirling is used instead of precession.

Fluctuations in the cone angle, superposed on precession (Fig. 5.3, b) are generally termed nutation. This is not encountered in most linear systems.

5.2.3 Small rotations of the spin axis

The relative motion of the x,y,z frame with respect to the stationary X,Y,Z frame can be described through a set of three angles θ , ϕ and ψ (Fig. 5.4), corresponding to three successive rotations. Note that the order is important, because the rotations are not commutative. In the following, the first rotation is about the transverse axis which first follows the spin axis in accord with the right-hand rule for rotations.

Fig. 5.4

It is supposed that ϕ and ψ are small precession angles, while tΩθ = , where the disk angular speed of rotation .const=Ω

Consider the following sequence of rotations [5.2]:

a) a rotation ϕ around the Y-axis, transferring the Z-axis into the 1z -axis and the X-axis into the 1x -axis ; the X,Y,Z frame becomes the 11 z,Y,x frame;


b) a rotation ψ around the 1z -axis which transfers the Y -axis into the 1y -axis and the 1x -axis into the x -axis; the 11 z,Y,x frame becomes the 11 z,y,x frame;

c) a rotation θ about the Ox axis: the 11 z,y,x frame becomes the z,y,x frame.

a b c

Fig. 5.5

The axes X,Y,Z have unit vectors K,J,I , the axes 111 z,y,x have unit vectors 111 k,j,i , and the axes x,y,z have unit vectors k,j,i .

a) Rotation ϕ (angular velocity ϕ& around OY) (Fig. 5.5, a)

The relation between the unit vectors is

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

≅⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

=⎭⎬⎫

⎩⎨⎧

1

1

1

1

11

cossinsincos

ki

ki

KI

ϕϕ

ϕϕϕϕ

.

In expanded form

[ ]⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

1

1

1

1

10010

01

kJi

TkJi

KJI

ϕ

ϕ

ϕ.

b) Rotation ψ (angular velocity ψ& around 1Oz ) (Fig. 5.5, b)

The relation between the unit vectors is

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡ −≅

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡ −=

⎭⎬⎫

⎩⎨⎧

11

1

11

cossinsincos

ji

ji

Ji

ψψ

ψψψψ

.

In expanded form


[ ]⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

1

1

1

1

1

1

1000101

kji

Tkji

kJi

ψψψ

.

c) Rotation θ (angular velocity Ωθ =& around Ox) (Fig. 5.5, c)

The relationship between the unit vectors is

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡ −=

⎭⎬⎫

⎩⎨⎧

kj

kj

θθθθ

cossinsincos

1

1 .

In expanded form

[ ]⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

kji

Tkji

kji

θ

θθθθ

cossin0sincos0001

1

1 .

The relationship between the unit vectors in the stationary and rotating frames is of the form

[ ]⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

kji

TKJI

.

where the transformation matrix

[ ] [ ][ ][ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−+−

==θθϕθθψ

θψθϕθψθϕ

θψϕ

cossinsincos

sincoscossin1TTTT .

The vector of the instantaneous angular velocity is

1kJi ψϕΩω && ++= .

In terms of the unit vectors of the x,y,z system of coordinates

kjiJ θθψ sincos −+= ,

kjk θθ cossin1 += , so that

( ) ( ) ( ) kji θϕθψθψθϕψϕΩω sincossincos &&&&& −++++= .

The components of the angular velocity in the rotor-fixed (mobile) x,y,z reference system are


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

−++

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

θϕθψθψθϕ

ψϕΩ

ωωω

sincossincos&&

&&

&

z

y

x

. (5.2)

These will be used in the expression of the disk kinetic energy.

5.3 Equations of motion for rotor components

In this section, the equations of motion of disks, shaft segments, bearings, seals and couplings are set up, together with the corresponding element matrices.

5.3.1 Thin disks

In the following, only axi-symmetric rotors (hence disks) are considered and x,y,z are principal directions of inertia.

5.3.1.1 Inertia and gyroscopic matrices of rigid disks

In the fixed frame X,Y,Z, the disk position is defined by two translations ( v and w) and two rotations (ϕ and ψ ) (Fig. 5.6, a), the displacements of the shaft cross-section at the disk attachment. The angles ϕ and ψ are approximately equal to the angular displacements collinear with the Y- and Z-axis, respectively.

Neglecting any unbalance, the kinetic energy of a rigid axi-symmetric disk is

( ) ( )22222

21

21

zzyyxxdd JJJwmT ωωω ++++= &&v , (5.3)

where Px JJ = and Tzy JJJ == are principal moments of inertia with respect to

the x,y,z coordinate frame fixed to the disk and dm is the mass of the disk, PJ is the axial (polar) mass moment of inertia and TJ is the diametral mass moment of inertia.

Substituting xω , yω , zω from (5.2) into (5.3), the kinetic energy becomes

( ) ( )

( ) ( )[ ]22

222

sincossincos21

21

21

θϕθψθψθϕ

ψϕΩ

&&&&

&&&

−+++

++++=

T

Pdd

J

JwmT v

or


( ) ( ) ( )2222222

212

21

21 ψϕψϕψϕΩΩ &&&&&& ++++++= TPd

d JJwmT v .

Neglecting the term 22

21 ψϕ&PJ , the kinetic energy becomes

( ) ( ) 22222

21

21

21 ΩψϕΩψϕ PPTd

d JJJwmT +++++= &&&&&v . (5.4)

In equation (5.4), the term ( )22

21 wmd && +v accounts for rectilinear

translation, ( )22

21 ψϕ && +TJ - for rotary inertia, ψϕΩ &PJ - for gyroscopic

coupling and 2

21 ΩPJ - for rotation around the spin axis.

a b

Fig. 5.6

Applying Lagrange’s equations, the equations of motion of the rigid circular disk are obtained in the form

yd

dTmT

t==⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂ vv

&&&d

d , (5.5, a)

zPT

ddMJJTT

t=−=

∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂ ϕΩψ

ψψ&&&

&dd , (5.5, b)

zd

dTmT

t==⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂ ww

&&&d

d , (5.5, c)

( ) ( ) yPT

dMJJT

t−=−−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−∂∂ ψΩϕ

ϕ&&&

&dd , (5.5, d)

0=∂∂v

dT , 0=∂∂w

dT , ( ) 0=−∂

∂ϕ

dT ,


where yT , zT and yM , zM are the components of the applied force and couple, respectively (Fig. 5.6, b).

In matrix form

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

y

z

z

y

P

P

T

d

T

d

MTMT

J

J

Jm

Jm

ϕ

ψΩ

ϕ

ψ

&

&

&

&

&&

&&

&&

&&

-w

v

-w

v

0000000

0000000

000000000000

.

Introducing the 12× state vectors in the X-Y and X-Z planes, respectively,

⎭⎬⎫

⎩⎨⎧

=ψvd

yu , ⎭⎬⎫

⎩⎨⎧−

=ϕwd

zu ,

and the corresponding 12× forcing vectors acting on the disk in the two planes

⎭⎬⎫

⎩⎨⎧

=z

ydy M

Tf ,

⎭⎬⎫

⎩⎨⎧−

=y

zdz M

Tf ,

equation (5.5) becomes

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

dz

dy

dz

dy

d

d

dz

dy

d

d

f

f

u

u

g

g

u

u

m

m

&

&

&&

&&

0

0

0

0Ω (5.6)

with the forcing right hand term including mass and static misalignment unbalance, interconnection forces and other external effects on the disk.

The submatrices

[ ] ⎥⎦

⎤⎢⎣

⎡=

T

dd

Jm

m0

0, [ ] ⎥

⎦

⎤⎢⎣

⎡=

P

d

Jg

000

, (5.7)

form the inertia and gyroscopic submatrices, respectively, of an axi-symmetric rigid disk.

5.3.1.2 Unbalance force vectors

For the analysis of the synchronous response of a rotor system, it is important to determine the vectors of mass unbalance and static angular misalignment of a rigid disk.


Mass unbalance vectors

Consider the disk shown in Fig. 5.7, a. The mass center G of the disk is located at an eccentricity aCG = from the geometric center C. With respect to the rotating frame x,y,z it is located at αcosaac = , αsinaas = (Fig. 5.7, b). At 0=t (at rest), α is the angle of CG with the Y-axis.

a b

Fig. 5.7

The unbalanced centrifugal force acting in G along CG is 2Ωamd . The

components along the (spinning) rotor-fixed reference system are 2Ωcd am and 2Ωsd am , and along the (inertial) fixed reference system are

.tamtamf

,tamtamf

cdsdz

sdcdy

ΩΩΩΩ

ΩΩΩΩ

sincos

sincos22

22

+=

−= (5.8)

The vector of unbalance forces (neglecting disk skewness) can be written

t

a

a

mta

a

mff

fc

s

ds

c

ddz

dyd Ω

Ω

Ω

ΩΩ

Ω

sin

0

0cos

0

02

2

2

2

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧−

+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

= . (5.9)

Permanent disk skew vector

A disk misaligned with its driving shaft receives active gyroscopic moments which force pitch changes in the disk analogous to the translations set up by centrifugal forces due to mass unbalance.

Consider a rigid disk (Fig. 5.8, a) which is not perpendicular to the shaft. There is a static angular misalignment τ between a principal axis of inertia of the disk and the shaft axis.


The disk skew can be defined by a rotating vector τ normal to the spin axis and to the line of maximum skew angle. At 0=t the vector τ makes an angle β with the rotating z axis, so that the disk skew vector has components

βττ cos=c and βττ sin=s in the rotating frame x,y,z (see also eq.(2.84) in Ch. 2).

The components of angular displacements (Fig. 5.8, b) become

( )( ) ,t

,t

z

y

βΩτψα

βΩτϕα

++=

+−=

cos

sin

where ϕ and ψ are elastic rotations.

a b Fig. 5.8

The corresponding angular velocities and accelerations are

( )( ) ,t

,t

z

y

βΩτΩψα

βΩτΩϕα

+−=

+−=

sin

cos&&

&&

( )( ).t

,t

z

y

βΩτΩψα

βΩτΩϕα

+−=

++=

cos

sin2

2

&&&&

&&&& (5.10)

From equations (5.5, b) and (5.5, d), replacing ϕ by yα and ψ by zα , we obtain

( )

( ) .MJJ

,MJJ

yzPyT

zyPzT

−=−−

=−+

αΩα

αΩα&&&

&&&

Substitution of (5.10) yields

( ) ( )

( ) ( ) ( ).tJJJJM

,tJJJJM

TPPTy

TPPTz

βΩτΩψΩϕ

βΩτΩϕΩψ

+−+−−=−

+−+−=

sin

cos 2

2

&&&

&&&

The vector of gyroscopic moments acting on the disk is

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

⎭⎬⎫

⎩⎨⎧−

+⎭⎬⎫

⎩⎨⎧

−=⎭⎬⎫

⎩⎨⎧−

ttJJM

MPT

y

z Ωβτβτ

Ωβτβτ

Ω sincossin

cossincos2 . (5.11)


The vector of unbalance forces and moments is

( )

( )

( )

( )

t

JJamJJam

t

JJamJJam

f

cPT

cd

sPT

sd

sPT

sd

cPT

cd

d Ω

τ

τΩΩ

τ

τΩ sincos 22

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−

−−−

+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−

−= . (5.12)

Note that the moment of inertia difference PT JJ − can be both negative (thin disk) and positive (thick disk). When 0<− PT JJ , the applied moment is restoring, acting to pitch the disk in the direction opposite the initial skew. When

0>− PT JJ , the applied moment increases the angular unbalance by pitching the thick disk further in the direction of the initial skew.

5.3.1.3 Flexible disks

Flexible disks can be modeled as two rigid disks connected by springs with rotational stiffness Rk (Fig. 5.9).

Fig. 5.9

The inner disk has four degrees of freedom, two translatios v , w , and two rotations ϕ , ψ . The inertia properties are m, PJ , TJ . The outer disk has only two rotational degrees of freedom, ϕ and ψ . It has only polar and diametral mass moments of inertia PJ and TJ . Its mass is lumped in the inner disk.

The corresponding 66× matrices [5.1] can be reduced to 44× matrices by a static condensation, selecting the coordinates of the inner disk as “active” and those of the outer disk as “omitted” (see Section 5.8.1.2).


5.3.2 Uniform shaft elements

Shaft segments are considered isotropic and axi-symmetric about the spin axis of the rotor. This presentation is restricted to uniform cross-section elements.

5.3.2.1 Timoshenko vs. Bernoulli beam elements

Consider a two-noded uniform shaft segment (Fig. 5.10) modeled by a Bresse-Timoshenko beam element.

Dropping the element index, the following notations are used: l - length of shaft element, E – Young’s modulus, G – shear modulus, IE - flexural rigidity,

sAG - effective shear rigidity, sA - reduced shear area, χ - shear coefficient, ρ - mass density of shaft material, Aρμ = mass per unit length, I – second moment of area of cross section, Iˆ ρμ = rotary mass per unit length.

Fig. 5.10

At a given node, the shaft has four degrees of freedom, two transverse displacements v and w , and two rotations ψ and ϕ , measured in the fixed coordinate system. Define two 14× sub-vectors of nodal displacements in the X-Y and X-Z planes, respectively,

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

j

j

i

i

syu

ψ

ψv

v

, ⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−

−=

j

j

i

i

szu

ϕ

ϕw

w

, (5.13, a)

and the corresponding vectors of nodal forces


⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

jz

jy

iz

iy

sy

MTMT

f , ⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−

−=

jy

jz

iy

iz

sz

MTMT

f . (5.13, b)

Note the “minus” sign of rotations and moments about the Y-axis whose positive signs are in accordance with the right-hand rule. The sign convention used here is that positive internal forces act in positive (negative) coordinate directions on beam cross sections with a positive (negative) outward normal.

In the Bernoulli-Euler beam theory, deformations due to shear are neglected. The Bresse-Timoshenko beam theory is based on the following hypotheses: a) planar cross sections remain undistorted and plane; b) warping is neglected; and c) an average shear strain is considered, independent of Y.

The planar section hypothesis introduces additional fake stiffness against warping. In some points, distortions due to shear are underestimated. The relationship for T is not correct. The solution is to use an “effective shear area”

AAs < . From energy considerations

∫∫ ∫ =⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

ll

xGATxA

G sA

d21dd

21 22τ ,

so that

( )

AA

AAs χ

τ

τ==

∫∫

d

d2

2

.

For a hollow steel shaft, with outer diameter D and inner diameter d, the shear coefficient is

( )

2

21033131

1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

=

DdDd..

χ .

Equations in the X-Y plane

At any point in the beam cross section, the axial displacement is proportional to the distance from the neutral axis

yu ψ−= .

The average shear strain (Fig. 5.11) is


vv ′+−=∂∂

+∂∂

= ψγxy

uxy .

The axial strain is

ψε ′−=∂∂

= yxu

x .

The bending moment is

∫ ′=−=A

zxz IEAyM ψσ d .

The equations in the X-Y plane are presented in Table 5.1.

Neglecting the ψμ &&ˆ term, a convenient static relationship can be established between v′ and ψ

ψψ ′′=′s

zGAEI-v , (5.14)

though, for the Bresse-Timoshenko beam, v and ψ are kinematically independent.

Fig. 5.11

Introducing the dimensionless variable

l

x=ξ ,

so that

ξ

ξξ d

d1dd

dd

dd

l==

xx,

the static relationship between v′ and ψ becomes [5.3]


Table 5.1

Bernoulli-Euler beam Bresse-Timoshenko beam

a) Equilibrium

.v 0

0

=+′−

=+′

&&μy

yz

T

,TM

.v 0

0

=+′−

=−+′

&&

&&

μ

ψμ

y

yz

T

,ˆTM

b) Elasticity

xyzz IEM κ= .AGT

,IEM

xysy

xyzz

γ

κ

=

=

c) Kinematics

.-v ψ

ψκ′=

′=

0

,xy .-v ψγ

ψκ

′=

′=

xy

xy ,

Elimination of xyκ Elimination of xyκ and xyγ

v ′′=′= zzz IEIEM ψ ( )ψψ-v′=

′=

sy

zz

AGT,IEM

Equations of motion

Elimination of zM and yT Elimination of zM and yT

0=+′ vv v &&μzIE ( )( ) .vv-

v-0

0=+′′′

=−′−′′&&

&&

μψψμψψ

s

sz

AG,ˆAGIE


2

2

dd

12dd1

ξψκψ

ξ−=

vl

(5.14, a)

where

212ls

z

AGIE

=κ . (5.15)

Equations in the X-Z plane

The axial displacement at any point is

zu ϕ= .

The average shear strain (Fig. 5.12) is

ww ′+=∂∂

+∂∂

= ϕγxz

uxz .

The axial strain is

ϕε ′=∂∂

= zxu

x .

The bending moment is

∫ ′=−=A

yxy IEAzM ϕσ d .

The equations in the X-Z plane are presented in Table 5.2.

Fig. 5.12

Neglecting the ϕμ &&ˆ term, the “static” coupling between w′ and ϕ is

( ) ( )″−−−=′′+−=′ ϕϕϕϕs

y

s

y

GAEI

GAEI

w (5.16)


Table 5.2

Bernoulli-Euler beam Bresse-Timoshenko beam

a) Equilibrium

.w 0

0

=−′

=+′

&&μz

zy

T

,TM

.w 0

0

=−′

=−+′

&&

&&

μ

ϕμ

z

zy

T

,ˆTM

b) Elasticity

xzyy IEM κ= .AGT

,IEM

xzsz

xzyy

γ

κ

=

=

c) Kinematics

.w′−=

′=ϕ

ϕκ ,xz .w′+=

′=ϕγϕκ

xz

xz ,

Elimination of xzκ Elimination of xzκ and xzγ

w ′′−=′= yyy IEIEM ϕ ( )w′+=

′=

ϕ

ϕ

sz

yy

AGT

,IEM

Equations of motion

Elimination of yM and zT Elimination of yM and zT

0=+′ ww v &&μyIE ( )( ) .ww

w0

0

=−′′+′

=−′+−′′

&&

&&

μϕ

ϕμϕϕ

s

sy

AG

,ˆAGIE


Introducing the dimensionless variable

l

x=ξ ,

the static relationship between w′ and ϕ becomes

( ) ( )2

2

dd

12dd1

ξϕκϕ

ξ−

−−=w

l (5.16, a)

where

212ls

y

AG

IE=κ , zy II = .

5.3.2.2 Coordinates and shape functions

Consider third-degree polynomials as approximating functions for the displacements and rotations in the X-Y plane

( )( ) .BBBB

,AAAA

012

23

3

012

23

3

+++=

+++=

ξξξξψ

ξξξξv

Substitution in the static “coupling” condition (5.14, a)

( ) ( )23012

23

3122

3 262

231 BBBBBBAAA +−+++=++ ξκξξξξξl

yields

03 =B , 323 ABl

= , 212 ABl

= , ⎟⎠⎞

⎜⎝⎛ += 310 2

1 AAB κl

,

so that ( )ξψ is of second degree.

The translation v and the rotation ψ are approximated in terms of the nodal displacements, using static shape functions:

( ) ( ) ( ) ( ) ( ) ⎣ ⎦ ( ) ( ) ( ) ( ) ( ) ⎣ ⎦ ,uN~N~N~N~N~

,uNNNNNsyji

syji

=+++=

=+++=

ψξξψξξξψ

ψξξψξξξ

4321

4321

ji

ji

vv

vvv (5.17)

where

⎣ ⎦ ⎣ ⎦4321 NNNNN = , ⎣ ⎦ ⎣ ⎦4321 N~N~N~N~N~ = .


An example of calculation is given here for ( )ξ3N . According to the general properties of shape functions, 3N has a unit value at coordinate 3 and is zero at coordinates 1, 2 and 4. The four boundary conditions yield a set of four equations

( ) 00 =v → 00 =A , ( ) 1=lv → 12

1 23 =+⎟⎠⎞

⎜⎝⎛ − AA κ ,

( ) 00 =ψ → 31 2AA κ

−= , ( ) 0=lψ → 023 23 =+ AA ,

wherefrom the four constants are obtained as

κ+

−=1

23A ,

κ+=

13

2A , κ

κ+

=11A , 00 =A ,

so that

( ) ( )κξξξκ

ξ ++−+

= 233 32

11N .

The shear-modified shape functions for displacements are

( ) ( )[ ]( ) ( ) ( )

( ) ( )( ) ( ) ( ) .N

,N

,N

,N

⎥⎦⎤

⎢⎣⎡ +−++−

+=

+−+

=

⎥⎦⎤

⎢⎣⎡ −++−

+=

−++−+

=

2324

323

2322

321

211

231

12

21

1

12311

1

ξξκξξκ

ξ

κξξξκ

ξ

ξξκξξξκ

ξ

ξκξξκ

ξ

ll

ll

(5.18)

For 0=κ , the above shape functions become the third-degree Hermite polynomials used for Bernoulli-Euler beams.

The shape functions for rotations are [5.3]

( ) ( )

( ) ( )[ ]( ) ( )

( ) ( ) .N~

,N~

,N~

,N~

κξξξκ

ξ

ξξκ

ξ

ξκξξκ

ξ

ξξκ

ξ

+−+

=

⎟⎠⎞

⎜⎝⎛ +−

+=

−++−+

=

⎥⎦⎤

⎢⎣⎡ −

+=

231

1

6611

1

13411

1

6611

1

24

23

22

21

l

l

(5.19)

For 0=κ


( ) ,NN~ ii ξξ

∂∂

=l

1 ( )41,..,i = .

It is useful to introduce a third set of shape functions defined by

⎣ ⎦ ⎣ ⎦ ⎥⎦

⎥⎢⎣

⎢−=

ξdd1 NN~N

l

which will be used in the derivation of the stiffness matrix.

Their expressions are

( ) ( )

( ) ( ) .NN

,NN

κκξξ

κκξξ

+==

+=−=

121

11

42

31l (5.20)

Similar relations hold for the displacements in the X-Z plane

⎣ ⎦ ⎣ ⎦ ,uN~

,uNsz

sz

=−

=

ϕ

w (5.21)

because of the similar coupling relationship (5.16, a) between w and ϕ− .

5.3.2.3 Inertia and gyroscopic matrices

For an infinitesimal uniform shaft element of length xd , the kinetic energy can be obtained from the similar expression (5.4) derived for a thin disk

( ) ( ) ( ) xIIwAT Ps d2

21

21

21d 22222

⎥⎦⎤

⎢⎣⎡ +++++= ψϕΩΩρψϕρρ &&&&&v ,

where

IIP 2= , Aρμ = , Iˆ ρμ = , sPPP JIˆ

l

1== ρμ . (5.22)

Integrating, the kinetic energy for the shaft element becomes

( ) ( ) ∫∫∫ +++++=

lll

&&&&&

0

2

0

22

0

22 d21d

2d

2xˆJx

ˆxwT P

sP

s ψϕΩμΩψϕμμ v .

a) The translatory energy is

( )∫ +=

l

&&

0

221 d

21 xwT s vμ .


Expressing velocities in terms of nodal coordinates and shape functions

⎣ ⎦ ⎣ ⎦TTsy

sy NuuN &&&& === Tvv ,

⎣ ⎦ ⎣ ⎦ sy

TTsy uNNu &&&&& == vvv T2 , ⎣ ⎦ ⎣ ⎦ s

zTTs

z uNNu &&& =2w ,

this energy can be written

⎣ ⎦ ⎣ ⎦

[ ]

⎣ ⎦ ⎣ ⎦

[ ]

sz

m

TTsz

sy

m

TTsy

s uxNNuuxNNuT

sT

sT

&

444 3444 21

&&

444 3444 21

&

ll

∫∫ +=00

1 d21d

21 μμ .

The translational mass submatrix of the shaft element

[ ] ( )⎣ ⎦ ( )⎣ ⎦∫=1

0

dξξξμ NNm TsT l (5.23)

is the same in the X-Y and X-Z planes.

b) The rotary energy is

( )∫ +=

l

&&

0

222 d

21 xˆT s ψϕμ .

Expressing angular velocities in terms of nodal coordinates and shape functions

⎣ ⎦ syuN~ && =ψ , ⎣ ⎦ s

zuN~ && −=ϕ ,

⎣ ⎦ ⎣ ⎦ sy

TTsy uN~N~u &&& =2ψ , ⎣ ⎦ ⎣ ⎦ s

zTTs

z uN~N~u &&& =2ϕ ,

this energy can be written

⎣ ⎦ ⎣ ⎦

[ ]

⎣ ⎦ ⎣ ⎦

[ ]

sz

m

TTsz

sy

m

TTsy

s uxN~N~ˆuuxN~N~ˆuT

sR

sR

&

44 344 21

&&

44 344 21

&

ll

∫∫ +=00

2 d21d

21 μμ .

The rotational mass submatrix of the shaft element


[ ] ( )⎣ ⎦ ( )⎣ ⎦∫=

1

0

dξξξμ N~N~ˆm TsR l (5.24)

is the same in the X-Y and X-Z planes.

c) The gyroscopic effect energy is

∫=l

&

03 dxˆT Ps ψϕΩμ ,

⎣ ⎦ ⎣ ⎦

[ ]

sy

g

TP

Tsz

s uxN~N~ˆuT

s444 3444 21

&

l

∫−=

0

3 dμΩ .

The gyroscopic submatrix of the shaft element

[ ] ( )⎣ ⎦ ( )⎣ ⎦ [ ]sR

TP

s mN~N~ˆg 2d1

0

== ∫ ξξξμ l . (5.25)

The total kinetic energy of the uniform shaft element is

[ ] [ ] [ ] 2

21

21

21 ΩΩ s

Psy

sTsz

sz

sTsz

sy

sTsy

s JuguumuumuT +−+= &&&&&

where [ ] [ ] [ ]s

RsT

s mmm += . (5.26)

The total mass submatrix of the uniform shaft element is

[ ]⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

s

ss

sss

ssss

s

mmmmmmmmmm

m

44

3433

242322

14131211

sym

. (5.27)

where

( ) RTsm αακκ 36140294156 211 +++= ,

( ) ( ) RTs ..m ακακκ ll 15351753822 212 −+++= ,

( ) RTsm αακκ 367012654 213 −++= ,


( ) ( ) RTs ..m ακακκ ll 15351753113 214 −+++−= ,

( ) ( ) RTs .m ακκακκ 222222 10545374 ll +++++= ,

( ) ( ) RTs .m ακκακκ 222224 5515373 ll −+−++−= ,

ss mm 1423 −= , ss mm 1133 = , ss mm 1234 −= , ss mm 2244 = ,

( )21420 κμα+

=l

T , ( )22 130 κμα+

=l

lˆR .

The gyroscopic submatrix of the uniform shaft element is

[ ]⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

s

ss

sss

ssss

s

gggggggggg

g

44

3433

242322

14131211

sym

. (5.28)

where

Rsg α7211 = , ( ) R

sg ακ l153212 −= , Rsg α7213 −= ,

ss gg 1214 = , ( ) Rsg ακκ 2222 10542 l++= ,

( ) Rsg ακκ 2224 5512 l−+−= ,

ss gg 1223 −= , ss gg 1133 = , ss gg 2334 = , ss gg 2244 = .

Substitution of sT in Lagrange’s equations yields

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

sz

sy

sz

sy

s

s

sz

sy

s

s

f

f

u

u

g

g

u

u

m

m

&

&

&&

&&

0

0

0

0Ω

or

[ ] [ ] sss fuGuM =+ &&& Ω , (5.29)

where the 88× inertia matrix [ ]sM is symmetric and the 88× gyroscopic matrix

[ ]sG is skew-symmetric .

5.3.2.4 Stiffness matrix

For an infinitesimal shaft element, the strain energy is


( ) xAGxx

IEU xzxys ddd

dd

21d 22

22

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

++⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛= γγϕψ .

Integrating and substituting the shear strains we obtain

( ) ( ) ( )[ ]∫∫ ′++−′+′+′=

ll

0

22

0

22 d2

d2

xwAG

xIEU s ϕψϕψ v .

a) The flexural energy is

( ) ( )∫∫∫ ′−+′=′+′=

lll

0

2

0

2

0

221 d

2d

2d

2xIExIExIEU ϕψϕψ .

Expressing rotations and curvatures in terms of nodal coordinates and shape functions

⎣ ⎦ syuN~=ψ , ⎣ ⎦ s

yuN~ °=′l

1ψ ,

where

( ) ( )ξdd

=° ,

⎣ ⎦ ⎣ ⎦ sy

TTsy uN~N~u °°=′

21l

2ψ ,

the bending energy can be written

⎣ ⎦ ⎣ ⎦

[ ]

⎣ ⎦ ⎣ ⎦

[ ]

sz

k

TTsz

sy

k

TTsy uN~N~IEuuN~N~IEuU

sB

sB

444 3444 21l

444 3444 21l ∫∫ °°°° +=

1

0

1

0

1 d21d

21 ξξ .

The stiffness submatrix for bending is

[ ] ∫ ⎥⎦

⎥⎢⎣

⎢⎥⎦

⎥⎢⎣

⎢=

1

0

ddd

dd ξ

ξξN~N~IEk

TsB

l. (5.30)

b) The shear energy is

( ) ( )∫∫ ′++′−=ll

0

2

0

22 d

2d

2xwAGxAGU ss ϕψ v

where


⎣ ⎦ ⎣ ⎦( ) ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ sy

sy

sy uNuNN~uNN~ =⎟

⎠⎞

⎜⎝⎛ −=′−=′− °

l

1vψ ,

⎣ ⎦ ⎣ ⎦( ) ⎣ ⎦ sz

sz uNuNN~ −=′−−=′+ wϕ ,

so that

⎣ ⎦ ⎣ ⎦

[ ]

⎣ ⎦ ⎣ ⎦

[ ]

sz

k

Ts

Tsz

sy

k

Ts

Tsy uNNAGuuNNAGuU

sS

sS

444 3444 21

l

444 3444 21

l ∫∫ +=

1

0

1

0

2 d21d

21 ξξ .

The stiffness submatrix for shear is

[ ] ⎣ ⎦ ⎣ ⎦∫=

1

0

dξNNAGkT

ssS l . (5.31)

The total strain energy of the uniform shaft element is

[ ] [ ] sz

sTsz

sy

sTsy ukuukuU

21

21

+= ,

where the stiffness submatrix is

[ ] [ ] [ ]sS

sB

s kkk += . (5.32)

or

[ ]⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−

+=

2

22

2

22

3

sym00

00000

4sym612

264612612

11

l

ll

l

l

lll

ll

lκ

κIEk s .

Substitution of U in Lagrange’s equations yields

[ ] [ ][ ] [ ]

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

sz

sy

sz

sy

s

s

f

f

u

u

k

k

0

0

or

[ ] sss fuK = ,

where [ ]sK is the 88× full stiffness matrix of the uniform shaft element.


5.3.2.5 Unbalance force vectors

Consider a linear distribution of the mass unbalance along the shaft element, between cLa , sLa at the left end, and cRa , sRa at the right end

( ) ( )( ) ( ) ,aaa

,aaa

sRsLss

cRcLsc

ξξξ

ξξξ

+−=

+−=

1

1 (5.33)

where LLcL aa αcos= , LLsL aa αsin= , RRcR aa αcos= , RRsR aa αsin= ,

La , Ra are the unbalance radii, and Lα , Rα are the unbalance phase angles.

The corresponding distributed unbalance forces are

( ) ( )t,at,p syy ξΩμξ 2= , ( ) ( )t,at,p s

zz ξΩμξ 2= , where

( ) ( ) ( )( ) ( ) ( ) .tatat,a

,tatat,asc

ss

sz

ss

sc

sy

ΩξΩξξ

ΩξΩξξ

sincos

sincos

+=

−=

The virtual work of these forces is

( ) ( )∫∫ +=ll

00

dd xpxpW zT

yT wv δδδ ,

⎣ ⎦

⎣ ⎦

44 344 21

l

44 344 21

l

sz

sy f

zTTs

z

f

yTTs

y pNupNuW ∫∫ +=

1

0

1

0

dd ξδξδδ .

The kinematically equivalent nodal forces are

⎣ ⎦∫=1

0

2 dξΩμ sy

Tsy aNf l , ⎣ ⎦∫=

1

0

2 dξΩμ sz

Tsz aNf l .

The shaft mass unbalance vector has the expression

⎣ ⎦

⎣ ⎦

⎣ ⎦

⎣ ⎦ ⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧−

+

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∫

∫

∫

∫t

aN

aN

t

aN

aN

f

f

sc

T

ss

T

ss

T

sc

T

sz

sy Ω

ξ

ξ

ΩμΩ

ξ

ξ

Ωμ sin

d

d

cos

d

d

1

0

1

021

0

1

02 ll .


An example of calculation is given below

( ) ( ) ( )[ ]

( ) ( ) ( )

( ) ( ) ( )[ ].aa

NaNa

aaNaN

cRcL

cRcL

cRcLsc

κκκ

ξξξξξξ

ξξξξξξ

2018404211201

dd1

d1d

1

01

1

01

1

01

1

01

++++

=

=+−=

=+−=

∫∫

∫∫

The vector of unbalance forces can be written

( )

[ ]

[ ]

[ ]

[ ] ⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

−+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

+= t

aa

a

aa

at

aa

a

aa

af

cR

cL

sR

sL

sR

sL

cR

cL

s ΩΩκ

Ωμ sincos1120

2l ,

where

[ ] ( ) ( )

( ) ( ) ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−+−++++++

=

ll

ll

κκκκ

κκκκ

565440422018

545620184042

a ,

so that

⎟

⎟⎠

⎞⎜⎜⎝

⎛

⎭⎬⎫

⎩⎨⎧ −

+⎭⎬⎫

⎩⎨⎧

= tQQ

tQQ

fy

z

z

ys ΩΩ sincos .

The unbalance force vector is of the form

( )

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

−−−−

+

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

+= ttf s Ω

νννννννν

Ω

νννννννν

κΩμ sin

cos1120

4

3

2

1

8

7

6

5

8

7

6

5

4

3

2

1

2

l

l

l

l

l

l

l

l

l , (5.34)

where


( ) ( ) cRcL aa κκν 201840421 +++= ,

( ) ( ) cRcL aa κκν 54562 +++= ,

( ) ( ) cRcL aa κκν 404220183 +++= ,

( ) ( ) cRcL aa κκν 56544 +++=− ,

( ) ( ) sRsL aa κκν 201840425 +++= ,

( ) ( ) sRsL aa κκν 54566 +++= ,

( ) ( ) sRsL aa κκν 404220187 +++= ,

( ) ( ) sRsL aa κκν 56548 +++=− .

5.3.2.6 Rotor modeling

Figure 5.13 illustrates an example of finite element modeling for a low pressure turbine rotor (M. L. Adams, 1980).

Fig. 5.13

For the disks integral with the shaft, note the way in which the outer diameter of each shaft element is established, in order to ensure a continuous stepwise variation along the rotor length, which is important in the calculation of shaft element stiffnesses. The remaining part of each disk is considered in the calculation of the disk mass and mass moments of inertia.


Fig. 5.14

For monoblock rotors with integral disks (Fig. 5.14), as used in high-pressure turbines, and for shrunk-on disks (Fig. 5.15), the portion of the disk which contributes to the shaft stiffness is sometimes determined using the empirical angle rule. The same approach is used to replace a portion of a shaft with a large diameter step with several steps of slowly increased diameter (Fig. 5.16).

Fig. 5.15

Very often different diameters are used for the stiffness matrix and the mass matrix of a rotor portion. For instance, the wirings of an electrical motor contribute only to the kinetic energy, but not to the potential energy. Therefore, the diameter used in the calculation of the mass matrix is greater than the diameter used in the stiffness matrix calculation.

Fig. 5.16


Conical shaft finite elements are available [5.4] but their presentation is outside the scope of this presentation. User supplied elements are also used for complicated geometries where the stiffness matrix is calculated by inverting the experimentally obtained flexibility matrix.

5.3.3 Bearings and seals

Radial bearings are usually modeled by translational stiffness and damping coefficients. Inertia effects are considered only for annular seals in centrifugal pumps.

5.3.3.1 Stiffness matrix of a radial bearing

The force-displacement relationship for a radial bearing (in the Y-Z plane) can be written

[ ]⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

wv

wv b

zzzy

yzyy

z

y kkkkk

TT

, (5.35)

where the elements of [ ]bk are stiffness coefficients, so that ijk is the spring force in the ith direction due to a unit displacement in the jth direction.

Fig. 5.17

Consider the ZY ′−′ - reference frame (Fig. 5.17), rotated through an angle α with respect to the ZY − reference frame. The transformation of displacements can be written

[ ]⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

=⎭⎬⎫

⎩⎨⎧

′′

wv

wv

wv

Rαααα

cossinsincos

.

The transformation of forces is


[ ]⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

=⎭⎬⎫

⎩⎨⎧

′′

z

y

z

y

z

y

TT

RTT

TT

αααα

cossinsincos

.

Dropping the ‘b’ index, the new force-displacement relation is

[ ] [ ][ ] [ ][ ][ ]⎭⎬⎫

⎩⎨⎧

′′

=⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

′′ −

wv

wv 1RkRkR

TT

RTT

z

y

z

y .

Because [ ] [ ]TTT =−1 , the transformed stiffness matrix is

[ ] [ ][ ][ ] ⎥⎦

⎤⎢⎣

⎡ −⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

==′cssc

kkkk

cssc

RkRkzzzy

yzyyT , (5.36)

where αcos=c and αsin=s .

The original non-symmetric stiffness matrix can be split into the sum of a symmetric and a skew-symmetric component:

[ ] [ ] [ ]434214342143421

as k

a

a

k

zzs

syy

k

zzzy

yzyy

kk

kkkk

kkkk

⎥⎦

⎤⎢⎣

⎡−

+⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡0

0, (5.37)

where

( )zyyzs kkk +=21 , ( )zyyza kkk −=

21 . (5.38)

The transformed stiffness matrix becomes

[ ] [ ] [ ] [ ]( ) [ ] [ ] [ ]asT

as kkRkkRk ′+′=+=′

where, denoting αcos=c and αsin=s ,

[ ] ( ) ( )( ) ( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

−+−+−

−+−++=′

sckcksksckcskksckcskkcskskck

kszzyysyyzz

syyzzszzyys 2

22222

2222, (5.39)

and [ ] [ ]aa kk =′ .

These results imply that:

a) the skew-symmetric part [ ]ak is independent of the rotation [ ]T , hence of the angle α ;

b) when

zzyy

skk

k−

=∗ 22tan α ,


the symmetric part [ ]sk is diagonal and the angles ∗α and 090+∗α define the principal directions of flexibility.

The symmetric matrix [ ]sk can be diagonalized through the coordinate transformation [ ]R , whereas the skew-symmetric matrix [ ]ak remains unchanged. This implies that a non-symmetric stiffness matrix can be transformed into a matrix in which the off-diagonal elements are skew-symmetrical.

In the special case when the diagonal elements are identical, two particular cases are of interest:

a) when the off-diagonal elements are skew-symmetrical, the transformed matrix is the same as the original matrix:

⎥⎦

⎤⎢⎣

⎡−

=⎥⎦

⎤⎢⎣

⎡ −⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡− 12

21

12

21

kkkk

cssc

kkkk

cssc

hence the bearing is isotropic.

b) when the off-diagonal elements are identical

[ ] ⎥⎦

⎤⎢⎣

⎡−

+=⎥

⎦

⎤⎢⎣

⎡ −⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

=′αα

αα2sin2cos

2cos2sin

212

221

12

21

kkkkkk

cssc

kkkk

cssc

k

and for 045=∗α

[ ] ⎥⎦

⎤⎢⎣

⎡−

+=′

21

21

00

kkkk

k ,

so that the bearing is orthotropic, with principal axes of elasticity at 045 and 0135 .

The following six important conclusions can be drawn [5.5]:

a) when zyyz kk ≠ , the bearing is anisotropic;

b) when zyyz kk −= and zzyy kk = , the bearing is isotropic;

c) when zyyz kk = and zzyy kk ≠ , the bearing is orthotropic, with principal

directions defined by ∗α and 090+∗α ;

d) when zyyz kk = and zzyy kk = , the bearing is orthotropic, with principal

directions at 045 and 0135 ;

e) when 0== zyyz kk and zzyy kk ≠ , the bearing is orthotropic, with horizontal and vertical principal directions;


f) when 0== zyyz kk and zzyy kk = , the bearing is isotropic;

Moreover, the condition zyyz kk ≠ (non-zero skew-symmetric coupling element ak ) is the major cause of the instability of an anisotropic system.

5.3.3.2 Fluid film bearings

Linearized hydrodynamic fluid film bearings are represented by the incremental force components obtained by a Taylor-series expansion of the force-displacement and force-velocity relations about an equilibrium configuration of the journal in the bearing [5.6]:

[ ] [ ]⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

wv

wv

&

&

4342143421bb c

bzz

bzy

byz

byy

k

bzz

bzy

byz

byy

z

y

cccc

kkkk

TT

. (5.40)

In expanded form

.

MTMT

cc

cc

kk

kk

y

z

z

y

bzz

bzy

byz

byy

bzz

bzy

byz

byy

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

ϕ

ψ

ϕ

ψ

&

&

&

&

-w

v

-w

v

000000000000

000000000000

In partitioned form

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

bz

by

bz

by

bzz

bzy

byz

byy

bz

by

bzz

bzy

byz

byy

f

f

u

u

cc

cc

u

u

kk

kk

&

& (5.41)

where

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡=

000b

ijbij

kk , [ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡=

000b

ijbij

cc , ( )z,yj,i = . (5.42)

In the following, bearing/pedestal inertia effects are neglected.

5.3.3.3 Annular seals

Short annular seals in centrifugal pumps are usually considered isotropic. The dynamic seal forces are represented by a reaction force / seal motion of the form [5.7]


⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

=⎭⎬⎫

⎩⎨⎧

wv

wv

wv

&&

&&

&

&M

CccC

KkkK

TT

z

y . (5.43)

The diagonal elements of their stiffness and damping matrices are equal, while the off-diagonal terms are equal, but with reversed sign. The cross-coupled damping and the mass term arise primarily from inertial effects. Cross-coupled stiffnesses arise from fluid rotation, in the same way as in uncavitated plain journal bearings. The dynamic coefficients of equation (5.43) are defined in Section 6.10.2.

The impeller-volute/diffuser interaction forces are generally modeled by an equation of the form

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

=⎭⎬⎫

⎩⎨⎧

wv

wv

wv

&&

&&

&

&

MmmM

CccC

KkkK

TT

c

c

z

y . (5.44)

Note the presence of a cross-coupled mass coefficient cm , in comparison with the liquid-seal model of equation (5.43). The sign of cm is negative, which implies that it is destabilizing for forward precession. The model with identical diagonal elements and skew-symmetric off-diagonal elements ensures radial isotropy.

The forces developed by centrifugal compressor seal labyrinths are at least one order of magnitude lower than their liquid counterparts. They have negligible added-mass terms and are typically modeled by a reaction force/motion model of the form

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

=⎭⎬⎫

⎩⎨⎧

wv

wv

&

&

CccC

KkkK

TT

z

y . (5.45)

Unlike the pump seal model of equation (5.43), the direct stiffness term is typically negligible and is negative in many cases.

Usually only translational coefficients are considered. Angular dynamic coefficients are used only for long annular clearance seals in multi-stage centrifugal pumps, where forces give rise to tilting shaft motions and couples produce linear displacements.

5.3.4 Flexible couplings

A flexible coupling can be modeled as an elastic element with isotropic translational stiffness Tk and rotational stiffness Rk between station i on one shaft and station j on the other, as shown in Fig. 5.18 [5.1].

The stiffness matrix of such a flexible coupling is of the form


[ ]

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

−−

−−

=

RRTT

RRTT

RRTT

RRTT

coupling

kkkk

kkkk

kkkk

kkkk

k . (5.46)

This simple type of coupling model allows modest amounts of relative transverse motion and angular misalignment between the centerlines of the components being coupled, but prevents any relative axial motion, so it does not apply to spline couplings.

Fig. 5.18

Inertia effects can be taken into account by including thin rigid disks at each of the connecting points.

5.4 System equations of motion

The equations of motion for the rotor-bearings-seals system are first obtained in second order form, by assembling the element matrices.


5.4.1 Second order configuration space form

It is convenient to use a global displacement vector whose upper half contains the nodal displacements in the Y-X plane, while the lower half contains those in the Z-X plane

T

Z

nn

Y

nnT

TT

,,,,,,,,,,,,,x444444 3444444 21

L44444 344444 21

L 11112211 ϕϕϕψψψ −−−= wwwvvv . (5.47)

Correspondingly, the upper half of the global forcing vector contains the nodal forces in the Y-X plane, and the lower half has the nodal forces acting in the Z-X plane

T

F

nynzyzyz

F

nznyzyzyT

Tz

Ty

M,T,,M,T,M,T,M,T,,M,T,M,f4444444 34444444 21

L444444 3444444 21

L 22112211

−−−= T .

(5.48)

One can write

⎭

⎬⎫

⎩⎨⎧

=ZY

x , ⎭

⎬⎫

⎩⎨⎧

=z

y

FF

f . (5.49)

By combining the component element equations, the assembled system equations of motion can be written

[ ] [ ] [ ] fxKxCxM =++ &&& , (5.50)

where

[ ] [ ] [ ][ ] [ ] ⎥⎦

⎤⎢⎣

⎡=

mm

M0

0, [ ] [ ] [ ]

[ ] [ ] ⎥⎦⎤

⎢⎣

⎡+

+=

zzzy

yzyy

kkkkkk

K ,

[ ][ ] [ ][ ] [ ]

[ ] [ ][ ] [ ] ⎥⎦

⎤⎢⎣

⎡−

+⎥⎥⎦

⎤

⎢⎢⎣

⎡=

00g

gcc

ccC

zzzy

yzyy Ω , (5.51)

⎭⎬

⎫

⎩⎨⎧

=ZY

x&

&& ,

⎭⎬⎫

⎩⎨⎧

=ZY

x&&

&&&&

are of order nN 4= , where n is the number of nodes.

The global inertia matrix is shown in Fig. 5.19. The global gyroscopic matrix is shown in Fig. 5.20. The global stiffness matrix is shown in Fig. 5.21.


Fig. 5.19

Fig. 5.20

Fig. 5.21


Matrices [ ]yzk , [ ]zyk and [ ]zzk resemble [ ]yyk . Bearing matrices are similar.

Fig. 5.22

For the rotor model from Fig. 5.22, the equations of motion have the form

Fig. 5.23

Fig. 5.24


The vectors of mass unbalance forces are

tFtFf sc ΩΩ sincos += (5.52)

where the vectors in the right hand side are shown in Fig. 5.24.

5.4.2 First order state space form

For computational purposes, the equation of motion (5.50) is transformed into the first order state space form. Introducing an auxiliary equation

[ ] [ ] 0=− xMxM && , (5.53)

equations (5.50) and (5.53) can be combined to give

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

⎭

⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡− 00

00

fxx

MK

xx

MMC

&&&

& (5.54)

or

[ ] [ ] pqBqA =+& , (5.55)

where the NN 22 × matrices [ ]A and [ ]B are real but non-symmetrical

[ ] [ ] [ ][ ] [ ] ⎥⎦

⎤⎢⎣

⎡−

=0MMC

A , [ ] [ ] [ ][ ] [ ] ⎥⎦

⎤⎢⎣

⎡=

MK

B0

0.

The resulting system of equations is non-self-adjoint.

Note that equations (5.50) and (5.53) can alternatively give

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

fx

x

CK

M

x

x

M

M 00

0

0

&&&

& (5.56)

but in the following only the form (5.54) will be used.

5.5 Eigenvalue analysis

Solving the rotordynamics eigenvalue problem, natural frequencies, damping ratios, precession modal forms and stability thresholds can be determined for damped gyroscopic systems.


5.5.1 Right and left eigenvectors

The eigenvalues and right eigenvectors are obtained by solving the homogeneous form of equation (5.55)

[ ] [ ] 0=+ qBqA & . (5.57)

Assuming a solution of the form

teλΦ Rq = ,

equation (5.57) can be written

[ ] [ ]( ) 0=+ RBA Φλ . (5.58)

There are Nr 2= eigenvalues rλ obtained from

[ ] [ ]( ) 0det =+ BAλ , (5.59)

and N2 right eigenvectors RrΦ satisfying the generalized eigenvalue problem

[ ] [ ] Rr

R AB ΦλΦ −= , ( )N,...,r 21= . (5.60)

For the transposed equation

[ ] [ ] 0=+ qBqA TT & ,

assume solutions of the form

teλΦ Lq = .

This yields

[ ] [ ]( ) 0=+ LTT BA Φλ . (5.61)

The eigenvalues are given by the equation

[ ] [ ]( ) 0det =+ TBAλ

which has the same solutions as those of equation (5.59).

Because the transposed of equation (5.61) is


[ ] [ ]( ) ⎣ ⎦0=+ BATL λΦ ,

LΦ are referred to as left eigenvectors, solutions of the eigenproblem

[ ] [ ] Lr

Tr

Lr

T AB ΦλΦ −= , ( )N,...,r 21= . (5.62)

The 12 ×N right and left eigenvectors satisfy the bi-orthogonality relations

[ ]

[ ] ⎩⎨⎧

≠=

==

⎩⎨⎧

≠=

==

srsr

B

srsr

A

rrsr

Rr

TLs

rrsr

Rr

TLs

for0for

for0for

βδβΦΦ

αδαΦΦ

(5.63)

so that

r

rr α

βλ −= . (5.64)

These bi-orthogonality relations between the modes of the original system and those of the transposed system can be used to uncouple the system equations.

5.5.2 Reduction to the standard eigenvalue problem

Equation (5.58) can be written in the form

[ ] RRR Φλ

Φ 1= (5.65)

where

[ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] ⎥

⎥⎦

⎤

⎢⎢⎣

⎡ −−=−=−−

−

0

111

IMKCKABR

is a NN 22 × non-symmetric real matrix.

This formulation has the drawback of giving the reciprocals of eigenvalues. The upper half of the eigenvectors yields the 1×N eigenvectors of the original problem.

An alternative is to use the form (5.56). For an assumed solution

teλux = , (5.66)

the homogeneous form of equation (5.56) becomes


[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

⎭

⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡ −+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡000

00

uu

CKM

uu

MM

λλλ (5.67)

or

[ ] [ ]

[ ] [ ] [ ] [ ][ ] [ ][ ] [ ]

0

000

11 =⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡−− −− u

uI

ICMKM

Iλ

λ .

Remember that nN 4= , where n is the number of nodes.

5.5.3 Campbell and stability diagrams

Equation (5.65) has N2 eigensolutions, where N is the order of the system global matrices. They are purely real for overdamped modes and appear in complex conjugate pairs for underdamped or undamped modes of precession.

In general, the complex eigenvalues are of the form

rrr ωαλ i+= , rrr ωαλ i−=∗ , (5.68)

and they are functions of the rotating assembly spin speed Ω .

The imaginary part rω of the eigenvalue is the damped natural frequency of precession (whirl speed or precession speed), and the real part rα is the damping constant. A stable mode requires a non-positive value for rα .

Often, the damping is expressed in terms of the damping ratio

r

rr ω

αζ −≅ ,

or in terms of the logarithmic decrement

r

rrr ω

απζπδ 22 −≅= .

It is common practice to plot both the precession natural frequencies and damping constants as a function of the rotor spin speed Ω . These plots are called whirl speed maps [5.1], and one is shown in Fig. 5.25.

In most practical applications, if the system becomes unstable, it is usually the first forward precession mode which yields the instability, while the remaining modes remain stable. In Fig. 5.25, the first mode is illustrated to become unstable at an onset speed of instability, oiΩ , also called a threshold speed, thΩ .

Plots of only the precession natural frequencies versus spin speed are commonly referred to as Campbell diagrams. When points are marked on the


natural frequency curves, with the corresponding values of the logarithmic decrement, the plots are called Lund diagrams. Plots of only the real part of the eigenvalues versus spin speed are called stability diagrams. Examples of Campbell diagrams and stability diagrams are given for selected rotor models in Sections 3.4 and 4.5 of Part I.

Fig. 5.25

A critical speed of order κ of a single-shaft rotor system is defined as a spin speed for which a multiple of that speed coincides with one of the system’s natural frequencies of precession. An excitation frequency line of equation

Ωκω = is included in Fig. 5.25. When Ω equals rΩ , the excitation rΩκ creates a resonance condition.

One approach for determining critical speeds is to generate the Campbell diagram, include all excitation frequency lines of interest, and graphically note the intersections to obtain the critical speeds associated with each excitation.

The complex conjugate eigenvectors are of the form

rrr bau i+= , rrr bau i−=∗ . (5.69)

The free precession solution can be written as the sum of two complex eigensolutions associated with the pair of eigenvalues and right eigenvectors

( )

( ) ( ) ( ) ( ) .baba

uutxt

rrt

rr

tr

trr

rrrr

rr

ωαωα

λλ

ii eiei

ee−+

∗

−++=

=+=∗

( )

( ) .tbta

batx

rrrrt

tt

r

tt

rt

r

r

rrrrr

ωωα

ωωωωα

sincose2

ii2eei

2eee2

iiii

−=

=⎟⎟⎠

⎞⎜⎜⎝

⎛ −+

+=

−−

(5.70)

Figure 5.26 shows the nature of the motion in an undamped precession mode (node and mode indices are dropped out).


Fig. 5.26

The factor trαe2 does not influence the mode shape, being a common factor for all coordinates. Therefore, the upper half of the modal vector can be approximated by

tutux rsrcr ωω sincos += ,

where

rrc uReau == , rrs uImbu −=−= .

This way, the orbit at any station becomes an ellipse instead of a spiral, which is taken into account by plotting incomplete (“open”) ellipses.

An element of the rth mode of precession has the form

( ) tututx rsrcj ωω sincos += ,

defining a harmonic motion with frequency rω .

The two components of the translational motion at any station are


,tt

,tt

rsrc

rsrc

ωωωω

sincossincoswwwvvv+=+=

(5.71)

defining the parametric equations of an ellipse.

Fig. 5.27

The resulting displacement vector is

( ) ( ) ( )tttr wv i+= .

Connecting the end points of the displacement vectors at all stations along a rotor (at a given time t), the mode shape is obtained, which is a curve in space (Fig. 5.27). Examples are given in Sections 3.4 and 4.5 of Part I.

The special case of conservative gyroscopic systems is worth mentioning. If a pair of bearing principal axes exists, for an undamped gyroscopic system, equation (5.50) becomes

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

⎭

⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡00

00

00

00

ZY

kk

ZY

gg

ZY

mm

z

y&

&

&&

&&.(5.72)

For an assumed solution of the form (5.66), because the system has pure imaginary eigenvalues, ωλ i= , so that equation (5.72) becomes [5.8]

[ ] [ ] [ ]

[ ] [ ] [ ]

⎭

⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

++

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

−00

ii

ii

2

2

IR

IR

z

yzzyy

mkggmk

ωΩωΩωω ,

which gives four real coupled matrix equations. It can be shown that the solutions of the equations are proportional to each other

RI yy β= , IR zz β−= ,

where β is a proportionality constant.

The modal vector can be written

( ) ( ) ( )

⎭⎬⎫

⎩⎨⎧

+=⎭⎬⎫

⎩⎨⎧

+−+

=⎭⎬⎫

⎩⎨⎧

+−+

=⎭⎬⎫

⎩⎨⎧

++

I

R

I

R

II

RR

IR

IR

zy

zy

zzyy

zzyy

ii1

ii1

ii

ii

βββ

ββ


and can be divided by ( )βi1+ .

Hence, conservative gyroscopic systems are described by pure imaginary eigenvalues, and eigenvectors are real in the X-Y plane and pure imaginary in the X-Z plane, respectively.

5.6 Unbalance response

The forced response of a rotor system can be determined either indirectly, in modal coordinates, or directly, in physical coordinates.

5.6.1 Modal analysis solution

In the case of synchronous excitation due to mass unbalance

tFf Ωie= , tXx Ωie= ,

equation (5.45) becomes

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

⎭

⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡− 0i0

00

iF

XX

MK

MMC

ΩΩ

or

[ ] [ ]( ) PQBA =+Ωi

where

⎭

⎬⎫

⎩⎨⎧

=X

XQ

Ωi,

⎭⎬⎫

⎩⎨⎧

=0F

P .

Assume a solution of the form

r

N

rr

RQ ηΦ∑=

=2

1

, (5.73)

where rη are modal coordinates.

Multiplying to the left by Tr

RΦ and considering the bi-orthogonality

relations (5.53), the rth decoupled equation is

[ ] [ ]( ) PBATr

Lrr

RTr

L ΦηΦΩΦ =+i


or

( ) PTr

Lrrr ΦηλΩα =−i .

The rth modal coordinate is

( )rr

Tr

L

rPλΩα

Φη

−=

i.

From equation (5.73), the vector of physical coordinates is

( ) PQrr

Tr

Lr

RN

rλΩα

ΦΦ

−=∑

=i

2

1

,

whose upper half is

( ) FXrr

Tr

Lr

RN

rλΩα

ΦΦ

−=∑

=i

2

1

, (5.74)

where rRΦ and r

LΦ are the upper halves of the corresponding modal

vectors.

5.6.2 Spectral analysis solution

Consider again equation (5.50). For a synchronous excitation

tFtFf sc ΩΩ sincos += , (5.75)

the steady-state response has the form

tXtXx sc ΩΩ sincos += . (5.76)

Substitution of (5.75) and (5.76) into (5.50) yields

[ ] [ ]( ) [ ] [ ] [ ] [ ]( ) .FXMKXC

,FXCXMK

ssc

csc

=−+−

=+−2

2

ΩΩ

ΩΩ (5.77)

This is a linear set of equations which can be solved with known routines. The “cos” and “sin” components are substituted back in equation (5.76). The two components of the translational motion at any station are given by equations of the form (5.71), wherefrom the elliptic orbits can be determined as shown in Section 5.7.


5.7 Kinematics of elliptic motion

In the following, the stationary rectangular coordinate frame is z,y,x .

5.7.1 Elliptic orbits

The simplest steady motion of a rotor (nodal) point is a planar motion with the precession frequency ω . Its displacement components in the y and z directions are

.tztzz,tytyy

sc

sc

ωωωω

sincossincos

+=+=

(5.78)

They define an elliptic orbit, as illustrated in Fig. 5.28.

Fig. 5.28

The ellipse results by composition of two harmonic motions of different amplitudes and phase angles


( )( ) ,tAtAtAz

,tAtAtAy

zzzzzz

yyyyyy

ωθωθθω

ωθωθθω

sinsincoscoscos

sinsincoscoscos

−=+=

−=+=

in two perpendicular directions.

It is seen that

( ) 2122scy yyA += ,

c

sy y

y−=θtan ,

( ) 2122scz zzA += ,

c

sz z

z−=θtan .

The parameters of the elliptic motion are generally a function of the spin speed Ω . Two particular cases are of interest [5.1]:

a) If zy θθ = , the y and z motions are in phase, and the orbit reduces to a straight line (Fig. 5.29).

Fig. 5.29

b) If cs zy −= ( )cz+ and sc zy += ( )sz− the y motion leads (lags) the z

motion by 090 , and the orbit reduces to forward (backward) circular motion (Fig. 5.30).

Fig. 5.30


The parametric equations (5.68) define an ellipse, which is inclined with respect to the y-z axes. Solving first for tωcos and tωsin as

( ) ( )( ) ( ) ,yzzyzyyzt

,yzzyyzzyt

scsccc

scscss

−−=−−=

ωω

sincos

and then eliminating the time, the orbit equation is obtained as

( ) ( ) ( ) ( ) 2222222 2 sccsscssccsc zyzyzyyzyzyzyyzz −=+++−+ . (5.79)

Equation (5.79) is more often expressed in terms of the major and minor semiaxes a and b, and the orbital inclination angle α (Fig. 5.31).

Fig. 5.31

In an 11 zy − (principal) coordinate system, with the axes along the ellipse axes, the motion is described as

( )( ),tbz

,tayαγωαγω

−+=−+=

sincos

1

1 (5.80)

where γ is a phase angle (inclination of radius vector at 0=t ) so that

12

12

1 =⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

bz

ay . (5.81)

The coordinate transformation

,zyz,zyy

αααα

cossinsincos

11

11

+=−=

(5.82)

leads to parametric equations of the form (5.78). By combining equations (5.78), (5.80) and (5.82), it is possible to obtain the four parameters a, b, α and γ in terms of cy , sy , cz and sz . This will be done in the following by a different approach.


5.7.2 Decomposition into forward and backward circular motions

Elliptical orbits may be represented with the aid of rotating complex vectors, by treating the y-z plane as a complex plane with a real axis along the y-axis and the imaginary axis along the z-axis.

Fig. 5.32

The resultant of a forward rotating vector fr and a backward rotating vector br is a rotating vector whose end moves along an elliptical orbit (Fig. 5.32):

( ) ( )bf tb

tf

tb

tf rrrrr θωθωωω +−+− +=+= iiii eeee . (5.83)

From equations (5.78)

( ) ( )

( ) ( ) ,zzz

,yyy

tts

ttc

tts

ttc

ωωωω

ωωωω

iiii

iiii

eei2

1ee21

eei2

1ee21

−−

−−

−++=

−++=

so that

( ) ( ) ( ) ( ) .zyzyezyzye

zyr

cssct

cssct

⎥⎦⎤

⎢⎣⎡ ++−+⎥⎦

⎤⎢⎣⎡ +−++=

=+=

−

21i

21

21i

21

i

ii ωω

It is seen that r is the sum of a forward rotating vector with complex amplitude fr , and a backward rotating vector with complex amplitude br , where


( ) ( )

( ) ( ) .erzyzyr

,erzyzyr

b

f

bcsscb

fcsscf

θ

θ

i

i

21i

21

21i

21

=++−=

=+−++= (5.84)

The amplitudes and phase angles are given by

( ) ( )

( ) ( ) .zy

zy,bazyzyr

,zyzy,bazyzyr

sc

csbcsscb

sc

csfcsscf

−+

=−

=++−=

++−

=+

=+−++=

θ

θ

tan22

1

tan22

1

22

22

(5.85)

The major semiaxis is

bf rra += , (5.86)

( ) ( ) ( ) 22222222222

41

21

csscscscscsc zyzyzzyyzzyya −−+++++++= .

The minor semiaxis is

⎥⎦

⎤⎢⎣

⎡=−=

sc

scbf zz

yya

rrb det1 . (5.87)

The inclination of the major axis (attitude angle) is

( )bf θθα +=21 . (5.88)

From the condition of phase coincidence

bf tt θωθωα +−=+= ∗∗ , fbt θθω −=∗2 ,

where ∗t is the time when P is in the point of maximum displacement amplitude.

The following useful relations can be established

( )2222

22tanscsc

scsc

zzyyzzyyt−+−

+=∗ω , (5.89)

( )2222

22tanscsc

sscc

zzyyzyzy−−+

+=α , (5.90)

The precession is forward when

bf rr > , 0>b , cssc zyzy > .


The precession is backward when

bf rr < , 0<b , cssc zyzy < .

The orbit is a straight line (degenerated ellipse) when

bf rr = , 0=b , cssc zyzy = .

5.7.3 Variable angular speed along the ellipse

Consider an ellipse (Fig. 5.33) described by the parametric equations

.tbz,tay

ωω

sincos

==

(5.91)

Fig. 5.33

Comparison with equations (5.71) yields

( )( ) ,tbtbtAz

,tatAy

zz

yy

ωπωθω

ωθω

sin2

coscos

coscos

=⎟⎠⎞

⎜⎝⎛ +=+=

=+=

so that

2

, 0 πθθ ==== zzyy bA,,aA .

The concentric circles method of constructing an ellipse is illustrated in Fig. 5.34. Two concentric circles are drawn with a center at O. The diameter of the large circle equals the given major axis a2 . The diameter of the small circle equals the given minor axis b2 . A diagonal which makes an angle tω with the y-axis intersects the large circle in M and the small circle in P. Point B on the ellipse is plotted by projecting downward from M to intersect with the horizontal line drawn from P.


Point A on the y axis corresponds to 0=t . Point B corresponds to t, and point C to ωπ 2+t . Note that points B and C correspond to a time interval of

ωπ 2 ( )090=∠MON while the angle BOC is larger than 090 .

Fig. 5.34

If point M moves along the circle of radius a with constant angular velocity ω , the point B moves along the ellipse with variable angular velocity.

Denoting AOB=θ , the angular position at time t is given by

tab

yz ωθ tantan == . (5.92)

The variation of θ as a function of tω is shown in Fig. 5.35. The deviation from the straight line indicates a variable angular velocity.

Fig. 5.35


Indeed, the angular velocity of the motion along the ellipse is

( )t

ab

tab

ttω

ωω

ωθωθθ

22

2

tan1

sec

dd

dd

⎟⎠⎞

⎜⎝⎛+

===& . (5.93)

The variation of θ& as a function of tω is plotted in Fig. 5.36. It is seen that θ& varies between ( )ωω ab=′ and ( )ωω ba=′′ so that ωωω ′′′= .

Fig. 5.36

In conclusion, the angular speed of the precession motion is not θ& - that of the point B along the elliptic orbit, but the speed ω of the points M and P along the generating circles of radii a and b, respectively.

5.8 Model order reduction

The initial finite element discretization of a rotor system needs a relatively large number of degrees of freedom (DOFs) for satisfactory accuracy. A reduction of the number of DOFs is sometimes necessary because, in actual applications, only a few lower modes are of concern, giving little justification for solving the complete equation of motion in the dynamic analysis.

5.8.1 Model condensation

Static and dynamic condensation techniques can be used to produce reduced models possessing eigensystems that approximate that of the original full system model.


5.8.1.1 Formalism of coordinate reduction

Consider equation (5.50) in the form

[ ] [ ] [ ] 1111 ×××××××

=++NNNNNNNNNNfxKxCxM &&& . (5.94)

A rectangular transformation matrix [ ]T is seeked, which relates the nN 4= elements (coordinates) of the vector x to a smaller number NL < of

elements (coordinates) of a vector u , so that

[ ] 11 ×××

=LLNNuTx . (5.95)

The transformation is time independent, so that

[ ] uTx = , [ ] uTx && = , [ ] uTx &&&& = . (5.96)

Substituting equations (5.96) in (5.94) and premultiplying by [ ]TT , the energy equivalence yields the reduced equations of motion

[ ] [ ] [ ] 1111 ×××××××

=++L

red

LLL

red

LLL

red

LLL

red fuKuCuM &&& , (5.97)

where

[ ] [ ] [ ][ ]TMTM Tred = , [ ] [ ] [ ][ ]TCTC Tred = , [ ] [ ] [ ][ ]TKTK Tred = ,

[ ] fTf Tred = .

A proper choice of [ ]T will drastically reduce the number of DOFs without altering the lower eigenfrequencies and the mode shapes of interest.

5.8.1.2 Guyan/Irons condensation

The basis for the Guyan/Irons reduction is to follow a standard procedure used in static structural analysis, namely, elimination of DOFs at which no forces are applied, whence the name of static condensation.

The coordinates (DOFs) are partitioned into two groups: a) the active (“master”, retained) coordinates, and b) the omitted (“slave”, discarded) coordinates, denoted by “a” and “o”, respectively.

Partitioning the equation (5.94) accordingly


[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

⎭

⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡

o

a

o

a

oooa

aoaa

o

a

oooa

aoaa

o

a

oooa

aoaa

ff

xx

KKKK

xx

CCCC

xx

MMMM

&

&

&&

&&.

(5.98)

Assuming 0=of , the static force-deflection relationship reduces to

[ ] [ ][ ] [ ]

⎭

⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡0a

o

a

oooa

aoaa fxx

KKKK

. (5.99)

The lower partition provides a static constraint equation

[ ] [ ] 0=+ oooaoa xKxK (5.100)

which can be written

[ ] [ ] aoaooo xKKx 1−−= . (5.101)

The original set of coordinates x can be related to the subset of active coordinates by the equation

[ ][ ] [ ] [ ] aa

oaoo

a

o

a xTxKK

Ixx

x =⎥⎦

⎤⎢⎣

⎡−

=⎭⎬⎫

⎩⎨⎧

= −1 . (5.102)

Equation (5.102) may be referred to as a Ritz transformation. The Ritz basis vectors, which are columns of the Ritz transformation matrix [ ]T , are the displacement patterns associated with unit-displacement of the respective a-coordinates while the o-coordinates are released

[ ] [ ] ∑=

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧==

L

jajj

aL

a

La xtx

xtttxTx

1

1

21 LL . (5.103)

The reduced set of equations has the form (5.97), where

axu = ,

[ ] [ ] [ ][ ] [ ]oaooaoaared KKKKK 1−−= , (5.104)

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ],KKMKK

KKMMKKMM

oaooooooao

oaooaooaooaoaared

11

11

−−

−−

+

+−−= , (5.105)

and [ ]redC has an expression similar to (5.105).


Replacing the dynamic relationship between active and omitted DOFs by a static relationship, the Guyan/Irons reduction is an incomplete extension of the Static condensation, with inherent loss of accuracy.

One exception is worth mentioning: a lumped-mass model, consisting of point masses at the nodes where translational displacements are defined (mass moments of inertia neglected). With all the rotational DOFs as o-DOFs and all the translational DOFs as a-DOFs,

[ ] [ ]0=ooM , [ ] [ ]0=oaM , [ ] [ ]0=aoM , [ ] [ ]aared MM =

and the Guyan reduction is accurate.

Drawbacks: a) misapplication can lead to serious modeling errors; b) destroys the banded form of matrices; and c) requires insight, experience and skill to partition the DOFs, though automatic procedures exist for the selection of active DOFs. In conclusion, the accuracy of the refined finite element model one tries very hard to achieve may be lost by the Guyan/Irons reduction.

5.8.1.3 Use of macroelements

Rotating shafts have variable cross sections and usually it is necessary to use a large number of finite elements to obtain a good model of the rotor. The number of elements can be reduced by introducing macro-elements [5.9]. Several short cylindrical elements can be treated as one element. Formally, this is done by a static condensation, treating the interior coordinates at the steps of the cross section as o-DOFs and the boundary coordinates as a-DOFs. This increases numerical economy without loss of accuracy in the results.

a b

Fig. 5.37

For the stepped shaft from Fig. 5.37, a, a macro-element is shown in Fig. 5.37, b. Considering only the motion in the Y-X plane, the 88× macro-element matrix has a banded form (Fig. 5.38). Reordering the nodal displacements, moving up the external DOFs selected as a-DOFs and moving down the internal DOFs selected as o-DOFs, destroys the banded form.


Fig. 5.38

Elimination of internal DOFs using the transformation (5.85) yields a condensed 44× matrix. This allows to maintain the banded structure of the system matrix (Fig. 5.39).

Fig. 5.39

5.8.1.4 Modal condensation

Consider the homogeneous part of the equation (5.50) of a damped anisotropic rotor system, written as

[ ] [ ] [ ]( ) [ ] [ ]( ) 0=++++ xKKxGCxM bsb &&& Ω , (5.106)

where [ ]bK and [ ]bC are the stiffness and damping matrices for bearings,

respectively, [ ]sK is the shaft stiffness matrix, [ ]G is the (shaft+disks) gyroscopic matrix, [ ]M is the mass matrix, and x is the 14 ×n state vector.

The corresponding complex eigenvalue problem yields the damped eigenfrequencies and the complex eigenvectors. For rotor systems with a large


number of DOFs, the complex eigenvalue procedure may run into numerical difficulties and may be time consuming.

One approach to avoid some of these problems is the modal condensation method. One variant is based on the analysis of the isotropic undamped non-gyroscopic part of equation (5.106) in the Y-X plane:

[ ] [ ] 0=++ YkkYm yy&& . (5.107)

The shaft is assumed symmetrical, rotor gyroscopic effects are neglected, bearing damping is neglected, and only an average bearing principal stiffness term is considered, usually the symmetric component defined in equation (5.37). This (neighboring) associate conservative system has planar undamped modes entering as columns in the modal matrix

[ ] [ ]n221 ΦΦΦΦ L= . (5.108)

A truncated modal matrix is used as the transformation matrix, using only the L lower modes of the system described by (5.107).

Retaining the first L columns of the matrix (5.108)

[ ] [ ]LΦΦΦΦ L21=∗ . (5.109)

The coordinate transformation is

[ ]

[ ] [ ][ ] [ ]

[ ] u

uu

uu

ZY

xz

y

z

y ∗∗

∗∗ =

⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

= ΦΦ

ΦΦ0

0 (5.110)

where u is the reduced state vector.

Substitution of (5.110) into (5.106) and pre-multiplication by [ ]T∗Φ yields the reduced set of equations of motion (5.97)


where

[ ] [ ] [ ] [ ]∗∗= ΦΦ MMTred , [ ] ff

Tred ∗= Φ ,

[ ] [ ] [ ] [ ]( ) [ ]∗∗ += ΦΦ GCC bTred , (5.111)

[ ] [ ] [ ] [ ]( ) [ ]∗∗ += ΦΦ bsTred KKK .

After determining the modal coordinates u , the physical coordinates are calculated from equation (5.110).

One can say that up to 10-12 modal vectors jΦ have to be used in order to accurately determine the first 2-3 eigenvalues whose imaginary parts fall within the operating speed range. It is possible to introduce diagonal modal damping values accounting for the external and internal/structural damping.

The modal method does not require any intuition on mass lumping, component mode selection, and iterative procedures to improve the transformation matrix. The only basic assumption is that the linear combination of the Ritz basis vectors obtained by considering the undamped isotropic rotor system constitutes a good approximation to the complex eigenvectors of the heavily damped anisotropic rotor system.

5.8.2 Model substructuring

The rotor-stator-foundation system can be subdivided into components or substructures, analyzing the components separately, and then coupling them to obtain the mathematical model of the full system. Methods of component mode synthesis (CMS) or substructure coupling for dynamic analysis are used to synthesize the system equations of motion from the characteristic displacement modes of the components.

In the CMS method, the displacement of any point in a component is represented as the superposition of two types of component modes: a) constrained normal modes, defining displacements relative to the fixed component boundaries, and b) constraint modes, produced by displacing the boundary coordinates. In addition, the coordinates of any component are classified as: a) boundary (junction) coordinates, if they are common to two or more components, and b) interior coordinates, if they do not interface with any other component.

The static constraint modes are determined by giving each of the boundary coordinates a unit displacement in turn, fixing all other boundary coordinates and allowing the internal coordinates to displace. The dynamic constrained normal modes are found by fixing all boundary coordinates and determining the free vibration modes of the constrained component.


The reduction of the number of DOFs of the system is provided by the constrained normal modes. The assumption is that the system response can be found accurately enough by retaining only a properly selected reduced number of constrained normal modes. Each component is transformed in terms of its constrained normal modes and assembled into a lower order set of system equations of motion.

The analysis of a full rotor-bearing-foundation system is beyond the aim of this presentation. The subsystem rotor-bearings will be considered separately to emphasize the steps in the CMS analysis of a substructure [5.9].

Consider a uniform shaft (Fig. 5.40) supported by three bearings. Neglecting coupling effects, consider only the vibration in the Y-X plane.

Fig. 5.40


The shaft is divided into 7 beam finite elements. The nodal coordinates 1v ,

2v ,…, 8v , 1ψ , 2ψ ,…, 8ψ are listed in the upper part of Fig. 5.40, a.

Let the boundary coordinates 1v , 4v and 8v be the a-DOFs, and the interior coordinates be the o-DOFs:

TaY 841 vvv= ,

ToY 8776655433221 ψψψψψψψψ vvvvv= .

Order the elements of the global displacement vector so that active DOFs are collected at the top

⎭

⎬⎫

⎩⎨⎧

=o

a

YY

Y . (5.112)

The equation of motion can be written in partitioned-matrix form as

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡

oy

ay

o

a

oooa

aoaa

o

aaa

o

a

oooa

aoaaFF

YY

kkkk

YYc

YY

mmmm

&

&

&&

&&

000

.

(5.113)

Matrices are shown in Fig. 5.41 where the shaded areas represent shaft elements while circles represent bearing elements.

Fig. 5.41

5.8.2.1 Constrained normal modes

If translational displacements in bearings (boundary coordinates) are blocked, then 0841 === vvv , 0=aY and equation (5.113) reduces to that of a conservative auxiliary system. The homogeneous form


[ ] [ ] 0=+ oooooo YkYm && (5.114)

yields the so-called “constrained normal modes”, i.e. the eigenvectors jΦ ( )131,....,j = of the rigidly supported (constrained) shaft (Fig. 5.40, b).

By truncation of these component modes, a transformation matrix is set up by using, for example, only 5 (out of 13) vectors in a reduced modal matrix

[ ] [ ]521 ΦΦΦΦ L=∗ . (5.115)

The coordinate transformation is written as

[ ] 15513113 ××

∗

×= qYo Φ (5.116)

where q is a column vector of constrained modal coordinates.

Equations (5.112) and (5.116) yield

(5.117)

5.8.2.2 Constraint modes

The static behavior of the rigidly supported shaft is described by the static part of equation (5.113), where 0=

oyF

[ ] [ ][ ] [ ]

⎭

⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡0

ay

o

a

oooa

aoaa FYY

kkkk

. (5.118)


The interior coordinates oY can be expressed in terms of the boundary coordinates aY using the lower part of equation (5.118)

[ ] [ ] 0=+ oooaoa YkYk .

This can also be written as

[ ] [ ] [ ] astataoaooo YYkkY Φ=−= −1 . (5.119)

The “constraint modes” are the columns of the matrix [ ] [ ] TTstataI ⎥⎦

⎤⎢⎣⎡ Φ

where

[ ] [ ] [ ]oaoostat kk 1−−=Φ . (5.120)

The static constraint modes are defined by producing a unit displacement of each boundary coordinate in turn, with all other coordinates blocked and all interior coordinates unconstrained and unloaded (Fig. 5.40, c). They represent global shape functions or Ritz vectors.

5.8.2.3 Combined static and modal condensation

The constrained normal modes [ ]∗Φ and the constraint modes [ ]statΦ are superposed to obtain a modal transformation of the component physical coordinates. Equation (5.117) becomes

(5.121)

or

[ ] uTY = . (5.122)


Substitution of (5.122) into (5.113) and premultiplication by [ ]TT yields the reduced set of equations of motion

(5.123)

which can be written as

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ] M&&& =

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡u

k

ku

cu

mm

mmredqq

redaaaa

redqq

redqa

redaq

redaa

0

0

00

0,

where

[ ] [ ] [ ][ ] [ ]oaooaoaaredaa kkkkk 1−−= ,

[ ] [ ] [ ] [ ]∗∗= ΦΦ ooTred

qq kk ,

[ ] [ ]aaredaa cc = ,

[ ] [ ] [ ] [ ]∗∗= ΦΦ ooTred

qq mm , (5.124)

[ ] [ ] [ ] [ ] [ ]( )oastatooTred

qa mmm += ∗ ΦΦ ,

[ ] [ ]Tredqa

redaq mm = ,

[ ] [ ] [ ] [ ] [ ]( ) [ ] [ ] [ ]aastataooastatooT

statredaa mmmmm +++= ΦΦΦ ,

Once the truncated component equations are developed for each component of the system, the next step in the analysis is to assemble them to form the truncated system equations. They are solved in terms of the system coordinate vector, composed of the boundary coordinates of the system and the retained constrained modal coordinates of each component.

Subsequent development may proceed in first order form.

Concluding, in the Craig-Bampton component synthesis methods, the displacement of any point in a component is represented as a superposition of two types of displacement modes: a) constrained normal modes – displacements


relative to the fixed component boundaries, and b) static constraint modes – displacements produced by displacing the boundaries.

The number of degrees of freedom for the whole system is reduced by truncating the number of constrained normal modes. Usually, overdamped or heavily damped modes are truncated first, then the highest frequency modes.

5.8.3 Stepwise model reduction methods

The static or Guyan reduction simply neglects the inertia terms associated with the omitted degrees of freedom (o-DOFs). In the Improved Reduced System (IRS) method, the inertial effects of o-DOFs are taken into account. Its robustness is dependent on the observability of the structure from the selected active degrees of freedom (a-DOFs). The Iterative IRS method (IIRS) is based on a serial elimination of o-DOFs and an automatic selection of the number and location of a-DOFs [5.10]. The IIRS method converges to a reduced model which reproduces a subset of the modal model of the full system.

In the equation of motion (5.50) of a damped rotor-bearing system, the size of matrices is N×4 , where N is the number of nodes in the finite element model. For a large complex system, this leads to a large size eigenvalue problem, while only the first several eigenproperties are of practical interest.

In order to reduce the size of the eigenvalue problem, the conservative non-gyroscopic homogeneous part of equation (5.50) is considered first

[ ] [ ] 0=+ xKxM && , (5.125)

where [ ]M and [ ]K are the symmetric parts of matrices [ ]M and [ ]K , respectively.

The physical coordinates in the vector x are eliminated one at a time. The criterion is the value of the ratio jjjj mk of the diagonal elements of [ ]M and

[ ]K . The DOF for which this ratio is highest is denoted xo and moved into the lower location of the displacement vector for elimination.

Equation (5.125) can be written

[ ] [ ] [ ][ ] [ ] [ ] [ ][ ] 0=+ xPPKPxPPMP TTTT && ,

where [ ]P is a permutation matrix.

It can be partitioned as


[ ] ⎣ ⎦

[ ] ⎣ ⎦

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡00

o

a

oooa

aoaa

o

a

oooa

aoaa

xx

KKKK

xx

MMMM

&&

&& (5.126)

where ox , ooK , and ooM are scalars, and ax is the column vector of the remaining a-DOFs. If several DOFs have the same ratio jjjj mk , then the one with the smallest index is considered first. If this ratio is greater than a cut-off frequency squared ω c

2 , then the corresponding DOF is eliminated.

5.8.3.1 Stepwise Guyan Reduction (SGR)

From the static part of equation (5.126), a constraint equation between the o-DOF and the column vector of a-DOFs is obtained as

⎣ ⎦ ⎣ ⎦ aSToaaoaooo xGxKKx =−= −1 .

The reduction to a-DOFs in the SGR method is defined by

[ ]

⎣ ⎦ [ ] aSaSToa

a

o

a xTxG

Ixx

=⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

, [ ] [ ] aS xTPx = .

After one reduction step, the SGR reduced equation (5.125) becomes

[ ] [ ] 0=+ aSTaST xKxM && , (5.127)

where

[ ] [ ] [ ] [ ] [ ][ ]STT

SST TPMPTM = , [ ] [ ] [ ] [ ] [ ][ ]STT

SST TPKPTK = .

Each reduction step is characterized by the product [ ][ ]STP . The whole process is described by a transformation matrix of the form

[ ] [ ][ ]( ) [ ][ ]( ) [ ][ ]( )nSSS TP.........TPTPT 21=

where n is the number of o-DOFs.

At the end of the stepwise elimination of o-DOFs, the full reduced matrices are

[ ] [ ] [ ][ ]TMTM Tred = , [ ] [ ] [ ][ ]TKTK T

red = , [ ] [ ] [ ][ ]TCTC Tred = ,(5.128)

where the same matrix [ ]T has been used to reduce in [ ]C the damping and gyroscopic matrices.

The physical coordinate reduction is defined by

[ ] redxTx = . (5.129)


After solving the reduced eigenvalue problem, the full size modal vectors can be recovered by back-expansion, using the inverse SGR method (ISGR) based on the transformation (5.129).

5.8.3.2 Stepwise Improved Reduction (SIR)

If the acceleration vector from equation (5.127)

[ ] [ ] aSTSTa xKMx 1−−=&& ,

and the acceleration scalar

⎣ ⎦ [ ] [ ] aSTSToaooo xKMKKx 11 −−=&&

are substituted into the lower part of equation (5.126), then a new constraint equation is obtained between the o-DOF and the column vector of a-DOFs

( )⎣ ⎦ aoao xGx 1=

where

( )[ ] ⎣ ⎦ ⎣ ⎦ ⎣ ⎦( ) [ ] [ ][ ]STSTSToaoooaoaoooa KMGMMKKG 111 −− +−−= . (5.130)

In the SIR method, the reduction to a-DOFs is defined by

[ ]

( )⎣ ⎦ [ ] aaoa

a

o

a xTxG

Ixx

11 =⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

, [ ] [ ] axTPx 1= .

After one reduction step, the reduced homogeneous equation of motion is

[ ] [ ] [ ] 0111 =++ aaa xKxCxM &&& ,

where

[ ] [ ] [ ] [ ] [ ][ ]111 TPMPTM TT= , [ ] [ ] [ ] [ ] [ ][ ]111 TPKPTK TT= ,

[ ] [ ] [ ] [ ] [ ][ ]111 TPCPTC TT= .

The SIR transformation matrix is of the form

[ ] [ ][ ]( ) [ ][ ]( ) [ ][ ]( )nTP.........TPTPT 12111= (5.131)

and the full reduced system matrices are given by (5.128).

Modal vector back-expansion is carried out using the Inverse SIR (ISIR) method based on the transformation (5.129) where [ ]T is given by (5.131).


5.8.3.3 Stepwise Iterated Improved Reduction (SIIR)

If substitution of accelerations ax&& and ox&& into equation (5.126) is repeated, for the subsequent iterations the constraint equation becomes

( )⎣ ⎦ ai

oao xGx 1+= ,

where

( )[ ] ⎣ ⎦ ⎣ ⎦ ( )⎣ ⎦( ) [ ] [ ][ ]iii

oaoooaoaooi

oa KMGMMKKG 111 −−+ +−−= .

The reduction to a-DOFs becomes

[ ]

( )⎣ ⎦ [ ] aiaioa

a

o

a xTxG

Ixx

=⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

, [ ] [ ] ai xTPx = ,

where the subscript i denotes the ith iteration. After one reduction step, the SIIR homogeneous equation of motion of the damped gyroscopic system is

[ ] [ ] [ ] 0=++ aiaiai xKxCxM &&& ,

where

[ ] [ ] [ ] [ ] [ ][ ]iTT

ii TPMPTM = , [ ] [ ] [ ] [ ] [ ][ ]iTT

ii TPKPTK = ,

[ ] [ ] [ ] [ ] [ ][ ]iTT

ii TPCPTC = . (5.132)

At each reduction step, the effect of the removed o-DOF is redistributed to all the remaining a-DOFs, so that the next reduction will remove the o-DOF with the highest jjjj mk ratio in the reduced mass and stiffness matrices. The

procedure is applied until the highest ratio jjjj mk is equal to or less than ω c2 . At

this point the a-DOFs represent the selected active DOFs of the reduced model.

Indeed, for sinusoidal excitation with frequency ω , the lower part of equation (5.126) yields

( ) ⎣ ⎦ ⎣ ⎦( ) .xMKMKKx aoaoaooooooo21121 1 ωω −−−=

−−− (5.133)

If the first parenthesis in (5.133) is approximated by the truncated binomial expansion

( ) ( )oooooooo MKMK 12112 11 −−− +≅− ωω (5.134)

and the terms in ω 4 are ignored, then a constraint equation is obtained as


⎣ ⎦ ⎣ ⎦ ⎣ ⎦( )[ ] .xGMMKKx aSToaoooaoaoo 21

o +−−= − ω (5.135)

When ax is a mode shape of the reduced conservative problem, equation (5.135) becomes equation (5.130). But equation (5.134) is valid only for frequencies ./MK oooo<<2ω This means that if a limit ω c

2 is established, then the elimination of o-DOFs can be done until the ratio oooo /MK is equal to or smaller than this value. In this way, the minimum number of a-DOFs is automatically determined, as well as their location.

For conservative non-gyroscopic systems, the SIIR method converges monotonically to a reduced model that preserves the lower eigenvalues and the corresponding reduced eigenvectors of the full system. For damped gyroscopic systems, the arbitrary reduction of damping and gyroscopic matrices in (5.132) may lead to better prediction accuracy of the SIR method for some eigenmodes.

After solving the reduced eigenvalue problem, equation (5.129) is used in the Inverse SIIR (ISIIR) method to expand the a-DOF vector to the size of the full problem, using the transformation matrix

[ ] [ ][ ]( ) [ ][ ]( ) [ ][ ]( )niii TP.........TPTPT 21= ,

where the subscript i is the number of iterations in the SIIR method.

A measure of the accuracy of the expanded mode shapes is given by the relative mode shape error

( ) ( ) ( )%FEMandedexpFEM 100absabs ⋅− ΦΦΦ .

Example 5.1

Figure 5.42 shows the single-disk multi-stepped shaft rotor system with two identical isotropic bearings of reference [5.8]. It was modeled with 76 DOFs, using 18 Timoshenko shaft elements including gyroscopic effects and neglecting internal damping [5.11]. The reduction procedure, applied with a cut-off frequency of 8000 rad/sec, selected 8 active DOFs: 3, 9, 25, 33, 41, 47, 63, 71.

The damped eigenfrequencies, computed at a spin speed rpm30000=n , are given in Table 5.3. The "true" values listed in the second column correspond to the full eigenvalue problem (76 DOFs). Columns three to five list natural frequencies computed using SGR, SIR and SIIR.. Note that the 8 translation DOFs are located at 4 nodes, the a-DOFs selected in one plane being selected in the other plane too.


Fig. 5.42

Columns six to eight list the relative error of expanded mode shapes by Inverse Stepwise Reduction. For some modes, the iterations in SIIR (and ISIIR) do not improve on the values in SIR (and ISIR).

Table. 5.3

Damped eigenfrequency, Hz Mode shape error, %

Mode “True” FEM SGR SIR SIIR

5 iter ISGR ISIR ISIIR 5 iter

1 246.01 246.78 246.27 246.21 2.04 1.46 1.32 2 296.49 296.91 296.81 296.74 1.54 1.82 1.66 3 774.00 777.26 773.00 772.79 3.59 1.07 1.23 4 808.31 808.38 807.32 807.20 2,16 0.91 1.12 5 1165.7 1287.3 1174.9 1166.8 24.96 7.57 3.44 6 1367.4 1425.4 1366.1 1369.7 14.71 0.69 3.35 7 1959.9 1997.6 1960.0 1958.7 3.59 1.26 0.57 8 2020.6 2054.1 2020.0 2020.7 3.29 0.61 0.53

Example 5.2

Figure 5.43 shows the four-disk three-bearing rotor from reference [5.12] with variable inner diameter. The rotor was modeled with 13 nodes (52 DOFs), using Timoshenko shaft elements with consistent mass and gyroscopic matrices.


Fig. 5.43

The reduction procedures applied with a cut-off frequency of secrad3000 selected 24 a-DOFs. Further reduction applied to select only 8

active DOFs, using the MK criterion, selected the four DOFs of node 1, and the translational DOFs of nodes 10 and 12 [5.11].

Table. 5.4


Mode “True” FEM SGR SIR SIIR ISGR ISIR ISIIR

1 46.391 46.419 46.395 46.394 0.28 0.10 0.07 2 59.573 59.611 59.580 59.578 0.50 0.13 0.09 3 178.44 179.77 178.42 178.40 4.71 0.17 0.10 4 179.41 180.76 179.39 179.38 4.83 0.19 0.07 5 380.29 399.07 380.75 380.40 18.5 3.33 1.31 6 441.37 469.27 443.79 441.66 34.2 9.59 3.11 7 461.39 507.28 463.35 459.26 43.5 13.8 1.21 8 463.96 508.54 465.16 461.70 46.7 14.1 1.70

Table 5.4 lists the damped eigenfrequencies computed at a spin speed of 3000 rpm for the first eight modes of precession. Again, the accuracy is very good for the SIIR method with only 5 iterations. Mode shape expansion by the ISIIR method gives excellent results.

Example 5.3

The reduction procedures have been applied to the rotating shaft of a vertical Kaplan hydraulic unit (Fig. 5.44) from [5.13]. The shaft was modeled with


14 Timoshenko elements (60 DOFs) and simplified equivalent constant properties for the three horizontal bearings [5.11].

Fig. 5.44

Using a cut-off frequency of 500 rad/sec, the selected a-DOFs were: 1, 9, 21, 29, 31, 39, 51, 59, six of them for the three large masses: turbine, generator and auxiliary rotor. The eight lowest eigenfrequencies, computed for an angular speed of 3000 rpm, are listed in Table 5.5 together with the relative error of the expanded mode shapes.

Table. 5.5


Mode “True” FEM SGR SIR SIIR ISGR ISIR ISIIR

1 25.337 25.357 25.368 25.371 1.21 1.49 1.61 2 28.762 28.884 28.799 28.796 2.91 1.61 1.58 3 38.092 38.681 38.084 38.084 9.54 0.38 0.71 4 38.386 38.946 38.389 38.379 10.3 1.50 0.75 5 45.026 45.779 45.040 45.045 7.94 1.37 1.68 6 47.195 48.015 47.245 47.226 8.49 2.59 2.02 7 65.724 66.776 65.774 65.705 7.43 2.38 0.90 8 68.203 69.263 68.203 68.198 6.88 1.20 0.95


References

5.1 Ehrich, F. F. (ed.), Handbook of Rotordynamics, McGraw Hill, New York, 1992.

5.2 Lalanne, M., Ferraris, G., Tran, D. M., Quéau, J. P., and Berthier, P., Comportement dynamique des rotors de turbomachines, I.N.S.A. Lyon, 1982.

5.3 Marguerre, K. and Wölfel, H., Mechanics of Vibration, Sijthoff & Noordhoff, Alphen aan den Rijn, 1979.

5.4 Genta, G. and Gugliotta, A., A conical element for finite element rotor dynamics, J. Sound Vib., vol.120, no. 1, p. 175-182, 1988.

5.5 Lee, C.-W., Vibration Analysis of Rotors, Kluwer Academic Publ., Dordrecht, 1993.

5.6 Someya, T. (ed), Journal-Bearing Databook, Springer, Berlin, 1988.

5.7 Childs, D., Turbomachinery Rotordynamics: Phenomena, Modeling and Analysis, Wiley, 1993.

5.8 Wang, W., and Kirkhope, J., New eigensolutions and modal analysis for gyroscopic/rotor systems, Part I: Undamped sytems, J. Sound Vib., vol.175, no.2, pp 159-170, 1994.

5.9 Gasch, R., and Knothe, K., Strukturdynamik, Band 2, Kontinua und ihre Diskretisierung, Springer, Berlin, 1989.

5.10 Shah, V. N., and Raymund, M., Analytical Selection of Masters for the Reduced Eigenvalue Problem, Int. J. Num. Methods in Engineering, vol.18, pp 89-98, 1982.

5.11 Radeş, M., Rotor-bearing model order reduction, Proc. 5th Int. IFToMM Conference on Rotor Dynamics, Vieweg, pp 148-159, 1998.

5.12 Rajan, M., Rajan, S. D., Nelson, H. D., and Chen, W. J., Optimal Placement of Critical Speeds in Rotor-Bearing Systems, ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design, vol.109, pp 152-157, 1987.

5.13 Gmür, T. C., and Rodrigues, J. D., Shaft Finite Elements for Rotor Dynamics Analysis, ASME Journal of Vibration and Acoustics, vol.113, pp 482-493, 1991.

6. FLUID FILM BEARINGS AND SEALS

This chapter is a brief introduction to hydrodynamic fluid film journal bearings and seals for rotating machinery. The presentation describes how the fluid film bearings and seals work and some of the basic principles that underline their operation. The focus is on their dynamic characteristics and the influence of different bearing and seal configurations on the dynamics of rotor-bearing systems.

6.1 Fluid film bearings

Fluid film bearings operate with a thin film of lubricant between two surfaces which are in relative motion. The lubricant may serve some or all of the following purposes: a) support the rotor load; b) reduce support friction; c) provide radial stiffness and radial squeeze-film damping; d) cool the bearing; and e) attenuate transmitted rotor vibrations.

Fluid film bearings may be classified into several main types according to: a) principle: hydrodynamic (self-acting), hydrostatic (externally pressurized), or hybrid (combination of the two); b) fluid: liquid or gas; c) flow regime: laminar or turbulent; d) geometry: plain cylindrical, four axial groove, elliptical (lemon), three-lobe, pressure dam, Rayleigh step, tilting pad, floating ring, others; and e) load relative direction: radial or axial.

In the following, attention will be restricted to hydrodynamic, incompressible, laminar, oil-lubricated journal bearings. In this case, the load carrying capacity derives from the hydrodynamic pressures generated in the lubricant film by the rotation of the journal. Their operation is based on the formation of an oil wedge which lifts the journal. Configurations giving rise to more wedges ensure better stability. Tilting pad designs are intrinsically stable. Solutions with Rayleigh steps have high load carrying capacity.


6.2. Static characteristics of journal bearings

Consider a horizontal bearing with a journal rotating at a constant angular speed Ω . Assume that the journal is loaded vertically and downwards by an external force of constant magnitude W (Fig. 6.1).

Fig. 6.1

The center JO of the loaded journal shifts from the center BO of the bearing. A convergent clearance space is formed between the journal and the bearing surface. Due to adhesion and viscosity, the lubricant is forced into the wedge-like clearance so that a hydrodynamic pressure p is generated in the fluid film.

The load carrying capacity of the bearing is generated by the pressure forces that compensate the friction forces produced in the viscous fluid by the relative motion of the journal and bearing surfaces. In the remaining divergent clearance space, the lubricant flow is usually cavitated by aeration (expulsion of air or gas) and the fluid film is ruptured.

The wedge film pressure p has an asymmetrical distribution with respect to the JBOO line. The resultant force F of the pressure p is balanced by the static external load W. The equilibrium position of the journal center is dependent on the load W and the spin speed Ω .

6. FLUID FILM BEARINGS 79

6.2.1 Geometry of a plain cylindrical bearing

Figure 6.2 illustrates the cross section of a "plain" journal bearing [6.1]. The bearing main dimensions are: BB RD 2= - inner bearing diameter, RD 2= - outer journal diameter, L - bearing length, RRC B −= - bearing clearance,

JBe OO= - eccentricity, ϕ - attitude angle and h - fluid film thickness.

Fig. 6.2

It is useful to introduce the following dimensionless geometrical parameters: L/D - length-to-diameter ratio; C/R - clearance ratio; Ce=ε - eccentricity ratio; and Chh = - normalized film thickness.

The journal is assumed to rotate anticlockwise with a constant angular speed NπΩ 2= (rad/sec), where N represents the journal spin speed in rps.

In the triangle MOO JB , CRB +=MO , hRJ +=MO , so that

( ) ( ) ( ) θcos2222 CReCRehR ++++=+ .

Expanding and discarding higher-order terms in e, Rh , and RC , the film thickness can be approximated by

( )θε cos1+≅ Ch , (6.1)

which describes the film shape to within 1% accuracy.

The minimum film thickness is

( )ε−= 1Chmin . (6.2)


6.2.2 Equilibrium position of journal center in bearing

The load carrying capacity of a bearing is conveniently characterized by a dimensionless duty parameter called the Sommerfeld number.

The mean pressure on the projected area of the journal is

LD

Wp =′

or, in dimensionless form,

22

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛′=

RC

LDNW

RC

Nppm μμ

,

where W is the radial load, N, and μ is the lubricant absolute viscosity, 2mNsec .

The Sommerfeld number is commonly defined as

21⎟⎠⎞

⎜⎝⎛==

CR

WLDN

pS

m

μ . (6.3)

A smaller value of S indicates a heavily loaded bearing with a lower rotational speed. In some publications the inverse of S is referred to as the Sommerfeld number.

a b Fig. 6.3 (from [6.2])


The locus of the static equilibrium position for the journal center is not a straight line, but a curve resembling a semi-circle (Fig. 6.3, a). Figure 6.3, b shows the static equilibrium loci for five values of DL .

The locus is defined by the eccentricity ratio 0ε and the attitude angle

0ϕ . Both parameters are functions of load and speed through the parameter S.

Fig. 6.4 (from [6.3])

For W=const. and increasing Ω , the journal center moves upwards the equilibrium locus. Beginning at the bottom of the bearing ( )10 =ε , when the rotor starts up, and rising as the speed increases, the journal reaches the concentric position ( )00 =ε in the limit, as the spin speed becomes infinitely large.

For .const=Ω and increasing W, the journal center moves downwards the equilibrium locus. Beginning in the concentric position ( )00 =ε , where the


bearing is unloaded, the journal center moves downwards as the load is increased, approaching the bottom of the bearing ( )10 =ε as the load goes to infinity.

Fig. 6.5 (from [6.2])

Fig. 6.6 (from [6.2])


Figure 6.4 gives the eccentricity ratio 0ε of a plain cylindrical bearing as a function of the Sommerfeld number, for three values of L/D. Figure 6.5 gives the attitude angle 0ϕ as a function of S, while Fig. 6.6 shoves 01 ε− vs. S.

For known bearing geometry, lubricant viscosity and static load, the equilibrium position, given by ( )S00 εε = and ( )S00 ϕϕ = , is a function of the speed only. Note that the force and the displacement are not in the same direction.

Although for horizontal shafts the load W is vertically down, the journal center shifts in an oblique equilibrium position. The journal bearing behaves as an asymmetric element, with cross-coupled as well as direct stiffnesses. The bearing is anisotropic, having different stiffnesses in different radial directions.

6.3 Dynamic coefficients of journal bearings

Consider that the journal moves in the bearing along an orbit around its steady-state equilibrium position (Fig. 6.7). The resultant reaction force of lubricant film has a variable magnitude and direction (it is no more vertical). Its components yF and zF are non-linear functions of the journal center displacements y and z, and its velocity components y& and z& :

( )z,y,z,yFF yy &&= , ( )z,y,z,yFF zz &&= .

For a small amplitude motion, the bearing reaction may be expressed by the first order Taylor series expansion of its components around the static equilibrium position (note the direction of the force components here, selected to avoid the “minus” sign in the following expressions):

.zcyczkykFF

,zcyczkykFF

zzzyzzzyzz

yzyyyzyyyy

&&

&&

++++=

++++=

0

0 (6.4)

The static reaction force components are WFy =0

and 00=zF . The

eight coefficients of the linearized force components are computed as the gradients in the static equilibrium position ( )0==== zyzy && :

0⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂=

yF

k yyy ,

0⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂=

zF

k yyz ,

0⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=y

Fk zzy ,

0⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=z

Fk zzz ,

0⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂=

yF

c yyy &

, 0⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂=

zF

c yyz &

, 0⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=y

Fc zzy &

, 0⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=z

Fc zzz &

.

They result from the solution of the lubrication equation.


The linearization of the bearing reaction forces has the advantage of decoupling the rotor and the bearings. Otherwise, the rotor equations must be integrated simultaneously with the lubrication equation.

Fig. 6.7

In matrix form, equation (6.4) can be written

⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

z

y

z

y

FFW

FF

ΔΔ

0.

The increments of the film force due to small movements around the position of static equilibrium are expressed in terms of stiffness and damping coefficients

[ ] [ ]⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

zy

czy

kzy

cccc

zy

kkkk

FF bb

zzzy

yzyy

zzzy

yzyy

z

y

&

&

&

&

ΔΔ

. (6.5)

The eight linearized stiffness and damping coefficients depend on the journal steady-state operating conditions, hence upon the rotational speed.

The dimensionless stiffness coefficients are defined as

[ ] [ ]b

zzzy

yzyyb kWC

KKKK

K =⎥⎦

⎤⎢⎣

⎡= , (6.6, a)

while the dimensionless damping coefficients are defined as

[ ] [ ]b

zzzy

yzyyb cW

CCCCC

C Ω=⎥

⎦

⎤⎢⎣

⎡= . (6.6, b)

For a given set of geometrical parameters and lubricant viscosity, these eight coefficients are functions of the Sommerfeld number S or the eccentricity


ratio. They are referred to as the eight dynamic bearing coefficients. Values of these coefficients are given in the book edited by Someya [6.1].

For flexible shafts or slightly tilted rigid journals, the journal axis may not be parallel to the bearing axis. The pressure distribution along the journal length gives rise to reaction moments. The corresponding Taylor series expansion defines four moment stiffness coefficients and four moment damping coefficients. Generally, compared to the radial coefficients, they are smaller by a factor of (2L/ l ) where l is the span adjacent to the bearing and L is the bearing length. Only for long bearings or for higher order shaft modes, the influence of moment coefficients may become significant.

The stiffness matrix of journal bearings is non-symmetric. This is the cause of rotor instability above a limiting running speed called the onset speed of instability. The unstable motion of journal bearings is called oil whirl and involves large-amplitude subsynchronous motion at the rotor critical speed. Vertical rotors without side loads may experience a motion consisting of a limit cycle whose frequency tracks at approximately one-half running speed which is also called oil whirl. At speeds above twice the rotor's natural frequency, the rotor subsynchronous motion stops tracking running speed and precesses at the natural frequency, motion that is called oil whip. These unstable motions are presented in more detail in Chapter 7.

Cavitation, compressibility, and turbulence are phenomena which complicate the modelling and analysis of hydrodynamic bearings. Their account is beyond the aim of this presentation.

6.4 Reynolds equation and its boundary conditions

In the following, certain facts relating to rotor-bearing interaction will be presented, in order to understand the methodology and simplifying assumptions used in the calculation of bearing dynamic characteristics.

First, the Reynolds equation for laminar flow is derived. It is the governing equation for the pressure field within a bearing with isoviscous flow. Second, by integrating this pressure field, the non-linear bearing forces acting upon the journal are determined. The boundary conditions for the pressure field distribution are shortly described. Third, the non-linear dynamic fluid film reaction force is expanded in a Taylor series and the terms of second and higher powers are disregarded. The stiffness and damping coefficients (6.5) are determined from the first order terms of this series [6.4].

Note that u, v , w are fluid velocity components in the x, y, z directions, and not displacements.


6.4.1 General assumptions

The following 13 assumptions are used to derive Reynolds’ equation from the general Navier-Stokes equations:

1. The lubricant is isotropic.

2. The lubricant is Newtonian. It obeys Newton's viscosity law of friction

γμτ &= , (6.7)

where τ is the shear stress in the fluid film, 2mN , γ& is the shear strain rate,

1/sec, and μ is the lubricant dynamic (absolute) viscosity, 2mNsec .

Sometimes μ is measured in centipoises (1cP = 31000541 −⋅. 2mNsec ).

3. The dynamic viscosity μ does not vary across the film thickness.

4. The fluid film is very thin compared to its length and width:

Rhmin << , ( ) 12 <<RC .

5. The flow is laminar and free of inertia:

( ) 1<<ReRC , ( )2300and0010 <≅ Re.RC

where the Reynolds number

μ

ρ CRe tw= or μ

ρ mint hRe w= (6.8)

is a measure of the ratio between the inertia forces and the viscous friction forces acting on a lubricant volume, ρ is the lubricant mass density, 3mkg , ΩRt =w is the journal surface velocity, m/sec, and C is the radial clearance, m.

6. There is no slip between lubricant and the journal/bearing surfaces.

7. The curvature of journal and bearing surfaces is neglected.

8. The deformation of journal and bearing is neglected.

9. The variation of bearing clearance in the circumferential direction is small.

10. The relative inclination of journal and bearing axes (hence the restoring moment) is neglected.

11. The fluid velocity component across the thickness of the film is small in comparison to the circumferential and axial components.


12. The pressure is constant across the film thickness.

13. First and second order velocity gradients in circumferential and axial directions are negligible compared to those across the thickness.

Supplementary assumptions:

14. Incompressible lubricant ( ).const=ρ .

15. Isothermal and isoviscous flow.

In order to determine the approximate bearing temperature, a simplified version of the energy equation is used. Because the predominant flow component is in the circumferential direction, usual assumptions include: a) neglecting the small contributions from the pressure gradients to both flow and friction, and b) ignorance of any heat conduction along the film or to the surfaces.

6.4.2 Reynolds’ equation

Reynolds’ equation is a greatly simplified version of the Navier-Stokes equations and the equation of continuity.

Fig. 6.8

Neglecting body forces and (both temporal and convective) inertia forces (see assumptions 4 and 5), the equilibrium of pressure forces and friction viscous forces acting on an elementary fluid volume (Fig. 6.8) yields

yz

p zy

∂

∂=

∂∂ τ

where yw

zy ∂∂

= μτ ,

so that

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=∂∂

yw

yzp μ , ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=∂∂

yu

yxp μ , 0=

∂∂

yp ,

According to assumption 3


xp

yu

∂∂

=∂∂

μ1

2

2,

zp

y ∂∂

=∂∂

μ1

2

2w , (6.9)

where the pressure gradients xp ∂∂ and zp ∂∂ are not functions of y.

For non-slip boundary conditions

0=y , 0=w , 0=v , 0=u , (stationary bearing)

hy = , hww = , th ∂∂=v , 0=u , (assumption 9)

the velocity field is

( ) hwhyhyy

zp

+−∂∂

=μ21w , ( )hyy

xpu −

∂∂

=μ21 .

The volume flow per unit bearing width, in the direction of motion, is

zphhyq h

h

z ∂∂

−== ∫ μ122d

3

0

ww , (6.10)

where the first term in the right hand side is the shear flow (Couette flow) and the second term is the pressure gradient flow (Poiseuille flow).

The volume flow per unit bearing width along the journal length is

xphyuq

h

x ∂∂

−== ∫ μ12d

3

0

. (6.11)

For an infinitesimal film volume of height h and area dzdx , the equation of continuity is

( ) ( ) ( ) 0=∂∂

+∂∂

+∂∂ h

tq

xq

z xz ρρρ . (6.12)

Substitution of zq and xq from (6.10) and (6.11) yields

( ) ( )ht

hzx

phxz

phz h ρρ

μρ

μρ

∂∂

+∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂ w

21

1212

33.

For an incompressible fluid ( .const=ρ ) the laminar-flow Reynolds equation in Cartesian coordinates (with time-dependent effects) becomes


th

zh

xph

xzph

zh

∂∂

+∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

21212

33 wμμ

. (6.13)

Replacing θRz = and Rh Ω=w , the Reynolds equation in circumferential coordinates becomes

thh

xph

xph

R ∂∂

+∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

θΩ

μθμθ 212121 33

2 , (6.14)

where ( )θε cos1+=Ch and θ is the angular coordinate defined in Fig. 6.1.

6.4.3 Boundary conditions for the fluid film pressure field

The circumferential boundary conditions commonly used for the plain cylindrical bearings in the case of steady state operation are shown in Fig. 6.9.

a b c

Fig. 6.9

a) The Full-Sommerfeld boundary condition ( π2 - film bearing) is anti-symmetrical with respect to πθ =' (Fig. 6.9, a):

( ) 0>x,'p θ for πθ << '0 ,

( ) ( )x,'px,'p θπθ −−= 2 .

A negative pressure of the same order as the positive one is generated for angles πθπ 2<< ' , though, because of the film rupture, the pressure is equal to the atmospheric one in this region. Large negative pressures do not exist in the diverging region and the predicted load capacity is zero what is physically unrealistic. This condition is seldom applied, except when higher pressure is supplied.

b) The Gümbel (Half-Sommerfeld) boundary condition (π -film bearing) replaces the negative pressure in the Sommerfeld boundary condition by an atmospheric pressure (Fig. 6.9, b):


( ) 0>x,'p θ for πθ << '0 ,

( ) 0=x,'p θ for πθπ 2≤≤ ' .

Although it is physically inappropriate, because the flow continuity is not satisfied at πθ =' , it is often used because of its simplicity. The errors introduced are small.

c) The Reynolds boundary condition (Fig. 6.9, c) imposes a zero pressure gradient at the point where the fluid film pressure equals the ambient pressure:

( ) 0>x,'p θ for *' θθ <<0 ,

( ) 0=x,'p θ for πθθ 2≤≤ '* ,

0=∂∂

'pθ

at *' θθ = .

6.5 Analytical solutions of Reynolds equation

Methods of solving Reynolds' equation include the finite difference method, the finite element method and the infinite series function method. In the following, some analytical solutions are presented.

6.5.1 Short bearing (Ocvirk) solution

In a very short cylindrical bearing, the Poiseuille flow component (x-component) in the axial direction becomes much larger than the circumferential one (θ -component). If the pressure-gradient contribution to the circumferential velocity w is neglected, the first term in Reynolds' equation (6.14) can be ignored:

thh

xph

∂∂

+∂∂

=∂∂

θΩ

μ 212 2

23. (6.15)

For .const=μ the equation (6.15) is easily integrated

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛−−=

thh

hxLp 21

43 3

22

θΩμ (6.16)

so that, along the x axis (Fig. 6.10),

( ) ( ) 022 =−= L,pL,p θθ .


The fluid film forces are calculated by integrating the pressure over the film domain. It is convenient to calculate the force components in the radial and tangential directions, εF and ϕF , respectively (Fig. 6.11).

Fig. 6.10 Fig. 6.11

The force components are:

∫ ∫−

=2

2

2

1

ddcos

L

L

'

'

x'R'pFθ

θ

ε θθ , ∫ ∫−

=2

2

2

1

ddsin

L

L

'

'

x'R'pFθ

θ

ϕ θθ , (6.17)

where ϕθθ −=' is measured from the JBOO line.

Substitution of the normalized film thickness

'Chh θε cos1+==

into the pressure equation (6.16) yields

( )[ ]''hR

xDL

CRp θεθϕεΩμ cos2sin2113 3

222&& −−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛= (6.18)

where

td

d1 εΩ

ε =& , td

d1 ϕΩ

ϕ =& ,

are the derivatives of ε and ϕ with respect to the dimensionless time tΩ .

The pressure is positive when

( ) 0cos2sin21 ≥−− '' θεθϕε && .


The film begins at 1'' θθ = and ends at 2'' θθ = where 0=p :

( )ϕεεθ

&

&

212tan 1

1 −=′ − , πθθ +′=′ 12 .

At 1θθ ′=' , the slope of the pressure profile must be positive:

01

>⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=θθθp ,

( ) ( ) 0cos

21sin2cos211

11 >′

−=′+′−

θϕεθεθϕε&

&& ,

so that ( )

( ) ( )2221221

21cosεϕε

ϕεθ&&

&

+−

−=′ ,

( ) ( )2221221

2sinεϕε

εθ&&

&

+−=′ ,

which define the quadrant for 1θ′ .

By substituting the pressure (6.18) into (6.17), the force components become

( )[ ]∫ −−⎟⎠⎞

⎜⎝⎛−=

'

'

''''h

SDLWF

2

1

3

2

dcoscos2sin2112θ

θ

ε θθθεθϕεπ && , (6.19, a)

( )[ ]∫ −−⎟⎠⎞

⎜⎝⎛=

'

'

''''h

SDLWF

2

1

3

2

dsincos2sin2112θ

θ

ϕ θθθεθϕεπ && . (6.19, b)

Under static loading conditions 0==ϕε && , the equilibrium position of the journal center is given by the coordinates 0ε and 0ϕ , 01 =′θ and πθ =′2 . Using this position, a coordinate system (Fig. 6.12) can be defined with the r-axis in the radial direction ( 0ε -direction) and the t-axis in the tangential direction ( 0ϕ -direction). The components of the pressure resultant along these axes are

( ) ( )( ) ( ) ⎭

⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

=⎭⎬⎫

⎩⎨⎧

ϕ

ε

ϕϕϕϕϕϕϕϕ

FF

FF

t

r

00

00

-cos-sin-sin-cos

. (6.20)

For motions with small amplitudes, εΔΔ Ce = in the r-direction and ϕΔεϕΔ 00 Ce = in the t-direction. A first order Taylor series expansion around the

static equilibrium position yields


ϕΔεϕε

εΔε

ϕΔεϕε

εΔε

&&

&& 0

000

000 ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+= rrrrrr

FFFFFF , (6.21, a)

ϕΔεϕε

εΔε

ϕΔεϕε

εΔε

&&

&& 0

000

000 ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+= tttttt

FFFFFF , (6.21, b)

In evaluating the derivatives, equations (6.19) must be differentiated with respect to the three variables ε , ε& and ϕ& , using Leibniz's rule for the differentiation of integrals.

Fig. 6.12

With 01 =′θ and πθ =′2 , corresponding to 0==ϕε && in the static equilibrium position, equations (6.19) may be integrated and, for 0εε = and

0ϕϕ = , the film forces are

( )( )22

0

20

2

01

40

ε

επε−

⎟⎠⎞

⎜⎝⎛== S

DLWFFr , (6.22, a)

( )( ) 232

0

02

01

0ε

εππϕ

−⎟⎠⎞

⎜⎝⎛== S

DLWFFt . (6.22, b)

From Fig. 6.12 it is seen that

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

=⎭⎬⎫

⎩⎨⎧

t

r

z

y

FF

FF

00

00

cossinsincos

ϕϕϕϕ

. (6.23)

At the static equilibrium, 0=zF and WFy = ,


00

0

41

tan20

0 εεπ

ϕ−

==r

t

FF

,

1sincos 0000 =+ ϕϕ

WF

WF tr ,

which implies that

( )220

20

01

4cos0

ε

εσϕ−

==WFr , (6.24, a)

( ) 232

0

00

1sin0

ε

επσϕ−

==WFt . (6.24, b)

Eliminating 0ϕ between equations (6.24), the modified Sommerfeld number becomes

( )

( )20

2200

220

2

116

1

επεε

επσ

−+

−=⎟

⎠⎞

⎜⎝⎛= S

DL , (6.25)

which defines the relationship between 0ε and S.

Equations (6.21) may be written in matrix form as

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧−

=⎭⎬⎫

⎩⎨⎧− ϕΔε

εΔϕΔεεΔ

&

&

000

0

tttr

rtrr

tttr

rtrr

t

r

t

r

CCCC

KKKK

WFWF

WFWF

. (6.26)

The dimensionless stiffness and damping coefficients in equation (6.26) are related to the actual coefficients by equations similar to (6.6). They are obtained from a differentiation of equations (6.20) and with substitution from equations (6.24):

( )( )

( )( ) W

FFW

FW

Krr

rr0

200

20

320

200

00 1

12

1

1811

εε

ε

ε

εεσεεε

−

+=

−

+=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

= ,

W

FFW

Ktr

tr0

00

11εϕε

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂= ,

( )( ) ( ) W

FFW

FW

Ktt

rt0

200

20

2520

20

00 1

21

1

2111

εε

ε

ε

επσεεϕ

−

+−=

−

+−=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂−=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

= ,


W

FFW

FW

Krt

tt0

000

111εεϕε

ε =⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

−= ,

rtr

rr KFW

FW

C 211

00−=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=εεε&&

,

( ) tt

rrtr K

W

FFW

FW

C 22

1

811 0

022

0

0

00−=−=

−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂=

εε

εσϕεϕεε&&

,

( ) tr

rtrt C

W

FFW

FW

C =−=−

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

−= 0

022

0

0

00

2

1

811εε

εσεεϕ

&&,

( ) tr

tttt K

W

FFW

FW

C 22

1

211 0

0232

000==

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂−=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

−=εε

πσϕεϕεϕ

&&.

As the r-t coordinate system changes orientation with the equilibrium position, and therefore, with the rotor speed, it is convenient to make a transformation into the fixed y-z coordinate system.

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

=⎭⎬⎫

⎩⎨⎧

zy

ΔΔ

ϕϕϕϕ

ϕΔεεΔ

00

00

0 cossinsincos

. (6.27)

The transformation of forces is shown in equation (6.23) so that the y-z dimensionless coefficients are obtained from the transformation

⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡

00

00

00

00

cossinsincos

cossinsincos

ϕϕϕϕ

ϕϕϕϕ

tttr

rtrr

zzzy

yzyy

KKKK

KKKK

.

The analytical form of the eight dynamic coefficients in y-z coordinates is

( ) ( ) ( )[ ]40

220

2220

0 2323214 επεππ

εε

−+++−

==Q

WkC

K yyyy ,

( ) ( ) ( )[ ]40

220

22200

0 232321

επεππεε

επ−+++

−==

QWkC

K yzyz ,

( ) ( )[ ]40

220

22200

0 1621

επεππεε

επ−−−

−−==

QWkC

K zyzy ,


( ) ( )[ ]20

220 1624 εππε −+== Q

WkCK zz

zz , (6.28)

( ) ( )[ ]40

220

22200

0 2481

2 επεππεε

επΩ+−+

−==

QW

cCC yy

yy ,

( ) ( )[ ] zyyz

yz CQW

cCC =−+== 2

022

0 1628 εππεΩ

,

( ) ( )[ ]20

22

0

020 162

12εππ

εεεπΩ

−+−

==Q

WcCC zz

zz ,

where

( ) ( )[ ] 23

20

2200 116

−−+= επεεQ .

Figure 6.13 shows an alternate form of the dimensionless stiffness and damping coefficients

σ2

jiji

KK = ,

σ2ji

jiC

C = , z,yj,i = (6.29)

as a function of 0ε for π -film (Gümbel) short bearings.

a b

Fig. 6.13 (from [6.5])

The short bearing solution proves to be a valid approximation for plain journal bearings with an L/D ratio less than 0.5 and 70.<ε . As this condition is satisfied in many practical applications, the Ocvirk solution is frequently employed in the analysis of rotor-bearing systems.


a b

Fig. 6.14 (from [6.5])

More often, the eight dynamic coefficients given by equations (6.28) are plotted versus the Sommerfeld number S for specified values of the L/D ratio. Figure 6.14 shows the plots for L/D = 0.25. For any specific application, it is more useful to plot the physical stiffness and damping coefficients ijk , ijc against the rotational speed Ω .

Example 6.1 Plots of the physical dynamic bearing coefficients are shown in Fig. 6.15

for a bearing with L = 20 mm, D = 80 mm, C = 0.05 mm, μ = 0.7 Ps, and W = 417.5 N.

Fig. 6.15 (from [6.5])


The static characteristics are given in Table 6.1 [6.5].

Table 6.1

Spin frequency, Hz S 0ε 0ϕ 1 0.1085 0.9512 0.2750 5 0.5423 0.8805 0.4178

10 1.0846 0.8292 0.5019 20 2.1692 0.7593 0.6049 30 3.2538 0.7079 0.6758 40 4.3384 0.6661 0.7316 50 5.4230 0.6305 0.7781 60 6.5076 0.5992 0.8184 70 7.5923 0.5714 0.8540 80 8.6769 0.5462 0.8859 90 9.7615 0.5232 0.9150

It is interesting to study the asymptotic behaviour of the bearing coefficients as the eccentricity tends to zero. From equation (6.25) one can obtain the asymptotic relation between 0ε and S given by

2

201

⎟⎠⎞

⎜⎝⎛≅

LDS

πε .

Then it can be shown that, as 0ε approaches zero (or equivalently S tends to infinity),

π82 ≅≅≅≅ zyyzyyzz CCKK ,

2

222 ⎟⎠⎞

⎜⎝⎛≅≅≅−≅

DLSCCKK zzyyzyyz π ,

C

Wkk yyzz π82 ≅≅ ,

3

8⎟⎠⎞

⎜⎝⎛≅−≅

CLDkk zyyz

Ωμπ , (6.30)

3

4⎟⎠⎞

⎜⎝⎛≅≅

CLDcc zzyy

μπ , ΩπC

Wcc zyyz8

≅≅ .

As 00 →ε ( ∞→S or 0→ΩW ), the direct stiffness terms, yyk and

zzk , become negligible compared with the cross-coupled stiffness terms, yzk and


zyk , the direct damping terms, yyc and zzc , tend to a non-zero limit, while the cross-coupled damping terms, yzc and zyc , tend to zero.

It is known that a journal bearing must have a radial restoring force in order to have a finite stability threshold. If the journal is operating at zero eccentricity, or if the fluid film is not allowed to cavitate, then the principal stiffness terms vanish, the journal is inherently unstable and the system will exhibit a half frequency whirl.

It is also known that the cross-coupled stiffness terms, yzk and zyk , are the major sources of instability. In order to generate instability, the cross-coupled term zyk must be negative. The largest degree of instability occurs when

zyyz kk −= . When yzk becomes negative, the system stability rapidly improves.

6.5.2 Infinitely long bearing (Sommerfeld) solution

The Sommerfeld solution to the Reynolds equation is obtained by considering 0=∂∂ xp , hence dropping the second term on the left in equation (6.14), and integrating with respect to θ to obtain the pressure field. A short presentation is given in Section 7.3.4.1.

The Sommerfeld boundary condition leads to an erroneous result (journal displacement always perpendicular to the direction of the static load W) so that the Gümbel boundary condition is usually used.

6.5.3 Finite-length cavitated bearing (Moes) solution

The previous asymptotic solutions have proven to be useful in the dynamic analysis of bearings: a) the Ocvirk (short) bearing solution - for small eccentricity ratios and very small L/D values, and b) The Sommerfeld (long) bearing solution - for large eccentricity ratios and large L/D values.

Using a weighted sum of the asymptotic solutions mentioned above, Moes and Childs [6.6] obtained an accurate finite-length analytic solution which is valid for general finite-length bearings at both large and small eccentricity ratios. The results for a cavitated π - bearing are given below.

A dimensionless force ∗0F is used

3

00 ⎟

⎠⎞

⎜⎝⎛=∗

RC

CLFFΩμ

(6.31)


where WF =0 is the static reaction load.

A bearing impedance vector is also defined, whose magnitude at the static equilibrium position is

0

00 ε

∗=

FZ .

In terms of the Sommerfeld number

00

2

0

12 ZC

RF

LDSεππ

Ωμ =⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛= . (6.32)

Figure 6.16 illustrates ( )0εSS = for some L/D values. For a given applied load and bearing, it can be used to determine 0ε from the value of S calculated from (6.32).

The equilibrium attitude angle 0ϕ is given by

0

201

0 314

tanε

εϕ

ba −

= −

where

B.a 1221+= , B.b 6031+= ,

( )2

201

−

⎟⎠⎞

⎜⎝⎛−=

DLB ε .

The equilibrium magnitude of the impedance vector is

232

020

0150

1

dGE.Z

+= ,

where

01 ξ−=d , 00 1221 Q.E += ,

( )0

000 14

6313ξ

η−

+=

Q.G , ( )2

00 1−

⎟⎠⎞

⎜⎝⎛−=

DLQ ξ ,

000 cosϕεξ = , 000 sinϕεη = .


Fig. 6.16 (from [6.3])

The physical stiffness and damping coefficients are given by

jiji KCFk 0= , jiji C

CFcΩ0= , z,yj,i = (6.33)

where the dimensionless stiffness coefficients are

000

sincos1 ϕεϕϕ

ε ∂∂

−=zzK ,

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+−= 000

cossin1 ϕεϕϕ

εzyK , (6.34)

00

00

sin1sin1 ϕε

ϕε Z

ZK yz ∂∂

+= ,

00

00

cos1cos1 ϕε

ϕε Z

ZK yy ∂∂

+= ,

and the dimensionless damping coefficients are


00

sin2 ϕαϕ

ε ∂∂

=zzC ,

00

cos2 ϕαϕ

ε ∂∂

=zyC , (6.35)

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+−= 00

00

sin1cos2 ϕα

ϕε Z

ZCyz ,

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−= 00

00

cos1sin2 ϕα

ϕε Z

ZCyy ,

where

αϕ

ϕα ∂∂

∂∂

=∂∂ ZZ ,

εη

ηεξ

ξε ∂∂

∂∂

+∂∂

∂∂

=∂∂ ZZZ ,

ϕη

ηϕξ

ξϕ ∂∂

∂∂

+∂∂

∂∂

=∂∂ ZZZ ,

( )20

02

20

21

1cos1234

εϕ

εεϕ

−⎥⎥⎦

⎤

⎢⎢⎣

⎡−

−=

∂∂

ba

bab ,

20

001

01

1sin2

1ε

εεπϕαϕ

−⎟⎠⎞

⎜⎝⎛ +−+=

∂∂ − ,

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−

+−=

∂∂ −

dDLE.

dG

GEZZ

23122

431 2

0200

20

20

0η

ξ,

0

20

20

20

01ηη GE

GZZ+

−=∂∂ ,

000 sincos ϕεϕεϕ

εξ

∂∂

−=∂∂ , 00 sinϕε

ϕξ

−=∂∂ ,

000 cossin ϕεϕεϕ

εη

∂∂

+=∂∂ , 00 cosϕε

ϕη=

∂∂ .

Note that zyyz CC ≠ .

The physical coefficients of the fluid film bearing from Example 6.1 are listed in Table 6.2.


Table 6.2

Freq. yyk yzk zyk zzk yyc yzc zyc zzc

[Hz] [N/m] ×710 [Nsec/m] ×510

10 8.4844 4.6560 0.6521 1.5066 1.0840 0.2292 0.2430 0.1333

20 5.8945 4.0755 0.3442 1.5749 4.8069 1.1958 1.2656 0.8750

30 4.7415 3.8018 0.1424 1.6261 3.0295 0.8249 0.8703 0.6978

40 4.0521 3.6379 -0.0188 1.6678 2.2017 0.6361 0.6692 0.6008

50 3.5814 3.5298 -0.1586 1.7033 1.7295 0.5208 0.5466 0.5387

60 3.2344 3.4553 -0.2854 1.7341 1.4269 0.4426 0.4636 0.4953

70 2.9658 3.4034 -0.4034 1.7614 1.2177 0.3858 0.4036 0.4631

80 2.7504 3.3677 -0.5153 1.7857 1.0652 0.3426 0.3579 0.4382

90 2.5734 3.3445 -0.6227 1.8076 0.9495 0.3085 0.3219 0.4184

6.6 Physical significance of the bearing coefficients

In order to better understand the physical significance of the eight linearized dynamic bearing coefficients, it is useful to calculate the work done along a closed orbit by the bearing forces [6.7].

First, the non-symmetric stiffness and damping matrices are decomposed into their symmetric (s) and skew-symmetric (a) parts

[ ] [ ] [ ]ba

bs

b kkk += , [ ] [ ] [ ]ba

bs

b ccc += , (6.36)

where

[ ] [ ] [ ] ⎟⎠⎞⎜

⎝⎛ +=

T

21 bbb

s kkk , [ ] [ ] [ ] ⎟⎠⎞⎜

⎝⎛ −=

T

21 bbb

a kkk , (6.37)

[ ] [ ] [ ] ⎟⎠⎞⎜

⎝⎛ +=

T

21 bbb

s ccc , [ ] [ ] [ ] ⎟⎠⎞⎜

⎝⎛ −=

T

21 bbb

a ccc . (6.38)

The damping matrix can be written

[ ] ( )( )

( )( ) ⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

+=

021

210

21

21

zyyz

zyyz

zzzyyz

zyyzyyb

cc

cc

ccc

cccc ,


[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

00

azy

ayz

szz

szy

syz

syyb

cc

cccc

c . (6.39)

The matrix [ ]bac is not a true damping matrix, in the dissipative sense. It

embodies a conservative gyroscopic-like force field which, in the absence of other true gyroscopic effects, produces the bifurcation of eigenfrequencies along forward and backward precession branches. Indeed, the incremental work done by the corresponding forces at any point on any orbit is zero:

⎣ ⎦⎭⎬⎫

⎩⎨⎧

−=⎭⎬⎫

⎩⎨⎧

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−=

tzty

yczczy

zy

cc

W azy

ayza

zy

ayz

dd

dd

00

d

T

&

&&&

&

&,

( ) ( ) ( ) 0dddd ≡−−=+−= tzyyzctzyctyzcW ayz

azy

ayz &&&&&&&& . (6.40)

This conservative property shows that the force vector ⎣ ⎦ is always perpendicular to the velocity vector . For most journal bearings the cross-coupling damping coefficients are equal and the skew-symmetric matrix vanishes.

The stiffness matrix can be written

[ ] ( )( )

( )( ) ⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

+=

021

210

21

21

zyyz

zyyz

zzzyyz

zyyzyyb

kk

kk

kkk

kkkk ,

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

00

azy

ayz

szz

szy

syz

syyb

kk

kkkk

k . (6.41)

The matrix [ ]bak is not a true stiffness matrix since it provides no radial

restoring force. It embodies a non-conservative circulatory force. The incremental work done by this force at any point on any orbit can be written as

⎣ ⎦⎭⎬⎫

⎩⎨⎧

−=⎭⎬⎫

⎩⎨⎧

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−=

zy

ykzkzy

zy

kk

W azy

ayz

T

azy

ayz

dd

dd

00

d ,

zfyfzykyzkW zyazy

ayz ddddd +=+−= . (6.42)

Because yf

zf zy

∂∂

≠∂

∂, Wd is not an exact differential. The work done

between two points is path dependent and thus the force is non-conservative.


The net energy exchange per cycle to the rotor resulting from [ ]bak is

∫ −−= )dd( zyyzkE ayzcycle . (6.43)

Splitting this integral into two line integrals between points A ( )minA yy = and B ( )maxB yy = (Fig. 6.17), and integrating the dy terms by parts, yields ( )12 zz > :

( ) yzzkEB

A

y

y

ayzcycle d2 12∫ −= . (6.44)

The integral equals the closed orbit area. It is positive for forward precession and negative for backward precession. Generally 0≥a

yzk for journal

bearings. Thus the [ ]bak effect represents “negative” damping on forward whirls

and positive damping on backward whirls. Only for straight-line (degenerated)

orbits, where yzyzB

A

B

A

y

y

y

y

dd 12 ∫∫ = , is the net cyclic exchange of energy zero.

Fig. 6.17

For a closed precession orbit described by the parametric equations

( )( ),tAz

,tAy

zz

yy

θω

θω

+=

+=

cos

cos (6.45)

the work done by the bearing forces (pay attention to the negative sign)


,zcyczkykF

,zcyczkykF

zzzyzzzyz

yzyyyzyyy

&&

&&

−−−−=

−−−−= (6.46)

over one cycle is

( ) ( ) tzFyFzFyFWT

zyzy ddd

0∫∫ +=+= && . (6.47)

The following integrals can be easily calculated:

( ) 0d0

=+− ∫ tzzkyykT

zzyy && ,

( ) ( ) ( )zyzyzyyz

T

zyyz AAkktzykyzk θθπ −−=+− ∫ sind0

&& ,

( ) ( )22

0

22 d zzzyyy

T

zzyy AcActzcyc +−=+− ∫ ωπ&& ,

( ) ( ) ( )zyzyzyyz

T

zyyz AAcctzycyzc θθωπ −+−=+− ∫ cosd0

&&&& .

The net energy exchange per cycle to the rotor system is

( ) ( )( ) ( ) ( ),AcAcAAcc

AAkkW

zzzyyyzyzyzyyz

zyzyzyyz

22cos

sin

+−−+−

−−−=

ωπθθωπ

θθπ (6.48)

where the positive term represents supplied energy, while negative terms represent dissipated energy.

With yzk positive, the cross-coupling stiffness coefficients can supply energy to the motion, hence provide “negative” damping.

The first term may also be written

( ) ( )( )( )( )( ) ( ) ,bakkyzzykk

AAAAkk

AAkkW

zyyzcscszyyz

zzyyzzyyzyyz

zyzyzyyzsup

−=+−−=

=−−=

=−−=

ππ

θθθθπ

θθπ

sincoscossin

sin

( ) ⎟⎠⎞⎜

⎝⎛ −−= 22

bfzyyzsup rrkkW π , (6.49)


where the elliptical orbit has been decomposed into two circular orbits, one with forward precession and radius fr , and one with backward precession and radius

br .

For 0>yzk , if bf rr > , 0>b , the precession is forward and

0>supW . Thus, it is the forward whirl component which is responsible for the “negative” damping.

6.7. Bearing temperature

In any specific application, the bearing geometry, the static load, the type of lubricant and its properties are given. To obtain the bearing coefficients at any selected speed Ω it is necessary first to determine the lubricant viscosity. This could be done by an iterative procedure [6.4].

6.7.1 Approximate bearing temperature

A temperature value is estimated and the corresponding viscosity value is found from a viscosity-temperature diagram for the lubricant (Fig. 6.18). So it is possible to compute the Sommerfeld number and thereafter to enter the tables of bearing data to find the following parameters:

- the dimensionless side flow

CLDN

QQπ

21= , (6.50)

where Q is the lubricant flow obtained by integrating the pressure gradient at the edge of the bearing;

- the dimensionless friction power loss

323 DLNPCP

μπ= , (6.51)

where P is the power loss by friction;

- the dimensionless temperature rise


2

⎟⎠⎞

⎜⎝⎛

=

CR

c

TT

v

Ωμ

Δ , (6.52)

The temperature rise TΔ through the lubricant film can be calculated from a simplified version of the energy equation, neglecting heat conduction.

For an infinitesimal axial film strip of thickness minh and width θdR , the heat balance yields

( )min

z hRR

RTqc

2

dd ΩμΩτθ

≅≅v ,

where vc is the specific heat per unit volume and T is the temperature.

Under the above assumptions zq is equal to the shear flow

ΩRhq minz 21

= . (6.53)

The heat balance equation reduces to

222

dd

minhR

cT μΩθ v= . (6.54)

The viscosity μ is a given function of the temperature T, for any particular lubricant. The temperature, hence, the viscosity variation along the film can be computed from

∫∫ =2

11

22

d2d

θ

θ

θΩμ min

T

Th

Rc

T

v.

However, it is common practice to assume .const=μ , equal to an average value, so that the total increase in temperature

∫=2

1

22 d2

θ

θ

θΩμΔminh

Rc

Tv

. (6.55)

By assuming that, say, 80 per cent of the friction heat is carried away by the lubricant, the operating temperature of the bearing, operT , results from a simple heat balance as


QP

CR

c.TT supoper πΩμ 480

2

⎟⎠⎞

⎜⎝⎛+=

v, (6.56)

where supT is the lubricant supply temperature.

The maximum temperature in the film is

TCR

cTTTT operopermax

2

⎟⎠⎞

⎜⎝⎛+=+=

v

ΩμΔ . (6.57)

If maxT differs from the estimated temperature, a new calculation should be performed using a viscosity value based on maxT and repeated until the two temperatures agree. Once the true Sommerfeld number has been obtained, the corresponding spring and damping coefficients can be found either from tables, usually requiring interpolation, or from formulae.

6.7.2 Viscosity-temperature relationship

The viscosity of lubricating oils is extremely sensitive to the operating temperature (Fig. 6.18). With increasing temperature, the viscosity of oils falls quite rapidly (up to 80% for a temperature increase of C250 ). It is important to know the viscosity at the operating temperature, because it determines the Sommerfeld number used in the estimation the bearing dynamic coefficients. The oil viscosity at a specific temperature can be either calculated from a viscosity-temperature equation or obtained from the viscosity-temperature ASTM chart.

Usually kinematic viscosity is measured in centistokes (cSt) and the dynamic (absolute) viscosity is measured in centipoise (cP). The water at C200 has a viscosity of 1.0020 cP. In SI units, a millipascal sec⋅ , or mPa s⋅ , is equal to a centipoise and smm1cSt1 2= .

Walther’s equation [6.8] has the form

( ) cTdba 1=+ν ,

where a, b, c, d are constants, T is the absolute temperature, K, and ν is the kinematic viscosity, sm2 . Kinematic viscosity is defined as the ratio of dynamic viscosity to fluid density, ρμν = . The typical density of a mineral oil is 850

3mkg .

The most widely used chart is the ASTM D341 viscosity-temperature chart, which is entirely empirical and is based on Walther’s equation.


In deriving the bases for the ASTM chart, logs were taken from Walther’s equation and it was considered that 10=d , yielding

( ) cTba 1loglog 1010 +=+ν .

Fig. 6.18 (from [6.2])

It has been found that if ν is in cS (centistokes), then 70.a ≅ . After substituting into the equation, the logs were taken incorrectly as


( ) Tca.cS 101010 log70loglog ⋅−′=+ν , (6.58)

where a′ and c are constants. To use this equation, the kinematic viscosity should be larger than 2 cS. For lower viscosities, the constant 0.7 increases according to relations in ASTM D341.

In the ASTM chart, the ordinate is ( )70loglog 1010 .cS +ν and the abscissa is T10log . Despite the mathematical error, the ASTM chart is quite successful and works well for mineral and synthetic oils under normal conditions. Figure 6.18 is a Walther-Ubbelohde chart for oils used in internal combustion engines. A similar chart exists for turbine oils [6.2].

The major factor in lubrication is that viscosity changes very strongly with temperature and pressure. A general rule is that the more viscous the oil the more susceptible it is to change. The thicker the oil the larger the viscosity-temperature derivative. The viscosity index (VI) is commonly used to indicate the relative decrease in viscosity with increasing temperature (Dean and Davis, 1929). The standard temperatures used in the oil industry are F1000 and F2100 ( )C99andC837 00. . The best paraffinic mineral oils (of Pennsylvania) were given a viscosity index of 100 and the poorest naphtenic oils (of Texas) a V.I. of zero. The American Society of Automotive Engineers (SAE) have divided oils into grades. A 5W oil has a maximum viscosity of 1200 centipoises at F00 and a minimum viscosity of 3.9 centipoises at F2100 . A 10W oil has a viscosity between 1200 and 2400 centipoises at F00 and a minimum viscosity of 3.9 centipoises at

F2100 . A 20W oil has a viscosity between 2400 and 9600 centipoises at F00 and a minimum viscosity of 3.9 centipoises at F2100 . An SAE 20 oil has a viscosity between 5.7 and 9.6 centistokes at F2100 . Furthermore SAE 30, SAE 40, SAE 50 oils have viscosities between 9.6-12.9, 12.9-16.8 and 16.8-22.7 centistokes, respectively.

Some oils, with polymers added to them, have high viscosity indices (about 150), and are called multigrade oils. This is because they are in one grade at the F00 end and in a higher grade at F2100 . As an example, a 10W/30 oil may have a viscosity of 2100 cP at F00 and 11.5 cS at F2100 . It falls into the 10W range at F00 and SAE 30 at F2100 . For this reason it is called 10W/30 [6.8].

It would be advantageous to produce a thick oil with the viscosity-temperature derivative of a thinner oil. To some extent (2-3 per cent) this can be done by the addition of polymer thickeners, often a methacrylate. The result is also a multigrade oil. An oil may be an SAE 10 at F1000 but SAE 30 at F2100 , hence the oil is designated SAE 10/30.


6.8 Common fluid film journal bearings

In the evolution of rotating machinery many different bearing configurations have been developed in order to achieve acceptable performance, most of then aiming to improve the rotor stability.

6.8.1 Plain journal bearings

The plain journal bearing presented so far (Fig. 6.2) is the simplest fixed-geometry design, suitable for highly-loaded or low-speed rotors which are not subject to rotordynamic instability. Low cost and ease of manufacture are the main advantages.

In the centered position, the bearing plain cylindrical shell is concentric with the journal surface, so there is no preload. Plain sleeve bearings have the highest cross-coupling of all bearings, making them the most destabilizing configuration.

6.8.2 Axial groove bearings

The most common cylindrical journal bearing (Fig. 6.19) is horizontally split with axial oil distribution grooves (or relief pockets) along each horizontal split line.

Fig. 6.19 Fig. 6.20

The four-axial-groove journal bearing (Fig. 6.20) incorporates four axial grooves, 090 apart, located at 045 from the vertical axis. There is no preload, but the design is more stable than the plain journal bearing for some applications.

Table 6.3 lists the dimensionless dynamic coefficients for a two-axial-groove bearing with 50.DL = as a function of Sommerfeld’s number [6.1]. Figure 6.21 presents the dimensionless dynamic coefficients for a bearing with

1=DL .


Table 6.3

Dynamic coefficients of two-axial-groove bearing, 50.DL =

Stiffness coefficients Damping coefficients

S yyK yzK zyK zzK yyC yzC zyC zzC

5.96 1.77 13.6 -13.1 2.72 27.2 2.06 2.06 14.9 4.43 1.75 10.3 -9.66 2.70 20.6 2.08 2.08 11.4 2.07 1.88 5.63 -4.41 2.56 11.2 2.22 2.22 6.43 1.24 2.07 4.27 -2.56 2.34 8.50 2.32 2.32 4.73 0.798 2.39 3.75 -1.57 2.17 7.32 2.24 2.24 3.50 0.517 2.89 3.57 -0.924 2.03 6.81 2.10 2.10 2.60 0.323 3.65 3.62 -0.427 1.92 6.81 2.08 2.08 2.06 0.187 4.92 3.88 0.0235 1.83 7.32 2.16 2.16 1.70 0.135 5.90 4.11 0.258 1.80 7.65 2.10 2.10 1.46

0.0926 7.35 4.46 0.527 1.78 8.17 2.05 2.05 1.24 0.0582 9.56 4.92 0.805 1.73 9.12 2.06 2.06 1.06 0.0315 13.8 5.76 1.24 1.72 10.6 2.03 2.03 0.846

0.00499 43.3 9.61 4.26 1.99 22.4 2.93 2.93 0.637

a b

Fig. 6.21 (from [6.9])

The average values used in simplified calculations [6.9]

( ) ( )

( ) ( ),KKCCC

,CCKKK

yzzyzzyyav

zyyzzzyyav

−++=

−++=

Ω

Ω

21

21

221

(6.59)


are plotted with broken lines.

6.8.3 Pressure dam bearings

A pressure dam bearing (Fig. 6.22) is basically a plain bearing modified to incorporate a central relief groove or scallop along the top half of the bearing shell, ending abruptly at a step. As the lubricant is carried around the bearing, it encounters the step that produces an increased pressure at the top of the journal (Fig. 6.23), inducing a stabilizing force downwards. This forces the rotor into the bottom half of the bearing to a greater eccentricity.

Fig. 6.22 Fig. 6.23

Pressure dam bearings have greater power consumption than plain bearings and are more expensive to manufacture because of the precise machining required to produce the correct dam geometry. They have high load capacity and are beneficial in improving rotordynamic stability.

An improved configuration incorporates a circumferential relief track cut into the lower pad, as shown schematically in Fig. 6.24. This groove reduces the

DL ratio of the load-carrying bearing segment and increases the bearing unit loading.

Fig. 6.24


Figure 6.25 presents the dimensionless dynamic coefficients for a pressure dam bearing with 1=DL as a function of Sommerfeld’s number. Numerical values of these coefficients are given in Table 6.4 [6.1].

Fig. 6.25 (from [6.1])

Table 6.4 Dimensionless dynamic coefficients of pressure dam bearing, 1=DL



1.72 4.59 8.55 -2.77 5.3 14.0 6.15 5.90 8.44 1.09 4.67 6.8 -0.832 4.19 10.8 4.38 4.37 5.18

0.742 4.72 5.79 -0.273 3.34 9.24 3.46 3.46 3.6 0.517 4.85 5.22 -0.0993 2.85 8.08 2.95 2.95 2.72 0.362 5.43 5.11 0.231 2.50 8.30 2.84 2.84 2.11 0.249 6.20 4.85 0.414 2.21 8.01 2.63 2.64 1.70 0.165 7.57 5.15 0.697 2.17 8.58 2.42 2.43 1.52 0.102 9.69 5.33 0.985 2.05 8.95 2.39 2.40 1.21 0.0558 14.10 6.28 1.38 1.99 10.8 2.33 2.34 1.01 0.0219 30.0 7.86 2.91 1.82 13.0 1.77 1.78 0.555


Table 6.5 Dimensionless dynamic coefficients of offset bearing, 50.DL = , 50.mp =



8.519 47.06 82.04 5.48 64.74 97.59 45.00 45.00 59.71 4.240 23.60 41.06 2.64 32.32 49.04 22.62 22.62 29.94 2.805 15.81 27.42 1.65 21.49 32.97 15.22 15.22 20.06 2.081 11.93 20.61 1.12 16.05 25.01 11.56 11.56 15.15 1.339 8.08 13.79 0.54 10.56 17.15 7.98 7.98 10.25 0.953 6.18 10.39 0.20 7.78 13.34 6.31 6.31 7.83 0.717 5.14 8.45 -0.05 6.15 11.29 5.43 5.43 6.51 0.555 4.63 7.20 -0.09 5.00 10.00 4.76 4.76 5.38 0.493 4.56 6.72 0.01 4.53 9.49 4.38 4.38 4.74 0.353 4.63 5.78 0.22 3.53 8.51 3.56 3.56 3.40 0.284 4.85 5.40 0.33 3.08 8.17 3.18 3.18 2.79 0.228 5.18 5.15 0.42 2.74 7.99 2.88 2.88 2.34 0.182 5.65 5.01 0.51 2.48 7.95 2.65 2.65 1.98 0.162 5.93 4.97 0.55 2.37 7.97 2.53 2.53 1.82 0.143 6.26 4.95 0.606 2.27 8.02 2.46 2.46 1.69 0.126 6.64 4.95 0.65 2.19 8.10 2.38 2.38 1.56

6.8.4 Offset halves bearings

The offset split journal bearing is shown in Fig. 6.26, a. It is preloaded in the horizontal direction by displacing the upper and lower halves transverse to the journal axis a slight amount d, usually about one half the radial clearance. Offset bearings have increasing load capacities as the offset is increased.

a b

Fig. 6.26


It is manufactured by offsetting the halves before finish boring so that when the halves are matched, the pad centers will be horizontally offset. The horizontal clearance is smaller than the vertical clearance. The vertical clearance equals the pad machined clearance. The horizontal diametral clearance is equal to the pad machined diametral clearance, minus twice the radial offset.

This bearing is more stable than the plain journal bearing, but there is still a tendency for instability and it is unidirectional.

Table 6.5 lists the dimensionless dynamic coefficients for an offset bearing with 50.DL = and C.d 50= as a function of Sommerfeld’s number [6.4].

6.8.5 Multilobe bearings

These consist of a number of fixed bearing segments bored to a larger radius than the bearing set clearance, thus creating a preload. While a grooved bearing consists of a number of partial arcs with a common center, the lobed bearing is made up of partial arcs whose centers do not coincide.

6.8.5.1 Bearing preload

Consider the journal in the concentric position (Fig. 6.27). The bearing pad (lobe), with machined bore radius PR , is centered at point pO . The bearing center

is JB OO ≡ . The set bore has a radius BR which is the distance from the bearing center to the pad surface at the point of minimum clearance.

Fig. 6.27


If PO coincides with BO , then BP RR = and the preload is zero. In this case, if the journal is centered in the bearing, the pad arc will be concentric with the journal surface. If the pad bore is larger than the set bore, i.e. BP RR > , the centers of curvature will be offset.

The preload factor pm is defined as

p

bp C

Cm −=1 , (6.60)

where RRC Pp −= is the machined clearance, usually taken as the representative

bearing clearance, and RRC Bb −= is the assembled clearance. If the pad machined bore is held at a fixed value, the preload is increased by moving the pad radially inward toward the shaft. The distance between the pad center of curvature and the bearing center

ppbpBP CmCCRRd =−=−= . (6.61)

The preload forces the oil to converge at each pad because of the reduced clearance at the midpoint of the pad. The result is an oil wedge effect.

6.8.5.2 Elliptical (lemon-bore) bearings

The elliptical bearing consists of two partial arcs (Fig. 6.28). The center of the bottom arc is above the bearing center, while the center of the upper arc is below the bearing center. This arrangement has the effect of preloading the bearing. The tight clearances at the minimum points create higher minimum temperatures, but the open clearances at the splits provide large amounts of cool oil.

a b

Fig. 6.28

In the elliptical bearing, the shells are first put together with shims, bored, and then assembled after removal of the shims, as shown in Fig. 6.29.


The pressure distribution on the journal is shown in Fig. 6.30 for a particular design. The wedge formed in the top half creates a downward pressure which suppresses the whirl in marginally unstable bearing designs. Also, the journal center eccentricity is increased, which results in increased stiffness.

Fig. 6.29 (from [6.1]) Fig. 6.30

Figure 6.31 presents the dimensionless dynamic coefficients for an elliptical (two-lobe) bearing with 50.DL = and 43=pm , for Reynolds boundary conditions, as a function of Sommerfeld’s number. Numerical values of these coefficients are given in Table 6.6 [6.1].

Table 6.6

Dimensionless dynamic coefficients of elliptical bearing, 50.DL = , 43=pm



5.06 474 72 -215 49.8 548 -157 -157 92.5 3.01 282 40.6 -128 29.7 319 -94.3 -94.1 54.4 2.04 186 26.5 -86.7 18.2 215 -63.8 -63.6 36.9 0.983 93.7 12.7 -43.2 9.79 106 -30.8 -30.7 17.9 0.477 46.6 6.83 -20.4 4.79 50.9 -15 -15 8.77 0.286 29.5 4.91 -12.3 3.24 33 -8.45 -8.42 5.45 0.197 22.4 3.88 -8.44 2.35 24.7 -5.76 -5.75 3.93

0.0999 15.6 3.82 -3.75 1.61 16.1 -2.13 -2.09 2.24 0.0504 14.7 4.36 -1.11 1.4 12.3 0.085 0.0901 1.32 0.0306 16.9 5.21 0.0379 1.49 11.2 0.582 0.586 0.978 0.0199 20.1 6.32 1.08 1.62 12.1 1.24 1.25 0.866 0.00976 29.8 9.04 2.59 1.93 15.8 2.63 2.66 0.84


a b

Fig. 6.31 (from [6.1])

6.8.5.3 Three-lobe and four-lobe bearings

Three lobe (Fig. 6.32, a) and four-lobe (Fig. 6.33) bearings have fixed segments (called lobes) bored to a larger radius than the bearing radius.

a b Fig. 6.32

There are also configurations with five or over ten lobes. The lobes can be converging and diverging across each segment. Tapered land designs have constantly converging segments.

The inner surface of the three-lobe bearing (Fig. 6.32, b) is composed of three arc segments whose centers are displaced from one another and from the common bearing center, providing a preload. The three lubricant wedges around the journal stiffen and stabilize the bearing, even at the centered position. Increasing the number of lobes, more lubricant wedges are formed around the journal, improving the stability. A wide range of preloads is possible by changing the pad and set bores.


Fig. 6.33

Figures 6.34 and 6.35 present the dimensionless dynamic coefficients for a three-lobe and a four-lobe bearing, respectively, with 50.DL = and 43=pm , for Reynolds boundary conditions, as a function of Sommerfeld’s number. Numerical values of these coefficients are given in Tables 6.7 and 6.8 [6.1].

a b

Fig. 6.34 (from [6.1])

a b

Fig. 6.35 (from [6.1])


Table 6.7

Dimensionless dynamic coefficients of three-lobe bearing, 50.DL = , 43=pm



1.27 73.2 36.4 -42.4 64.8 88.4 -3.45 -3.31 85.6 0.625 38.4 18.2 -23.4 30.9 48.0 -2.52 -2.44 44.0 0.299 21.9 8.52 -12.5 14.2 26.2 -2.16 -2.12 20.5 0.187 17.1 5.56 -8.56 8.92 19.2 -1.80 -1.77 12.2 0.131 15.3 4.28 -6.88 6.48 16.0 -1.67 -1.67 9.04 0.095 14.8 3.47 -5.68 5.04 13.84 -1.52 -1.50 6.92 0.070 15.8 3.03 -4.72 4.24 12.5 -1.17 -1.29 5.20 0.0516 16.0 2.93 -4.04 3.55 11.8 -0.944 -0.972 4.28 0.0374 17.8 3.03 -3.32 3.10 11.4 -0.688 -0.688 3.56 0.0256 20.8 3.44 -2.41 2.70 11.6 -0.298 -0.288 2.58 0.0204 23.2 3.86 -1.92 2.444 12.0 -0.128 -0.112 2.28

Table 6.8

Dimensionless dynamic coefficients of four-lobe bearing, 50.DL = , 43=pm



1.24 64.6 43.9 -44.2 65.0 98.5 -0.068 0.14 98.4 0.614 33.2 22.4 -22.2 32.6 50.7 0.036 0.14 49.3 0.293 18.7 11.8 -11.3 16.5 26.7 0.136 0.192 24.2 0.182 14.7 8.19 -7.33 11.1 19.1 0.0967 0.132 15.3 0.123 13.5 6.32 -5.62 8.52 15.3 0.196 0.224 11.3 0.0853 13.6 5.36 -4.61 6.92 13.6 0.288 0.308 8.89 0.0593 14.6 4.81 -3.53 5.85 12.7 0.236 0.256 6.38 0.04 16.5 4.53 -3.08 5.04 12.5 0.376 0.392 5.32

0.0256 19.8 4.56 -2.62 4.32 13.0 0.492 0.508 4.35 0.0147 25.9 5.14 -1.74 3.61 14.6 0.56 0.58 2.70 0.0103 31.4 5.82 -1.36 3.07 16.4 0.776 0.792 2.28


6.8.5.4 Pressure dam multi-lobe bearings

The performance of a series of non-cylindrical bearings such as elliptical, offset-halves, orthogonally displaced and multi lobe bearings is improved by incorporating pressure dams. Figure 6.36 shows a four-lobe pressure dam bearing.

Fig. 6.36 (from [6.10])

Figures 6.37 show the circumferential variation of fluid film pressures at the center line of a conventional four-lobe bearing and a four-lobe pressure dam bearing, for two extreme eccentricity ratios.

a b

Fig. 6.37 (from [6.10])

In the ordinary four-lobe bearing (Fig. 6.37, a), lobes 1 and 4 are either cavitated over most of the surface or have low pressures. When pressure dams are


incorporated in the lobes 1 and 4 (Fig. 6.37, b), peak pressures are generated at the dam locations, followed by a steep fall in the pressures. These pressures are distributed over a much wider range of the surface and their peaks are much higher than those arising in the conventional design. There is also a comparative decrease of the peak pressures is the lower lobes, where relief tracks are used. The generation of high pressures in the upper half of the bearing improves its stability. The distribution of pressures throughout the circumference ensures a stable operation of the bearing.

6.8.6 Tilting pad journal bearings

The tilting-pad journal bearing consists of several individual journal pads that can pivot in the bore of a retainer. The tilting pad is like a multilobe bearing with pivoting lobes, or pads. The most common tilting-pad bearing arrangements have four or five pads.

Fig. 6.38 Fig. 6.39

Figure 6.38 illustrates the rocker pivot arrangement. This design is the simplest and least expensive to manufacture. The pad design allows tilting motion in the circumferential direction but none axially. Since the support contact is a line, the pivot stresses may be high if a good alignment is not achieved.

The spherical point pivot arrangement is illustrated in Fig. 6.39. A spherical button is mounted in either the pad or the housing and pivots on a hardened flat disk in the opposite member. This allows tilting in all directions.

In the spherical surface pivot design (Fig. 6.40), the pad load is transmitted into the housing through a ball and socket arrangement. This allows the pad to pitch in the conventional manner but can also accommodate shaft misalignment. The spherical pivot tends to be more durable than the rocker pivot, as the surface contact of the ball-and-socket joint has a lower unit loading than the line contact of the rocker-pivot. The sliding motion between the ball and socket helps avoiding fretting. Like the fixed-geometry bearings, there is a thin layer of babbitt metal (50 to 125 mμ ) applied to the bearing surface to protect the shaft.

Recently, flexible (flexural) pad bearings are of increased attention.


Fig. 6.40 Fig. 6.41

Tilting pads may be mounted offset. Pad offset is a measure of how far along the pad arc the pivot is located (Fig. 6.41)

pad

pivotOffsetθθ

= . (6.62)

An offset of 0.5 corresponds to a centered pivot, where the pivot is positioned symmetrically between the leading and trailing edges. Offsetting the pivots in the direction of rotation to values grater than 0.5 can raise the minimum film thickness, and hence the load capacity. A common offset value is about 0.6. Application of offset pivots is limited to unidirectional operation.

The value of offset is meaningful only when the bearing is preloaded. Preload can be accomplished by boring the pad arcs to a larger diameter than the clearance diameter. Typical preload values range from 0 to 0.5, with the most common being about 0.3. An increase in preload will increase the stiffness, but will result in a reduction of the damping in the bearing.

a b

Fig. 6.42 (from [6.11])

The primary advantage of this design is that each pad can pivot independently to develop its own pressure profile. This feature significantly reduces the cross-coupled stiffness. The pressure distributions on the journal of a four-pad bearing with rigid pivots are shown for a vertical load in Fig. 6.42, a and


for a load on an arbitrary angle in Fig. 6.42, b. The bearing has 76220.DL = ,

50.mp = , arc length 072 , pad angle 045 and offset 0.5.

Figures 6.43, a and b present the dimensionless dynamic coefficients of a four-pad tilting pad bearing with load between pads (LBP), as a function of Sommerfeld’s number, for Reynolds boundary conditions. The bearing has

50.DL = , 32=pm , pad angle 080 and offset 0.5. The bearing is isotropic, with equal direct stiffness and damping coefficients, respectively. It is intrinsically stable, having zero cross-coupled stiffness coefficients.

a b

Fig. 6.43 (from [6.1])

The pressure distributions on the journal of a five-pad bearing are shown in Fig. 6.44. The bearing has 1=DL , offset 0.5, pivot angle 036 , and no preload. Figure 6.44, a is for a LOP (load on pad) bearing. Figure 6.44, b is for the same bearing with load between pads (LBP). The attitude angle is zero in both cases.

a b c

Fig. 6.44 (from [6.11])


Figure 6.44, c shows the pressure distribution for the load vector between the pad and the center of the oil groove. In this case, since the load is not symmetric with respect to the pads, the bearing attitude angle is not zero and the cross-coupling stiffness and damping coefficients are not zero.

Table 6.9 lists the dimensionless dynamic coefficients for a five-pad tilting pad bearing with 50.DL = and 43=pm , for load between pads and Reynolds boundary conditions, as a function of Sommerfeld’s number [6.1].

Table 6.9 Dimensionless dynamic coefficients of five-pad bearing, 50.DL = , 43=pm , LBP



16.1 2010 0 0 2010 1230 0 0 1230 6.42 803 0 0 803 490 0 0 490 3.21 401 0 0 401 245 0 0 245 1.57 196 0 0 196 120 0 0 120 0.515 66.1 0 0 65.2 40.2 0 0 39.8 0.243 32.8 0 0 31.6 19.7 0 0 19.2 0.120 18.3 0 0 16.5 10.5 0 0 9.85

0.0763 14.5 0 0 11.7 7.76 0 0 6.72 0.0484 12.9 0 0 9.18 6.33 0 0 4.93 0.0356 12.7 0 0 8.31 5.82 0 0 4.17 0.0264 13.0 0 0 7.95 5.59 0 0 3.72 0.0187 13.8 0 0 8,00 5.60 0 0 3.48 0.0128 15.7 0 0 8.78 5.80 0 0 3.41

The load between pad (LBP) configuration is used when the most important features are higher load capacity and better synchronous response. In this case the damping by the effective support area is larger than for LOP bearings. Heavy rotors running at relatively low speeds (low Sommerfeld number) have LBP tilting pad bearings.

Light rotors running at high speeds predominantly have load on pad (LOP) bearings. This configuration provides asymmetry in the support, which enhances the stability, which is the most important required characteristic.

For a bearing with five 060 tilting pads, centrally pivoted, with 50.DL = , 9550.BL = (where B is the pad arc length), no preload and pad

inertia neglected, the dimensionless stiffness and damping coefficients are


shown as functions of the Sommerfeld number for a LBP bearing in Fig. 6.45, a and for a LOP bearing in Fig. 6.45, b.

a b

Fig. 6.45 (from [6.12])

For the LOP bearing, a sudden reduction in the coefficients yyK and zzK appears when approaching the condition of pad resonance. This does not entail a loss of the overall bearing stiffness, since at resonance the pad inertia cannot be neglected, and the cross-coupling terms become important and replace the direct-coupling terms. However, in this case the bearing loses its inherently stable characteristic. Pad resonance occurs in general at high Sommerfeld numbers, where the minimum film thickness is large and the damping is high.

6.8.7 Rayleigh step journal bearings

The stepped pad bearing (Rayleigh pad bearing) is known as the bearing with the highest load capacity amongst all other possible bearing geometries. This is based on an observation of Lord Rayleigh (1918) that, when side leakage is neglected (infinitely wide bearing), the step film shape has the greatest load capacity in a slider bearing lubricated with an incompressible fluid.

Figure 6.46, a illustrates the journal in a concentric position. In this case, the number of steps placed around the bearing has to be limited to one in order to obtain a non-zero load capacity. The main parameters are rC - the radial clearance


in the ridge region, sC - the radial clearance in the step region, β - the angle subtended by the step region, γ - the angle subtended by the ridge region, and δ - the angle subtended by the lubrication groove.

Figure 6.46, b depicts the eccentric Rayleigh step journal, where Φ is the attitude angle and Γ is the eccentricity orientation angle. A pad is defined as the portion of a bearing included in the angle subtended by the ridge, step, and lubrication groove. Each pad acts independently since the pressure is reduced to ambient at the lubrication supply grooves. Figure 6.46, c shows a schematic view of the geometry of a four-pad Rayleigh step bearing.

a b c Fig. 6.46 (from [6.13])

In a single step Rayleigh journal bearing, a positive pressure is developed completely around the journal. The pressure distribution is triangular for a radius-to-length ratio equal to zero, with the peak value at the step and there is an optimal step configuration yielding maximum load capacity. Side leakage is usually more pronounced than in other designs. Side rails are placed at the ends of the bearing to restrict the axial flow and increase the load capacity. The study of the dynamic characteristics of the incompressibly lubricated Rayleigh step journal bearing is under development.

6.8.8 Floating ring bearings

The floating ring bearing finds application in turbochargers, where the rotor is light and runs at very high speeds. It is basically a plain cylindrical journal bearing, but with a loose bushing inserted in the clearance space between the journal and the bearing housing (Fig. 6.47). There are two lubricant films, an inner film identified by subscript 1, and an outer film identified by subscript 2. Each film can be considered as a plain journal bearing.


The oil-film formed between the ring and the bearing housing provides damping which is thought to be useful in improving the rotor stability as it spins in the inner film.

Fig. 6.47

Almost all rotors of automotive turbochargers exhibit strong sub-harmonic vibrations that cause noise (rumble) and rotor-stator rub. Their study is the subject of intense actual research efforts.

6.9 Squeeze film dampers

Squeeze film dampers are employed to improve the vibration characteristics of rotors by adding damping to either stabilize otherwise unstable rotors or to reduce synchronous rotor amplitudes. These are a special case of a hydrodynamic journal bearing where the rotation speed is zero and the hydrodynamic pressure forces are produced by the lubricant being squeezed between the journal and the bearing surface.

6.9.1 Basic principle

Squeeze-film dampers (SFD) are non-linear elements that are widely used in high-speed rotor-dynamic systems to attenuate vibrations and transmitted forces. The journal of a SFD is formed by a ring (sleeve) that is tightly fitted (fixed) on to the outer race of a rolling-element bearing (or the bush of a sleeve bearing). The SVD itself is an oil-film filling the annular clearance between the journal and the


bearing housing. The “journal” is prevented from rotating by some kind of mechanical restraint, but it is free to orbit within the annular clearance.

Fig. 6.48 (from [6.14])

An SFD is also often used on its own, between a bearing and its housing, to reduce vibrations while running through the rotor critical speeds. Rotation of the inner member of the SFD is in this case prevented by antirotation pins or dogs. The damper becomes part of a series structure of rotor, damper and housing flexibility.

Unlike the hydrodynamic journal bearing, an SFD is inherently stable, i.e. it cannot introduce self-excited vibrations. However, in the presence of rotating out-of-balance excitation, the forced periodic vibrations of a squeeze-film damped rotating system can become unstable, resulting in either amplitude jump phenomena or undesirable non-synchronous frequency components that are not simple integer multiples of the rotational speed.

In practical applications, an SFD is used either with or without a parallel retainer spring (Fig. 6.48). Retainer springs help to attenuate non-linear effects, especially if preloaded to centralize the journal in its housing. It typically consists of an annulus with a clearance of 25 to 500 μm that contains the oil film and surrounds the outer race of the rolling element bearing. The squirrel cage constrains the rolling element bearing from rotating, preventing the shaft to reach the oil film. It also acts as a light retainer spring that centers the journal in the damper clearance.

Other designs of squeeze film dampers do not incorporate squirrel cages; they rely instead on a dog mechanism to constrain the outer race of the rolling element from rotating. In such dampers the journal bottoms down in the damper clearance if no precession occurs.

The stationary oil film is the main characteristic of squeeze film dampers and is, in fact, the only difference between a squeeze film damper and a journal bearing, both of which are hydrodynamic bearings. The oil film in journal bearings rotates due to the rotation of the rotor even if there is no precession. The rotating


oil creates the static load-carrying capacity of the journal bearing and contributes to its stiffness. But oil rotation also causes instabilities such as oil whirl and oil whip.

In contrast, in squeeze film dampers the oil does not rotate if the rotor is not precessing. These dampers therefore have no static load-carrying capacity and no stiffness. The oil film dampens motion only when the rotor whirls. If there is no precession (and no retainer spring) the journal would bottom down in a squeeze film damper. Because the oil film is stationary, oil whirl and oil whip do not occur in squeeze film dampers.

6.9.2 SFD design configurations

A locally sealed damper with end hole feed/drain and radial piston rings is shown in Fig. 6.49, a. In such a damper, tightly sealed in the axial direction, the oil will flow circumferentially when squeezed.

Fig. 6.49 (from [6.15])

An open-ended damper, sometimes referred to as a globally sealed damper, with end groove feed/drain and axial face seals is shown in figure 6.49, b. Because the damper is not sealed in the axial direction, the oil can flow axially when squeezed.


A different configuration for feeding oil is the central circumferential feed groove; it is most commonly found in dampers without end seals.

Other squeeze film damper designs are used with rolling element bearings, where the centering spring is generally eliminated, but antirotation tabs or keys are provided. Common end sealing arrangements include: a) radial O-ring seals; b) piston ring seals; and c) side O-ring seals.

6.9.3 Squeeze film stiffness and damping coefficients

The stiffness sfK and damping coefficient sfC of the squeeze film damper can be derived from a solution of the Reynolds equation for a nonrotating journal bearing.

For a squeeze film with no end flow (Fig. 6.49, a), assuming laminar flow, full cavitation, a circular synchronous precession orbit and an inertialess oil film, the stiffness and damping coefficient are found to be [6.16]

( )( )

1 2

24223

3

εε

ωεμ

−+=

rsf

C

LRK , (6.63)

( )( ) 1 2

1221223

3

εε

μπ

−+=

rsf

C

LRC , (6.64)

where R - squeeze film radius, L - axial length of damper, rC - radial clearance, μ - oil dynamic viscosity, ε - eccentricity ratio (orbit radius / radial clearance), ω - whirl speed.

For relatively short dampers ( )50.DL < with open ends, assuming unrestricted end flow (Fig. 6.49, b), the damper stiffness is

( ) 1

2223

3

ε

ωεμ

−=

rsf

C

LRK (6.65)

and the damping coefficient is

( ) 1 2

2323

3

ε

μπ

−=

rsf

C

LRC . (6.66)

For the uncavitated film, 0=sfK and


( ) 1

2323

3

ε

μπ

−=

rsf

C

LRC . (6.67)

For pure radial squeeze motion, 0=sfK and

for cavitated film

( )[ ]( )( )

1

12 cos 2523

213

ε

εεπμ

−

+−=

−

rsf

C

LRC , (6.68)

for uncavitated film

( )( )

1

12 2523

23

ε

εμπ

−

+=

rsf

C

LRC . (6.69)

Note that the above stiffness and damping coefficients apply for a circular orbit of radius εrC . Dampers have no equilibrium position for an applied static load in the sense of a journal bearing. The "stiffness" developed by a damper results from its orbital motion due to a rotating load and could more properly be defined with a cross-coupled damping coefficient.

For dampers operating at high speeds and high values of the squeeze-film Reynolds number

μωρ 2 Re rC

= , (6.70)

where ρ is the oil density, fluid inertia forces become significant. Inertia or "added mass" coefficients can be obtained dividing the fluid inertia forces to the radial acceleration εω rC2 . The direct inertia coefficient is

( )12

1 12

2

3−

−−= β

εββρπ

rsf C

LRM , (6.71)

and the cross-coupled inertia coefficient is

⎟⎠⎞

⎜⎝⎛

+−

+⎟⎠⎞

⎜⎝⎛−=

εε

εερ

11ln12

7027

3

rsf C

LRm , (6.72)

where

( ) 2121 εβ −= . (6.73)


6.9.4 Design of a squeeze film damper

If a squeeze film damper is added to an existing compressor design to cure a subsynchronous vibration problem, a stability analysis is carried out based on a finite element model or a transfer matrix model of the rotor-bearing system.

Input parameters are damper radius, R, damper length, L, oil dynamic viscosity, μ , and rotational angular speed, Ω . Usually, an aerodynamic cross-coupling stiffness is also considered. A practical highest level of cross-coupling for design studies is 1.75·107 N/m [6.16]. Output parameters are the radial clearance,

rC , and the eccentricity ratio, 0ε , which produce support stiffness and damping coefficients able to eliminate the instability.

A computer program helps solving the damped eigenvalue problem and calculating the logarithmic decrement (or simply the real part of eigenvalues) as a function of design parameters.

A typical stability map is shown in Fig. 6.50, where the real part of the eigenvalue of the unstable mode is plotted as a function of support damping. Each curve corresponds to a given value of support stiffness. Separate maps are plotted for several assumed values of the aerodynamic cross-coupling stiffness. The rotor and bearing characteristics remain unchanged.

Fig. 6.50 (from [6.16])

Curves crossing the zero level of eigenvalue real part define stiffnesses for which the system can be stable. As the support stiffness is increased, the system


becomes less stable, so that there is a threshold value thK above which the system is always stable (the curves do not intersect the zero real part value).

On the other hand, for curves crossing the horizontal zero level axis, there is an intermediate range of support damping values for which the system is stable for a given value of support stiffness. For damping less than a certain value, c1, or larger than another value, c2, defined by the crossing points, the system is again unstable so that the actual value of support damping has to be selected between the two limits.

Thus, linearized stability maps as that from Fig. 6.50 provide information on the support characteristics needed to promote stability in a given rotor-bearing system.

Fig. 6.51 (from [6.16]) Fig. 6.52 (from [6.16])

In order to further determine the damper design parameters, two more diagrams are needed illustrating the support stiffness (Fig. 6.51) and damping (Fig. 6.52) versus eccentricity, for different values of the radial clearance.

First, the diagram from Fig. 6.51 is used. Drawing a horizontal line for the value thK , the values of 0ε at the crossing points with curves rC =const. are determined.

Based on these values, the dotted line thK =const. is drawn in Fig. 6.52, which sets an upper border for the range of possible stable operation. Next, two horizontal lines are drawn in Fig. 6.52 corresponding to c1 and c2, the limits of variation of the support damping for stable operation. The area between these two


limits and below the thK curve defines the design region of the damper. The design parameters rC and 0ε are selected to correspond to points within this region.

6.10 Annular liquid seals

Plain (ungrooved) annular seals are joints with the inner cylinder (the rotor) rotating in a tight fluid annulus with respect to the outer cylinder (the stator). For a centered seal, there is an axial velocity due to the axial pressure gradient (leakage flow) and a circumferential velocity due to the shaft rotation and shaft-fluid friction. In the "bulk-flow" model [6.17], these velocities are constant across the film, and shear stresses are accounted for only at the boundaries, at the shaft and the seal stator.

Fig. 6.53 (from [6.18])


For an eccentric seal, the difference between the pressure gradient on the tight and loose sides provides a restoring force, which opposes the shaft displacement, yielding a large direct stiffness. This is the Lomakin effect.

For a whirling shaft, a tangential force will appear normal to the shaft radial displacement, yielding a cross-coupled stiffness.

In the "short seal" solution, the pressure-induced tangential velocity is neglected. An improved treatment accounts for the development of circumferential flow due to the shear forces along the seal.

In the following, the short seal solutions of Black and Childs are presented.

6.10.1 Hydrostatic reaction. Lomakin effect

The steady axial pressure distribution at any circumferential location within a plain non-serrated seal is shown in Fig. 6.53, a for a simple case without shaft rotation. The pressure drops at the seal entrance are shown in Fig. 6.53, b.

The pressure at the inlet of the seal is sp and that at the seal exit is ep . As flow accelerates into the seal, a pressure drop occurs due to the Bernoulli effect. An additional decrease at the inlet region occurs due to the gradual growth of a turbulent flow velocity field into its fully developed form. The remainder of the pressure drop through the seal is very nearly linear.

The seal differential pressure can be calculated using Yamada's leakage relationship [6.19] for flow between concentric rotating cylinders

( )2

212Vp ρσξΔ ++= , (6.74)

where ξ is the entrance loss factor, ρ is the fluid density, V is the average fluid velocity, and σ is a friction loss factor defined by

rCLλσ = . (6.75)

In the above, L is the seal length, rC is the radial clearance, and λ is a friction factor, defined by Yamada to be the following function of the axial and circumferential Reynolds numbers, aR , cR :

83

241

8710790

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

−

a

ca R

RR.λ , (6.76, a)


2ν

ra

CVR = , νΩ r

cCRR = , (6.76, b)

where ν is the fluid kinematic viscosity, R is the seal radius and Ω is the rotor angular speed, rad/sec. The friction law definition of equation (6.75) fits the Blasius equation for pipe friction [6.17].

If the shaft moves downward within the seal, the velocity at the top of the shaft, topV , increases, because of the larger gap through which the fluid flows. The

flow velocity at the bottom, botV , decreases because of the smaller gap height there. Thus, pressure drops at the seal entrance are flow-velocity dependent and differ from top to bottom as shown in Fig 6.53, b. At the top, the entrance pressure drop increases. At the bottom, the entrance pressure drop decreases. Therefore, a net increase in pressure at the bottom of the seal produces an upward force tending to restore the shaft to its original centered position.

A similar situation occurs when the lateral velocity of the shaft changes.

The combined effect of the inlet loss and axial-pressure gradient accounts for the large direct stiffness which can be developed by annular seals. This physical mechanism for the direct stiffness was first explained by Lomakin (1958).

The actual pressure distributions shown in Figs 6.53, a and b are modified by shaft rotation and geometric variations within the seal, but the underlying principles governing the pressure buildup within a seal remain the same.

The RCr ratio for seals is generally on the order of 0.003 versus 0.001 for bearings. These enlarged clearances, combined with a high pressure drop across the seal, mean that flow is highly turbulent.

Most analyses for pump seals have used "bulk-flow" models for the fluid within the seal. In such models, the variation of fluid velocity components across the clearance is neglected. Average (across the clearance) velocity components are used – hence the bulk-flow designation. Bulk-flow models neglect the shear stress variation for the fluid within the clearance and only account for shear stresses at the boundaries of the model [6.14].

6.10.2 Rotordynamic coefficients

Dynamic seal forces are normally represented by a reaction force / seal motion model of the form [6.17]:

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

=⎭⎬⎫

⎩⎨⎧−−

zy

Mzy

CccC

zy

KkkK

FF

z

y

&&

&&

&

& + , (6.77)


where y and z are fixed coordinate displacements of the shaft and the dots are derivatives with respect to time. This model is valid for small motion about a centered position.

Short annular seals are usually considered isotropic. The diagonal terms of their stiffness and damping matrices are equal, while the off-diagonal terms are equal, but with reversed sign. The cross-coupled damping and the mass term arise primarily from inertial effects. Short seal solutions are obtained by neglecting the pressure-induced circumferential flow, while including the shear-induced flow.

Cross-coupled stiffnesses arise from fluid rotation, in the same way as in uncavitated plain journal bearings. For example, moving the seal rotor horizontally rightwards, a converging section is created in the lower half and a diverging section in the upper half of the seal (shaft rotates counterclockwise). The pressure is elevated in the converging section and depressed in the diverging section, yielding a reaction force upwards. This force acts normal to the radial displacement, having a destabilizing effect.

For long annular clearance seals, like those used to break down large pressure differences in multi-stage pumps, angular dynamic coefficients are introduced, because the rotor is acted upon by couples, and forces give rise to tilting shaft motions, and moments produce linear displacements.

The dynamic coefficients of equation (6.77) can be expressed in the following form:

Stiffness coefficients:

( )4 22203 TK Ωμμμ −= , 2 13 /Tk Ωμμ= ; (6.78)

Damping coefficients:

TC 13 μμ= , 223 Tc Ωμμ= ; (6.79)

Inertia coefficients: 2

23 TM μμ= ; (6.80)

where

λΔπμ pR

3 = (6.81)

and the mean passage time is

VLT = . (6.82)


The quantities 0μ , 1μ and 2μ depend on the friction loss coefficient σ , and the entry loss coefficient ξ . Their formulae are given below for three slightly different annular seal models.

1. Black's model [6.20], [6.18]

( )( )

21 1

2

2

0σξσξμ

+++

= '

( ) ( )( ) ( )( )3

4322

121

331 1 333 2332 11σξ

σσξσξξσξμ++

+++++++=

... , (6.83)

( ) ( ) ( )( ) ( )( ) 4

4322

221

331 1 2 21 1121 330σξ

σσξσξξσξξμ++

++++++−+=

.. .

For finite-length seals, Jensen developed the following formulae to account for finite RL ratios:

12

00 2801−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+=

RL.μμ ,

12

11 2301−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+=

RL.μμ ,

12

22 0601−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+=

RL.μμ ,

where 0μ , 1μ and 2μ are given by (6.83), and the "bar" quantities have to be used in equations (6.78) to (6.80) instead of those without bar over letter.

2. Jensen's model [6.21]

In a further development, the coefficients 0μ , 1μ and 2μ were defined in terms of the following additional parameter

122

871

87

−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

a

c

a

cRR

RRβ (6.84)

which accounts for a circumferential variation in λ due to a radial displacement perturbation from a centered position. They can be calculated as

.251 2

0 DBAσμ = ,


2

22

161

65

85

31

65 2

DB

FECAA ++⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛ ++

=σξσ

μ , (6.85)

( )3

2

261

21 251

61

31 2

DB

FEC.AA ++++⎟⎠⎞

⎜⎝⎛ −

=σξσξ

μ ,

where

ξ+=1A , 21 σξ ++=B , 750751 β..C −= ,

σCAD += , 32 σCAE = , 43 σCF = .

Fig. 6.54 (from [6.21])


Plots of 0μ , 1μ and 2μ from equations (6.85) are provided in Fig. 6.54 as a function of β and σ , for 50.=ξ . These coefficients are comparatively insensible to anticipated variations of the entrance loss factor ξ .

Black also examined the influence of inlet swirl on seal coefficients. In previous analyses, a fluid element entering a seal was assumed to instantaneously achieve the half-speed tangential velocity 2ΩR . In a later paper he demonstrates that a fluid element must travel a substantial distance axially along the seal before asymptotically approaching this limiting velocity. The practical consequence of this swirl effect is that predictions of the cross-coupling terms, k and c, may be substantially reduced. This applies especially for interstage seals, in which the inlet tangential velocity is negligible.

3. Childs' model [6.22]

Childs completed a seal analysis based on Hirs' turbulent lubrication model [6.23]. In his analysis he assumed completely developed turbulent flow in both the circumferential and axial directions. The short seal dynamic coefficients derived by Childs are given by equations (6.78)-(6.80) but the coefficients 0μ , 1μ and 2μ are

( )σξ

σμ21

12 02

0 ++−

=mE ,

61

2212 2

1 ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++

++= EBE

σσξσμ , (6.86)

61

212 ⎟⎠⎞

⎜⎝⎛ +

++= E

σξσμ ,

where

( ) 121

σξξ

BE

+++

= , ( ) 141 02 mbB ++= β ,

411

2b+=β ,

ΩRV

RRb

c

a ==2

, (6.87)

and 0m is an exponent entering the loss factor expression ( )750.mo ≅

21

20

0

0

411

m

ma b

Rn

+

⎟⎠⎞

⎜⎝⎛ +=λ . (6.88)


The short seal dynamic coefficients derived by Childs are similar to the previously mentioned solution by Black when a one-half shaft speed swirl velocity is assumed.

Childs considered the influence of the initial swirl 0V . In this case, the seal dynamic coefficients are written in the following form:

Stiffness coefficients:

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡+−=

4 12

22

03 aTK μΩμμ , ( ) 2

213Tak Ωμμ += ; (6.89)

Damping coefficients:

TC 13 μμ= , ( ) 2323 Tac Ωμμ += ; (6.90)

Inertia coefficients: 2

23 TM μμ= , (6.91)

where

( )[ ]01 1 ma ++= βσ , (6.92)

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +−−⎟

⎠⎞

⎜⎝⎛ +

++= −− aa

aaE

aVa e 1

21e1 1

2122

2

20

1 σξσ ,

( )⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛ ++−⎟

⎠⎞

⎜⎝⎛ −+

++= − 1 1

21e1 1

2122 2

02 a

EaBEB

aVa a

σσξσ ,

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−−⎟

⎠⎞

⎜⎝⎛ ++

++=

−− e

21e1 1

21

212

20

3 aaE

aVa

aa

σξσ .

For 0V =0, 321 aaa == =0, and one obtains equations (6.78)-(6.80).

From equation (6.77), a circular shaft orbit of radius A and precession frequency ω yields radial and circumferential force components

( )AMcKFr 2ωω −+−= , ( ) ACkF ωθ −= .

The circumferential force can produce unstable operating regimes.

Example 6.2

An annular neck ring seal has the following geometric properties: length L =50 mm, radius R =75 mm ( )31=DL , radial clearance rC =0.25 mm.


At operating conditions N =1200 rpm, the pressure difference across the axial length of the seal is pΔ =1.38·106 Pa. The water properties are: dynamic viscosity μ =4.14·10-4 Ns/m2, mass density ρ =979 kg/m3.

Assume initially that the flow velocity is V =25 m/s.

The initial axial Reynolds number is

559292 ,CVR ra ==

μρ .

The circumferential Reynolds number is

572 5 ==

μρΩ r

cCRR .

Solving for σ gives

=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

83

2

41 8710790

a

c/

ar RR

R.

CLσ 1.217.

Considering ξ =0.1, from the equation for pΔ a new value of V is obtained

21

212

⎟⎟⎠

⎞⎜⎜⎝

⎛++

=σξρΔ /pV =26.77 m/s.

The process is repeated iteratively until the mean flow velocity is obtained as 59228.V = m/s for which 1741.=σ .

The final value of the axial Reynolds number is =aR 33,807 showing that the flow is fully turbulent.

Substituting ξ =0.1 and σ =1.193 into the equations (6.83) gives

0μ =0.1414, 1μ =0.35, 2μ =0.05287, and also 3μ =5.6·107.

The mean passage time is

VLT = =1.749·10-3 s.

The dynamic coefficients (6.78)-(6.80) have the following values

K =7.1139·106 N/m, k =2.0604·106 N/m,


C =32793 Ns/m, c =1153 Ns/m, M =9.176 Ns2/m.

Note that c

MΩ =1, so that M and c tend to cancel for synchronous

precession at the running speed and the radial reaction force component rF is given only by the direct stiffness K.

The whirl frequency ratio is Ω

ΩC

kw = =0.5 corresponding to the average

tangential velocity within the seal .R 2Ω The result is the same as for a plain journal bearing. For nωΩ = , the rotor's first critical speed, the tangential reaction

force component θF becomes destabilizing at the speed nn

th .ωωΩ 2

50== . This

shows that, theoretically, a seal becomes destabilizing at running speeds in excess of about twice the rotor's first critical speed. Because in seals wΩ can be decreased, the on-set speed of instability thΩ is increased.

6.10.3 Final remarks on seals

The above seal analyses are restricted to a) plain seal surfaces; b) constant inlet loss coefficients, and c) fully developed inlet circumferential flow velocity. The axial flow velocity is assumed to be much greater than the velocity in the circumferential direction. The predominant effect of variation in the inlet swirl velocity is to alter the cross-coupled stiffness coefficients. These coefficients are reduced in magnitude as the swirl velocity is reduced.

In fact, under proper values of seal geometry, axial pressure drops and inlet swirl velocity, the signs of the cross-coupled stiffness coefficients are reversed; cross-coupling promotes stability rather than instability. Experiments with spiral serrated seals, for axial Reynolds numbers ranging from 20,000 up to 40,000, revealed that long serrated seals possess negative direct stiffness. In calculations, correction factors on the seal length in a plain seal have to be used to account for grooving in serrated seals.

Dynamic coefficients in equation (6.77) are almost skew-symmetric, even for volute pumps, although such results would be expected only for rotationally symmetric configurations. In contrast to bearings, seals develop significant direct stiffness in the centered position, independent of the rotation speed, from a hydrostatic mechanism.

Damping coefficients of seals are relatively large. As a result, the majority of centrifugal pumps operate with no evidence of rotor dynamic problems. For close clearance rings, hydrodynamic mass effects have been seen to be strong.


For synchronous rotor excitation and whirling, it was shown that the seal restoring force increases with the square of the speed. It has been thought of as being generated by a fictitious mass, called the Lomakin mass. However, a geometric mass correction is correct only for pure synchronous whirling. When the virtual mass is greater than the actual geometric mass, the critical speed is suppressed. Such an approach would be correct only for systems in which the Lomakin mass and the geometric mass occur at the same locations. For systems in which this does not occur, the Lomakin effect should be treated as a stiffness and hence as a function of speed.

6.11 Annular gas seals

Annular gas seals, or labyrinth seals, have a leakage-control role in centrifugal compressors. Forces developed by gas seals are roughly proportional to the pressure drop across the seals and the fluid density within the seal. Because of the density dependence, gas seals have had a great impact on high-pressure compressors for gases with high molecular weight.

Fig. 6.55 (from [6.14])

Several usual forms of labyrinths are shown in Fig. 6.55.

Labyrinths can have a negative effect on the smooth running of compressor rotors. Two basically different mechanisms are possible.

1. The first case occurs if unbalance induced precession causes contact between labyrinth tabs and the shaft. Wherever a possible contact of the labyrinth


leads to a marked heating and hence deformation of the rotor, it is advantageous to have the labyrinth strips located on the shaft. The cooling induced by the strips protects the shaft against a local rise in temperature.

2. The second case is less apparent but more important. Any eccentricity causes a change of flow in the labyrinth and induces reaction forces that act back on the shaft and thus may diminish the rotor system damping. If this damping is sufficiently reduced, self-induced rotor vibrations occur that render the operation of the compressor impossible.

Fig. 6.56 (from [6.17])

Figure 6.56 illustrates a typical sealing arrangement for a last stage compressor impeller. The eye-packing seal limits return-flow leakages down the front of the impeller. The shaft seal restricts leakage along the shaft to the preceding stage. The eye packing and shaft seal configurations are called "see-through" or half-labyrinth seals (see Fig. 6.55).

In a throughflow compressor, leakage flow through the balance drum is returned to the inlet. The balance drum is a long seal which absorbs the full pressure differential of the compressor. The balance drum seal configuration is called an interlocking or full labyrinth (see Fig. 6.55).

For a back-to-back compressor, the balance drum absorbs the pressure drop between the last stage of the compressor and the last stage of the initial series of impellers, i.e., about one half of pressure drop per machine. For the same inlet and discharge pressures, the average density is higher in the center labyrinth of a back-to-back machine than in the balance drum labyrinth of a series machine.

Back-to-back compressors are more sensitive to the forces from the central labyrinth than are series machines to forces from the balance-drum labyrinth, and


this is mainly because of the larger deflections at the middle span for the first critical mode.

The forces developed by compressor seal labyrinths are at least one order of magnitude lower than their liquid seal counterparts. They have negligible added-mass terms and are typically modeled by the following reaction force/motion model:

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

=⎭⎬⎫

⎩⎨⎧−−

zy

CccC

zy

KkkK

FF

z

y

&

&. (6.93)

The diagonal terms of their stiffness and damping matrices are equal, while the off-diagonal terms are equal, but with reversed sign, which denotes isotropy. Unlike the pump seal model of equation (6.77), the direct stiffness term is typically negligible and is negative in many cases.

The coefficients in equation (6.93) are dependent on the gas density, giving rise to "load dependent" instabilities. This leads to an "onset power level of instability" instead of the known "onset speed of instability" encountered for rotors in hydrodynamic bearings, which exhibit speed-dependent instabilities.

Fig. 6.57 (from [6.9])

Tests have shown that for "see-through" labyrinth seals: a) cross-coupled stiffness coefficients increase directly with increasing inlet tangential velocities; b) seal damping is small but must be accounted for to obtain reasonable rotordynamic predictions; and c) as clearances decrease, teeth-on-rotor seals become less stable and teeth-on-stator seals become more stable.

Stabilizing solutions include the swirl brakes or swirl webs at the entrance to labyrinth seals (Fig. 6.57). This device uses an array of axial vanes at the seal inlet to destroy or reduce the inlet tangential velocity.

Another approach for stabilizing compressors involves a "shunt line" and entails rerouting gas from the compressor discharge and injection flow into one of


the early cavities in the balance-piston labyrinth (Fig. 6.58). Usually, the flow from the back side of the last stage impeller enters the labyrinth with a high tangential velocity which yields a high destabilizing force. The shunt flow forces the gas outward along the back face of the impeller, reducing the seal tangential velocity.

Fig. 6.58 (from [6.9])

Of all the labyrinth seals, the destabilizing forces are maximum for the balance piston labyrinth because it is the longest seal and has the highest pressure drop. For a throughflow machine it also has the highest gas density.

Some success has been obtained in improving the stability of compressors by replacing labyrinth seals with honeycomb seals. The latter develop larger destabilizing coefficients but much higher direct damping values. Brush seals, using wires in contact with a ceramic coating on the shaft, have a sharply reduced leakage flow as compared to a labyrinth or honeycomb seal.

6.12 Floating contact seals

Oil seals are frequently used in high speed multistage compressors to minimize the leakage of the process gas into the atmosphere. For hazardous gases, tightness is an important requirement. This is achieved with liquid seals in conjunction with floating ring seals.


Fig. 6.59 (from [6.9])

6.12.1 Design characteristics

If the desired pressure drop across the seal ring is adequate, only one seal ring is employed. Otherwise, a double floating ring combination is used, with sealing oil feed in the middle, as illustrated in Fig. 6.59.

The seal segments consist of outer and inner seals which are spring-loaded against each other. The preload of the spring causes the lapped external faces of the seal segments to be in contact with the seal cartridge housing. An antirotation pin is provided to prevent rotation of the seal segments.

Oil is introduced between the rings at a pressure slightly above compressor suction pressure and then leaks axially along the shaft. The inner ring seals against process gas leakage, while the outer ring limits seal oil leakage to the outside. Most of the oil is recovered.

Heat dissipation is critical to the operational security of oil seals. This is ensured by the liquid flow in the relatively wide seal ring clearances, typically 50 to 80 μm, which entails high leak rates. These can be lowered providing the floating ring with a return thread.


Fig. 6.60 (from [6.24])

A single ring seal arrangement, used in a 5-stage compressor for refinery service, is depicted in Fig. 6.60, where: 1 – oil bushing, 2 – diaphragm, 3 – sleeve (impeller), and 4 – stator.

Fig. 6.61 (from [6.24])


The "trapped bushing seal", shown in Fig. 6.61, is a "dual" bushing which encompasses both the inner and outer seal in one ring. One can distinguish: 1 - shaft, 2 - sleeve with interference fit under bushing, 3 – stator, 4 – stepped dual bushing, 5 – bushing cage, 6 – nut, 7 – shear ring, 8 – spacer ring.

The compact design allows shorter bearing spans for higher critical speeds. The sleeve 2 under bushing protects the shaft and simplifies assembly and disassembly. It requires only a jack/puller bolt ring. The spacer ring 8 is fitted at initial assembly, so there is no field fitting of parts.

This arrangement allows the bushing to freely track the shaft motion. Seal induced hydrodynamic forces are dissipated in seal motion and not applied to the shaft. The seal is also insensitive to wear on the axial faces [6.24].

a b

Fig. 6.62 (from [6.25])

Figure 6.62 shows a typical cone seal arrangement. Fig. 6.62, a is the original design which experienced severe vibrations, and Fig. 6.62, b is the improved design, obtained by reducing the sealing length and the width of the lapped face, and increasing the steady holding force of the springs.

An arrangement intended to solve the seal centering problem is shown in Fig. 6.63, where a tilting-pad bearing element is incorporated in the high-pressure seal ring.

Possible design improvements to the high-pressure ring are: a) decreasing the seal face area, in order to decrease the unbalanced axial force, hence the friction force; b) cutting circumferential grooves in the seal land region, to reduce the effective axial hydrodynamic length of the ring; and c) increasing the seal clearance.


Fig. 6.63 (from [6.9])

6.12.2 Seal ring lockup

Compressors with oil seal rings are often subject to subsynchronous vibrations. The seal rings are designed to "float", in the sense that they orbit with the shaft as it vibrates (rotation is prevented by some pin or other device which still allows radial ring motion). If the seal ring orbits with the shaft, the oil film force, and thus the dynamic coefficients, between the shaft and ring are very small.

In certain circumstances, the seal rings may be prevented from moving. This is known as the "seal ring lockup". It causes large oil film forces to act between the shaft and ring, which may produce large stiffness and damping coefficients.

The building up of the cross-coupled stiffnesses by the seal ring lockup, in a machine with tilting pad bearings, can be described as follows [6.26]:

1. For a non-rotating shaft, the oil ring is resting on the top of the shaft. As the rotational speed increases, the oil seal fluid film supports the weight of the oil seal ring.

2. The pressure drop across the seal ring increases with the machine speed. This in turn increases the axial force due to unbalanced pressure, and correspondingly, the radial friction force acting on the seal ring (given by the coefficient of friction times the axial force). When the radial friction force acting on the seal exceeds the oil film radial force, the seal will lock up.

3. As the shaft speed continues to increase, the (tilting pad) bearing oil film forces tend to force the shaft upward in the bearing (the bearings are unloaded). The seal ring oil film forces tend to oppose this because the shaft is now taking an eccentric position within the fixed seal ring.


4. Unbalance forces inherent in rotor make the shaft undergo dynamic orbits. These dynamic forces in turn produce dynamic oil film forces in the lubricant between the shaft and fixed seal ring. If the combination of static and dynamic forces is larger than the radial friction force, the oil seal ring will move. If the dynamic forces are not large enough to overcome the available radial friction force at a given speed, the seal ring is likely to lockup. This can happen at some speed well below running speed.

5. As the shaft speed increases, the oil film forces in the bearings will lift the shaft in the bearing. This will create an eccentric position of the shaft relative to the locked up oil seal ring. This in turn will produce large cross-coupling stiffness terms acting on the shaft due to the oil film in the seal ring.

6.12.3 Locked-up oil seal rotordynamic coefficients

In approximate calculations, it is assumed that the seal ring is locked up with the shaft centered and behaves as a non-cavitated concentric plain sleeve bearing, and there is no axial pressure gradient. Oil seal rings are short (0.05<L/D<0.2) and their flow is generally laminar, non-cavitating.

For this centric case, the cross-coupled stiffnesses are [6.24]

2 4

233

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=−=

DL

CRL

CLDkk

rrzyyz μΩπμΩπ , (6.94)

where Ω - shaft angular speed of rotation, rad/sec, D – shaft diameter, R – shaft radius, L – axial seal length, rC - oil seal ring radial clearance, μ - fluid dynamic viscosity.

As coefficients are proportional to 3L , a considerable reduction in destabilization can be achieved by providing the seal with circumferential grooves, as shown in Fig. 6.63.

For a concentric seal, the direct stiffnesses yyk and zzk are negligible. The direct damping coefficients are

yzyy kc 2Ω

= , yzzz kc 2Ω

= . (6.95)

Note that these properties are inversely proportional to 3rC , hence highly

sensitive to variations in clearance. Thus, enlarged clearances at the seals would allow operation with low vibrations even when the seals are locked up.

The eccentric values of cross-coupled terms are known to be several times the value for the centered position.


There are more accurate calculation schemes to determine the seal dynamic coefficients, for various assumptions about the seal lockup eccentricity [6.17]. However, the results are sensitive to the assumptions about whether lockup results in increased or decreased journal loading in rotor bearings.

Stable operation of a rotor is possible even if the seal locks up and the seal clearances are abnormally large. Due to eccentric lock-up of the seal ring, direct stiffness terms are developed. Reduced journal loading will raise the natural frequencies of the rotor because the effective bearing span is reduced transferring the load to the seals. In this case, the frequency of the non-synchronous instability is higher than the calculated rigid bearing critical speed.

A quasi-static method of determining the seal ring lockup conditions has been used with good results [6.26]. It involves matching the radial friction force to the seal ring weight, wherefrom the ring initial lockup speed is calculated first. As the shaft speed increases, it lifts off of the bottom pad of the tilting pad bearings. Based on known relationships for tilting pad bearings, the shaft position relative to the bearing center line is determined, wherefrom the oil seal ring eccentricity and attitude angle can be calculated for a load capacity equal to the oil seal ring weight. Otherwise, because the static attitude angle depends on the friction force, it is, strictly speaking, indeterminate. Selection of the friction coefficient is a difficult task.

The subsequent calculation of rotordynamic coefficients can be done using the Ocvirk short-bearing model [6.17]. The relationship of the static eccentricity ratio 0ε to the modified Sommerfeld number

60

22

WW

DL

CRN

WDLS *

rs =⎟

⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=μ (6.96)

for a short bearing is illustrated in Fig. 6.64, a. Figure 6.64, b shows the rotordynamic coefficients for a short seal with an assumed attitude angle of 90o

[6.9].

The seal has no direct stiffness for this attitude angle, and the cross-coupled stiffness and direct damping terms rise sharply with decreasing sS , i.e. increasing static eccentricity ratio. The increase of yyc and zzc with 0ε can be advantageous at low speeds, when the rotor is traversing a critical speed; however, the increase in yzk and zyk can lead to rotor instability at running speeds of the order of twice the critical speed.

For five values of the static eccentricity ratio 0ε , the dimensionless oil seal coefficients are given in Table 6.10 [6.27].

The physical (dimensional) coefficients are then calculated as


r

*j,ij,i C

WKk = , r

*j,ij,i C

WCc Ω

= , ( )z,yj,i = (6.97)

where

60

22

2

3⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛==

DL

CRDL

CNLRW

rr* Ωμμπ (6.98)

and 602 NπΩ = .

a b

Fig. 6.64 (from [6.9])

Table 6.10

0ε yzK zyK yyC zzC

0.01 3.14 -3.14 6.28 6.28 0.1 3.28 -3.19 6.38 6.57 0.2 3.76 -3.33 6.67 7.51 0.5 9.67 -4.84 9.67 19.35 0.8 92.1 -14.54 29.1 184.2

Note that zyyz KK −≠ and zzyy CC ≠ so that the locked-up eccentric seal is anisotropic.


The direct stiffness and damping coefficients are not as important as the cross-coupled coefficients because the seals are close to the bearings. Bearing direct stiffness and damping coefficients are much larger and tend to dominate the seal coefficients. Tilting pad bearings have no cross-coupled coefficients when symmetric loading occurs, so the oil seal cross-coupled coefficients are very significant.

A simple breakdown seal can have an axial force as high as ptD Δπ 2

.

For a seal pressure drop pΔ = 6.89·106 Pa, a diameter D=165 mm, and a face width t = 6.35·10-3 m, the axial locking force is approximately 1.11·104 N. For three seals and assuming a coefficient of friction of 0.1, the radial load capacity could be as high as 3.34·103 N, a value equal to a typical bearing load.

It is easy to see that, unless a special pressure-balanced ring seal is employed in medium-weight rotor, high-pressure machines, the bearings and seals will interact as rotor supports. In heavy rotor, lower pressure machines this interaction is less likely.

Thus, the high pressure can indirectly cause instability in otherwise stable machines, apart from influencing the gas density, by locking oil breakdown seals axially such as to cause interaction with the stable rotor / tilting pad bearing system.

In conclusion, oil-buffered seals should be designed to act as sealing elements only and should not be part of the dynamic system of rotor and bearings. A pressure-balanced seal should be a design requirement, and means for causing seal lock-up should be eliminated from the compressor design.

References

6.1 Someya T. (ed), Journal-Bearing Databook, Springer, Berlin, 1988.

6.2 Constantinescu, V. N., Nica, Al., Pascovici, M. D., Ceptureanu, Gh., Nedelcu, Şt., Lagăre cu alunecare, Editura tehnică, Bucureşti, 1980.

6.3 Rieger, N. F., Vibrations of Rotating Machinery, Vibration Institute, Illinois, 1977.

6.4 Lund, J. W., Rotor-Bearing Dynamics, Ossolineum, 1979.

6.5 Lee, C.-W., Vibration Analysis of Rotors, Kluwer Acad. Publ., Dordrecht, 1993.

6.6 Childs, D., Moes, H., and van Leeuwen, H., Journal bearing impedance descriptions for rotordynamic applications, Journal of Lubrication Technology, pp.198-219, 1977.


6.7 Adams, M. L., Insights into linearized rotor dynamics, Part 2, Journal of Sound and Vibration, vol.112, nr.1, pp.97-110, 1987.

6.8 Cameron, A., Basic Lubrication Theory, 3rd ed., Ellis Horwood Ltd., Chichester, 1981.

6.9 Ehrich F.F. (ed), Handbook of Rotordynamics, Mc Graw Hill, New York, 1992.

6.10 Bhushan, G., Rattan, S. S., and Mehta, N. P., Effect of pressure dams and relief-tracks on the performance of a four-lobe bearing, IE (I) Journal-MC, vol.85, pp.194-198, 2005.

6.11 Chen, W. J. and Gunter, E. J., Introduction to Dynamics of Rotor-Bearing Systems, Trafford, Victoria, 2005.

6.12 Lund, J. W., Spring and damping coefficients for the tilting-pad journal bearing, ASLE Transactions, vol.7, no.4, pp.342-352, 1964.

6.13 Hamrock, B. J., and Anderson, W. J., Incompressibly lubricated Rayleigh step journal bearing, NASA TN D-4873, 1968.

6.14 Gasch, R., Nordmann, R., and Pfützner, H., Rotordynamik, 2. Auflage, Springer, Berlin, 2002.

6.15 El-Shafei A., Squeeze film dampers: effective damping devices for rotating machinery, Vibrations, vol.5, nr.3, pp.8-11, 1989.

6.16 Gunter E. J., Barrett L. E., and Allaire P. E., Stabilization of turbomachinery with squeeze film dampers – Theory and applications, Proc. Conf. "Vibrations in Rotating Machinery", I. Mech. Eng., pp.291-300, 1976.

6.17 Childs D., Turbomachinery Rotordynamics: Phenomena, Modelling and Analysis, Wiley, 1993.

6.18 Barrett L. E., Turbulent flow annular pump seals: A literature review, Shock and Vibration Digest, vol.16, nr.2, pp.3-13, 1984.

6.19 Yamada Y., Resistance of flow through an annulus with an inner rotating cylinder, Bull. J.S.M.E., vol.5, nr.18, pp.302-310, 1962.

6.20 Black H. F., Effects of hydraulic forces in annular pressure seals on the vibrations of centrifugal pump rotors, J. Mech. Eng. Sci., vol.11, nr.2, pp.206-213, 1969.

6.21 Black H. F. and Jensen D. N., Dynamic hybrid properties of annular pressure seals, J. Mech. Eng. Sci., Proc. Inst. Mech. Eng., vol.184, pp.92-100, 1970.

6.22 Childs D. W., Dynamic Analysis of Turbulent Annular Seals Based on Hirs' Lubrication Equation, J. of Lubrication Technology, Trans. ASME, vol.105, pp.437-444, 1983.


6.23 Hirs G. G., A Bulk-Flow Theory for Turbulence in Lubricant Films, J. of Lubrication Technology, Trans. ASME, pp.137-146, 1973.

6.24 Emerick M. F., Vibration and Destabilizing Effects of Floating Ring Seals in Compressors, NASA CP 2250, pp.187-204, 1982.

6.25 Kirk R. G. and Simpson M., Full Load Shop Testing of 18000 HP Gas Turbine Driven Centrifugal Compressor for Offshore Platform Service: Evaluation of Rotor Dynamic Performance, NASA CP 2409, pp.1-13, 1985.

6.26 Allaire P. E. and Kocur J. A. Jr., Oil Seal Effects and Synchronous Vibrations in High-Speed Compressors, NASA CP 2409, pp.205-223, 1985.

6.27 Kirk R. G., The Impact of Rotor Dynamics Analysis on Advanced Turbo Compressor Design, Proc. Conf. "Vibrations in Rotating Machinery", I. Mech. Eng., University of Cambridge, 15-17 Sept. 1976, pp.139-150, 1977.

7. INSTABILITY OF ROTORS

Under certain operational conditions, the rotor precession orbit grows suddenly, accompanied by a change in the whirl frequency to a nonsynchronous value. With rare exceptions, this frequency is that of the lowest natural frequency of the rotor system, i.e. the frequency of the first critical speed. In most cases it proves impossible to pass through this threshold of instability without endangering the mechanical integrity of the machine.

Limiting the analysis to the stability of the static equilibrium position of the rotor axis, the system can be assumed to be linear. At the threshold of instability, called the “onset speed of instability”, the real part of an eigenvalue becomes zero. Nonlinearities have little or no influence on this speed limit. The linear model contains the governing parameters which explain the physical mechanism of the instability. Only when it is of interest to explore the behavior beyond the onset speed of instability does it become necessary to include nonlinearities, which is beyond the aim of this presentation.

This chapter is concerned with instabilities of the flexural precession motion produced by hydrodynamic forces in bearings and seals, interaction of bladed disks and impellers with fluid film forces, dry friction, internal shaft damping and rotor asymmetry. Some of them are self-excited instabilities, in which the whirling is always at a subsynchronous frequency equal to an eigenfrequency of the system. Others are produced by parametric excitation and can be synchronous, sub- or super-synchronous. They are more like forced vibrations and some can even be “driven through” the speed range of instability (i.e., a higher speed can be found where the rotor regains its stability).

7.1 Whirling of rotating shafts

In the case of whirling, rotors are destabilized by tangential reaction forces which are normal to a radial displacement and in the direction of shaft rotation. The destabilizing forces are in the direction of the whirling motion, and in opposition to


the external damping force, which tends to inhibit the whirling motion. Often these forces are approximated as being proportional to the radial deflection (Fig. 7.1). The proportionality factor is a cross-coupled stiffness coefficient, since it relates the force magnitude to a deflection normal to the force.

In matrix form

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

−=⎭⎬⎫

⎩⎨⎧

zy

kk

ff

yz

yz

z

y0

0, (7.1)

where the cross-coupling terms are skew-symmetric

yzzy kk −= . (7.2)

As shown in Section 6.6, as the tangential force Tf acts in the direction of the whirling, it supplies energy to the motion, hence provides “negative” damping.

Fig. 7.1

If the free motion is decomposed into two whirls, each in an Archimedean spiral (Fig. 5.26), one with the direction of rotation and the other against it, at speeds below the onset speed of instability both spirals are of decreasing amplitude. At speeds above the onset speed of instability, in almost all cases, the whirl in the direction of rotation is of exponentially increasing amplitude (hence unstable) while that in opposite direction is of decreasing amplitude. It is the forward whirl component which is responsible for the “negative” damping.

For an idealized rotor with a single lumped mass m and shaft stiffness k, whirling with a speed ω at an instantaneous radius r, the forces acting on the mass are (Fig. 7.2) the centrifugal force 2ωrm , the elastic restoring force rk , the radial

damping force ( )trc dd , the radial inertia force ( )22 dd trm , the tangential damping force rcω , the Coriolis force ( )trm dd2 ω and the destabilizing force Tf . The equations of motion are written using d’Alembert’s principle [7.1].

The radial force balance yields

7. INSTABILITY OF ROTORS 163

0dd

dd 2

2

2=−++ ωrmrk

trc

trm , (7.3)

and the tangential force balance gives

0dd2 =−+ Tfrc

trm ωω , (7.4)

where the destabilizing force is proportional to the radial deflection

rkf yzT = , (7.5)

and yzk is the cross-coupled stiffness.

Fig. 7.2

The solution takes the form

tλrr e0= . (7.6)

For the system to be stable, the coefficient of the exponent

ωω

λm

ckyz

2−

= (7.7)

must be negative, giving the requirement for stable operation as

ωckyz < (7.8)

or, in dimensionless form,


ζωω

22 =<mc

m

kyz (7.9)

where ζ is the damping ratio.

At the onset of instability 0=λ , so that the whirl speed at onset is found, from equation (7.3), to be

mk

=ω . (7.10)

The whirling speed at the onset of instability is the rotor natural frequency, irrespective of the shaft rotational speed. Even if the rotational speed Ω increases, the whirl speed remains equal to the rotor natural frequency so that the whirling is a subsynchronous motion.

Beyond the onset speed of instability, 0>λ and the solution (7.6) describes an exponential spiral, hence the name of rotor whirling. Actually, nonlinearities in the rotor system dissipate energy more rapidly with increasing amplitudes than predicted by a linear model. Hence, whirl amplitudes increase sharply with time but are generally finite, achieving a steady-state limit cycle with a closed orbit.

Equation (7.9) shows that, increasing the stator system damping ratio, the onset speed of instability is pushed to a higher speed. Rotor stability is enhanced by increasing the external damping, or equivalently, by introduction of anisotropy into the bearing supports (see Section 3.1.3). The sensitivity of rotors to destabilizing forces increases with rotor flexibility. Avoidance can be achieved by increasing the system stiffness (and critical speeds).

7.2 Instability due to rotating damping

The instability due to internal rotor friction is analyzed in Sections 2.3.2, 2.3.3 and 3.1.3 (Part I) for a single-disk rotor. The threshold speed marking the transition from stability to instability is given for viscous damping and is shown to be raised by anisotropic support stiffness.

In 1980 appeared the remarkable paper of Stephen Crandall [7.2] giving a physical explanation of the fact that a mechanism of energy dissipation can cause instability. Based on a simple two-dimensional model, Crandall’s paper gives a clear insight into the destabilizing mechanism of damping in rotating parts and permits an elementary physical determination of the stability limit. The following presentation reproduces most of the content of that paper.


7.2.1 Planar rotor model

Consider the planar model of a disk mounted on a flexible shaft (Fig. 7.3). The rigid ring is forced to turn at an angular speed Ω by an external source. A mass particle is suspended elastically from the ring by four massless linear springs. The mass m represents the central disk of the Laval-Jeffcott model and the springs represent the massless flexible shaft of stiffness k, independent of the rotational speed. Gravity in the plane of the figure is neglected.

The elastic force system for small displacements is circularly isotropic. The equilibrium position is with the mass at the origin. When the mass has a radial displacement r, in any direction within the plane, the elastic restoring force is rk , directed back toward the origin.

Fig. 7.3 (from [7.2]) Fig. 7.4 (from [7.2])

The motion is conveniently expressed in terms of either stationary coordinates y, z , or rotating ring-fixed coordinates η , ζ , as in Fig. 2.14.

In stationary coordinates, the equations of motion for the mass particle have the form (2.1). They are decoupled and independent of Ω . All natural motions are linear combinations of two independent modes which have the same natural frequency mkn =ω . The two independent modes may be taken as rectilinear oscillations along two diameters or as a pair of circular precession modes, one counter-clockwise and the other clockwise.

The equations of motion with respect to the rotating axes η , ζ are

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡ −+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡00

00

0220

00

2

2

ζη

ΩΩ

ζη

ΩΩ

ζη

mkmk

mm

mm

&

&

&&

&&.

(7.11)


7.2.2 Qualitative effect of damping

Consider now the introduction of isotropic linear viscous damping to the rotor model of Fig. 7.3. Damping with respect to the stationary axes (external damping) is represented in Fig. 7.4 by the four dashpots whose outer extremities are fixed in the stationary reference frame. For small motions of m, the system of dashpots develops a circularly isotropic damping force. The resultant dashpot force acts in the opposite direction of the absolute velocity of m. If m is forced to travel in a circular orbit of radius r with counter-clockwise angular speed nω , the damping force ns rc ω acts tangent to the circle in a clockwise sense.

Damping of the relative motion with respect to the rotating system is represented in Fig. 7.5 by the four dashpots whose outer extremities are fixed in the rotating ring.

Fig. 7.5 (from [7.2])

“For small motions of m, this set of dashpots develops a circularly isotropic damping force which acts in opposition to the velocity of m relative to the rotating system. In particular, if m is forced to travel in a circular orbit of radius r with counter-clockwise angular speed nω as viewed from the stationary axes, it will appear to have a counter-clockwise angular speed Ωω −n in the rotating frame. The damping force generated by the rotating dashpots is ( )Ωω −nr rc acting tangentially to the circle in a clockwise sense.”

“The effects of these damping mechanisms on the free motion of the rotor can be argued qualitatively as follows. Suppose that the damping forces are very small in comparison with the spring forces. This implies that a free orbit will undergo only a small change during one period nωπ2 as a result of the damping. If initially the orbit is a counter-clockwise circle of radius r, the preceding discussion shows that when the rotor rotation angular speed Ω is subcritical, i.e. less than the natural frequency of the free motion nω , both the stationary and rotating dashpots act to retard the motion of m. The rotor does work on the dashpots and the total


energy of the rotor orbit is diminished. If initially the orbit is a superposition of a counter-clockwise whirl and a clockwise whirl, a similar argument shows that the radii of both components are diminished after a period. Any rotor orbit generated by an accidental disturbance will thus be damped out and the system is stable.”

“When the system rotates supercritically, the rotating dashpots do work on the rotor and add energy to the rotor orbit. This destabilizing action can be seen by returning to Fig. 7.5 and considering the case of a counter-clockwise whirl of radius r at the absolute angular speed nω when nωΩ > . The relative motion now is a backward (i.e., clockwise) whirl with angular speed nωΩ − . The resultant dashpot force acts tangentially to the circular path in a forward (i.e., counter-clockwise) sense with magnitude ( )nr rc ωΩ − . The supercritical rotation of the system thus acts to drag the rotor forward around its orbit through the rotating dashpots.”

“This action may be seen in an even clearer fashion if, instead of the four dashpots in Fig. 7.5, it is imagined that the interior of the ring is filled with a massless viscous fluid which rotates with the ring and acts to retard the relative motion of the mass m with respect to the ring. When the mass has a circular orbit in the same absolute sense as the ring rotation, but at a slower speed, the viscous drag pulls the mass forward and adds energy to its orbit.”

Fig. 7.6 (from [7.2])

The model in Fig. 7.6 (Pippard, 1978) is essentially equivalent. The rotor is modeled by a conical pendulum with a heavy bob suspended in the gravity field by a string of length l . The natural modes are forward and backward conical whirls with the same natural frequency lgn =ω (for small cone angles). Damping is modeled by allowing the pendulum bob to dip into a glass of water. If the glass is on


a variable speed turntable, damping with respect to a rotating frame can be demonstrated. “If the fluid is caused to rotate more slowly than the pendulum bob in its circular orbit, the bob experiences a retarding force and sinks toward the vertical. But if the fluid rotates faster than the bob, it urges the bob onwards and causes its orbit radius to increase”.

“Returning to the planar rotor model with both stationary and rotating damping, a forward whirl at absolute angular speed nω will be retarded by the stationary damping force and will be urged on by the rotating damping force when the rotation is supercritical, nωΩ > . Whether the net effect is to remove or to add energy to the whirl orbit, depends on which force is larger. Neutral stability occurs when the forces are equal. There is then no change in the orbit energy and the free motion persists indefinitely. The quantitative determination of the stability borderline is discussed in the following section, taking account of the frequency dependence of the damping mechanisms.”

7.2.3 Whirl speeds of rotor with rotating damping

For some of the systems considered later in this chapter, is useful to study the variation of whirl speeds as a function of the rotational speed Ω .

7.2.3.1 In rotating coordinates

In the rotating coordinates, the equations of motion for the planar rotor model of Fig. 7.3 (perfectly balanced shaft) with the rotating damping elements of Fig. 7.5 are

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡ −+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡00

00

22

00

2

2

ζη

ΩΩ

ζη

ΩΩ

ζη

mkmk

cmmc

mm

r

r&

&

&&

&&. (7.12)

Note that the rotating damping is represented by both a normal damping term and by a quasi-gyroscopic term.

Trying solutions of the form

tλ

ζη

ζη

e0

0

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

,

we obtain

( )

( ) .mkcmm

,mmkcm

r

r

02

02

022

0

0022

=−+++

=−−++

ζΩλληλΩ

ζλΩηΩλλ (7.13)

The condition to have nontrivial solutions is


( ) ( ) 02 2222 =+−++ λΩΩλλ mmkcm r , or

( ) ( ) 02i 2222 =−−++ λΩΩλλ mmkcm r ,

λΩΩλλ mmkcm r 2i22 ±=−++ , (7.14)

( )[ ] ( )[ ] 02i2i 2222 =−+−+−+++ ΩλΩλΩλΩλ mkmcmmkmcm rr .

Denoting

n

rr m

cω

ζ2

= , mk

n =ω , (7.15)

the characteristic equation becomes

( )[ ] ( )[ ] 02i22i2 222222 =−+−+−+++ ΩωλΩωζλΩωλΩωζλ nnrnnr .

From the first factor it may be shown that, if 1<<rζ , the roots are

( ) ( )( ) ( ),

,

nnr

nnr

ΩωΩωζλ

ΩωΩωζλ

+−+−=

−+−−=

i

i

2

1 (7.16, a)

and from the second factor

( ) ( )( ) ( ),

,

nnr

nnr

ΩωΩωζλ

ΩωΩωζλ

−−−−=

+++−=

i

i

4

3 (7.16, b)

The whirl radius is

( ) ( ) ( )

( ) ( ) ( ) .CC

CC

ttt

ttt

nnnr

nnnr

⎥⎦⎤

⎢⎣⎡ ++

+⎥⎦⎤

⎢⎣⎡ +=

+−++−

−−−−−

ΩωΩωΩωζ

ΩωΩωΩωζρ

i4

i3

i2

i1

eee

eee (7.17)

The amplitudes of the second pair of whirls obviously decay with time, whereas the time variation of the amplitudes of the first pair of whirls depends upon the value of the imposed rotational speed of the shaft Ω relative to the value of the natural frequency of the system when not rotating, nω . When Ω is less than nω the whirls decay and when Ω is greater than nω they grow exponentially with time; the system is unstable.

From the first equation (7.13) the mode shapes are defined by


λΩ

Ωωλωζληζ

22 222

0

0 −++= nnr ,

which for the two pairs of roots yields

i210

0 −=⎟⎟⎠

⎞⎜⎜⎝

⎛

,ηζ , i

430

0 +=⎟⎟⎠

⎞⎜⎜⎝

⎛

,ηζ . (7.18)

7.2.3.2 In stationary coordinates

In the stationary coordinates, the equations for the same damped system are

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡00

00

00

00

00

zy

cc

zy

kk

zy

cc

zy

mm

r

r

r

rΩ

Ω&

&

&&

&&. (7.19)

In (7.19) the rotating damping is also represented by both a normal damping term and a circulatory term defined by a skew-symmetric matrix.

Trying solutions of the form

t

zy

zy λe

0

0

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

, (7.20)

the condition to have nontrivial solutions is

( ) ( ) 0222 =+++ Ωλλ rr ckcm ,

or

( ) ( ) 0i 222 =−++ Ωλλ rr ckcm ,

which can be written

( )( ) 0ii 22 =+++−++ ΩλλΩλλ rrrr ckcmckcm . (7.21)

For mkcr 2<< , the roots are

( ) nnr, ωΩωζλ i21 ±+−= , ( ) nnr, ωΩωζλ i43 ±−−= , (7.22)

with the notations (7.15).

This analysis masks the importance of the direction of Ω . For Ω in the positive direction some of these roots do not apply.

As shown in Section 2.3.2.4

when nωω = , ( ) nnr ωΩωζλ i+−−= , (7.23)


and when nωω −= , ( ) nnr ωΩωζλ i−+−= , (7.24)

therefore the whirl radius is

( ) ( ) tttt nnrnnr BAr ωΩωζωΩωζ ii eeee −+−−−+= , (7.25)

which is exactly the same motion as found previously in the rotating coordinate system.

7.2.3.3 Whirl speeds

In equation (7.17), the first pair of whirls consists of a co-rotating and a counter-rotating whirl of speeds ( )Ωω −± n relative to the moving coordinate system. Relative to the fixed coordinate system, they are forward whirls of speed nω and nωΩ −2 respectively. The second pair consists of co-rotating and counter-rotating whirls respectively, of speeds ( )Ωω +± n relative to the moving coordinate system, and backward whirls of speeds nωΩ +2 and nω− relative to the fixed coordinate system.

a b

Fig. 7.7 (after [7.3])

The whirl speeds, that is the imaginary parts of the eigenvalues, are shown as functions of the rotational speed Ω in Fig. 7.7. They are the natural angular frequencies of precession. The values Rω in Fig. 7.7, a are relative to the rotating coordinate system and the values Sω in Fig. 7.7, b are relative to the stationary coordinate system. Note that Ωωω += RS for positive Ω only.

At first sight this is a surprising result. As we have assumed light damping, the natural frequencies do not depend on the magnitude of damping. For zero


damping we expect the result obtained for undamped rotors with whirl speeds nω± . Figure 7.7, b indicates two extra natural frequencies which are speed dependent. They result from the second factor in equation (7.21). This can be obtained from the first factor, replacing Ω by Ω− , implying that it is associated with the rotor rotating in a reverse direction. In this case Ωωω −= RS .

This confusion is due to the formulation of system equations in real notations. When real notations are used, the information over Ω− is folded into that over Ω+ . Using complex notation instead of real notation elucidates the character of the additional eigensolutions [7.4]. The complex notation can distinguish a rotor rotating in the positive direction from one in the negative direction. The real notation always gives complex conjugate solutions (the equations of motion have real coefficients), not making any difference in the rotating directions.

On the other hand, it is difficult to interpret physically modal vectors with complex vector components such as (7.18). Introducing the complex notation, the complex modal vectors are of the form 00 iζη + , and become zero for the modes associated with 3λ and 4λ . This implies that those modes exist merely as a pure mathematical consequence, but they do not physically exist. In other words, they are never excited by any realistic excitation forces. This can be simply shown calculating the forced response, for example, to a constant force rotating with a constant angular speed with respect to the moving coordinate system [7.3].

7.2.4 Quantitative effects of damping

For the planar model of the rotor, we can define a stationary loss factor ( )ωη s by taking the ratio of the tangential drag force, due to stationary damping, to

the radial spring force when the mass particle executes a circular orbit with radius r at the speed ω . When nωω = , the backward drag force sD due to the stationary damping is

( ) rkD nss ωη= . (7.26) “Similarly, a rotating loss factor ( )ωη r is defined as the ratio of the

tangential drag force, due to rotating damping, to the radial spring force when the mass particle executes a circular orbit with radius r at a speed ω , with respect to the rotating system. In a forward whirl at absolute speed nω , the speed with respect to the system rotating at speed Ω is Ωω −n . Thus, when the system rotates

supercritically, the drag force rD due to the rotating damping is forward and has the magnitude

( ) rkD nrr ωΩη −= (7.27)


for an orbit of radius r. “

“If sr DD = , we have a steady orbit of fixed radius r. If sr DD < , there is a net retardation and the orbit energy decreases. If sr DD > , there is net acceleration and the orbit energy increases. “

“Alternatively, the stability is decided by the relative magnitudes of the rotating and stationary loss factors as indicated in Fig. 7.8. The stability borderline condition occurs when the loss factors are equal

( ) ( )nsnr ωηωΩη =− . (7.28)

The forward whirl is unstable for supercritical rotation speeds Ω at which the loss factor of rotating damping is greater than the loss factor of stationary damping.”

Fig. 7.8 (from [7.2])

The running speeds Ω which satisfy (7.28) depend crucially on the frequency dependence of the rotating damping mechanism and on the magnitude of the stationary damping.

7.3 Whirl in hydrodynamic bearings

A well-known form of rotor instability, also referred to as oil-whirl or fractional frequency whirl, is a self-excited whirl induced by the nonlinear forces generated in the oil wedge of hydrodynamic journal bearings. Oil whip occurs at the frequency corresponding to the lowest critical speed and maintains that frequency while the rotation speed increases. Both oil whirl and oil whip are superimposed on the synchronous precession due to mass unbalance. Oil whirl amplitudes are usually small. Oil whip amplitudes are large and usually exceed bearing clearance. The onset of oil whip should trigger emergency shutdown of the machine.


7.3.1 Oil-whirl and oil-whip phenomena

The following description is adapted from [7.5].

When the shaft starts rotating with a slowly increasing rotating speed, small amplitude synchronous precession (denoted 1X) is observed all along the rotor axis. This is produced by inherent rotor residual unbalance. At low rotating speeds, this precession is stable, an impulse perturbation of the rotor causes a short time transient whirling motion, and the same precession pattern is re-established (Fig. 7.9).

At higher rotation speeds (usually below the first critical speed), the forced synchronous precession is not the only regime of vibration. Along with 1X precession, oil whirl appears.

Oil whirl is the rotor lateral forward subharmonic precession around the bearing center at a frequency close to half the rotation speed. In this range of rotating speeds the rotor behaves as a rigid body. The amplitudes of oil whirl are usually much higher than those of the synchronous precession. They are, however, limited by the bearing clearance and the fluid non-linear forces.

Fig. 7.9 (after [7.5])

With increasing rotation speed, the pattern of precession remains stable. The oil whirl “half” frequency follows the increasing rotating speed, maintaining the

21≈ ratio with it. The magnitude of precession radii remains nearly constant and usually high. At the bearing, the precession radius may cover nearly all bearing clearance. In the considered range of rotating speed, the bearing fluid dynamic effects clearly dominate. The forced synchronous precession represents a small fraction of dynamic response, as the half-spectrum indicates (Fig. 7.10).


When the increasing rotation speed approaches the first critical speed 1cΩ ,

i.e. the first natural frequency of the rotor, oil whirl suddenly becomes unstable and disappears, being suppressed and replaced by increasing synchronous precession. The forced precession dominates, reaching the largest radius at the resonance frequency corresponding to the mass/stiffness/damping properties of the rotor. The bearing fluid dynamic effects now yield priority to the elastic rotor mechanical effects.

Above the first critical speed, the synchronous precession decays, and the bearing-fluid forces come back into action. With increasing rotating speed, shortly after the first critical speed, oil whirl occurs again. The previously described pattern repeats. The width of the rotating speed range in which the synchronous precession magnitude dominates depends directly on the amount of rotor unbalance: the higher the unbalance, the wider this range [7.4].

Fig. 7.10 (from [7.5])

When the rotor speed reaches about twice the first synchronous critical speed 12 cΩ , the half-speed oil whirl frequency reaches the value of the first natural frequency of the rotor. The oil whirl pattern is replaced by oil whip – a subharmonic forward precession motion of the rotor. The rotor continues whirling violently with the frequency and the mode shape corresponding to the first critical speed, even if the speed increases further. The rotor precession “locks” on the frequency of the first synchronous critical speed even if the rotating speed is increased.


In this range of high rotating speed, the shaft cannot be considered rigid. Its flexibility causes a strong coupling of the rotor/bearing system. The rotor mass and stiffness parameters become the dominant dynamic factors. The amplitude of oil whip journal precession is limited by the bearing clearance, but the shaft orbit radius may become very high, as the shaft whirls at its natural frequency, i.e. in resonant conditions. The above described phenomena may take slightly different forms in various machines provided with fluid-film bearings and/or seals.

Since its discovery (B. L. Newkirk and H. D. Taylor, 1925) much progress has been made in explaining oil whirl and oil whip but the phenomenon is still imperfectly understood. “What is clear is that when a rotor is whirling at less than the average velocity of the oil film, the oil drags on the rotor and generates a force opposing the rotor motion. This is positive damping. When the rotor whirl exceeds the average velocity of the oil film, the oil pushes on the rotor. This is negative damping. At the onset of negative damping the rotor becomes unstable and a perturbation will cause the bearing journal to initiate a subsynchronous whirl of ever-increasing amplitude. As the journal motion grows large, the damping becomes positive, the amplitude decreases, and the motion settles into a stable limit cycle. This is oil whirl. This scenario is consistent with oil whirl being a nonlinear phenomenon of the Van der Pol type” [7.6].

“What is less clear is how oil whirl is transformed into oil whip. Apparently, as the lowest critical speed is approached, the limit cycle frequency synchronizes with

1cΩ , and stays locked to 1cΩ , as long as Ω is greater than

approximately 1

2 cΩ . As Ω continues to increase, it is possible for oil whip to be suppressed and for oil whirl to reappear” [7.6].

7.3.2 Half-frequency whirl

Consider a lightly loaded journal bearing, i.e., the pressure developed in the film is insignificant, the journal center operates close to the bearing center and the eccentricity is very small compared to the radial clearance. Since the pressure induced flow is assumed to be negligible, the velocity profile of the film in the clearance space is linear, with a maximum value ΩR at the journal surface, as shown in Fig. 7.11.

The flow into the wedge of the journal bearing is

( )eCRLqi += Ω21 . (7.29)

The flow out of the wedge is

( )eCRLqo −= Ω21 . (7.30)


If pressure is developed in the film, when the bearing is operating under steady conditions, the flow-in is reduced and the flow-out is increased by the pressure induced flow, which balances iq and oq to maintain the flow continuity. However, if the load is small, in the absence of pressure a small whirl velocity is induced to maintain the flow balance.

Fig. 7.11 (after [7.7])

If the instantaneous whirl velocity is ω for the journal center JO , then the induced velocity is ωe as shown in Fig. 7.11. By lifting off the journal from its steady state position, the film volume increases by

ωeRLV 2= , (7.31)

where RL2 is the projected area of the bearing. Therefore

( ) ( ) ωΩΩ eRLeCRLeCRL 221

21

+−=+ (7.32)

and

Ωω21

= . (7.33)

Hence the rotor tries to whirl at a frequency of half the speed of rotation to maintain the flow balance.

If Ωω21

> , the outward flow is more and therefore pressure is developed in

the film and the bearing becomes stable. If however Ωω21

< , the flow-in is more,

the bearing loses its load carrying capacity and continues to whirl, in order to create


more space for the excessive oil coming into the wedge. The rotor thus loses the load carrying capacity and becomes unstable. The frequency of whirl in the rotors under such conditions is observed to be around 0.46 to 0.48 rotation speed.

7.3.3 Onset speed of instability

It can be shown that, for describing the onset of instability, a journal bearing can be represented by a single effective spring coefficient and a single effective damping coefficient. These coefficients can be calculated from the eight dynamic bearing coefficients. However, the two coefficients derived in this manner depend on the precession frequency, such that the effective damping coefficient is negative for small frequencies and becomes positive for higher frequencies. The frequency at which the damping becomes zero is called the instability frequency. This can be found as a function of speed as shown by the corresponding curve in Fig. 7.12. The synchronous precession line crosses the natural frequency curve at the first critical speed (where the rotation speed is equal to the system natural frequency). The half frequency whirl line crosses the curve at the instability threshold known as oil whip.

Fig. 7.12 (from [7.8])

For frequencies less than the instability frequency (i.e. in the region below the curve), the effective damping is negative, and for frequencies greater than the instability frequency the damping is positive. The effective spring coefficient of the bearing together with the flexibility of the rotor determines the resonant frequencies of the rotor-bearing system. Since the bearing stiffness is a function of speed, the resonant frequencies become speed dependent. The lowest of these resonant frequencies is shown by the curve labelled ‘system eigenfrequency’. This curve intersects the curve for the instability frequency at a speed denoted as ‘instability threshold speed’.


Assume that the rotor is subjected to a small disturbance. It will then tend to precess at its lowest natural frequency. However, if the rotor is running below the instability threshold speed, the bearings provide positive damping and the precession motion dies out. As the speed is increased the damping available for the precession diminishes until it becomes equal to zero at the onset speed of instability. Attempting to increase the speed beyond the threshold speed causes the damping to become negative, so that any imposed disturbance is amplified and the system is unstable.

From Fig. 7.12 it is seen that if the rotor mass is increased or the shaft is made more flexible, the system natural frequency is lowered, whereby the intersection between the two frequency curves moves to the left and the threshold speed is reduced. Conversely, if external damping is present in the system (for instance, in the supports) the curve of the film instability frequency is lowered, thereby raising the threshold speed. The latter characteristic suggests means by which an otherwise unstable system may be stabilized and also explains why some machinery, notably liquid pumps and high-pressure gas compressors with liquid seals, may operate stable well above the theoretical threshold speed. The damping capacity of the liquid passing through the impeller or seals acts as external damping to the system and stabilizes the rotor.

7.3.4 Crandall’s explanation of journal bearing instability

In a second remarkable paper of Stephen Crandall [7.9] a simple explanation of the journal-bearing instability is based on a heuristic hypothesis. A fluid-filled journal bearing is viewed as a pump circulating fluid around the annular space between the journal and the bearing. A small whirling motion of the journal generates a wave of thickness variation progressing around the channel. It is hypothesized that the fluid flow drives the whirl whenever the mean of the pumped fluid velocity is greater than the peripheral speed of the thickness-variation wave. For non-cavitating long bearings, the hypothesis predicts the instability onset correctly for unloaded bearings but gradually overpredicts the onset speed as the load is increased. The following presentation reproduces most of the content of Crandall’s paper to make it available to our students.

7.3.4.1 Sommerfeld’s analysis

For simplicity, the discussion is centered on the case of a full circular bearing with two-dimensional non-cavitating flow.

The idealized case treated in 1904 by Sommerfeld [7.10] is sketched in Fig. 7.13. A journal of radius R rotates at constant speed Ω within a full circular bearing of radius CR + , where the radial clearance C is very small in comparison to R. The annular space between journal and bearing is filled with an incompressible fluid lubricant with uniform viscosity μ . The fluid flow is taken to be two-dimensional,


i.e. axial flow is assumed to be negligible. In addition it is assumed that cavitation does not occur. The width of the bearing normal to the plane of the figure is L.

“Sommerfeld’s principal result is that in the equilibrium position under a vertical load W the journal is eccentrically displaced by a distance e in the horizontal direction. “

Fig. 7.13 (from [7.9])

“Because of the eccentricity e, the film thickness h varies with position θ around the annular space. For RC << , the approximate relation is

( )θεθ cos1cos +=+= CeCh , (7.34)

where Ce=ε is the eccentricity ratio.”

“Under the assumptions of laminar viscous flow with no pressure variation across the thickness of the film, the only possible flow patterns that satisfy the requirements of Fluid Mechanics in a uniform channel of thickness h are combinations of the two basic patterns shown in Fig. 7.14. Here z is distance along the channel and y is distance across the channel, with hy <<0 .”

Fig. 7.14 (from [7.9])

“The fluid velocity (in the z-direction at position y) is denoted by w . The volume flow rate across a section of the channel of width L normal to the plane of the figure is denoted by Q and zp ∂∂ is the pressure gradient along the channel. For the linear profile at the left of Fig. 7.14, the pattern depends on the parameter 0w ,


which is the velocity of the upper channel wall (the velocity of the lower channel wall is taken to be zero). It can be shown that

hy

0ww = , 20hLQ w= , 0=

∂∂

zp . (7.35)

For the parabolic profile at the right of Fig. 7.14, the pattern depends on the parameter A which is a velocity whose magnitude is four times the peak velocity in the profile or six times the average velocity.

⎟⎟⎠

⎞⎜⎜⎝

⎛−= 2

2

hy

hyAw ,

6hLAQ = , 2

2h

Azp μ

−=∂∂ . (7.36)

While the relations (7.35) and (7.36) are strictly correct for steady state flows with constant values of the parameters h, 0w , and A, Reynolds’ theory of lubrication extends them to apply to slow variations, both in time t and space z, of these parameters.

In application to the bearing of Fig. 7.13, the component flows of Fig. 7.14 are superposed, with θRz = and ΩR=0w , with h given by (7.34), and the undetermined parameter A to be fixed by the requirements of continuity, 0=∂∂ θQ , and uniqueness of pressure, ( ) ( )πθθ 2+= pp . The value of A so determined is

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−

++

−= 3

cos11

216

2

21 θεε

εΩRA (7.37)

and the total volume flow rate is

2

21

21

εεΩ

+

−= CLRQ . (7.38)

The pressure gradients of the component flows in Fig. 7.14 are superposed and integrated to obtain the pressure distribution ( )θp acting on the journal. The resultant of the pressures acting on width L of the journal is a force, acting vertically upwards through the journal center JO , of magnitude

( )( ) 2122

2

112

12εε

εΩμπ−+

⎟⎠⎞

⎜⎝⎛=

CRRLW . (7.39)

The viscous shearing stresses acting on the boundary of the journal also produce a resultant force on the journal, but its magnitude is smaller than the pressure resultant (7.39) by a factor of order RC , and thus may be neglected. The force (7.39) must then be equal and opposite to the applied load W for the journal to be in equilibrium.”


“The remarkable property of the Sommerfeld bearing model is that the equilibrium displacement is at right angles to the applied load. This characteristic implies that an unloaded bearing is unstable with respect to slow forward whirling motions of the journal.

To see this, imagine that an external agent moves the center of the journal JO of Fig. 7.13 in a circular path of radius e in the counter-clockwise sense about

the bearing center BO . When the journal passes through the position shown in Fig. 7.13, if the motion is slow enough, the lubricant flow and pressure distribution will be very the same as for the equilibrium configuration shown there. This means that the fluid will be exerting a force very nearly equal and opposite to W on the journal. This force, in the same direction as the velocity of the journal center JO in its circular path, does positive work on the whirling motion. The Sommerfeld bearing thus promotes whirling instability for very slow forward whirling speeds. To consider more rapid whirling speeds it is necessary to extend the Sommerfeld analysis to include motion of the journal center JO .”

7.3.4.2 Whirling stability of unloaded bearing

“According to (7.39), the equilibrium position for an unloaded bearing ( )0=W has zero eccentricity ( )00 == e,ε . In this configuration, the parameter 1A of equation (7.37) vanishes and the lubricant flow pattern is simply the linear profile shown on the left of Fig. 7.14.”

Fig. 7.15 (from [7.9])

“The bearing kinematics are displayed in Fig. 7.15 for the case where the journal center JO whirls at the steady angular speed ω in a circle of radius 0r about the equilibrium position in which JO and BO coincide. Note that the diametrally


opposite sections of maximum and minimum film thickness advance around the bearing at the speed ω . At the location θ , the film thickness is

( ) ( )trCt,h ωθθ −+= cos0 . (7.40)

Note that the dependence of h on space, θ , and time, t, is that of a progressive wave circling the annular channel with an angular phase speed ω or a linear phase velocity ωR .

The explanation of Newkirk and Taylor [7.11] is based on an application of the continuity requirement to the flow in a channel whose thickness varies according to (7.40). The continuity requirement applied to a differential arc of length θdR is

0=∂∂

+∂∂

thL

RQθ

. (7.41)

Newkirk and Taylor assumed that the lubricant flow retains the linear profile of Fig. 7.14 so that 2hLRQ Ω= which reduces (7.41) to

02

=∂∂

+∂∂

thh

θΩ . (7.42)

When ( )t,h θ from (7.40) is substituted in (7.42), the result is that continuity cannot be satisfied unless

2Ωω = . (7.43)

Newkirk and Taylor took this to provide analytical verification of the half-frequency whirl phenomenon which they observed in a vertical shaft running in a bearing with plentiful oil supply. It also appeared to explain the “oil resonance” peak in response when the rotation speed was twice the natural frequency of the system. The simple result (7.43) was less satisfactory in explaining the oil-whip phenomenon where the whirling frequency remains at the natural frequency as the rotation speed Ω is increased, although it was noted that the onset of whipping always occurred at speeds equal to or greater than twice the natural frequency.”

In Crandall’s heuristic hypothesis [7.2] the rotating journal is considered to be a pump impeller which maintains the lubricant flow pattern with linear profile when the journal is unloaded and centered. The fluid velocity varies linearly from

0=w at the bearing to ΩR=w at the journal. This flow can be decomposed into a mean flow with uniform velocity 2ΩR=w and no vorticity plus a residual flow with zero mean velocity and large vorticity. It is assumed that the mean flow is available to encourage (or discourage) small whirling perturbations of the centered configuration. If a whirl involving a thickness variation wave of the form (7.40) is imposed, Crandall’s hypothesis postulates that energy will be pumped into the whirl


if the mean flow velocity 2ΩR is greater than the phase velocity ωR of the whirl around the periphery of oil film. Conversely, energy will be removed from the whirl if the phase velocity of the whirl is greater than the mean velocity of the lubricant. Neutral stability occurs when these velocities are equal, i.e. when 2Ωω = . This hypothesis thus “explains” half-frequency whirls and oil whip for a system which has an unloaded bearing with plentiful oil supply.

If the system provides little restraint on the journal, its motion will be primarily determined by the fluid film forces acting on it. When a low frequency whirl ( )2Ωω < is accidentally started, energy will be pumped into the whirl, accelerating the whirl until ω gets sufficiently close to 2Ω that the energy pumped into the whirl just balances the system energy losses in a steady “half-frequency” whirl.

If the system provides considerable restraint on the journal and only permits appreciable whirling motion at a natural frequency, the fluid film in the bearing will remove energy from accidental whirls at a natural frequency ω whenever ωΩ 2< . However, when ωΩ 2> the fluid film will pump energy into any such whirl. Whether or not oil whip occurs depends on whether the energy supplied by the fluid film is sufficient to overcome the system losses.

Both of the previous explanations are incomplete in the sense that they do not make use of all the physical requirements involved. Both arguments use the flow pattern of the underlying steady centered flow and both use the kinematics of a small whirling perturbation. The Newkirk and Taylor argument makes explicit use of the continuity requirement for the perturbation but neither explanation explicitly invokes any consideration of the pressure developed in the oil film.

Historically, the first complete dynamic analysis of the whirling stability of an unloaded bearing was given in 1933 by Robertson [7.12]. The development which follows is essentially just a linearized version of Robertson’s analysis.

We consider the whirl defined by the eccentricity 0r ( )Cr <<0 and the frequency ω to be a small perturbation of the underlying centered rotation in which the flow distribution is simply

CzRΩ=w , Cz <<0 . (7.44)

When the journal is whirling, the flow pattern will be a superposition of the two profiles of Fig. 7.14. To fit the conditions of Fig. 7.15 we take

⎟⎟⎠

⎞⎜⎜⎝

⎛−+= 2

2

hy

hyA

hyRΩw , (7.45)

where ( )t,h θ is given by (7.40) and A is to be determined from the continuity requirement (7.41) and the requirement of single-valued pressure, ( ) ( )πθθ 2+= pp .


Using a linear perturbation analysis, we neglect terms of order ( )20 Cr in

comparison with terms of order Cr0 and find

( )tCr

RA ωθωΩ−⎟

⎠⎞

⎜⎝⎛ −−= cos

26 0 . (7.46)

The corresponding total volume flow rate for the lubricant film is

( )tLRrLhRQ ωθωΩΩ −⎟⎠⎞

⎜⎝⎛ −−= cos

22 0 ,

( )tLRrLCRQ ωθωΩ −−= cos2 0 . (7.47)

The pressure distribution is obtained by integrating the pressure gradient given in (7.36). To first order in Cr0

( ) ( )tC

rRpt,p ωθωΩμθ −⎟

⎠⎞

⎜⎝⎛ −=− sin

212 3

020 . (7.48)

The resultant force acting on the journal due to these pressures is directed at right angles to the journal displacement 0r and has the magnitude

⎟⎠⎞

⎜⎝⎛ −= ωΩμπ

212 3

3

0 CRLrF . (7.49)

The sense is such that when ωΩ >2 , F has the same direction as the instantaneous velocity JV of the journal center. The rate at which the fluid film forces do work on the journal (i.e., the power flow into the whirl) is

⎟⎠⎞

⎜⎝⎛ −= ωΩωμπ

212 2

03

3r

CRLVF J . (7.50)

This analysis shows that the amplitude A of the component flow with parabolic profile, the pressure in the fluid film, the resultant force on the journal, and the power flow into the whirl all are proportional to the factor ( )ωΩ −2 . Low speed whirls are encouraged and high speed whirls are discouraged. The whirling frequency of neutral stability is 2Ωω = .

These results can be compared with the two simplified arguments considered previously. The assumption in the Newkirk and Taylor argument that the velocity profile remains linear is equivalent to assuming that the parameter A in (7.36) vanishes, which according to (7.46) implies that ω must equal 2Ω . Furthermore the vanishing of A implies an absence of pressure gradient and consequently an


absence of resulting force so that at the particular whirl frequency 2Ωω = the fluid film neither retards nor advances the whirl. This is the neutral stability condition.

Crandall’s hypothesis that the mean flow drives the whirl whenever the mean fluid velocity is greater than the phase velocity of the whirl is essentially a qualitative statement equivalent to the quantitative statement represented by equation (7.50). It happens that for an unloaded bearing the frequency of neutral whirl is given correctly by the heuristic hypothesis. For loaded bearings it is difficult to see how the Newkirk and Taylor argument can be extended to predict any other frequency of neutral whirl than 2Ωω = . Crandall’s hypothesis was however easily extended. It no longer predicts the exact frequency of neutral whirl but it provides useful approximations for moderate loads.

A linear perturbation analysis [7.9] shows that the neutral stability whirl frequency

( )( )

Ωε

εω 22

22

19

13

−−

−+= (7.51)

varies from 2Ωω = at 0=ε for an unloaded bearing, to 3Ωω → when the load approaches infinity and the eccentricity ratio ε approaches unity. The variation of

Ωω according to (7.51) is represented by the curve labeled A in Fig. 7.16.

Fig. 7.16 (from [7.9])

A computation based on Crandall’s hypothesis [7.9] yields the curve B in Fig. 7.16. It provides a useful approximation of the whirl frequency at the neutral stability for values of the steady state eccentricity ratio that are less than 50.=ε .


7.3.5 Stability of linear systems

For equations of motion with real coefficients, the characteristic equation

( ) 012

21

10 =+++++= −−−

nnnnn aa......aaaf λλλλλ , (7.52)

has n roots of the general form iii ωαλ i+= . Repeated roots are not considered here.

If any 0>iα , the system will be unstable. The types of motion associated with the position of a root in the complex plane (Fig. 7.17) are summarized in Table 7.1.

Fig. 7.17 (after [7.3])

Table 7.1

Position of root in Fig. 7.17

Type of motion

1 stable; damped oscillatory

2 stable; damped aperiodic

3 oscillatory

4 unstable; divergent (distance increases

exponentially with time)

5 unstable; oscillatory with increasing amplitude,

referred to as dynamically unstable

It is often of interest to decide whether or not a system is stable without having to solve its characteristic equation. In 1877, E. J. Routh established an


algorithm for determining the number of roots of a real polynomial which lie in the right half-plane. Independently, in 1895, A. Hurwitz gave a second solution. The determinant inequalities obtained by him are known as the Routh-Hurwitz stability conditions.

Consider the matrix with real elements

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−−

−−−−

−−

−−− nnnn aaaa

aa|aaa|

a|a|aa||||a|a

322212

02345

0123

01

0

00

0000

LLLL

LLLL

LL

LL

LL

. (7.53)

Construct the following n determinants indicated by broken lines

11 aD = , (7.54)

032123

012 aaaa

aaaa

D −== , (7.55)

( ) ( )0321305411

345

123

01

3

0aaaaaaaaaa

aaaaaa

aaD −+−−== , (7.56)

nnn

n

a..aa

.aaaa

.aaaa

.aa

D

L

MM

L

L

L

2212

2345

0123

01 00

−−

= . (7.57)

If nr > or 0<r then 0=ra . A necessary and sufficient condition for all roots of ( ) 0=λf to lie in the left half-plane is that all partial determinants be positive 0>iD . If 0>na , it is only necessary to test the determinants from 1D to

1−nD as 1−= nnn DaD .


The coefficients ra may be expressed in terms of the roots iλ of equation (7.52), for example

∑=

−=n

iiaa

101 λ , ∑∑

= =

=n

ij

n

jiaa

1 102 λλ ( )ji ≠

and so on. By considering the forms of these expressions it is easy to show that all ia have the same sign as 0a if all the roots have negative real parts. The only way

in which any of the coefficients may be zero or take the opposite sign to 0a is for one or more of the roots to have positive real parts.

Hence, a necessary, but not sufficient, condition for all roots of ( ) 0=λf to lie in the left half-plane is that all ra ( )n,...,,r 21= be positive.

7.3.6 Instability of a simple rigid rotor

Consider a mass 2m attached at the middle of a rigid shaft running in two identical fluid film bearings. Due to symmetry, the bearings share the rotor load equally, having the same operating eccentricity and dynamic coefficients. Only the translatory whirl is of interest.

7.3.6.1 Threshold speed and unstable whirl frequency

For constant load and rotation speed, the journal center maintains a steady state equilibrium position in the bearing clearance, defined uniquely by the Sommerfeld number (6.3)

2

2⎟⎠⎞

⎜⎝⎛=

CR

WLDS μ

πΩ

(7.58)

which represents the operating conditions. The equation of motion with respect to the equilibrium state is written, considering each journal having a mass m, as

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

z

y

zzzy

yzyy

zzzy

yzyy

ff

zy

kkkk

zy

cccc

zy

mm

&

&

&&

&&

00

. (7.59)

Substituting solutions of the form

tyy λe0= , tzz λe0= , (7.60)

into the homogeneous part of equation (7.59) we obtain


⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++++++

00

0

02

2

zy

kcmkckckcm

zzzzzyzy

yzyzyyyy

λλλλλλ

. (7.61)

For nontrivial values of 0y and 0z , we obtain the characteristic equation

( ) ( ) ( ) ( ) 022 =++−++++ yzyzzyzyzzzzyyyy kckckcmkcm λλλλλλ . (7.62)

This is a fourth order algebraic equation of the form

0432

23

14

0 =++++ aaaaa λλλλ , (7.63) where

20 ma = ,

( )zzyy ccma +=1 ,

( ) ( )zyyzzzyyzzyy cccckkma −++=2 , (7.64)

( )yzzyzyyzyyzzzzyy kckckckca +−+=3 ,

zyyzzzyy kkkka −=4 .

The Routh-Hurwitz conditions of stable precession are

01 >a , 02 >a , 03 >a , 04 >a , (7.65)

03021 >− aaaa , (7.66)

0421

230321 >−− aaaaaaa . (7.67)

At the onset of instability, the root of equation (7.63) is purely imaginary

ωλ i= , (7.68)

where ω is the whirl frequency.

Substituting (7.68) into equation (7.63) yields

( ) 0i 33

142

24

0 =+−++− ωωωω aaaaa . (7.69) Equating the real and imaginary parts to zero, we find

042

24

0 =+− aaa ωω , (7.70)

( ) 0213 =− ωω aa . (7.71)

From equation (7.71) we obtain the whirl frequency at the stability limit


th

aa

ΩΩ

ω=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

1

32 . (7.72)

Substituting (7.72) into equation (7.70) yields

0421

230321 =−− aaaaaaa . (7.73)

This means that at the rotation speed thΩΩ = the stability boundary is encountered and the unstable whirl frequency is given by equation (7.72).

Usually, the threshold speed and the whirl frequency are expressed in terms of the eight dimensionless bearing coefficients

,cW

CC,kWCK jijijiji

Ω== z,yj,i = (7.74)

where C is the radial clearance, gmW = is the static load on the bearing and Ω is the rotational angular speed.

Denoting

zzyy CCA +=1 , zzyy KKA +=2 , zyyzzzyy CCCCA −=3 , (7.75)

( )yzzyzyyzyyzzzzyy KCKCKCKCA +−+=4 , zyyzzzyy KKKKA −=5 ,

and substituting (7.74) and (7.75) into (7.64), we obtain 2

0 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

gWa , 1

2

1 ACg

WaΩ

= , 3

3

2

2

2 ACWA

CgWa ⎟

⎠⎞

⎜⎝⎛+=

Ω,

42

2

3 ACWaΩ

= , 5

2

4 ACWa ⎟

⎠⎞

⎜⎝⎛= . (7.76)

Substituting now (7.76) into equations (7.73) and (7.72) we obtain the dimensionless onset speed of instability

4215

21

24

4312

AAAAAA

AAACgth

−+=

Ω (7.77)

and the dimensionless unstable whirl frequency

th

AA

Cgth

ΩΩ

ω

=⎟⎟⎠

⎞⎜⎜⎝

⎛=

1

42

(7.78)


In practical calculations, the Sommerfeld number is computed at the operating speed. If the threshold speed is higher than the operating speed, the system is stable; otherwise, it is unstable.

7.3.6.2 Critical rotor mass

The threshold of instability can be defined by means of general design charts in which both the rotor and the bearing parameters are included.

If the rotor is assumed to be rigid, then the rotor mass is the only rotor parameter. In the case of laminar incompressible fluid-film bearings, the dimensionless unstable whirl frequency is only a function of the Sommerfeld number S (and also of the eccentricity ratio ε ). Hence it is possible at any Sommerfeld number (or eccentricity ratio) to calculate the journal mass required for the whirl motion to exist.

In dimensionless form, the journal mass for a rigid rotor may be expressed as WmC 2ω . Equation (7.78) can be written

( )

zzyy

yzzyzyyzyyzzzzyy

CCKCKCKCKC

KWmC

+

+−+==

2ω , (7.79)

where K is the dimensionless rotor mass per bearing.

Fig. 7.18 (from [7.8])


For a rotor operating in incompressible plain cylindrical bearings, a curve giving the relationship between K and ε (Fig. 7.18) defines the threshold of instability. This curve can be used to determine the onset speed of instability once it has been found how the eccentricity ratio ε depends on the rotor speed.

Equation (7.77) can be written (dropping the index th)

225

3

25

3

421521

24

4312

KKAA

AK

AKA

K

A

AAAAAA

AAAWmC

+−=

−+=

−+=

Ω,

225

3

KKAA

AKW

mC+−

=Ω

or using (7.58)

225

322

KKAA

AK

CRLD

WmCS

+−=

⎟⎠⎞

⎜⎝⎛μ

π . (7.80)

The dimensionless critical rotor mass can be expressed as a function of the Sommerfeld number [7.8] as

225

32 2

1KKAA

AKS

CRLD

WmC+−

=

⎟⎠⎞

⎜⎝⎛ π

μ

. (7.81)

Figure 7.19 shows a stability design chart for a plain journal bearing [7.8]. The dimensionless critical mass ( )2CRLDmWC μ is plotted as a function of the Sommerfeld number S for two values of the ratio DL . A rotor is stable for operating conditions at the left of the curves.

Fig. 7.19 (from [7.8])


Similar design charts are shown in Figs. 7.20 giving the dimensionless critical rotor mass at onset of instability as a function of S. Figure 7.20, a applies to elliptical bearings operating with an incompressible lubricant in a laminar regime, while Fig. 7.20, b applies to partial arc, centrally loaded bearings.

a b

Fig. 7.20 (from [7.8])

The procedure for determining the threshold speed is the following. Calculate the dimensionless rotor mass ( )2CRLDWCm μ . Enter the appropriate design chart and read off the corresponding value of S. The speed corresponding to this Sommerfeld number is the threshold speed, i.e. the rotor speed at the onset of instability. If the operating speed is higher than the threshold speed the design must be revised.

The charts assume the rotor and the pedestals to be rigid. However, the threshold speed with a flexible rotor and flexible pedestals can be calculated on the basis of the given design charts.

7.3.7 Instability of a simple flexible rotor

Consider a symmetric rotor consisting of a shaft with a stiffness k2 and a lumped central mass m2 . The shaft is supported at the ends in fluid film bearings. The angular speed of rotation is Ω . External damping is provided at the rotor mass and at the bearings, with damping coefficients ec2 and sc , respectively. Although practical rotors are considerably more complex, this simplified model contains the essential features of the problem and admits closed form solutions [7.13].


7.3.7.1 Rotor with external damping

For small amplitudes, with y, z, the coordinates of the rotor mass and 11 z,y the coordinates of the journals, the equations of the free motion can be written

as

.zy

cc

zy

kkkk

zy

cccc

zy

kk

zy

kk

zy

cc

zy

mm

s

s

zzzy

yzyy

zzzy

yzyy

e

e

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−=

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−=

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡

1

1

1

1

1

1

1

1

00

00

00

00

00

&

&

&

&

&

&

&&

&&

(7.82)

The eight bearing coefficients depend on the particular bearing geometry and on the operating conditions, expressed through the Sommerfeld number S.

Considering the plain cylindrical journal bearing as a representative example, and assuming it to be short ( )21<DL and operating at sufficiently high speed such that

122

>⎟⎠⎞

⎜⎝⎛ S

DL , (7.83)

then the dynamic bearing coefficients are approximately given by (6.30)

CWkk yyzz π

82 ≅≅ , Ωπ C

Wcc zyyz8

≅≅ ,

2322 22 ⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛≅≅

DL

CRL

CWS

DLcc zzyy μπ

Ωπ , (7.84)

yyzyyz cCWS

DLkk

2

22 Ωπ =⎟

⎠⎞

⎜⎝⎛≅−≅ .

On the threshold of instability, the motion is purely harmonic with frequency ω , whereby equations (7.82) can be written as

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−++

+−+

00

iiii

1

1

zy

XckckckXck

zzzzzyzy

yzyzyyyy

ωωωω

, (7.85)

where


( )s

e

e ccmkcmkX ω

ωωωω i

ii

2

2−

+−+−

−= , (7.86)

( ) ( )( ) ( ) ( ) ( ) ⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡+

+−−

+−

−−−= s

e

e

e

e ccmk

ck

cmk

cmmkkX222

2

222

222i

ωωω

ωω

ωωω .

This value should be equal to the root of the determinant of equation (7.85) which, by means of (7.84) becomes

( )( ) ( )

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−−

+±++= 2

22

41

1i21

yz

zzyyyzyyzzyy k

kkcckkX

ωω . (7.87)

As ω is equal to Ω21 or less, the second term inside the square root is of

the order of ( ) 241 and can be ignored. Thus, with yyyz ck2Ω

= from (7.84), X is

approximately equal to

⎟⎠⎞

⎜⎝⎛ ±+=

2i ΩωyycKX , (7.88)

where the average bearing stiffness

( )CWkkK zzyy π

621

=+= . (7.89)

Equating the real and imaginary parts of equation (7.86) and (7.88) and disregarding the term ( )2

ecω , the whirl frequency is determined as

( )KkmKk+

== 0ωω , (7.90)

which is the natural frequency of the system when the shaft is supported in bearings with stiffness K.

The condition for stability is

yyes cck

kKc ⎟⎠⎞

⎜⎝⎛ −>⎟

⎠⎞

⎜⎝⎛ +

+ 00

2

0 21 ωΩωω ,


( )23

00

2

0 2 ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛−>⎟

⎠⎞

⎜⎝⎛ +

+DL

CRLc

kkKc es μπωΩωω . (7.91)

The whirl motion is a forward precession.

In the absence of external damping, the relation (7.91) shows that the rotor is unstable when the speed exceeds twice the first critical speed. This is true only for sufficiently large values of the Sommerfeld number, just as the inequality altogether is based on several assumptions. Hence, (7.91) primarily serves the purpose of estimating the magnitude of external damping required for stabilization.

It should be added that, instead of providing external damping which acts directly on the journal, a much more effective method of stabilizing hydrodynamic bearing whirl is to mount the bearing in a flexible, damped support. By proper tuning it is possible to eliminate the instability completely [7.13].

Fig. 7.21 (after [7.14])


7.3.7.2 Rotor without external damping

If the external damping in the simple flexible rotor is neglected, introducing 0=ec and 0=sc in (7.86), then substituting X in (7.85) and not limiting the

analysis to plain short bearings, we obtain

4215

21

24

431

4

1

213

2

211 AAAAAA

AAA

AAC

S

AACg

st

th

−++=

δπ

Ω (7.92)

where kgm

st =δ is the static deflection of the shaft.

Figure 7.21 shows stability charts computed based on equation (7.92) for a circular and an elliptical bearing, with 80.DL = and 31012 −⋅= .RC , the latter having a horizontal radial clearance three times the vertical radial clearance.

Practical measures taken to enhance stability above thΩ are, amongst others, increasing bearing clearance, decreasing lubricant viscosity, shortening bearing length or, equivalently, making a central circumferential oil groove, increasing shaft diameter or shortening shaft length/bearing span, increasing bearing damping and bearing stiffness anisotropy, adding squeeze film dampers and changing to multi lobe or tilting pad bearings.

Fig. 7.22 (from [7.15])


7.3.8 Instability of complex flexible rotors

Complex flexible rotors can be modeled by finite elements. The equations of motion contain the system matrices which are assembled from the element matrices, as shown in Chapter 5.

The study of the free precession leads to an eigenproblem with speed dependent eigenvalues. The Campbell diagram (Fig, 7.22, a) displays the natural frequencies of precession as a function of the rotational speed. The stability diagram (Fig. 7.22, b) shows the real part of the eigenvalues versus the rotational speed. The ‘onset speed of instability” is determined at the intersection of such a curve with the speed axis. The problem is fully illustrated in Chapter 4.

7.4 Interaction with fluid flow forces

Any whirl motion of the shaft in rotating machinery (turbines, compressors, pumps) affects the flow field of the working fluid, thereby setting up differential forces and moments. Thus, a coupling exists between the motion of the shaft and the aerodynamic/hydrodynamic reaction forces which could be destabilizing.

One mechanism for such a coupling, postulated by Thomas [7.16] and Alford [7.17], applies to axial flow machines where a radial displacement of the wheel center in a stage generates a transverse force, proportional to the displacement. It is known as “steam whirl” or “blade-tip-clearance” effect.

a b c

Fig. 7.23 (after [7.14])

7.4.1 Steam whirl

In the stage of an axial flow machine (Fig. 7.23, a), a radial deflection (downwards) will close the radial clearance on one side (bottom) and open the


clearance on the opposite side (top). The closer clearance zone is expected to operate more efficiently and to be more highly loaded than the open clearance zone.

There will be an increased horizontal force 1F at the bottom and an opposite decreased horizontal force 3F at the top, with vertical forces 2F and 4F of intermediate magnitudes (Fig. 7.23, b). A resultant net tangential force F is generated (Fig. 7.23, c) which, if larger than the external damping force, can induce a whirl in the direction of rotor rotation (i.e., forward whirl).

The transverse force is taken proportional to the radial displacement of the wheel center. The coefficient of proportionality is of the form [7.13]

Hr

T

Hh 2d

d00

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

−=ηη

χ , (7.93)

where η is the stage efficiency, T is the stage torque, r is the pitch (mean blade) radius, H is the vane height, h is the tip clearance, and subscript ‘o’ refers to the concentric condition.

Consider a symmetric rotor consisting of a shaft with a stiffness k2 and a lumped central mass m2 . The shaft is supported at the ends in bearings with stiffness Bk such that the undamped natural frequency of the system is obtained from (7.90) substituting BkK = . The angular speed of rotation is Ω . External damping is provided at the rotor mass and at the bearings, with damping coefficients

ec2 and Bc , respectively.

Assuming the central rotor mass to be a turbine stage, with y, z, the coordinates of the rotor mass and 11 z,y the coordinates of the shaft at the supports, the equations of the free motion are

.zy

kk

zy

cc

zy

kk

zy

kk

zy

zy

cc

zy

mm

B

B

B

B

e

e

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−=

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−=

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡−

+⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡

1

1

1

1

1

1

00

00

00

00

00

00

00

&

&

&

&

&&

&&

χχ

(7.94)

Neglecting second order terms in χ , ec and Bc , the characteristic equation becomes


0i2

2 =±+

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

++ χλλB

BB

Be kk

kkckk

kcm . (7.95)

Setting λ equal to ωi at the threshold of instability, the whirl frequency is found to be equal to (7.90)

( )B

Bkkm

kk+

== 0ωω ,

while the stability condition becomes

χωω >⎟⎟⎠

⎞⎜⎜⎝

⎛+

+ BB

e ckk

kc 0

2

0 . (7.96)

At the onset of instability, the whirl motion is forward precession when χ is positive, and backward precession when χ is negative.

If the aerodynamic load torque T of the single stage increases with speed, then equation (7.96) shows that there may be an onset speed of instability above which the inequality is no longer satisfied. This threshold speed is load dependent, so that the instability starts at a certain level of power. It can be raised by either stiffening the shaft or increasing the damping. It is known that increasing the bearing support stiffness usually reduces the effective damping coefficient and often has a very small effect on the critical speed. Thus, in practice, the parameter most feasible to modify for improved stability is the bearing support damping.

Indications are that the proposed form of rotor-fluid flow interaction is in qualitative agreement with practical experience, but the particular expression for χ , equation (7.93), only covers one special mechanism for the interaction. Although the equation seems to yield values of the right order of magnitude, there is evidence to suggest that, in addition to torque, such parameters as flow and pressure level may be of equal importance [7.13].

7.4.2 Impeller-diffuser interaction

Hydraulic forces acting on the rotor of a centrifugal pump arise in two different ways. First, when the shaft center is fixed, radial forces occur as a result of distribution of static pressure and fluid momenta. These forces are termed steady excitation forces. Second, when the shaft center also moves as a result of precession, additional forces are generated due to the interaction of impeller and surrounding fluid. These are dynamic forces and are described by dynamic coefficients.


As the rotor moves eccentrically within the pump, significant forces are developed a) in the annular seals; b) between the impeller shroud and the stator; c) between the impeller back disc and the stator; and d) between impeller and diffuser. When a shaft is displaced laterally in a ring seal, a strong restoring force is set up in the clearance space. This hydrostatic force increases with the pressure drop and the amplitude of shaft lateral motion. In effect, the clearance acts as a spring. The phenomenon is known as the Lomakin effect (Section 6.10.1).

For a whirling shaft, a destabilizing tangential force appears normal to the shaft radial displacement, yielding a cross-coupled stiffness. This is approximately proportional to the average tangential velocity of the fluid. Wide variations occur in the inlet swirl velocity, depending upon where annular seals are located in the pump. Swirl brakes are commonly used upstream of seals to sharply reduce this velocity. A deliberately roughened stator also reduces the average tangential velocity.

Recent research revealed that impeller shrouds develop significant destabilizing forces. Conservation of momentum considerations for leakage flow that proceeds radially inward, along the impeller shroud, can generate high tangential velocities for the flow entering the neck ring and balance piston seals. Enlarged clearances of the exit seal, due to wear or damage, increase the tangential velocity over the shroud and into the seal. Lateral forces on pump impellers can also have a significant dynamic effect.

Forces are exerted on the impeller as a result of the asymmetry of flow caused by the volute or diffuser and the motion of the impeller. Supersynchronous forces are generated in volute pumps at the vane passing frequency. In turn, the diffuser pumps can generate greater forces in the subsynchronous range, depending on the casing tip clearance. Subsynchronous forces with frequency components from 30% to 80% of running speed can be attributed to partial flows or associated with destabilizing forces. The impeller/casing interaction can be described with dynamic force coefficients.

Self-excited vibrations are common in centrifugal compressors, apparently with greater frequency in back-to-back designs and in high density and high-speed industrial units. Actually, a large percentage of high speed compressors will sustain some low-level subharmonic vibrations even during normal operation. Some of these units develop instability problems. The vibration frequency observed is normally the fundamental bending natural frequency of the rotor.

Self-excited vibrations due to aerodynamic forces are twofold: a) vibrations where the destabilizing forces are function of the whirling motion of the rotor; they are produced by labyrinth seals and asymmetric blade loading of the impeller; and b) vibrations where the main flow is destabilized by hydrodynamic or viscous forces which are independent of the resulting mechanical vibration, such as those produced by rotating stall and/or surge.


Aerodynamic forces arising from the interaction of the impeller with the driven fluid introduce both direct and cross-coupled stiffnesses. In a radial flow machine, the change in force due to the clearance variations is negligible compared to that generated by the change in fluid momentum in an eccentric impeller, since the former is perpendicular to the main flow path.

Forced vibrations have also been recorded in all high-density compressors. They have the following typical behaviors: a) they appear relatively near to surge and are very stable in amplitude; b) asynchronous frequency is very low (10 percent of rpm); and c) asynchronous amplitude depends on the tip speed and gas density.

7.5 Dry friction backward whirl

Dry friction backward whirl is a self-excited vibration in systems with rotor-to-stator contact. The rotor is in continuous contact with the stator, slipping continuously on the contact surface and whirling backward at a non-synchronous frequency which depends on damping and the ratio between rotor and stator stiffnesses.

7.5.1 Rotor-stator rubbing

If the rotor contacts the stator, a wide variety of different motion types are possible, from synchronous forward whirl, sub- and super-harmonic precession, to backward whirl and chaotic motion. The contact problem is highly non-linear even if the rotor and the stator are linear.

The phenomenon of backward whirl was recognized the first time by Newkirk [7.18] investigating radial shaft rubbing. In an experiment, the rubbing became severe and due to the large friction force the rotor started rolling violently in the inner surface of a rigid stator ring. The rolling did not take place until the shaft rubbed very hard on the stator ring.

Despite the strong non-linearity due to contact, rotor unbalance can cause purely synchronous precession. However, in some circumstances, the synchronous motion may become unstable and the rotor motion may turn into a non-synchronous precession which can be very distructive. The backward whirl may emerge even in speed ranges where the synchronous forward precession is stable.

A rub occurrence in a 600 MW turbo generator which was totally destroyed during backward whirl is described in [7.19]. Damages resulting from rotor-to-stator contact are reported by insurance companies to be the most frequently failures in steam turbines and to cover up to 22% of all indemnification payments in the field.


Dry friction whirl is mainly due to inadequate or improper lubrication in close clearance machinery. Den Hartog gave a simplified description of the mechanism causing this effect in his well-known book [7.21]. The same explanation was presented in [7.22]. An analytical solution taking into account rotor unbalance, slip and friction between rotor and stator, the mass, damping and stiffness properties of the rotor is presented in [7.19].

Fig. 7.24 (from [7.20])

A recent investigation supported by experimental work is reported in [7.20]. Figure 7.24 illustrates the displacement of a rotor without slip at the inner surface of its housing, at several consecutive times. It can be seen that while the rotor rotation is only 60 degrees, the whirl motion completes one whole cycle.

7.5.2 Dry friction whirl

Basically, the dry whirl is the result of an unequalized Coulomb friction force appearing at a point on the periphery of a shaft or rotor when it is deflected for some reason against the side of a clearance annulus (Fig. 7.25). Since the friction force, fF , is approximately proportional to the radial component, N, of the contact force, we have the preconditions for instability.

The friction force fF can be resolved into a couple CFf and a parallel force of equal magnitude through the (deflected) shaft center. The couple acts merely as a brake on the shaft, which is supposed to be driven at uniform speed Ω , requiring some increase in the driving torque and is inconsequential. The force fF through the center of the shaft, however, drives it in a direction tangent to the bearing inner surface. The direction of fF changes with the position of the shaft in bearing or guide, so that the shaft will be driven around, as indicated by the dotted


circle. It will be noticed that the shaft is driven around the clearance in a direction opposite to that of its own rotation (counterwhirl). As the point of contact moves around the periphery of the housing or guide, so does the force remain tangential to the precession circle, thus perpetuating the whirl.

Fig. 7.25 Fig. 7.26

This backward precession can arise if the contact friction force is large enough to prevent slipping between the rotor and stator. A no-slip condition at the point of contact requires the rotor to precess at a speed RΩ related to the shaft rotational speed through the kinematic requirement (Fig. 7.26)

ΩΩCr

R −= , (7.97, a)

where r is the radius of the contacting shaft and C is the radial clearance. Since annular clearances even at non-seal locations are very small compared to the radius of the rotor, this results in extremely high frequencies of the backward precession which cannot be reached in reality, with slipping motion and flexible contact surfaces. However, still very high super-synchronous frequencies and high forces can occur which can cause serious damage. Note that backward precession induces alternating stresses in the shaft which can lead to fatigue failures.

If the rotor is running with slip, a different expression can be established for the whirl speed. The normal force is equal to the difference between the centrifugal force and the elastic restoring force CkCmN R −= 2Ω . In the tangential direction, the viscous damping force is Red CcF Ω= and the friction force is NFf μ= , where μ is the coefficient of friction.

To maintain continuous contact, the resulting radial force must be directed outward. The condition CkCm R >2Ω yields

mk

R >Ω ,


i.e. the absolute value of the whirl frequency must be higher than the rotor undamped natural frequency.

In order to sustain the whirl, the resulting tangential force must be in the direction of the friction force, 0≥+ df FF ,

( ) 02 ≥+− ReR CcCkCm ΩΩμ .

This condition leads to the following quadratic equation in the whirl frequency

02 ≥−+mk

mc

Re

R Ωμ

Ω

with the acceptable solution

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+−−= 2

21

μζ

μζΩ ee

R mk , (7.97, b)

where mk

cee 2=ζ .

This simple explanation is based on a single-disk symmetrical rotor model in which the plane of rubbing coincides with the plane containing the center of the lumped mass. In the general case, however, rubbing in one plane may cause a large amplitude whirl in another, with the whirl amplitude not bounded by the rubbing clearance.

Although Den Hartog’s explanation seems to indicate that all cases of clearance rubbing would initiate uncontrolled rotor vibration, such is not the case in actual machinery, mainly due to stator flexibility and inherent damping in the system. It is considered that, since there must be some minimum value of the coefficient of friction at the rubbing surface, below which the vibration cannot be sustained, it is probable that, given a minimum damping level, vibration would decay regardless of the degree of stator stiffness present.

Whether backward whirl with slip or pure rolling occurs, is determined by the friction angle. Below a minimum friction angle backward whirl is impossible.

7.6 Instability due to asymmetric factors

The purpose of this section is to present the peculiar theoretical characteristics of unsymmetrical rotors in relatively simple quantitative terms. Shaft orthotropy produces a peak of resonant vibration about half the regular critical speed and, for small damping, a range of possible unstable behavior between the two


critical speeds. Moreover, the unbalance response is a function of the angle at which the unbalance happens to lie in the rotor. Rotors having disks with unequal diametral moments of inertia (e.g., two-bladed small airplane propellers, wind turbines and fans) are dynamically unstable above a certain speed and some of these may return to a stable condition at a sufficiently high speed, depending on the particular magnitudes of the gyroscopic coupling and the inertia inequality.

7.6.1 Parametric excitation

A separate class of rotordynamic instabilities is characterized by conditions in the machine which are described by differential equations with variable coefficients. Since the coefficients are made up of the parameters of the rotor-bearing system, this type of instability is referred to as “parametric excitation”. In contrast to the self-excited instabilities, in which the whirling is always at a subsynchronous frequency equal to an eigenvalue of the system, parametric excitations produce whirling which can be synchronous, subsynchronous, or supersynchronous, depending on how the parameters vary in each particular case. Some of these “instabilities” are more like a critical speed or forced vibration, than like a true instability, in both their mathematical form and their behavior. Indeed, some can even be “driven through”, i.e. a higher speed can be found where the rotor regains its stability. Sometimes, however, the “speed range of instability” is quite wide. In the following two simple cases are presented: a) a shaft with unequal stiffnesses in orthogonal directions, and b) a disk with unequal dimetral moments of inertia in orthogonal directions.

7.6.2 Shaft anisotropy

A shaft may have unsymmetrical flexibility due to slots cut in it, such as keyways and armature winding slots, or due to manufacturing imperfections. For shafts with dissimilar stiffnesses in two mutually perpendicular direction (stiffness orthotropy) a range of rotational speeds exists in which the shaft precession is unstable. For such systems with circularly isotropic supports it is more convenient to work in a coordinate system which rotates with the principal axes of the shaft cross section, thus avoiding time varying coefficients in the equations of motion.

7.6.2.1 Free precession of an undamped vertical shaft

Consider the Laval-Jeffcott rotor having a central mass and a cross-section with principal second moments of area 1I and 2I and rigid bearings (Fig. 7.27, a). The analysis is carried out using a coordinate system η , ζ rotating at the constant angular speed of the shaft, Ω , positive in the counter-clockwise direction. It is


assumed that the displacements η and ζ are parallel to the principal axes of inertia (Fig. 7.27, b).

Let the bending stiffness in the direction−η be ηk and that in the direction−ζ be ζk . For the shaft in Fig. 7.27, ζη kk < . The axis−η along which

the stiffness is a minimum is referred to as the “weak axis”.

Denoting

( )ζη kkk +=21 , ( )ηζΔ kkk −=

21 , (7.98)

the principal stiffnesses can be written

kkk Δζ += , kkk Δη −= , (7.99)

and a degree of dissymmetry can be defined as

ηζ

ηζΔμkkkk

kk

+

−== . (7.100)

a b

Fig. 7.27

If the shaft were isotropic, the equations of motion in stationary coordinates would be (2.3)

.θtemzkzm

,θtemykym

) (sin

) ( cos

02

02

+=+

+=+

ΩΩ

ΩΩ

&&

&& (7.101)

Using the coordinate transformation (2.35)

,tΩtΩz,tΩtΩy

cossin sin cos

ζηζη

+=−=

(7.102)

equations (7.101) can be written in rotating coordinates as (2.47)


.θΩemζΩmk ηΩmζm

,θΩemηΩmkζΩmηm

022

022

sin)(2

cos)( 2

=−++

=−+−

&&&

&&& (7.103)

For an orthotropic shaft, equations (7.103) become

.θΩemζΩmk ηΩmζm

,θΩemηΩmkζΩmηm

022

022

sin)(2

cos)( 2

=−++

=−+−

ζ

η

&&&

&&& (7.104)

7.6.2.1 Free precession. Whirl speeds

For a vertical rotor having a central perfectly balanced mass, consider the homogeneous part of equations (7.104)

,ζΩ ηΩζ

,ηΩζΩη

0)(2

0)( 222

22

=−++

=−+−

ζ

η

ω

ω

&&&

&&& (7.105)

where

mkη

ηω =2 , mkζ

ζω =2 , (7.106)

are bending natural frequencies.

With exponential solutions of the form

te ληη 0= , te λζζ 0= , (7.107)

the equations (7.105) transform into

.ζΩ Ω

,ζΩηΩ

0)(2

02)(

0222

0

00222

=−++

=−−+

ζ

η

ωληλ

λωλ (7.108)

In order to have non-trivial solutions, the determinant of the displacement coefficients must vanish. This condition yields the characteristic equation

( ) 02)( )( 2222222 =+−+−+ λΩωλωλ ζη ΩΩ , or

0)( )( )2( 222222224 =−−+++− ΩΩΩ ζηζη ωωλωωλ . (7.109) Thus

)( )()2(41

)2(21

22222222

222221

ΩΩΩ

Ω,

−−−++±

±++−=

ζηζη

ζη

ωωωω

ωωλ. (7.110)


For the shaft in Fig. 7.27, ζη ωω < . The variation of 2λ as a function of 2Ω is shown in Fig. 7.28, a.

a b

Fig. 7.28 (from [7.23])

When ηωΩ < and Ωωζ < , both roots 2λ are negative, so that λ is purely imaginary,

111 iωλλ ±=∗, , 211 λω = , 222 iωλλ ±=∗, , 2

22 λω = , (7.111)

and the rotor is always stable.

When ζη ωΩω << , the root 21λ is positive

111 αλλ ±=∗, , 211 λα = , (7.112)

such that one of the values of λ is positive real and the rotor becomes unstable.

The variation of 1ω , 2ω and 1α as a function of Ω is illustrated in Fig. 7.28, b.

On the threshold of instability, the real part of λ is zero and, as 2λ is real, the imaginary part of λ must also be zero. Thus the threshold speeds may be determined by setting 0=λ in equation (7.109). They are

ηωΩ =1c , ζωΩ =2c , (7.113)


and the rotor is unstable between these two speeds. The unstable motion in this range could have been predicted from equation (7.109) using the Routh-Hurwitz criterion by observing that the coefficient 4a is negative.

7.6.2.2 Whirl speed diagrams

Figure 7.29, a gives the natural frequencies Rω as observed in the rotating coordinate system, as a function of Ω . Figure 7.29, b shows a similar plot of the natural frequencies Sω relative to the stationary z,y coordinate system. Both figures are computed for a system with a degree of dissymmetry 280.=μ .

a b

Fig. 7.29 (from [7.3])

An interesting phenomenon is associated with the point P in Fig. 7.29, b. It defines a whirl component which does not precess ( )0=Sω at the rotational speed

2nωΩ = . It could be expected that such a motion could be excited by a stationary excitation such as gravity acting on a horizontal rotor. In Fig. 7.29, a this point is on the line Ωω −=R . Indeed, a force fixed in the stationary frame is seen as rotating backwards at the speed Ω− in the rotating reference frame. This is analyzed in Section 7.6.2.5.

The asymptotes are like the whirl speed lines in Fig. 7.7, drawn for a rotor with a rotationally isotropic shaft. Although four quadrants are shown, only the right half-plane (where 0>Ω ) is required.


Fig. 7.30

Fig. 7.31

A whirl speed diagram represented in the right half plane only (for positive Ω ) is shown in Fig. 7.30. In order to amplify the instability region, the diagram is computed for an irealistic degree of dissymmetry 52830.=μ [7.23]. The 45 degree


dotted line Ωω =S shows the two critical speeds corresponding to the synchronous precession.

Often, the whirl speed diagram is represented only in the first quadrant, as in Fig. 7.31. The aliasing of positive and negative frequencies requires great care and some expertise in the interpretation of the diagram.

7.6.2.3 Stability chart

Denoting

2

2220

ζη ωωω

+= , (7.114)

the precession is unstable in the region ζη ωΩω << or at rotational speeds

μωΩμω +<<− 11 00 . (7.115)

Figure 7.32 is a stability chart showing the regions of stable and unstable precession as functions of the degree of dissymmetry μ and the rotational speed Ω . The limit curves are parabolas. The unstable range increases with the degree of dissymmetry.

Fig. 7.32 (after [7.14])

7.6.2.4 Unbalance response

In rotating coordinates, the unbalance response without damping is described by equations (7.104). The steady state solutions are

022

2cosθ

ΩωΩη

η −= e , 022

2sin θ

ΩωΩζ

ζ −= e . (7.116)

The unbalance gives rise to a displacement constant in time, of magnitude


22 ζηρ += . In the stationary coordinate system, the disk center has a synchronous precession along a circle of radius ρ=r . This depends on both the speed Ω and the orientation angle of the center of gravity 0θ . A more detailed presentation is given in Section 7.6.2.6 for rotors with external damping.

7.6.2.5 Gravity loading

As can be seen from the natural frequency plots (Section 7.6.2.2), the precession of a perfectly balanced rotating asymmetrical shaft exhibits an additional peculiarity when its axis is horizontal. The gravitational force induces a secondary resonance.

In stationary coordinates

gmf y = , 0=zf . (7.117)

The transformation of forces from stationary to rotating coordinates is

.tΩftΩff

,tΩftΩff

zy

zy

cossin

sin cos

+−=

+=

ζ

η (7.118)

Substituting (7.117) into (7.118), the gravity loading can be expressed in rotating coordinates as

tΩgmf cos=η , tΩgmf sin −=ζ . (7.119)

Including in equations (7.104) the gravitational force gm instead of the unbalance forces we obtain

.tgmζΩmk ηΩmζm

,tgmηΩmkζΩmηm

Ω

Ω

ζ

η

sin)(2

cos)( 22

2

−=−++

=−+−

&&&

&&& (7.120)

With solutions of the form

tΩG cosηη = , tΩG sin ζζ = , (7.121)

we obtain

,gζΩ Ω

,gζΩηΩ

GG

GG

−=−+−

−=−−

)2(2

2)2( 222

222

ζ

η

ωη

ω (7.122)

and

( )22222

22

2

4

ζηζη

ζ

ωωΩωω

ω

+−

−−=

ΩgηG , (7.123, a)


( )22222

22

2

4

ζηζη

η

ωωΩωω

ω

+−

−=

ΩgζG . (7.123, b)

A resonance will be observed when

222

222

21

sΩ Ωωω

ωω

ζη

ζη =⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+= , (7.124)

that is, at approximately one half of the mean of the two critical speeds.

Indeed, when the two stiffnesses differ only slightly, i.e. 0ωωη = , ( ) 01 ωεωζ += ,

( )

( )

( )[ ]( )( ) ( )εω

εεω

ε

εω

ωω

ωωΩ

ζη

ζη +≅+

+≅

++

+=

+= 1

41421

112

1

2

20

20

2

220

22

222s ,

or

22

12

12

0 ζη ωωεωΩ

+=⎟

⎠⎞

⎜⎝⎛ +≅s . (7.125)

Thus a sub-harmonic resonance, caused entirely by the gravitational force, occurs in rotors with asymmetric shafts, known as a secondary critical speed.

Fig. 7.33 (from [7.4])

In the neighborhood of the secondary critical speed, both the numerator and the denominator of the gravity forced precession are of the same order of magnitude. The steady state precession radius depends upon the ratio ( ) ( )2222 ΩΩωω ηζ −− s . Therefore, the rate of amplitude build-up at this speed is slow.


A diagram showing the whirl amplitude as a function of the rotational speed in a forced precession produced by both out-of-balance and gravity is shown schematically in Fig. 7.33.

7.6.2.6 Dissymmetric shaft with external damping

The following presentation is reproduced from the classical paper by H. D. Taylor [7.24].

Fig. 7.34

Figure 7.34 illustrates the variation of the static deflection of a horizontal shaft, if it is rolled slowly, the deflection being a maximum when the weak axis is vertical (position A) and a minimum when the weak axis is horizontal (position C). For intermediate positions (as at S) the shaft center traverses a circular path through the extreme points, and makes two complete circuits of this path per revolution of the shaft.

Figure 7.35 shows the shaft section position at several consecutive times.

Fig. 7.35


When the shaft is running, a mechanical unbalance rotating at twice shaft speed will produce a double-frequency vibration of the shaft, with a resonant peak at a shaft speed of about half the average fundamental critical speeds.

The unbalance response of a simple rotor with external viscous damping is described in rotating coordinates by equations of the form

( )( ) ⎭

⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−+

−−−+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡ −+

⎭⎬⎫

⎩⎨⎧

0

0222

00

022

0

0

0

sin cos

1 2 21

2 2 22

θθ

Ωζη

ΩωμΩωζΩωζΩωμ

ζη

ωζΩΩωζ

ζη

ee

e

e

e&

&

&&

&&

(7.126) where the damping ratio

02 ω

ζmce

e = . (7.127)

The unbalance response curves have different shapes according to the relative values of the degree of dissymmetry with respect to the damping ratio. When

eζμ > , the precession radius of the shaft may become infinite, in spite of the presence of damping. For shafts with moderate dissymmetry ( )eζμ < , there is apparently no infinite precession radius but a number of other peculiarities.

Fig. 7.36

In the case of moderate dissymmetry, a set of unbalance response curves is given in Fig. 7.36. These are drawn for 050.=μ and ..e 070=ζ Note the importance of a new variable, the orientation angle 0θ of the center of gravity with


respect to the week axis. The peak response amplitude apparently varies by five to one on a rotor running with the same amount of unbalance over the same speed range, with the same damping and the same degree of dissymmetry and with changes only in the angular position of unbalance. The highest peak appears when the unbalance is 045 ahead of the weak axis, in the direction of rotation. The lowest peak appears when the unbalance is 045 behind this axis ( )0

0 45−=θ .

Fig. 7.37

This suggests the use of a polar diagram, Fig. 7.37, to show the variation of the peak amplitude with the angle of unbalance. It appears that this amplitude may vary considerably with the direction of rotation since the most sensitive point for unbalance ( 045 ahead of the weak axis) becomes the least sensitive ( 045 trailing) if the rotation is reversed.

In can be shown that the stability condition is

041 20

222

2

20

2≥+−⎟

⎟⎠

⎞⎜⎜⎝

⎛−

ωΩζμ

ωΩ

e . (7.128)


Fig. 7.38 (after [7.14])

A stability chart is shown in Fig. 7.38, where the limit curves have equations

( )2

20

22220

22121 ⎟

⎟⎠

⎞⎜⎜⎝

⎛−−−=

ω

ωωζζ

ωΩ ζη

ee m . (7.129)

With appropriate damping levels and especially with a minimum of shaft dissymmetry it seems possible to avoid the unstable operation.

7.6.3 Asymmetric inertias

We shall restrict our study to a light symmetrical elastic shaft rotating with a constant angular speed in identical symmetrical bearings and carrying a centrally mounted rigid disk. As the disk is supported such that its angular motions are uncoupled from the translational motions and it spins about one of its principal axes of inertia, we shall examine the small amplitude oscillations about the other two principal axes.

7.6.3.1 Euler’s equations

Assume a rigid disk with its center of mass pivotally mounted at point O, the origin of a stationary set of axes X, Y, Z (Fig. 7.39).

It is convenient to use a coordinate system ζηξ ,, which rotates with the disk, in which the moments of inertia and the products of inertia are constant. If this system coincides with the principal axes of inertia of the disk, the products of inertia are zero and the moments of inertia are its principal moments of inertia 1J , 2J , 3J .


If we let the triad of unit vectors k,j,i of axes ζηξ ,, coincide and rotate with the disk principal axes of inertia 1, 2, 3, the angular momentum about its center of mass is given by

kJjJiJK 332211 ωωω ++= , (7.130)

where 1ω , 2ω , 3ω are components of the absolute angular velocity of the disk along the principal axes of inertia.

The angular velocity, ω , of the triad of unit vectors ,k,j,i relative to a stationary coordinate system must be

kji 321 ωωωω ++= . (7.131)

According to the theorem of angular momentum, the external torque is equal to the time rate of change of K

Kt

Kt

KM ×+∂∂

== ωdd , (7.132)

332211

321332211

ωωωωωωωωω

JJJ

kjikJjJiJM +++= &&& . (7.133)

If the torque acting on the disk has components 1M , 2M , 3M , along the principal axes of inertia

kMjMiMM 321 ++= . (7.134)

Equating the corresponding components from (7.133) and (7.134) we obtain the scalar equations

( ) 3232111 ωωω JJJM −−= & , (7.135, a)

( ) 1313222 ωωω JJJM −−= & , (7.135, b)

( ) 2121333 ωωω JJJM −−= & , (7.135, c)

known as Euler’s dynamical equations.

7.6.3.2 Equations of angular motion

If the axis−ξ is along the axis of rotation (spin axis) and the −η and axis−ζ are along perpendicular diameters, we can introduce the following

notations


1JJ P = , 2JJ =η , 3JJ =ζ , (7.136)

.const== 1ωΩ , 2ωωη = , 3ωωζ = , (7.137)

01 =M , 2MM =η , 3MM =ζ . (7.138)

Equations (7.135, b) and (7.135, c) become

( )( ) .MJJJ

,MJJJ

P

P

ζηηζζ

ηζζηη

ωΩω

ωΩω

=−−

=−−

&

& (7.139)

Let the angular position of the disk relative to the stationary coordinate system be defined by

kj γβσ += (7.140)

where β and γ are the components of the angle between the shaft axis and the bearing axis in the rotating system.

Fig. 7.39

The time rate of change of σ is

kkjj &&&&& γγββσ +++= , (7.141) where

kjijj ΩΩΩ =×=×=& , jkikk ΩΩΩ −=×=×=& ,

so that ( ) ( ) kj βΩγγΩβσ ++−=& . (7.142)

But kj ζη ωωσ +=& . (7.143)


The components of the absolute angular velocity along the diametral axes are

.

,

βΩγω

γΩβω

ζ

η

+=

−=

&

& (7.144)

Substituting (7.144) into (7.139) yields

( ) ( ) ( )( ) ( ) ( ) ,MJJJ

,MJJJ

P

P

ζηζ

ηζη

γΩβΩβΩγ

ΩβΩγγΩβ

=−−−+

=+−−−&&&&

&&&& (7.145)

or

( ) ( )( ) ( ) .MJJJJJJ

,MJJJJJJ

PP

PP

ζηζηζ

ηζζηη

γΩβΩγ

βΩγΩβ

=−+−−−

=−+−−+2

2

&&&

&&& (7.146)

If the angular displacements β and γ are opposed by moments

βη KM −= , γζ KM −= , (7.147)

where K is a torsional stiffness, equations (7.146) become

( ) ( )[ ]( ) ( )[ ] .JJKJJJJ

,JJKJJJJ

PP

PP

0

0

2

2

=−++−−−

=−++−−+

γΩβΩγ

βΩγΩβ

ηζηζ

ζζηη

&&&

&&& (7.148)

7.6.3.3 Whirl speeds in rotating coordinates

The equations of motion (7.148) can now be rewritten in terms of dimensionless parameters

( ) ( ) ( )[ ]( ) ( ) ( )[ ] ,

,

0121

012122

0

220

=−−−+−−−

=−−−+−++

γεαΩωβΩαγε

βεαΩωγΩαβε

&&&

&&& (7.149)

where

( )ζη

αJJ

J P

+=

21

is a gyroscopic coupling factor, (7.150)

ζη

ζηεJJJJ

+

−= is an inertia inequality factor, (7.151)


( )ζη

ωJJ

K

+=

21

20 is a reference (fictitious) natural frequency. (7.152)

With exponential solutions of the form

te λββ 0= , te λγγ 0= , (7.153)

the equations (7.149) transform into

( ) ( )[ ] ( )

( ) ( ) ( )[ ] . Ω

,Ω

0112

0211

022

02

0

0022

02

=−−−+++−−

=−+−−−++

γεαΩωλεβλα

γλαβεαΩωλε (7.154)

In order to have non-trivial solutions, the determinant of the angular displacement coefficients must vanish. This condition yields the characteristic equation

02 32

24

1 =++ aaa λλ , (7.155) where

21 1 ε−=a ,

( ) ⎥⎦⎤

⎢⎣⎡ −+−+= 2

21 222

02 ααεΩωa , (7.156)

( )[ ] ( )[ ]εαΩωεαΩω +−−−−−= 11 220

2203a .

The eigenvalues are

1

31222

4321 aaaaa

,,,−±−

±=λ . (7.157)

The conditions for stability are

01 >a , 02 >a , 03 >a . (7.158)

The first two conditions are satisfied for

210 << ε and 20 ≤<α , (7.159)

and the last condition requires

( )[ ] ( )[ ] 011 220

220 >+−−−−− εαΩωεαΩω . (7.160)


Let denote the critical speeds

εα

ωΩ

+−=

10

1 , εα

ωΩ

−−=

10

2 . (7.161)

Case I. When PJ is the largest moment of inertia, αε <+1 , the condition (7.160) is satisfied and the system is stable for all Ω . The whirl speeds relative to a coordinate system which rotates with the rotor are represented as a function of the rotational speed in Fig. 7.40, a. It is easy to show that for this case 031

22 >− aaa ,

that is 2λ is real and negative. The system is oscillatory.

a b c

Fig. 7.40

Case II. When PJ is the intermediate moment of inertia, εαε +<<− 11 , the system is stable at speeds 1ΩΩ < and unstable for 1ΩΩ > . If there is one critical speed, motion will be stable below this speed and unstable above it (Fig. 7.40, b).

Case III. When PJ is the smallest moment of inertia, εα −<1 , the system is unstable at speeds 21 ΩΩΩ << and stable for 1ΩΩ < and ΩΩ <2 . If there are two critical speeds, motion will be stable below the lower and above the higher speed, but it will be unstable between the two critical speeds (fig. 7.40, c). For a given α , introducing an inertia asymmetry ε tends to lower 1Ω and to rise 2Ω , increasing the speed range of instability. This applies also to overhung rotors, with disks or propellers supported by rigid shafts, pivotally mounted at one end, for which the transverse moments of inertia with respect to the pivot (including a mass times length squared) are larger than the axial mass moment of inertia.


Fig. 7.41 (after [7.25])

Figure 7.41 gives a 3D diagram of the stability boundary of undamped rotors with non-circular shafts as a function of the parameters α and ε , and the dimensionless speed 0ωΩ [7.25] for Case III.

7.6.3.4 Influence of external damping

In the presence of external viscous damping, equations (7.49) become

( ) ( ) ( )[ ]( ) ( ) ( )[ ] ,

,

ee

ee

012221

021221

22000

022

00

=−−−+++−−−

=−−−−+−+++

γεαΩωβΩωζγωζβΩαγε

γΩωζβεαΩωγΩαβωζβε

&&&&

&&&&(7.162)

where

m

cee

02ωζ = . (7.163)

The condition of stability (7.160) becomes

( )[ ] ( )[ ] ( ) 0211 2220

220 >++−−−−− ΩζεαΩωεαΩω e (7.164)

and the critical speeds (7.161) at the ends of the unstable region become

( ) ( ) 2222021

1421

1

εαζζζα

ωΩ

+−+±−−

=

eee

, . (7.165)


As shown in the stability map of Fig. 7.42, with (stationary) external damping the unstable range ( )12 ΩΩ − is narrower and shifted at higher rotational speeds.

Fig. 7.42 (after [7.14])

7.6.4 Finite element analysis of asymmetric rotors

When the rotor is asymmetric and the bearings are axially anisotropic, the equations of motion have periodic coefficients with respect to both a rotating and a stationary reference frame. The latter is used in the following. A general approach based on a variant of Hill’s infinite determinant method [7.26] is presented for rotors modeled by finite elements.

7.6.4.1 Element matrices in the stationary reference frame

In the reference frame rotating with the rotor, the mass matrix of an asymmetric element can be written in the form

[ ] [ ] [ ][ ] [ ] ,

mm

M r⎥⎥⎦

⎤

⎢⎢⎣

⎡=

2

1

00

where [ ]1m and [ ]2m are the coherent mass submatrices corresponding to the coordinates in the mutually perpendicular planes 1 and 2, respectively [7.27].

In the stationary reference frame attached to the stator, the mass matrix is obtained by the congruence transformation

[ ] [ ] [ ] [ ]Trs RMRM = (7.166)


in which the orthogonal rotation matrix

[ ] ⎥⎦

⎤⎢⎣

⎡−

=tttt

R cos sin sin cos

ΩΩΩΩ

,

where Ω is the rotational speed of the rotor, assumed to be constant.

Matrix (7.166) has the form

[ ] [ ] [ ] [ ][ ] [ ] [ ] ⎥

⎦

⎤⎢⎣

⎡−

+=

tmmtmtmtmm

Mdmd

ddms 2cos 2sin

2sin 2cosΩΩ

ΩΩ (7.167)

where

[ ] [ ] [ ]( ),mmmm 2121

+= [ ] [ ] [ ]( ).mmmd 21

21 −=

Matrix (7.167) can also be written

[ ] [ ] [ ] [ ] tMtMMM css 2cos 2sin 220 ΩΩ ++= (7.168)

or

[ ] [ ] [ ] [ ] tts MMMM 2i +

20 2i

2 e e ΩΩ+

−− ++= (7.169)

where

[ ] [ ] [ ]( ) ,MMM sc i21

222 +=− [ ] [ ] [ ]( ).MMM sc 222 i21

−=+

Analogously, the combined gyroscopic and damping matrix is

[ ] [ ] [ ] [ ] tts CCCC 2i +

20 2i

2 e e ΩΩ+

−− ++= (7.170)

and the stiffness matrix for an asymmetric shaft is

[ ] [ ] [ ] [ ] tts KKKK 2i +

20 2i

2 e e ΩΩ+

−− ++= . (7.171)

The submatrices entering the matrices (7.169)-(7.171) are presented Chapter 5.

7.6.4.2 Solution of the equations of motion with harmonic coefficients

The equation of motion of the finite element model containing asymmetric components can be written in matrix form (dropping the index s)

( )[ ] ( )[ ] ( )[ ] ( ) tFutKutCutM =++ &&& . (7.172)

In (7.172) matrices are real and periodic, with period Ωπ2=T :


( )[ ] ( )[ ]TtMtM += , ( )[ ] ( )[ ]TtCtC += , ( )[ ] ( )[ ]TtKtK += .

Equation (7.172) can be transformed in a set of differential equations of first degree [7.28] which, when ( ) 0=tF , can be written

( ) ( )[ ] ( ) 0=− txtAtx& , (7.173)

where ( )[ ] ( )[ ]TtAtA += .

The differential equations with periodic coefficients have solutions of the form

( ) ( ) trtx ktk k )()( e λ= (7.174)

where ( ) ( ) Ttrtr kk += )()( is the eigenvector associated with the complex eigenvalue

.kkk ωαλ i+= (7.175)

The eigenproblem is solved representing the matrix ( )[ ]A t and the

eigenvectors ( ) tr k )( by the Fourier series

( )[ ] [ ] tm

mmAtA ie Ω∑

∞

−∞=

= , tn

n

kn

k rtr i)()( e )( Ω∑∞

−∞=

= . (7.176)

Substitution of (7.176) into equation (7.173) yields

[ ] 0e e e ie i)( i i)( i)( =−+ ∑∑∑∑∞

−∞=

∞

−∞=

∞

−∞=

∞

−∞=

tn

n

kn

tm

mm

tn

n

kn

tn

n

knk rArnr ΩΩΩΩ Ωλ ,

equation which must be satisfied independently for all frequencies nΩ .

Canceling the coefficients of the terms tn ie Ω , for each value n (harmonic balance), leads to a hyper eigenvalue problem (of infinite dimension) of the form

[ ]

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]

0

2i

i

i

2i

2

1

0

1

2

012

1012

21012

2101

210

=

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

+

−

−

−

+

+

−

−

++

−++

−−++

−−+

−−

...r

r

r

r

r...

.....................

...IAAA.........

...AIAAA......

...AAAAA...

......AAIAA...

.........AAIA...

.....................

I

)k(

)k(

)k(

)k(

)k(

k

Ω

Ω

Ω

Ω

λ


or [ ] [ ]( ) 0 =− )k(

k qBIλ . (7.177)

Matrix [ ]B , with constant elements (time invariant), has an infinite number of eigenvectors (and eigenvalues), which are not all linearly independent. In order to completely describe the homogeneous solution, only 2N eigenvectors (and eigenvalues) are required, where N is the dimension of the matrices of the original system.

Indeed

tln

n

kn

tltn

n

kn

tk rrtx kk )( i

-=

)( ) i( i

-=

)()( e e e e )( ΩΩλΩλ −∞

∞

+∞

∞∑∑ == .

It follows that

)( i i ΩωαΩλλ ll kkkl ++=+= , ( ),...,,,,,,...,l 3210123 +++−−−=

is also an eigenvalue of the matrix [ ])(tB , and the corresponding eigenvector is

Ωlkl qq i )()( e −= ,

so that only 2N eigenvectors are linearly independent [7.28].

In practical applications, the infinite Fourier series is truncated to the first p harmonics, hence to ( )12 +p terms.

This leads to a finite eigenproblem

[ ] [ ]( ) 0 ˆqBI )k(k =−λ (7.178)

of dimension 2 2 1N p( )+ , where 2N is the number of degrees of freedom.

From the 2 2 1N p( )+ eigenvalues (and eigenvectors) of the matrix [ ]B only the basis 2N eigenvalues (and eigenvectors) are calculated [7.28]. Generally, the only major difficulty in the solution of the eigenproblem (7.178) is the inversion of the matrix [ ]Ms to obtain the state space form (especially for systems with large dissymmetry).

References

7.1 Ehrich, F. F. (ed.), Handbook of Rotordynamics, McGraw Hill, New York, 1992.


7.2 Crandall, S. H., Physical explanations of the destabilizing effect of damping in rotating parts, Rotordynamic Instability Problems in High-Performance Turbomachinery, NASA CP 2133, 1980, pp 369-382.

7.3 McCallion, H., Vibration of Linear Mechanical Systems, Longman, London, 1973.

7.4 Lee, C.-W., Vibration Analysis of Rotors, Kluwer Academic Publ., Dordrecht, 1993.

7.5 Muszynska, A., Whirl and whip – rotor/bearing stability problems, Instability in Rotating machinery, NASA CP 2409, 1985, pp 155-177.

7.6 Nelson, F. C., A review of the origins and current status of rotor dynamics, Proc. 6th Int. Conference on Rotor Dynamics, Sydney, Australia, Sept 30 –Oct 4, 2002, pp 745-751.

7.7 Rao, J. S., Rotor Dynamics, Wiley Eastern Ltd., New Delhi, 1983.

7.8 Sternlicht, B. and Rieger, N. F., Rotor stability, Proceedings of the Institution of Mechanical Engineers., vol.182, Pt.3A, 1967-1968, pp 82-99.

7.9 Crandall, S. H., Heuristic explanation of journal bearing instability, Rotordynamic Instability Problems in High-Performance Turbomachinery, NASA CP 2250, 1982, pp 274-283.

7.10 Sommerfeld, A., Zur hydrodynamischen Theorie der Schmiermittelreibung, Zeitschrift für Mathematik und Physik, vol.50, 1904, pp 97-155.

7.11 Newkirk, B. L. and Taylor, H. D., Shaft whipping due to oil action in journal bearings, General Electric Review, vol.28, 1925, pp 559-568.

7.12 Robertson, D., Whirling of a journal in a sleeve bearing, Philosophical Magazine, S.7, vol.15, no.96, 1933, pp 113-130.

7.13 Lund, J. W., Some unstable whirl phenomena in rotating machinery, Shock and Vibration Digest, vol.7, no.6, 1975, pp 5-12.

7.14 Gasch, R., and Pfützner, H., Rotordynamik, Springer, Berlin, 1975.

7.15 Fritzen, C. P., and Nordmann, R., Influence of parameter changes to stability behavior of rotors, Rotordynamic Instability Problems in High-Performance Turbomachinery, NASA CP 2250, 1982, pp 284-306.

7.16 Thomas, H.-J., Instabile Eigenschwingungen von Turbinenläufern, angefacht durch die Spaltströmungen in Stopfbuchsen und Beschaufelungen, AEG-Sonderdruck Z10/5729, 1958.

7.17 Alford, J. S., Protecting turbomachinery from selfexcited whirl, Journal of Engineering of Power, Trans. ASME, Series A, vol.87, no.10, 1965, pp 333-344.

7.18 Newkirk, B. L., Shaft rubbing, Mechanical Engineering, vol.48, 1926, pp 830-832.


7.19 Ehehalt, J., Hochlenert, D., Markert, R. and H. I. Weber, Approximate description of backward whirl at rotor-stator-contact, Advances in Vibration Control and Diagnostics (N. Bachschmid and P. Pennacchi, eds), Polimetrica Int. Sci. Publ., Monza, Italy, 2006.

7.20 Bartha, A. R., Dry Friction Backward Whirl of Rotors, Dissertation ETH No. 13817, ETH Zürich, 2000.

7.21 DenHartog, J. P., Mechanical Vibrations, 4th ed., McGraw-Hill, 1956.

7.22 Ehrich, F. F., The dynamic stability of rotor/stator radial rubs in rotating machinery, Journal of Engineering for Industry, Trans.ASME, Series B, vol.91, no.4, 1969, pp 1025-1028.

7.23 Krämer, E., Maschinendynamik, Springer, Berlin, 1984.

7.24 Taylor, H. D., Critical-speed behavior of unsymmetrical shafts, Journal of Applied Mechanics, June 1940, pp A71-A79.

7.25 Brosens, P. J. and Crandall, S. H., Whirling of unsymmetrical rotors, Journal of Applied Mechanics, Trans.ASME, Series E, vol.28, no.3, 1961, pp 355-362.

7.26 Xu, J., Aeroelastik einer Windturbine mit drei gelenkig befestigten Flügeln, Fortschrittsberichte VDI, Reihe 11, No.185, VDI-Verlag, Düsseldorf, 1993.

7.27 Genta, G., Whirling of unsymmetrical rotors. A finite element approach based on complex coordinates, Journal of Sound and Vibration, vol.124, no.1, 1988, pp 27-33.

7.28 Gasch, R. Knothe, K., Strukturdynamik, Springer, Berlin, 1989.

Selected Bibliography

7.29 Biezeno, C. B., and Grammel, R., Technische Dynamik, Bd.3, Springer, Berlin, 1953.

7.30 Bolotin, V. V., Nonconservative Problems of the Theory of Elastic Stability, Pergamon Press, London, 1963.

7.31 Dimentberg, F. M., Flexural Vibrations of Rotating Shafts, Butterworth, London, 1961.

7.32 Gasch, R., Nordmann, R., and Pfützner, H., Rotordynamik, 2. Auflage, Springer, Berlin, 2002.

7.33 Krämer, E., Dynamics of Rotors and Foundations, Springer, Berlin, 1993.

7.34 Thomas, H. J., Thermische Kraftanlagen, 2.Aufl., Springer, Berlin, 1985.

7.35 Tondl, A., Some Problems of Rotor Dynamics, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1965.


7.36 Vance, J. M., Rotordynamics of Turbomachinery, Wiley, New York, 1988.

7.37 Crandall, S. H. and Brosens, P. J., On the stability of rotation of a rotor with rotationally unsymmetric inertia and stiffness properties, Journal of Applied Mechanics, Trans.ASME, Series E, Dec 1961, pp 567-570.

7.38 Hori, Y., A theory of oil whip, Journal of Applied Mechanics, Trans.ASME, Series E, vol.26, no.2, 1959, pp 189-198.

7.39 Inagaki, T., Kanki, H., and Shiraki, K., Response analysis of a general asymmetric rotor-bearing system, Journal of Mechanical Design, vol.102, no.1, 1980, pp 147-157.

7.40 Iwatsubo, T., Stability problems on rotor systems, Shock and Vibration Digest, vol.11, no.3, 1979, pp17-26.

7.41 Jei, Y.-G., and Lee, C.-W., Modal characteristics of asymmetrical rotor-bearing systems, Journal of Sound and Vibration, vol.162, no.2, 1993, pp 209-229.

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Index Annular seals 34

− gas 147 − liquid 137

Approximate bearing temperature 107 Attitude angle 79

Bearing 77 − clearance 79 − dynamic coefficients 83 − − dimensionless 84 − − physical significance 103 − preload 118 − static characteristics 78 − static equilibrium locus curve 80 − temperature 107

Black’s seal model 141 Bulk flow seal models 139

Campbell diagram 44 Childs’ seal model 142 Component mode synthesis 62 Concentric circles method 54 Conservative gyroscopic system 46 Constrained normal modes 64 Constraint modes 65 Crandall’s model 165, 179 Critical rotor mass 192 Cross-coupled stiffness 162

Damping matrix 34 Dissymmetric shaft 216 Double-frequency vibration 217 Dry friction backward whirl 203

Eccentricity 79 Effective shear area 14 Eigenvalue analysis 40 Elliptic orbit 49

Finite-length cavitated bearing 99 Flexible couplings 35

− disk 12 Fluid film bearings 77

− axial groove 112 − elliptical 118 − floating ring 129 − four-lobe 121 − multilobe 117 − offset halves 116 − plain 112 − pressure dam 114 − − multi-lobe 122 − Rayleigh step 128 − three-lobe 120 − tilting pad 124 Fractional frequency whirl 173 Full-Sommerfeld boundary conditions 89

Gravity loading 214 Guyan reduction 69 Gümbel boundary conditions 89 Gyroscopic matrix of − rigid disk 9 − shaft element 24, 38

Half-frequency whirl 176 Half-Sommerfeld boundary conditions 89

Impeller-diffuser interaction 201 Improved reduction 10 Instability of rotors 161 Internal rotor friction 164 Iterated improved reduction 71

Jenssen’s seal model 141

Kinetic energy of − axi-symmetric disk 7 − shaft element 21

Lagrange equations 1 Left eigenvectors 42 Lomakin effect 138

Macroelements 59 Mass matrix of − rigid disk 9 − shaft element 23, 38

DYNAMICS OF MACHINERY

234

− unsymmetric element 226 Mass unbalance vectors 39

− of rigid disk 10 − of shaft element 28

Modal analysis 47 Model condensation 56

− combined static and modal 66 − Guyan-Irons 57 − modal 60 − static 58

Model order reduction 56, 68 Moes’ solution 99

Nutation 3

Ocvirk bearing 90 Oil whip 174

− whirl 174 Onset of instability speed 43, 178

Permanent disk skew vector 10 Planar rotor model 165 Precession

− backward 3 − forward 3

Reference frames 3 Reynolds boundary conditions 90

− equation 87 Right eigenvectors 41 Rotating damping 164 Rotor-stator rubbing 203 Routh-Hurwitz stability conditions 188

Seals 77 − annular gas 147 − annular liquid 137 − bushing 152 − floating contact 150 − labyrinth 147 − locked-up 154

Seal ring lockup 154 Secondary critical speed 215 Shaft

− anisotropy 207 − finite elements 16, 18

Shape functions 19, 21 − for rotations 20 − shear-modified 20

Shear coefficient 14 − energy 25

Short bearing 90 Squeeze film dampers 130 − design 134 − stiffness and damping coefficients 133 Sommerfeld bearing 99, 179 − number 80 − − modified 156 Spectral analysis 48 Stability chart 135, 197, 213, 225 − criterion 187 Steam whirl 199 Stepwise model reduction 68 Stiffness matrix of − fluid film bearing 34 − radial bearing 31 − shaft element 26, 38 Strain energy of shaft element 25 Substructure coupling 62

Unbalance response 47 Unsymmetric inertias 219

Viscosity-temperature relationship 109 − − chart 110

Walther’s equation 109 Whirl speed diagrams 171, 211

M Rades - Dynamics of Machinery 2

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