A Modeling Study of the Sensitivity of Natural
Frequency of Vibration to Geometric Variations
in a Turbine Blade
by
Daniel A. Snyder
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING in MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
December 2010
(For Graduation August 2011)
© Copyright 2010
by
Daniel A. Snyder
All Rights Reserved
ii
Contents
Contents ............................................................................................................................ iii
List of Tables ..................................................................................................................... v
List of Figures ................................................................................................................... vi
Acknowledgement ........................................................................................................... vii
Abstract ........................................................................................................................... viii
1. Introduction and Background ...................................................................................... 1
1.1
1.2
1.3
High Cycle Fatigue ............................................................................................. 1
1.1.1
Zero-Mean Stresses ................................................................................ 1
1.1.2
Non-zero Mean Stresses ......................................................................... 2
1.1.3
Modified Goodman Diagram ................................................................. 2
Vibratory Response ............................................................................................ 2
1.2.1
Design Philosophy ................................................................................. 3
1.2.2
Frequency Prediction ............................................................................. 3
1.2.3
Campbell Diagram ................................................................................. 3
The Problem........................................................................................................ 5
1.3.1
1.4
Defining a Design Space ........................................................................ 5
Previous Work on this Topic .............................................................................. 6
2. Theory and Methodology ............................................................................................ 8
2.1
2.2
Monte Carlo Simulation ..................................................................................... 8
2.1.1
Geometric Parameter Scheme ................................................................ 8
2.1.2
Blade Root Geometry........................................................................... 10
2.1.3
Tip Shrouds .......................................................................................... 10
2.1.4
Generating Random Variable Combinations ....................................... 11
Modal Analysis ................................................................................................. 13
2.2.1
Execution of Computations .................................................................. 14
3. Results and Discussion .............................................................................................. 16
iii
3.1
Mode Shape Identification ................................................................................ 16
3.2
General Trends in Response ............................................................................. 16
3.3
3.2.1
First Mode Shape ................................................................................. 16
3.2.2
Second Mode Shape ............................................................................. 16
3.2.3
Third Mode Shape ................................................................................ 17
3.2.4
Fourth Mode Shape .............................................................................. 17
3.2.5
Fifth Mode Shape ................................................................................. 17
Regression Functions ........................................................................................ 17
3.3.1
Linear Regression................................................................................. 17
3.3.2
Nonlinear Regression ........................................................................... 18
3.3.3
Partial Least Squares Regression ......................................................... 19
4. Conclusion ................................................................................................................. 20
5. References.................................................................................................................. 21
6. Appendices ................................................................................................................ 22
6.1
Appendix 1: Turbine Blade Parametric Model ................................................. 22
iv
List of Tables
Table 1: Geometric Parameters and their Ranges ............................................................ 12
v
List of Figures
Figure 1: Modified Goodman Diagram ............................................................................. 2
Figure 2: Campbell Diagram ............................................................................................. 4
Figure 3: Parametric Model of Fictitious Turbine Blade................................................... 9
Figure 4: Generic Tip Shroud Design .............................................................................. 10
Figure 5: Airfoil Section Definition ................................................................................ 11
Figure 5: Top View of Fictitious Parametric Turbine Blade Model ............................... 22
Figure 7: View of Fictitious Turbine Blade Root Attachment Geometry ....................... 23
Figure 8: Close-up View of Turbine Airfoil Section Definition ..................................... 23
Figure 10: Random Turbine Blade – 1 ............................................................................ 24
Figure 11: Random Turbine Blade – 2 ............................................................................ 25
Figure 12: Random Turbine Blade – 3 ............................................................................ 26
Figure 13: Random Turbine Blade – 4 ............................................................................ 27
Figure 14: Random Turbine Blade – 5 ............................................................................ 28
Figure 15: Mode Shape 1 at 299 Hz ................................................................................ 29
Figure 16: Mode Shape 2 at 712 Hz ................................................................................ 29
Figure 17: Mode Shape 3 at 1267 Hz .............................................................................. 30
Figure 18: Mode Shape 4 at 1694 Hz .............................................................................. 30
Figure 19: Mode Shape 5 at 2905 Hz .............................................................................. 31
vi
Acknowledgement
Thanks to Jeff Beattie for help with ANSYS and for giving me the concept for this
project. Thanks to Grant Reinman for consultation on statistical methods such as partial
least squares regression and principal component analysis. Thanks to Pratt & Whitney
for providing me with an environment that fosters research and new ideas.
vii
Abstract
A Monte Carlo simulation is used to investigate the effect of geometric variation in
a turbine blade on its natural frequencies of vibration. First, a parameter scheme is
selected and implemented in a parametric 3D solid model. The parameters chosen are
those that affect section properties of the airfoil at three different sections throughout the
airfoil. Other parameters control other geometric features in the turbine blade. A total
of 37 geometric parameters are independently varied to test the response of the first 5
natural frequencies of the turbine blade model. The material model is isotropic with an
elastic modulus of 30E+6 psi and a Poisson Ratio of 0.3 and a density of .289 lb/in 3. A
random set of values for the parameters is generated using a Latin Hypercube function
implemented in Matlab.
A total of 50 combinations of 37 independently varied
parameters result in 50 sets of natural frequencies. The impact of each geometric
parameter on each natural frequency is quantified by calculating a linear regression.
The linear regression function approximates each natural frequency as a linear function
of each independent variable (geometric parameter).
This result shows that the
fundamental mode of vibration is affected mainly by the moment of inertia of the blade
about the fixity point. All parameters that tend to increase the moment of inertia of the
blade tend to decrease the first natural frequency. The second natural frequency is also
bound to a similar relationship with mass moments because its mode shape is also a
bending mode. The third frequency is associated with a torsional mode shape. The
parameters that increase airfoil root stiffness increase this frequency. The parameters
that decrease the polar moment of inertia of the tip tend to increase this frequency also.
The importance of this type of study is to explore and understand the implications of
varying parameters and combinations of parameters in the design space. Further design
studies like this could allow the design engineer to tune certain frequencies out of
undesirable ranges while leaving other modes frequencies unchanged.
viii
1. Introduction and Background
This project deals with prediction of the effects of manufacturing/geometry
variation on turbine blade vibration. It is important to prevent resonant vibration of
turbine blades during operating conditions. Turbine blades experience unsteady forcing
functions at many different frequencies. If the blade experiences an excitation frequency
equal to its natural frequency, the blade will usually fail very quickly in a failure mode
known as high-cycle fatigue. This fatigue mode is characterized by relatively low
fluctuation in stress and very high frequency of fluctuation. This failure mode embrittles
the material and causes it to crack in regions of high steady stress. Characterization of
materials in this failure mode gives rise to the Goodman Diagram.
1.1 High Cycle Fatigue
High cycle fatigue is the failure mode of a material experiencing fluctuating stress.
Compared to yielding or rupture failure in which stress is said to be steady or
unchanging, fluctuating stresses cause materials to fail by becoming brittle and cracking.
The characterization of materials in this failure mode involves testing materials subject
to alternating stresses of different means and different amplitudes. In an alternating
stress condition, the mean stress is defined as the average stress over a long period of
time. For stresses whose variation is sinusoidal over time, the mean stress is simply the
average of the maximum and minimum stress.
1.1.1
Zero-Mean Stresses
Testing materials with an alternating stress that has a zero mean allows the engineer
to measure how many cycles before the material fails. Typically, these data are plotted
on a chart with number of cycles on the horizontal axis and stress amplitude on the
vertical axis. The number of cycles is typically on a logarithmic scale for plotting
purposes since number of cycles to failure can become exponentially larger for low
stress levels. Some materials, such as steels, have a stress level below which fatigue
failure will never occur – this is called the endurance limit.
1
1.1.2
Non-zero Mean Stresses
In cases when the temporal average of an alternating stress is not zero, the fatigue
life is reduced compared to the same amplitude of alternating stress with a zero mean.
When data are collected from material tests in which the mean stress is not zero, the data
are plotted on a two dimensional plane with level curves of fatigue life. The plane of
interest shows the average, or steady stress on the horizontal axis and the alternating
stress amplitude on the vertical axis. There are several methods of data fitting a curve of
constant cyclic life. The most common is called the Modified Goodman Line.
Figure 1: Modified Goodman Diagram
1.1.3
Modified Goodman Diagram
The Modified Goodman Diagram has two lines plotted on the stress plane. The
lines delimit the state of stress below which the material will not fail (assuming a certain
probabilistic certainty). The Goodman line plotted on the x-y plane of mean stress and
alternating stress is described by the equation:

m 

 u 
a  e 1 
[1]
Understanding material response to alternating stress with nonzero mean is required
when designing mechanical components that experience vibration. The next section

deals with vibratory response of turbine blades.
1.2 Vibratory Response
Turbine blades have curved surfaces used to redirect the flow of fluid in order to
extract work from it. Jet engines use turbine blades, positioned behind turbine vanes, to
extract work from the flow of very hot combustion products of fuel and air. The turbine
vanes are static and act as nozzles to direct the flow toward the blades at the proper
direction and flow rate. When the blades see the flow, its intensity fluctuates as is goes
between nozzles and wakes. In general, the frequency of excitation is related to the
rotational speed of the turbine. The excitation frequency is also related to multiplicative
factors related to number of disturbances in the airflow around the turbine. For example
2
such factors can be: number of upstream vanes (nozzles) adjacent to the blade row,
number of downstream vanes adjacent to the blade row, difference between number of
upstream and downstream vanes, number of fuel nozzles in the combustor, and many
other geometric features.
1.2.1
Design Philosophy
In the design of a turbine blade, the design engineer tries to minimize the number of
times when the blade will experience an excitation frequency equal to one of its natural
frequencies at a given running condition. One can never prevent all resonant excitations
but can try to place them at engine operating conditions that are not used for long
periods of time. The major operating conditions to at which the design engineer must
avoid resonance are idle, take-off, climb, and cruise. Those operating conditions may all
be associated with different engine operating speeds, each of which creates the potential
to excite a resonant mode of the turbine blades.
1.2.2
Frequency Prediction
The typical method of predicting natural frequency of a given blade design is to use
Finite Element Analysis. The natural frequencies and mode shapes (eigenvalues and
eigenvectors) can be numerically approximated and used in the design iterations to
prevent resonant excitations. In this type of finite element analysis, the equation to be
solved is that of 3-dimensional vibration. It is multidimensional extension of the one
dimensional vibration equation. The equation is of the form:
Ý kx  0
mxÝ
[2]
Where in the case of a single degree of freedom system, m is mass, k is stiffness,
and x is scalar displacement. In the case of multiple degrees of freedom, x is a vector of

displacements while k is a matrix of stiffness (between every mass).
1.2.3
Campbell Diagram
Using modal analyses at several operating conditions (with different temperatures
and rotation speeds) the engineer can produce a Campbell Diagram, Figure 2. The
Campbell diagram simply plots natural frequencies versus engine operating speed. The
horizontal axis shows the engine speed (in Rev/min). The vertical axis shows the modal
3
frequencies (in Hertz).
The Campbell Diagram shown uses fabricated data for
illustrative purposes. The horizontal, drooping lines are the natural frequencies of the
blade. The straight lines of various integer slopes are the excitation frequencies caused
at certain engine speeds. The slope of the line is called the Engine Order. This is an
integer factor equal to a rotational symmetry or repetition found somewhere in the path
of the turbine blades such as the number of nozzles before the blade or the number fuel
nozzles in the combustion chamber. Any repeated feature that the blade will see in its
travel around the engine represents a potential for resonant excitation. If an engine order
line crosses a natural frequency line at an operating speed it is said to be a “resonant
crossing.” This is a visualization tool that allows the engineer to easily see what modes
of vibration will be excited at which engine speeds. The following diagram shows an
example of a Campbell Diagram with 7 natural frequencies and 15 engine orders of
concern.
Figure 2: Campbell Diagram
4
Again, the horizontal axis is the engine rotational speed (in RPM) and the vertical
axis is the vibration frequency in Hertz. The conversion from RPM to Hertz is 1/60
(revolutions per minute to cycles per second).
The natural frequency lines are not perfectly horizontal. Usually the lines are not
perfectly horizontal because natural frequency will vary depending on engine speed and
temperature. Although engine speed and temperature are not directly related, they are
often highly correlated. For the purposes of the Campbell Diagram, we assume a
constant known temperature for each operating condition. For turbine blades, the natural
frequency lines usually droop with higher engine speed because thermal softening
effects overtake stress-stiffening effects. For fan blades, the lines of natural frequency
typically increase slightly with respect to engine operating speeds because temperature
increase is small and stress stiffening effects overtake.
1.3 The Problem
The difficulty is that when a resonant crossing is predicted, it is up to the intuition of
the engineer to know what geometric properties of the blade to change in order to affect
the natural frequency desirably.
As an added complication, changing one natural
frequency desirably may adversely affect another natural frequency.
Without a
comprehensive analysis of the entire design space, one can never fully understand the
practical limitations of tuning turbine blade airfoils.
1.3.1
Defining a Design Space
The question posed in this study is: can one accurately and quantitatively
characterize the effect each geometric parameter, or combination thereof, has on natural
frequencies, or combinations thereof? In order to do this, one first needs to define the
design space. The design space is comprised of defined, controllable parameters that
affect the shape of the turbine blade. Some simple examples are: height, thickness,
aspect ratio, etc.
Every design feature in the turbine blade can have a numerical
parameter associated with it. In order to fully understand the design space, the engineer
must devise a way to test every region of the design space equally.
5
For a simple two-variable design space, assuming there are absolute maxima and
minima constraining each variable, the design space is rectangular and has four corners.
For higher dimensional design spaces, it is not immediately obvious how to explore the
boundaries and interior regions of the space.
1.4 Previous Work on this Topic
In a paper published by J. M. Brown and R. Grandhi, a similar study was performed
on fan blade airfoils (Brown & Grandhi, 2008). In this study, a population of fan blades
was measured using a coordinate measuring machine (CMM). The machine measures
the 3D Cartesian position of a point on the surface of the object given an approach
orientation. The machine can repeat this measurement for many different points around
the airfoil. The data collected was then made to have a zero-mean by subtracting the
mean value from each variable.
The interpretation of this zero-mean data is the
“deviation” from an average airfoil. Zero represents a point being equal to the average
position and positive or negative represents deviation from the average. The variations
to be measured were caused by random manufacturing variation. To study the effect of
this variation on the natural frequency of the airfoils, a large number of realistic sets of
deviation variables were to be generated.
Many of the deviation measurements of the airfoils would be highly covariant. This
is because the airfoil, while deviating from an average population, still remains smooth.
Points adjacent to one another on the airfoil surface had high covariance. The authors of
this paper projected the measured variable space of high covariance into an orthogonal
variable space by means of principal component analysis. This is a statistical technique
that determines orthogonal linear combinations of variables that most highly explain the
variance in the data not explained by precedent variables combinations. The technique
involves simply finding the eigenvectors of the covariance matrix of the dataset. The nth
eigenvector projects the old variable space into the nth new variable. A matrix whose
rows are the eigenvectors of the covariance matrix forms the transformation matrix that
transforms the old, highly covariant variable space into a new set of independent
(orthogonal) variables. In many cases, the majority of the variation in the data is
explained using a small number of orthogonal variables. The measure data set may have
6
thousands of dimensions but the majority of the variance can be explained by a much
smaller number of dimensions or variables.
This is referred to as “reduced order
modeling.” Using this technique, Brown and Grandhi were able to randomly create
realistic combinations of variables that represented plausible airfoils.
In this case,
plausible means that the deviations were random but the randomly generated airfoils
were still as smooth as the measured ones.
These randomly selected deviations representing realistic airfoils were then input
into a low fidelity finite element analysis to determine the perturbation of the natural
frequency of the airfoil. The result of the study was that the natural frequency of the
airfoils was significantly affected by manufacturing variation.
Brown and Grandhi’s paper illustrates that it is possible to characterize
manufacturing variation and to determine its effect on responses such as natural
frequency.
7
2. Theory and Methodology
2.1 Monte Carlo Simulation
In order to explore the design space affecting the modal response of a turbine blade,
Monte Carlo simulation will be used in this study. In this simulation, many geometric
parameters will be varied randomly to see their independent effect on the desired
response – frequency in this case. Required for this type of analysis is a 3D solid model
of a turbine blade using a certain parameter scheme. A scheme of parameters controlling
the shape of a turbine blade model is not unique. The size and shape of its features could
be defined in many different ways.
2.1.1
Geometric Parameter Scheme
For this analysis, the turbine blade will be constructed between two fixed points in
space representing the inner and outer flow path surfaces. An airfoil will be defined
between these two points using three cross-section curves as seen in Figure 5. There
will be a section at the inner radius, outer radius, and half way in between. Each airfoil
cross-section curve will be defined by its leading-edge and trailing-edge points. Other
parameters defining the airfoil will be its maximum thickness at the middle, section
curvature, leading edge diameter, trailing edge diameter, axial chord length, true chord
length, and several other parameters fully defining the airfoil section.
Since there will be three airfoil sections, spline surfaces used to connect the sections
into a solid airfoil will be second-degree (quadratic) in the vertical direction. Using
more sections could give extra flexibility to the airfoil but can also lead to reversals in
the airfoil shape. Using three sections allows for a maximum of one reversal over the
whole airfoil. A reversal is when one part of the airfoil reverses direction on its way up
the airfoil. A five-section airfoil could reverse direction four times. Typically, the
interpolation spline degree is limited to 3 making it a natural cubic interpolating spline.
The following figures show the turbine blade model that will be used. The appendix
also contains additional information about how the turbine blade geometry can be
morphed using randomly selected values for parameters.
8
Figure 3: Parametric Model of Fictitious Turbine Blade
9
2.1.2
Blade Root Geometry
The turbine blade will have a root at the bottom and a tip-shroud at the top. The
root will be defined by several parameters, not all of which will need to be varied in this
analysis. The main effect that the root will have on the mode frequencies will be due to
its mass. Its stiffness will not cause very much variation in the frequency. The airfoil
stiffness will be a significant driver of frequency variation.
2.1.3
Tip Shrouds
The tip of the airfoil will be attached to a tip-shroud (see Figure 4 and Figure 6).
This is a design feature typically used to reduce endwall losses in a turbine. Airfoils
without shrouds (unshrouded airfoils) exhibit differential motion between the outer gaspath surface and the airfoil tip. A shroud is like an outer gas-path that moves with the
airfoil because it is attached. There is no differential motion between the airfoil and the
endwall so the aerodynamic losses are reduced. The shroud can also be used as a
vibratory friction damper. Each shroud can be made to interlock with adjacent shrouds
and cause frictional damping. While I will include this design feature in the model, I
will not be analyzing the variation in damping effectiveness because it is beyond the
scope of this analysis. In the vibratory analysis, I will not model any displacement
boundary condition at the shroud interface. This vibratory analysis will be simulating
the result of a single blade “ping” test with the blade constrained in a root fixture.
Figure 4: Generic Tip Shroud Design
10
2.1.4
Generating Random Variable Combinations
In order to generate random sets of parameters for the solid models, matlab will be
used to create a latin hypercube design space. Matlab implements this using the function
lhsdesign(N,P). N is the number of samples and P is the number of variables. The
output of the function is uniformly distributed from 0 to 1. Each variable is uncorrelated
to every other variable. While uniformly distributed data may not be as realistic as
normally distributed data, uniform sampling tests all regions of the design space equally.
Such is the goal of Latin Hypercube sampling. I will define a maximum and minimum
value for each parameter I want to vary. Then I will transform the random numbers to
be between the minimum and maximum for each geometric variable that I have defined
using a linear equation. The combinations of geometric parameters will be used to
create unique 3-D models of turbine blades representing that point in the design space.
The parametric model must be rigorously tested for robustness given highly variable
input parameters. In the case that a certain variable is causing a high instance of model
failure, the parameter range is reduced.
The process of updating the parametric
computer model can be done automatically with macros in Unigraphics NX6.
Figure 5: Airfoil Section Definition
Many of the parameters in the model control airfoil section properties. Some also
control non airfoil features such as tip shroud shape and size, root thickness, airfoil
11
length, and so on. The parameters used in the solid model are shown in Table 1 along
with their mean, minimum, and maximum values.
Mean
0
0
0
0
0
0
1.4
1.4
1.15
0.5
0.5
0.5
10
16
0.25
0.25
0.25
0.07
0.07
0.07
0.02
0.75
0.38
0.25
0.25
0.25
17
1.6
0.08
0.2
0.05
0.025
0.025
0.025
1.85
1.85
1.85
airfoil_x_position_mid
airfoil_x_position_root
airfoil_x_position_tip
airfoil_y_position_mid
airfoil_y_position_root
airfoil_y_position_tip
axial_chord_mid
axial_chord_root
axial_chord_tip
cmax_mid
cmax_root
cmax_tip
flowpath_id_rad
flowpath_od_rad
hmax_mid
hmax_root
hmax_tip
led_mid
led_root
led_tip
mateface_gap
neck_length
neck_width
peak_chord_percentage_mid
peak_chord_percentage_root
peak_chord_percentage_tip
platform_angle
rim_width
shroud_bump_height
shroud_bump_length
shroud_thickness
ted_mid
ted_root
ted_tip
true_chord_mid
true_chord_root
true_chord_tip
Min
-0.03
-0.03
-0.03
-0.03
-0.03
-0.03
1.3
1.3
1.1
0.4
0.4
0.4
9.5
15.5
0.2
0.2
0.2
0.04
0.04
0.04
0.01
0.65
0.36
0.2
0.2
0.2
14
1.5
0.05
0.15
0.04
0.015
0.015
0.015
1.8
1.8
1.8
Max
0.03
0.03
0.03
0.03
0.03
0.03
1.5
1.5
1.2
0.6
0.6
0.6
10.5
16.5
0.3
0.3
0.3
0.1
0.1
0.1
0.03
0.85
0.4
0.4
0.4
0.4
20
1.7
0.11
0.25
0.06
0.035
0.035
0.035
1.9
1.9
1.9
Table 1: Geometric Parameters and their Ranges
2.1.5
Explanation of Parameter Scheme
They are listed here in alphabetical order. The first six are positioning parameters
for the airfoil sections. They move the airfoil sections in the X and Y directions
independently. There are six of them because there are three sections in the airfoil.
12
The next parameter is airfoil axial chord. This is the distance front-to-back of the
airfoil section when viewed from the side. Axial refers to the direction of the engine
axis. There are three of these parameters; there is one for each airfoil section.
The next set of three parameters is called “cmax” here. It stands for “camber.”
What is meant by camber is the curvature of the airfoil section. In the defining section
(See Figure 5), the parameter controls the distance from the center of the airfoil max
thickness circle to the line tangent to the leading edge and trailing edge arcs. A larger
camber distance causes the minimum moment of inertia to increase. The maximum
moment of inertia, which is perpendicular to the minimum, is not changed very much.
The next two parameters are flowpath inner radius and flowpath outer radius.
Airfoil length is the difference between these two parameters. Airfoil radial position is
related to the sum of these parameters. The frequencies should depend on airfoil length
but should be independent of airfoil radial position.
A more intelligent parameter
scheme would define airfoil length separately from airfoil radial position.
The next three parameters are “hmax.” This refers to airfoil max thickness for each
airfoil section. It is defined in the airfoil section as the diameter of the arc tangent to the
concave and convex side of the airfoil. Increasing this parameter stiffens the airfoil and
adds mass to the airfoil.
The next three parameters are “led”. This stands for Leading Edge Diameter. It is
the diameter of the leading edge arc for each section. It adds stiffness and mass to the
leading edge of the airfoil.
The next parameter is “mateface_gap” which refers to the clearance between the
platforms of two adjacent blades. Its range is set to be very small (between 0.010” and
0.030”).
The next parameter is
2.2 Modal Analysis
After that, every blade will undergo modal analysis implemented using the Finite
Element approach.
I will implement the analysis using ANSYS, a finite element
analysis software package widely used in the industry. Modal analysis in its most basic
sense solves the homogeneous differential equation:
13
M XÝÝ K X  0
[3]
In this equation, X is a vector of the displacements of all the degrees of freedom of
the mass system. K is the stiffness matrix that relates the displacements to one another.

M is the mass matrix and it is a diagonal matrix that contains all the masses associated
with each displacement. In three dimensions, each mass will have three degrees of
freedom. Each entry in the displacement vector, X, is one of those displacements. The
continuous material of the turbine blade will be discretized into a finite number of
elements that have mass and stiffness. Inside each element, the continuous displacement
gradient is approximated as a simple, low order function continuous at the element
boundaries. Continuity in spatial derivatives of displacement is not necessarily enforced
at element boundaries.
The finite element software requires certain inputs from the engineer. The first is
the geometry of the object to be analyzed.
The engineer also inputs the material
properties, namely mass density, elastic modulus and Poisson ratio. For this analysis, all
the properties will be assumed as homogeneous and isotropic. Homogeneous means that
the properties are constant throughout the material. Isotropic means that the non-scalar
properties do not depend on orientation. Pertinent to this analysis, the elastic modulus
and Poisson ratio will not vary with respect to orientation. Some turbine blades are
made of anisotropic materials such as single crystal nickel superalloys. Anisotropic
material properties will significantly affect the vibratory response of the turbine blade.
In the scope of this analysis, we will assume the blades are made of an “equiaxed,”
isotropic material. This type of material has a random crystal structure throughout. The
anisotropy of any individual crystals is averaged out over the entire sample of material.
The finite element software program automatically generates the stiffness and mass
matrices (and damping matrices if required). Although, this analysis will not include
any internal or external damping effects.
2.2.1
Execution of Computations
The modal analysis was performed using Unigraphics NX6 Advanced Simulation
with ANSYS 12.0 as a solver. The model is prepared in NX Simulation. This includes
meshing the model into a finite element model, applying displacement fixity conditions
14
at the root, and defining material properties. The material used for this analysis was a
simple structural steel. The material does not actually matter since the desired outcome
is just the geometric sensitivity. Material variation affects all results uniformly and
therefore cancels out when comparing frequencies relative to one another.
The finite element modes were meshed using NX Simulation.
The meshes
contained between 25,000 and 35,000 nodes. When the mesh was submitted to ANSYS
for solution, the solution took 30 to 60 seconds to solve the first 5 mode shapes. The 50
parametric model updates and finite element solutions were performed manually due to
the quickness of the solution.
A more investigative approach could explore more
combinations of parameters by using an automation macro.
15
3. Results and Discussion
The parameter scheme chosen for the turbine blade proved to be robust. A wide
range of parameters produced stable, realistic turbine blade geometries. Out of 50
parameter combinations of 37 parameters, 100% resulted in valid part geometries. This
was due, in part, to the limited range of the parameters. The parametric blade model was
tested extensively for robustness.
Parameter ranges were determined based on the
tendency of the model to produce valid geometries. Examples of failure would be:
airfoil falling off platform; shroud falling off airfoil tip; airfoil to platform blend failure.
3.1 Mode Shape Identification
Under special conditions of symmetry, mode shape identification can be difficult
and natural frequencies can be very close to one another. All of the natural frequencies
of this turbine blade were spaced out enough that there was no confusion about which
mode was associated with which frequency.
For the purposes of this analysis it is
assumed that the mode shapes, when sorted by frequency, are the same for every blade.
3.2 General Trends in Response
3.2.1
First Mode Shape
The first mode of vibration (see Figure 14) is a bending mode. The airfoil is tending
to bend about the root section’s minimum moment of inertia. The analysis shows that
the parameters that affect root stiffness have a large impact on the first mode frequency.
The first mode frequency is also affected by parameters that affect tip mass. Parameters
that affect tip section stiffness have very little impact on this mode frequency.
3.2.2
Second Mode Shape
The second mode of vibration was also bending about the root (see Figure 15), but
in contrast to the first mode, the deflection was in the perpendicular direction. The blade
is tending to bend about the root section’s maximum moment of inertia. The frequency
is correspondingly higher due to the increased stiffness in that direction. The tip mass
affects this frequency similarly to the first mode frequency.
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The chord length of the
airfoil root section also causes increased stiffness in this vibration mode thereby
increasing the frequency.
3.2.3
Third Mode Shape
The third mode of vibration is a torsional mode (see Figure 16). This means that the
vibration is characterized by twisting of the airfoil tip. This frequency is affected by the
torsional stiffness of the root and midsection of the airfoil. The frequency is also
affected by tip rotational moment of inertia.
3.2.4
Fourth Mode Shape
The fourth mode is not easy to describe (see Figure 17). It exhibits motion in the tip
shroud and the airfoil trailing edge. The two motions of the shroud and the airfoil
trailing edge are opposing one another as if to counterbalance each other’s effect on the
structure. This could be considered a second-order bending mode. The frequency is
affected by many geometric factors. The linear regression in the following section will
reveal the trends more clearly than one can discern from simply looking at the response
matrix.
3.2.5
Fifth Mode Shape
See Figure 18 for a displacement plot of the fifth mode. The fifth mode shape
shows much of its deflection in the corner of one side of the tip shroud. That motion is
counterbalanced by motion in the trailing edge and leading edge of the airfoil. This
could be considered a third-order bending mode combined with some twisting of the tip
shroud.
3.3 Regression Functions
3.3.1
Linear Regression
The first step in interpreting the frequency results is to see which parameters most
highly affect the frequencies. One approach to this is to create a linear regression. The
geometric parameters will be the predictors while the calculated frequencies are the
responses. Matlab can generate coefficients for a linear regression for each response and
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calculate the residual for each data point. The residual is the difference between the
actual response and the predicted response using the linear regression at an observation
(data point). If the regression is performed using the predictor variables as dimensional
quantities, then the coefficients will have units of response/predictor. In this case it
would be Hertz/Inch. This means that the response should change by a certain number
of Hertz for every inch of variation in the predictor variable. For some of the geometric
parameters, one inch is a fairly acceptable variation. For other parameters, one inch of
variation is inconceivable and would cause the model to fail.
To account for this, the predictors can be transformed to a new space depending on
their acceptable ranges of variation.
In this new space, zero would represent the
minimum value for the parameter and one would represent the maximum value for the
parameter. This is referred to as non-dimensionalizing.
The linear influence coefficient resulting from the dimensional and non-dimensional
predictor variables is quite different.
3.3.1.1 Pareto Analysis
The regression coefficients represent the strength of influence each variable has on
the response.
To create the Pareto Chart, sort the magnitude of all the influence
coefficients and plot the values on a bar chart. The typical result is that a small number
of the parameters will have a large percentage of the influence on the response.
3.3.2
Nonlinear Regression
The responses may be linear over a small range of predictor variables, but the
approximation may break down over a larger domain of the variables. The next step
after linear regression is nonlinear regression. Instead of just regressing the responses
with the individual variables, we also use combinations of two or more variables.
FIPER can do this automatically. One can choose to use combinations of different
variables and combinations of the same variable (the variable squared). The result of
this regression can also be plotted on a Pareto Chart to visually show the influence of
variables and combinations thereof.
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3.3.3
Partial Least Squares Regression
Another regression technique particularly well suited to this situation is called
partial least squares regression. Herman Wold and his son Svante Wold developed PLS
regression in the early 1980’s. It has also been called “Projection to Latent Structures”
but the name Partial Least Squares has been more often used.
It finds linear
combinations of X that most highly explain variability in linear combinations of Y.
See the following equation
….
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4. Conclusion
I conclude that this work has proven that… It can be used in the future to predict
the frequency response of a design space.
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5. References
Abdi, H. (2003). Partial Least Squares (PLS) Regression. In M. B. Lewis-Beck,
Encyclopedia of Social Sciences Research Methods. Thousand Oaks, CA: Sage
Publishing.
Brown, J. M., & Grandhi, R. V. (2008). Reduced-Order Model Development for Airfoil
Forced Response. International Journal of Rotating Machinery , 2008, 1-12.
Hastie, T., Tibshirani, R., & Friedman, J. (2008). The Elements of Statistical Learning
(2nd Edition ed.). Stanford, CA, USA: Springer.
Norton, R. L. (2006). 6.11 - Designing for Fluctuating Uniaxial Stresses. In R. L.
Norton, Machine Design, an Integrated Approach 3rd Edition (pp. 356-360).
Upper Saddle River, NJ, USA: Prentice Hall.
Petrov, E. P. (2008). A Sensitivity-Based Method for Direct Stochastic Analysis of
Nonlinear Forced Response for Bladed Disks with Friction Interfaces. Journal of
Engineering for Gas Turbines and Power , 130 / 022503-1.
Richardson, M. H. (1997). Is It a Mode Shape or an Operating Deflection Shape? Sound
& Vibration Magazine (30th Anniversery).
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6. Appendices
6.1 Appendix 1: Turbine Blade Parametric Model
Figure 6: Top View of Fictitious Parametric Turbine Blade Model
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Figure 7: View of Fictitious Turbine Blade Root Attachment Geometry
Figure 8: Close-up View of Turbine Airfoil Section Definition
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Figure 9: Random Turbine Blade – 1
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Figure 10: Random Turbine Blade – 2
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Figure 11: Random Turbine Blade – 3
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Figure 12: Random Turbine Blade – 4
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Figure 13: Random Turbine Blade – 5
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Figure 14: Mode Shape 1 at 299 Hz
Figure 15: Mode Shape 2 at 712 Hz
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Figure 16: Mode Shape 3 at 1267 Hz
Figure 17: Mode Shape 4 at 1694 Hz
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Figure 18: Mode Shape 5 at 2905 Hz
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