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Vibration Prediction of Bladed Disks Coupled by
Friction Joints
Malte Krack, Loic Salles, Fabrice Thouverez
To cite this version:
Malte Krack, Loic Salles, Fabrice Thouverez. Vibration Prediction of Bladed Disks Coupled by Friction
Joints. Archives of Computational Methods in Engineering, Springer Verlag, 2017, 24 (3), pp.589-636.
�10.1007/s11831-016-9183-2�. �hal-01825517�
Noname manuscript No.
(will be inserted by the editor)
Vibration Prediction of Bladed Disks Coupled by Friction
Joints
Malte Krack · Loic Salles · Fabrice Thouverez
Received: date / Accepted: date
Abstract The present review article addresses the vi-
bration behavior of bladed disks encountered e. g. in air-
craft engines as well as industrial gas and steam tur-
bines. The utilization of the dissipative effects of dry
friction in mechanical joints is a common means of the
passive mitigation of structural vibrations caused by
aeroelastic excitation mechanisms. The prediction of
the vibration behavior is a scientific challenge due to
(a) the strongly nonlinear and non-uniform contact in-
teractions involving local sticking, sliding and liftoff, (b)
topological complexity of the coupled structure, and (c)
the multi-disciplinary character of the problem associ-
ated with the need to account for structural mechanical
as well as fluid dynamical effects. The purpose of this
article is the overview and discussion the current state
of the art of vibration prediction approaches. The mod-
eling approaches in this work embrace the description
of the rotating bladed disk, the contact modeling, the
consideration of aeroelastic effects, appropriate model
reduction techniques and the exploitation of the rota-
tionally periodic nature of the problem. The simulation
approaches cover the direct computation of periodic,
steady-state externally forced and self-excited vibra-
M. Krack
Institute of Aircraft Propulsion Systems, University of
Stuttgart, Pfaffenwaldring 6, 70569 Stuttgart, Germany E-
mail: malte.krack@ila.uni-stuttgart.de
L. Salles
Vibration University Technology Centre, Department of Me-
chanical Engineering, Imperial College London, Exhibition
Road, London SW7 2AZ, UK
E-mail: l.salles@imperial.ac.uk
F. Thouverez
´
Ecole Centrale de Lyon, Laboratoire de Tribologie et Dy-
namique des Systmes, 36 avenue Guy de Collongue, 69134
Ecully Cedex, France
E-mail: fabrice.thouverez@ec-lyon.fr
tions using the high-order harmonic balance method,
the formulation of the contact problem in the frequency
domain, methods for the solution of the governing al-
gebraic equations and advanced simulation approaches,
including the concept of nonlinear modes.
Keywords friction damping · structural dynamics ·
turbomachinery · contact mechanics · cyclic symmetry ·
harmonic balance · continuation
Nomenclature
Scalars, sets
N
0
initial normal load
g
n,0
initial normal gap
H set of (temporal) harmonics
m engine order
m
0
fundamental engine order
M set of relevant engine orders
DL
Dynamic Lagrangian penalty coefficient
Ω
rot
rotational speed
θ inter-blade phase angle
Vectors
f
a
aerodynamical forces
f
ae
, F
ae
aerodynamical external forces (time
domain, frequency domain)
f
ai
aerodynamical interaction forces
f
c
, F
c
global contact forces (time domain,
frequency domain)
2 Malte Krack et al.
Vectors
g contact gaps
λ, Λ local contact forces (time domain,
frequency domain)
u, U vector of (generalized) coordinates
(time domain, frequency domain)
p pressure
Matrices
B interface coupling matrix
D damping matrix
G
ai
aeroelastic transfer matrix
H dynamic compliance matrix
I identity matrix
K matrix of velocity proportional forces
M mass matrix
S dynamic stiffness matrix
T matrix of component modes
W
n
s
discrete Fourier matrix for n
s
samples
∇ frequency domain derivative matrix
Numbers
n
c
number of contact points
n
d
number of (generalized) coordinates
n
fe
number of finite element nodal
degrees of freedom
n
fe,s
... per sector
n
if
number of interfaces
n
r
number of component modes
n
s
number of sectors
Subscripts, superscripts
( )
n
associated to the normal
direction
( )
t
associated to the tangential
direction
(n)
( ) associated to sector n
C
( ) in the coordinates of the
continuous contact interface
c
( ) in the coordinates of the
discrete contact interface
fe
( ) in the physical degrees of
freedom of the finite element model
r
( ) in the generalized coordinates
of the component modes
tw
( ) in traveling wave coordinates
Operators
( )
∗
complex conjugate
={( )} imaginary part
<{( )} real part
( )
+
pseudo inverse
( )
H
Hermitian transpose
( )
T
transpose
N
A
null space of matrix A
1 Introduction
1.1 Engineering relevance of friction damping of
bladed disks
Fig. 1: Examples for High-Cycle-Fatigue failures of
bladed disks: (a)-(b) debris of the first stage aero engine
compressor rotor and its blades [100], (c) fracture of an
aero engine low pressure turbine rotor blade [4]
In the ongoing quest for improved efficiency, mod-
ern turbomachines are driven near their structural me-
chanical limits. The successful operation of these ma-
chines depends largely on the structural mechanical
integrity of the rotating components, owing to their
comparatively high loading [152]. Bladed disks undergo
high mechanical stress during operation. Static stresses
are caused by thermal loads, static fluid pressures and
rotation-induced centrifugal loads. Mechanical vibra-
tions, caused by additional dynamic loads of different
origins, lead to dynamic stresses. Depending on the
static stress level, sustained and high dynamic stresses
can lead to high cycle fatigue (HCF). This type of fa-
tigue leads to substantial life cycle costs and presents a
major safety issue. A primary goal of the design process
is to ensure the structural mechanical integrity. Hence,
vibrations are a central concern in the design of aircraft
engines as well as industrial gas and steam turbines.
Vibration Prediction of Bladed Disks Coupled by Friction Joints 3
resonances
flutter
forced NSV
Fig. 2: The schematic Campbell diagram illustrates
natural frequencies, frequencies of synchronous excita-
tion (integer multiples mΩ
rot
, depicted as so-called load
lines), and representative vibration regimes
Vibration mechanisms
Two of the most important vibration mechanisms of
bladed disks are of aeroelastic nature: (a) forced re-
sponse and (b) flutter [152].
(a) Synchronous forced response In the case of forced
response, the dynamical loads are caused by the rota-
tion of the blades through the circumferentially inho-
mogeneous pressure field. Pressure inhomogeneities are
caused by aerodynamical blade row interactions and
non-uniform inflow conditions. An example for blade
row interaction are wakes from upstream stator vanes
which leads to the so-called nozzle-excitation. Non-uniform
inflow conditions are caused by asymmetries in the flow
path due to e. g. struts or casing ovality. In steady op-
eration, the resulting inhomogeneous pressure field is
essentially time-invariant in the non-rotating frame of
reference. From the perspective of the rotating bladed
disk, this pressure field takes the form of a wave trav-
eling with rotor speed. This results in dynamic load-
ing with frequencies being integer-multiples of the ro-
tational frequency, hence the term synchronous exci-
tation. Under the condition of resonance; i. e. , if an
excitation frequency coincides with a natural frequency
of the structure, the forced vibration response can reach
particularly high levels. Resonance coincidences are com-
monly identified using the Campbell diagram, see Fig. 2.
(b) Flutter Flutter refers to the unstable aeroelastic in-
teraction of a vibrating structure with the surrounding
fluid flow. In turbomachinery, flutter is mainly caused
by the cascade effect; i. e. , the aerodynamical interfer-
ence among the blades within a blade row. If this in-
teraction is unstable, the blades receive energy from
the unsteady flow as a consequence of their vibration,
which represents a self-excitation mechanism. This pos-
itive feedback leads to continuously increasing vibration
level, until nonlinear effects come into play or the struc-
ture fails. Form and frequency of the vibration are usu-
ally similar to one of the structure’s normal modes of
vibration. In contrast to forced response, the oscillation
frequency is generally not an integer-multiple of the ro-
tational speed [152,101].
Other vibration mechanisms of aeroelastic nature in-
clude vortex shedding and rotating instabilities that oc-
cur largely under the condition of partial loading. These
unsteady mechanisms cause non-synchronous forced or
self-excited vibrations. Besides aeroelastic mechanisms,
mechanical effects can lead to vibrations of the bladed
disk. Common examples are the rubbing between blades
and the casing, vibrations due to torsional or lateral
dynamics of the rotor shaft, and extreme events such
as ingestion of birds or ice (foreign object damage) or
structural failure of individual blades (domestic object
damage).
Means of vibration reduction
Fig. 3: Common types of friction joints [143]: (a) roots
joints, (b) tip shrouds, (c) underplatform dampers, (d)
damper wires, (e) damper pins
Different strategies are pursued to avoid and miti-
gate vibrations of bladed disks. An important approach
is to avoid the excitation of resonant vibrations and
aeroelastic instabilities in the operating range. This can
be achieved by adjusting the system’s dynamical char-
acteristics (e. g. blade counts, natural frequencies) with
appropriate design measures. If resonance coincidence
cannot be completely avoided, one has to make sure
that the vibration response remains within tolerable
bounds. This can be accomplished by mitigating the
4 Malte Krack et al.
excitation level of the associated wave lengths in the
relevant frequency range. An important means to avoid
flutter is intentional mistuning, i. e. , a deliberate vari-
ation of the blade-to-blade properties. If excessive ex-
citation by resonant forcing or flutter still cannot be
avoided for all important modes in the relevant oper-
ating range, one may have to increase the damping of
the system. The total damping D
total
of the system is
composed as,
D
total
= D
aerodynamical
+D
material
+ D
joints
| {z }
D
mechanical
+D
rest
. (1)
In the case of resonant forced response, an increase of
the total damping can lead to a significant reduction of
the vibration levels. In the case of flutter, the aerody-
namical damping, D
aerodynamical
is negative, and addi-
tional damping can lead to the complete suppression or,
if nonlinear effects become important, the stabilization
of vibrations in so-called limit cycles.
Mechanical damping, D
mechanical
, can be grouped into
material damping, D
material
, which is usually compara-
tively small, and the damping due to mechanical joints,
D
joints
. Dissipation can also have non-aerodynamical
and non-mechanical, e. g. electromagnetic origin. Such
dissipation mechanisms are accounted for in D
rest
.
The most important sub-group of joint damping, D
joints
,
is friction damping, which refers to the dissipative ef-
fects related to dry frictional local sliding in mechan-
ical joints. Friction damping is certainly the most es-
tablished damping technology of bladed disks. A major
drawback of friction damping is that it comes at the
cost of wear effects. It should be remarked that many
damping technologies with successful applications in
other fields, cannot cope with the harsh environment
(high temperatures, high centrifugal stresses, corrosive
gases), and the strictly limited design space. Also, the
application of active or semi-active vibration control
strategies is hampered by the requirement of fail-safe
operation. Noteworthy alternatives to friction damping
include piezoelectric shunt damping [58,178], eddy cur-
rent damping [88,87], viscoelastic material damping of
coatings [177,60], and impact or particle damping.
Friction damping takes place in mechanical joints that
are either inherent to bladed disks, such as the ones
between blades and roots, or can be introduced ad-
ditionally, e. g. in the form of underplatform dampers.
Some of the most common forms of mechanical joints
are illustrated in Fig. 3. Besides damping, additional
joints also increase the elastic coupling among adjacent
blades. This changes the structure’s modal characteris-
tics and is in fact sometimes the primary motivation for
the introduction of these joints. By means of an appro-
priate design of mechanical joints, a considerable miti-
gation of vibrations and resulting dynamic stresses can
be achieved. This, in turn, leads to an improved struc-
tural reliability and decreases fatigue-related costs. Fur-
thermore, this can lead to an increased feasible blade
design space and an extended range of tolerable oper-
ating conditions, and, thus, contributes to an increased
efficiency of the turbomachine.
1.2 Scientific complexity of the topic
The design of bladed disks with mechanical joints relies
on a profound understanding of the relevant physical
phenomena and adequate tools for the assessment of
the structural dynamic characteristics. Important char-
acteristics in this context are the vibration level, the res-
onance frequencies and the mechanical damping. Dur-
ing the design phase, these measures are determined
largely by means of vibration prediction, whereas tests
are many carried out for validation. The vibration pre-
diction is particularly difficult due to the following as-
pects.
(a) Nonlinearity The contact interactions in mechani-
cal joints represent strongly nonlinear phenomena. This
nonlinearity leads to a strong coupling of different time
and length scales. More specifically, the local stick, slip
and liftoff phenomena in the mechanical joints, occur-
ring on relatively short time and length scales, have
considerable effects on the global vibration behavior
of bladed disks, occurring on much longer time and
length scales, and vice-versa. Moreover, the dependence
on the vibration level needs to be taken into account in
the analysis of structural dynamic characteristics, such
as resonance frequencies, effective damping and deflec-
tion shape. Also, nonlinearity can give rise to phenom-
ena such as co-existence of multiple stable vibration
states, and steady-state vibrations that exhibit signif-
icant frequency components not present in the excita-
tion spectrum. Suitable simulation methods are often
based on iterative, numerical procedures which are com-
paratively time-consuming.
(b) Model order Turbomachinery bladed disks exhibit
blades with generic, three-dimensional profiles and of-
ten consist of a large number of components, possibly
including additional devices such as friction dampers.
Hence, spatial discretization is commonly carried out
using finite elements. Moreover, the different compo-
nents have a number of extended contact interfaces
where nonlinear contact interactions may take place.
A fine spatial discretization is required to accurately
resolve the local contact interactions and the dynamic
![Fig. 17: Illustration of the spatiotemporal nature of a traveling wave: (a) q as a function of the sector number and the temporal phase similar to [113], (b) temporal phase as a bijective function of time](/figures/fig-17-illustration-of-the-spatiotemporal-nature-of-a-2jy9a1po.webp)
