Sensitivity of Aeroelastic Properties of an Oscillating LPT Cascade
Nenad Glodic
Licentiate Thesis 2013
Department of Energy Technology Division of Heat and Power Technology
Royal Institute of Technology 100 44 Stockholm, Sweden
TRITA KRV Report 13/05 ISSN 1100/7990
ISRN KTH/KRV/13/05-SE ISBN 978-91-7501-809-6
© 2013 Nenad Glodic
ABSTRACT
Modern turbomachinery design is characterized by a tendency towards thinner, lighter and highly loaded blades, which in turn gives rise to increased sensitivity to flow induced vibration such as flutter. Flutter is a self-excited and self-sustained instability phenomenon that may lead to structural failure due to High Cycle Fatigue (HCF) or material overload. In order to be able to predict potential flutter situations, it is necessary to accurately assess the unsteady aerodynamics during flutter and to understand the physics behind its driving mechanisms. Current numerical tools used for predicting unsteady aerodynamics of vibrating turbomachinery components are capable of modeling the flow field at high level of detail, but may fail in predicting the correct unsteady aerodynamics under certain conditions. Continuous validation of numerical models against experimental data therefore plays significant role in improving the prediction accuracy and reliability of the models.
In flutter investigations, it is common to consider aerodynamically symmetric (tuned) setups. Due to manufacturing tolerances, assembly inaccuracies as well as in-service wear, the aerodynamic properties in a blade row may become asymmetric. Such asymmetries can be observed both in terms of steady as well as unsteady aerodynamic properties, and it is of great interest to understand the effects this may have on the aeroelastic stability of the system.
Under certain conditions vibratory modes of realistic blade profiles tend to be coupled i.e. the contents of a given mode of vibration include displacements perpendicular and parallel to the chord as well as torsion of the profile. Current design trends for compressor blades that are resulting in low aspect ratio blades potentially reduce the frequency spacing between certain modes (i.e. 2F & 1T).
Combined modes are also likely to occur in case of the vibration of a bladed disk with a comparatively soft disk and rigid blades or due to tying blades together in sectors (e.g. in turbines).
The present investigation focuses on two areas that are of importance for improving the understanding of aeroelastic behavior of oscillating blade rows.
Firstly, aeroelastic properties of combined mode shapes in an oscillating Low Pressure Turbine (LPT) cascade were studied and validity of the mode superposition principle was assessed. Secondly, the effects of aerodynamic mistuning on the aeroelastic properties of the cascade were addressed. The aerodynamic mistuning considered here is caused by blade-to-blade stagger angle variations
The work has been carried out as compound experimental and numerical investigation, where numerical results are validated against test data. On the experimental side a test facility comprising an annular sector of seven free- standing LPT blades is used. The aeroelastic response phenomena were studied in the influence coefficient domain where one of the blades is made to oscillate in three-dimensional pure or combined modes, while the unsteady blade surface pressure is acquired on the oscillating blade itself and on the non-oscillating neighbor blades. On the numerical side, a series of numerical simulations were carried out using a commercial CFD code on a full-scale time-marching 3D
viscous model. In accordance with the experimental part the simulations are performed using the influence coefficient approach, with only one blade oscillating.
The results of combined modes studies suggest the validity of combining the aeroelastic properties of two modes over the investigated range of operating parameters. Quality parameters, indicating differences in mean absolute and imaginary values of the unsteady response between combined mode data and superposed data, feature values that are well below measurement accuracy of the setup.
The findings of aerodynamic mistuning investigations indicate that the effect of de- staggering a single blade on steady aerodynamics in the cascade seem to be predominantly an effect of the change in passage throat. The changes in steady aerodynamics are thereby observed on the unsteady aerodynamics where distinctive effects on flow velocity lead to changes in the local unsteady pressure coefficients. In order to assess the overall aeroelastic stability of a randomly mistuned blade row, a Reduced Order Model (ROM) model is introduced, allowing for probabilistic analyses. From the analyses, an effect of destabilization due to aero-asymmetries was observed. However the observed effect was of moderate magnitude.
Keywords: flutter, aeroelastic response, combined modes, bending-torsion flutter, mode superposition, aerodynamic asymmetries, ROM, probabilistic analysis
SAMMANFATTNING
Modern design av turbomaskiner karakteriseras av en tendens till tunnare, lättare och högbelastade skovlar, som i sin tur är känsligare för flödesinducerade vibrationer så som fladder. Fladder är ett självexciterat och självbärande instabilitetsfenomen som kan leda till ett strukturellt haveri p.g.a. utmattning eller överbelastning. För att kunna prediktera potentiella fladdersituationer är det nödvändigt att noggrant studera den instationära aerodynamiken under fladder och att försöka förstå fysiken bakom dess drivande mekanismer. Aktuella numeriska verktyg som används för prediktering av den instationära aerodynamiken orsakad av de vibrerande turbinkomponenterna är kapabla att modellera strömningsfältet på en hög detaljnivå. Kontinuerlig validering av numeriska modeller mot experimentell data spelar en viktig roll för att förbättra predikteringsnoggrannhet av modeller.
I fladderundersökningar, är det vanligt att anta ett aerodynamiskt symmetriskt system. På grund av tillverkningstoleranser, onoggrannhet vid montering samt slitage, kan de aerodynamiska egenskaperna i en skovelrad bli asymmetriska.
Sådana asymmetrier kan observeras i både stationära samt instationära aerodynamiska egenskaper. Det är av stort intresse att förstå vilka effekter detta kan få på systemets aeroelastiska stabilitet.
Vibrationsmoder av realistiska skovelprofiler är nästan alltid kopplade d.v.s.
innehållet i en given vibrationsmod inkluderar förskjutningar parallella och vinkelräta mot kordan samt vridning av profilen. Nuvarande designtrender för kompressorskovlar resulterar i låga spann -korda förhållanden som potentiellt minskar frekvensskillnaden mellan vissa moder (t.ex.. 2F & 1T). Kombinerade vibrationsmoder kan sannolikt inträffa vid vibration av en rotorskiva med en jämförelsevis mjuk skiva och styva skovlar eller på grund av i sektorer kopplade skovlar (t.ex. i turbiner).
Den här undersökningen fokuserar på två områden som är av betydelse för att förbättra förståelsen av de aeroelastiska egenskaperna i oscillerande skovelgitter.
För det första undersöks de aeroelastiska egenskaperna av kombinerade moder i ett oscillerande lågtrycksturbinsgitter och giltighet av superpositionsprincipen av moder bedöms. Vidare undersöks vilka effekter de aerodynamiska asymmetrierna har på de aeroelastiska egenskaperna i gittret. De aerodynamiska asymmetrierna anses här orsakade av profilvinkelvariationer mellan individuella skovlar.
Arbetet omfattar både experimentella undersökningar och numeriska simulationer, där de numeriska resultaten har validerats mot provdata. På den experimentella sidan används en provanläggning som innefattar en annulär sektor av sju fristående lågtrycksturbinskovlar. Det aeroelastiska gensvaret i sektorgittret studeras med hjälp av influenskoefficienter, där en av skovlarna oscilleras i tredimensionella, rena eller kombinerade moder medan det instationära trycket på skovelytan uppmäts både på den oscillerande skoveln och på de icke-oscillerande angränsande skovlarna. På den numeriska sidan, genomförs en serie av simuleringar med en kommersiell CFD-kod som använder en fullskalig 3D viskös modell. I enlighet med de experimentella proven görs också simuleringar i influenskoefficientdomänen, där endast en av skovlarna oscilleras.
Resultaten av undersökningarna av kombinerade moder tyder på att för det undersökta området av driftparametrar är det giltigt att kombinera aeroelastiska egenskaper hos två oscillationsmoder. Kvalitetsparametrar som indikerar skillnader i de absoluta och imaginära värdena på den instationära aerodynamiska responsen mellan de uppmätta värdena av kombinerade moder och de superponerade rena moderna har värden som är betydligt lägre än mätnoggrannhet i systemet.
Resultaten av aerodynamiska mistuning-undersökningar tyder på att påverkan av ändringar i skovelns profilvinkel på den stationära aerodynamiken i kaskaden verkar främst vara en effekt av areaförändringen i skovelpassagens trängsta sektion. Förändringarna observerades för den instationära aerodynamiken där särskiljande effekter av flödeshastighet leder till förändringar i de lokala instationära tryckkoefficienterna. För att bedöma den totala aeroelastiska stabiliteten hos en slumpmässigt aerodynamiskt mistuned skovelrad, introducerades en modell med reducerade frihetsgrader (Reduced Order Model - ROM), vilket möjliggjort probabilistiska analyser. Från analyserna har en påverkan av destabilisering på grund av aero-asymmetrier observerats men den observerade effekten är av måttlig storlek.
Nyckelord: fladder, aeroelastiskt gensvar, kombinerade oscillationsmoder, superposition, aerodynamiska asymmetrier, ROM, probabilistisk analys
PREFACE
The thesis is based on the following papers:
1 Glodic, N., Bartelt, M., Vogt, D. M., Fransson, T.H., 2009
„Aeroelastic Properties of Combined Mode Shapes in an Oscillating LPT Cascade”
Paper presented at ISUAAAT2009
2 Glodic, N., Vogt, D. M., Fransson, T.H., 2011
“Experimental and Numerical Investigation of Mistuned Aerodynamic Influence Coefficients in an Oscillating LPT Cascade”
Paper presented at ASME Turbo Expo 2011
The involvement of Dr. Damian Vogt and Prof. Torsten Fransson in the above publications consisted of problem formulation and discussion of results. The contribution of Mr. Michael Bartelt in Paper 1 consisted of supervised numerical computations and experimental testing during his M. Sc. Thesis work. For all publications the underlying material was part of the work elaborated in this thesis.
ACKNOWLEDGMENTS
The present study has been funded by the Swedish Energy Agency, Siemens Industrial Turbomachinery AB, GKN Aerospace (former Volvo Aero Corporation), and the Royal Institute of Technology through the Swedish research program TURBOPOWER, the support of which is gratefully acknowledged.
I would like to express my gratitude to Prof. Torsten Fransson at the Chair of Heat and Power Technology at KTH, for giving me opportunity to perform this work at the department and for sharing his enthusiasm regarding aeroelasticity research.
Endless thanks go to my supervisor Assoc. Prof. Damian Vogt at KTH, for his enthusiastic and valuable support without which this study would not be possible.
To the Turbovib project members from GKN Aerospace (former Volvo Aero), Siemens Industrial Turbomachinery AB, Department of Solid Mechanics and Marcus Wallenberg Laboratory for Sound and Vibration Research at KTH, for their support and productive discussions. It was a pleasure to be a part of such a highly competent team.
Special thanks to my colleague Jens Fridh for introducing me to the exciting world of experimental aerodynamics.
I would like to thank my colleagues from the Aeromech group and others at the HPT Division for fruitful discussions, mutual motivation and some excellent fishing tours.
I would also like to express my gratitude to the technicians in the lab, Stellan Hedberg, Christer Blomqvist, Leif Pettersson, Mikael Schullström and Göran Arntyr, for providing great technical support and for all interesting and stimulating discussions that we had during our coffee brakes.
Finally, I would like to express my sincere and deepest gratitude to my wife Vera and to my two beautiful sons, Stefan and Simon, for supporting me and being a warm light through tough times.
TABLE OF CONTENTS
ABSTRACT ... 1
SAMMANFATTNING ... 3
PREFACE ... 5
ACKNOWLEDGMENTS ... 7
TABLE OF CONTENTS ... 9
LIST OF FIGURES ... 11
LIST OF TABLES ... 14
NOMENCLATURE ... 15
1 BACKGROUND ... 19
FLUTTER IN TURBOMACHINERY ... 19
1.1 MODE SHAPES SENSITIVITY ... 23
1.2 EFFECTS OF MISTUNING ... 26
1.3 1.3.1 Aerodynamic Mistuning ... 28
2 FUNDAMENTAL CONCEPTS ... 31
CLASSICAL APPROACH IN FLUTTER STABILITY PREDICTIONS ... 31
2.1 GOVERNING EQUATIONS ... 31
2.2 UNSTEADY AERODYNAMIC FORCES ... 32
2.3 STABILITY PARAMETER ... 34
2.4 AERODYNAMIC INFLUENCE COEFFICIENTS ... 35
2.5 REDUCED ORDER MODEL FOR ADDRESSING AERODYNAMIC MISTUNING ... 36
2.6 3 OBJECTIVES ... 39
4 INVESTIGATION METHODOLOGY ... 40
5 EXPERIMENTAL INVESTIGATIONS ... 41
DESCRIPTION OF TEST SETUP ... 41
5.1 5.1.1 Test Object ... 41
5.1.2 Experimental Facility ... 41
5.1.3 Coordinate Systems ... 44
5.1.4 Convention of Blade Oscillation and Stagger Angles ... 45
5.1.5 Measurement Equipment ... 47
5.1.5.1 Steady-state measurement setup ... 47
5.1.5.2 Unsteady measurement setup ... 48
EXPERIMENTAL TESTING STRATEGY ... 53
5.2 6 NUMERICAL INVESTIGATIONS ... 55
NUMERICAL METHOD ... 55
6.1 6.1.1 Governing equations ... 55
6.1.2 RANS and Turbulence modelling ... 57
DESCRIPTION OF NUMERICAL MODEL ... 60
6.2 SIMULATION APPROACH ... 62
6.3 6.3.1 Aerodynamic mistuning simulations ... 63
7 RESULTS ... 65
STEADY-STATE RESULTS ... 65
7.1 7.1.1 Influence of Aerodynamic Asymmetries on Steady Blade Loading ... 66
UNSTEADY RESPONSE-NUMERICAL PREDICTIONS VS.TEST DATA ... 70
7.2 7.2.1 Axial Bending ... 70
7.2.2 Circumferential Bending ... 73
7.2.3 Torsion ... 74
7.2.4 Impact of Flow Velocity ... 76
COMBINED MODE SHAPES ... 78
7.3 7.3.1 Linear Superposition of Modes ... 78
7.3.2 Axial Bending-Torsion ... 79
7.3.2.1 Effects of Flow Velocity ... 82
7.3.2.2 Effects of Flow Incidence ... 85
7.3.2.3 Mode Combination at Different Bending-to-Torsion Amplitude Ratios .. 87
7.3.2.4 Three-Dimensional Effects on Mode Combination ... 87
7.3.3 Circumferential Bending-Torsion ... 88
7.3.3.1 Effects of Flow Velocity ... 91
EFFECTS OF AERODYNAMIC MISTUNING ON AEROELASTIC RESPONSE ... 94
7.4 7.4.1 Unsteady Response- Mistuned Case ... 94
7.4.2 Mistuned Aerodynamic Influence Coefficients... 100
7.4.2.1 Impact of Reduced Frequency ... 102
7.4.3 Mistuned Aeroelastic Model- Probabilistic Analysis ... 104
8 SUMMARY ... 111
CONCLUSIONS ... 111
8.1 RECOMMENDATIONS FOR FUTURE WORK ... 113
8.2 9 REFERENCES ... 115
LIST OF FIGURES
Figure 1.1: Collar’s triangle of forces ... 19
Figure 1.2: Cut-away of a Rolls-Royce Trent 1000 engine (picture courtesy of Rolls-Royce) ... 20
Figure 1.3: A typical Campbell diagram ... 20
Figure 1.4: Aerodynamic coupling in a turbomachine blade row (Vogt, 2005) ... 22
Figure 1.5: Nodal diameter patterns and corresponding instantaneous cascade geometries (adapted from Vogt, 2005) ... 23
Figure 1.6: Mode shape sensitivity plot at low subsonic Mach number; shaded areas mark stable regions; values display aerodynamic damping parameters (Vogt and Fransson, 2007) ... 24
Figure 1.7: The effect of aerodynamic detuning on the critical reduced frequency with the elastic axis and center of gravity located at 40% of the chord (Hoyniak and Fleeter, 1986) ... 26
Figure 1.8: Effect of mistuning on flutter (Kaza and Kielb, 1983) ... 27
Figure 2.1: Investigated orthogonal mode shapes ... 32
Figure 2.2: Response and excitation correlated by the phase angle ... 34
Figure 2.3: Characteristic stability curve (S-curve) ... 36
Figure 2.4: Schematic illustration of aerodynamic coupling influences in the cascade assuming lumped parameter model ... 37
Figure 5.1: LPT rotor blade row (including rotor disk) ... 41
Figure 5.2: Schematic drawing of the air supply system ... 42
Figure 5.3: Test facility and sketch of the test module ... 43
Figure 5.4: Inlet and outlet sidewalls; Vogt (2005) ... 43
Figure 5.5: Global test rig coordinate system ... 44
Figure 5.6: Local on-blade coordinate system ... 45
Figure 5.7: Orthogonal oscillation modes ... 46
Figure 5.8: Definition of stagger angle and de-staggering convention ... 46
Figure 5.9: Distribution of the pressure taps on the blade surface (non-oscillating blade) ... 47
Figure 5.10: Non-oscillating instrumented blades ... 48
Figure 5.11: Oscillating blade used for the unsteady pressure measurements .... 49
Figure 5.12: Dynamic calibration setup ... 50
Figure 5.13: Example of a transfer characteristic ... 50
Figure 5.14: Laser signal calibration setup ... 52
Figure 6.1: Modeled sector cascade ... 60
Figure 6.2: Different investigated mesh types; Mårtensson (2005) ... 61
Figure 6.3: Time history of unsteady response on the surface of the oscillating blade ... 62
Figure 7.1: Steady blade loading at midspan section on blade 0; M2=0.4 ... 65
Figure 7.2: Impact of blade 0 de-staggering on passage throats ... 66
Figure 7.3: Influence of aerodynamic asymmetries on steady blade loading; low subsonic case (M2=0.4); 50% span ... 67
Figure 7.4: Influence of aerodynamic asymmetries on steady blade loading; high subsonic velocity (M2=0.8); 50% span ... 68
Figure 7.5: Experimental data vs. CFD predictions; Steady blade loading at midspan; low subsonic velocity (M2=0.4); ... 69
Figure 7.6: Unsteady blade surface pressure at midspan; axial bending; k=0.3; M2=0.4 ... 70
Figure 7.7: Spatially resolved unsteady force component at midspan section on blades -1, 0 and +1 (imaginary and real); IBPA=0deg ... 71 Figure 7.8: Nominal aerodynamic force influence coefficients for blades -1, 0 and +1; axial bending; k=0.3; M2=0.4 ... 72 Figure 7.9: Nominal aerodynamic influence coefficients for blades -1, 0 and +1 in complex plane ... 72 Figure 7.10: Unsteady blade surface pressure at midspan; circumferential
bending; k=0.3; M2=0.4 ... 73 Figure 7.11: Nominal integrated aerodynamic influence coefficients for blades -1, 0 and +1; circumferential bending; k=0.3; M2=0.4 ... 74 Figure 7.12: Unsteady blade surface pressure at midspan section; torsion; k=0.3;
M2=0.4 ... 75 Figure 7.13: Nominal aerodynamic influence coefficients for blades -1, 0 and +1;
torsion; k=0.3; M2=0.4 ... 76 Figure 7.14: Comparison of measured and predicted unsteady loading on blades - 1, 0 and +1 at different velocity levels (M2=0.4 & M2=0.8); axial bending mode;
k=0.1 ... 77 Figure 7.15: Nominal aerodynamic influence coefficients at two different velocity levels; axial bending mode; k=0.1 ... 78 Figure 7.16: Unsteady blade surface pressure at midspan, experimental data;
combined axial bending-torsion mode at R=1; k=0.1 and M2=0.4; ... 79 Figure 7.17: Differences in unsteady pressure data obtained from direct testing of combined mode and result from pure mode superposition ... 80 Figure 7.18: Numerical results of combined axial bending-torsion mode at M2=0.4 and k=0.1 ... 81 Figure 7.19: Variation of quality parameters with reduced frequency; axial bending- torsion; low subsonic velocity M2=0.4; experimental data (left) & numerical results (right); ... 82 Figure 7.20: Unsteady blade surface pressure at midspan, experimental data;
combined axial bending-torsion mode at k=0.1 and M2=0.8 ... 83 Figure 7.21: Numerical results of combined axial bending-torsion mode at M2=0.8 and k=0.1 ... 84 Figure 7.22: Variation of quality parameters with reduced frequency; axial bending- torsion; high subsonic velocity M2=0.8 ... 84 Figure 7.24: Variation of quality parameters with inflow incidence; blade 0 ... 86 Figure 7.25: Unsteady response at midspan on blade 0; combined axial bending- torsion mode; at k=0.3 and M2=0.4; amplitude ratio R=0.5 (left) and R=2 (right) . 87 Figure 7.26: Unsteady blade surface pressure at 10% span on blades ±1,
experimental data; combined axial bending-torsion mode at k=0.1 and M2=0.4 .. 87 Figure 7.27: Unsteady blade surface pressure at 90% span on blades ±1,
experimental data; combined axial bending-torsion mode at k=0.1 and M2=0.4 .. 88 Figure 7.28: Unsteady blade surface pressure at midspan, experimental data;
combined circumferential bending-torsion mode at R=1; k=0.1 and M2=0.4; ... 89 Figure 7.29: Numerical results of combined circumferential bending-torsion mode at M2=0.4 and k=0.1 ... 90 Figure 7.30: Variation of quality parameters with reduced frequency circumferential bending-torsion; low subsonic velocity M2=0.4; ... 90 Figure 7.31: Unsteady blade surface pressure at midspan, experimental data;
combined circumferential bending-torsion mode at R=1; k=0.1 and M2=0.8; ... 91 Figure 7.32: Numerical results of combined circumferential bending-torsion mode at M2=0.8 and k=0.1 ... 92
Figure 7.33: Variation of quality parameters with reduced frequency circumferential bending-torsion; high subsonic velocity M2=0.8; ... 93 Figure 7.34: Unsteady pressure data at midspan on blades ... 95 Figure 7.35: Numerical results for unsteady blade surface pressure distribution;
axial bending, k=0.3 and M2=0.4; de-stagger angle ∆γ=-2.5°, 0°, 2.5° ... 96 Figure 7.36: Variation of Cp amplitudes at specific locations on blades -1, 0 and +1; axial bending, k=0.3 and M2=0.4; ... 96 Figure 7.37: Unsteady pressure response at midspan on blades -, 0 and +1; axial bending; k=0.1 and M2=0.8; de-stagger angle ∆γ=-2.5°, 0°, 2.5° ... 97 Figure 7.38: Unsteady pressure response at midspan on blades -, 0 and +1;
circumferential bending; k=0.3 and M2=0.4; de-stagger angle ∆γ=-2.5°, 0°, 2.5° . 98 Figure 7.39: Unsteady pressure response at midspan on blades -1, 0 and +1;
torsion; k=0.3 and M2=0.4; de-stagger angle ∆γ=-2.5°, 0°, 2.5° ... 99 Figure 7.40: Mistuned influence coefficients for blades -1, 0 and +1; axial bending;
M2=0.4 & k=0.3; experimental data (black colored) and numerical results (blue colored); ... 100 Figure 7.41: Mistuned influence coefficients for blades -1, 0 and +1; circumferential bending; M2=0.4 & k=0.3; experimental data (black colored) and numerical results (blue colored); ... 101 Figure 7.42: Mistuned influence coefficients for blades -1, 0 and +1; torsion;
M2=0.4 & k=0.3; experimental data (black colored) and numerical results (blue colored); ... 102 Figure 7.43: Mistuned influence coefficients vs. reduced frequency; axial bending;
M2=0.4; experimental data ... 102 Figure 7.44: Mistuned influence coefficients vs. reduced frequency; circumferential bending; M2=0.4; experimental data ... 103 Figure 7.45: Mistuned influence coefficients vs. reduced frequency; torsion;
M2=0.4; experimental data ... 104 Figure 7.46: Variation of the influence coefficients due to de-stagger on various blades in the cascade; numerical results; axial bending at k=0.3; L1 ... 105 Figure 7.47: Effect of random mistuning on flutter stability; 20% of blades de-
staggered; axial bending at k=0.3 & M2=0.4; numerical results ... 106 Figure 7.48: Cumulative probability of least stable mode; axial bending ... 106 Figure 7.49: Effect of random mistuning on flutter stability; 20% of blades de-
staggered; axial bending at M2= 0.8 & k=0.1; numerical results ... 107 Figure 7.50: Cumulative probability of least stable mode; axial bending at k=0.1 &
M2=0.8 ... 107 Figure 7.51: Effect of random mistuning on flutter stability; 20% of blades de-
staggered; circumferential bending at k=0.3; numerical results ... 108 Figure 7.52: Cumulative probability of least stable mode; circumferential bending ... 108 Figure 7.53: Effect of random mistuning on flutter stability; 20% of blades de-
staggered; torsion at k=0.3; numerical results ... 109 Figure 7.54: Cumulative probability of least stable mode; torsion ... 109
LIST OF TABLES
Table 5.1: Key design parameters ... 41 Table 5.2: Test conditions ... 53 Table 7.1: The impact of aerodynamic asymmetries on minimum stability ... 110
NOMENCLATURE Latin Symbols
A blade oscillation amplitude, per-degree basis for 3D consideration, per millimeter (bending) and per radian (torsion), respectively, for 2D consideration (stability parameter)
c blade chord
ae
ck, aerodynamic stiffness matrix element
ae
cg, aerodynamic damping matrix element cp static pressure coefficient
ref s ref
ref s
p p p
p c p
, ,
0 ,
A
cˆp, normalized unsteady pressure coefficient,
ref dyn
p A p
c p
,
ˆ ˆ
;
cv specific heat at constant volume df infinitesimal force component dm infinitesimal moment component
ds infinitesimal arcwise surface component, per unit span e direction vector torsion mode (radial)
Fˆ complex force vector
G damping matrix
hˆ complex mode shape vector h oscillation i amplitude i imaginary unit, i 1 i internal (thermal) energy
k reduced frequency
K stiffness matrix
M mass matrix
M Mach number
m mass
m
n, blade indices
n normal vector to surface element
N number of blades
p pressure
p mean pressure
p~ time-varying perturbation pressure pˆ unsteady pressure amplitude (complex) R bending-to-torsion amplitude ratio
Q modal displacement vector
r distance from center of torsion to force realization point t time, time of flight
T oscillation period
u absolute outflow velocity
w v
u ,, Cartesian velocity components
cycle
W work per cycle; positive if the fluid is transferring work to the structure unstable situation
z y
x ,, Cartesian coordinates
X displacement vector
Greek Symbols
de-stagger angle
torsion orthogonal mode coordinate
circumferential orthogonal mode coordinate
angular coordinate in cylindrical coordinate system
eigenvalue
axial orthogonal mode coordinate
density
interblade phase angle;
phase angle
rotational frequency, 2f
pressure ratio
stability parameter
Subscripts
0 total
1 cascade inlet
2 cascade outlet
ae aerodynamic
avg average
ax axial
damping related to damping
disturbance related to disturbance
dyn dynamic
ic influence coefficient
outlet cascade outlet
phase phase of complex quantity
ref reference
TWM traveling wave mode
torsion direction
circumferential bending direction
axial bending direction
Abbreviations
arc normalized arcwise coordinate; negative branch on suction side, positive on pressure side
avg average
CFD computational fluid dynamics COT center of torsion deg degree
HCF high cycle fatigue
HPT Heat and Power Technology FEM finite element method
IBPA interblade phase angle Im imaginary part of complex number INFC influence coefficient
KTH Kungliga Tekniska Högskolan (Royal Institute of Technology)
LE leading edge
LPT low-pressure turbine
PS pressure side
PSI Pressure System Inc.
Re real part of complex number
SS suction side
TE trailing edge
TWM traveling wave mode
1 BACKGROUND Flutter in Turbomachinery
1.1
Modern aircraft engine design is characterized by a tendency towards thinner, lighter and more highly loaded blades. At the same time, developments in power generation turbines are towards greater power density and larger turbines. These trends give rise to increased sensitivity for flow induced vibrations and result in increasing challenges concerning structural integrity of the engine.
Flutter is an aeroelastic instability phenomenon that involves self-excited and self- sustained vibrations of a structure in a flow field. The vibration motion of a blade, initialized by small aerodynamic disturbances in the flow, causes an unsteady pressure field on the blade surface, which results in unsteady aerodynamic forces.
These unsteady aerodynamic forces on the blade can either dampen the vibrations or they can feed energy into the structure leading to rapid increase in the vibration amplitude during each cycle. Such a situation will finally lead to extreme blade oscillations and, in a short time period, result in material failure. As such, a correct prediction of the aeroelastic properties represents one of the paramount challenges in the turbomachinery aeromechanical design process.
The assessment of flutter involves three different disciplines: fluid mechanics, structural dynamics and solid mechanics. They are embraced by a common discipline, denoted ‘dynamic aeroelasticity’. The need for these different disciplines can be explained by considering the existence of flutter. Referring to Collar’s Triangle (Collar, 1946) in Figure 1.1, flutter arises from an interaction between the unsteady aerodynamic forces resulting from the fluid, the elastic forces of the structure and the inertial forces. During flutter, the balance between these forces is disturbed, resulting in a condition where the fluid feeds energy into the structure leading to larger vibration amplitudes and consequently inducing even larger aerodynamic forcing.
Figure 1.1: Collar’s triangle of forces
The occurrence of flutter in turbomachines is usually limited to slender and long blades that are highly loaded. Typically this situation can be found in fans, low pressure compressors (LPC) and low pressure turbines (LPT). The location of these areas in a typical jet engine is depicted in Figure 1.2.
Figure 1.2: Cut-away of a Rolls-Royce Trent 1000 engine (picture courtesy of Rolls-Royce)
The unsteady pressure fields on blade surfaces may also be induced by external sources. Such blade excitations may result in forced vibrations. The excitation frequency is typically synchronized with the rotational speed of the engine, since the most common forced vibration problems are caused by the relative motion between rotating and non-rotating parts. Resonant vibrations occur if the frequency and shape of the excitation coincides with the natural frequency of a structural mode of the rotor. Such occurrences are traditionally identified by means of a Campbell diagram (an example of which is shown in Figure 1.3) showing the variation of excitation- and natural frequencies with rotational speed.
Figure 1.3: A typical Campbell diagram
Fan LPC LPT
Potential resonant vibration problems can occur at crossings between the engine order lines and the natural frequency lines. Since the occurrence of flutter does not require external excitation, such as engine order, it can appear off the excitation lines.
From basic flutter studies in the field of aeronautics, it is known that flutter stability of isolated aircraft wings is mainly influenced by a ratio between the wing mass and the mass of the surrounding air inside a circle with the radius of half chord.
This ratio, defined as mass ratio , for wings has typically values of 1 5. Mass ratio in turbomachines reaches larger values (typically in a range from 100 to 600) and is therefore not of the same importance as for wing flutter. On the other hand, it was found that the ratio between the blade oscillation frequency, the blade chord and the flow velocity has a great influence on flutter stability in turbomachinery blade rows. This ratio is known as the reduced frequency k and is defined by the time t needed for a fluid particle to travel across the blade chord c divided by the oscillation period ܶ.
u fc T
k t 2
Eq. 1-1
Flutter occurs below a certain critical reduced frequency; for turbomachinery blades typical values li in the range between 0.1 and 1. Previous studies of fundamental bending and torsion mode shapes conducted by Panovsky and Kielb (2000) indicated that each of these modes becomes unstable for reduced frequency in the range of 0.2 and 0.3. Vogt (2005) has shown that an increase in the reduced frequency tends to lead to an increase in response magnitude and that it has a stabilizing effect on the travelling wave mode stability.
From Eq. 1-1, one can notice that for a constant oscillation frequency the value of the critical reduced frequency is approached as flow velocity increases. For small values of reduced frequency the flow has a quasi-steady character and for large values unsteady effects become dominant (Srinivasan, 1997). Controlling the reduced frequency has therefore become an important issue in the aeroelastic design process.
The flutter behavior in turbomachines is predominantly affected by the interaction of the whole blade assembly instead of only a single blade. Oscillations of each individual blade in the blade row will induce an aerodynamic response both on the blade itself and on its adjacent blades. An isolated blade might show no tendency to flutter, but when included in a blade row, this assembly might become aeroelastically unstable. The interaction between blades is denoted as aerodynamic coupling and is an important factor when the aerodynamic response of the blades in an oscillating blade row is assessed. Aerodynamic coupling between the blades in a blade row is schematically illustrated in Figure 1.4.
Figure 1.4: Aerodynamic coupling in a turbomachine blade row (Vogt, 2005) The coupling is such that all blades have an influence on each other, but the coupling influence is strongest between adjacent blades. Flutter studies on turbine blade rows have indicated that the reference blade and its immediate neighbours are dominant contributors to the unsteady behavior of cascades (Panovsky and Kielb, 2000, Vogt, 2005), not taking into account acoustic resonance effects.
Acoustic resonance effects are highly significant if there is an empty duct adjacent to the blade row (e.g. fan blades adjacent to the engine intake duct). The oscillation of the blades causes a fluctuating pressure field that propagates up- and downstream of the blade row and can induce acoustic resonance in the adjacent ducts. Acoustic resonance flutter is almost exclusively of relevance in empty ducts adjacent to the considered blade row, as adjacent blade rows tend to suppress resonant behavior (Whitehead, 1973). Wu et al. (2003) highlighted the relevance of aeroacoustic flutter for a high bypass-ratio jet engine where the aeroacoustic properties of the inlet duct triggered flutter of the fan, observed as sharp and local drop in the flutter stability margin referred to as “flutter bite”.
While investigating mechanisms for wide-chord fan blade flutter, Vahdati et al.
(2011) were able to identify two types of flutter: stall flutter (driven by flow separation) and acoustic flutter (driven by intake acoustics). The latter occurs due to wave reflections and was found to be independent of flow separation. It was also shown that acoustic flutter can significantly intensify stall flutter if they occur at the same speed (“flutter bite”). The use of blade frequency (stiffening of the blade) for flutter stability control does not seem to be effective method for suppression of acoustic flutter.
Assuming that all blades on a turbomachinery component (such as rotor) vibrate in the same mode, at the same frequency and at the same amplitude, a so-called travelling wave mode is obtained (Crawley, 1988). From a flutter stability point of view this assumption represents the least stable condition and therefore tends to be over-conservative. A parameter that characterizes the phase lag between two adjacent blades is referred to as interblade phase angle. It translates into the nodal diameter as follows:
Forward travelling wave mode
N
ND
2 Eq. 1-2
Backward travelling wave mode
N ND N
2 , Eq. 1-3
where is the interblade phase angle, N is the number of blades and ND is the nodal diameter which gives the order of the travelling wave. Each nodal diameter pattern induces a pair of travelling waves (forward and backward travelling wave).
The resulting mode is referred to as travelling wave since it resembles a wave around the circumference of the blade row. An example of nodal diameter patterns and corresponding cascade geometries is shown in Figure 1.5.
Nodal diameter pattern Blade row Nodal diameter pattern Blade row
Nodal diameter ND=1 Nodal diameter ND=3
Figure 1.5: Nodal diameter patterns and corresponding instantaneous cascade geometries (adapted from Vogt, 2005)
Mode Shapes Sensitivity 1.2
Blades in a bladed disk are structurally coupled through the disk and can even be connected to each other through shrouds i.e. shrouded bladed disks. This coupling will greatly affect the vibration properties of the assembly. Srinisivan (1997) pointed out that bladed disk assemblies will experience system modes influenced by the characteristics of individual blade vibration, the support structure (including part-span and tip shrouds), the operating conditions, damping and the extent of the mistuning among blades.
When considering unshrouded bladed disks, one can identify two possible types of modes. The first includes blade dominated modes and corresponds to the situation where long, flexible blades are mounted on a comparatively stiff disk. To assess blade dominated modes one can concentrate on analyzing single blade modes.
−100 0 100
−300
−200
−100 0 100 200 300 400
−100 0 100
−300
−200
−100 0 100 200 300 400
+ - +
+ - +
-
-
The second type includes disk dominated modes and occurs when small and stiff blades are mounted on a comparatively flexible disk.
In the classical approach to flutter stability prediction, the aerodynamic and structural parts are treated separately (Marshall & Imregun, 1996). Since the critical mode shape of the setup is not known a priori, one needs to predict aerodynamic properties for a large number of possible mode shapes. If linearity and non-deforming blade sections are assumed one could span a mode space by investigating three orthogonal modes since linear superposition of modes can be employed when investigating coupled modes.
The influence of the mode shape on flutter stability has been investigated by many researchers. The conclusion from studies conducted by Panovsky et al. 1997 and Vogt and Fransson (2004), on a low pressure turbine blade in rigid body oscillations showed that the blade mode shape could have a significant impact on aeroelastic stability. The study has also shown that certain mode shapes are more sensitive to reduced frequency changes than others. The recommendation was to include mode shape sensitivity studies in aeroelastic design phase, which previously was mainly concentrated on studying the reduced frequency of the blade.
Panovsky and Kieb (2000) developed a graphical method to assess flutter stability on a two dimensional section of a cascade. The method is based on representing any possible pure rigid body mode as torsion mode around a respective centre of torsion. Modes of bending character (i.e. translational movement of blade) are represented by a center of torsion far away from the reference blade whereas modes of torsion character feature a center of torsion close to the reference blade.
An example of a mode shape sensitivity plot (referred to as a stability map or tie- dye plot) is shown in Figure 1.6, where stable and unstable regions related to the torsion axis location are presented.
Figure 1.6: Mode shape sensitivity plot at low subsonic Mach number; shaded areas mark stable regions; values display aerodynamic damping parameters
(Vogt and Fransson, 2007)
Figure 1.6 indicates that mode shape sensitivity is large for torsion type modes as the stability gradients are comparatively large in proximity to the reference blade.
Studies conducted by Chernysheva et al. (2001), Kielb et al. (2003) and Vogt (2005) indicate that stability maps have similar appearance for different types of turbine blade geometries.
Srinivasan (1997) explains that vibratory modes of realistic blade profiles are almost always coupled i.e. the contents of a given mode of vibration include displacements perpendicular and parallel to the chord as well as torsion of the profile.
Current design trends for compressor blades that are resulting in low aspect ratio blades potentially reduce the frequency spacing between certain modes (e.g. 2F &
1T) leading to mode coalescence (Mårtensson et al., 2008). However, coalescence flutter is not a high risk item in the design if a reasonable frequency separation can be established. For the investigated highly loaded transonic fan design a 2% frequency separation allows avoiding aeroelastic mode coupling effects.
Another occurrence of combined modes is given by the vibration of bladed disks when having a comparatively soft disk and rigid blades. In such a case, bending and torsion modes are coupled by vibration kinematics and feature a phase lag between the modes of 90°.
Coupled bending-torsion modes in fans and compressor rotors were investigated analytically by Bendiksen and Friedman (1980) using a 2D section model .The outcome was that flutter boundaries were considerably affected by introducing coupling between the bending and torsional degrees of freedom. At the same time, there was no appreciable tendency for the bending and torsional frequencies to coalesce as flutter approaches, except at very low reduced frequencies. Coupling between bending and torsion was also found to reduce sensitivity of Single Degree of Freedom (SDOF) torsional flutter boundaries to the elastic axis location.
Generally the combination of bending and torsion modes is achieved under a certain phase lag between involved modes. The effect of the phase angle between modes has been addressed by Försching (1991) from a quasi-steady viewpoint.
He identified that a phase lag of 90° represents the most unfavorable situation from a stability point of view, as the work produced by the fluid on the structure during the cycle will have positive sign.
Försching (1994) pointed out that single mode kinematic coupling in bending and torsion is one of the important structural dynamic features of beam-like turbomachinery blades, which significantly influences their flutter stability. Since the mass ratio of turbomachinery blades is generally much higher than that of an aircraft wing, flutter frequencies and the corresponding mode shapes can be considered to be unaffected by the aerodynamic loading. As such, aeroelastic bending/torsion coalescence flutter coupling between two modes, typical for classical aircraft wing flutter, is less important in turbomachinery blades. He also mentioned that the frequency ratio for the first bending/first torsion mode is typically in the range of 0.3-0.7 and that aerodynamic coupling between these two modes, causing a classic coalescence flutter, is unlikely to occur. Mode coupling
can however occur between 2nd bending and 1st torsion modes since the frequency ratio often falls in the range 0.8-1.2.
Hoyniak and Fleeter (1986) showed that, for the investigated unstable supersonic twelve-bladed rotor cascade, bending-torsion coupling effects were found to be significant for a bending-torsion natural frequency ratio between 0.6 and 1.2.
Outside this region results are approximately the same as found for the single degree of freedom analysis i.e. similar to results from torsion mode analysis as presented in Figure 1.7. This indicates that for the investigated cascade configurations the bending mode is not coupling with the torsion mode at higher natural frequency ratios.
Figure 1.7: The effect of aerodynamic detuning on the critical reduced frequency with the elastic axis and center of gravity located at 40% of the chord (Hoyniak and
Fleeter, 1986)
Effects of Mistuning 1.3
The aeroelastic stability behavior of turbomachinery blades is sensitive to small variations in natural frequencies and mode shapes between individual blades in a blade row. Due to manufacturing uncertainty and tolerances, turbomachinery blades in bladed-disks (also in blisks) will always differ from one another. Most often when talking about mistuning one is referring to structural mistuning which is defined as blade to blade differences in blade mass, stiffness and damping properties. Mistuning will lead to changes in natural frequencies and mode shapes and can have a major impact on the aeroelastic properties of cascades.
Structural mistuning typically has an adverse effect on forced response but a beneficial effect on flutter stability. For forced response, mistuning will cause frequency split and mode localization, which in its turn results in having multiple
peaks in the resonant response curve and a magnification of the response amplitude. Mistuning sensitivity was shown to be strongly depended on structural coupling in bladed disks (Bladh et. al., 1999).
In the case of flutter, mistuning breaks the symmetry of a perfect travelling mode and by this tends to suppress flutter. The typical influence of structural mistuning on flutter stability can be observed in root locus plot depicted in Figure 1.8 (Kaza and Kielb, 1983).
Figure 1.8: Effect of mistuning on flutter (Kaza and Kielb, 1983)
The complex eigenvalues here present solution to the eigenvalue problem derived from the aeroelastic differential equation of motion. The real part of the eigenvalue in this case represents damping and determines stability. The imaginary part specifies the ratio of the flutter frequency to the reference frequency. One can observe the tendency of moving the least stable modes to the stable region when alternating mistuning is introduced. It can be also noticed that higher level of alternating mistuning cause locus split i.e. the modes separate into high and low frequency groups. The degree of separation increases with the level of mistuning.
In analysis of effects of mistuning on flutter stability both structural and aerodynamic coupling mechanisms can play a significant role. Several high fidelity mistuning models have been developed but the emphasis has been put on structural coupling. A few recent publications described high fidelity mistuning models with both structural and aerodynamic coupling (Senturier et al., 2000; Kielb et al., 2004; Kielb, R. et al., 2005; Sladojevic et al., 2006).
By comparing the stabilizing effects of mistuning predicted using only aerodynamic coupling and the results from models using aerodynamic and FMM structural coupling, Kielb et al. (2004) showed that the stabilizing effect of mistuning is over predicted in cases where only aerodynamic coupling is included. It is recognized that the beneficial effect of structural mistuning on flutter stability is inhibited by the addition of structural coupling effects. This behavior is caused by the increased spread in tuned frequency due to structural coupling, making the system less sensitive to mistuning.
Previous studies have shown that different mistuning patterns can have very different effect on flutter stability. Crawley and Hall (1985) showed that alternating blade frequencies is a robust optimum arrangement. Finding optimal mistuning patterns for increasing flutter stability was the target of several different studies.
Kaza and Kielb (1983) concluded that alternate blade mistuning may be nearly optimal in increasing flutter stability of shroudless fans. A more recent probabilistic flutter study of a mistuned bladed disk, conducted by Kielb et al. (2006), presented a method for identifying beneficial structural mistuning patterns.
Försching (1994) indicated that when concerning practical feasibility of optimal blade mistuning as an inverse design procedure, the difficulty is not only in deciding the level of mistuning required, but also in the complexity of the mistuning pattern and the number of different blades which must be manufactured and kept in inventory.
1.3.1 Aerodynamic Mistuning
Although considerable effort has been put into research on structural mistuning effects, there is still a need to address another type of mistuning, known as aerodynamic mistuning or aerodynamic asymmetries. Aerodynamic mistuning refers to aerodynamic non-uniformities due to geometric asymmetries in blade cascades. Although no structural properties of blades are changed, both steady and unsteady loads on blades are affected. Asymmetries may appear as a result of the engine manufacturing process within the frames of manufacturing tolerance or assembly inaccuracies, as well as in-service wear or due to damages.
An early study on the effects of aerodynamic mistuning carried out by Hoyniak and Fleeter (1986), looked into how passage-to-passage differences (particularly differences in circumferential blade spacing) in the flow field could affect flutter stability. It was shown that the effect of aerodynamic mistuning is dependent on the location of the elastic axis and the center of gravity. Alternating blade spacing was found to have stabilizing effects on coupled bending-torsion unstalled supersonic flutter. One of the results from this study is presented in Figure 1.7 showing the effect of the aerodynamic detuning on the critical reduced frequency.
Sladojevic et al. (2006) investigated the influence of stagger angle variation on aerodynamic damping and frequency shift for a model without structural coupling.
The studied blade represented a layer extracted from the fan blade used in civil aero engines and modal properties were artificially created. The investigated mode shape was a mixture of bending and torsion designed with aim to cause instability in lower modes. Modal frequencies were insensitive to stagger angle changes.
Stagger angle variations up to 0.5 degrees did not affect the stability of the system. Large alternating stagger angle variations up to 2 degrees were found to have a destabilizing effect. The opposite behavior was found for random patterns i.e. random variations of stagger angle have in this case slightly increased the overall damping of the system.
Kielb et al. (2007) addressed the phenomenon of aerodynamic asymmetries on a probabilistic basis. By perturbing the aerodynamic coupling coefficient between individual blades in the blade rows both intentional (i.e. symmetry groups) and
random asymmetries could be studied. It was found that random aerodynamic perturbation could have a destabilizing effect while single blade and alternating perturbations tend to suppress flutter.
Stüer et al. (2008) investigated numerically the impact of aerodynamic mistuning (in this case alternating chord length) on the aeroelastic stability of highly staggered low pressure turbine blades with supersonic exit flow. Due to structural coupling in the investigated configuration, there was no net gain in stability when mistuning was introduced.
Ekici et al. (2008) investigated the effect of aerodynamic asymmetries caused by alternate blade-to-blade spacing and alternate staggering on the aeroelastic stability of a linear cascade. It was found that alternating spacing improved the stability of the system, which is in line with results presented by Fleeter and Hoyniak (1986). Alternating staggering, on the other hand, was shown to have either stabilizing or destabilizing effect depending on the direction of miss- staggering.
Vogt et al. (2009) studied the influence of aerodynamic asymmetries on mode shape sensitivity of an oscillating LPT cascade. The asymmetric aerodynamic perturbation was applied in a generic manner, using perturbation data acquired at negative incidence off-design operation of the setup. It was identified that mode regions that showed greater dependence from asymmetries were torsion-bending types of modes with torsion centre away from the blade pressure side.
The scope of previous studies on effects of aerodynamic mistuning has so far been limited to the above described cases. There is a strong need for investigation of additional cases before general conclusions can be made concerning the effect of aerodynamic mistuning on flutter stability in turbomachines. Keeping this in mind, the present study aims to determine the level of change in aeroelastic properties of an oscillating LP turbine blade row when changing the blade-to-blade stagger angle. For the first time ever, as far as the author is aware, the perturbed aerodynamic influence coefficients are obtained directly employing both experimental testing and numerical simulations.
2 FUNDAMENTAL CONCEPTS Classical Approach in Flutter Stability Predictions 2.1
The classical approach in aeroelastic stability predictions assumes a decoupling of the structure and flow domains (Marshall & Imregun, 1996). Due to the high mass ratios which are characteristic of turbomachines, the structural terms are comparatively large in comparison to the aerodynamic damping term. Structural and aerodynamic parts can be treated as decoupled i.e. on the structural side, free vibration analysis can be used for determination of the mode shapes without introducing aerodynamic loads, while unsteady aerodynamic forces are obtained from pure unsteady aerodynamic analysis with the blade oscillating in the eigen mode shapes obtained from the structural calculation. A similar approach will be employed in the present work, with the exception that the mode shapes of the oscillations are artificial 3D rigid body mode shapes. Flutter stability analysis in the context of the present work is carried out with a goal of determining whether the unsteady flow field has a destabilizing or stabilizing character, rather than the stability analysis for entire flow-structure system.
Governing Equations 2.2
The aeroelastic system presents a system of differential equations describing the interaction between a structure and a fluid. Basic aeroelastic equations of motion state a balance between structural and aerodynamic forces as
M
X
G
X
K
X Fae(t)
, Eq. 2-1 where
M , G, K are modal mass, modal damping and modal stiffness matrices,
X is modal coordinate vector and ܨሺݐሻ is unsteady modal force vector consisting of) ( )
( )
(t F t F t
Fae damping disturbance , Eq. 2-2
where Fdamping(t) stands for the aerodynamic damping forces due to the motion of the blade in the surrounding fluid and Fdisturbance(t) contains forces due to the upstream and downstream disturbances. In flutter applications the only forces of interest are ones induced by the motion of the blade, thusFdisturbance(t)0.
The usual approach to solve the aeroelastic equations is to introduce modal coordinates as following
X(t)
Qeit Eq. 2-3where is the mode shape, Q modal displacement and is frequency. The aeroelastic equation can now be expressed in frequency domain as
2 Mm i Gm Gae Km Kae
Q 0 Eq. 2-4where
G
ae denotes the aerodynamic damping matrix and K
ae the aerodynamic stiffness matrix respectivelyThe aeroelastic equation can now be recognized as a complex eigenvalue problem where eigenvalues are describing the stability of the system.
Unsteady Aerodynamic Forces 2.3
In the present investigation it has been assumed that the blades are oscillating in three orthogonal mode shapes: axial bending, circumferential bending and torsion.
The direction of the bending modes and torsion axis is defined in Figure 2.1.
Figure 2.1: Investigated orthogonal mode shapes
The harmonic motion of the blade can be expressed by complex vector
hˆ hˆ,hˆ,hˆ
describing a 2D representation of an arbitrary mode shape.Verdon (1987) showed that for small perturbations the unsteady pressure on a blade surface, resulting from an applied harmonic blade motion, can be represented as a harmonic oscillation around a steady mean pressure as
)
ˆ (
) , ( ) , ,
~( ) , ( ) , ,
(x y t p x y p x y t p x y p ei t p h
p Eq. 2-5
where p( yx, ) is the steady mean pressure, ~p(x,y,t) the respective time-varying perturbation part and pˆis the complex amplitude of pressure perturbation due to harmonic oscillation.
A complex unsteady pressure coefficient was introduced by Vogt (2005) and is defined as complex amplitude of unsteady pressure normalized by the reference dynamic head pref,dyn and the oscillation amplitudeA:
ref dyn
p A p
c p
,
ˆ ˆ
Eq. 2-6
The reference dynamic head is taken as the difference between the total and static pressures upstream of the cascade.
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
10%
50%
90%
COT
Analogue to that, the unsteady normalized force can be expressed as
ref
pdyn
A f F
,
ˆ ˆ
Eq. 2.7
The unsteady normalized force can be obtained by integration of the infinitesimal local unsteady force components around the blade profile. Considering an orthogonal system of three modes the infinitesimal normalized force components per surface element ds are defined as
ds n c f
dˆ ˆp Eq. 2-8
ds n c f
dˆ ˆp Eq. 2-9
ds e c r m
dˆ (ˆp) Eq. 2-10
and from these the normalized unsteady force is obtained as
df ds fˆ ˆ
Eq. 2-11
For a system of three orthogonal modes assessment of mode shape sensitivity would include integration of the normal forces for all three orthogonal modes and the resulting matrix contains the force component in all three directions, as follows:
f f f
f f f
f f f
Fˆ Eq. 2-12
The first index refers to the mode shape causing the force, and the second to the direction in which the force is acting. As already mentioned in the previous section, any arbitrary rigid body mode can be represented using torsion mode representation i.e. by defining the location of a center of torsion close to the reference blade if the oscillation mode has a torsion character or by locating the center of torsion infinitely far away from the blade if motion is of translational character (bending modes).
The unsteady aerodynamic force for an arbitrary rigid-body mode is then obtained as
i Ti
i h F h
Fˆ ˆ ˆ ˆ Eq. 2-13
