Stiffness Characteristics of Airfoils Under Pulse Loading
Dissertation
Presented in Partial Fulfillment of the Requirements for the Degree
Doctor of Philosophy in the Graduate School of The Ohio State University
By
Kevin Eugene Turner, M.S.
Graduate Program in Mechanical Engineering
The Ohio State University
2009
Dissertation Committee:
Michael Dunn, Advisor
June Lee
Daniel Mendelsohn
James Williams
c Copyright by
Kevin Eugene Turner
2009
Abstract
The turbomachinery industry continually struggles with the adverse effects of contact rubs between airfoils and casings. The key parameter controlling the severity of
a given rub event is the contact load produced when the airfoil tips incur into the
casing. These highly non-linear and transient forces are difficult to calculate and their
effects on the static and rotating components are not well understood. To help provide this insight, experimental and analytical capabilities have been established and
exercised through an alliance between GE Aviation and The Ohio State University
Gas Turbine Laboratory. One of the early findings of the program is the influence of
blade flexibility on the physics of rub events.
The core focus of the work presented in this dissertation is to quantify the influence of airfoil flexibility through a novel modeling approach that is based on the
relationship between applied force duration and maximum tip deflection. This relationship is initially established using a series of forward, non-linear and transient
analyses in which simulated impulse rub loads are applied. This procedure, although
effective, is highly inefficient and costly to conduct by requiring numerous explicit
simulations. To alleviate this issue, a simplified model, named the pulse magnification model, is developed that only requires a modal analysis and a static analyses to
fully describe how the airfoil stiffness changes with respect to load duration. Results
from the pulse magnification model are compared to results from the full transient
ii
simulation method and to experimental results, providing sound verification for the
use of the modeling approach.
Furthermore, a unique and highly efficient method to model airfoil geometries was
developed and is outlined in this dissertation. This method produces quality Finite
Element airfoil definitions directly from a fully parameterized mathematical model.
The effectiveness of this approach is demonstrated by comparing modal properties
of the simulated geometries to modal properties of various current airfoil designs.
Finally, this modeling approach was used in conjunction with the pulse magnification
model to study the effects of various airfoil geometric features on the stiffness of the
blade under impulsive loading.
iii
Dedicated to my lovely wife and kids.
Thanks for the boundless love and support.
iv
Acknowledgments
I wish to thank my advisor, Mike Dunn, for his prolonged patience, technical
expertise and encouragement, all of which made this dissertation possible.
A special thanks goes to Dennis Corbly, Steve Manwaring, Larry Bach and Darin
Ditommaso whose support and career guidance has been exceptional.
I am grateful to the many co-workers I’ve had the opportunity to work with
over the past 10 years, I owe my professional success to you. Particularly to Andy
Blair, Jim Griffiths, Sunil Sinha, Joel Kirk, Bruce Dickman, Alan Turner, Chuck
Orkiszewski, Sanjiv Tewani, Dan Mollmann, Al Storace and Tod Steen for their
wealth of knowledge and support during this long journey.
I also wish to thank Corso Padova and Jeff Barton for their vast experimental and
technical assistance.
A loving thanks goes to my wife, Bobbie, whose support, encouragement and love
have been my guiding light. To my son Mason and daughter Madison, thanks for
your continued support and giving daddy time to work, I have a lot of fishing trips
and tea parties to make up.
This research has been funded by the General Electric Company.
v
Vita
1977 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Born — Columbus, Ohio
2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S. Mechanical Engineering,
The Ohio State University,
Columbus, OH
2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.S. Mechanical Engineering,
The Ohio State University,
Columbus, OH
2000 - Present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Engineer, GE Aviation,
Cincinnati, OH
Fields of Study
Major Field: Mechanical Engineering
vi
Table of Contents
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvi
Chapters:
1.
2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
1.2
1.3
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1
4
6
6
7
Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.1
2.2
2.3
11
12
15
16
19
Problem Statement and Industry Need
Previous Work . . . . . . . . . . . . .
Research Goals . . . . . . . . . . . . .
1.3.1 Experimental . . . . . . . . . .
1.3.2 Analytical . . . . . . . . . . . .
Introduction
Formulation .
Mean Surface
2.3.1 Twist
2.3.2 Lean .
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21
24
26
28
30
32
35
35
37
39
44
45
49
Pulse Stiffness Determination . . . . . . . . . . . . . . . . . . . . . . . .
54
3.1
3.2
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54
55
56
58
64
65
67
69
71
73
Rub Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
4.1
4.2
4.3
91
92
97
2.4
2.5
2.6
2.7
3.
3.3
3.4
4.
5.
Introduction . . . . . . . . . . . .
Tip Stiffness . . . . . . . . . . . .
3.2.1 Static Loading . . . . . . .
3.2.2 Pulse Loading . . . . . . .
Implementation . . . . . . . . . . .
3.3.1 Pulse Magnification . . . .
3.3.2 Pulse Width Normalization
κ-Model . . . . . . . . . . . . . . .
3.4.1 Coefficients . . . . . . . . .
3.4.2 Results . . . . . . . . . . .
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Max Rub Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparisons to Experimental Data . . . . . . . . . . . . . . . . . .
Airfoil Geometry Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.1
5.2
5.3
6.
2.3.3 Bow . . . . . . . . . . . . . .
2.3.4 Camber . . . . . . . . . . . .
Domain Constraints . . . . . . . . .
2.4.1 Span and Chord . . . . . . .
2.4.2 Sweep . . . . . . . . . . . . .
2.4.3 Flow Path Angles . . . . . .
Thickness . . . . . . . . . . . . . . .
2.5.1 Span-wise Variation . . . . .
2.5.2 Chordal Variation . . . . . .
Model Discretization . . . . . . . . .
Application to Common Geometries
2.7.1 High Pressure Compressor . .
2.7.2 Fan . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Effect on Inflection Point . . . . . . . . . . . . . . . . . . . . . . . 106
Effect on Tip Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . 110
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.1
6.2
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Future Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . 116
viii
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Appendix A: Additional Spatial Tip Force Profiles . . . . . . . . . . . . . . . 120
ix
List of Tables
Table
Page
2.1
Geometry Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
2.2
HPC Natural Frequency Comparisons . . . . . . . . . . . . . . . . . .
48
2.3
Fan Natural Frequency Comparisons . . . . . . . . . . . . . . . . . .
52
3.1
Best Fit Coefficients for κ . . . . . . . . . . . . . . . . . . . . . . . .
78
4.1
Comparisons of Maximum Measured Tangential Forces To Predicted . 103
5.1
Effect of Geometry Parameters on the Inflection Point
5.2
Effect of Geometry Parameters on Static Tangential Tip Stiffness . . 111
5.3
Effect of Geometry Parameters on Static Radial Tip Stiffness . . . . . 113
x
. . . . . . . . 109
List of Figures
Figure
Page
1.1
Commercial Jet Engine, CFM56-7 . . . . . . . . . . . . . . . . . . . .
3
1.2
Compressor Spin Pit Facility, a) Above Ground, b) Below Ground . .
6
1.3
CSPF Rub Components, a) Blade and Casing, b) Rub Shoe . . . . .
7
1.4
CSPF Key Instrumentation, a) LMU, b) Airfoil Strain Gauges . . . .
8
2.1
Bladed Rotor Example . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2
Process Flow for Model Generation . . . . . . . . . . . . . . . . . . .
14
2.3
Bladed Rotor in Engine Coordinates . . . . . . . . . . . . . . . . . .
16
2.4
Airfoil Mean Surface . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.5
Twist Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.6
Effect of Twist on the Tangential Airfoil Stiffness . . . . . . . . . . .
19
2.7
Lean Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.8
Effect of Lean on Radial Airfoil Stiffness . . . . . . . . . . . . . . . .
21
2.9
Bow Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.10 Camber Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.11 Detailed Schematic of Camber . . . . . . . . . . . . . . . . . . . . . .
25
xi
2.12 Airfoil Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.13 Chord and Span Definition . . . . . . . . . . . . . . . . . . . . . . . .
29
2.14 Schematic of a Bladed Disk Rotor . . . . . . . . . . . . . . . . . . . .
29
2.15 Sweep (linear) Definition . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.16 Sweep (exponential) Definition . . . . . . . . . . . . . . . . . . . . . .
31
2.17 Inner and Outer Flow Path Angle Definition . . . . . . . . . . . . . .
33
2.18 Tip and Root Thickness Definition . . . . . . . . . . . . . . . . . . .
36
2.19 Minimum and Maximum Thickness Definition . . . . . . . . . . . . .
38
2.20 Discretized Airfoil Mean Surface . . . . . . . . . . . . . . . . . . . . .
40
2.21 Asymmetric Cross-Section Feature
. . . . . . . . . . . . . . . . . . .
42
2.22 Node Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2.23 Comparison of HPC Geometries, Tangential View, a) Flat Plate, b)
Simulated, c) Actual . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.24 Comparison of HPC Geometries, Tip View, a) Flat Plate, b) Simulated, c) Actual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.25 Comparison of HPC Geometries, Axial View, a) Flat Plate, b) Simulated, c) Actual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
2.26 Comparison of HPC Modeshapes, a) Flat Plate, b) Simulated, c) Actual 49
2.27 Comparison of Fan Geometries, Isometric View, a) Flat Plate, b) Simulated, c) Actual . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2.28 Comparison of Fan Geometries, Tip View, a) Flat Plate, b) Simulated,
c) Actual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2.29 Comparison of Fan Geometries, Axial View, a) Flat Plate, b) Simulated, c) Actual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
xii
2.30 Comparison of Fan Modeshapes, a) Flat Plate, b) Simulated, c) Actual 52
3.1
Schematic of Tip Loading . . . . . . . . . . . . . . . . . . . . . . . .
56
3.2
Static Tip Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.3
Pulse Width Definition . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.4
Dynamic Tip Loading . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.5
Series of Tip Deflection Curves . . . . . . . . . . . . . . . . . . . . .
62
3.6
Build Up of Pulse Deflection Curve . . . . . . . . . . . . . . . . . . .
63
3.7
Tangential Deflection Curve Versus Pulse Width for a Small Flat Plate 64
3.8
Tangential Stiffness Curve Versus Pulse Width for a Small Flat Plate
65
3.9
κθ Versus Pulse Width for a Small Flat Plate . . . . . . . . . . . . .
67
3.10 κθ Versus τ for a Small Flat Plate . . . . . . . . . . . . . . . . . . . .
69
3.11 Effect of µ and η on κ . . . . . . . . . . . . . . . . . . . . . . . . . .
71
3.12 κθ Curve for a Small Flat Plate Under Various Loadings . . . . . . .
74
3.13 κr Curve for a Small Flat Plate Under Various Loadings . . . . . . .
74
3.14 Distribution of µθ for a Small Flat Plate . . . . . . . . . . . . . . . .
75
3.15 Distribution of ηθ for a Small Flat Plate . . . . . . . . . . . . . . . .
76
3.16 Effect of Fr and Fθ on µθ for a Small Flat Plate . . . . . . . . . . . .
77
3.17 Effect of Fr and Fθ on ηθ for a Small Flat Plate . . . . . . . . . . . .
77
3.18 Predicted κθ with Error Bars for a Small Flat Plate . . . . . . . . . .
79
3.19 Predicted κr with Error Bars for a Small Flat Plate . . . . . . . . . .
80
xiii
3.20 Distribution of Eθ for a Small Flat Plate . . . . . . . . . . . . . . . .
81
3.21 Distribution of Er for a Small Flat Plate . . . . . . . . . . . . . . . .
81
3.22 Comparison of κθ for a Small and Large Flat Plate . . . . . . . . . .
82
3.23 Comparison of κr for a Small and Large Flat Plate . . . . . . . . . .
83
3.24 Effect of Rotational Speed on K s . . . . . . . . . . . . . . . . . . . .
84
3.25 Effect of Rotor Speed on Fundamental Airfoil Frequency . . . . . . .
85
3.26 Effect of Rotor Spin on κθ for a Small Flat Plate . . . . . . . . . . . .
86
3.27 Effect of Rotor Spin on κr for a Small Flat Plate . . . . . . . . . . . .
87
3.28 Comparison of κθ for Simulated HPC Airfoil and Small Flat Plate
Under Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.29 Comparison of κr for Simulated HPC Airfoil and Small Flat Plate
Under Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.30 Comparison of κr for Simulated HPC Airfoil Using Specific Coefficients 90
4.1
Maximum Possible Fθ for a Radial Incursion of 0.001 in . . . . . . . .
94
4.2
Maximum Possible Fθ for a Radial Incursion of 0.005 in . . . . . . . .
94
4.3
Maximum Possible Fθ for a Radial Incursion of 0.015 in . . . . . . . .
95
4.4
Maximum Possible Fθ for a Radial Incursion of 0.03 in . . . . . . . .
96
4.5
Maximum Possible Fθ for Fr = 150lbf
. . . . . . . . . . . . . . . . .
97
4.6
Maximum Possible Fr for Fθ = 150lbf
. . . . . . . . . . . . . . . . .
98
4.7
Rub Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
4.8
Pulse Width for CSPF Experiments . . . . . . . . . . . . . . . . . . . 101
4.9
Rub Force Map for Various Radial Incursion Depths . . . . . . . . . . 102
xiv
4.10 Comparison of Predicted Feθ to Measured . . . . . . . . . . . . . . . . 103
5.1
Effect of Geometry Parameters on the Inflection Point
5.2
Effect of Geometry Parameters on Static Tangential Tip Stiffness . . 111
5.3
Effect of Geometry Parameters on Static Radial Tip Stiffness . . . . . 112
xv
. . . . . . . . 108
Nomenclature
Abbreviations
GTL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Gas Turbine Laboratory
HPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . high-pressure compressor
TE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . trailing edge
LE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . leading edge
GEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Electric Aviation
DOE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . design of experiments
FE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finite Element
FEA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finite Element Analysis
CSPF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressor Spin Pit Facility
LSPF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Large Spin Pit Facility
LMU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .load measuring unit
deg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . degree
NRMSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . normalized root mean square error
ANSYS1 . . . . . . . . . . . . . . . . . . . . . . . . . . . Commercially available finite element software
1
R
ANSYSis
a registered trademark of SAS IP, Inc.
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LS-DYNA2 . . . . . . . . . . . . . . . . . Commercially available explicit finite element software
MATLAB3 . . . . . . . . . . . . . . . Commercially available programmable analysis software
Symbols
c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . chord
s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . span
β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . stagger angle
ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dihedral angle (lean)
φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . inner flow path angle
Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . time dependent force scalar
θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . outer flow path angle
λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . linear sweep angle
λs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . exponential sweep parameter
AR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . aspect ratio
T R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ratio of edge thickness to max thickness
T T R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ration of tip thickness to root thickness
b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . bow parameter
τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dynamic pulse width
κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pulse magnification parameter
f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . frequency
2
R
LS-DYNAis
a registered trademark of Livermore Software Technology Corp.
3
R
MATLABis
a registered trademark of The Mathworks Inc.
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E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . error function
G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . scalar function describing mean surface
x, y, z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cartesian coordinates
r, θ, z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cylindrical coordinates
→
−
N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . normal vector field
th . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . airfoil thickness
σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . stress
F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . force
Fe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . normalized force
ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rotational speed
t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . distance through airfoil thickness
P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . number of nodes
∆t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pulse width
δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . radial incursion
δe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . normalized radial incursion
T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . period
ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . minimization function
µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . coefficient on log term
η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . coefficient on exponential term
ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . camber angle
xviii
bi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . unit vector in x direction
b
j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . unit vector in y direction
b
k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . unit vector in z direction
R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . radius
S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mean surface
Units
fps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ft/s
mils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .in·10−3
psi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lb/in2
rpm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .rev/min
ms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s·10−3
Subscripts
root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . airfoil inner radius, mid-chord
edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . airfoil edge, LE or TE
tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . airfoil outer radius, mid-chord
twist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . airfoil twist
sweep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . airfoil sweep
lean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . airfoil lean
bow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . airfoil bow
camber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . airfoil camber
xix
p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . node index
θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .tangential
r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . radial
c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .casing
Superscripts
d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dynamic
s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . static
xx
Chapter 1: Introduction
1.1
Problem Statement and Industry Need
Countless advancements in the field of turbomachinery design have been made
over the years, making today’s machines more efficient, more powerful and more reliable than predecessor machines still in operation. However, despite the numerous
improvements, engineers and scientists are continually working to further improve
upon nearly flawless designs. In order for manufactures of turbomachinery to stay
competitive in today’s marketplace, they are pushing the design envelope even further,
trying to stay ahead of the competition, or simply satisfy ever increasing restrictions
handed down by the certifying agencies and/or governmental mandates. Due to today’s political and economical environment, significant emphasis is being placed on
making turbomachines cheaper to own and more environmentally safe to operate. A
key design consideration for making such improvements is minimizing the operational
clearances between the rotating airfoils and their matching stationary casings. By reducing these clearances, engineers are able to increase cycle efficiencies and reduce
fuel consumption. There is a limit, however, to the amount the operational clearances can be reduced. This limit is controlled by many complex and interdependent
factors that must be taken into consideration and accounted for by designers. While
1
making clearances as tight as possible is desirable from an efficiency or aerodynamic
perspective, it is unfavorable from a durability or mechanical perspective since the
probability of blade-to-case rubbing increases with the decreasing clearance. These
countering objectives bring about a need for balancing the risks and the benefits
associated with reducing operational clearances.
Although this delicate balancing between risks and benefits exists for the turbomachinery industry as a whole, it is the jet engine industry that toils the most over
this particular design concept. The reason being is simple; there is a heightened
level of risk and reward for air-based propulsion machines, such as the one shown in
figure 1.1, over ground-based machines, such as power generation and pumping machines. Additional benefits to be realized by jet engines come about due to the added
restrictions associated with larger and more diverse operational envelopes, more severe weight restrictions, higher dependency on expensive and exotic materials, and
broader reliability and safety concerns. As for the risk side of things, there are many
compounding factors that make the design and operation of jet engines more complex
and more susceptible to blade-to-case rubbing. It is for these reasons that the research
described within, focuses on jet engine specific hardware and operation. However, it
is worth noting, the technology and understanding brought about by this research is
applicable to all forms of turbomachinery.
One of the most difficult challenges for engineers is being able to accurately predict
and quantify operational, or hot, clearances. During operation, clearances change
depending on many factors, which include but are not limited to, the rotational speed
of the rotors, thermal state of the rotors and casings, altitude, maneuvers, transient
operation effects, environmental effects, manufacturing tolerances, erosion, corrosion
2
Figure 1.1: Commercial Jet Engine, CFM56-7
and other forms of engine deterioration. All the stated factors and others, need to be
taken into consideration by engineers when they set static, or cold, clearances. Static
clearances refer to the size of the gap between the rotor blades and their corresponding
casings while the engine is not running. From this point, designers predict the amount
of closures the engines will see over a wide range of operational states and back out
cold clearances with conservative margin built in to maintain suitable operation.
The rationale behind this research is to gain an understanding of what happens
when blade-to-case closures exceed the operational clearances and rubbing ensues.
The expectation is this; having a working knowledge of the rub phenomena will
expand the design space available to designers and allow them to reduce their margins
on blade closures and/or increase the durability and reliability of their designs. As the
market continues to move in the direction of more performance, better fuel economy
and lower cost of ownership, jet engine manufactures will continue to push the design
envelope further and further along the direction of tighter clearances to a point where
3
rubs can no longer be avoided. Designers can make this feasible by discovering ways of
making engines more tolerant of blade-to-case rubbing. The research being conducted
at The Ohio State University Gas Turbine Lab (OSU-GTL), under the auspices of
GE Aviation, will play a vital role in this transition.
1.2
Previous Work
Since the dawn of the jet age, over half a century ago, blade-to-case rubbing has
been an issue for jet engine designers and manufacturers. The common remedy has
been to open clearance to a point where blade rubs can be avoided. Although this
remedy is an effective one, it does come with a price, namely reduced performance and
efficiency. In today’s marketplace, where fractions of a percent in efficiency make or
break a design, the cost associated with opening clearances to avoid rubs is becoming
less and less appealing. As a result, research in the field of blade-to-case rubbing is
still in its infancy, and in many ways is being pioneered by Padova et al. [7, 8, 9] and
their experimental work at the OSU-GTL.
Despite the relatively recent need for an expanded understanding of blade rubs,
research in this field, albeit sparse, has taken place since the beginning of the jet
age. It was not until the 1980’s that a significant level of concentrated effort was
given to the field of blade-to-case rubs. Even then, most of the work was focused
more on the rotor dynamic response and not on the blade and case dynamics. That
changed in the late 1980’s when A. Muszynska began to branch away from studying
rotor response and started considering more local blade and casing dynamics as well.
Muszynska’s survey [6] written in 1989 provides an exhaustive literature review of
4
rub-related vibration and reveals a sudden surge in research on blade rubs in the
1980’s and the primary focus on rotor dynamics.
Since Muszynska’s survey was written, several noteworthy papers have appeared in
the literature that deal with blade rub related topics. On the experimental side there
has been very little work done beyond the efforts of Padova et al. mentioned earlier,
to study blade rubs under realistic engine conditions. Most notable is Ahrens et al. [1]
who published work on experimentally measuring contact loads, which consisted of
rotating an artificially stiff flat plate at speeds much lower than used during common
engine operation. In addition, two theses were written to enhance the experimental
efforts of Padova et al. at the OSU-GTL, Young [12] and Ferguson [2] both developed
linear inverse techniques to back out true rub loads from indirect measurements taken
in the CSPF.
On the analytical side, there have been numerous papers that have discussed
the impact of blade-to-case rubs on the dynamics of rotor systems, but very few
on the localized blade and casing dynamics. Ahrens et al. [4] discussed a modeling
approach to predict contact loads using a flat plate formulation that provided a great
supplement to the experimental testing they did. Garza [3] completed a thesis in
which he describes a full transient and non-linear rub simulation using LS-DYNA.
His simulation work was focused on predicting the dynamic behavior of the test
blade used in the OSU-GTL compressor rub facility. Likewise, a thesis by Turner [11]
describes the development of a new, more efficient, simulation technique to predict
rub-induced blade dynamics and compared results of his predictions to experimental
results obtained using the CSPF. Finally, Sinha [10], published his work on simulating
the blade dynamics using a very sophisticated Timoshenko beam formulation.
5
1.3
1.3.1
Research Goals
Experimental
The scope and objective of this research is an important component of the scope
and objectives of the work being done both experimentally and analytically at OSU
to understand blade-to-case rub phenomena. OSU GTL’s compressor spin pit facility
(CSPF), shown in figures 1.2 and 1.3, is uniquely suited to experimentally simulate
blade tip rubs in a controlled environment. This facility has a demonstrated capability
of running actual engine hardware at true engine speeds. The instrumentation, shown
in figure 1.4, used during these rub experiments include: blade strain gauges, casing
load cells and casing accelerometers. The resulting measurements, in conjunction
with other facility instrumentation, are sufficient to characterize both the stationary
and rotational component responses.
Figure 1.2: Compressor Spin Pit Facility, a) Above Ground, b) Below Ground
6
Figure 1.3: CSPF Rub Components, a) Blade and Casing, b) Rub Shoe
The CSPF has been exercised many times and a considerable database of responses
has been generated. Key design parameters that will allow direct impact to product
design are included in the database. These parameters include, rotational speed,
incursion depth, several common tip geometries and various casing treatments, as
detailed by Padova et al. [9]. Additional capabilities, in the form of a larger spin-pit
facility (LSPF), have been commissioned at the OSU-GTL and will become a prime
resource as the rub program continues.
1.3.2
Analytical
The experimental efforts cover only half of the program’s scope. The other half
pertains to expanding analytical capabilities. The existence of the vast experimental
database is vital for advancing the predictive capacity of existing Finite Element
Analysis (FEA) codes and the development of other codes. Extensive validation
and correlation efforts have been possible by comparing analytical results with the
experimental results [2, 11, 12].
7
Figure 1.4: CSPF Key Instrumentation, a) LMU, b) Airfoil Strain Gauges
8
The scope of the research associated with this dissertation work is much more
focused than the broad perspective just given. The key objective of this research
is to understand the stiffness characteristics of the airfoils as they are subjected to
impulsive loads at their tips during a rub event. This aspect of blade behavior is
important in the context of blade-to-case rubbing because in many situations the
rub loads produced are like impulse loads. The rubs that occur in turbomachines
are often short lived because the blade can quickly loose contact with the casing.
The loss of contact can be a result of the instigating issue being eradicated, dynamic
deflection of the contacting structures, or removal of contacting materials. Whatever
the cause may be, the fact remains that rub loads can effectively be simulated using
pulse conditions.
An early discovery of the rub research program was the importance of blade flexibility in the physics of the high-speed contacts. The experiments revealed that
the airfoil could withstand relatively large (much larger than expected) amounts of
interference with no signs of plastic deformation or material removal. For these experiments, the blades were run into a solid, near rigid, metal shoe that also showed no
signs of material removal. Therefore, the only logical explanation for how the system
relieved the interference was for the blade to elastically deflect out of the way. More
importantly, the rub loads being measured during these events were much lower than
would be required to statically deflect the blade radially to relieve the interference.
For that reason, it was surmised that the flexibility of the blade must be considerably
different during the dynamic rub event than it is during a static loading event.
With this knowledge comes the need to be able to analytically explain the flexibility of the airfoils during simulated rub events. Much of the analytical work done
9
in the past has either relied on a static loading assumption and/or has been based on
simplified geometries such as beams or plates. The ineffectiveness of using a static
load has already been discussed and proven through testing. The effectiveness of
assuming simplified geometry, however, needs to be investigated further and is one
of the key objectives of this work. In Chapter 2 of this paper, a novel modeling
approach will be introduced that will allow for common airfoil geometry features.
The approach, based on mathematically representing all the geometric features, will
instantly produced quality FE models that structurally represent actual airfoils used
in the compressive stages of axial-flow turbomachines. Later on, in Chapter 5, results from a series of parametric studies using the modeling approach will be given
to explain the effect of each geometric parameter on blade stiffness.
Being able to efficiently and effectively build FE models of various airfoil shapes
is one matter, being able to efficiently and effectively solve the models to understand
the transient stiffness behavior is another matter all together. The core element
of this research work is to develop a simplified modeling approach to understand the
effect of pulsive-type loads on the stiffness behavior of turbomachinery airfoils. Chapter 3 describes the development of this analysis technique, which is later validated in
Chapter 4, using data from the OSU-GTL CSPF.
10
Chapter 2: Model Development
2.1
Introduction
One of the key objectives of this work is to establish how certain geometric parameters and loading conditions affect an airfoil’s ability to generate loads while rubbing
into the casing structure. There are two major advantages to doing so, a) the maximum possible tip loads can be determined quickly for an existing airfoil geometry,
and b) engineers can design airfoils to be more rub tolerant by knowing the relationships between geometric parameters and maximum possible rub loads. To make
this possible, a large number of analyses are required not only to study an individual
parameter’s affect, but also to study how the interaction of various parameters affect
the rub characteristics of a blade. This type of study is most effectively handled statistically using the Design of Experiments (DOE) approach and explained in detail
in Chapter 5. Suffice it to say for now, that to meet the objectives of this research,
the ability to model and analyze a large number of airfoil geometries and loading
conditions is required.
There are essentially two methods available for completing the DOE described
above, a) an analytical approach, or b) a numerical approach. Each approach has
certain advantages and disadvantages, but in the end only the latter is feasible. The
11
analytical method would certainly provide efficiency, but would dramatically limit the
types of geometries and loading conditions that could be analyzed. In order to study
the affects of the various complex geometric features that today’s airfoil designers use,
numerical simulation is mandatory. As part of this work, two common, commercially
available Finite Element Analysis (FEA) codes will be used to analyze each blade
geometry. The first, ANSYS, will be used to perform a modal analysis to determine
natural frequencies and modeshapes. The second, LS-DYNA, is well suited for highly
transient analyses and will be used as such to determine the transient airfoil responses
to simulated rub loads.
Even with the help of the FEA codes, there still exists a significant problem,
generating the numerous FE models required to carry out the DOE. There is an
insufficient number of blade designs in existence to satisfy the needs of the study,
so the capability to generate numerous blade geometries with varying features and
loading conditions needs to be established. The development of this capability, along
with some applications of this capability toward common geometries is the focus of
the current chapter.
2.2
Formulation
The airfoil geometries of today turbomachines are very complex and include various 2 and 3-dimensional features. These features have been introduced over the years
for their aerodynamic benefit and often come with structural consequences that need
to be fully understood through various structural analyses. Historically, many of these
analyses would have been completed analytically using either beam or plate theory,
but the extreme complexity of current airfoils, such as the one shown in figure 2.1,
12
have rendered many of these analyses ineffective. As a result, engineers rely heavily
on numerical analyses, such as Finite Elements, to determine the structural characteristics of their blade designs. With this, comes the often tedious work of building
solid models of the complex geometries and discretizing them.
Figure 2.1: Bladed Rotor Example
A technique has been developed to dramatically reduce the effort required to
produce quality FE models using common airfoil design parameters. The proposed
technique is based on a fully mathematical representation of the blade geometry and
allows representative FE models to be generated instantly. The methodology bypasses
the need for generating solid models and produces a discretized model directly. Since
the approach relies on mathematically describing the airfoil geometry, there is a limit
to what features can be included. For instance, internal cooling passages used in
some cooled turbine blades would be difficult to include using this approach. The
13
goal for this work is to produce FE models that capture the key structural properties
of common solid airfoil geometries, the models are not intended to capture internal
geometries or represent the aerodynamic properties. However, the general approach
is robust and can be expanded to include more geometric features to satisfy additional
needs.
Figure 2.2: Process Flow for Model Generation
The general approach, from supplying inputs to generating FE models, is outlined
in figure 2.2 and explained in detail in the following sections of this chapter. There are
several geometric parameters that are supplied as inputs to the model and include:
a) twist, b) lean, c) bow, d) camber, e) span, f) chord, g) inner and outer flow path
definition, h) sweep, and i) thickness variation. There are several other inputs that
14
describe the loading to be applied to the model and include magnitude and duration
of tip forces and rotational speed. Aside from the advantage of producing realistic
blade geometries instantly, this approach also has other advantages such as it can be
easily implemented into any design process and placed in a DOE loop to provide a
means for studying the various effects of each design parameter.
2.3
Mean Surface
The first step in building an airfoil model is to find an equation for the mean
surface. The mean surface is a surface that defines the general shape of the airfoil
and lies roughly mid-way between the pressure and suction surfaces of the airfoil.
The mean surface can be quite complex when considering all the key design features
used in today’s turbomachines. To overcome the complexity, this approach takes
advantage of superposition. By isolating each geometric feature, a relationship can
be formed between the main parameters defining the geometric feature and the airfoil
coordinates. When this relationship has been established for each geometric feature,
as seen in equation 2.1, they are summed up to produce one equation for the airfoil
mean surface.
S = z(x, y) = ztwist + zlean + zbow + zcamber
(2.1)
The mean surface is defined using an engine-based coordinate system, as shown in
figure 2.3, in which the x-axis is aligned with the engine axis, the y-axis is aligned in
the span-wise (radial) direction and the z-axis is aligned in the tangential direction.
In the figure of the bladed rotor, the mean surface would correspond to the airfoil
aligned with the y-axis. The mean surface, S, is valid for all x and y and will need
15
to be trimmed using known boundaries, as is described in § 2.4. An illustration of a
generic mean surface is shown in figure 2.4.
Figure 2.3: Bladed Rotor in Engine Coordinates
The goal of this research is to understand how the stiffness characteristics of an
airfoil are affected by pulse loading. With that in mind, several major geometric features that are likely to influence the flexural tendencies of an airfoil were incorporated
into the model. Of those features, listed in § 2.2, only four of them have a role in
defining the mean surface and they include: a) twist, b) lean, c) bow, and d) camber.
How these geometric attributes are parameterized and related to airfoil coordinates
will be explained in the next four subsections.
2.3.1
Twist
The first geometric feature to be considered is twist. Twist, also referred to as
stagger, is a term that refers to the relative angle between the airfoil and the engine
16
Figure 2.4: Airfoil Mean Surface
axis. Figure 2.5 provides an illustration for how twist looks when applied to a meshed
flat plate. The figure shows three distinct views: a) axial-radial (xy) plane on the
left, b) radial-tangential (yz) plane on the right, and c) axial-tangential (xz) plane on
the top. An equivalent figure will be given for each design feature in their respective
subsections.
Twist can greatly effect the stiffness characteristics of a cantilevered airfoil by
offsetting the minimum stiffness axis from the tangential direction. For events such
as case rubs, the blade tip can see high tangential loads due to friction and plowing,
which tend to oppose the rotation of the blade. Through this misalignment of the
minimum stiffness axis and the tangential direction, the tangential flexibility of the
airfoil becomes coupled with the axial flexibility causing a reduction in the blade’s
ability to resist motion in the axial direction, while increasing the resistance in the
tangential direction. In the limit when β becomes 90 deg, the area moment of inertias
17
relative to the axial and tangential directions will be inverted, giving the maximum
possible tip stiffness in the tangential direction. In practice, twist angles can become
quite large, reaching values of 60 deg in today’s 3-dimensional fan blade designs.
Figure 2.5: Twist Definition
Twist is defined by the angle β(y) and is allowed to vary linearly across the span.
To fully define β, two parameters are needed, βroot and βtip , which define the relative
angle at the root (y = 0) and at the blade tip (y = s) respectively. The twist
parameter, β, is further defined in equation 2.2. Permitting the twist, or stagger,
angle to vary radially in a linear fashion, provides a good means to structurally
represent the twist seen in most common airfoil geometries.
β(y) =
βtip − βroot
y + βroot
s
18
(2.2)
Figure 2.6: Effect of Twist on the Tangential Airfoil Stiffness
The mean surface is impacted by the twist angle as shown in figure 2.6 for a given
radial location. Since the twist is represented by an angle only, the mean surface
varies linearly with respect to the axial direction. This linearity will change later on
when other features such as camber are included. The formula for the mean surface as
a function of twist is given in equation 2.3. Combining equation 2.2 and equation 2.3
yields the relationship of the mean surface to the stagger at the tip and root locations
of the airfoil.
ztwist (x, y) = tan (β(y)) x
ztwist (x, y) = tan
2.3.2
βtip − βroot
y + βroot x
s
(2.3)
(2.4)
Lean
The next geometric feature to consider is lean, or dihedral. Lean is a term to
describe the tilt of the airfoil in the tangential direction and is illustrated in figure 2.7.
19
Lean is another design feature that is frequent in airfoil designs and it is represented
in the model through the mean surface. Lean is a relatively simple feature to include
and can have significant effects on the stiffness behavior of an airfoil. The maximum
stiffness axis of a cantilevered airfoil is aligned with the longitudinal direction, as
shown in figure 2.8. Implementing lean effectively offsets the stiffest axis of the blade
relative to the radial direction and moves it toward the axis of minimum stiffness
in the tangential direction. This offset can become important when considering the
radial loading that is applied during an incursion of the blade tip into the structural
casing. As a result of the misalignment, the blade’s ability to resist motion will be
reduced in the radial direction and increased in the tangential direction.
Figure 2.7: Lean Definition
Lean is described by the parameter ν, which is the angle of the mean surface
relative to radial, in the direction of rotation. Lean is constant for all x and all
y, resulting in a linear relationship of the mean surface relative to y, as given in
20
Figure 2.8: Effect of Lean on Radial Airfoil Stiffness
equation 2.5. In practice, lean values for rotating airfoils are rather small, not getting
much above 10 deg.
zlean (y) = tan(ν)y
2.3.3
(2.5)
Bow
The next geometric feature to consider is bow, which is a special type of lean.
Bow refers to the offset applied to a mid-span (40% span) location of the blade in
the tangential direction. Furthermore, the offset is eliminated at the root and tip
locations producing a bow-like shape to the airfoil in the radial-tangential (xy) plane.
Figure 2.9 shows what bow looks like when applied to a simple flat plate. Bow is not
a true design parameter per se, rather it is an extention to a commonly used practice
of applying lean only on the lower radial portion of the airfoil. This geometric feature
21
was isolated from lean to better facilitate the incorporation of this important design
aspect.
Figure 2.9: Bow Definition
Bow is being considered in the development of this model based on its ability to
effect blade stiffness. Under tip-rub loading, the stiffness characteristics generated by
the non-linear bow shape are similar to those generated by lean, but do differ in two
distinct ways. First of all, the impact on stiffness is isolated to the radially inward
portion of the airfoil, in close proximity of the fixity to ground. At this location, the
strain energy due to bending is the highest and subsequently becomes more sensitive
to stiffness changes. Secondly, unlike pure lean where slopes are generally low, the
slope (or dihedral) of the bow at the root section can be relatively high, further
accentuating the impact on the blade’s ability to resist motion due to tip loading.
zbow (y) = c1 y 5 + c2 y 4 + c3 y 3 + c4 y 2 + c5 y + c6
22
(2.6)
Including the non-linear bow feature into the model is a bit more complicated
than the previous linear features. To simplify the process as much as possible a
polynomial function is used to represent the profile of the bow relative to the y-axis.
As shown in equation 2.6, a 5th order polynomial is used to provide enough degreesof-freedom to satisfy the six desired shape conditions shown in equation 2.7. The
first three conditions guarantee that the prescribed offset of 0, b, and 0 are obtained
at the root, 40% span and tip locations, respectively. The parameter b is described
as the maximum offset value in inches. The maximum bow location is set at 40%
based on observing common designs and to assure a suitable bow shape since the 5th
order polynomial becomes ill-shaped for apex locations above 40%. The last three
conditions given in equation 2.7 guarantee that the bow profile has zero slope at the
root, tip and maximum offset locations. Applying the six conditions to equation 2.6
yields the system of equations provided in equation 2.8. These equations are left in
matrix form and are setup to solve for the six polynomial coefficients c1 , c2 , ...c6 .
zbow (0) = 0
zbow (.4s) = −b
zbow (s) = 0
dzbow
|y=0 = 0
dy
dzbow
|y=.4s = 0
dy
dzbow
|y=s = 0
dy
23
(2.7)
−1 
 
c1
(.4s)5 (.4s)4 (.4s)3 (.4s)2
 c2   (s)5

(s)4
(s)3
(s)2 

 
=
 c3   5(.4s)4 4(.4s)3 3(.4s)2 2(.4s)  
5(s)4
4(s)3
3(s)2
2(s)
c4


−b
0 

0 
0
(2.8)
c5 = c6 = 0
2.3.4
Camber
The final geometric feature to include in the mean surface formulation is camber.
Camber, as shown in figures 2.10 and 2.11, refers to the slope the leading edge and
the trailing edge make relative to the axial direction. Camber is used heavily in
the aerodynamic design of airfoils as a means to control the direction of the fluid
flow at the ingress and egress of a given blade row. More often than not, designers
fine tune the amount of camber applied to the leading edge independently from the
trailing edge, resulting in different ingress and egress angles. However, to simplify
the inclusion of camber into the model it is assumed that these angles are equal.
In doing so, the number of required parameters is kept to minimum and a simple
quadratic function can be used to describe the camber. It is believed that although
these assumptions do not satisfy the aerodynamic behavior of the blade, they do
adequately capture the mechanical behavior of the blade.
Like the previous three geometric features discussed, camber can also have a large
influence on the stiffness characteristics of an airfoil. Unlike the previous three, however, camber does not rely on redirecting the principal stiffness directions to do so.
Instead, camber increases the area moment of inertia and therefore increases the airfoil’s resistance to motion, particularly in the tangential direction. In practice, camber
is typically greatest at the root and diminishes as radius increases. As a result, much
24
Figure 2.10: Camber Definition
like bow, the stiffness altering effects of camber are concentrated at the airfoil root
where they can have the largest impact on reducing the blade’s bending flexibility.
Figure 2.11: Detailed Schematic of Camber
For inclusion in the model, camber is allowed to vary linearly along the span of
the blade. Therefore two parameters, ζroot and ζtip , are required, which specify the
camber at the blade root and tip, respectively. At each radial location, the camber
25
offsets the mean surface in the z-direction by imposing a quadratic profile relative to
the x-axis. This profile is defined to be zero at the two edges of the blade and equal
to the c/2 tan(ζ) at mid-chord. As depicted in figure 2.11, the parameter ζ is defined
as the angle between the axial direction and a line drawn between the edge of the
mean surface and the apex located at mid-chord. This definition leads to equation 2.9
for the mean surface when only camber is applied. The entire equation for the mean
surface is obtained by incorporating equations 2.4, 2.5, 2.6, and 2.9 into equation 2.1,
resulting in equation 2.10
zcamber = tan
ζtip − ζroot
y + ζroot
s
c 2 2
− x
2 c
2
X
βtip − βroot
y + βroot x + tan(ν)y +
S = z(x, y) = tan
c(6−p) y p
s
p=5
ζtip − ζroot
c 2 2
y + ζroot
− x
+ tan
s
2 c
(2.9)
2.4
(2.10)
Domain Constraints
The previous section focused on generating a mathematical representation of the
airfoil mean surface, which is defined for all x and all y. In order to make use of
this surface to build an airfoil model, domain boundaries need to be established for
x and y. A graphical representation of this concept is given in figure 2.12. Several
new geometrical features will be discussed in this section that will provide the means
for determining the domain constraints, they include: a) span, b) chord, c) inner flow
path angle, and d) outer flow path angle.
26
Figure 2.12: Airfoil Boundary
Much like the process used to generate the mean surface equation, the boundary
constraints will also be built up using superposition. Equations 2.11 and 2.12 represent the constraints on x and y, respectively. There are three geometric features that
define the x-domain, sweep, twist and camber. In § 2.3.1 and § 2.3.4 it was explained
how twist and camber are used to define the mean surface. In this section it will
be shown these parameters are also needed to define the x-domain to assure volume
compatibility. For the y-domain, there are three additional geometric features, span,
inner flow path angle and outer flow path angle that define the airfoil extent.
− (xsweep1 + xsweep2 + xtwist + xcamber ) ≤ x(y) ≤ − (xsweep1 + xsweep2 − xtwist − xcamber )
(2.11)
yφ ≤ y(x) ≤ s − yθ
27
(2.12)
2.4.1
Span and Chord
The first two features to discuss are arguably the most fundamental, span (s) and
chord (c), which provide the overall size of the airfoil and are essential to determining
the x and y domains. Up to this point, span and chord have been referenced several
times, but no formal definition of these parameters has been provided. Figure 2.13
shows what span and chord are for a simple flat plate. Essentially, span refers to
the height, or length of the airfoil and chord refers to the width. When considering
more realistic blade geometries, the definition of these two parameters change slightly.
Span is still the length of the blade, but since the length of most airfoils vary relative
to the axial direction, span is defined as the radial distance between the blade root
and blade tip at the leading edge location only. The leading edge has been chosen for
simplicity since it is typically the longest part of the blade in axial-flow compressive
rotor stages. Chord is defined as the tangential distance between the leading edge
and the trailing edge. In this modeling approach it has been assumed that chord is
constant along the entire airfoil span. In practice, this assumption holds up quite
well, although allowing the chord to taper as a function of span would be a relatively
easy addition to this model. Figure 2.14 shows a more appropriate representation of
span and chord for realistic type airfoils. It is common practice to specify either the
span or chord dimension and an aspect ratio. Aspect ratio is defined as the ratio of
span to chord as shown in equation 2.13.
AR =
28
s
c
(2.13)
Figure 2.13: Chord and Span Definition
Figure 2.14: Schematic of a Bladed Disk Rotor
29
2.4.2
Sweep
The next geometric feature to include when discussing domain constraints is
sweep. Sweep is described as the tilting of the airfoil in the axial direction. Forward sweep, in which the tip of the airfoil is forward of the root, is most common in
today’s blade designs. There are two types of sweep explained in this subsection, one
linear and one based on an exponential function. Including both types in the model
formulation allow for a much wider range of geometries that can be simulated with
this general approach. Figures 2.15 and 2.16 show how the two types of sweep look
when applied on a simple flat plate. By itself, moderate levels of sweep are not expected to have a significant impact on the tangential or radial stiffness characteristics.
However, when included in conjunction with other design parameters such as twist,
sweep can become an important contributor in determining the blade flexibility.
Figure 2.15: Sweep (linear) Definition
30
Figure 2.16: Sweep (exponential) Definition
The linear sweep feature is very common in past designs and continues to be used
today. The parameter λ1 is used to define the linear sweep feature and refers to the
angle formed between the radial direction and the airfoil edge in the axial-radial (xy)
plane. The formula describing the sweep is given in equation 2.14. The linear sweep
is applied to the entire span of the airfoil, providing an axial offset at the blade tip
equal to s tan(λ1 ). The sweep is applied uniformly across the axial extent of the blade
due to the assumption that chord remains fixed along the entire length of the airfoil.
xsweep1 = tan(λ1 )y
(2.14)
The exponential form of sweep is applied in a similar fashion. The key distinguishing factor for the second form of sweep is that the axial offset varies exponentially
with blade height. The purpose of including this style of sweep is to allow the model
to capture a relatively new geometric feature being used today in which sweep is applied more heavily near the tip of the blade than it is at the root. Equation 2.15 gives
31
the relationship between the exponential sweep and axial blade boundaries. For the
exponential sweep, the defining parameter is λ2 and is a measure of the axial offset
of the blade tip in units of length. Again, the sweep is applied uniformly across the
blade chord to be compatible with the constant chord assumption.
xsweep2 =
2.4.3
es
λ2
(ey − 1)
−1
(2.15)
Flow Path Angles
The final geometric feature needed to fully define the domain constraints are the
inner and outer flow path angles, φ and θ, respectively. In compressive stages of axialflow turbomachines, which can include fan, booster (or low pressure compressor), and
high-pressure compressor stages, the flow path converges as the fluid is pressurized.
To accommodate the converging flow paths, the radial height of each blade must
become a function of axial position. To include this effect in the model, both the root
and the tip of the airfoil are allowed to vary linearly across the axial extent of the
blade. As seen in figure 2.17, φ defines the angle by which the blade root converges
and θ defines the converging angle of the blade tip. These two parameters, along with
span, control the radial extent of the blade model. Equations 2.16 and 2.17 provide
the relationship between the blade height and axial position.
c
tan(φ)
yφ = x +
2
(2.16)
c
yθ = s − x +
tan(θ)
2
(2.17)
32
Figure 2.17: Inner and Outer Flow Path Angle Definition
The two flow path parameters are being included in the model because they are
widely used in airfoil design and become an easy add-on to the model. In addition,
they can impact the stiffness characteristics of the blade by effectively shortening the
blade over portions of the chord. Furthermore, when considering dynamic loading on
the blade, for instance during a rub event, the modal characteristics of the blade can
be significantly affected by the tip and root convergences.
The final piece to describing the airfoil boundary constraints is to include twist
and camber. These two geometric features are unique in that they affect the mean
surface shape and play a role in defining the axial airfoil boundaries. As can be seen
in figure 2.5, the axial extent of the blade decreases with increasing span. This is a
result of forcing the blade volume to remain constant while the twist is applied. In
other words, the width of the blade is not allowed to change so the axial extent of the
blade must shrink as the twist angle increases. This is mathematically represented in
equation 2.18.
33
xtwist
c
= cos
2
βtip − βroot
y + βroot
s
(2.18)
A similar scenario occurs when camber is applied. As the amount of camber is
increased, the axial extent of the blade must reduce if volume is to remain constant.
A major difference is that when twist is applied, the axial extent decreases but the
chord remains constant because the blade is now just facing a different direction. With
camber, the reduction in axial extent is caused by a reduction in chord. Unfortunately,
accounting for the axial contraction due to camber is not as straight forward as it was
for twist. The only way to assure a consistent volume is to keep the arc length of the
mean surface equal to the chord length. The equation for calculating the arc length
is given in equation 2.20, while the relationship between axial reduction and camber
is provided in equation 2.19. There is not a closed-form solution for the variable χ,
therefore it will need to be determined iteratively.
xcamber = χ −
Z
χ
c=
s
1+
−χ
∂zcamber
∂x
2
Z
dx = 2
0
χ
s
c
2
16
1 + 2 tan2
c
(2.19)
ζtip − ζroot
y + ζroot x2 dx
s
(2.20)
The airfoil boundary can now be described in full by including equations 2.14,
2.15, and 2.18 into equation 2.11 and by including equations 2.16 and 2.17 into equation 2.12. Equation 2.21 provides the axial limits as a function of y while equation 2.22
provides the radial limits as a function of x.
34
βtip − βroot
c
y + βroot + χ −
s
2
(2.21)
λ2
c
βtip − βroot
c
y
x(y) ≤ − s
(e − 1) + tan(λ1 )y − cos
y + βroot − χ −
e −1
2
s
2
λ2
c
x(y) ≥ − s
(ey − 1) + tan(λ1 )y + cos
e −1
2
2.5
x+
c
c
tan(φ) ≤ y(x) ≤ s − x +
tan(θ)
2
2
(2.22)
Thickness
In § 2.3 it was explained how to develop an equation to represent the mean surface
of an airfoil. Then in § 2.4 the details of how to bound the mean surface was described.
In this section the final geometric features will be introduced that will add thickness to
the bounded mean surface, resulting in a fully defined 3-dimensional airfoil geometry.
To characterize the thickness everywhere on the blade, four parameters will be used to
allow the thickness to vary along the span-wise direction and the chord-wise direction.
The four parameters include: a) throot , b) thedge , c) T R, and d) T T R and will be
explained in detail in the following two subsections. Once again, the principle of
superposition will be used to include the two independent thickness contributors, as
is portrayed in equation 2.23.
th(x, y) = th(x) + th(y)
2.5.1
(2.23)
Span-wise Variation
In every airfoil design, the thickness varies along the length of the blade, with the
thickest section positioned at the root and the thinnest section positioned at the tip.
35
This tapering of thickness is an important design feature in controlling the static and
dynamic stresses that are generated during operation. The thickness profile is also
very influential in determining the inertia and stiffness characteristics of the blade,
which is why it is included in the model. The thicker an airfoil is, especially near
the root, the stiffer the airfoil will be in bending. Likewise, the thicker an airfoil
is, especially near the tip, the larger the inertia of the airfoil will be. Both of these
effects will change the blade’s resistance to motion under tip loading and need to be
considered in the model.
Figure 2.18: Tip and Root Thickness Definition
As illustrated in figure 2.18, the airfoil thickness is allowed to vary linearly along
the length of the blade. The span-wise variation is described by two parameters,
throot and T T R, which describe the maximum thickness at the root of the blade and
the contraction at the tip, respectively. In this model, as will be seen in § 2.5.2,
the maximum root thickness occurs at mid-chord. The contraction parameter, as
36
provided in equation 2.24, is defined as the ratio of tip thickness to root thickness.
Equation 2.25 then describes the linear thickness variation as a function of y, going
from throot at y = 0 to thtip at y = s.
thtip
throot
(2.24)
throot
(T T R − 1)y
s
(2.25)
TTR =
th(y) =
2.5.2
Chordal Variation
The last thickness feature to consider is the thickness variation across the width
of the airfoil. Much like the thickness taper along the length of the blade, designers
also include taper from the the maximum thickness location (somewhere near midchord) out towards to the leading and trailing edges. Unlike the span-wise variation,
though, the chord-wise variation can not be approximated well by a linear distribution. In practice, designers carefully craft the chordal thickness profile to achieve
maximum efficiency, resulting in very elaborate profiles that are difficult to describe
mathematically. To overcome this difficulty, several assumptions are made and include: a) leading and trailing edges have equal thickness, b) the maximum thickness
always occurs at mid-chord, and c) the thickness varies with x2 . A view of how the
chordal thickness variation looks when applied to a flat plate is given in figure 2.19.
The figure reveals that the thickness profile is applied only on the suction surface (SS)
and not the pressure surface (PS). This is to better represent true airfoil geometry.
This asymmetric feature will be explain in more detail in § 2.6.
37
Figure 2.19: Minimum and Maximum Thickness Definition
There is one additional parameter, T R, required to describe the chord-wise thickness variation. This parameter is defined as the ratio of maximum (or mid-chord)
thickness to minimum (or edge) thickness. This ratio, given in equation 2.26 is valid
for all y, but is specified at the root section for convenience. In actuality, the leading
and trailing edges of airfoils can be quite thin and rounded, However, to facilitate
discretization of the model, a finite thickness (i.e. T R > 0) needs to be specified for
the edges.
TR =
thedge
throot
(2.26)
The thickness distribution as a function of x is given in equation 2.27. This equation is based on the three assumptions previously listed. The variation is quadratic
with a thickness of throot occurring at the mid-chord (x = 0) and a thickness of
thedge occurring at the two edges (x = ±c/2). Summing up the chord-wise and spanwise thickness variations, as shown in equation 2.28, gives a complete description of
38
the airfoil’s thickness. This equation is valid for the x and y domains provided in
equations 2.21 and 2.22.
th(x) =
th(x, y) =
2.6
4
2
(T R − 1)x + 1 throot
c2
4
throot
2
(T T R − 1)y
(T
R
−
1)x
+
1
th
+
root
c2
s
(2.27)
(2.28)
Model Discretization
Thus far, chapter 2.1 has been focused on developing a series of analytical equations to define the geometry of a parameterized airfoil. The focus now turns toward
converting the analytical representation into a discretized form to make numerical
simulation such as FEA possible. Discretization is a process, like meshing, by which
objects are broken up into small, discrete pieces. In a typical modeling process, the
airfoil geometry definition would come from a solid model. That solid model would
then be converted into a FE model via a meshing routine. This process is effective
and grants the designers a near limitless design space in which to work. However, the
process is not always efficient. For complex geometries, such as those used in turbomachinery blades, the meshing process can be tedious and costly. Since the proposed
modeling technique replaces the solid modeling phase with a purely analytical account
of the geometry, a unique opportunity exists to eliminate the meshing step altogether
when converting to a discretized model. As a result, the approach outlined in this
section, along with the previously outlined approach for geometry development, can
be used to instantly generate parametrically defined FE airfoil models.
39
Figure 2.20: Discretized Airfoil Mean Surface
The first step in converting the analytical geometry to a FE model is to discretize
the bounded mean surface developed in § 2.3 and in § 2.4. As seen in figure 2.20, a
grid of points that lie on the mean surface are chosen within the boundaries defined
in equations 2.21 and 2.22. There is an identical number of points in each row, and
all points in given row are equally spaced. Furthermore, there are an equal number
of rows in each column, and all rows are evenly spaced across the length of the blade.
This provides a uniform grid from which to build the model. Row indices are indicated
by l, while the index m denotes the column. In the figure, lmax and mmax are used
for the maximum number of rows and columns, respectively.
The next step is to determine the surface normal direction at each grid point on
the mean surface. The fact that the mean surface is analytically defined, means
the normal field can be determine analytically as well, for all x and y. This is
accomplished first by converting the mean surface into the scalar function, G, as
40
shown in equation 2.29. Having the mean surface represented in this fashion facilitates
the use of the gradient function, as given in equation 2.30. Finally, the normal vector
→
−
field, N , is calculated using equation 2.31.
G = S(x, y) − z
2
X
βtip − βroot
= tan
y + βroot x + tan(ν)y +
c(6−p) y p
s
p=5
ζtip − ζroot
c 2 2
+ tan
y + ζroot
− x −z
s
2 c
∇G =
(2.29)
∂Gb ∂G b ∂G b
i+
j+
k
∂x
∂y
∂z
(2.30)
→
−
∇G
N =
k∇Gk
(2.31)
The normal vector at each grid point dictates the direction along which nodes will
be placed. The next step is to determine the distance, t, between each node and the
mean surface. Before that can be accomplished, the number of nodes through the
thickness needs to be decided. Theoretically, this number can be anything greater
than unity. Practically, however, this number needs to be as small as possible to help
with element aspect ratio, but large enough to satisfy the accuracy requirements of
the FE analysis. Determining the required number is case dependent and beyond the
scope of this work, but a good rule of thumb is to include at least three nodes through
the thickness to capture stress gradients and proper bending characteristics.
Each node should be equally distributed through the thickness, and lie at a distance from the mean surface calculated from equation 2.32. This equation uses the
41
thickness equation 2.28 as input. The index n is used to define node plane, which
will range from 1 to the number of nodes through the thickness. Equation 2.32 is
setup to provide the asymmetric cross-section shown in figure 2.21. The asymmetric
shape maintains the required thickness distribution, but provides a more realistic airfoil shape by clearly defining a suction (convex) surface and a pressure surface. The
pressure surface is forced to lie at a distance equal to half the minimum thickness for
each grid row, this assures that the mean surface will become the mid-surface at the
two airfoil edges.
min(thxyl )
min(thxyl )
≤ tl,m,n ≤ thl,m −
2
2
(2.32)
Figure 2.21: Asymmetric Cross-Section Feature
The next step is to create all the nodes. The process is defined by equation 2.33,
and illustrated in figure 2.22. This equation solves for the Cartesian coordinates of
each node, using the index p to denote node number. The y coordinate includes an
offset of rroot to convert to engine coordinates, as shown in figure 2.14. Until this
point, it was assumed that airfoil root lied on the engine axis. This assumption is
42
valid for model construction, but becomes invalid during subsequent FE analyses that
include rotor spin.
Figure 2.22: Node Projection
D
E
hx, y, zip = (xl,m + tl,m,n · Nbi ) , yl,m + tl,m,n · Nbj + rroot , zl,m + tl,m,n · Nbk
(2.33)
The final step to generating a FE airfoil model is meshing. All element vertices
(nodal coordinates) have been generated for solid brick elements. Since the conversion
of these nodal coordinates into elements is dependent on the FEA software being used,
a description of the element creation process will be omitted. It is recommended that
readers reference their FEA software manual to understand the code specific element
formulation and nodal indexing for 8-noded hexahedral elements.
43
2.7
Application to Common Geometries
Thus far, Chapter 2 has dealt with developing a method to quickly build FE
models that simulate realistic airfoils used in compressive turbomachinery rotors. The
remainder of the chapter will be focused on applying the modeling technique to two
vastly different airfoil designs and comparing how well they capture the geometric and
modal characteristics. The geometric comparisons are strictly qualitative with visual
inspections being the primary source. The modal comparisons, however, provide a
good quantitative measure of how well the simulated geometry captures the dynamic
properties of the actual geometry. Modal comparisons are based on inspections of the
modeshapes and frequencies.
Parameter
rroot
AR
s
θ
φ
throot
TTR
TR
βroot
βtip
ν
b
λ
ζroot
ζtip
HPC Geometry
Fan Geometry
Units Flat Plate Simulated Flat Plate Simulated
in
7.8
7.8
8.3
8.3
1.5
1.5
2.1
2.1
1.5
1.5
18.2
18.2
in
deg
0.0
6.0
0.0
9.0
0.0
0.0
0.0
23.3
deg
in
0.1
0.1
0.6
0.6
1.0
0.6
1.0
0.3
1.0
0.1
1.0
0.2
deg
0.0
33.0
0.0
-10.0
deg
0.0
49.0
0.0
62.0
0.0
6.0
0.0
11.0
deg
in
0.0
0.0
0.0
0.0
deg
0.0
2.0
0.0
2.4
deg
0.0
12.0
0.0
25.0
deg
0.0
9.0
0.0
5.0
Table 2.1: Geometry Parameters
44
The two blade designs chosen for this study truly span the continuum of current
designs. The first example is a high-pressure compressor blade from a power generation machine. The airfoil is relatively small, coming from a latter stage of the
compressor. The design is quite dated and only consists of simple 2-dimensional features. The second example is a fan blade from a high bypass commercial engine.
This airfoil has a more modern design that takes advantage of complex 3-dimensional
geometric features. Table 2.1 provides a list of all the model parameters used to simulate the example designs. Provided in the table as well, are the equivalent parameter
values that one might use if a simple flat-plate model is desired.
2.7.1
High Pressure Compressor
The first comparisons will be for the HPC airfoil. Figures 2.23, 2.24, and 2.25
provide views in the tangential direction, radially inward direction from the tip, and
axial direction, respectively. These views were chosen because they tend to highlight
the various geometric features included in the design. In each figure, three different
geometries are shown so that the simulated geometry and the equivalent flat plate
can be compared to the actual geometry. Obviously the flat plate is not expected to
capture the geometry well, but it is interesting to see how much they differ, especially
since flat plates continue to be a strong area of research and are frequently used to
simulate airfoil static and dynamic properties. From looking at the figures of the
HPC airfoil, it can be seen how well the simulated model approximates the actual
geometry. Aside from the root fillet, there are not many visible differences between
the two.
45
Figure 2.23: Comparison of HPC Geometries, Tangential View, a) Flat Plate, b)
Simulated, c) Actual
Figure 2.24: Comparison of HPC Geometries, Tip View, a) Flat Plate, b) Simulated,
c) Actual
46
Figure 2.25: Comparison of HPC Geometries, Axial View, a) Flat Plate, b) Simulated,
c) Actual
An excellent way to compare the structural characteristics of one geometry to
another is to compare their modal properties. This quantitative evaluation was completed and the results are given in table 2.2 and figure 2.26 for the HPC airfoil. Again,
the comparisons are being made between the simulated geometry, an equivalent flat
plate and the actual geometry. A modal analysis was run using the ANSYS software in which the airfoils were clamped in all directions at the root. In addition,
bench conditions were replicated by applying room temperature and no rotation to
the models.
The study includes the first 10 modes of the airfoils, chosen because they span
the typical frequency range that these airfoils would likely encounter in operation.
Table 2.2 shows the percent frequency difference values for the flat plate and simulated
airfoil in columns two and three respectively. The simulated blade does an excellent
job of predicting the natural frequencies of the airfoil with a maximum error of 6%
47
for the first 10 modes. The flat plate, in contrast, is off in frequency by as much as
41% and misses modes 5 and 10 altogether.
The modeshapes for all three blades are given in figure 2.26. The modeshapes for
the flat plate are given in the first row, the simulated airfoil in the second row, and
the actual airfoil modeshapes are given in the last row. The modeshapes, determined
numerically, are given in bi-color (scarlet and gray) fashion to highlight the nodal
lines. The flat plate results provide a nice way to characterize the airfoil modes by
commonly known plate modes. Aside from modes 5 and 10, the flat plate modes are
easily recognized in the simulated and actual airfoil modes. The simulated blade does
a near flawless job in capturing the modeshapes of the true airfoil and validates the
use of the simulated blade for further structural analyses.
Mode
1
2
3
4
5
6
7
8
9
10
% Error
Flat Plate Simulated
-13.7
4.7
-5.8
-4.9
19.2
2.9
8.4
0.1
n/a
-3.0
26.4
-0.3
31.1
-1.1
41.6
-1.7
31.8
1.8
n/a
-6.2
Table 2.2: HPC Natural Frequency Comparisons
48
Figure 2.26: Comparison of HPC Modeshapes, a) Flat Plate, b) Simulated, c) Actual
2.7.2
Fan
This chapter concludes with a comparison of how well the modeling approach
proposed in this dissertation work can simulate a modern fan blade design. As mentioned earlier, this airfoil is taken from a commercial jet engine and takes advantage
of various 3-dimensional design features. Figures 2.27, 2.28, and 2.29 show an isometric view, tip view, and an axial view, respectively. Again, each view was chosen
to highlight the key geometric features used to design this airfoil. It can be seen by
looking at the figures just how well the simulated geometry mimics the actual.
Like with the HPC blade, the first 10 modes of the fan blade were also evaluated
and compared. Table 2.3 gives the natural frequency errors for the simulated geometry
and an equivalent flat plate. In this case, the simulated fan blade does not do as well
as the simulated HPC blade did, but is still within 18% of actual, impressive given the
tremendous complexity of this airfoil and the minimal effort put toward optimizing
49
Figure 2.27: Comparison of Fan Geometries, Isometric View, a) Flat Plate, b) Simulated, c) Actual
Figure 2.28: Comparison of Fan Geometries, Tip View, a) Flat Plate, b) Simulated,
c) Actual
50
Figure 2.29: Comparison of Fan Geometries, Axial View, a) Flat Plate, b) Simulated,
c) Actual
the model parameter values. The flat plate, on the other hand, does not do well at
all, missing three modes and off by as much as 160%.
Figure 2.30 gives the modeshapes. Unlike with the HPC blade, the fan blade
modeshapes are not easily mapped back to the plate modes. The HPC geometry was
relatively simple and consisted of mostly 2-dimensional features that did not distort
the plate modes. The fan blade, however, includes strong 3-dimensional features that
blend and couple the various plate modes to produce very complex modal deflections.
Despite this complexity, the simulated blade does an exceptional job capturing the
true deflection patterns.
It is impressive how well the modeling approach developed in this dissertation work
can approximate the actual geometry and modal characteristics. In addition, building
the simulated geometry took a fraction of the time it would take to build a simple
flat plate model. There are many uses for a technique like this one, that can instantly
51
Mode
1
2
3
4
5
6
7
8
9
10
% Error
Flat Plate Simulated
0.5
-0.3
109.2
3.1
141.2
11.8
-46.1
-4.5
120.4
-2.6
147.7
5.9
159.5
18.6
n/a
10.5
n/a
8.9
n/a
5.2
Table 2.3: Fan Natural Frequency Comparisons
Figure 2.30: Comparison of Fan Modeshapes, a) Flat Plate, b) Simulated, c) Actual
52
generate new parameterized airfoil designs. Having this capability facilitates many
types of analyses, such as statistical blade design studies, that are near impossible
with traditional modeling approaches.
53
Chapter 3: Pulse Stiffness Determination
3.1
Introduction
The most important element of an airfoil-to-case rub event are the contact forces
generated. The magnitude, direction and duration of these forces dictate how the
rotating and static structures will be affected during a given rub event. Although
the vast majority of rubs are harmless and result in nuisance maintenance actions,
the potential for bigger problems does exists. Engineers are aware of these potential
issues and work hard to avoid them. In doing so, they make many conservative
assumptions related to the physics of the rub events. Atop the list of assumptions
lies those that are related to rub forces. Engineers have had little success in making
robust predictions of the rub loads due to the extreme complexity of the high-speed
rub mechanics. As a result, there is a strong need in industry to improve the rub load
predictive capabilities.
One of the key features controlling the rub forces is the flexibility of the airfoil.
This flexibility can have dramatic effects on the rub physics and is one of the aspects
of blade rubs that makes it unique. Most theory surrounding high-speed sliding
contacts and machining relies on at least one (often both) of the contacting mediums
to be rigid. This rigid condition is rarely met in actual blade rub situations where
54
blades are thin plate-like structures and casing structures often represent thin-walled
cylinders. Furthermore, engineers rely heavily on soft abradable materials on the
casing to reduce the rub loads, making them much softer than the metal casing itself.
The importance of blade and case flexibilities is well known in industry and various
techniques exists to allow engineers to approximate them. A common approach, used
for its simplicity, is to model the contact between the blade and case with one or more
springs. The stiffness of these springs can then be adjusted to satisfy a particular
modeling need. These springs account for a variety of flexibilities and determining
the appropriate spring stiffness is not well established. The focus of this chapter is to
explore a method by which one can determine airfoil equivalent contact stiffness.
3.2
Tip Stiffness
A relationship between the force applied to an airfoil and the ensuing defection of
the airfoil needs to be generated in order to determine airfoil equivalent stiffness. Since
this work is focused on blade rubs, it makes sense to concentrate on understanding
how the blade deflects under rub-like (pulse) loads. The purpose of this section is
to describe what tip stiffness is and how it is calculated for static and pulse loading
conditions.
Figure 3.1 gives a simplified schematic of the forces that will be considered in this
chapter. As a result of the blade-to-case contact, forces will be imposed on the tip
of the blades that are in contact and on the casing structure. The digram shows a
→
−
resultant force F being applied in the radial-tangential plane. The resultant force
has a component in the radially inward direction, acting normal to the casing, and
a component in the tangential direction, acting to oppose rotation. The radial, or
55
normal, force is generated due to the blade pushing into the casing and results in a
compressive load on the blade tip. The tangential, or lateral, force is generated due
to the rub friction and plowing of casing material. It is assumed that no forces are
generated in the axial direction. In reality there are axial forces generated during
rub events due to the geometric features of the blade. These forces are often ignored
based on the assumption that they are small compared to the tangential and radial
forces, and will likewise be ignored in this work.
Figure 3.1: Schematic of Tip Loading
3.2.1
Static Loading
In this subsection static loading conditions will be considered. Static loads are
the simplest type of loads and will help in setting the foundation for what is to come.
56
Figure 3.2 shows how the static loads are implemented. Both loads are distributed
evenly over the tip of the airfoil and tracked by the total integrated load as shown in
equation 3.1, where P is the total number of nodes on the tip. As a result only two
parameters, Fθ and Fr , are required to fully describe the loading condition.
Figure 3.2: Static Tip Loading
F =
P
X
Fp
(3.1)
p=1
Figure 3.2 also defines how the radial and tangential deflections are tracked, denoted by ur and uθ , respectively. For a simple flat plate, all tip nodes will move
uniformly. For actual airfoils, however, that is not always the case and therefore
nodal displacements are averaged across the blade tip and used to determine tip stiffnesses. Equation 3.2 provides the means for calculating the tip stiffness in the radial
direction and is defined simply as the ratio of radial force magnitude over the average
57
tip deflection in the radial direction. A similar relationship exists for the tip stiffness
in the tangential direction and is given in equation 3.3. Since cylindrical coordinates
are being used, the tangential deflection becomes the product of the nodal radius and
the change in angular position (uθ = rdθ).
Fr
ur
(3.2)
Fθ
Fθ
=
uθ
rdθ
(3.3)
Kr =
Kθ =
At this point, one could make an initial estimate of the stiffness characteristics
of an airfoil under tip-rub loading. Assumptions can be made about the relative
magnitude of the radial and tangential force components, using either a friction or
plowing model, reducing the tip loading down to a single force magnitude parameter.
A force versus deflection curve (stiffness curve) can then be generated by varying
the load magnitude through a series of elastic, large deflection static analyses and
plotting the applied force versus the resulting radial and tangential displacements.
3.2.2
Pulse Loading
The previous subsection defined tip stiffness and how it is calculated under simple static loading. Furthermore, a description was given as to how one could take
the approach and generate a the force versus deflection curve needed in subsequent
analyses. Although simple and easy to implement, this process makes an assumption
that blade dynamics can be ignored when determining blade deflection. Which, as
shown by the experimental results of Podova et al. [7, 9], is a bad assumption under
certain situations such as impulsive rub loads. In this subsection, that assumption
58
is eliminated by considering pulse load effects using an extension of the static load
method.
Pulse loads refer to forces that are applied over a short amount of time, less than
0.1 s, or 100 ms. Although that seems like a very short time window, it can easily
represent 30 or more revolutions in some turbomachines, certainly long enough to
include an entire rub event. For the compressor rub testing being done at the OSUGTL, a single rub only lasts 0.9 ms. The variable ∆t is used to define the pulse width
in time, as shown in figure 3.3.
Figure 3.3: Pulse Width Definition
For this analysis, several key assumptions are made about the tip forces. First of
all, the radial and tangential forces are allowed to vary with axial position, x. For
simplicity, a uniform profile as given in equation 3.5, is used throughout the work
summarized in this paper. The total force is simply the sum of all nodal loads in
the respective direction. Two other spatial profiles, linear and quadratic, have been
59
worked out and are included in the Appendix. The second assumption is that the
load does not vary across the thickness of the blade. The tip thickness of most fan
and compressor blades is relatively thin, justifying the use of this assumption. The
final assumption is that the pulse load is quadratic with respect to time. The tip
force can be represented as the product of a spatial varying component F (x) and a
time varying component Φ(t), as shown in equation 3.4, where the scalar function
Φ(t) is defined in equation 3.6.
Figure 3.4: Dynamic Tip Loading
F (x, t) = Φ(t)F (x)
60
(3.4)
F (x) =
P
X
Fp
(3.5)
t
1−
t
∆t
(3.6)
p=1
4
Φ(t) =
∆t
With the inclusion of a time dependent force comes the need to move away from
a simple static analysis and solve for the tip deflections through a transient analysis. The FEA code LS-DYNA was chosen to perform the transient analyses because
it includes an explicit based solver that is ideal for short-duration loading events
such as impacts. Unlike implicit solvers, the time-marching explicit solver allows for
stress wave propagation in the airfoil, a feature that can prove to be important in
determining tip deflection behavior under pulse loading.
In addition to the need to switch analysis types, the introduction of time dependent loading also creates a need to redefine tip stiffness. Under static loading,
tip stiffness was defined in § 3.2.1 as the ratio of applied tip load to resulting tip
deflection. Using this same approach for the transient loading case would result in
a time dependent stiffness determined by calculating the force-to-deflection ratio at
each time step. Although feasible, this does generate several undesirable issues. First
of all, time dependent results would be difficult to incorporate into many of the subsequent analyses that the stiffness values are being developed for. Secondly, pulse
loading would create stiffness that varied from infinity to zero over the short time
window, making the result very difficult to utilize and highly reliant on loading conditions. During a pulse, the load ramps up very quickly, much quicker than the airfoil
can respond, resulting in infinite stiffness at the onset of the loading. The load then
returns to zero within milliseconds, driving the stiffness to zero as well. This much
61
variation in such a short time span, provides a stiffness that is of no real value. Finally, having a time dependent stiffness would be overly cumbersome because the tip
stiffness would now become a function of t and ∆t.
To overcome these issues, tip stiffness needs to be redefined. The goal is to allow
the stiffness to be dependent on the pulse width only, providing information about
the sensitivity of the blade stiffness to loading duration. This information can then
be used by engineers to decide what a suitable stiffness assumption is for a given rub
situation. Figures 3.5 and 3.6 provide an illustration of the proposed approach to
developing a ∆t dependent stiffness. The method is analogous to the widely used
shock spectrum approach [5].
Figure 3.5: Series of Tip Deflection Curves
62
The first step is to solve a series of transient analyses where only the width of the
pulse load is adjusted. The range of ∆t’s needed will be discussed later in § 3.3.2, but
for now assume the range is sufficiently large to adequately define the relationship of
tip deflection to pulse width. Figure 3.5 shows a series of four curves that represent
the time dependent tip deflection for four different pulse widths, ∆t1...4 . From each
of these curves only one value is important, the max deflection, denoted by u1...4 in
the figure. Since the interest is in determining stiffness values, care should be taken
to pick the max displacement that occurs only during the time window that the load
is being applied. In the vast majority of the cases, the max displacement will occur
during the load application, but as will be seen later, the peak deflection does not
always align in time with the peak load.
Figure 3.6: Build Up of Pulse Deflection Curve
The next step is to plot the max tip deflections, u1...4 , versus their corresponding
pulse width values, ∆t1...4 , as illustrated in figure 3.6. Although the data in this
illustration is fictitious, it does help to clarify the process, and in fact does contain
63
most of the important features seen in the non-fictitious data. Figure 3.7 shows the
product of this procedure, in the tangential direction, when applied to the small
flat plate introduced in chapter 2. As can be seen, these curves are non-linear and
contain an apex, an interesting feature that will be explored further in the next
section. Finally, the deflection curve can be inverted and scaled by the max applied
force to produce a pulse width dependent stiffness curve, like the given in figure 3.8.
Figure 3.7: Tangential Deflection Curve Versus Pulse Width for a Small Flat Plate
3.3
Implementation
The previous section focused on defining tip stiffness and how to determine a static
and pulse version. The static tip stiffness, denoted as K s moving forward, is not a new
concept and this general approach is widely used across the engineering disciplines.
What is new is incorporating the effect that dynamic pulse loading can have on the
64
Figure 3.8: Tangential Stiffness Curve Versus Pulse Width for a Small Flat Plate
stiffness of a cantilevered airfoil. This dynamic (or pulse) stiffness, denoted as K d ,
differs from traditional dynamic stiffness definitions that are frequency based. In
§ 3.2.2 a shock spectrum based method was proposed to determine the tip stiffness as
a function of pulse width. In this section the consequences of the formulation will be
discussed and consideration will be given into the application of the said approach.
Throughout the section, a stationary flat plate will be used as the working medium
to develop and explain the new ideas. Later on in § 3.4 the ideas will be expanded
to include simulated airfoil geometry and rotational effects.
3.3.1
Pulse Magnification
One of the issues associated with the pulse stiffness method is being able to easily
compare one geometry to another. As with any stiffness, K d is directly proportional
to the applied load. In the static situation this is not a concern since the force
65
versus deflection curves for various geometries can be compared easily on a single
2-dimensional plot. In the case of the pulse stiffness, however, the additional variable
∆t requires either a 3-dimensional view of the results or a series of 2-dimensional plots.
Either way, the ability to quickly compare geometries is impeded by the addition of
pulse width. A related issue is the lack of a well established baseline from which to
compare the pulse stiffness values.
There is a simple and effective solution to both the aforementioned issues, pulse
magnification. Pulse magnification, denoted by the Greek symbol κ, is a term being
introduced to describe the amplification, or attenuation, of the stiffness characteristics
of an airfoil as a result of the pulse loading. As shown in equation 3.7 for the tangential
and radial directions, κ is simply the ratio of the pulse stiffness to the static stiffness.
Taking this ratio has three main purposes: a) it eliminates the magnitude of the input
force from the list of variables, b) it provides an immediate sense of how the pulse
loading affects the stiffness as compared to a static load, and c) it supplies a common,
normalized scale, from which all geometries can be compared.
usθ
Kθd
=
Kθs
udθ
us
Kd
κr = rs = dr
Kr
ur
κθ =
(3.7)
(3.8)
Figure 3.9 provides the κ curve for the example flat plate case introduced in § 3.2.2.
Several interesting features are immediately noticeable from this figure that facilitate
the segmenting of the curve into three distinct regions. Starting from the left, the
first region includes a rapid decrease in the pule magnification. Following the rapid
66
Figure 3.9: κθ Versus Pulse Width for a Small Flat Plate
decrease is a short region in which κ reaches a minimum of about 50%, signifing
that the pulse loading acts to attenuate the stiffness in this region of pulse widths.
The third and final region includes the remainder of the curve, in which κ tends
toward unity as the pulse width increases. In fact, the pulse magnification factor
does converge to unity for a sufficiently large ∆t. This convergence makes intuitive
sense since in the limit as ∆t goes to infinity, the pulse load becomes a purely static
load.
3.3.2
Pulse Width Normalization
The fact that the pulse magnification curve can be segmented into distinct regions
based on the pulse width begs the question as to why? Certainly it makes sense that
for large ∆t’s the curve becomes unity, that reason was just explain. The answer for
the other two regions has to do with airfoil inertia and dynamic response. For really
67
small pulse widths, the blade’s inertia becomes a dominant player in the response of
the blade, impeding its movement. In the limit as ∆t goes to zero, the blade will not
respond to the input, causing the stiffness, and therefore κ, to tend toward infinity.
Moreover, there has to be a point where the input pulse will tend to excite the
natural frequencies of the blade, which is precisely what causes the pulse magnification
to reach a minimum that is much below unity. Since there is a component of loading
in line with the lateral direction of the blade, the first bending mode can easily be
excited by the pulse. Furthermore, the load is being applied at the tip of the blade
where most of the kinetic energy associated with the first bending mode is located,
resulting in an even higher sensitivity of this mode to the input pulse. It is possible
that other modes, especially higher order bending modes, could be excited by the
pulse tip force. These natural frequencies, however, tend to be much higher and
would require extremely short pulse widths.
τ=
∆t
= ∆t · f1
T1
(3.9)
In addition to providing a clean explanation of the behavior of κ, the inherent
tendency of the blade to respond in the first bending (flexural) mode provides a convenient way to normalize ∆t. As given in equation 3.9, a new dimensionless variable τ
can be introduced that is the ratio of the pulse width to the period of the fundamental
bending mode. By introducing this normalization, all attributes that affect natural
frequencies, such as geometry, material properties, rotation, and temperature, can be
normalized out and each blade/loading combination can be compared on a common
scale.
68
Figure 3.10: κθ Versus τ for a Small Flat Plate
Figure 3.10 places the example flat plate κ curve on the dimensionless time scale.
Log scale is used so that large τ ’s can be considered. This is important to see when
the pulse magnification value reaches unity, which is around 100. The log scale also
provides a zoomed look at the region surrounding τ = 1, where κ reaches a minimum.
3.4
κ-Model
In § 3.3 it was explained how the concept of pulse stiffness could be fashioned
to support the engineering decisions that need to be made by providing a common
platform from which to compare all designs. One major issue still remains however,
which is that there is a large amount of analysis required to generate the pulse magnification curves. For instance, to generate figure 3.10, a modal analysis was run
using ANSYS to get the fundamental natural frequency, a static loading case was run
using LS-DYNA implicit to determine baseline, and finally 50 full transient analyses
69
were completed using LS-DYNA explicit to determine max tip deflections for various pulse widths. Not to mention all the post-processing steps required to convert
the raw analysis results into the final form. Furthermore, all steps except the modal
analysis would need to be repeated for each load level to determine the pulse stiffness
characteristics as a function of applied load. Even when placed in a full automation
loop, as was done for this work, the analysis is costly. The focus of this section is on
eliminating the need for the expensive analysis by developing a mathematical model
to predict the pulse magnification curve.
lim κ = ∞
τ →0
(3.10)
lim κ = 1
τ →∞
The first step in building the mathematical model is to consider how the boundaries of the function behave. Equation 3.10 provides the bounds that are based on the
previous discussion in § 3.3.1 and in § 3.3.2. The second step is to take into account
the general shape of the pulse magnification curve and choose terms that allow for the
required behavior. For instance, the asymptotic behavior of κ requires a combination
of a log term, an exponential term and an inverse of τ . Through deliberation and
trial and error, it was concluded that the best model for pulse magnification is the
one given in equation 3.11. This equation is able to capture the general shape of the
curve and satisfies the limits given in equation 3.10.
κ(τ ) = µ
ln(τ )
+ ηe−τ + 1
τ
70
(3.11)
3.4.1
Coefficients
There are two coefficients, µ and η, in equation 3.11 that need to be determined
before the model is complete. To understand the sensitivity of the function to each
coefficient, derivatives relative to each one were taken and plotted in figure 3.11. The
coefficient µ is the most influential in determining the shape of the κ curve for τ < 1.
The coefficient η, on the other hand, is responsible for fine tuning the shape around
the inflection point (τ = 1) and providing the gradual transition from the inflection
point to where the pulse magnification becomes unity.
Figure 3.11: Effect of µ and η on κ
ψ=
n
X
(b
κi − κi )2
i=1
71
(3.12)
A best fit approach is used to determine the coefficients µ and η. The objective
function ψ, given in equation 3.12, is defined as the sum of the squared errors. In this
equation, κ is defined as the truth, and is generated from the full transient analysis
method. The estimator is defined as κ
b and is given by equation 3.11. Finally, n is
the total number of data points available from the transient solution method. The
objective function is then minimized, as shown in equation 3.13, to determine the best
possible values for µ and η. The minimization is done by taking the derivatives of
the objective function with respect to each coefficient and setting them equal to zero,
as given in equation 3.14. This yields two equations from which the best fit values
for the coefficients can be determined. The system of equations is provided in matrix
form in equation 3.15. The resultant error from the minimization process can be
defined in several different ways. A common practice is to calculate the Normalized
Root Mean Square Error (NRMSE), as given in equation 3.16.
"
min [ψ] = min
n
X
#
(b
κi − κi )2
(3.13)
i=1
∂ψ
∂ψ
=0=
∂µ
∂η
(3.14)
 P ln(τi ) P
i


P −2τi
−τi
(e
)
(e
)
i
i
τi
µ

P ln(τi ) 2
P ln(τi ) P −τi
η
)
i
i
i (e
τi
τi
P
P −τi !
(e )
i (κi − 1)
P i ln(τi ) = P
i (κi − 1)
i
τi
qP
n
E=
i=1
(b
κi − κi )2 /n
max(κ) − min(κ)
72
(3.15)
(3.16)
3.4.2
Results
So far this section has been focused on developing the κ-model (equation 3.11)
and the tools needed to determine the best coefficients to use. In an ideal situation,
a single set of coefficients could be determined that would allow the model to predict
the pulse stiffness characteristics for any airfoil under any loading. In this situation,
all that would be required to determine the pulse stiffness curve for a given geometry
would be a modal and static analysis. The modal analysis would provide the first
flex frequency to determine τ and the static analysis would provide the scalar to
convert κ into a pulse stiffness, as shown in equation 3.17. For the remainder of this
section, focus will be turned toward the application of the model and determining the
feasibility of a unified set of coefficients.
K d (τ ) = K s · κ(τ )
(3.17)
The first step in determining a common set of coefficients is to understand how
the magnitude of applied load affects the pulse magnification curve. This step was
completed using the small flat plate model introduced in Chapter 2 to simulate a
small HPC blade. The load in each direction was varied in 100 lbf increments from 0
to 500 lbf, resulting in 36 loading combinations. For each loading condition a static
analysis was run to determine the static tip stiffness and 50 transient analyses were
run, each at a different pulse load width. The pulse width varied from 0.1 ms to a
value equal to a τ of 100. These runs, in addition to determining the fundamental
natural frequency, resulted in nearly 1,800 analyses. The outcome of those runs in
the tangential and radial directions are provided in figures 3.12 and 3.13, respectively.
73
Figure 3.12: κθ Curve for a Small Flat Plate Under Various Loadings
Figure 3.13: κr Curve for a Small Flat Plate Under Various Loadings
74
There is minimal variation in κ throughout the range of τ ’s, especially considering
the broad range of load amplitudes applied. There is virtually no spread in the results
to the right of the inflection point (τ > 1). To the left of the inflection point, the
variation steadily increases as τ tends to zero. Even at the far left of the range, the
variation is low enough to warrant a single set of coefficients. The minimization procedure outlined in § 3.4.1 was employed on each curve in both directions to determine
the best set of coefficients. Histograms of µθ and ηθ are given in figures 3.14 and 3.15,
respectively. The range for both coefficients are relatively small, -0.28 to -0.42 for µθ
and -1.25 to -1.31 for ηθ . Normal distributions are fit to the histograms for reference
and the corresponding mean and sigma values are given in the captions.
Figure 3.14: Distribution of µθ for a Small Flat Plate, (µθ = −0.343, s = 0.035)
To further understand the variation in the coefficients and how they are influenced
by the tip loading, color maps of each coefficient were created (figures 3.16 and 3.17).
75
Figure 3.15: Distribution of ηθ for a Small Flat Plate, (η θ = −1.28, s = 0.0118)
The amplitude of the applied radial force is plotted along the vertical axis and the
amplitude of the tangential force is plotted along the horizontal axis. The coefficient
value is plotted on the color scale. Despite the small disparity in coefficient values,
the maps are helpful in determining the sensitivities to tip loading. The µ coefficient
is inversely proportional to the tangential load, while insensitive to radial load. On
the other hand, the η coefficient is proportional to radial load, while fairly insensitive
to tangential load. There is a discontinuity in ηθ at a tangential load value of 400 lbf
and a radial load of 200 lbf. It is believed that this discontinuity is purely a numerical
anomaly brought on by a small scale and round-off error. The discontinuity is singular,
despite the broad appearance given by the linear interpolation scheme.
Table 3.1 gives the final model coefficients for the radial and tangential directions and represents the culmination of all 1800 runs. The best-fit coefficients differ
76
Figure 3.16: Effect of Fr and Fθ on µθ for a Small Flat Plate
Figure 3.17: Effect of Fr and Fθ on ηθ for a Small Flat Plate
77
somewhat between the two directions, with the radial values being about twice as
large as the tangential values. The physical reason for this is not entirely clear, but
mathematically it has to do with the fact that the pulse magnification curves in the
radial direction are flatter through the inflection point and grow to infinity faster to
the left of the inflection point. The more interesting question is, however, why do
the radial and tangential curves follow the same general trend? The reason has to do
with the inherent coupling between the radial and tangential tip deflections and the
relative stiffness of the tip in each direction. The blade is relatively stiff in the radial
direction, so it will move very little in the radial direction due to a pure radial load.
In the tangential direction, however, the blade is not nearly as stiff and will move
much more in that direction under a tangential load of an equal amplitude. Under
combined loading, the tangential load acts initially to deflect the tip in the tangential
direction, effectively reducing the longitudinal (radial) stiffness, allowing the radial
force to produce a higher radial deflection than it would otherwise be able to without
its tangential counterpart. Furthermore, the radial and tangential deflections are geometrically coupled. The majority of the radial deflection is a result of the blade tip
moving in the tangential direction. As the blade bends over in the lateral (tangential)
direction, the resulting tip radius is reduced, assuring consistency in blade length.
Component
µ
η
Tangential (θ) -0.3 -1.3
Radial (r)
-0.6 -2.2
Table 3.1: Best Fit Coefficients for κ
78
Figures 3.18 and 3.19 show the effectiveness of the tuned κ-model in predicting
the κ curves given in figures 3.12 and 3.13. The plots show the predicted curve in
black and the transient simulation results are represented by the red error bars. The
tangential predicted curve does a great job representing the median of the simulation
results. In the radial direction, however, the agreement is not as good and for low τ
values the curve crosses though the low end of the result range. This misalignment
is intentional and provides a better fit through the inflection point, where having an
optimal fit is of primary interest.
Figure 3.18: Predicted κθ with Error Bars for a Small Flat Plate
Figures 3.20 and 3.21 give the distribution of the NRMSE error values for the
tangential and radial directions, respectively. The error represents the difference
between the predicted κ curve and the full transient simulation results. The errors
79
Figure 3.19: Predicted κr with Error Bars for a Small Flat Plate
are low with a max error of about 1.1% in the tangential direction and 5.5% in the
radial direction.
The optimized set of coefficients were determined using the small flat plate geometry. Before they can be deemed sufficient, their ability to predict other geometries
must be verified. Figures 3.22 and 3.23 show a comparison between the small HPC
equivalent flat plate, the large fan equivalent flat plate, and the model prediction.
The combined loading condition of 200 lbf in the radial and 200 lbf in the tangential direction was chosen to base the comparison on. The model’s prediction of the
tangential curve is flawless. Furthermore, the benefit of the selected normalization
becomes clear. The two geometries are radically different in size and modal characteristics, yet due to the normalization, they reduce to identical pulse magnification
curves. In the radial direction, the model does a good job splitting the difference
80
Figure 3.20: Distribution of Eθ for a Small Flat Plate, E θ = 0.00953, s = 0.000693
Figure 3.21: Distribution of Er for a Small Flat Plate, E r = 0.039, s = 0.005
81
between the two geometries for τ < 1. The agreement is great through the inflection
point, where the need for a good match is the largest.
Figure 3.22: Comparison of κθ for a Small and Large Flat Plate
The next model validation step considers the effect of rotation. The range of
rotational speeds seen in turbomachines is quite large, spanning from a few hundred
rpm to 20,000 rpm or so in common applications. For smaller machines, the rotational
speeds can get much higher and is often set by the rotor’s largest airfoil tip radius.
Equation 3.18 gives the relationship between rotational speed, Ω in rpm, to the blade
tip velocity, Vtip in fps and tip radius, rtip in inches. It is a common practice in
turbomachinery design to keep the tip velocities subsonic to avoid various noise,
structural and aerodynamic issues. For this study, it is assumed that the tip velocities
do not exceed 1,300 fps, providing a good upper bound that is compatible with most
turbomachines.
82
Figure 3.23: Comparison of κr for a Small and Large Flat Plate
Ω=
360 Vtip
π rtip
(3.18)
Rotation is an extremely important parameter in airfoil design because it can
have dramatic effects on several key aspects of the design, including, among others:
radial expansion, mean stresses, airfoil stiffness and natural frequencies. Relative
to the pulse magnification model, the airfoil stiffness and resonant frequency effects
are the most important. As the airfoil is undergoing rotation, it is subjected to a
large centrifugal acceleration field that causes the blade to deflect in various ways.
These deflections generate static stresses that tend to increase the stiffness of the
blade, a term often referred to as stress-stiffening. Furthermore, if the deflection due
to rotation is large enough at the blade tip, then a non-linear effect, termed spinsoftening, can act to decrease the stiffness of the airfoil. The net effect however, is
83
an increase in blade stiffness due to rotation, as seen in figure 3.24. This figure gives
the relationship between the static tip stiffness and rotational speed for the small
flat plate example. The stiffness, and fundamental natural frequency as shown in
figure 3.25, increases with the square of the rotational speed (Ω2 ). This relationship
can be exploited to further reduce the analysis burden of understanding how pulse
stiffness varies with rotational speed.
Figure 3.24: Effect of Rotational Speed on K s
As shown in equations 3.19 and 3.20, the pulse stiffness can be transformed into
a function of rotational speed. By knowing how stiffness and frequency vary with
speed, then a single analysis could be performed at zero speed and subsequently
scaled, based on curves similar to those given in figures 3.24 and 3.25, to estimate
the pulse stiffness at any speed. The impact of rotational speed on the static stiffness
and the fundamental natural frequency of an airfoil is highly dependent of geometry,
84
Figure 3.25: Effect of Rotor Speed on Fundamental Airfoil Frequency
which needs to be explored in detail before the concept of speed scaling can be used.
Understanding the geometry dependence is beyond the scope of this work and will
left for future consideration.
τ (Ω) =
∆t
T1 (Ω)
K d (Ω) = K s (Ω) · κ (τ (Ω))
(3.19)
(3.20)
Figures 3.26 and 3.27 show how well the κ-model does at capturing the pulse
magnification for the small flat plate in the tangential and radial directions respectively. The three curves included in each figure correspond to the Ω = 0 rpm case,
the Ω = 16, 192 rpm case and the predicted curve. The model does an excellent job
in the tangential direction, where there is no variation in κ across the extremely large
85
speed range. Again, in the radial direction, the model does a great job through and
to the right of the inflection point. For τ < 1, the variation between the zero and max
speed cases increases and the model suffers a bit. However, the match is still suitable,
especially considering the 16,192 rpm speed range that the model is representing.
Figure 3.26: Effect of Rotor Spin on κθ for a Small Flat Plate
The final validation for the κ-model will be focused on how well it can predict the
pulse magnification of a realistic airfoil geometry. For this study the simulated HPC
blade introduced in Chapter 2 will be compared against the small flat plate and the
model. The results of this study are given for the tangential and radial directions,
respectively, in figures 3.28 and 3.29. The range of τ for the simulated geometry is
much less than for the flat plate and was set based on the measurements completed
in the CSPF and to reduce the number of transient analyses required. More on the
comparison to experimental data and the down-select in the range of τ will be given
86
Figure 3.27: Effect of Rotor Spin on κr for a Small Flat Plate
in Chapter 4. Once again, the model has proven to be a viable option for determining
the pulse stiffness characteristics. The agreement in the tangential direction is near
flawless, while the radial direction continues to capture the inflection point very well.
In this chapter the concept of pulse stiffness was introduced and it was shown how a
simple model can be used to predict the pulse stiffness characteristics of a cantilevered
airfoil. The model was tested to see how well it does when applied to several different
geometries and a complete range of rotational speeds. The model has proven to be
a feasible method and dramatically reduced the amount of analysis required. Unlike
with the full simulation method, where many fully non-linear, transient analyses are
needed, no transient analyses are required whatsoever. All that is required is a single
elastic, large deflection static loading analysis and a modal analysis, each of which
only take seconds using today’s computers, not hours like a comparable transient
87
Figure 3.28: Comparison of κθ for Simulated HPC Airfoil and Small Flat Plate Under
Rotation
Figure 3.29: Comparison of κr for Simulated HPC Airfoil and Small Flat Plate Under
Rotation
88
simulation may take. In the tangential direction, the model is robust and can be used
with confidence. In the radial direction, however, some care is needed when applying
the κ-model, especially for small ∆t’s. When the pulse width is smaller than the
period of the first bending mode of the airfoil, the model captures the general trend
well, but can differ in magnitude from actual.
(κ2 − 1)τ2 − (κ1 − 1) ln(τ2 )
τ1
µ = κ1 −
−
1
τ2 e(τ1 −τ2 ) − ln(τ2 )
ln(τ1 )
η=
(κ2 − 1)τ2 − (κ1 − 1) ln(τ2 )
τ2 e−τ2 − e−τ1 ln(τ2 )
(3.21)
(3.22)
In most cases, this will not be a major drawback because the most important piece
of information, the minimum stiffness, lies at the inflection point where the model
always does well. If however, accurate radial pulse stiffness information is needed for
τ < 1, then a specific set of coefficients (µ and η) can be determined quite easily on
a case-by-case basis. Doing so would require running two transient analyses, one at
the smallest τ desired and another near the inflection point. Equations 3.21 and 3.22
can be used to determine the specific coefficients, where τ1 , κ1 and τ2 , κ2 represent
the coordinates of the two solution points. Figure 3.30 shows the results of using the
specific coefficient approach to improve the prediction of the simulated HPC airfoil.
With the specific coefficients (µ = −.55 and η = −1.73) the κ-model does well at
predicting the radial pulse magnification.
89
Figure 3.30: Comparison of κr for Simulated HPC Airfoil Using Specific Coefficients
90
Chapter 4: Rub Loads
4.1
Introduction
In Chapter 3 a novel approach to determining the stiffness of cantilevered airfoils
under impulsive loads was outlined. The approach is based on the ability to represent
the transient force versus deflection information in such a way that all geometries,
rotational speeds and force magnitudes can be explained by a single relationship. The
fundamental use for this methodology is to provide information about the blade tip
deflection characteristics. This information can then be used in subsequent analyses
to determine the tip loads that develop during blade-to-case rubs. To put it another
way, the intent of the approach is not to predict the magnitude, direction or duration
of a given rub event. For that to happen, much more information about the system as
a whole is needed. For instance, dynamic properties of the rotor system and the casing
structure, material and failure properties of the contacting structures, incursion rates
and depths, relative velocities, and other aspects of the system need to be known to
make an accurate assessment of the rub loads. That is not to say however, that with
a few key assumptions the approach can not be used to help make judgment about
what type of loads to expect. The focus of this chapter is exactly that, using the κ
91
modeling approach to gain insight into the loads that can develop during an blade
rub.
4.2
Max Rub Loads
The physics associated with blade tip rubs is extremely complex and it is not
reasonable to expect a simple model such as the pulse magnification model to predict
the rub phenomena well. However, there are some ways that the approach can be
used to provide a glimpse into what might be happening during certain rub events.
In this section, a series of logical concepts and assumptions will be made that will
allow the pulse magnification model to be used to gain knowledge about the airfoil
rub loads.
It is through a highly simplified view of a blade-to-case rub that this concept
begins. During a rub event, the blade, which is moving at a high rate of speed, comes
into contact with the stationary casing. As the two entities make contact, only one
thing can result, something has to move out of the way. What that something is and
how it moves out of the way is what makes up the complexity of the problem.
There are several ways that the blade and/or casing structures can relieve the
interference: a) through deflection, b) through material removal, or c) through failure. In the vast majority of rub events, there is no significant failures associated
with either the blade or the casing, so the latter cause is dismissed in the interest
of broad generalization and is reserved for independent study. The second cause,
material removal, is commonly seen and in fact is often relied upon as a mitigating
design feature. More often than not, the material removal is localized to the casing
structure where engineers commonly apply a layer of abradable material designed
92
to abrade away and reduce the rub loads and minimize blade material loss. Unlike
failure, material removal is quite common and therefore cannot be dismissed based
on broad generalization. Instead, its dismissal is based solely on the fact that the
pulse magnification model is not a suitable approach for describing material removal.
The only cause left is deflection, which is a subject that the pulse magnification
model can help with. Deflection, whether elastic or plastic, can occur through several
means, including: a) dynamics of the rotor system, b) dynamics of the casing structure, or c) dynamics of the blade. The rotor dynamics is certainly a viable source of
deflection. However, relative to the airfoil, the dynamics of the rotor system is primarily detected as pure rigid body motion, something that the κ-model is not equipped
for. Casing dynamics is another viable and highly likely source of deflection. Again,
however, this would not be a practical use of the pulse magnification model since it
knows nothing about the casing structure, including its dynamic properties. Blade
dynamics, on other hand, is an area that the model can shed some light on.
So far in this section, the key assumptions have been established and justifications
for them have been given. In summary, the pulse magnification model can provide
information about the level of loads associated with a rub event between an elastic
airfoil and a rigid casing. Furthermore, there can be no material removal, material
failure or any significant rigid motion of the rotor. Fortunately, these assumptions are
perfectly valid for the CSPF at the OSU-GTL. As a result, some coarse predictions
using the κ-model will be compared to experimental results obtained using the CSPF
in § 4.3. For the remainder of this section, the concept behind how to make the
predictions will be discussed.
93
Figure 4.1: Maximum Possible Fθ for a Radial Incursion of 0.001 in
Figure 4.2: Maximum Possible Fθ for a Radial Incursion of 0.005 in
94
Going back to the simple view of a blade-to-case rub, as the elastic blade incurs
into the rigid casing, it must deflect radially inward an amount equal to or greater
than the interference depth, δ. For this study, any radial movement beyond what is
required to relieve the interference is meaningless because as soon as the blade looses
contact with the casing, the external tip load due to rubbing is lost as well. There
will also be tangential deflection, but that has no bearing on determining when the
blade will loose contact and therefore is of no concern in this study as well. What is
important is the radial deflection along with the radial and tangential loads, precisely
what is represented by the κr curves. As a result, the pulse magnification curve in the
radial direction can be used to determine the loading conditions required to achieve
a desired radial deflection at various pulse widths. Based on the fact that the load is
removed as soon as the blade looses contact, the radial deflection can be set equal to
the amount of interference (ur = δ).
Figure 4.3: Maximum Possible Fθ for a Radial Incursion of 0.015 in
95
Figure 4.4: Maximum Possible Fθ for a Radial Incursion of 0.03 in
Figures 4.1, 4.2, 4.3, and 4.4 show, using the small flat plate model, how this concept can be used to determine the maximum radial and tangential load combination
for a fixed incursion depth of 1, 5, 15 and 30 mils respectively. In each figure there are
three lines corresponding to three different pulse widths. As expected, the maximum
possible tip forces increase with increasing incursion. What may not be expected is
the flatness of each curve, suggesting that the radial load has much less influence on
the radial deflection than does the tangential load. The reason for this was discussed
at length in § 3.4.2. The τmin curve is consistently the highest and for incursion
depths greater than or equal to 0.015 inches, the maximum allowable tangential force
is above the 500 lbf limit used to develop the pulse magnification curve for the small
flat plate. This suggests that at the small pulse widths, the blade inertia is preventing
the blade from deflecting, and therefore, requires much larger forces to achieve the
96
Figure 4.5: Maximum Possible Fθ for Fr = 150lbf
desired incursion depth. The next highest curve is the τmax curve, which is approximately steady state. Finally, the lowest curve in each figure is the one associated
with a τ = 1. This suggests that when the pulse width is equal to the period of the
first bending mode, it takes relatively little force to reach the desired incursion depth.
These plots clearly highlight how exaggerated the tip loads can be when steady state
is assumed. Figures 4.5 and 4.6 show the same data reformatted to highlight how the
maximum tangential and radial force varies with respect to incursion depth.
4.3
Comparisons to Experimental Data
Throughout the previous section, it was describe how to use the pulse magnification curves to gain insight into the magnitude of radial and tangential pulse loads
required to reach a desired radial deflection, or incursion. The process involves four
key variables, radial force amplitude, tangential force amplitude, radial incursion and
97
Figure 4.6: Maximum Possible Fr for Fθ = 150lbf
pulse width. In the previous section, the focus was on establishing the relationship
between three of the variables when one was fixed. In many situations, engineers
only know the value of one of the variables, so the results in the previous section are
valuable in providing ranges of loads to expect. While this is a great and worthwhile
extension of the κ-model, it does not lend itself well to validation. Fortunately, the
rub research program at OSU provides a unique opportunity to gain useful data that
can be used to validate the approach. The data was acquired through a series of
controlled tip rub experiments in the CSPF. The focus of this section is to utilize the
OSU rub data to quantitatively validate the pulse magnification model.
There are numerous aspects of the CSPF that make it unique, but what makes
the facility ideal for validating the pulse magnification model is the ability to control many of the parameters associated with high-speed compressor blade rubs. Of
98
particular use to this study, is the capability to control incursion depth and rub duration while measuring the radial and tangential rub loads. Figure 1.4 shows the key
instrumentation layout including the load cells mounted on the backside of the rub
shoe. There are three load cells and each measure the transient loads for all three
axes. The load cells are aligned in a cylindrical coordinate system such that each
of them measure loads in the radial, tangential and axial directions. Since the load
cells are separated from the rub surface by the relatively massive rub shoe assembly,
the load cells do not directly measure the rub load. However, the stator system is
sufficiently stiff that the measured loads are expected to be highly representative of
the true sliding contact loads. For purposes of validation, the measured radial loads
will be used as an input into the pulse magnification model and the tangential loads
will be used to compare to the predicted tangential loads.
Figure 4.7: Rub Geometry
99
α(δ) = 2 cos
−1
Rc2 + δ 2 − (Rroot + s)2
2Rc δ
(4.1)
With the method of setting the experimental radial and tangential load values
established, the focus turns to setting the incursion depth . The CSPF is designed
such that the load measuring unit (LMU) is moved radially inward by a set rate of
travel to a desired depth. The amount of incursion, δ, is determined by the amount
of travel of the LMU minus the radial gap, G0 , between the rub shoe and the tip of
the rotating airfoil. As shown in figure 4.7, the radius of the blade tip, Rroot + s, is
less than the radius of the casing, Rc . As a result, the arc of the rub, α, becomes
a function of the incursion depth and can vary from 0 deg to the full extent of the
shoe, or 90 deg. Equation 4.1 provides the relationship between the rub arc and the
incursion depth.
τ (δ) =
30α(δ)
∆t(δ)
=
T1
πΩT1
(4.2)
The final parameter to establish experimentally is the rub duration, or pulse width,
∆t. The rub duration, as shown in equation 4.2, is controlled by the set incursion
depth and the rotational speed, Ω. The faster the rotor is spinning, the shorter the
amount of time is required to travel across the rub arc. Figure 4.8 shows how the
rub arc and rub duration vary with respect to the incursion depth. The curve was
generated for a rotation speed of about 16,000 rpm, a common speed used during the
test series, and spans the range of incursion depths tested thus far.
With all the pieces in place, comparisons between predicted tangential loads, that
are based on the κ-model, and experimental results can be made. Figure 4.9 shows
a predicted rub load map of normalized tangential rub loads, Feθ , versus normalized
100
Figure 4.8: Pulse Width for CSPF Experiments
e The normalization
radial rub load, Fer , for various normalized incursion depths, δ.
factor for the rub loads is based on the static force applied to the load cells during
installation, while the normalization factor for the incursion depth is based on the
maximum tip thickness of the airfoil. The rub map was generated using data from a
full suite of transient runs and not taken directly from the pulse magnification model.
The rub load map is a convenient way to present the predictions, from which
comparisons to experimental results can be made. In addition, the map provides
information about the relationship between the tangential rub load and how the
radial rub load changes with incursion depth. Each line on the map is fairly linear
and their slopes can be used as inputs into various rub models. For instance, in a
Coloumb friction based rub model, the map would provide an equivalent coefficient
of friction. The horizontal lines associated with the smallest two incursion depths are
101
Figure 4.9: Rub Force Map for Various Radial Incursion Depths
artificial and represent the lower bound of tangential loads included in the transient
analysis. Inside the bounds, however, the slopes are relatively insensitive to incursion
depth.
Figure 4.10 gives the comparison between the predicted tangential rub loads and
the measured tangential loads. Two sets of predictions are included, the full transient
method is shown with circles and the κ-model results are shown with squares. At
first, it appears the model does a better job at matching the data than does the
full transient solution method. However, this method is based on determining the
max possible load and is expected to over predict the actual loads. This is the case
with the full transient method, it consistently over predicts the measured load, but
follows the same slope. Where as the κ-model spans the data by over predicting the
lower loads and under predicting the higher loads, resulting in a flatter slope. It is
102
Figure 4.10: Comparison of Predicted Feθ to Measured
anticipated that this difference in slope can be corrected if specific coefficients were
used instead of the generic set given in table 3.1.
δe
0.03
0.07
0.11
0.12
0.14
Feθmeasured
0.83
1.55
2.03
2.38
2.15
Full Simulation Method
κ-Model
predicted
predicted
Feθ
% Difference Feθ
% Difference
1.25
50.6
1.25
50.6
1.53
-1.3
1.30
-16.1
2.30
13.3
1.80
-11.3
2.43
2.1
1.88
-21.0
2.65
23.3
2.08
-3.3
Table 4.1: Comparisons of Maximum Measured Tangential Forces To Predicted
The measured loads were taken from a series of 10 runs where the blade was
rubbed into a steel shoe without any abradable material. Several repeats at each
103
incursion depth were made and are represented in figure 4.10. When the measurements are reduced to only the max measured for each incursion depth tested, the
comparison between predicted and measured values, as shown in table 4.1, improves.
The max difference of 50% occurs at a very small incursion depth where the accuracy
of the measurement is questionable. Aside from that outlier, the remainder of the
predictions are within 25% for the full transient method and the κ-model method.
Although this level of accuracy is not usually celebrated, it is quite remarkable
considering a) the sheer simplicity of the model, b) the extreme complexity of the
highly transient and non-linear rub events, c) the long list of assumptions required
to allow the results of the κ-model to be compared to measured rub load data, and
d) the fact that this method was developed to merely understand how pulse widths
affect blade deflection behavior, not to give deterministic load values. This level of
accuracy is far better than what most engineers have to work with and could easily
and effectively be incorporated into their design and analysis procedures. With only
a simple modal analysis and a single static load analysis required, the cost and effort
required to achieve this accuracy is minimal.
104
Chapter 5: Airfoil Geometry Effects
5.1
Introduction
The final element of this research project is to explore what effect various airfoil
geometry features have on the stiffness characteristics. This is an important aspect of
the work contained in this dissertation because it provides guidelines that can be used
in either the design of new airfoils or the study of existing ones. The strict demands
placed on the performance and efficiency of today’s turbomachines makes avoiding
blade-to-case rubs impossible, and as the demands increase, so to will the frequency
of blade rub events. As a result, engineers must consider these rub events as part
of their design procedures and make certain that the engine system can tolerate the
expected rubs. There are several common ways that engineers go about doing this,
applying abradable materials to the casings, thermal based clearance controls, and
general structural design procedures that assure structural integrity, to name a few.
Another concept that could be employed to mitigate the potential hazards of
a blade rub event is to design the airfoils such that they produce more tolerable
rub characteristics. There are several issues with this approach that cause it to be
widely under utilized, and they include: a) the difficulty of predicting the rub loads,
b) inability to efficiently model the system, and c) cost. By eliminating these issues,
105
the pulse magnification model provides a unique opportunity to consider the geometric
features of an airfoil as part of the overall rub tolerant system design process. This
general design philosophy can be best explained using an extreme example. If the
only concern airfoil designers had was to minimize the rub loads produced when
uncontrollable rub events occurred, then the designer would likely make the blades
out of a material like rubber that is very flexible. Unfortunately, blade designers
have many other structural and aerodynamic requirements that they must consider
and satisfy. That is not to say though, that the designers could not consider blade
flexibility under rub load conditions and tailor their designs such that they remain as
flexible as possible and still satisfy their other design criteria.
The focus of this chapter is to introduce the idea of using the available geometric
features of an airfoil to purposely affect the tip stiffness characteristics during static
and pulse loading. The pulse magnification model will be used to understand how
each of the design parameters introduced in Chapter 2 change the flexibility of the
blade tip. The study has been conducted using the HPC and fan geometries, provided
in § 2.7, as two individual baselines from which the modified geometries are compared.
Only the direct, or main, effect of each design parameter is considered, the interaction
effects of the design parameters have not been considered for this study.
5.2
Effect on Inflection Point
As explained in Chapter 3 and validated in Chapter 4, the entire pulse stiffness
characteristic of any static or rotating airfoil can be described for all pulse widths using
the κ-model. The model, given in equation 3.11 and table 3.1, relies on knowing the
fundamental bending frequency of the airfoil and the static tip stiffness under the
106
desired radial and tangential loading. As a result, there is no need in this study to
expand out the entire pulse magnification curve. Only the parameter’s affect on the
position of the inflection point (τ = 1) and the magnitude of the static tip stiffness
in each direction will be considered. The magnitude of the radial and tangential
loads used in this study are 100 lbf in each direction. This loading combination
was chosen because it is a relatively low magnitude and is likely an acceptable load
for most blade designs and can therefore be used confidently to study the stiffness
characteristics of each blade geometry in the study. In addition, the analysis was
completed for stationary airfoils for simplicity purposes. However, as shown in § 3.4.2,
the stationary results can be expanded to include rotational effects as well.
Of the geometric parameters given in table 2.1, rroot , s, and b where not varied as
part of this study. The root radius, for non-rotating airfoils, has no effect on either the
tip stiffness or the fundamental bending mode frequency and therefore was eliminated
from the list of variables. Airfoil span was also fixed since the aspect ratio was used to
vary overall airfoil size. The bow parameter was eliminated because neither the HPC
nor fan airfoil example contained bow. The remainder of the parameters were varied
one at a time to see how the pulse magnification model is affected. The variation
spanned from 50% of baseline to 150% of baseline in 25% increments. In some cases,
for instance βtip on the fan blade, this wide variation exceeded the bounds of the
common airfoil design space. Nevertheless, these values were premitted in this study
since the sole purpose is to identify trends and allowing for the uniform variation
offers a convenient way to compare all parameters on a universal scale.
Figure 5.1 provides the results from the inflection location study. There are 12
plots in the figure, one for each of the geometric parameters varied. In each plot
107
Figure 5.1: Effect of Geometry Parameters on the Inflection Point (Change in ∆t vs.
Change in Parameter, Relative to Baseline)
108
there are two lines, one for the HPC style blade design and one for the fan style
blade design. The horizontal axes gives the ratio of the geometric parameters to their
baseline value and the vertical axes provide the scaled value of ∆t corresponding to
the inflection point (τ = 1), which are also normalized by the baseline. Each plot has
been put on identical scales to highlight the parameters with the largest impact on
the location of the inflection point.
∆t/∆tbaseline = 50%
∆t/∆tbaseline = 150%
Parameter HPC Airfoil Fan Airfoil HPC Airfoil Fan Airfoil
AR
1.04
1.04
1.07
0.98
1.03
1.04
0.97
0.97
θ
n/a
1.08
n/a
0.89
φ
throot
1.73
1.80
0.71
0.70
TTR
0.92
0.95
1.05
1.03
1.00
1.00
1.00
1.00
TR
βroot
0.98
1.00
1.08
1.01
βtip
0.88
0.77
1.15
1.12
1.00
1.01
1.00
0.99
ν
λ1
1.00
1.01
1.00
0.99
ζroot
1.09
1.19
0.92
0.88
1.03
1.02
0.97
0.98
ζtip
Table 5.1: Effect of Geometry Parameters on the Inflection Point
By far, the thickness of the airfoil root, throot , is the dominant variable. This
result comes as no surprise since the frequency of the fundamental bending mode is
highly sensitive to the stiffness of the root portion of the airfoil, where the strain
energy is the highest. Similarly, the additional mass is concentrated at the blade hub
where there is little kinetic energy. Both of these conditions will drive up the first
flexural mode frequency and drive down the inflection point. The root thickness would
109
definitely be a good parameter to use to set the inflection point near the expected
rub duration. Compared to root thickness, the effect of the other 11 variables are
minimal, tip twist and root camber being the next two in line. Values for inner flow
path angle, φ, are not given for the HPC blade since the baseline had a zero value.
The affect on inflection point is very consistent between the HPC style blade and the
fan style blade, which is ideal for establishing a unified set of design guidelines. For
convenience, tabulated results are given for the extreme variation points in table 5.1.
5.3
Effect on Tip Stiffness
The second and final aspect to consider when determining geometric affects on
blade pulse stiffness is the static tip stiffness, F s . The static tip stiffness is the most
crucial part of the pulse magnification model because it is ultimately what sets the
magnitude of the pulse stiffness. Figures 5.2 and 5.3, along with tables 5.2 and 5.3,
provide the results for the tangential stiffness and radial stiffness, respectively. The
format of figures and tables are identical to the ones provided in the previous section.
The vertical axes now correspond to the ratio of static tip stiffness to the baseline
static tip stiffness.
For the most part, the radial and tangential trends are the same due to the
coupling described in § 3.4.2. Likewise, the trends are similar between the HPC and
fan blade types, with the root thickness being the dominating variable controlling the
tip stiffness. The root thickness, once again, is the most influential parameter for the
same reason given for the frequency effect. This comes as no surprise since, under
rub-type loading, the primary deflection occurs in the tangential direction, similar
to the primary bending mode of the airfoil. Other influential parameters controlling
110
Figure 5.2: Effect of Geometry Parameters on Static Tangential Tip Stiffness (Change
in Kθs vs. Change in Parameter, Relative to Baseline)
s
s
K s /Kbaseline
= 50%
K s /Kbaseline
= 150%
Parameter HPC Airfoil Fan Airfoil HPC Airfoil Fan Airfoil
AR
2.31
2.71
0.53
0.64
θ
0.94
0.92
1.06
1.09
φ
n/a
0.87
n/a
1.22
throot
0.38
0.14
3.14
3.13
TTR
0.58
0.65
1.33
1.32
TR
0.96
0.95
1.03
1.05
βroot
0.84
1.05
1.19
0.95
βtip
1.07
1.50
0.89
1.02
ν
1.00
1.02
1.00
0.98
λ1
1.00
0.99
1.00
1.01
ζroot
0.79
0.65
1.24
1.47
ζtip
0.95
0.96
1.05
1.04
Table 5.2: Effect of Geometry Parameters on Static Tangential Tip Stiffness
111
tip stiffness are aspect ratio, root camber, and the ratio of edge thickness to max
thickness. It is interesting to note that for the majority of the parameters, the effect on
tip stiffness is non-linear, which highlights the need to model true geometry features.
With more simplified modeling approaches that rely on linearity, the estimated change
in stiffness due to these parameters could be erroneous. This further reinforces the
importance of a feature based modeling approach like the one introduced in Chapter 2.
Figure 5.3: Effect of Geometry Parameters on Static Radial Tip Stiffness (Change in
Krs vs. Change in Parameter, Relative to Baseline)
112
s
s
K s /Kbaseline
= 50%
K s /Kbaseline
= 150%
Parameter HPC Airfoil Fan Airfoil HPC Airfoil Fan Airfoil
AR
1.37
0.93
0.34
1.16
θ
0.90
0.77
1.09
1.08
n/a
0.92
n/a
1.04
φ
0.32
0.12
3.90
2.25
throot
TTR
0.21
0.46
1.21
1.37
0.95
0.90
1.03
0.96
TR
0.49
0.99
1.04
0.85
βroot
βtip
1.20
2.12
0.26
0.79
1.00
0.87
0.97
0.75
ν
λ1
0.99
0.84
1.00
0.99
ζroot
0.52
0.48
2.36
1.64
1.17
0.83
0.85
1.01
ζtip
Table 5.3: Effect of Geometry Parameters on Static Radial Tip Stiffness
113
Chapter 6: Conclusion
6.1
Conclusions
An efficient method had been developed to study the effect of dynamic pulse loading on the stiffness behavior of cantilevered airfoils. When applied to turbomachinery
blades, the approach becomes an effective way to describe the deflection behavior of
the blade during a tip rub event, which frequently produces impulsive loads on the
blade tip. The primary intent of the analysis technique is to provide valuable airfoil
tip stiffness information that can be used as an input into subsequent rub simulations
where a good estimate of the contact stiffness is often needed. The method is not
intended, however, to be a predictive tool for understanding rub loads. The approach
takes into consideration the blade and how it performs under pulse loading. To accurately describe rub loads, attention must be given to the casing structure and to the
mechanics of the high-speed contact. Although, as shown in Chapter 4, under the
right conditions and assumptions the approach can be used to gain insight into the
amount of load a particular blade could produce when it comes into contact with a
relatively rigid structure. In fact, the approach was used to estimate the maximum
possible rub forces for the CSPF experiments and the results were quite impressive,
providing a conservative upper bound to what was realized from measurements.
114
This approach, named pulse magnification because it describes how impulsive
loads alter the tip stiffness as compared to static loading, requires only two simple,
static analyses to fully define the pulse stiffness characteristics for an entire range
of pulse widths. Equivalent analyses, as used to validate the pulse magnification
model, require many transient simulations that can be very costly. To put things into
perspective, a set of nearly 1,800 transient analyses was required to describe the pulse
stiffness characteristics of a single blade across an appropriate range of pulse widths
and load amplitudes. This analysis set took several days to complete, even when
running 30 analyses in parallel on individual multi-processor machines. Using the
pulse magnification model, however, a single static load analysis and a single modal
analysis is all that is required, taking minutes not days.
One of the unique and beneficial aspects of this work is that it does not rely
on simplified beam or plate models that have become common place in rub related
research. Actual turbomachinery airfoils seen in today’s machines are geometrically
complex and include many 2 and 3-dimensional features that look nothing like a beam
or plate. The beam and plate models provide an excellent way to approximate many
airfoil characteristics and their simplicity should be taken advantage of whenever
possible. On the other hand, there is a limit to what these modeling approaches can
explain. It was shown in Chapter 5 that the influence of several geometric features
on tip stiffness can be quite significant. For instance, camber, can nearly triple the
stiffness of an airfoil when allowed to vary over an acceptable range.
To support the development and testing of the pulse magnification model, and
to make sure all the geometric effects on stiffness could be included, an FE model
building algorithm was created and outlined in Chapter 2. The algorithm is based
115
on a purely mathematical representation of the airfoil, resulting in a very efficient
method to create blade FE models. The algorithm includes a large set of geometric
features that are common in today’s designs. This methodology was verified by comparing overall shape and modal properties of simulated blades versus actual blades,
for two vastly different designs. The development and validation has been focused on
geometries seen in compressive stages of axial-flow turbomachines, but the approach
could easily be generalized to include expansion stages as well. The advantage of
this modeling technique far surpasses the ability it gives to study tip stiffness effects.
Since the structural properties of airfoils are so well captured, this technique can be
used for any general study where effects on structural properties are required.
6.2
Future Opportunities
Methods described in this dissertation are sound and can be directly applied to
many situations. However, as with any new method, additional opportunities exists
to further validate and improve them. As for the FE model building algorithm,
the amount of improvements are near limitless. Any geometric feature that can be
described mathematically, can be included. There are no inherent restrictions in the
general approach that limit what geometric features can be included and the use of
superposition makes it easy to incorporate them.
As for the pulse magnification model developed as part of this dissertation work,
there are certainly improvements that could be made as part of follow-on work. To
begin with, the inclusion of axial stiffness could be made. It is common practice to
ignore the axial component because it is believed to be insignificant when compared
to the radial and tangential components. However, with certain geometric features
116
such as twist, the axial and tangential deflections become coupled and the axial deflection can become as large as the tangential. Secondly, the inclusion of force dependent
coefficients could be included to further reduce the variation seen in the pulse magnification model. It has been shown that the variation can become significant for τ < 1,
especially in the radial direction. The dominant cause of this is a loss of linearity
between force amplitude and resulting tip deflection. Despite the non-linearity, the
general trend of the model remains valid. Therefore, by allowing the two model coefficients to vary with the amplitude of applied force, a more sophisticated and accurate
model could be developed. Thirdly, the effects of the applied boundary conditions
should be studied and accounted for to improve accuracy of the model. Currently, it
is assumed that the localized deflection at the airfoil tip is sufficiently separated from
the root, where the fixities are applied. In many cases this is likely true, but could
become invalid under certain circumstances. Fourthly, the effect of partial chord rubs
on the tip deflection behavior could be investigated. For development purposes, the
loading was evenly distributed along the tip of the blade. In many rub events, however, only a portion of the tip actually makes contact. Finally, the entire model is
based on the assumption that the blade deflects elastically. It would be beneficial to
understand how plasticity affects the pulse magnification model.
117
Bibliography
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rubbing, IMechE, 2000, pp. 259–263.
[2] J. Ferguson, A moving load finite element-based approach to determining blade
tip forces during a blade-on-casing incursion in a gas turbine engine, Master’s
thesis, The Ohio State University, Columbus, OH, 2008.
[3] J. Garza, Tip rub induced blade vibrations: Experimental and computational
results, Master’s thesis, The Ohio State University, Columbus, OH, 2006.
[4] J. Jiang, J. Ahrens, H. Ulbrich, and E. Scheideler, A contact model of a rotating rubbing blade, Proceedings of the 5th International Conference on Rotor
Dynamics (Darmstadt), 1998, pp. 478–489.
[5] Leonard Meirovitch, Fundamentals of vibration, McGraw-Hill, 2001.
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[7] C. Padova, J. Barton, M. Dunn, and S. Manwaring, Experimental results from
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119
Appendix A: Additional Spatial Tip Force Profiles
Linear Tip Force Profile
F (x) =
P
X
(mxp + F0 ) = m
p=1
P
X
xp + P · F 0
p=1
F (X1 ) = 0
m
F0
−1 P
=
xp P
X1
1
p
F
0
Quadratic Tip Force Profile
F (x) =
P
X
P
P
X
X
a1 x2p + a2 xp + a3 = a1
x2p + a2
x p + P · a3
p=1
p=1
p=1
F (X1 ) = 0
F (X2 ) = 0
  P 2 P
−1 

a1
F
p xp
p xp P
 a2  =  X12
X1
1   0 
2
a3
0
X2
X2
1

120