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Dynamics of Machinery
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Mircea Rade 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-lagre 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 Rade
Prefa
Lucrarea se bazeaz pe partea a doua a cursului de Dinamica mainilor predat din 1993 studenilor Filierei Engleze a Facultii de Inginerie n Limbi Strine (F.I.L.S.) la Universitatea Politehnica Bucureti. Coninutul cursului s-a lrgit n timp, pornind de la un curs postuniversitar organizat ntre 1985 i 1990 n cadrul Catedrei de Rezistena materialelor i continuat pn n 2007 la cursurile de masterat n specialitatea Sigurana i Integritatea Mainilor. Capitole din curs au fost predate din 1995 la cursurile de studii aprofundate i masterat organizate la Facultatea de Inginerie Mecanic i Mecatronic.
Dinamica mainilor a fost introdus n planul de nvmnt al F.I.L.S. n 1993. Pentru a susine cursul, am publicat Dynamics of Machinery la U. P. B. n 1995, urmat de Dinamica sistemelor rotor-lagre n 1996 i Rotating Machinery n 2005, ultima coninnd materialul ilustrativ utilizat n cadrul cursului.
Cursul are un loc bine definit n planul de nvmnt, urmrind: a) descrierea fenomenelor dinamice specifice mainilor; b) modelarea sistemelor rotor-lagre i analiza acestora cu metoda elementelor finite; c) narmarea studenilor cu baza fizic necesar n rezolvarea problemelor de vibraii ale mainilor; i d) familiarizarea cu metodele de supraveghere a strii mainilor i diagnosticare a defectelor.
Fiind un curs predat unor studeni a cror limb matern nu este limba englez, au fost reproduse expresii i fraze din lucrri scrise de vorbitori nativi ai acestei limbi. Pentru studenii F.I.L.S. s-a definit i ilustrat n detaliu terminologia specific limbii engleze.
n prima parte se descriu fenomenele de baz din dinamica rotorilor, rspunsul dinamic al rotorilor simpli n lagre rigide i lagre elastice, precum i principalele etape ale unei analize de dinamica rotorilor. n aceast a doua parte se prezint modelarea cu elemente finite a sistemelor rotor-lagre, lagrele hidrodinamice i etanrile cu lichid i gaz, precum i instabilitatea rotorilor. n partea a treia se trateaz lagrele cu rulmeni, echilibrarea rotoarelor, msurarea vibraiilor pentru supravegherea funcionrii mainilor i diagnosticarea defectelor, standarde i recomandri privind limitele admisibile ale vibraiilor mainilor, precum i elemente de dinamica mainilor cu mecanism biel-manivel i vibraiile conductelor aferente. Nu se trateaz vibraiile paletelor, discurilor paletate i ale roilor centrifugale.
Iulie 2009 Mircea Rade
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 Reynoldsequation 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 Crandalls 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 Lagranges 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
1 . (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 (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
DYNAMICS OF MACHINERY 4
The motion in which the Ox axis traces out a cone (Fig. 5.3, a), with angular velocity , 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> , while in backward precession 0
5. FEA OF ROTOR-BEARING SYSTEMS 5
b) a rotation 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 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 (angular velocity & around OY) (Fig. 5.5, a) The relation between the unit vectors is
=
1
1
1
1
11
cossinsincos
ki
ki
KI
.
In expanded form
[ ]
=
=
1
1
1
1
10010
01
kJi
TkJi
KJI
.
b) Rotation (angular velocity & around 1Oz ) (Fig. 5.5, b) The relation between the unit vectors is
=
11
1
11
cossinsincos
ji
ji
Ji
.
In expanded form
DYNAMICS OF MACHINERY 6
[ ]
=
=
1
1
1
1
1
1
1000101
kji
Tkji
kJi
.
c) Rotation (angular velocity =& around Ox) (Fig. 5.5, c) The relationship between the unit vectors is
=
kj
kj
cossinsincos
1
1 .
In expanded form
[ ]
=
=
kji
Tkji
kji
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
[ ] [ ][ ][ ]
+
==
cossinsincos
sincoscossin1TTTT .
The vector of the instantaneous angular velocity is
1kJi && ++= . In terms of the unit vectors of the x,y,z system of coordinates
kjiJ sincos += , kjk cossin1 += , so that
( ) ( ) ( ) kji sincossincos &&&&& ++++= . The components of the angular velocity in the rotor-fixed (mobile) x,y,z reference system are
5. FEA OF ROTOR-BEARING SYSTEMS 7
++
=
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 ( and ) (Fig. 5.6, a), the displacements of the shaft cross-section at the disk attachment. The angles and 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
( ) ( )2222221
21
zzyyxxdd JJJwmT ++++= &&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 x , y , z from (5.2) into (5.3), the kinetic energy becomes
( ) ( )( ) ( )[ ]22
222
sincossincos21
21
21
&&&&
&&&
+++
++++=
T
Pdd
J
JwmT v
or
DYNAMICS OF MACHINERY 8
( ) ( ) ( )2222222212
21
21 &&&&&& ++++++= TPdd JJwmT v .
Neglecting the term 2221 &PJ , the kinetic energy becomes
( ) ( ) 2222221
21
21 PPTdd JJJwmT +++++= &&&&&v . (5.4)
In equation (5.4), the term ( )2221 wmd && +v accounts for rectilinear
translation, ( )2221 && +TJ - for rotary inertia, &PJ - for gyroscopic
coupling and 221 PJ - for rotation around the spin axis.
a b
Fig. 5.6
Applying Lagranges equations, the equations of motion of the rigid circular disk are obtained in the form
yd
dTmT
t==
vv
&&&dd , (5.5, a)
zPT
ddMJJTT
t==
&&&&d
d , (5.5, b)
zd
dTmT
t==
ww
&&&dd , (5.5, c)
( ) ( ) yPTd
MJJTt
==
&&&&d
d , (5.5, d)
0=v
dT , 0=w
dT , ( ) 0=
dT ,
5. FEA OF ROTOR-BEARING SYSTEMS 9
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
&&&&
&&&&&&&&
-w
v
-w
v
0000000
0000000
000000000000
.
Introducing the 12 state vectors in the X-Y and X-Z planes, respectively,
{ }
= vd
yu , { } = wdzu , 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
0 (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
[ ] = Tdd Jmm 0 0 , [ ] = Pd Jg 0 00 , (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.
DYNAMICS OF MACHINERY 10
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 cosaac = , sinaas = (Fig. 5.7, b). At 0=t (at rest), is the angle of CG with the Y-axis.
a b
Fig. 5.7
The unbalanced centrifugal force acting in G along CG is 2amd . The components along the (spinning) rotor-fixed reference system are 2cd am and
2sd am , and along the (inertial) fixed reference system are
.tamtamf
,tamtamf
cdsdz
sdcdy
sincos
sincos22
22
+==
(5.8)
The vector of unbalance forces (neglecting disk skewness) can be written
{ } { }{ } taa
mta
a
mfff
c
s
ds
c
ddz
dyd
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 between a principal axis of inertia of the disk and the shaft axis.
5. FEA OF ROTOR-BEARING SYSTEMS 11
The disk skew can be defined by a rotating vector normal to the spin axis and to the line of maximum skew angle. At 0=t the vector makes an angle with the rotating z axis, so that the disk skew vector has components
cos=c and 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
++=+=
cos
sin
where and are elastic rotations.
a b Fig. 5.8
The corresponding angular velocities and accelerations are
( )( ) ,t
,t
z
y
+=+=
sin
cos&&&&
( )( ).t
,t
z
y
+=++=
cos
sin2
2
&&&&&&&&
(5.10)
From equations (5.5, b) and (5.5, d), replacing by y and by z , we obtain
( )
( ) .MJJ,MJJ
yzPyT
zyPzT
==+
&&&&&&
Substitution of (5.10) yields
( ) ( )
( ) ( ) ( ).tJJJJM,tJJJJM
TPPTy
TPPTz
++=
++=sin
cos 2
2
&&&&&&
The vector of gyroscopic moments acting on the disk is
( )
+
=
ttJJMM
PTy
z
sincossin
cossincos2 . (5.11)
DYNAMICS OF MACHINERY 12
The vector of unbalance forces and moments is
{ } ( )( )
( )( )
t
JJamJJam
t
JJamJJam
f
cPT
cd
sPT
sd
sPT
sd
cPT
cd
d
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 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 , . The inertia properties are m, PJ , TJ . The outer disk has only two rotational degrees of freedom, and . 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. FEA OF ROTOR-BEARING SYSTEMS 13
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 Youngs modulus, G shear modulus, IE - flexural rigidity,
sAG - effective shear rigidity, sA - reduced shear area, - shear coefficient, - mass density of shaft material, A = mass per unit length, I second moment of area of cross section, I = 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 and , 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
v
v
, { }
=j
j
i
i
szu
w
w
, (5.13, a)
and the corresponding vectors of nodal forces
DYNAMICS OF MACHINERY 14
{ }
=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 22 ,
so that
( )
AA
AAs
==
d
d2
2
.
For a hollow steel shaft, with outer diameter D and inner diameter d, the shear coefficient is
( )2
21033131
1
++=
DdDd..
.
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 = . The average shear strain (Fig. 5.11) is
5. FEA OF ROTOR-BEARING SYSTEMS 15
vv +=+
= xy
uxy .
The axial strain is
== y
xu
x .
The bending moment is
==A
zxz IEAyM d .
The equations in the X-Y plane are presented in Table 5.1.
Neglecting the && term, a convenient static relationship can be established between v and =
s
zGAEI-v , (5.14)
though, for the Bresse-Timoshenko beam, v and are kinematically independent.
Fig. 5.11
Introducing the dimensionless variable
lx= ,
so that
dd1
dd
dd
dd
l== xx ,
the static relationship between v and becomes [5.3]
DYNAMICS OF MACHINERY 16
Table 5.1
Bernoulli-Euler beam Bresse-Timoshenko beam
a) Equilibrium
.v 00
=+=+&&y
yz
T
,TM
.v 00
=+=+
&&&&
y
yz
T
,TM
b) Elasticity
xyzz IEM = .AGT,IEM
xysy
xyzz
==
c) Kinematics
.-v =
=0
,xy .-v
==
xy
xy ,
Elimination of xy Elimination of xy and xy
v == zzz IEIEM ( )-v==
sy
zz
AGT,IEM
Equations of motion
Elimination of zM and yT Elimination of zM and yT
0=+ vv v &&zIE ( )( ) .vv-v-
00
=+=
&&&&
s
sz
AG,AGIE
5. FEA OF ROTOR-BEARING SYSTEMS 17
22
dd
12dd1
=
vl (5.14, a)
where
212 lsz
AGIE= . (5.15)
Equations in the X-Z plane
The axial displacement at any point is
zu = . The average shear strain (Fig. 5.12) is
ww +=+
= xz
uxz .
The axial strain is
== z
xu
x .
The bending moment is
==A
yxy IEAzM d .
The equations in the X-Z plane are presented in Table 5.2.
Fig. 5.12
Neglecting the && term, the static coupling between w and is
( ) ( )=+= s
y
s
y
GAEI
GAEI
w (5.16)
DYNAMICS OF MACHINERY 18
Table 5.2
Bernoulli-Euler beam Bresse-Timoshenko beam
a) Equilibrium
.w 00
==+&&zzy
T
,TM
.w 00
==+
&&&&
z
zy
T
,TM
b) Elasticity
xzyy IEM = .AGT,IEM
xzsz
xzyy
==
c) Kinematics
.w==
,xz
.w+==
xz
xz ,
Elimination of xz Elimination of xz and xz
w == yyy IEIEM ( )w+==
sz
yy
AGT
,IEM
Equations of motion
Elimination of yM and zT Elimination of yM and zT
0=+ ww v &&yIE ( )( ) .www
0
0
=+=+
&&&&
s
sy
AG
,AGIE
5. FEA OF ROTOR-BEARING SYSTEMS 19
Introducing the dimensionless variable
lx= ,
the static relationship between w and becomes
( ) ( )22
dd
12dd1
=wl (5.16, a) where
212 lsy
AGIE= , 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
+++=+++=
v
Substitution in the static coupling condition (5.14, a)
( ) ( )230122331223 262231 BBBBBBAAA ++++=++ l yields
03 =B , 32 3 AB l= , 212 AB l= ,
+= 310 2
1 AAB l ,
so that ( ) is of second degree. The translation v and the rotation are approximated in terms of the
nodal displacements, using static shape functions:
( ) ( ) ( ) ( ) ( ) { }( ) ( ) ( ) ( ) ( ) { },uN~N~N~N~N~
,uNNNNNsyji
syji
=+++==+++=
4321
4321
ji
ji
vv
vvv (5.17)
where
4321 NNNNN = , 4321 N~N~N~N~N~ = .
DYNAMICS OF MACHINERY 20
An example of calculation is given here for ( )3N . 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 121 23 =+
AA ,
( ) 00 = 31 2 AA= , ( ) 0=l 023 23 =+ AA ,
wherefrom the four constants are obtained as
+= 12
3A , += 13
2A , += 11A , 00 =A ,
so that
( ) ( ) +++= 233 321 1N . The shear-modified shape functions for displacements are
( ) ( )[ ]( ) ( ) ( )( ) ( )( ) ( ) ( ) .N
,N
,N
,N
++++=
++=
+++=
+++=
2324
323
2322
321
211
231
12
21
1
12311
1
ll
ll (5.18)
For 0= , 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~
++=
++=
+++=
+=
231
1
6611
1
13411
1
6611
1
24
23
22
21
l
l
(5.19)
For 0=
5. FEA OF ROTOR-BEARING SYSTEMS 21
( ) ,NN~ ii = l
1 ( )41,..,i = . It is useful to introduce a third set of shape functions defined by
= dd1 NN~N l
which will be used in the derivation of the stiffness matrix.
Their expressions are
( ) ( )( ) ( ) .NN
,NN
+==+==
121
11
42
31 l (5.20)
Similar relations hold for the displacements in the X-Z plane
{ } { },uN~,uN
sz
sz
==w
(5.21)
because of the similar coupling relationship (5.16, a) between w and .
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 d2212121d 22222 +++++= &&&&&v , where
IIP 2= , A = , I = , sPPP JI l1== . (5.22)
Integrating, the kinetic energy for the shaft element becomes
( ) ( ) +++++=lll
&&&&&0
2
0
22
0
22 d21d
2d
2xJx
xwT P
sP
s v .
a) The translatory energy is
( ) +=l
&&0
221 d2
1 xwT s v .
DYNAMICS OF MACHINERY 22
Expressing velocities in terms of nodal coordinates and shape functions
{ } { } TTsysy NuuN &&&& === Tvv , { } { }syTTsy uNNu &&&&& == vvv T2 , { } { }szTTsz uNNu &&& =2w ,
this energy can be written
{ } [ ]
{ } { } [ ]
{ }szm
TTsz
sy
m
TTsy
s uxNNuuxNNuT
sT
sT
&444 3444 21
&&444 3444 21
&ll
+=00
1 d21d
21 .
The translational mass submatrix of the shaft element
[ ] ( ) ( ) = 10
d NNm TsT l (5.23)
is the same in the X-Y and X-Z planes.
b) The rotary energy is
( ) +=l
&&0
222 d2
1 xT s .
Expressing angular velocities in terms of nodal coordinates and shape functions
{ }syuN~ && = , { }szuN~ && = , { } { }syTTsy uN~N~u &&& =2 , { } { }szTTsz uN~N~u &&& =2 , this energy can be written
{ } [ ]
{ } { } [ ]
{ }szm
TTsz
sy
m
TTsy
s uxN~N~uuxN~N~uT
sR
sR
&44 344 21
&&44 344 21
&ll
+=00
2 d21d
21 .
The rotational mass submatrix of the shaft element
5. FEA OF ROTOR-BEARING SYSTEMS 23
[ ] ( ) ( ) =1
0
d N~N~m TsR l (5.24)
is the same in the X-Y and X-Z planes.
c) The gyroscopic effect energy is
=l&
03 dxT Ps ,
{ } [ ]
{ }syg
TP
Tsz
s uxN~N~uT
s444 3444 21
&l
=0
3 d .
The gyroscopic submatrix of the shaft element
[ ] ( ) ( ) [ ]sRTPs mN~N~g 2d10
== l . (5.25) The total kinetic energy of the uniform shaft element is
{ } [ ] { } { } [ ]{ } { } [ ]{ } 221
21
21 sPsysTszszsTszsysTsys JuguumuumuT ++= &&&&&
where [ ] [ ] [ ]sRsTs 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 36140294156 211 +++= , ( ) ( ) RTs ..m ll 15351753822 212 +++= , ( ) RTsm 367012654 213 ++= ,
DYNAMICS OF MACHINERY 24
( ) ( ) RTs ..m ll 15351753113 214 +++= , ( ) ( ) RTs .m 222222 10545374 ll +++++= , ( ) ( ) RTs .m 222224 5515373 ll +++= , ss mm 1423 = , ss mm 1133 = , ss mm 1234 = , ss mm 2244 = ,
( )21420 +=l
T , ( )22 130 += ll
R .
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 7211 = , ( ) Rsg l153212 = , Rsg 7213 = ,
ss gg 1214 = , ( ) Rsg 2222 10542 l++= , ( ) Rsg 2224 5512 l+= , ss gg 1223 = , ss gg 1133 = , ss gg 2334 = , ss gg 2244 = . Substitution of sT in Lagranges 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
0
or
[ ]{ } [ ]{ } { }sss fuGuM =+ &&& , (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
5. FEA OF ROTOR-BEARING SYSTEMS 25
( ) xAGxx
IEU xzxys ddd
dd
21d 22
22
++
+
= .
Integrating and substituting the shear strains we obtain
( ) ( ) ( )[ ] ++++=ll
0
22
0
22 d2
d2
xwAG
xIEU s v .
a) The flexural energy is
( ) ( ) +=+=lll
0
2
0
2
0
221 d2
d2
d2
xIExIExIEU .
Expressing rotations and curvatures in terms of nodal coordinates and shape functions
{ }syuN~= , { }syuN~ = l1 , where
( ) ( )dd= ,
{ } { }syTTsy uN~N~u = 21l2 , the bending energy can be written
{ } [ ]
{ } { } [ ]
{ }szk
TTsz
sy
k
TTsy uN
~N~IEuuN~N~IEuU
sB
sB
444 3444 21l
444 3444 21l +=
1
0
1
0
1 d21d
21 .
The stiffness submatrix for bending is
[ ] =1
0
ddd
dd
N~N~IEkT
sB l . (5.30)
b) The shear energy is
( ) ( ) ++=ll
0
2
0
22 d2
d2
xwAGxAGU ss v where
DYNAMICS OF MACHINERY 26
( ){ } { } { }sysysy uNuNN~uNN~ = == l1v , ( ){ } { }szsz uNuNN~ ==+ w ,
so that
{ } [ ]
{ } { } [ ]
{ }szk
Ts
Tsz
sy
k
Ts
Tsy uNNAGuuNNAGuU
sS
sS
444 3444 21l
444 3444 21l +=
1
0
1
0
2 d21d
21 .
The stiffness submatrix for shear is
[ ] =1
0
dNNAGk TssS l . (5.31)
The total strain energy of the uniform shaft element is
{ } [ ] { } { } [ ]{ }szsTszsysTsy ukuukuU 2121 += , where the stiffness submatrix is
[ ] [ ] [ ]sSsBs kkk += . (5.32) or
[ ]
+
+=2
22
2
22
3
sym00
00000
4sym612
264612612
11
l
ll
lllllll
l
IEk s .
Substitution of U in Lagranges 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. FEA OF ROTOR-BEARING SYSTEMS 27
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
+=+=
1
1 (5.33)
where LLcL aa cos= , LLsL aa sin= , RRcR aa cos= , RRsR aa sin= ,
La , Ra are the unbalance radii, and L , R are the unbalance phase angles. The corresponding distributed unbalance forces are
( ) ( )t,at,p syy 2= , ( ) ( )t,at,p szz 2= , where
( ) ( ) ( )( ) ( ) ( ) .tatat,a
,tatat,asc
ss
sz
ss
sc
sy
sincos
sincos
+==
The virtual work of these forces is
( ) ( ) +=ll
00
dd xpxpW zT
yT wv ,
{ } { }
{ } { } 44 344 21
l44 344 21l
sz
sy f
zTTs
z
f
yTTs
y pNupNuW +=1
0
1
0
dd .
The kinematically equivalent nodal forces are
{ } =1
0
2 d syTsy aNf l , { } =1
0
2 d szTsz 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
sin
d
d
cos
d
d
1
0
1
021
0
1
02 ll .
DYNAMICS OF MACHINERY 28
An example of calculation is given below
( ) ( ) ( )[ ]
( ) ( ) ( )
( ) ( ) ( )[ ].aaNaNa
aaNaN
cRcL
cRcL
cRcLsc
2018404211201
dd1
d1d
1
01
1
01
1
01
1
01
++++=
=+=
=+=
The vector of unbalance forces can be written
{ } ( )[ ][ ]
[ ][ ]
+
+= taa
a
aa
at
aa
a
aa
af
cR
cL
sR
sL
sR
sL
cR
cL
s sincos1120
2l ,
where
[ ] ( ) ( )( ) ( )
++++++++
=ll
ll
565440422018
545620184042
a ,
so that
{ } { }{ } { }{ }
+
= t
tQQ
fy
z
z
ys sincos .
The unbalance force vector is of the form
{ } ( )
+
+= ttfs
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
5. FEA OF ROTOR-BEARING SYSTEMS 29
( ) ( ) cRcL aa 201840421 +++= , ( ) ( ) cRcL aa 54562 +++= , ( ) ( ) cRcL aa 404220183 +++= , ( ) ( ) cRcL aa 56544 +++= , ( ) ( ) sRsL aa 201840425 +++= , ( ) ( ) sRsL aa 54566 +++= , ( ) ( ) sRsL aa 404220187 +++= , ( ) ( ) sRsL aa 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.
DYNAMICS OF MACHINERY 30
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
5. FEA OF ROTOR-BEARING SYSTEMS 31
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 with respect to the ZY reference frame. The transformation of displacements can be written
[ ]
=
=
wv
wv
wv
R
cossinsincos
.
The transformation of forces is
DYNAMICS OF MACHINERY 32
[ ]
=
=
z
y
z
y
z
y
TT
RTT
TT
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 [ ] [ ][ ][ ]
== cs
sckkkk
cssc
RkRkzzzy
yzyyT , (5.36)
where cos=c and sin=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
+
=
00
, (5.37)
where
( )zyyzs kkk += 21 , ( )zyyza kkk = 21 . (5.38) The transformed stiffness matrix becomes
[ ] [ ] [ ] [ ]( ) [ ] [ ] [ ]asTas kkRkkRk +=+= where, denoting cos=c and sin=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 ;
b) when
zzyy
skk
k=
22tan ,
5. FEA OF ROTOR-BEARING SYSTEMS 33
the symmetric part [ ]sk is diagonal and the angles and 090+ 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
[ ]
+=
=
2sin2cos
2cos2sin
212
221
12
21
kkkkkk
cssc
kkkk
cssc
k
and for 045= [ ]
+=
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 and 090+ ; 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;
DYNAMICS OF MACHINERY 34
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
=
+
&&&&
-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
[ ] = 00 0bijb
ijkk , [ ] = 00 0
bijb
ijcc , ( )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]
5. FEA OF ROTOR-BEARING SYSTEMS 35
+
+
=
wv
wv
wv
&&&&
&&
MCccC
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
DYNAMICS OF MACHINERY 36
[ ]
=
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. FEA OF ROTOR-BEARING SYSTEMS 37
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
,,,,,,,,,,,,,x 444444 3444444 21 L44444 344444 21 L 11112211 = 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
[ ] [ ] [ ][ ] [ ]
=m
mM
00
, [ ] [ ] [ ][ ] [ ]
+
+=zzzy
yzyy
kkkkkk
K ,
[ ] [ ] [ ][ ] [ ] [ ] [ ][ ] [ ] +
=0
0g
gcc
ccC
zzzy
yzyy , (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.
DYNAMICS OF MACHINERY 38
Fig. 5.19
Fig. 5.20
Fig. 5.21
5. FEA OF ROTOR-BEARING SYSTEMS 39
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
DYNAMICS OF MACHINERY 40
The vectors of mass unbalance forces are
{ } { } { } tFtFf sc 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 [ ] [ ] [ ][ ] [ ]
= 0M
MCA , [ ] [ ] [ ][ ] [ ]
=
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. FEA OF ROTOR-BEARING SYSTEMS 41
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
{ } { } te Rq = , equation (5.57) can be written
[ ] [ ]( ){ } { }0=+ RBA . (5.58) There are Nr 2= eigenvalues r obtained from [ ] [ ]( ) 0det =+ BA , (5.59)
and N2 right eigenvectors { }Rr satisfying the generalized eigenvalue problem [ ]{ } [ ]{ }RrR AB = , ( )N,...,r 21= . (5.60) For the transposed equation
[ ] { } [ ] { } { }0=+ qBqA TT & , assume solutions of the form
{ } { } te Lq = . This yields
[ ] [ ]( ){ } { }0=+ LTT BA . (5.61) The eigenvalues are given by the equation
[ ] [ ]( ) 0det =+ TBA which has the same solutions as those of equation (5.59).
Because the transposed of equation (5.61) is
DYNAMICS OF MACHINERY 42
{ } [ ] [ ]( ) 0=+ BATL , { }L are referred to as left eigenvectors, solutions of the eigenproblem
[ ] { } [ ] { }LrTrLrT AB = , ( )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
(5.63)
so that
r
rr
= . (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 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 1N eigenvectors of the original problem.
An alternative is to use the form (5.56). For an assumed solution
{ } { } teux = , (5.66) the homogeneous form of equation (5.56) becomes
5. FEA OF ROTOR-BEARING SYSTEMS 43
[ ] [ ][ ] [ ]
{ }{ }
[ ] [ ][ ] [ ]
{ }{ }
{ }{ }
=
+
000
00
uu
CKM
uu
MM
(5.67)
or
[ ] [ ]
[ ] [ ] [ ] [ ][ ] [ ][ ] [ ]
{ }{ } { }00
0011 =
u
uI
ICMKM
I .
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 i+= , rrr i= , (5.68) and they are functions of the rotating assembly spin speed .
The imaginary part r of the eigenvalue is the damped natural frequency of precession (whirl speed or precession speed), and the real part r is the damping constant. A stable mode requires a non-positive value for r .
Often, the damping is expressed in terms of the damping ratio
r
rr
,
or in terms of the logarithmic decrement
r
rrr
22 = .
It is common practice to plot both the precession natural frequencies and damping constants as a function of the rotor spin speed . 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, oi , also called a threshold speed, th .
Plots of only the precession natural frequencies versus spin speed are commonly referred to as Campbell diagrams. When points are marked on the
DYNAMICS OF MACHINERY 44
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 of a single-shaft rotor system is defined as a spin speed for which a multiple of that speed coincides with one of the systems natural frequencies of precession. An excitation frequency line of equation
= is included in Fig. 5.25. When equals r , the excitation r 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
( ){ } { } { }
{ } { }( ) ( ) { } { }( ) ( ) .babauutx
trr
trr
tr
trr
rrrr
rr
ii eiei
ee+
++==+=
( ){ } { } { }
{ } { }( ) .tbtabatx
rrrrt
tt
r
tt
rt
r
r
rrrrr
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).
5. FEA OF ROTOR-BEARING SYSTEMS 45
Fig. 5.26
The factor tre2 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 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 sincos += , defining a harmonic motion with frequency r .
The two components of the translational motion at any station are
DYNAMICS OF MACHINERY 46
,tt
,tt
rsrc
rsrc
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, i= , so that equation (5.72) becomes [5.8]
[ ] [ ] [ ]
[ ] [ ] [ ]{ } { }{ } { }
{ }{ }
=
++
00
ii
ii
2
2
IR
IR
z
yzzyy
mkggmk
,
which gives four real coupled matrix equations. It can be shown that the solutions of the equations are proportional to each other
{ } { }RI yy = , { } { }IR zz = , where 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
5. FEA OF ROTOR-BEARING SYSTEMS 47
and can be divided by ( )i1+ . 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 ie= , { } { } tXx ie= , equation (5.45) becomes
[ ] [ ][ ] [ ]
[ ] [ ][ ] [ ]
{ }{ }
{ }{ }
=
+
0i0
00
iF
XX
MK
MMC
or
[ ] [ ]( ){ } { }PQBA =+i where
{ } { }{ }
=
XX
Q i , { }{ }{ }
=
0F
P .
Assume a solution of the form
{ } { } rNr
rRQ
==
2
1
, (5.73)
where r are modal coordinates. Multiplying to the left by { }TrR and considering the bi-orthogonality
relations (5.53), the rth decoupled equation is
{ } [ ] [ ]( ){ } { } { }PBA TrLrrRTrL =+i
DYNAMICS OF MACHINERY 48
or
( ) { } { }PTrLrrr =i . The rth modal coordinate is
{ } { }( )rr
Tr
L
rP
= i .
From equation (5.63), the vector of physical coordinates is
{ } { } { }( ) { }PQ rrTr
Lr
RN
r
== i
2
1
,
whose upper half is
{ } { } { }( ) { }FX rrTr
Lr
RN
r
== i
2
1
, (5.74)
where { }rR and { }rL 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 sincos += , (5.75) the steady-state response has the form
{ } { } { } tXtXx sc sincos += . (5.76) Substitution of (5.75) and (5.76) into (5.50) yields
[ ] [ ]( ){ } [ ]{ } { }[ ]{ } [ ] [ ]( ){ } { }.FXMKXC ,FXCXMK ssc csc =+
=+2
2
(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. FEA OF ROTOR-BEARING SYSTEMS 49
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 . Its displacement components in the y and z directions are
.tztzz,tytyy
sc
sc
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
DYNAMICS OF MACHINERY 50
( )( ) ,tAtAtAz
,tAtAtAy
zzzzzz
yyyyyy
sinsincoscoscos
sinsincoscoscos
=+==+=
in two perpendicular directions.
It is seen that
( ) 2122 scy yyA += , c
sy y
y=tan ,
( ) 2122 scz zzA += , c
sz z
z=tan .
The parameters of the elliptic motion are generally a function of the spin speed . Two particular cases are of interest [5.1]: a) If zy = , 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
5. FEA OF ROTOR-BEARING SYSTEMS 51
The parametric equations (5.68) define an ellipse, which is inclined with respect to the y-z axes. Solving first for tcos and tsin as
( ) ( )( ) ( ) ,yzzyzyyzt
,yzzyyzzyt
scsccc
scscss
==
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 (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
,tay
+=+=
sincos
1
1 (5.80)
where 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
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, and in terms of cy , sy , cz and sz . This will be done in the following by a different approach.
DYNAMICS OF MACHINERY 52
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 tbtftbtf rrrrr ++ +=+= iiii eeee . (5.83) From equations (5.78)
( ) ( )( ) ( ) ,zzz
,yyy
tts
ttc
tts
ttc
iiii
iiii
eei2
1ee21
eei2
1ee21
++=
++=
so that
( ) ( ) ( ) ( ) .zyzyezyzyezyr
cssct
cssct
+++
+++==+=
21i
21
21i
21
i
ii
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
5. FEA OF ROTOR-BEARING SYSTEMS 53
( ) ( )( ) ( ) .erzyzyr
,erzyzyr
b
f
bcsscb
fcsscf
i
i
21i
21
21i
21
=++=
=+++= (5.84)
The amplitudes and phase angles are given by
( ) ( )( ) ( ) .
zyzy,bazyzyr
,zyzy,bazyzyr
sc
csbcsscb
sc
csfcsscf
+==++=++=+=+++=
tan22
1
tan22
1
22
22
(5.85)
The semi-major axis is
bf rra += , (5.86)
( ) ( ) ( ) 2222222222241
21
csscscscscsc zyzyzzyyzzyya +++++++= .
The semi-minor axis is
==sc
scbf zz
yya
rrb det1 . (5.87)
The inclination of the major axis (attitude angle) is
( )bf += 21 . (5.88) From the condition of phase coincidence
bf tt +=+= , fbt =2 , where t is the time when P is in the point of maximum displacement amplitude.
The following useful relations can be established
( )222222tanscsc
scsc
zzyyzzyyt +
+= , (5.89)
( )222222tanscsc
sscc
zzyyzyzy+
+= , (5.90)
The precession is forward when
bf rr > , 0>b , cssc zyzy > .
DYNAMICS OF MACHINERY 54
The precession is backward when
bf rr < , 0
5. FEA OF ROTOR-BEARING SYSTEMS 55
Point A on the y axis corresponds to 0=t . Point B corresponds to t, and point C to 2+t . Note that points B and C correspond to a time interval of
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 , the point B moves along the ellipse with variable angular velocity.
Denoting AOB= , the angular position at time t is given by t
ab
yz tantan == . (5.92)
The variation of as a function of t is shown in Fig. 5.35. The deviation from the straight line indicates a variable angular velocity.
Fig. 5.35
DYNAMICS OF MACHINERY 56
Indeed, the angular velocity of the motion along the ellipse is
( )t
ab
tab
tt
22
2
tan1
sec
dd
dd
+===& . (5.93)
The variation of & as a function of t is plotted in Fig. 5.36. It is seen that & varies between ( ) ab= and ( ) ba= so that = .
Fig. 5.36
In conclusion, the angular speed of the precession motion is not & - that of the point B along the elliptic orbit, but the speed 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. FEA OF ROTOR-BEARING SYSTEMS 57
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
DYNAMICS OF MACHINERY 58
[ ] [ ][ ] [ ]
{ }{ }
[ ] [ ][ ] [ ]
{ }{ }
[ ] [ ][ ] [ ]
{ }{ }
{ }{ }
=
+
+
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
{ } { }{ }[ ]
[ ] [ ] { } [ ]{ }aaoaooa
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).
5. FEA OF ROTOR-BEARING SYSTEMS 59
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.
DYNAMICS OF MACHINERY 60
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 &&& , (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
5. FEA OF ROTOR-BEARING SYSTEMS 61
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 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)
[ ] { } { } { }[ ]L L21= . (5.109) The coordinate transformation is
{ } { }{ } [ ] { }{ } [ ] [ ][ ] [ ] { }{ } [ ]{ }uuuuuZYx zyzy =
=
=
=
0
0 (5.110)
where { }u is the reduced state vector. Substitution of (5.110) into (5.106) and pre-multiplication by [ ]T yields
the reduced set of equations of motion (5.97)
DYNAMICS OF MACHINERY 62
where
[ ] [ ] [ ] [ ]= MM Tred , { } [ ] { }ff Tred = , [ ] [ ] [ ] [ ]( ) [ ] += GCC bTred , (5.111) [ ] [ ] [ ] [ ]( ) [ ] += 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 { }j 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.
5. FEA OF ROTOR-BEARING SYSTEMS 63
The reduction of t
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