An Approach on Tuning Frequency of a Rotating Blade
Loc Quang Duong, PhDUniversity of Connecticut, 2013
Blade failures, either in turbines or impellers, are always undesirable phenomena, as they tend to be catastrophic events. In multi-stage engine design, it is difficult – or impossible – to design a rotating component free of resonance. But it is imperative to tune the interfered frequencies outside of the operating speed range of the system, in order to avoid high cycle fatigue. This dissertation develops a methodology for accomplishing this design task. It is based on structural perturbations in the form of material redistribution, which impacts both the mass and stiffness of the component. This methodology is applied to two fundamentally different systems: turbine blades and impeller blades.
Loc Quang Duong – University of Connecticut, 2013
i
An Approach on Tuning Frequency of a Rotating
Blade
Loc Quang Duong
B.S., University of Western Ontario, 1984
M.S., University of Connecticut, 2004
A Dissertation
Submitted in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
at the
University of Connecticut
APPROVAL PAGE
Doctor of Philosophy Dissertation
An Approach on Tuning Frequency of a Rotating Blade
Presented by
Loc Quang Duong, B.S., M.S.
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ACKNOWLEDGMENTS
First, I would like to express my deep gratitude towards Professor K. Kazerounian, who served as my major advisor and provided me with tireless assistance, guidance, advice, wisdom and encouragement not only through this work but since the day I met Professor Kazem in my first class in Kinematics twelve years ago at the Pratt & Whitney site. I also would like to offer my gratitude to my co-major advisor, Professor K. Murphy, who gives me immense direction, motivation and particularly inspiration in the world of Vibrations through his course ME362 in 2000. Additionally, I would like to extend my appreciation to Professors E. Jordan and H. Ilies for their advice and participation as members of my advisory committee.
I am grateful to my employer UTC, Pratt & Whitney and Hamilton Sundstrand for generously supporting me in my graduate study pursuit.
CONTENTS
ACKNOWLEDGMENTS ………. iii
List of Figures………. viii
List of Symbols………..… xi
CHAPTER 1
Introduction ………... 1 1.1 Background ……….. 1 1.2 Overview ………. 2 1.2.1 Blade Resonance ……… 31.2.2 Impeller Blade-Disc Resonance ……….. 5
1.2.3 Squeeze Film Damper Performance Deterioration ……….. 7
1.2.4 Asymmetric Bearing Support Stiffness ………... 8
1.2.5 Rotor Thermal Bow ……… 9
1.2.6 Rotor Imbalance Induced by Blade Walking ………. 10
1.2.7 Rotor Rubbing ……… 11
v
CHAPTER 3
Dynamic Characteristics of Structures ………..…… 17
3.1 Spring–mass System ………. 17
3.2 Beam ………. 19
CHAPTER 4
Aero-mechanical Equation 4.1 Free Vibration ………... 224.1.1 Eigenvalue and Eigenvector ... 22
4.1.2 Physical Properties- Participation Factor ………... 23
4.2 Force Vibration ……… 26
4.3 Case Study ……… 31
CHAPTER 5
A. Eigenvalue Perturbation 5.1 The Perturbation of a Simple Eigenvalue ……….. 345.2 Mechanical System Eigenvalue Perturbation ……… 35
5.2.1 Evaluation of λoi ……….. 37
5.2.2 Evaluation of oi ……… 38
5.3 Node, Anti-node and Maximum Frequency Shift ………. 41
B. Guided Tuning Turbine Blade Method (GTTB) ……….…... 43
CHAPTER 6
Simulation and Validation
6.1 Finite element Analysis ……… 45
6.2 Laser Vibrometry Test ……….……. 50
6.2.1 Test Setup ……….... 50
6.2.2 Test Measurements ………..……. 53
6.3 Kt Contour Optimization ………...… 55
Paper 1
Guided Tuning of Turbine Blade (GTTB): A Practical Method To Avoid Operating At Resonance, JVA VIB-12-1129 ……….. 57Paper 2
An Approach On Tuning The Frequency Of A Rotating Blade, DETC2008-49442 ………... 68CHAPTER 7
Bi-Frequencies Tuning
7.1 Spring-mass System ……… 79 7.2 Stiffness Decoupling ……… 81 7.2.1 Nodal Diameter ………. 817.2.2 Finite Element Simulation ………. 82
7.3 Laser Vibrometry Test ……….. 90
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7.3.2 Test Measurements ……… 91 7.4 Eigenvalue Loci Curve Veering ………... 92
Paper 3
An Approach on Bladed-Disc Tuning, DETC2009-97325 ………….……… 98
CHAPTER 8
Closure
... 112List of Figures
Figure 1.1a. FEA Turbine Blade Complex Bending Mode ………..…. 3
Figure 1.1b. FEA Turbine Blade Campbell Diagram …….……….…. 4
Figure 1.1c. Turbine Blade HCF Failure ……….…. 4
Figure 1.2a. FEA Impeller 3 Nodal Diameter Mode Shape ………...…….….. 6
Figure 1.2b. Impeller 3Nodal Diameter Failure (back-face) ………...….…. 6
Figure 1.3. Squeeze Film Damper Cavitation ……… 7
Figure 1.4. Six Beams Squirrel Cage ………. 8
Figure 1.5. Radial Stiffness Variation ……… 8
Figure 1.6. Engine Vibration (1E) at Critical Speed at Different Restart Times …… 9
Figure 1.7. Interface Friction Schematic ………... 10
Figure 1.8. Blade Walking Phenomenon ………... 10
Figure 1.9. Housing Support of Variant Thickness ……… 11
Figure 1.10. Rubbing Model ……….. 12
Figure 3.1. 1D Spring-mass System ………... 17
Figure 3.2. Beam Simply Supported at Two Ends ………. 20
Figure 3.3. Beam Mode Shapes ………. 20
Figure 4.1. 1D Spring-mass System ……….. 24
Figure 4.2. Mach Number Contour ………. 26
Figure 4.3. Pressure Field ……….. 27
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Figure 4.5. Physical Representation of the Harmonic Motion and Unsteady Pressure 30
Figure 4.6. Blade and Velocity Diagram ………. 32
Figure 4.7. Campbell Diagram ……… 33
Figure 4.8. FEA Mode Shape ………. 33
Figure 4.9. Laser Vibrometry Test .………. 33
Figure 4.10. Non-intrusive Test Measurements ………... 33
Figure 5.1. 2ndBending Mode ……….….. 42
Figure 6.1. Blade Model with Sub-volumes at Primary Anti-node ………… ………. 47
Figure 6.2. FEA Blade Model ………....… 48
Figure 6.3. Mode Shape Corresponding to fn ……….. 48
Figure 6.4. Campbell Diagram ………... 49
Figure 6.5. Broach Block Fixture ………... 50
Figure 6.6. Broach Slot ………... 50
Figure 6.7. Broach Block Fixture and Laser Vibrometer ………. 51
Figure 6.8. Grid of Scan Points ………... 51
Figure 6.9. Average Frequency Response Spectrum ………... 52
Figure 6.10. Shaker Setup ……….. 52
Figure 6.11. Shaker Test Results ……….……….……… 53
Figure 6.12. Laser Vibrometry – Mode Shape of Baseline ………. 53
Figure 6.13. Laser Vibrometry – Mode Shape of Tuned Blade ……….. 54
Figure 6.14. FEA Contour Optimization ……...……….. 55
Figure 6.15. Tuned Blade ………...………...… 55
Figure 6.17. Blade Mid-Radial Span Pressure Distribution ………...… 56
Figure 7.1. Parallel and In-series Spring-mass System ………... 79
Figure 7.2. Impeller Cyclic Symmetric Sector ……… 83
Figure 7.3. 3D Finite Element Fully Expanded Model …...………..…... 83
Figure 7.4. Un-tuned Nodal Diameter Map ... 84
Figure 7.5. Finite Element 3Nodal Diameter Disc Mode Shape corresponding to λn … 85 Figure 7.6. Finite Element 3Nodal Diameter Disc Mode Shape corresponding to λn+1 .. 85
Figure 7.7. Inter-blade Coupling Effect ……….. 85
Figure 7.8. Inter-blade De-coupling Effect ……...………..……. 88
Figure 7.9. Tuned Nodal Diameter Map ………..…………...………..…... 89
Figure 7.10. Impeller Grid of Scan Points …..………..……….………….. 90
Figure 7.11. Laser Vibrometry-3ND Mode Shape, λn………. 91
Figure 7.12. Laser Vibrometry-3ND Mode Shape, λn+1 ……….. 91
Figure 7.13. Spring-Mass System ………... 93
Figure 7.14. Two Spring-Mass System Curve Veering of Eigenvalue Loci …...……. 94
Figure 7.15. Spring-Mass Cyclic System ……….……….… 95
Figure 7.16. Spring-Mass Cyclic System Curve Veering of Eigenvalue Loci ……… 95
Figure 7.18. Un-tuned Impeller Blade ….…….……….……… 97
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List of Symbols
Roman symbols and abbreviations A system matrix
B strain shape functions, number of blades C damping matrix
D elastic matrix
e radial offset, exponential E Young modulus, engine order EO excitation order Ff frictional force FN radial force FT tangential force G skew-symmetric matrix h radial clearance
i 1, imaginary unit of complex number im imaginary
k, K element stiffness, stiffness matrix Kr radial stiffness
L length m, M mass element, mass matrix
[N] shape function ND nodal diameter p parameter P pressure, force
{P} column vector of coefficient factor q generalized coordinate
qo uniform distributed load
r shaft geometric center radial displacement Re real
t thickness
T period of vibration, temperature
V velocity v, V volume v left eigenvector x, X displacement Y displacement u displacement
CFD computational fluid dynamics FEA finite element analysis
xiii
Greek symbols
ρ density ε coefficient θ angle
δ delta, variation of, Kronecker delta ∆ Increment of
∫ integral of λ eigenvalue
Ф eigenvector
CHAPTER 1.
Introduction
1.1 Background
In a typical gas turbine engine operation, ambient air entering the engine through an intake is pressurized by a compressor for delivery to the combustor. Combustion of the aviation fuel with compressed air in the combustor yields high temperature, high pressure gas. This gas enters the turbine section where fluid energy is extracted and converted into mechanical work through direct interaction with the blades. The compressor and the turbine modules are connected by means of mechanical joints to form the engine rotor supported by a bearing system. With the quest for higher performance, lighter rotor weight and longer bearing support span, the gas turbine rotor in aircraft application is becoming more and more flexible, usually operating as a supercritical rotor whose dynamic characteristics require a low level of mass imbalance. Vibration of the rotor is undesirable since it could potentially lead to accelerated engine deterioration and eventually premature engine failure. Besides mass imbalance, other causes of synchronous shaft whirl and vibration to the rotor vibration are:
2 Rotor thermal bow imbalance Turbine blade walking imbalance Rotor rub
The first two items forms the core of this work which involves the study of tuning method to shift the interfered frequencies out of the operating speed range to avoid blade high cycle fatigue failure that could eventually induce the following:
a) Excessive rotor imbalance gives rise to strong engine vibration that could damage the mounting system connecting the engine to the airframe resulting in engine separation.
b) In the case of a compressor airfoil, the released blade could flow through the diffuser to the combustor and damage its liners resulting in engine compartment fire, posing a serious flight safety issue.
c) The failed airfoil could potentially knock off other adjacent airfoils, resulting in a sudden rotor seizure that may shear the rotor shaft and subsequently engine over-speed.
d) The failed blade could damage the rotor assembly leading to disc burst.
Therefore, not only from the cost of replacement but also from the aspect of structural integrity and safety, this demonstrates that it is extremely imperative to tune the excited vibration modes out of the operating speed range.
1.2 Overview
1.2.1 Blade Resonance
Rotor high vibrations often results from rotating component operating at or near resonance. In the flow field, the forced responses of the rotor blades arise from the non-steady pressure distribution in the tangential direction as the gas flow past obstacles such as stator vanes present upstream of the gas path. As a result of the aerodynamic excitation, frequencies occur at integer multiples of the vane passing frequency. Blade resonance results if its natural frequencies coincide with the engine excitation order and thus are said to be synchronous with the rotor rotational speed. Figure 1.1a illustrates the complex bending mode of a turbine blade, having a maximum modal deflection located about the mid span of the blade trailing edge. If this blade interferes with an excitation source in the operating speed range as represented by point A in Figure 1.1b, blade Campbell diagram, the blade would be in a state of resonance and would eventually failed by high cycle fatigue as shown in Figure 1.1c.
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Figure 1.1b. Turbine Blade Campbell Diagram
Figure 1.1c. Turbine Blade HCF Failure
1.2.2 Impeller Bladed-Disc Resonance
An impeller is a radial compressor normally consisting of two sets of blades of different chord lengths. In the quest for high power density to weight ratio, the impeller is required to be operated at higher compression ratio and higher rotational speed but with minimum weight, which translates into thinner blade and disc thicknesses. Eventually, this yields little margin to dynamic stresses. Consequentially, impeller operating under resonant conditions usually results in catastrophic failure. In most cases, the excitation sources arise from non-uniform pressure distributions caused by fluid flow around obstacles. A simple example is the flow through the impeller diffuser vanes.
When a diffuser with N vanes is present at the impeller exducer end in a gas flow path, the impeller blades would experience N wakes during one revolution of the rotor of engine order one, symbolized as E. Thus, NE represents the source of impeller excitation. As consequence of disc-blade coupling, the bladed disc would operate at resonance if its nodal diameter mode shape coincides with the frequencies of the unsteady pressure field at the operating speed and provided that the following equation is satisfied:
NE = α*n ± ND, (1.1)
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Figure 1.2a. FEA Impeller 3Nodal Diameter Modal Shape
Figure 1.2b. Impeller 3 Nodal Diameter Failure (back-face)
1.2.3 Squeeze Film Damper Performance
Basically, the squeeze film damper (SFD) is a plain journal bearing with zero spin whose design is required to be sufficiently robust to limit deflection amplitudes not only during transient conditions but also during steady state operational conditions. A number of parameters are involved in the design of a SFD. The main ones are: SFD geometry (radius, length, clearance), SFD assembly (such as end seal type, oil feed and discharge scheme, shaft alignment), and the SFD operational conditions (such as oil pressure, oil viscosity). The performance of the SFD is of paramount importance in providing adequate damping to control the rotor vibration level and hence the overall engine structural integrity. Commonly, the performance of a SFD is affected by cavitation. In most aircraft application, piston rings are incorporated in this SFD design to reduce leakage. As a consequence of wear, rotor misalignment, and the local opening of the seal ring itself, oil leakage would occur along with the entrainment of air into the damper. With non-uniform and low oil distribution, the SFD is prone to cavitation. As the air bubble collapses, local increase in pressure causes pitting on SFD land as shown in Figure 1.3, leading to deterioration of the damping capacity. Cavitation, a non-linear
behavior,is difficult to quantify analytically since classical theory based upon Reynold’s
equation failed to predict the pressure field due to 2 phase flow nature resulted from air entrapment.
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1.2.4 Asymmetric Bearing Support Stiffness
Squirrel cage is used to center the rotor. It is a structurally flexible device to provide a required stiffness to the bearing support in order to avoid critical speed of the rotor in the operating speed range. The radial stiffness of the squirrel cage is determined by the beam dimensions (length, cross-section area) and the number of beams. The higher the number of beams, the more uniform radial stiffness distribution around the circumference is achieved.
Figure 1.4. Six beams Squirrel Cage
Figure 1.5. Radial Stiffness Variation (Six Beams Squirrel Cage)
0.95 1 1.05 1.1
0 60 120 180 240 300 360
Angular Position (deg)
1.2.5 Rotor Thermal Bow
In aviation application, the auxiliary power unit (APU), a gas turbine engine, is used to generate electricity and compressed air. The APU is usually installed at the tail cone of the aircraft and the engine air inlet is connected to a ducting system having opening on the aircraft fuselage. In some instances, due to the asymmetry of the air inlet duct system and the constraints imposed upon the engine compartment, the heat soak-back after the engine shut-down creates a non-uniform thermal distribution on the rotor. As a consequence of deflection due to shaft thermal bow, the center of mass is moved away from the center of rotation. In conjunction with rotor imbalance, rotor thermal bow leads to higher imbalance effect which in turns results in excessive radial excursions causing rubbing at close clearance stations during hot start up not only as the rotor crosses the critical speeds but also at operating speed until its thermal equilibrium is re-established. Figure 1.6 illustrates the vibration level of an engine with asymmetric inlet air system as the rotor crosses the critical speed at different restart times.
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1.2.6 Rotor Imbalance Induced By Blade Walking
The mechanism of blade walking is discussed by Duong and al. in [5]. It is a phenomena associated with the axial turbine blade resulting from its relative displacement within the rotor assembly either upstream or downstream of the gas flow path. In general, this phenomenon occurs as a consequence of excessive transient thermal gradient between blade and disc, in the presence of certain combustion residuals acting as solid lubricant, such as SiO2 and CaCO3 found in sandy environment, resulting in an equivalent
mechanical driving force, P, exceeding the blade lock resistant force. As illustrated in Figure 1.7, it begins with the micro-slip over a small portion of the blade and disk fir-tree interface, called slip zone. The growth of micro-slip over the entire interface yields macro-slip, the onset of blade walking.
The dislocation of the blades increases the rotor assembly imbalance.
Figure 1.7. Interface Friction Schematic
Figure 1.8. Blade Walking Phenomenon
1.2.7 Rotor Rubbing
Rubs between rotor and stationary stages are undesirable and sometimes can lead to grave consequences. Rub induces rotor instability as it can induce different physical phenomena such as thermal bow; imbalance due to initial permanent deflection. A particular case of rub induced instability due to stiffness variation is presented. As shown in Figure 1.9, the lab seal runner is press-fitted onto the housing having non-uniform thickness. To ensure the true geometric center of the housing assembly line, boring assembly is performed.
Figure 1.9. Housing Support of Variant Thickness
In the event of rotor excursions, rubbing between the labyrinth seal at its runner is modeled as shown in Figure 1.10 with Kr (Ө) representing the stator radial stiffness as
function of angular position defined by the XY coordinate system positioned at the housing center.
Let r be the shaft geometric center radial displacement:
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Figure 1.10. Rubbing Model
Let h and r are the radial clearance and shaft geometric center radial displacement, respectively. Assuming at the rubbing contact surface elastic deformation occurs and the generated friction force follows Coulomb’s law the following force components are obtained: ) ( ) (r h K 0 r N F rrhh (1.3) and FTFN , (1.4)
where FN is the normal force, FT is the tangential force, and μ is the coefficient of friction
between the stator and rotor.
Resolving these forces in the Cartesian coordinate XY, we have:
FXFTsinFNcos, (1.5)
FYFTcosFNsin. (1.6)
Equations 1.5 and 1.6 represent the non-linear forces appeared on the RHS of the rotor equations of motion (1.7 and 1.8), affecting the rotor dynamic performance.
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CHAPTER 2.
Dissertation Objective and Outline
Blade failure resulted either from the turbine or from the impeller blades is always an undesirable phenomenon since it could lead to catastrophic consequences. In a multi-stage engine, it is usually difficult or sometimes almost impossible to design a rotating component free of interference with all the potential excitation forces. Structurally, it is imperative to reduce the vibratory stresses to below the allowable limit in order to prevent failure due to high cycle fatigue. In principle, vibratory stresses due to interfered frequencies could be reduced either by one of the three following methods:
1) Reduce the magnitude of the excitation sources. 2) Introduce damping to dissipate vibration energy.
3) Tune the excited vibration modes out of the operating speed range. With additional necessary constraints such as:
a) Minimum perturbation to other frequencies and mode shapes since these modes are clear of interference with respect to other excitation engine orders. b) Minimum effect on performance and blade structural integrity itself together with no modification of upstream hard-wares, for instance the number of vanes on the stator or the number of fuel nozzles on the combustor.
eigenvalue whose mode shape is mainly associated with the airfoil, the under-platform damping is not an efficient method for reducing airfoil dynamic stresses. The third method implies the shifting of the interfered frequencies out of the operating speed range to avoid high cycle fatigue failure due to resonance is deemed to be the most efficient method, but it also represents a very challenging task since it is required not only to minimize perturbation to other frequencies and mode shapes which are clear of interferences, but also to not modify upstream hardware.
The first objective of this work is to develop an efficient and not computationally intensive method on the tuning of an interfered frequency of a turbine blade subjected to excitation generated by unsteady pressure in the operating speed range through classical perturbation theory with direct finite element method application. The Guided Tuning Turbine Blade Method (GTTB) is developed in response to this need for an effective and fast methodology. This method is validated by test measurements.
The second objective is to extend the single blade tuning approach to the simultaneously tuning of two inter-blade coupling adjacent natural frequencies corresponding to a particular nodal diameter out of the running speed range. The tuning process is subjected to no modification on upstream or downstream hardware, such as stator vanes, and to minimum perturbation to other frequencies since these modes are clear of interference with respect to other excitation engine orders.
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modal tuning is obtained through trial and error. This work presents the eigenvalue perturbation as a practical approach in modal tuning of rotating components.
The thesis is organized as follows:
Chapter 3 discusses the physical foundation for modal tuning by examining the dynamic behavior of two simple mechanical systems, discrete and continuous systems. A spring-mass system represents the former while a beam represents the latter. Chapter 4 starts with the investigation of the standard aeromechanical equation of an airfoil in which homogenous and particular solutions are studied. In Chapter 5, the perturbation of eigenvalue matrix and the derivative of eigenvalue and eigenvector are reviewed. From the structural mass and stiffness perturbation platform, the Guided Tuning Turbine Blade method (GTTB) is developed and presented. Chapter 6 illustrates the application of the GTTB method in conjunction with finite element method in the tuning of an interfered natural frequency of an axial turbine blade. The study is validated by laser vibrometry test. Chapter 6 ends with two published papers documenting the work on the first part of the thesis objective.
CHAPTER 3.
Dynamic Characteristics of Structures
The physical behaviors of two simple systems, a spring-mass and a beam, are examined. The former is subjected to stiffness perturbation while the latter experiences mass perturbation. It is demonstrated that through mass and stiffness perturbations, the eigen-value of change could be changed while causing no appreciable disturbance to other eigen-values.
3.1 Spring-mass System
Representing a discrete system, the spring-mass consists of four masses connected by five springs and fixed at two ends as shown in Figure 3.1.
Figure 3.1. 1D spring-mass system
Let k1= k2 = k3= k4 = 4 units, k5= 10 units, m1= m5 =1 unit and m2= m3 = 2 units. Then,
the stiffness and mass matrices are:
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Solving the equations of motion system natural frequency with corresponding mode shape expressed in column vector is obtained as following:
Freq = 773 . 14 0 0 0 0 586 . 9 0 0 0 0 549 . 4 0 0 0 0 093 . 1 Ф1 = 1366 . 0 4406 . 0 5039 . 0 2918 . 0 Ф2 = 2169 . 0 5126 . 0 3576 . 0 4144 . 0 Ф3 = 0837 . 0 0924 . 0 3416 . 0 8617 . 0 Ф4 = 9630 . 0 1860 . 0 0388 . 0 0229 . 0 (3.2)
Suppose it is asked to increase the value of λ4 of 14.773Hz to a value ≥ 16.50 Hz and
subjected to the following constraints:
1) only spring stiffness is allowed to be changed. 2) minimum disturbance to other eigenvalues.
An arbitrary change of k1 from 4 to 2 units yields un-satisfactory results as indicated
Thus k1 is not the right parameter. As observed from the mode shape given in (a), the
dominant eigenvector component corresponding to λ4 is associated with k4 and k5.
However, for minimum disturbance to other eigenvalues, k5 is the major control parameter. Thus increasing the spring stiffness k5 from 10 to 12 units, the requirements are satisfied by comparing (3.4) to (3.1).
Freq = 650 . 16 0 0 0 0 596 . 9 0 0 0 0 629 . 4 0 0 0 0 125 . 1 Ф1 = 1172 . 0 4358 . 0 5093 . 0 2963 . 0 Ф2 = 1845 . 0 5244 . 0 3494 . 0 4145 . 0 Ф3 = 0627 . 0 1003 . 0 3433 . 0 8604 . 0 Ф4 = 9738 . 0 1582 . 0 0270 . 0 0125 . 0 (3.4) 3.2 Beam
Representing a simple continuous system, the rectangular cross-section beam with dimensions L=20 in, H= 0.5 in and W= 1.0 in is shown in Figure 3.2. The beam is simply supported at two ends.
Consider the beam is made of steel with modulus of elasticity and density of 29.4.E6lb/in and 0.297 lb/in3, respectively.
The beam’s first two natural frequencies are: λ1 = 115 Hz
If a mass of 0.50 lbs is added to beam at the mid span, the first bending mode is reduced from 115 Hz to 99.5 Hz. Conversely, if a small mass of 0.02 lbs is removed from the mid-span, the first bending mode frequency is increased to 117.1Hz. However, in both cases the second bending mode virtually remains unchanged. Thus, within the consideration of the first two eigenvalues, one would be able to perturb the first natural frequency without disturbing the second one.
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CHAPTER 4.
Aero-mechanical Equation
In the flow field, the forced responses of the rotor blades arise from the non-steady pressure distribution in the tangential direction as the gas flow past obstacles such as stator vanes present upstream of the gas path. As a result, the aerodynamic excitation frequencies occur at integer multiples of the vane passing frequency. Blade resonance results if its natural frequencies coincide with the engine excitation orders. In general, the standard blade aeromechanical equation of motion is expressed as follows:
[M] { x } + [C]{ x } + [K]{x} = P(t) , (4.1) where the dot denotes derivatives with respect to time, {x} is structural displacement vector, [M] is structural mass matrix, [C] is structural damping matrix, [K] is structural mass matrix, and P(t) is aerodynamic force.
4.1 Free Vibration
4.1.1 Eigenvalue and Eigenvector
The first step of a dynamics analysis is to determine the structure natural frequencies and corresponding mode shapes in the interested frequency domain.
Neglecting the damping, the in-vacuo eigenvalues λi and eigenvectors i for a free
vibration problem, are evaluated from the determinant of:
[M] { x } + [K] {x} = 0. (4.2) By pre-multiplying the above equation by M-1, the following terms are obtained:
M-1K = A (a system matrix) And (4.2) can be written as:
[I]{ x } + [A]{x} = 0. (4.3) Assuming harmonic motion x = - λx, then equation (4.3) becomes:
[A – λI]{x} = 0 (4.4) The root λi of the characteristic equation, |A - λI|, are the eigenvalues.
And by substituting λi back into (4.3), corresponding eigenvector, i, is obtained.
In addition to obtaining λi and i, a free vibration analysis would also reveals the
physical behavior of the structure through its participation factor.
4.1.2 Physical Properties - Participation Factor
In a system with selected modes in consideration, the participation factor for the mode i is defined as:
Гi = Pi / mii , (4.5)
where
{Pi} = []T [M] {1} / []T {M] [], (4.6)
where {P}is column vector of coefficient factor of considered modes, [] is mode shape, {1} is column vector of one, and mii is effective mass.
With a mass normalized mode shape, the effective mass matrix is a unit one:
[mii] =[]T[M][] = [I]. (4.7)
Then (4.6) becomes:
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And the corresponding effective modal mass meff,i for mode i, that is the fraction of the
total static mass associated with that mode, is given by:
mii = {Pi}2/ mii . (4.9)
Consider a discrete spring-mass consists of four masses connected by five springs and fixed at two ends as shown in Figure 4.1.
Figure 4.1. 1D spring-mass system
Assuming k1= k2 = k3= k4 = 4 units, k5= 10 units, m1= m5 =1 unit and m2= m3 = 2 units,
then by solving the equations of motion one obtains the system natural frequency with its corresponding mode shape:
Then from (4.6) and (4.8), the participation factors (equation 4.5) and the effective mass (equation 4.9) are calculated:
{ Гi} = 6989 . 0 4370 . 0 1200 . 0 3035 . 2 and m11 = 5.3061; m22 = 0.0144; m33 = 0.1910; m44 = 0.4885.
The sum of the effective mass, Σmii, is just equal to the total mass of the system which is
6 units in this example. As observed, the first mass has the highest absolute participation factor and modal effective mass followed by the fourth mass. Thus, the modal participation factor and the modal effective mass provide a method for assessing the physical behavior of a vibration mode. In other words, it is a measure of the system response in a particular direction at a given natural frequency. A mode with relatively high effective mass has higher propensity to be excited by a forcing function than a mode with low effective mass.
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4.2 Force Vibration
As stated above, the standard blade aeromechanical equation of motion is given by (4.1). A study of the flow field to obtain the unsteady pressure distribution P(t), in the time domain, is not in the scope of this study. In essence, CFD simulations are used to determine the flow field characteristics. The starting point of the CFD analysis is the generation of the flow model of the domain to be analyzed. A finite number of points, connected by mesh are used to represent the continuous domain. At each of these points, a flow solver will solve the three basic governing conservations equations for mass, momentum and energy to determine the fluid flow field physical properties such as pressure, velocity, density and temperatures in the time domain as summarized in Figures 4.2 and 4.3.
Figure 4.3. Pressure Field
Through Fourier transform, the forcing function P(t) is decomposed into a sum of components each of which has a particular frequency ω, referred as Engine Orders. The subsequent step is to map the CFD pressure mesh onto the FE model. Each FE surface element is associated with a pressure node of the CFD mesh.
Figure 4.4. Blade Pressure Distribution in Frequency Domain
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To evaluate the response of the blade structure to the excitation source, a reduced modal model is needed. Consider x(t) as a linear combination of limited number of orthogonal mode shapes:
X =
k
k qk = [], (4.11)
where and q are normal modes and normal or modal coordinates, respectively.
Neglect the damping effect and substitute (4.11) into the equation of motion (4.1) and pre-multiply by T. The force vibration of the blade expressed in normal coordinates is: []T[M][]{q } + []T[K][]{q} = []T {p(ω)}, (4.12) where {p(ω)} is the Fourier transform of {p(t)}.
With a mass normalized mode shape:
[]T[M][] = [I],
and []T[K][] = [ω2]. (4.13) Equation [4.12] becomes:
[I]{q} + [ω2
]{q} = []T {p(ω)}. (4.14) With this reduced decoupled system, the motion response of each blade mode to each component of the excitation sources can be evaluated independently. The term on the RHS of equation (4.14) represents the modal force. This is a measure of how much the physical loading is transformed into the modal coordinate. The blade steady state response could be obtained by any ODE solver.
would have no contribution to the mode vibration if it is aligned in the physical coordinate with a zero participation factor.
The level of interaction between fluid and blade could be measured from the energy transferring from fluid to blade. The aerodynamic work per cycle, W, of blade motion is taking as the time integration of the product of the pressure and the blade velocity (X ) in one period of displacement over the blade area. W 0T PnXdA]dt
[ . (4.15) Representing the time dependence of oscillatory function by exp(iωt) and assume the motion is harmonic of Engine order ω, the modal displacement written in the complex form is:
X = Re{(X)(iωt) }. (4.16) With the complex velocity expressed as:
X = iω (X)(iωt) . (4.17) Similarly, the unsteady pressure is written as:
P = (Pr)(iωt+ φ) + i(Pim)(iωt+φ) . (4.18)
The real part of pressure will be in phase with the displacement, and the blade stability is determined by the phase difference between the imaginary pressure, i(Pim)(iωt+φ), and the
velocity, X .
30
Figure 4.5. Physical Representation of the Harmonic Motion and Unsteady Pressure
(Pr + Pim)e(iωt+φ)
4.3 Case Study
The aero-mechanical interaction is demonstrated in this case study. A blade natural frequency is excited if and only if there is positive energy transferred from fluid to the blade.
A blade shown in Figure 4.6, whose Campbell diagram illustrated in Figure 4.7, has mode N interfered with two Engine orders in the operating speed range. Engine order 13 corresponds to the immediate upstream stator while Engine order 16 is resulted from the immediate downstream nozzle. Modal analysis indicated that the nature of mode N is of stiff-wise bending (Figure 4.8) involving the dominant effective mass (close to 1.) in the axial direction (X axis) and rotation about Z axis. The coordinate system is shown in Figure 4.6.
The laser vibrometry test of mode N is shown in Figure 4.9, illustrating blade TE response. CFD analysis provides the un-steady pressure field. This pressure is mapped onto the FE grid for forced response analysis.
It was calculated that the center of the 13EO unsteady pressure is located on the pressure side of the blade profile about one quarter from the blade leading edge (LE) in the engine axis direction, while the one for the 16 EO from downstream effect is center about one fifth of the blade axial length measuring from the trailing edge (TE).
32
per cycle blade motion is calculated and found close to zero for the 13EO excitation while a weak value is obtained for the 16EO.
An Engine Non-intrusive test measurement (NSMS) using fiber optics to measure blade vibrations was performed and the results confirmed the analytical prediction as shown in Figure 4.10. A small blade deflection was recorded on the 16EO line while no signal was measured at the expected location on the 13EO line. In this particular case the interferences of mode N shown in the Campbell diagram (Figure 4.7) are considered to be acceptable.
Figure 4.7. Campbell Diagram
Figure 4.8. FEA Mode Shape Figure 4.9. Laser Vibrometry Test
Figure 4.10. Non-intrusive Test Measurements
0 500 1000 1500 2000 2500 3000 3500 4000 4500 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 rpm (normalized) F re q u e n c y ( H z ) 26EO 23EO
operating speed range
34
CHAPTER 5.
A. Eigenvalue Perturbation
5.1 The Perturbation of a Simple Eigenvalue
Consider an eigenvalue problem
Au = λ u, (4.1)
Where A is an (n x n) real matrix, λ is an eigenvalue with corresponding eigenvector, u. The eignvalue is determined from the determinant of the characteristic equation
det(A – λI) = 0. (4.2) Consider the eigenvalue for the transposed matrix
ATv = λ v, (4.3)
Since
det(AT– λI) = det(A – λI). (4.4) The eigenvalue value for the matrix AT can be written as
vTA = λvT, (4.5) where v is the left eigenvector corresponding to the eigenvalue λ.
Assume matrix A depends upon a vector of real parameter p = (p1,…,pn).
Then equation (4.2) gives
λ
det(A(p) – λI) ≠ 0, (4.6)
Applying implicit function theorem equation (4.6) can be expressed as
The derivative of the eigenvalue can be expressed in terms of the left hand vector v0.
Consider a point p0 in the parameter space p and assume λ0 is the simple eigenvalue of A0
= A(p0), then the derivative of (4.2) with respect to pi yields
pi A u0 + A0 pi u = p λ i u0 + λ0 pi u , (4.8)
Re-arranging equation (4.8) gives
(A0– λI) pi u = ( p λ i I - pi A ) u0 , (4.9)
The solution to the equation (4.9) exits if and only if
vT0( p λ i I - pi A ) u0 = 0, (4.10)
where vT0is the transposed of the left eigenvector corresponds to λ0. Therefore equation
(4.10) gives p λ i = vT0 pi A u0 / (vT0 u0). i=1,…,n (4.11)
5.2 Mechanical System Eigenvalue Perturbation
The undamped, homogenous equation of motion is
[M]{ x } + [K]{x} = 0. (4.12) where [M] and [K] are system mass and stiffness matrices, {x} is displacement vector, and the dot denotes the derivative with respect to time.
Assume the structure stiffness and mass matrices are perturbed by [δK] and [δM], respectively, we have:
36
[M] = [Mo] + [δM] . (4.14)
The subscript (o) denotes to the unperturbed system and subscript (i) represents the ith mode. For small perturbation (ie.[δK] « [K] and [δM] « [M]) the perturbed eigenvalue and eigenvector are expressed as
λi = λoi+ δλoi , (4.15)
and
i = oi + δoi . (4.16)
With the symmetric and positive definite of the mass and stiffness matrices, the orthogonal properties give:
oiT [Mo] oi = 1, (4.17) Toi [Ko] oi = λoi = ωoi2 . (4.18)
1) Evaluation of λoi
To solve for
[K] i = λi [M] i (4.19)
Substitute (4.16), (4.17) and (4.18) into (4.19)
([Ko] + [δK])( oi + δoi ) = (λoi + δλoi)([Mo] + [δM])( oi + δoi ) , (4.20)
Expanding the terms
[Ko] oi + [Ko] δoi + [δK] oi + [δK]δoi = λoi [Mo] oi + λoi [Mo] δoi +
δλoi[Mo] oi + δλoi[δM]oi + δλoi[Mo] δoi + λoi[δM] δoi +
δλoi[Mo] δoi + δλoi [δM] δoi . (4.21)
[Ko] δoi + [δK] oi = λoi [Mo] δoi + λoi [δM] oi + δλoi[Mo] oi (4.22)
One computational approximation method to calculate eigenvector derivative is to expand the derivative as a series of eigenvectors, thus for the ith eigenvector:
δi = oj N 1 j εij (4.23) where εij is a coefficient.
Substitute (4.23) into (4.22) gives:
[Ko] oj N 1 j εij + [δK]i = λoi [Mo] oj N 1 j εij + λoi [δM] oi + δλoi [Mo] oi . (4.24)
Applying the orthogonal properties by left multiplying byToi, equation (4.24) gives
38 But from equation (4.17)
T
oi [Mo] oi = 1,
Therefore
δλi = Toi [δK] oi - λoi Toi [δM] oi . (4.29) 2) Evaluation of oi
Take the derivative of equation (4.17), Toi [Mo] oi = 1, with respect to the design
variable (such as mass) if there is a constraint imposed on the eigenvector [16], the following is obtained.
(Toi)’ [Mo] oi + Toi [Mo]’ oi + Toi [Mo] i’= 0 , (4.30)
where the prime denotes derivative, thus
Toi [Mo] oi’= -(1/2) Toi [Mo]’ oi . (4.31)
To obtain ii substitute (4.23) into (4.31) and taking into account the normalization with
respect to the mass matrix.
N 1 j εij T oi [Mo]oj = -(1/2) T oi [Mo]’ I . (4.32)
For the case i = j
ii = -(1/2) Toi [Mo]’ oi . (4.33)
For the case i ≠ j
Substitute (4.33) into the equation {([K] – λoi [M]) oi =0} after taking its derivative and
then multiplying the resulting equation by Toj.
N 1 j εij T oj ([K] – λoi [M]) oj + T oj([δK] - λoi [δM]) oi– δλoi T oj[M]) oi = 0 . (4.35)
Due to M-orthogonal, the last term is equal to zero with T oj [K] oj = λoj T oj [M]) oj , and T oj [M] oj =1 . Then: ij = (Toj{[δK] - λoi [δM]}oi)/ (λoj - λoi) . i ≠ j (4.36)
In summary , equations (4.15) and (4.16) are expressed in terms of the unperturbed system as mass and stiffness is perturbed.
λi = λoi + (Toi [δK] oi - λoi Toi [δM] oi) (4.37) i = oi - (1/2) {Toi [δM] oi }oi + j1,N(ji) ( T oj{[δK] - λoi [δM]} oi) oj / (λoj - λoi) (4.38)
Through partial derivative of (4.37) and (4.38), the sensitivity of λi and i with respect to
δ[K] and δ [M] perturbations is studied.
a.1) kkl i kkl i = kkl (oi + ( T oi [δK]oi - oi T oi [δM]oi) ) .
With the symmetric [K] matrix, change in kkl termwill change klk, thus
kkl i = (2-kl)oi(k) oi(l) , (4.39)
40 Similarly, for changes in the mass,
mkl i mkl i = mkl (
oi + (Toi[δK] oi - oi T oi [δM] oi) ) , and mkl i = (2 -kl)
oioi(k) oi(l) . (4.40) b.1) kkl i kkl i = kkl (
1,N(j i) j (Toj [δK] oi)oj/ (
oj -
oi) ) , kkl i =
1,N(j i) j (2–kl){oj(k)oj(l)}oj/(
oj-
oi) . (4.41) b.2) mkl i mkl i = mkl (-(1/2){oi [T Mo]’oi}oi -
1,N(j i) j (
oi Toj[δM]oi)oj/ (
oj -
oi)) , mkl i = (- 1/2)(2 – kl) (oi(k)oi(l))oi -
1,N(j i) j
oi(2-kl)(oj(k)oj(l))oj/(
oj-
oi) . (4.42)In terms of frequency, for a structural system with distinct eigenvalues in the range of interest, the shift of λi either in the direction of increasing or decreasing amplitude can be
corresponding mode shape. The eigenvalue perturbation bounds are limited by the size and strength of its own mode shape as illustrated in equations (4.39) and (4.40).
While the perturbation of eigenvalue λi depends solely on the perturbation of [K] and [M]
matrices, the perturbation in the eigenvector i related to eigenvalue λi from the
perturbation of [K] and [M] depends not only from the coefficient corresponding to λi but
also from those of the other eigenvalues. As seen in equations (4.41) and (4.42), the sensitivity of eigenvector i is amplified if the difference between λi andλi+1 or λi-1 is
small.
Numerically, the sensitivity stability of the eigenvalue matrix depends on the mathematical properties of the matrix itself. From Bauer-Fike theorem, the eigenvalues of a matrix A are very sensitive to perturbation if all of its eingenvalues are smaller than the largest element [11]. Fortunately, this is not the case of real and symmetric, or complex Hermitian of common structural application.
5.3 Nodes Antinodes and Maximum Frequency Shift
Neglecting the damping, the in-vacuo eigenvalues λi and eigenvectors i for a free
vibration problem, are evaluated from the determinant of the homogeneous equation of motion
42
given mode the vibration will have nodes or places where the displacement is always equal to zero. The place where displacement is at maximum is called the primary node. The locations where displacements are relatively smaller than the primary anti-node are called the secondary anti-anti-nodes.
Consider the 1D beam of length L, whose mode shape is given by:
y = sin (
L ix
) (5.2)
where i denotes frequency index, i =1, 2, 3, etc…
For the first bending mode, the anti-node is located at the beam mid-span where the maximum beam deflection occurs. For the second bending mode, as shown below, a node is associated with the beam mid-span while the two anti-nodes of equal strength are located at one fourth and three fourth of the beam length from the left end.
Figure 5.1. 2nd Bending Mode.
In a two-dimensional system, the nodes are represented by lines where displacements are always equal to zero, while the primary anti-node becomes line where maximum displacement occurs.
The perturbation of the ith eigenvalue, λi, is maximized as the elements of the matrices
[K] and [M] are perturbed at the locations corresponding to the anti-nodes of its own mode shape, i. Equations (4.39) and (4.40) yield
k
kk i
= oi(k) oi(k) , (5.3)m
kk i
=
oioi(k) oi(k) . (5.4)B. Guided Tuning Turbine Blade Method (GTTB)
Based upon structural perturbations to the mass and stiffness at critical locations, as described previously from 5.1 to 5.3, the Guided Tuning of Turbine Blades method (GTTB) is developed.
The methodology is as follows:
(i) The various engine order excitations are identified.
(ii) Using a finite element description of the original blade geometry (pre-tuning), the free vibration eigenvalue problem is solved. This gives the natural frequencies and mode shapes.
(iii) A Campbell diagram is created, identifying all resonant conditions but especially the one(s) inside the operating speed range. The resonant conditions inside the operating range are referred to as the offending frequencies.
44
(v) A small amount of material is added to or removed from the blade at the maxima of the offending mode (see the previous step). This changes the mass and stiffness matrices and, hence, impacts the natural frequencies. But because it takes place at a modal peak, the change in the eigen-solution is most noticeable in the offending frequency/mode; there is a smaller change in the other frequencies/modes. The process returns to step (ii) and the process continues until the offending frequency is pushed outside of the operating range.
The application of the GTTB method is illustrated in Paper 1 at the end of Chapter 6.
C. Summary
The perturbation of eigenvalue λn depends solely on the perturbation of [K] and [M]
matrices. The corresponding mode shape given by n consists of regions with zero
displacements called nodes and regions with displacements called anti-nodes with the primary anti-node as the zone of highest deflections. Therefore, in order for the blade with distinct frequency λn with respect to its neighbor eigenvalues, to have maximum
perturbation (frequency shift), the elements of the [δM] and [δK] matrices must contain non-zero terms at index locations corresponding to the anti-nodes terms of the eigenvector n. Thus, the maximum frequency shift of a distinct eigenvalue is bounded
CHAPTER 6.
Simulation and Validation
6.1 Finite Element Analysis
A simple turbine blade was modeled with tetrahedron elements to provide a better understanding on the perturbation tuning process as shown in Figure 6.2. Normally, the mechanical damping properties are small and have no effect on varying the blade resonance frequency. Therefore, the density and modulus of elasticity are the only two material properties of interest in this sensitivity study.
The studied blade has modulus of elasticity of 30E6psi and density of 0.283 lb/in3. A modal analysis was performed with the blade fixed in all degrees of freedom at the fir-trees (at the hub). The results illustrated in Figure 6.3 indicate that the structure has distinct eigenvalues. Moreover, one mode n interferes in the operating speed range.
Suppose it is required to shift this resonant frequency λn from thecurrent interference
position (point A on Fig.6.4) to a position outside of the operating speed range (point B) with minimum perturbation on other eigenvalues such as frequency f1, shown in Figure
6.4, which is currently clear of resonance as indicated by point C. From perturbation theory, it is known that a Hermitian system of distinct eigenvalues (equations 4.39 & 4.40), δλn is linear proportional to the perturbation of [δK] and [δM].
46
each lumped mass representing the structure continuum. The mode shape given by n
consists of regions with zero displacements called nodes and regions with displacements called anti-nodes and the primary anti-node is the zone of highest deflections. Therefore, in order for the blade with distinct frequency λn with respect to its neighbor eigenvalues,
to have maximum perturbation (frequency shift), the elements of the [δM] and [δK] matrices must contain non-zero terms at index locations corresponding to the anti-nodes terms of the eigenvector n. Thus, the frequency shift of a distinct eigenvalue is bounded
by its associated eigenvector. Practically, for minimum aerodynamic disturbance to the airfoil, the focus of perturbation is concentrated on the primary anti-node.
As in any FE application, to ensure the integrity of the analysis, the structural mesh must be maintained when a perturbation is introduced. This is approached by dividing the volume associated with the primary anti-node into sub-volumes such that number of nodes and elements remain unchanged while individual sub-volume material properties, density and modulus of elasticity vary.
From finite element determination, the element stiffness (k) and mass {m} are defined as:
{k} = ∫BT
DBdv (6.1)
{m} = ∫ ρ[N]T[N]dv (6.2)
where dv is an element volume, ρ is the material density, [N] are the shape functions, B are the strain shape functions, and D is the elastic matrix.
From the baseline modal analysis, the eigenvalue λn and its corresponding mode shape n
are determined. This primary node corresponding to the mode shape n is found to be
sub-volumes v1,v2 and v3 with corresponding consistent mass matrices m1, m2 and m3 as
shown in Figure 6.1. Numerically through FEA, the stability, the smoothness and the bounds of the perturbed solutions of equations (4.39) to (4.42) can be studied independently by either modifying only the mass matrix (through density property) or the stiffness matrix (through Young modulus, [D] matrix) at a time. Let’s consider the physical case in which the mass is reduced from the primary anti-node. The solutions of subsequent reduction simultaneously to zero of density and modulus of elasticity in v1, v2
and v3 demonstrate that:
1) The process of tuning a frequency out of the interference zone is feasible
2) The tuned eigenvalue and its corresponding eigenvector vary on the perturbed interval smoothly as indicated by variation of λn from point A to point B.
3) The frequency shift is bounded by the size of the anti-node of interest. If we define ψ as the influence factor, the eigenvalue sensitivity is:
ψi = ( λj - λoi )/ λoi (6.3)
The analysis indicates that, for a distinct eigenvalue system in the interested frequency domain, as used in this investigation, the eigenvalue sensitivity of the other modes in particular to those adjacent to the tuned frequency could be on the order of less than one percent.
Figure 6.1. Blade Model with Sub-volumes at Primary Anti-node.
V1 V2
48
Figure 6.2. FEA Blade Model..
Figure 6.3. Mode Shape Corresponding to fn.
Secondary anti-nodes
50
6.2 Laser Vibrometry Test 6.2.1 Test Setup
A block fixture is used to support the blade. The block is machined so that the blade fir-tree can slide into the fixture (Figure 6.6). The blade fir-fir-trees are pressed against the fixture through bolt loads applied at the bottom of the block (Figure 6.5), the loading is calibrated to simulate the centrifugal effect.
Figure 6.5. Broach Block Fixture.
Figure 6.6.Broach Slot.
The setup for scanning laser vibrometry is shown in Figure 6.7. The blade is excited by sound pressure emitted from a speaker in the frequency range between 5 – 30 kHz. The laser measures the velocity of each scan point (Figure 6.8) corresponding to the sound
Broach block fixture
Load bar and load bolts
pressure input The average frequency response functions (FRF) is processed by the laser vibrometer. A sample plot is illustrated in Figure 6.9 .The FRF’s at each scan point is used to generate the deflection shape of the part at each frequency of interest.
Figure 6.7. Broach Block Fixture and Laser Vibrometer.
Figure 6.8. Grid of Scan Points.
52
Figure 6.9. Average frequency Response Spectrum.
The verification of crack initiation site and the high cycle fatigue capability is determined through subjecting the blade to an equivalent excitation source at the frequency of interest. The blade, block fixture and support structure are clamped to an isolator table and a speaker is placed near the blade and used for excitation as shown in Figure 6.10.
Figure 6.10. Shaker Setup.
6.2.2 Test Measurements
Figure 6.11. Shaker Test Results
Excitation Frequency = Blade Natural Frequency Crack Initiates & Propagates from Primary Anti-node Location.
Figure 6.12. Laser Vibrometry – Mode Shape of Baseline Mode Shape at Point A (Fig. 6.4).
54
6.2.3 Kt Contour Optimization
To reduce stress concentration effect, the modified trailing edge profile needs to be smoothly re-contoured.
Figure 6.14. FEA Contour Optimization.
Figure 6.15. Tuned Blade.
Configuration 1 Configuration 2 Configuration 3
56 Axial span N o rm a liz e d P re s s u re Pressure side Suction side LE TE
Figure 6.16. Tuned Contoured Blade Mode Shape Point B (Fig. 6.4).
Since the blade profile is changed during this redesign, one should re-examine the flow characteristics of the blade to insure the re-design has not negatively impacted its aerodynamic performance. Since a typical airfoil is designed for peak pressure ratio between pressure side and suction side on the first two third of the blade chord, it is expected that modification to the blade trailing edge would not have significant impact on the blade performance.
Paper 1
Journal of Vibration and Acoustics, Vib-12-1129
GUIDED TUNING OF TURBINE BLADES (GTTB):
A PRACTICAL METHOD TO AVOID OPERATING AT RESONANCE Loc Duong
Hamilton Sundstrand Power Systems, San Diego, CA 92186
Kevin D. Murphy Kazem Kazerounian
University of Connecticut, Storrs, CT 06269
ABSTRACT
In gas turbine applications, forced vibrations of turbine blades under resonant – or nearly resonant – conditions are undesirable. Usually in airfoil design procedures, at least the first three blade modes are required to be free of excitation in the operating speed range. However, not uncommon a blade may experience resonance at other higher natural frequencies. In an attempt to avoid resonant oscillations, the structural frequencies are tuned away from the excitation frequencies, by changing the geometry of the blade. The typical iterative design process – of adding and removing material though re-stacking the airfoil sections – is laborious and in no way assures an optimal design. In response to the need for an effective and fast methodology, the Guided Tuning of Turbine Blades method (GTTB) is developed and presented in this paper. A practical tuning technique, GTTB is based on structural perturbations to the mass and stiffness at critical locations, as determined by the methodology described herein. This shifts the excited natural frequency out of the operating speed range, while leaving the other structural frequencies largely undisturbed. The methodology is demonstrated here in the redesign of an actual turbine blade. The numerical results are validated experimentally using a laser vibrometer. The results indicate that the proposed method is not computationally intensive and renders effective results that jibe with experiments.
1. INTRODUCTION
58
Preliminary airfoil design is usually tuned for the first three or four natural frequencies to avoid resonance with these low engine orders. However, a blade may experience resonance atvane passing frequency, a high engine order. This excited natural frequency is required to be shifted outside of the operating speed range without largely disturbing the other frequencies.
In the open literature there has been a good deal of work on mistuning frequency/mode calculations [1-3]. These commonly deal with random structural perturbations away from the idealized, symmetric system, including the use of intentional mistuning patterns to mitigate the harmful effects of random mistuning. And, while these works are relevant, they do not suggest how to avoid the problem of resonance.
There are several options for avoiding large amplitude vibrations associated with resonance. The operating speed may be changed to stay away from a resonance condition, though this is a very restrictive option. Alternatively, a platform damper may be attached to the blade to reduce the mean amplitude (and stress level) [4-6]. Such dampers are widely employed in industry. However, they become less effective at high frequency.
Alternatively, one may redesign the blade itself, as recently suggested in reference [7]. This moves the structural frequencies away from the engine order excitation, avoiding resonance. This is the approach commonly taken by industry, though the specifics of their redesign procedures are not publically available. It was claimed in patents [8-9] that blade tuning was achieved through filling a recess at the blade tip with a material, which is different from that of the blade material. Journal publications that develop clear redesign methodologies are scares. This void in the open literature is a part of the motivation for the present paper. The idea here is to develop and demonstrate a consistent methodology that “tunes” the blade geometry such thatthe natural frequency changes and pushes the resonance condition out of the operating speed range as seen in figure 1. The task of tuning the blade is further complicated by the additional constraints: Redesigning the geometry changes all of the frequencies (eigenvalues). So, while a certain geometry change might move the ith frequency out of the operating range – it might also (inadvertently) move the jth frequency into the operating range. This undermines the tuning process. As such, the goal is to move the desired frequency while leaving the others largely unchanged. In figure 1, this would mean moving point A to A’, while not changing the fundamental frequencies represented byB, C, D, and E appreciably.
The preliminary airfoil geometry, tuned for fundamental modes, was developed to insure good aerodynamic performance. As such, changes to the geometry in this tuning process should not drastically impact the lift/drag characteristics of the blade. It is also assumed that modification to the upstream hardware, for instance the
number of vanes on stator or the number of fuel nozzles, is not an option.
