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94-GT-293
lished in an ASME Journal. Papers are available from ASME for 15 months
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Copyright © 1994 by ASME
A NUMERICAL METHOD FOR THE PREDICTION
OF BLADED DISK FORCED RESPONSE
Marc Berthillier, Marc Dhainaut, Franck Burgaud,
and Vincent Gamier
Snecma
Moissy Cramayel, France
ABSTRACT
A numerical method has been developed to predict the forced
response of bladed disks due to a wake excitation from upstream
blade rows. The structure is modelled by a 3D finite element
mesh of a bladed disk segment. Using cyclic symmetry, this
model provides a modal base for the rotating structure. The
aerodynamic damping of the vibratory modes and the excitation
pressures on the blades due to the propagation of upstream flow
defects are computed separately using the same 3D unsteady
Euler analysis software. A modal response solution of the
aeromechanical system is then performed.
This analytical methodology has been used to study the forced
response of an experimental high pressure compressor blisk. The
results are analysed and compared with actual rig tests.
[M]
[M S I
[mp ]
[E]
[E] *
[Qp, ]
Structural mass matrix
Structural mass matrix for 13 t
Modal mass matrix for (3 t
Expanded Fourier matrix
Complex conjugate of [E]
Mode shape matrix for f
{f9 (t)}
Gust response unsteady aerodynamic forces
f
Gust response unsteady aerodynamic forces for
blade numberj
Fourier component of { f g (t)} for harmonic V
Motion dependent unsteady aerodynamic forces
{ f s (t)}
{fs}
{ fm(t)}
{.ç (t)} Traveling wave representation of {f" (t)} for Q,
{ f (ç2 )} Centrifugal forces at speed of rotation S2
{ f}
NOMENCLATURE
{q(t)}
Structural damping matrix
Structural damping matrix for (3t
Modal damping matrix for (3
Aeroelastic matrix for fi t
Modal aeroelastic matrix for (3
Total stiffness matrix for rotation speed S2
[K]
[K Isr ]
Total stiffness matrix for rotation speed S2 and li t
Modal stiffness matrix for rotation speed S2 and (3,
[k b ]
[KE] pClassical stiffness matrix for Q=0 and 13 t
[C]
(Co I
[c p I
[A ]
[a ]
'
[Kc (6)]^ Geometric stiffness matrix for (3t [KS] itSupplementary centrifugal stiffness matrix for 0 1
Steady aerodynamic forces
Physical displacements
{q^, (t)}
Physical displacements for sector number j
Traveling wave representation of the
displacements
Traveling wave displacements for I 3
{
Modal coordinates for fi t
f qj (t) 1
{q (t)}
}
;
{q^t }
N
Nw
Nh
Mode shape j for (3 t
Number of blades
Number of wakes
Number of harmonics
Presented at the International Gas Turbine and Aeroengine Congress and Exposition
The Hague, Netherlands — June 13-16, 1994
This paper has been accepted for publication in the Transactions of the ASME
Discussion of it will be accepted at ASME Headquarters until September 30, 1994
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(3 1
Interblade phase angle = Nl
S2
U)
6
e,J
m3
4
Rotation speed in rad/s
Eigen frequency of the structure
Steady stress distribution
Excitability measure
Modal mass for modej
Critical damping ratio
will restrict our study to a wake excitation that can be created by a
variety of sources including support struts and stator blade rows.
Aerodynamic damping is computed via an Euler 3D code. In
addition, an equivalent viscous modal damping can be taken into
account. The structure is supposed to be perfectly tuned and
cyclic symmetry can be used to determine the dynamic behavior
of the structure starting from a finite element model of one sector
of the bladed disk. An application of this procedure has been
made for an experimental high pressure compressor blisk. The
computations have been fitted to actual rig tests.
INTRODUCTION
It is well known that high frequency fatigue is a major cause of
failure in axial compressor components. It occurs when vibration
takes place at or near resonant conditions of operation. In the
design process, Campbell diagrams are used to predict the
occurrence of the resonant conditions in the operating range of
the compressor. Unfortunately, it is not always possible to avoid
resonant conditions in some stages. Therefore, based on prior
experiences, an estimation of the response and the dynamic stress
is usually made to determine the acceptability of a blade design.
This approach may lead to erroneous conclusions and it becomes
necessary for the designer to have a tool that can predict more
accurately the forced response and the dynamic stress of the
components.
A large experimental effort is devoted to the subject of blade
forced response. Although these experiments are very useful to
assess and explain the effects of parameters such as steady blade
loading, shape and amplitude of aerodynamic forcing functions,
reduced frequency (see for example Fleeter (1992)), they cannot
provide quantitative answers in real life cases. This is why (apart
from issues of design cycle time and cost) accurate predictions of
blade forced response in the jet engine industrial environment can
only be achieved through the use of numerical methods.
A substantial part of the numerical analyses of bladed disk
forced response reported in the literature concentrate on the
structural point of view. Irretier and Omprakash (1991) studied
the effects of transient phenomena in the forced response but with
a very simple excitation. Wildheim (1979) found all the resonant
conditions for a rotating periodic structure excited by upstream
flow defects. On the aerodynamic side, many methods (i.e. flow
model and numerical algorithm), of varying degrees of
complexity, have been proposed to compute the motion
dependent and gust response aerodynamic forces. These have
been recently reviewed by Verdon.
A complete definition of the resonant vibratory response of the
engine bladed disks includes a coupled characterization of the
structural dynamics, of the aeromechanical damping and of the
external excitation. An early application of a 2.5D unsteady Euler
code to perform a parametric study of a stator blade row forced
response was given by Joubert and Ronchetti (1989). More
recently, rather precise aeromechanical studies have been
conducted by Chiang and Kielb (1992) and Murthy and Stefko
(1992), with simple 2D aerodynamical computations. Such an
approach has been used by Tatakis and Stockton (1993) to study
the influence of airfoil geometric and aerodynamic parameters on
forced response.
This paper presents a full 3D numerical approach of the dynamic
response of bladed disks. As the dominant source of excitation
can be attributed to the presence of a nonuniform inlet flow, we
THEORY
Dynamic equations of motion
The general aeroelastic response of a bladed disk is given by the
following equations :
[M]{q'(t)}+[C]{q'(t)}+[K]{q(t)}
={fs(t)}+{f (t)}
(1)
,
where the vectors { q ( t ) } and {f8 (t )} are the displacements
and forces for all the sectors of the stage. If the upstream flow
defects responsible for the excitation have a cyclic symmetry of
order Nit, (e.g. wakes from Nw identical and equally spaced
struts), the exciting forces { f a (t)} may be expressed as a Fourier
series of Nh harmonics, all multiples of N. Thus, for blade j :
lf5(t)1=Y,4 {f8}e
(Vn, N^(i n)]
(2)
with p = 1, Nh
where v = N w p
In the following, the notation Re(-) will be omitted.
As we suppose that all the sectors of the bladed disk are
identical, we can take advantage of the traveling wave
representation of the displacement (see Crawley 1988 ). This can
be written as :
{q(t)} = [E]{q,(t)}
(3)
where [E] is the expanded Fourier matrix with:
[Ell] ... [EJN]
and [Ek,]=
[E] =
[ENI] ... [ENN
i 1 j-lXk-f)
diag e
(
]
Substituting (3) into equation (1) and premultiplying by [E] - '
with [E]-' = _!_[E] where [E] * is the complex conjugate of
N
[E]:
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We may wish to reduce the size of the different systems more,
by projecting into a modal base, obtained from the undamped
homogeneous form of (6). The displacement {qp, (t)} can be
expressed as :
[E] -' [M][E]{qo(t)}+[E]_ ' [C][E]{qe(t)}+[E] -' [K][E]{ge(t)}=
[E] - '{ fg(t)}+[E] - '{f (t)}
(4)
As [M], [K], [C] are block circulant matrices, the matrices of
equation (4) are block diagonal (see Davis 1979).
For exemple,
[E]-'[M][E] = diag[Mp,]
and,
{qp}={....{qp1},...}
{q9, ( t) } = [QR, ]{ X9, }e 'v n r
where [Q,^, J is the mode shape matrix and {X 1 } the modal
coordinates. Assuming linearity, we can express the motion
dependent forces as:
l = - D/2......,+D/2
D = N if N even
D = N-1 ifNodd
{J (t)}=[A,,]{Xe,}ervn,
The development of the right hand term [E] - '{ fg(t)} shows
that a single interblade phase angle 0, contained between
[-7c, it] is excited by the harmonic v and is given by:
(3r = d with I = —v [N]
1e
1
,+ D]
-2
(7)
(8)
Substituting (7) and (8) into (6) and premultiplying by the
transpose conjugate of [Q,] results in the modal equations :
(5)
(— (v 92)2[ma,]+iv c4c,]+[k1]){X1}=
[a ] + {fs }+[a p ]{X,,}
This relation may be represented graphically by:
(9)
Where [inn,] and [kni ] are the diagonal modal matrices of the
structure for the selected eigenmodes.
v
[CO, Iis the structural modal damping matrix
[a,, ]
is the modal aeroelastic matrix.
The resolution of (9) gives the amplitude of the different modes
for the interblade phase angle (3 1 . The total response of the
structure is obtained by adding the different contributions for the
different interblade phase angles excited. The stresses are
obtained on any point of the structure using the modal stresses
distributions.
Mechanical model
-^.4
D/2
A finite element mesh is made for one blade and one sector of
the disk. In order to reduce the time spent on that task an
automatic generation of mesh is possible. A static analysis can be
performed for (3 o with the following general equation:
I
([KE] +[Kc( 6 )]+[K5]]p o {qR0 (t)} = {f(Q 2 )} + {.fas}
The system of equations (4) can then be reduced to Nh separated
systems of equations of smaller size, each one corresponding to
an interblade phase angle 03, :
where [KE ], [K o (6)], and [Ks ] are respectively the classical,
geometric, and supplementary centrifugal stiffness matrices (for a
more complete description, see Ferraris, Henry, Lalanne, and
Trompette, 1983).
[M9,]t4e^(t)1 + ^C9,]149,( t )I + ^K9,^lg9t( t )J =
{ f 8 }e Nn,
+If9r(t)I (6)
The stress distribution 6 is initially unknown, thus an iterative
procedure is used to perform the static solution. Once 6 is
known, the stiffness matrix [Kn,] of the structure for the rotation
speed f and for any ii i can be constructed. For all the interblade
phase angles excited by the gust induced forces, the following
dynamic system of equations can be solved.
Of course if any dissymmetry was present in the structure, or if
friction forces between the blades were added, some coupling
between the different interblade phase angles would have to be
taken into account.
3
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gust response harmonic of interest. It is useful to recognize this
fact, because numerically resolving the (unneeded) higher spatial
harmonics present in the original wake shape would require a
much finer grid than necessary, resulting in a direct computing
cost penalty. In the application considered in this paper however,
the grid used could perfectly resolve all the harmonics necessary
to reconstruct the original signal, so they were all kept. The Euler
equations are integrated in time until convergence to periodicity
is reached, which usually requires around 10 periods of wake
passing. The unsteady pressures on the blades are then Fourier
transformed and the first five harmonics are interpolated from the
aerodynamic mesh on to the structural mesh, thus providing the
gust response unsteady aerodynamic force for the forced response
mechanical computation.
1 w 2 [M9, ]+[KK, ])Jgat I = {0}
—
Thus the mode shapes {q p, } and all the terms of the matrices
[Q] and [k,] can be determined.
Unsteady Aerodynamic model
This model provides an estimate of the flow forces acting on the
blades. The first step is to determine the compressor operating
points corresponding to the frequency crossings of interest. For
each of these cases, we then have to perform a steady
computation to initialize the flowfield, followed by two unsteady
computations, one to determine the motion dependent unsteady
aerodynamic forces { f'" (t) } , the other to determine the gust
response unsteady aerodynamic forces { f g (t) } . All three
computations are done using the same 3D unsteady Euler code
(Gerolymos (1992)). This solver numerically integrates the 3D
unsteady Euler equations discretized in finite volume
formulation, using Runge Kutta schemes. A single interblade
channel is considered for the computations, making use of the
chorochronic periodicity of the flow field perturbed by harmonic
disturbances imposed at the boundaries (i.e. blade motion and
upstream vorticity). Convergence of the unsteady computation
towards time and space periodicity is estimated using residuals on
the first harmonic of pressure at the pitchwise boundaries. A
periodicity residual of 1% is considered satisfactory.
To compute the motion dependent forces {f'"(t)}, we use
mode shapes determined by the mechanical model, interpolated
from the structural mesh on to the aerodynamic mesh around the
blade. From this is derived a distribution of movement for the
aerodynamic mesh, which conforms with the blade vibration. The
unsteady Euler equations are integrated in time on the moving
grid, with a fixed time lag (corresponding to the interblade phase
angle of the vibratory mode) imposed at the permeable pitchwise
boundaries. Time integration usually covers a span of 10
vibration periods which is sufficient to establish spatial and time
periodicity, as assessed by the periodicity residuals. As the
vibration amplitude is small with respect to the blade chord, only
the first harmonic of the unsteady pressures on the blades are of
interest. Those computations are used to build the matrix [A,,]
for each interblade phase angle 13, and thus to obtain the modal
aeroelastic matrix after multiplication by [Q
1
APPLICATION
Description of experiment
The goal of this test series was to compare the resonant response
of a blisk rotor, composed of 72 blades, shown in figure 1, with
that found for rotors of conventional technology. Only the results
obtained on the blisk rotor will be presented here and the
experimental values of blade damping and stress compared with
the numerical prediction.
The compressor used (figure 2) is composed of three blade rows
(IGVs, rotor and stator), and is representative of the rear stages in
civil HP compressors. A section of the compressor is represented
on figure 3. The aerodynamic forcing function is provided by the
wakes of cylinders inserted vertically in the flowpath and equally
spaced around the circumference. The number of cylinders is
variable and can be adjusted in order to excite the blisk in a
resonant condition, for the first bending and first torsion modes,
with several engine harmonics. Also, three different cylinder
diameters were tried. Since existing correlations for wake
shedding are tailored for blade profiles and are not valid for
cylinders, the flow defect due to the cylinder wake was measured
immediately upstream of the rotor at four heights in the annulus :
20%, 40%, 60% and 80%. The experimental data was fitted with
analytical functions for Fourier transformation, subsequent
discretization and reintroduction at the upstream flow boundary.
An example of a data fit is given in figure 4. Note that the fit was
performed at the nominal operating point and with one cylinder in
the annulus, and expressed as a percentage of total pressure loss.
The same fit was then used for all other flow conditions and
number of cylinders. For evaluation of blade forced response,
strain gages were placed on seven blades; all results given
subsequently (unless otherwise stated) are an average of the
results of the individual blades. The Campbell diagram of the
blisk is given in figure 5. In the following, we will concentrate on
the resonant conditions 1T/2252.
The experiment itself was comparatively standard. The
compressor map was first measured and is given in figure 6.
When looking for resonant conditions, the compressor was
revved without changing the throttle setting, which defines an
operating line. This line and the operating point corresponding to
the 1T/2252 crossing are also shown on figure 6. Gage data was
recorded continuously then analysed off-line to yield blade
response and damping.
]
To determine the gust response forces {p8 (t)}, we need an
estimate of the flow defects present at the upstream boundary of
the domain. In the case of wakes being shed by upstream blade
rows, this is usually obtained through correlations giving a spatial
total pressure distribution (depth and width) as a function of the
upstream blade row loss coefficient, solidity and distance from its
trailing edge. This total pressure distribution is then Fourier
transformed in the tangential direction, at each radial station of
the computational grid. Usually, only the spatial harmonic
corresponding to the excitation being studied is used to specify
the upstream boundary condition of the computation, since in the
case of small perturbation amplitude, it is the main driver of the
4
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Mechanical model
A finite element mesh of the blisk has been performed for one
sector (figure 7) to predict stage frequencies and blade motions
using cyclic symmetry. The mesh density on the blade has been
adapted for stress predictions. The interblade phase angles being
taken into account in the computations are given by equation (5).
If we consider the resonance 1T/2252, we have to study separately
the Fourier order 1=-22, +28, +6 respectively for the first three
harmonics of the excitation force.
As we can see on the Campbell diagram, there is no mode
excited by the frequencies 4452 and 6652 for the rotation speed S2
chosen here. Thus we will only consider in this case the first
harmonic of the excitation force associated with the Fourier order
1=-22. For this Fourier order, the eigenvalues and eigenmodes
analysis show very little coupling between blade and disk motion
for the first three modes. The mode shape of the mode IT is
represented on figure 8.
For the first three modes, a good correlation on the frequencies
has been obtained. As the relative gap on the frequencies between
the lT/1B and 1T/2B modes is respectively 247% and 177%, we
can neglect in the computations the dynamic coupling between
the IT mode and its neighbouring modes. Although the program
calculates general problems, the system (9) to be solved is
reduced here to a single degree of freedom equation for the IT
mode.
Aerodynamical model
As mentioned earlier, the unsteady 3D Euler code uses H-type
meshes. For this application, a mesh size of
120(axial)x21(radial)x25(tangential) was used in order to
adequately resolve the unsteady aerodynamic phenomena present
in the flowfield, especially close to the leading edge. Figure 9
gives the mesh section at 90% blade height.
A steady computation was performed at the operating point
corresponding to the crossing of interest. Since the Euler
computation cannot reproduce exactly the compressor map, it was
preferred to reproduce mass flow and rotational speed (for
accurate upstream relative velocity) and allow a slight
descrepancy on the pressure ratio.
The steady computation was used to initialize two unsteady
computations ; one to determine the aerodynamic damping on the
modes excited by the wakes, the other to determine the gust
response unsteady aerodynamic forces due to the wakes of 22
upstream cylinders. Of course, a certain amount of mistuning
between blades was observed in the experiment, but the
computations were performed at only one vibration frequency /
rotational speed since the effect of varying this parameter over
such a small range was deemed to be second order. The frequency
used for the first torsion mode was 5082 Hz which corresponds to
a reduced frequency of 1.1, based on blade tip chord. The
corresponding rotational speed was 13900 RPM, for interaction
of the IT mode with the 22 upstream wakes.
15 periods of the unsteady phenomenon were simulated, both
for flutter and gust response. Unsteady convergence was adequate
after 10 periods but it was prefered to push on for another 5
periods to check that convergence was continuing to improve.
CPU time for each of these computations was roughly two hours
on a CRAY YMP computer - an implicit residual smoothing
technique added to the four step Runge Kutta scheme allowed the
use of large CFL numbers and thus reduced the number of
iterations per period.
Results and comparison
As the structure studied here is a blisk , there is no energy
dissipation at the root of the blades. The only sources of damping
are the internal friction in the material and the aerodynamic
damping due to the aerodynamic motion dependent forces. As the
former is very small for conventional steel materials, it will be
neglected in the following study. We can thus compare the
computed aerodynamic damping for the 1T/2252 resonant
condition with the total damping measured at that point during
the test and assumed here to be only aerodynamical.
The aerodynamical damping is computed by solving the
eigenvalue problem associated with the homogeneous form of
equation (9). The critical damping ratio is = 0.54 %. An
estimate of the experimental damping is given by the frequency
bandwidth method applied to the frequency response of a strain
gage obtained from the experimental Campbell analysis shown in
figure 10. The critical damping ratio obtained is E, = 0.36%.
The agreement between calculations and experiments, although
not excellent, can be considered as satisfactory if we consider the
lack of precision associated with the method we used to
determine the experimental damping.
The excitation forces can be analysed using several
characteristic numbers. The first one is the force magnitude :
v
f s } . Here at the crossing point 1 T/22-Q,
11 f 8II ={
f^
I1f811=0.011 daN.
The second one is the excitability measure for harmonic v and
{__ }T I
2
{}
mode j defined as e
_
T
r lTwhere
{qq, } is the mode shape j for the phase angle (31.
e,, is a real number between 0 and 1, and is independant of the
mode shape normalization. It is a measure of the compatibility of
the force distribution with the mode shape. For our case study,
e2n.=0.004 ; the excitation forces are mainly located on the lower
part of the blade (see figures 11a & 11b).
If we want to compare the severity of different excitations for
i}T{ fs}^
different modes, we can use the ratio
m
, where mj is
;
the modal mass for mode j.
The normalization of each mode shape is taken to induce a
maximum stress equal to 1 on the blade.
After the separate analysis of aerodynamical damping and
excitation forces, we performed the complete resolution of
equation (9). The stresses obtained are compared to the
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measurements made with six different strain gages placed on
different blades. The discrepancy between computations and
measurements is of the order of the magnitude of the blade to
blade variation. It is due to the mistuning, the lack of precision on
the strain gage locations, and the incertaintees of the
measurements. An illustration is given on figure 12 for the strain
gage location shown in figure 13. Only the extremum measured
values are represented. The difference in frequency between the
different peaks is mainly due to blade mistuning. For the
computations, the frequency of the model has been updated on
the mean of the experimental distribution. To draw the numerical
curve, we reasonably extended, on a narrow frequency band, the
unsteady aerodynamic pressures computed for the speed of
rotation and blade frequency of the resonant condition 1T/2252.
The difference between maximum and minimum experimental
stresses is rather large (40%). As our model does not take into
account the blade to blade mistuning, it is significant to consider
the average experimental stress. The numerical result and the
average experimental stress level were found to match perfectly.
The good agreement between experiments and calculations on the
dynamic response implies that the estimate of the gust response
unsteady forces is also of the right order of magnitude. Overall,
the numerical method can be considered to give satisfactory
agreement with the data for the case studied, at reasonable cost.
CONCLUSION
We have presented an aeromechanical method to predict the
forced response of bladed disks to aerodynamic excitation from
upstream wakes. The method is fully three dimensional, based on
a finite element model of the bladed disk coupled with a timemarching non-linear Euler solver. This degree of modelling is
probably the most elaborate which can be used in an industrial
environment (where reasonably fast and trouble free methods are
essential) for the next couple of years. The current method has
been thoroughly checked, and used in this study to predict the
aerodynamic damping, gust excitation and forced response of the
1T/2252 crossing of an HP compressor blisk. The predictions are
borne out by experimental results on the configuration studied :
orders of magnitude are well predicted and are within blade to
blade dispersion. It is therefore thought that this method is
adequate for selecting or improving blade configurations with
respect to aeromechanics. The method is currently being used to
predict the blade response at other crossings of the configuration
described here and of fans subjected to inlet or downstream
distortion.
Crawley, E.F. : "Aeroelastic Formulation for Tuned and Mistuned
Rotors" AGARD Manual on Aeroelasticity. in Axial-Flow
Turbomachines, vol 2, Structural Dynamics and Aeroelasticity,
1988.
Davis, P.J.: Circulant Matrices, 1979, Wiley-Interscience.
Ferraris, G. , Henry, R. , Lalanne, M. , and Trompette, P. Frequencies and mode shapes of rotating bladed axisymmetric
structures. Application to a jet engine. Advanced vibrations of
bladed disc assemblies, ASME, 1983, 10 p.
Fleeter, S. . "Forced Response Unsteady Aerodynamic
Experiments". AIAA Paper 92-0144.
Gerolymos, G.A.: "Advances in the Numerical Integration of the
3-D Euler Equations in Vibrating Cascades". ASME Paper 92GT-170.
Gerolymos, G.A.: "Coupled 3-D Aeroelastic Stability Analysis of
Bladed Disks". ASME Paper 92-GT-171.
Irretier, H. and Omprakash, V. : "Numerical Analysis of the
Transient Responses of Bladed Disks". ASME 1991 DE-Vol 36,
Machinery Dynamics and Element Vibrations.
Joubert, H. and Ronchetti, V. "Aerodynamic Study of Forced
Vibrations on Stator Rows of Axial Compressors". AGARD CP
468 : "Unsteady Aerodynamic Phenomena in Turbomachines",
August 1989.
Murthy, D.V.,and Stefko, G.L. "FREPS : A Forced Response
Prediction System for Turbomachinery Blade Rows". AIAA
Paper 92-3072.
Tatakis, S. and Stockton, R.: "Structural and Aerodynamic
Influences to Airfoil Forced Response" ASME Paper 93-GT243.
Verdon, J.M. : "Review of Unsteady Aerodynamic Methods for
Turbomachinery Aeroelastic and Aeroacoustic Applications".
AIAA Journal, Vol. 31, N°2, pp 235-250, February 1993.
Wildheim, S.J. : "Excitation of Rotationnally Periodic
Structures". ASME Journal of Applied Mechanics Vol 46, pp
878-882, 1979.
ACKNOWLEDGEMENTS
The authors would like to acknowledge the contribution of Marc
Loubet for running a thorough experiment and providing high
quality data, which is essential for the correlation and validation
of this analytical method.
REFERENCES
Chiang, H.D. and Kielb, R.E. : "An Analysis System for Blade
Forced Response" ASME Paper 92-GT-172.
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C.
..
a
j
!
Is
^
Figure 2 : Test rig
Figure 1: Blisk geometry
FREQ. (Hz)
0852 (RD1)
Figure 3 : Section of the test rig
wake disturbance
7052(RDE)
1.01
i
PT/PREF
b
ji
0.ss
0.98
0.96
-6
-4
-2
0
2
630
2T
2F
I'
0.97
-8
\Ji
2sM
IT
4
6
8
57f2
410
22f
9Q
1 F
tangential spacing
— data -a- analytical model
nuuu
10 000
Figure 5 : Campbell diagram
Figure 4 : Comparison of the wake modeling and data
7
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15 000 S2 (tr/mn)
J X 1.3
Surge line Operating point
1T/2252
1.2
04eY0.95
0.90
1.1
Nn
0.80
0.70
2.5
3.0
3.5
4.0
Dsrd
Figure 7 : Finite element mesh of a blisk sector
Figure 6 : Compressor map
1.000
1.085
1.018
0.951
0.883
0.816
0.749
0.661
0.614
0.546
0.479
0.412
0.344
0.277
0.21
0.142
0.075
0.008
Figure 8 : Mode shape on the blade for the IT mode
Figure 10 : Experimental Campbell analysis for
the 1T/22) crossing
Figure 9 : Aerodynamic mesh at 90% span
8
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2 .Poo
•
VALOUR
. 66.
leading edge
`000
V AL E U R
•
835.7
786.4
736.9
687.5
638
855.6
808.6
761.6
714.5
667.4
620.4
573.3
526.2
479.2
432.1
385
337.9
290.9
243.8
196.7
49.7
02.6
539.1
489.6
440.2
390.7
341 .3
291.8
242.4
192.9
Ia3.S
44.6
trailing edge
Figure 1 la: Module of the excitation forces
suction surface
.E6.
trailing edge
-.. _. .
_.
.=
as
'.,
Figure 11 b : Module of the excitation forces
pressure surface
.....
1
a 0.8
a
N
a 0.6
I
0
0.4
0.2
0
14000
14200
14400
14600
Rotational Speed (rpm)
14800
G ;
leading edge
Comp sveu Muupsuess
Min exp wens
Z
_ sa ..
15000
Figure 12 : Comparison between computational data
stress and experimental data stress
Figure 13 : Strain gage lopation
9
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