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Journal of Sound and Vibration 412 (2017) 389–409
Contents lists available at ScienceDirect
Journal of Sound and Vibration
journal homepage: www.elsevier.com/locate/jsvi
Characterisation and calculation of nonlinear vibrations in gas
foil bearing systems–An experimental and numerical
investigation
Robert Hoffmann * , Robert Liebich
Chair of Engineering Design and Product Reliability, Berlin Institute of Technology (TU Berlin), Str. d. 17. Juni 135, 10623 Berlin, Germany
article info
abstract
Article history:
Received 9 January 2017
Revised 27 September 2017
Accepted 28 September 2017
Available online XXX
This paper states a unique classification to understand the source of the subharmonic vibrations of gas foil bearing (GFB) systems, which will experimentally and numerically tested.
The classification is based on two cases, where an isolated system is assumed: Case 1
considers a poorly balance rotor, which results in increased displacement during operation and interacts with the nonlinear progressive structure. It is comparable to a DuffingOscillator. In contrast, for case 2 a well/perfectly balanced rotor is assumed. Hence, the only
source of nonlinear subharmonic whirling results from the fluid film self-excitation. Experimental tests with different unbalance levels and GFB modifications confirm these assumptions.
Furthermore, simulations are able to predict the self-excitations and synchronous and
subharmonic resonances of the experimental test. The numerical model is based on a
linearised eigenvalue problem. The GFB system uses linearised stiffness and damping
parameters by applying a perturbation method on the Reynolds Equation. The nonlinear bump structure is simplified by a link-spring model. It includes Coulomb friction
effects inside the elastic corrugated structure and captures the interaction between single
bumps.
© 2017 Published by Elsevier Ltd.
Keywords:
Gas foil bearings
Nonlinear vibrations
Linearised method
1. Introduction
Gas foil bearings (GFBs) have successfully been introduced into small turbo machinery for more than 40 years, e.g. air cycle
machines, turbo compressors, turbochargers and compressors of fuel cells. Major advantages of compliant foil bearings are low
drag friction, high speed operation, high temperature endurability and the omission of an oil system, [1]. In a bump type gas
foil bearing the elastic bearing wall comprises a bump and a top foil made of thin sheet metal. Both foils are fixed with the
bearing sleeve, e.g. by spot welds. Due to the eccentrically rotating bearing journal a fluid dynamic pressure field p(z,  ) is
generated in the aerodynamic wedge and deforms the elastic structure h(z,  ) and an optimal film thickness is achieved, see
Fig. 1. Thus, higher load capacities compared to rigid gas bearings are generated, [2]. The deformation of the foils may activate
sliding contacts inside the elastic structure delivers additional damping and improves the dynamic behaviour compared to rigid
gas bearings. Nevertheless, the low viscosity of the air film results in an overall low damping level, which is still a key issue
because the poor damping ability may result in nonlinear vibrations. Those can significantly affect the rotor dynamic perfor-
* Corresponding author.
E-mail addresses: robert.hoffmann@posteo.de (R. Hoffmann), robert.liebich@tu-berlin.de (R. Liebich).
https://doi.org/10.1016/j.jsv.2017.09.040
0022-460X/© 2017 Published by Elsevier Ltd.
390
R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
mr
nOSSV
Nomenclature
Abbreviations
CG
Center of Gravity
BP
Balance Plane
FE
Finite Element
GFB
Gas Foil Bearing
ODE
Ordinary Differential Equation
OSSV
Onset Speed of Subharmonic Vibration
RE
Reynolds Equation
Latin
bs
c
c0
ct
cd
cij
ct
ei
e0
Δei
f
fi
f OSSV
ho
shim width
radial bearing clearance
nominal radial bearing clearance
Petrov-model parameter
structural damping
linearised damping value i, j = x, y
slope parameter Petrov-Model
journal displacement i = x, y
zero order journal displacement
perturbed journal displacement i = x, y
frequency
eigenfrequency for mode i = 1, 2 … n
onset speed of subharmonic vibration; frequency
perturbed journal displacement i = x, y
bearing reaction force vector
friction force vector
pressure force vector
unbalance force vector
film thickness
vertical displacement link-spring-model
dynamic vertical displacement link-springmodel
bump height
rigid term of the film thickness
compliant term of the film thickness
zero order film thickness
perturbed film thickness i = x, y
perturbed compliant film thickness term i = x,
y
operation film thickness
j
k1
k2
kd
keq
kij
kt
ks
l
lb
lr
ls
ΔlCG,i
complex number j = −1
interaction spring stiffness link-spring-model
spring stiffness link-spring-model
dynamic structural stiffness
equivalent bump stiffness (keq = Ab /K)
linearised stiffness value i,j = x, y
Petrov-model stiffness parameter
static structural stiffness
bearing length
half bump length
shaft length
shim length
distance from CG to bearing midline i = F, R
Δei
fB
ff
fp
fU
h
Δh
Δĥ
hb
hr
hc
h0
hi
hc,i
√
p
pa
p0
pi
r
sb
Δs
t
tb
tf
u
w
x
xs
xbot , xup
u
y
z
Ab
A
C
CB
Da
G
E
F
Fr , Fl
F bot , F up
F b,x
Fs
Fx
Fp
̂
Fp
rotor mass
onset speed of subharmonic vibration; rotor
speed
pressure
ambient pressure
zero order pressure
perturbed pressure i = x, y
rotor displacement (magnitude)
bump pitch
shim thickness
time
bump thickness
top foil thickness
displacement vector
loading vector w = {W x ,W y }T
Cartesian coordinate
displacement Petrov-model
horizontal displacements link-spring model;
bottom and up
system vector; dynamic structural model
Cartesian coordinate
Cartesian coordinate
Bump surface (Ab = sb l)
system matrix dynamic link-spring-model
damping matrix
linearised GFB damping matrix
nominal shaft diameter
gyroscopic matrix
Young’s modulus
force
right and left normal force link-spring-model
friction force link-spring-model
horizontal beam lever force link-spring-model
interaction force link-spring-model
horizontal reaction force
bump load
Ji
K
dynamic bump load (amplitude)
moment of inertia i = x, y, z
stiffness matrix
( )
K
bump compliancy K =
KB
KBump
ΔL
L′
linearised GFB stiffness matrix
bump stiffness matrix
horizontal displacement link-spring-model
modified horizontal displacement link-springmodel
mass matrix
normal contact force
normal contact force link-spring-model
bump number
shim number
bearing journal radius
bump radius
ambient temperature
journal rotational speed U = RΩ
unbalance front (F) and rear (R)
bearing load i = x, y
M
N
N bot , N up
Nb
NS
R
Rb
Ta
U
UF , UR
Wi
2sb
E
lb
tb
3
(
1 − 2
)
R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
Greek



i
i
a



*
Ω
ΩOSSV

bump angle link-spring-model
attitude angle
structural loss factor  = c/kd
damping ratio mode i = 1, 2 … n
eigenvalue i = i ± j∗i mode i = 1, 2 … n
dynamic viscosity
friction coefficient
Poisson’s ratio
angular excitation frequency ( = 2 f )
undamped eigen angular frequency
angular rotor speed (Ω = 2 n)
angular onset speed of subharmonic vibrations
density

s
0
p
Skripts
i
i
391
circumferential coordinate
shim position
bump angle
scaling factor
bump index i = 1, 2, 3 …
index i = 1, 2, 3 … or direction i = x, y
Mathematical symbols
≈
approximately equal
≡
is defined as
̇ = d(…)∕ dt time derivation
(…)
Fig. 1. GFB with a dynamic pressure field.
mance. Experimental and numerical investigations of GFBs have shown nonlinear vibrations, where sub- and superharmonic
vibrations, jumps and bifurcations are present, [2–14].
For reducing nonlinear vibrations to achieve better dynamic performance and increased load capacity several passive methods and devices have been introduced. Structural modifications, e.g. variable stiffness distributions along the axial or/and the
circumferential direction, influence the fluid film and stiffens the system, enhance damping and improve the dynamic performance, [3,15]. Using a thin layer of a visco elastic material [16] or a copper coating [17,3] between top and bump foil increases
damping as well. Inserting metal shims between the bearing sleeve and bump structure affect the bearing clearance c( ), which
has a major impact on the aerodynamic wedge, and the preload of the elastic structure [8,10,18]. In addition, the wedges due
to shimming enhance the fluid film character, resulting in a higher stiffness and damping compared to the same GFB without
shimming, [19].
However, in the past the source of these vibrations has not been analysed in detail. Instead a lot of numerical and experimental work has been undertaken for better performance and increasing load capacity. A detailed explanation including experimental and numerical valid results is still missing. In this paper, a classification of nonlinear vibrations in a gas foil bearing system
will be delivered and tested in experiments and simulation. Rotor dynamic coast down tests with different balance grades will
be used for analysing the vibrations. A numerical method based on a linearised system will be used for evaluating the onset
speed of subharmonic vibration (OSSV) and subharmonic resonance. A correct and robust estimation of the OSSV and nonlinear
effects is significant for the whole design process and is essential to avoid additional costs.
2. Theoretical classification of nonlinear vibrations in gas foil bearing systems
The first attempt of explaining the nonlinear vibration is shown in Ref. [9]. The authors assigned the source toward the
elastic structure. Due to the progressive force displacement character of the bearing structure the system behaves like a forced
Duffing oscillator. In contrast, numerical [20–22] and experimental investigations [2,23] have shown nonlinear vibrations in
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R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
well-balanced systems, those are caused by a fluid film self-excitation.
Considering Fig. 2 two sources (Case 1 and Case 2) of nonlinear vibrations of an isolated system (e.g. no base excitation, no
internal excitation due to electric forces nor fluid-film forces except the bearings are included) can be stated to understand the
complexity of GFB systems.
2.1. Case 1: direct forcing/low balance grade
Low balance grades result in a forced rotor vibration. Due to increased rotor amplitudes, the lubrication film is highly
squeezed and the nonlinear elastic structure is more displaced. Because of the progressive force displacement character of
the lubrication film and the underlying elastic structure the system behaves like a duffing oscillator. Nonlinear vibrations e.g.
sub- and superharmonic vibrations and resonances, jumps and bifurcations can be generated, [24–27].
2.2. Case 2: self-excitation/high balance grade and poor bearing design
Nevertheless, high balance grades will result in self-excitation, characterised by a Hopf-Bifurcation, if the bearing design is
not appropriate. Furthermore, it is important to note that a self-excitation is independent of any kind of external load excitation
e.g. unbalance forces. Hence, the onset of self-excitation vibration (Hopf-Bifurcation) is therefore not driven by the unbalance
and may results limit cycle if damping is sufficient. However, if the rotor amplitudes may significantly amplified by the selfexcitation the lubrication film and the underlying elastic structure will be more squeezed and the duffing system characteristics
will clearly appear. Hence, nonlinear vibration effects due to the elastic structure as described in case 1 are possible as well. If
the rotor is well balanced the only source of nonlinear vibrations can be assigned to a fluid film self-excitation. However, it is
important to note, that the system is well isolated from surrounding excitations. In Refs. [10,28] experiments have shown some
vibrations due to base excitation in passenger cars.
3. Theoretical analysis of the nonlinear vibration
Eq. (1) describes a nonlinear rotor system. Where the global matrices for mass M, damping C and stiffness K are linear and
delivered by the rotor structure. Gyroscopic effects are captured by G(Ω).
̈ + [ − (Ω)]̇ +  = U (Ω, t) + B (̇ , , Ω) + 
(1)
The right side of Eq. (1) includes the unbalance vector fU , the nonlinear bearing reaction force vector fB and the loading
vector w due to mass forces of the rotor itself. For evaluating the nonlinear vibrations this nonlinear system can be solved
Fig. 2. Source classification of nonlinear vibration for a GFB system.
R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
393
Fig. 3. Evaluation of the self-excitation:  i vs. Ω vs. ∗i .
in time domain, [29–31]. Due to the complex elastic structure and the fluid film the bearing multiphysics model is very time
consuming, particularly if a nonlinear structural model including friction is applied [32].
Considering a well-balanced rotor system, where fU is neglected, the source of nonlinear vibrations is caused by the fluid
film self-excitation, see Fig. 2. The self-excitation (Hopf-Bifurcation) of the system emerges, when the onset speed of subharmonic vibrations Ω0 (OSSV) is reached. This bifurcation is characterised by an eigenvalue i = i ± j∗i of the Jacobi matrix of
the autonomous system, where the damping ratio  i of the mode i eigenvalue becomes zero, [33]. As shown in Ref. [21], an
autonomous system can be approximated with sufficient accuracy by a linearised system. This frequency domain method is
very fast in calculation and agrees well with the results of a time domain analysis, [21]. A good correlations for the simplified
linearised frequency domain method are achievable for equivalent bump stiffness (keq = Ab /K f ) of approx. keq = 0.8 MN/m, [34].
However, very soft and highly loaded structures may results in higher differences between both stability analysis methods, [20].
̈ + [(Ω, ) − (Ω)]̇ + (Ω, ) = 
(2)
In detail, the frequency domain method uses linearised bearing parameters (stiffness B (Ω, ) ⊆  and damping B (Ω, ) ⊆ )
for a linear eigenvalue problem, where a homogeneous linear ODE-system of Eq. (2) needs to be solved. Note that the system
depends on the rotation speed Ω and on the excitation frequency  (due to the compressible gas of the bearings). Therefore,
the eigenvalues i (Ω, ) = i (Ω, ) ± j∗i (Ω, ) are functions of the rotational speed Ω and the excitation frequency . The
intersection, where the real part of an eigenvalue becomes zero ( i (Ω0 , ) = 0) and the imaginary part is equal to the excitation
frequency (∗i (Ω0 , ) = ), states the point of a self-excitation, see Fig. 3.
The bearing parameters (KB (Ω, ) and CB (Ω, )) are calculated by using the infinitely small perturbation method of Lund’s
approach [35]. Where the numerical parameter identifications use complex nonlinear structural models, which take bump interactions and frictional contacts into account [19,32,36,37,38]. Furthermore, a unique assumption of distinguishing between a
static and a dynamic structural stiffness is used [39]. This assumption has been successfully tested on a first generation GFB [37],
where experimental results from Ref. [40] were used for validation.
4. Theoretical model for linearised bearing parameters
In this section the numerical model for calculating the linearised bearing parameter is given, which is mainly based on the
work [37]. Fig. 1 shows a compliant structure of a GFB in an inertial coordinate system (x, y and  ). A turning journal with an
angular speed Ω and a centre displacement ex and ey generates a forced slip stream with a film thickness of h(z,  ). It results in a
dynamic pressure field p(z,  ), which produces a reacting force fB . An equilibrium condition is reached if the sum of the loading
vector w = {W x , W y }T and the reacting force vector fB is zero.
l
B = R
∫0 ∫0
2
(
p(z,  ) − pa
)
{
cos 
sin 
}
dzd
(3)
An integration of the pressure field along the axial and circumferential directions yields a reacting force vector fB , Eq. (3). It acts
under the attitude angle  . Note that the axial direction z is related to the bearing length l. Due to the pressure field, an elastic
deformation of the foil structure is given by hc (z,  ) and is calculated by a structural model as shown below.
h(, z) = c( ) + ex cos( ) + ey sin( ) + hc (z,  ) + ho (Ω)
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
(4)
hr ( )
Eq. (4) describes a perfectly aligned journal, expansion effects due to temperature gradients and centrifugal forces are neglected.
Note that these effects should be taken into account if thin journal walls and high temperature operations are present, which is
neglected in this work. The film thickness is composed of a rigid term hr ( ), including the clearance c( ) and the journal centre
displacements, of a compliant term hc (z,  ) and of a term ho (Ω). The last term takes journal diameter changes due to centrifugal
394
R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
forces into account, see Appendix A. Note that the clearance is a theoretical parameter, which can be estimated experimentally.
Further, structural modifications, e.g. shimming may affect the clearance in circumferential direction, see Refs. [8,10,18,19]. The
pressure field p(z,  ) is calculated by solving the Reynolds Equation (RE). Due to low fluid film shear losses thermal effects in
experiments and simulation are neglected for the lubrication film model.
(

p
ph3
z
z
)
1 
+ 2
R 
(
p
ph

)
3
=
Ωa  (ph)

2
+ 12a
 (ph)
t
(5)
Eq. (5) considers a compressible, isothermal and isoviscous fluid. It links both pressure field and film thickness under the presence of journal rotation speed U = RΩ. The following boundary conditions are applied: p(z = 0,  ) = p(z = l,  ) = pa and p(z,
 = 0) = p(z,  = 2 ) = pa . If the pressure falls underneath the ambient pressure condition p( , z) < pa , a top foil lift-off will
occur and sub ambient pressure regions (p( , z) < pa ) will reach ambient level (p = pa ). Thus, sub ambient pressures are set
to p = pa . Bearing parameters are calculated by using Lund’s approach [35], which is based on a perturbation method. In the
equilibrium state of a given static load w under steady speed condition (Ω = const.) a harmonic perturbation is superimposed
with a frequency  and infinitely small eccentricity (Δex,y ≪ c0 ), where c0 is the nominal clearance. In addition, the pressure
field and the film thickness are affected by perturbations, where i = x, y.
e = e0 + Δei ejt
(6)
p = p0 + Δei pi ejt
(7)
(
)
h = h0 + Δei hi + hc,i ejt
(8)
Substituting Eq. (6)–(8) into Eq. (5), while neglecting terms of higher order, generates zero and first order RE, which have to be
successively solved.
[
kxx
kxy
kyx
kyy
]
[
+ j
cxx
cxy
cyx
cyy
]
⏟⏞⏞⏟⏞⏞⏟
⏟⏞⏞⏟⏞⏞⏟
B ()
B ()
l
2
= −R
∫0 ∫0
2
[
px cos 
py cos 
px sin 
py sin 
]
dzd
(9)
The perturbed pressure fields px and py are used to calculate the linearised stiffness and damping matrices (Eqs. (2) and (9)).
The calculation delivers complex matrix elements, where the real part results in the stiffness kij and the imaginary part in the
damping part jcij (i, j = x, y). A more detailed explanation is given in Ref. [41].
4.1. Structural model
A detailed description of the two nonlinearities caused by the fluid film and the elastic structure is necessary in order to
achieve valid results. Hence, a simple as well as detailed model needs to be introduced. It includes following features and
assumptions:
• Top foil displacement including axial and circumferential components.
• Nonlinear frictional contacts between top- and bump-foil and between bump-foil and housing are taken into account by
using the Coulomb model for static and dynamic analyses.
Each bump i can interact with its surrounding the preceding i − 1 and the subsequent bump i + 1.
The applied bump load F p is concentrated on the top centre of the bump.
Bumps are reduced towards rigid segments linked by pivots, while bump interactions are transmitted by linear springs.
Bump deformation along the axial direction is assumed to be constant, while longitudinal deformations of bumps are
neglected.
• No bump-foil separation from the housing is possible.
• All deformations are elastic.
•
•
•
•
The stationary pressure field p0 of the zero order RE results in a static bump force F p,0 , which preloads the bump structure.
Fp with an excitation frequency  additionally acts on the elastic structure small hysteresis
If a dynamic bump force amplitude ̂
due to sliding contacts will generated for each bump. The small dynamic hysteresis has significantly higher slopes compared
to a static load path (dashed line) as shown in Fig. 5. If the sticking phase is longer compared to the slip phase, the slope
hysteresis is increased and the local stiffness of the bump will increase as well. Extended stick phases are generated by decreased
displacements and force amplitudes.
The two structural stiffness ks and kd have to be calculated, where the static stiffness ks is used for the zero order RE (static
stationary case) and the dynamic stiffness kd are applied to solve the perturbed first order RE (dynamic case), which has been
shown in Ref. [39]. In addition, damping cd due to frictional contacts inside the structure is considered for the perturbed first
order RE, it is based on the enclosed local hysteresis, see Fig. 5. However, both models are using the same FE structure, where a
realistic structure (Fig. 4 (a)) is reduced to an equivalent model (Fig. 4 (b)): The top foil is described by a thin 2D plate, where
R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
395
3D effects due to normal tensions are neglected. These finite plate elements are linked to equivalent nonlinear
(
) bump springdamper elements, calculated by the structural models. Both the static and dynamic bump matrix Bump p , f are a function
of the loading state of the pressure field fp and the frictional contact forces ff . In addition, the top foil’s Young’s-modulus E is
increased by a factor of 4 to obtain appropriate sagging effects between adjacent bumps [42].
4.1.1. Structural bump model
For describing the elastic bump strip an equivalent link-spring model is used to describe the structural behaviour; it is
mainly based on the work of [38]. Fig. 6 (a) shows the free punch and the kinematic of the pivot. The kinematic of the linkspring-structure is described by the horizontal displacement ΔLi and the angle  i .
i
√
(
ΔL =
2Rb sin
√
tan( i ) =
(  ))2
0
2
( (
( ))
)2
( )
− Rb 1 − cos 0 − Δhi − Rb sin 0
a1 − (a2 + ΔLi )2
,
a2 + Δ L i
with
2
a1 = 4R2b sin
( )
0
2
, a2 = Rb sin(0 )
i
= Fsi+1 + Fbi ,x − Fsi − Fbi+,x1 ≡ Fxi
Fbot
(10)
(11)
(12)
The force equilibrium Eq. (12) is given for a bump segment I. The contact forces between bump/housing (index bot) (and
)
bump/top foil (index up) are included. Where beam lever forces for the i and i + 1 bump segments are given by Fbi ,x = Fri ∕ tan i
( )
and Fbi+,x1 = Fli ∕ tan i as well as the interaction forces Fsi = 2k1 ΔLi and Fsi+1 = 2k1 ΔLi+1 due to horizontal displacement ΔLi and
i
ΔLi+1 of the linear spring k1 , based on Castigliano theorem [38]. The right- and leftward forces include the friction force Fup
and
the concentrated bump loading force Fpi .
Fri =
Fpi
2
−
i
Fup
2
tan( i )
and Fli =
Fpi
2
+
i
Fup
2
tan( i )
(13)
4.1.2. Static structural bump model (ks )
For the static structure model the motion state is checked by the force equilibrium Eq. (12), where the sign-function of the
coulomb friction is considered.
i
Fup
= sign(ẋ up )Fpi ,0  i
(
)
i
and Fbot
= sign(ẋ bot ) · Fri + Fli+1  i
Fig. 4. GFB Structure: (a) real structure; (b) model (bump spring model and top foil plate model).
Fig. 5. Structural hysteresis of the elastic bump structure. Stationary loading (dashed line) and dynamic perturbation results in local hysteresis for each bump.
(14)
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R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
Fig. 6. Linked-spring-model: (a) Free punch of a single interacting bump segment I; (b) kinematic of a rigid bump lever; (c) kinematic for a stick-stick condition.
Table 1
Kinematic cases link-spring-model.
Motion
Case
rightward
leftward
stick
i
Fbot
< Fxi
i
−Fbot
> Fxi
i
i
−Fbot
≥ Fxi ≤ Fbot
Introducing the contact forces of Eq. (14) into the Eqs. (12) and (13) the motion states by considering the force equilibrium (Eq.
(12)) are used for identifying the kinematic of a single bump i, which depends on the conditions of its neighbored bumps. Three
kinematic motion cases are possible, see Table 1.
However, if both segments are in a stick condition the kinematic of the bump will change (Fig. 6 (c)) and delivers a higher
stiffness, see Fig. 5. Finally, using the motion states the stiffness ks can be calculated by simple equations, see Appendix B.
4.1.3. Dynamic structural bump model
To calculate the dynamic hysteresis of a bump structure, Eq. (12) has to be solved in the time domain, while varying Δh or F p .
For higher excitation frequencies the Gross-sliding regime becomes smaller [43]. Hence, the Stribeck effects are reduced and the
frequency has no significant impact on the structural hysteresis, which has been experimentally shown [12]. Thus, the Stribeck
effect is neglected and only Coulomb friction is used. Unfortunately, the Coulomb friction that considers the sign-function leads
to numerical problems. The introduction of the dynamic friction model of Petrov and Ewins [44] avoids that problem and a time
domain analysis is possible. This approach has been successfully introduced in GFB structures already by Le Lez et al. [32]. The
frictional forces due to acting normal force N can be described by a relative deflection (x − xs ) of a simple brush model with the
stiffness kt (Fig. 7 (a)).
(
)
F = kt x − xs ≡  N sign(ẋ S )
(15)
To overcome unsteadiness problems of the sign-function, an approximation is applied by using an arctangent-function:
sign(ẋ S ) ≈
2

arctan(ct ẋ S )
(16)
The approximation includes ct , which is a model parameter that controls the slope of the arctangent-function (Fig. 7 (b)). A
numerical investigation with a multiple parameter variation (Young’s-modulus, bump number, load level, and kt -Parameter)
shows, that applying a value of ct ≥ 6 × 106 s∕m in a kt range of 4 × 105 to 1 × 106 N/m the numerical results of a dynamic
hysteresis become independent of the slope parameter ct [37]. However, applying too high ct values increases significantly the
R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
397
Fig. 7. (a) Schematic brush model; (b) impact of the form parameter ct on the sign-approximation.
solution time. Hence, ct = 6×106 s/m is chosen.
(
(
dF
1
F
= kt ẋ − tan
dt
ct
2 N
))
(17)
Finally, a time derivation of Eq. (15) and substitution of Eq. (16) yields to Eq. (17).
(
(
1
i
Ḟ up
= kt ẋ iup − tan
ct
i
 Fup
i
2 Nup
))
(
(
1
i
and Ḟ bot
= kt ẋ ibot − tan
ct
i
 Fbot
i
2 Nbot
))
(18)
i
i
Note that the normal force is described by Nup
= Fup
and the normal force at the bump segment is based on the vertical righti
= Fri + Fli+1 . Applying this method on a GFB structure, the bump/top foil contact (index up)
and leftward bump lever forces Nbot
and bump segment/housing contact (index bot) are given by Eqs. (18). A time derivation of the force equilibrium of a single
bump segment and substituting the dynamic friction force results to:
d
dt
(
Fp
tan( )
)
i
(
) d
−1
+1
= 2k1 2ẋ ibot − ẋ ibot
− ẋ ibot
+
(
Fp
tan( )
dt
)
i+1
i
i
i+1
+ 2Ḟ bot
+ Ḟ up
+ Ḟ up
(19)
Furthermore, the kinematic relationships 2ΔLi = xi−1 − xi+1 and 2xiup = xi + xi−1 are used. Due to low masses of the GFB strucbot
bot
bot
bot
ture inertia effects are neglected. The equation is a nonlinear differential equation system and can be written in matrix form.
( )−1
̇ bot =  xbot ̇
(20)
For more details of the structural models including a validation, which is based on numerical and dynamic results, refer to [34].
For evaluating the dynamic structural parameter kd and cd a dynamic force F p (t) is used for perturbing the elastic structure with
an excitation frequency .
(
)
Fp (t) = Fp,0 + ̂
Fp sin(t) = Fp,0 · 1 + p sin(t)
(21)
Eq. (21) is used for excitation of the dynamic structural bump model of Eq. (20). For solving Eq. (20) in time domain a differential
equation solver, Matlab’s ode23s(…), is applied. It is a single-step solver for stiff problems and based on a modified Rosenbrock
formula of order 2, see Ref. [45]. Calculating the dynamic hysteresis, as illustrated in Fig. 5, the vertical dynamic displacement
Δĥ i of the bump and the dynamic bump forces ̂
Fpi are used for the estimation of the dynamic stiffness kid and the damping cdi for
each bump-spring-damper i.
Note the dynamic amplitude ̂
Fp is based on scaling the static load F p,0 by using a factor  p , which has been firstly shown by
Ref. [39]. Increasing the scaling factor towards 1 yields kd → ks . In contrast, decreasing  p towards small values results in very
small hysteresis with a higher gradient, thus the dynamic stiffness kd of the bumps increases, see Fig. 8. Investigations have
shown, that the influence of the scaling factor  p on the linearised bearing parameters for a lowly loaded GFB < 30 N, based on
the geometry of Table 3, has no significant influence, see Refs. [34,37]. In addition, in Ref. [39] the authors achieved with a value
of  p = 0.5 good correlations between simulation and test.
However, it is imported to note that beside the rotational speed Ω and the excitation frequency  the scaling factor  p has
an impact on the linearised bearing parameters as well.
For more details of the structural models including a validation, see Refs. [19,37,46]. Finally, the overall calculation process
is shown in Fig. 8, which includes the zero and first order solutions of the RE and the schematic results of the structural bump
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R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
Fig. 8. Model routine for linearised bearing parameter calculation. Including the impact of static and dynamic structural stiffness.
Table 2
Unbalance cases.
Unbalance
static
momentum
UF
UR
0 g mm
0 g mm
6 g mm
6 g mm
12 g mm
12 g mm
0 g mm
0 g mm
9 g mm
9 g mm
12 g mm
12 g mm
models. This procedure is inspired by the work of [39].
5. Results and discussion
5.1. Rotor dynamic test rig
A small rotor dynamic test rig as illustrated in Fig. 9 (a) and (b) is used to examine the effect of different unbalance levels
(Table 2) and structural GFB modifications on the nonlinear vibrations. The solid rigid shaft 9 (c) is supported by two identical
GFBs as shown in Fig. 9 (a). The GFBs have been modified by inserting three metal shims (25 μm thickness), see Fig. 10 (b).
Thus, the clearance is varied along circumferential direction by this modification, see Fig. 10 (c). The overall data of the test rig
configuration is listed in Table 3.
In operation the rotor is driven by a centered impulse air turbine. Axial forces due to the turbine are supported by axial thrust
pins, see Fig. 9 (a) and (b). Experimental and numerical modal analysis have shown that for the operation range of 0–65 krpm
the housing structure and the rotor can be assumed as rigid bodies, [14]. Hence, in further rotor dynamic stability and modal
analysis the housing is neglected.
Coast down tests from maximum operation speed are carried out, where the rotor speed and the displacements close to the
front and rear bearing positions have been simultaneously recorded. During deceleration the pressure valve is closed to avoid
additional turbine induced excitations. Waterfall diagrams
√including backward and forward whirling are generated by applying
FFT’s and using the magnitude of the displacements r = x2 + y2 . The experimental data are based on the works [14], where
detailed discussions and information are given. However, the main experimental results of the test will be shown in this work.
R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
399
Fig. 9. Schematic view of the test rig: (a) Cross section of the test rig; (b) photography of the test rig; (c) rotor cross section including nomenclature.
Fig. 10. Assembly position of the GFB: (a) Case without shimming; (b) Case with shimming including a rotation of 45◦ ; (c) impact of shimming on the radial displacement
of the elastic structure along the circumferential direction (shim thickness Δs = 25μm, three shims blue bar marks). (For interpretation of the references to colour in this
figure legend, the reader is referred to the web version of this article.)
5.2. Case 2: evaluation of self-excitation
No additional unbalance is added for evaluating the effect of self-excitations in a well balanced system. The balance grade
of the rotor has been measured and identified with lower than G0.4 (DIN-ISO 1940-1). Waterfall diagrams recorded at the front
bearing (F) for the bearing without shims (a) and with a shim modification are illustrated in Fig. 11.
The test without shims have a high number of nonlinear effects: At (1) the self-excitation due to the fluid film is present. It
excites the cylindrical rigid eigenmode with the subharmonic frequency f 1 . Due to the nonlinear behaviour of the progressive
system the subharmonic order 1/3Ω excites the system and results in subharmonic resonances for the first mode (2) and the
second conical mode (3). At the resonance cases (2) and (3) the system bifurcates and results in jumps towards lower (2) and
higher (3) frequencies. In particular the jump at the second subharmonic resonance (3) amplified the second conical mode
shape. In addition, lots of frequency modulations are shown, which is typical for a Duffing-oscillator.
In comparison, the bearing with a shim modification reduces significantly the nonlinear vibrations as shown in Fig. 11 (b).
The OSSV point (1) is shifted towards higher onset speeds (nOSSV ) and higher frequencies (f OSSV ). Comparing the amplitude level
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R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
Table 3
Data of the GFB and shaft configuration, Fluid properties and structural model parameters.
Gas Foil Bearing Configuration
bearing radius, R
bearing length, l
bump height, hb
bump thickness, tb
bump pitch, sb
bump number, Nb
half bump length l0
top foil thickness, tf
Young’s modulus, E
Poisson’s ratio, 
foil material
shim thickness, Δs
shim length, ls
shim width, bs
number of shims, Ns
shim position (measured from the spot weld),  s, i
shim material
19.050 mm
38.100 mm
0.50 mm
0.1 mm
4.572 mm
26
1.778 mm
0.1 mm
2.07 × 1011 N/m2
0.3
Inconel X-750
25 μm
38.1 mm
10 mm
3
67.5◦ ,187,5◦ and 307.5◦
Steel (1.1274)
Shaft Configuration
shaft material
rotor mass, mr
polar moment of inertia, Jz
transverse moment of inertia, Jx , Jy
shaft Length, lr
distance from CG to bearing midline, ΔlCG, i i = F, R
nominal shaft diameter, Da
nominal clearance bearing F, c0,F
nominal clearance bearing R, c0,R
42CrMo4 (1.7225)
2.148 kg
568.425 mm2 kg
6775,878 mm2 kg
212 mm
72.5 mm
38 mm
55 μm±6 μm
50 μm±6 μm
Fluid Properties
ambient pressure, pa
ambient temperature, T a
viscosity,  a
1 × 105 Pa
293 K
1.95×10-5 Pas
Structural Model Parameters
smoothing parameter, ct
contact stiffness, kt
contact stiffness (Shimming), kt
friction coefficient, 
6 × 1010 s/m
5 × 105 N/m
2 × 106 N/m
0.1
of the self-exciting cylindrical mode of Fig. 11 (a) and (b) a significant reduction due to shims is shown. The waterfall digram
of the shim case (Fig. 11 (b)) show no additional frequency modulations compared to the case without shims (Fig. 11 (a)). In
section a detailed explanation is given by the linearised bearing parameters.
5.3. Case 1: low balance grade/direct forcing
In Figs. 12 and 13 the results for additional unbalance tests (static and momentum unbalance; Table 2) are shown. The
unbalances are added at the balance planes (BP) of the shaft close to the front and rear bearing position, see Fig. 9 (c), by
attaching small bolts and washers.
As shown in the waterfall diagrams of Figs. 12 and 13 the nonlinear subharmonic vibrations are amplified by an increased
unbalance of the rotor system. Multiple frequency modulations and subharmonic orders (1/2Ω, 1/3Ω and 1/4Ω) are present.
In addition, synchronous (1Ω) subharmonic resonances (1/2Ω, 1/3Ω and 1/4 Ω) are present, which are highlighted by white
cross-marks in the waterfall diagrams. Interestingly, for the momentum unbalance cases (Fig. 12) and the highest value of the
static unbalance case (Fig. 13 (b)) a subharmonic 1/2Ω order with a subharmonic resonance for the cylindrical mode shape is
present. It shifted the OSSV towards lower rotor speeds compared to the self-excitation, as shown for the case 2 in Fig. 11 (a). In
this case the forced nonlinear behaviour of the unbalance and the progressive system results in the OSSV with a subharmonic
1/2Ω order, which is typical for asymmetrical and progressive force-displacement-relationship, see Refs. [27, p.136].
5.4. Numerical evaluation
In this part the self-excitations of case 2 and subharmonic resonances are calculated with the linearised model of Eq. (2),
where the rotor is modelled with 3D Timoshenko beam elements. Note that structural damping for the rotor is neglected. Applying linearised bearing parameters, which are calculated with the mentioned method (section 4) stability and modal analysis, can
R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
401
Fig. 11. Rotor Speed vs. Frequency vs. Amplitude (Coast down test, without additional unbalance and measured at the front GFB.): (a) Test without shims; (b) test with
shims (Δs = 25μm).
be carried out. The bearing configurations for the front and rear bearing (F and R) are listed in Table 3. Due to the symmetrical
shaft design the static bearing load generated by the rotor weight for each bearing is set to 10.54N. The dynamic parameters are
calculated for synchronous (Ω = ) and subharmonic excitations in a frequency range of 5–300 Hz. Rotor speeds up to 65 krpm
for evaluating the operation range are considered. Due to the shim modification the clearance is affected along the circumferential direction by the contour of Fig. 10 (c). Furthermore, the front and rear bearing are rotated 45◦ counter clockwise compared
to the unmodified bearings, as shown in Fig. 10 (a) and (b). Structural effects on damping and stiffness have been experimentally
examined and the kt value is increased compared to the bearing without shims, which yields a higher dissipation rate for the
elastic structure.
5.4.1. Self-excitation (case 2)
For evaluating the OSSV due to a fluid film self-excitation the rotor dynamic tests of case 2 is used for comparison. In Fig. 14
the stability contour plot for the bearing configuration without shimming (a) and with shims (b) is illustrated. The intersection
between the zero damping ratios and the self-excited eigenfrequency is moved towards higher rotor speeds and eigenfrequency
for the shimmed bearing configuration.
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R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
Fig. 12. Momentum unbalance test: Waterfall digram of coast down test for the front bearing without shimming. (a) Additional unbalance U i = 9 gmm (i = F, R); (b)
additional unbalance U i = 12 gmm (i = F, R).
Finally, the linearised simulation with the onset speed of instability (n0 ) and the eigenfrequency of the self-excitation (f 1,0 )
of the unstable cylindrical mode are in good agreement with the measured OSSV points (nOSSV and f OSSV ) of Fig. 11, see Table 4.
The reason for improved stability of the shimmed GFB can be explained by the bearing parameters, see Fig. C.16 and C.17.
First, considering only the higher differences of the cross-coupling stiffness kxy − kyx (Fig. C.16 (e) and (f)) the system with shims
would have a higher tendency of instability, [47, p.1]. Second, the shim configuration shows a softening effect due to smaller
differences cxy − cyx (Fig. C.17 (e) and (f)), [48, p.1]. However, due to the shims the direct stiffness values kxx and kyy (Fig. C.16
(a) - (e)) and the direct damping coefficients cxx and cyy (Fig. C.17 (a) - (e)) are increased compared to the bearing configuration
without shimming. In addition, a more pronounced anisotropic stiffness behaviour kxx > kyy is present. It significantly gains
higher onset speeds of instability due to self-excitation.
5.4.2. Subharmonic and synchronous resonances
As shown in Fig. 11 the shimmed bearing configuration has subharmonic resonances and nonlinear effects, e.g. frequency
modulations. Hence, the subharmonic resonances due to the nonlinear system will be tested by using the linearised model of
R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
403
Fig. 13. Static unbalance test: Waterfall diagram of coast down test for the front bearing without shimming. (a) Additional unbalance U i = 6 gmm (i = F, R); (b) additional
unbalance U i = 12 gmm (i = F, R).
Eq. (2).
In Fig. 15 the Campbell diagram for the forward whirl eigenfrequencies of the first two rigid body modes is shown. The results
are based on a numerical modal analysis of Eq. (2), where rotor speeds between 10krpm and 60krpm are considered and a set of
three different excitation orders are applied: A synchronous 1Ω and subharmonic excitation orders of 1/2Ω, 1/3Ω and 1/4Ω. Due
to the influence of the excitation order on the linearised bearing parameters, different eigenfrequencies for constant excitation
orders 1Ω (green) and subharmonic excitations of 1/2Ω (blue), 1/3Ω (red) and 1/4Ω (black) are illustrated in different colours.
The intersections between the eigenfrequencies of a constant excitation order and the corresponding excitation lines (dashed
line) yield the critical speeds of synchronous and subharmonic resonances.
In Table 5 the calculated intersections are listed. Comparing this values with the cross-marks of the waterfall diagrams
(Figs. 11–13) a good correlation is present. In addition, the simulation shows the same effect as the measurements of case 1 (low
balance grade/direct forcing), that the subharmonic resonance of the 1/2Ω order emerges before the fluid film self-excitation of
the GFB without shims, see Table 4.
In Table 6 the relative error between the simulated and measured resonances for the four test cases with an additional unbalance is listed. No clear trend is shown. However, most of the data show positive relative errors due to overpredicted resonance
for the simulated excitation orders. The discrepancies between simulation and experiments can be addressed to the simplifica-
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R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
Fig. 14. Stability digram (rotor speed vs. exciation frequency vs. damping ratio  ), self-exciting eigenfrequency f 1 and the rotor first 1Ω synchronous excitation (dotted
green line): (a) without shims; (b) with shims (Δs = 25μm). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of
this article.)
Table 4
Self-excitation comparision: Simulation vs. measurement.
Simulation
Measured
Δs
Frequency f 1,0
Rotor Speed n0
rel. Error f 1,0
rel. Error n1,0
0μm
25μm
137.16 Hz
222.17 Hz
16 977 rpm
34 466 rpm
0.32%
3.30%
−4.58%
−9.83%
Δs
Frequency f OSSV
Rotor Speed nOSSV
rel. Error f 1,0
rel. Error n1,0
0μm
25μm
136.72 Hz
214.84 Hz
17 754 rpm
37 854 rpm
–
–
–
–
Fig. 15. Campbell diagram of the forward whirling 1. and 2. mode under synchronous excitation 1Ω and subharmonic excitation 1/2Ω, 1/3Ω and 1/4Ω. GFB configuration
without shims (Δs = 0).
tion of small infinite small perturbations in the linearisation process. Increased unbalance levels may significantly activate the
elastic structure and reduces the local stiffness of the bump structure, as indicated in the force displacement hysteresis of Fig. 8.
Hence, the onset speed of subharmonic resonances is lowered. Finally, it should be highlighted that the linearisation is limited
to well-balanced systems and small displacements.
R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
405
Table 5
Critical Speeds for the 1. mode and 2. mode under synchronous excitation 1Ω and
subharmonic excitation 1/2Ω, 1/3Ω and 1/4Ω. GFB configuration without shims (Δs = 0).
Mode
1Ω
1/2Ω
1/3Ω
1/4Ω
1
2
–
13 281 rpm
15 503 rpm
22 985 rpm
25 171 rpm
34 060 rpm
35 351 rpm
47 172 rpm
Table 6
Relative error of the resonance speeds between simulation and experiments for the 1. mode and 2. mode under synchronous
excitation 1Ω and subharmonic excitation 1/2Ω, 1/3Ω and 1/4Ω. GFB configuration without shims (Δs = 0).
6. Conclusion
The paper has presented an experimental and numerical investigation of the nonlinear vibrations of a rigid rotor supported
by GFBs. For characterising the nonlinear vibrations of a rotor in GFBs an isolated system has been assumed. Two sources have
been identified: Case 1 is based on a direct forcing due to low balance grades, which results in nonlinear vibration generated by
the high displacements and the interaction with the nonlinear progressive force-displacement characteristic of the lubrication
film and the bearing structure. A variation of the unbalance state indicated, that higher unbalance values result in increased
forced displacements, which amplifying the nonlinear vibration character (jump, frequency modulations, subharmonic orders
and resonances). Due to the subharmonic resonance of the 1/2Ω order the OSSV is at lower rotor speeds compared to the selfexcitation of case 2.
In contrast, considering a well-balanced rotor system of case 2 (lower than G0.4) the fluid film self-excitation has been
identified as source of subharmonic whirling. However, the nonlinear force-displacement character of the bearing structure
amplified nonlinear effects, e.g. frequency modulations, subharmonic resonances, jumps, etc. Furthermore, shims improved the
dynamic performance and shifted the self-exciting towards higher values. The stiffness and damping characteristics according
the simulation results are improved compared to the base bearing without shimming.
For evaluating the self-excitation, synchronous and subharmonic resonances a linear system has been used. The GFBs are
based on linearised stiffness and damping values. Therefore, a perturbation method related on Lund’s [35] work has been
applied. The model couples the fluid film simulation with a nonlinear structural model including frictional contacts and bump
interactions. The onset of self-excitation of case 2 has been calculated with good agreement compared to the experimental tests.
Furthermore, a good correlation between simulation and measurement for evaluating the onset synchronous and sub harmonic
resonances has been found.
In future work, additional numerical investigation including time domain analysis will be carry out for proving the nonlinear
character in detail. In addition, a base excitation will be considered in the shown vibration characterisation of Fig. 2. Therefore,
experimental and numerical tests will be carried out.
Appendix A. Bearing journal model
The centrifugal forces of rotating bearing journal affect the radial clearance. Term ho (Ω) is based on the Timoshenko Theory
for a rotating disc.
[
ho (Ω) = −
A1 =
(
) D2
)
(
)
Da BJ Ω2 (
2
a
1 − BJ A1 + 1 − BJ A2 − 1 − BJ
16EBJ
4
)
D2a (
3 + BJ ;
4
A2 =
]
)
D2a (
3 + BJ
4
(A.1)
(A.2)
EBJ is the Young’s modulus,  BJ the Poisson’s ratio and BJ the density of the bearing journal (BJ).
Appendix B. Static structural bump stiffness ks
For estimating the static stiffness kis of a single bump i the vertical displacement Δhi0 of the bump based on the zero order
system is used.
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R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
Appendix B.1. Right- and leftward moving state
kis
(
Fpi ,0
=
=
Δhi0
)
2 ΔLi − ΔLi+1 k1 + kis+1 C2i
(B.1)
C1i
C1i and C2i are given by:
(
C1i
1
= Δhi0
2
C2i
1
= Δhi0+1
2
1
( ) −
tan  i
(
1
(
tan 
i
)
(
( ))
1 −  i tan  i
) +  i+1
i+1
)
(
(B.2a)
( ))
1 +  i+1 tan  i
(B.2b)
The stiffness of the free Bump i = N b is given by:
N
N
ks b
=
Fp,b0
N
Δh0 b
2ΔLNb k1
=
(B.3)
N
C1 b
N
Where C1 b is described by:
N
C1 b
1
N
= Δh0 b
2
(
1
(
tan  Nb
) −
Nb +1
)
(
(
1 −  Nb tan  Nb
))
(B.4)
Appendix B.2. Stick condition
If both bump segments are in stick condition the stiffness is calculated with:
(
kis =
Fpi ,0
Δhi0
=
(
i0
2 2Rb sin
2
)
− L ′ k2
Li
(
mit g2 =
)
L′2 + Rb − Δhi0
(
g2
(B.5a)
)2
− R2b
)
,
i
2L′ Rb − Δh0
(B.5b)
Where the length L′ is delivered by:
√
L′ =
(
R2b + Rb − Δhi0
)2
(
)
( )
− 2Rb Rb − Δhi0 cos 0
(B.6)
Appendix C. Linearised bearing parameters
Figs. C.16 and C.17 show the results of the front bearing configuration. Due to small variations of the nominal clearance c0
comparable results for the rear bearing configurations are delivered.
R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
407
Fig. C.16. Impact of structural modification on the linearised stiffness k ( ,  = x, y) vs. excitation frequency (f s ) vs. rotor speed (n). Front bearing (F), static
vertical loading W x = 10.54 N. Self-excited eigenfrequency of the first mode f 1 (red solid line); self-excitation point (OSSV) marked by the red point. Bearing
without shimming: (a) kxx ; (c) kyy ; (e) (kxy − kyx ). Bearing with shimming: (b) kxx ; (d) kyy ; (f) (kxy − kyx ) (vertical dashed line = OSSV of the bearing without
shims).
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R. Hoffmann and R. Liebich / Journal of Sound and Vibration 412 (2017) 389–409
Fig. C.17. Impact of structural modification on the linearised damping c ( ,  = x, y) vs. excitation frequency (f s ) vs. rotor speed (n). Front bearing (F), static
vertical loading W x = 10.54 N. Self-excited eigenfrequency of the first mode f 1 (red solid line); self-excitation point (OSSV) marked by the red point. Bearing
without shimming: (a) cxx ; (c) cyy ; (e) (cxy − cyx ). Bearing with shimming: (b) cxx ; (d) cyy ; (f) (cxy − cyx ) (vertical dashed line = OSSV of the bearing without shims).
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