Assessment of the Sensitivity of Natural
Frequency of Vibration with Respect to Variation
in Geometric Parameters 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, Thesis 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
List of Equations .............................................................................................................. vii
Acknowledgement .......................................................................................................... viii
Abstract ............................................................................................................................. ix
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 ................................................................................. 4
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............................................................................. 9
2.1.3
Tip Shrouds ............................................................................................ 9
2.1.4
Generating Random Variable Combinations ......................................... 9
Modal Analysis ................................................................................................. 10
2.2.1
Execution of Computations .................................................................. 11
iii
3. Results and Discussion .............................................................................................. 12
3.1
Mode Shape Identification ................................................................................ 12
3.2
General Trends in Response ............................................................................. 12
3.3
Regression Functions ........................................................................................ 12
3.3.1
Linear Regression................................................................................. 12
3.3.2
Nonlinear Regression ........................................................................... 13
3.3.3
Partial Least Squares Regression ......................................................... 13
4. Conclusion ................................................................................................................. 14
5. References.................................................................................................................. 15
6. Appendices ................................................................................................................ 16
iv
List of Tables
No table of figures entries found.
v
List of Figures
Figure 1: Modified Goodman Diagram ............................................................................. 2
Figure 2: Campbell Diagram ............................................................................................. 4
Figure 3: Diagram of Parametric Turbine Blade Model .................................................... 9
vi
List of Equations
Equation 1: One Dimensional Vibration Equation ............................................................ 3
Equation 2: Free Vibration Equation with Arbitrary Degrees of Freedom ..................... 10
vii
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.
viii
Abstract
This is what I did, what I found and why it was significant.
ix
1. Introduction and Background
This project deals with prediction of 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].
1.1 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 lines are drawn by…
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 (not moving) 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
2
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 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. The equation is of the form:
Ý kx  0
mxÝ
Equation 1: One Dimensional Vibration Equation

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).
3
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 [2]. This 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 frequencies (in
Hertz). The natural frequencies of the blade are plotted as horizontal lines on the graph.
Then, straight lines of various integer slopes are plotted over top of the horizontal lines.
These sloped lines represent the excitation frequency caused at a certain engine speed.
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.
This could be the number of nozzles before the blade or the number of nozzles after the
blade. Any repeated feature that the blade will see in its travel around the engine is an
engine order. If an engine order line crosses a natural frequency line at an operating
speed it is said to be a “resonant crossing.”
Figure 2: Campbell Diagram
The Campbell Diagram above is showing fabricated data for illustrative purposes.
The horizontal, drooping lines are the natural frequencies of the blade. The sloped lines
are the engine orders. Again, the horizontal axis is the engine rotational speed (in RPM)
4
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 they 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 research 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 all 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.
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.
5
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. 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
thousands of dimensions but the majority of the variance can be explained by a much
smaller number of dimensions or variables.
6
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 & Grandhi, 2008)
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 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. 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.
8
Figure 3: Diagram of Parametric Turbine Blade Model
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. 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 gas-path surface and
the airfoil. 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
losses are eliminated. 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. It is beyond the scope of this analysis. In the vibratory
analysis, I will include a surface normal constraint at the tip shrouds to simulate the true
engine running condition but I will not model it with friction.
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
9
lhsdesign(N,P). I will define a maximum and minimum value for each parameter I want
to vary. Then I will transform the random variables from a normalized (0 to 1)space into
dimensional variables between the minimum and maximum for each geometric variable.
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 process of updating the
parametric computer model can be done automatically with macros in Unigraphics NX6.
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:
M XÝÝ K X  0
Equation 2: Free Vibration Equation with Arbitrary Degrees of Freedom

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 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 and elastic modulus. 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 will not vary
10
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.
The software program automatically generates the stiffness and mass matrices (and
damping matrices if required). This analysis will not include any internal or external
damping effects.
2.2.1
Execution of Computations
The large number of finite element solutions to be generated will require automation
for practicality. To this end, iSight FIPER will be used to automate the process of
updating blade geometry in Unigraphics NX6 and performing the modal analysis in
ANSYS.
FIPER is a program best suited to Monte Carlo simulations and DOE’s
(Design of Experiment). Both of these techniques involve applying combinations of
variables to a system and recording its response. FIPER will store the predetermined set
of randomly selected geometric parameters and insert them into a text file in the format
of NAME = VALUE. This format of data file is used by Unigraphics NX6 to update
parameters internal to the part file.
After the part file is updated with the parameters it will be passed to a directory
where ANSYS will be called from a command line. A control file will be used to
execute macro commands. Displacement boundary conditions are applied to certain
named faces in the model – CONTACT for example. The material properties are
assigned to the body of a certain name - BLADE for example. See appendix **** for
the ANSYS macros used.
11
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 2000
parameter combinations of 58 parameters, *********% resulted in valid part
geometries. Examples of failure include: blade falling off platform or the tip shroud
unable to fully cover airfoil.
3.1 Mode Shape Identification
Since geometry configurations are changing so drastically, some of the mode shapes
of vibration changed dramatically. This led to the problem of how to identify modes.
One approach is to sort the frequencies by numerical value. One problem with this
approach is that the frequencies of vibration sometimes switched order for modes with
natural frequencies close to one another. ***I solved this problem by…***
3.2 General Trends in Response
Certain variables appear to affect the frequencies more than others. *** caused the
first mode of vibration to …
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 multivariate linear
regression.
The geometric parameters will be the predictors while the calculated
frequencies are the responses. FIPER can automatically create linear regression for each
response and calculate the residual for each data point. The residual is the difference
between the actual response at an observation and the predicted response using the linear
regression. Before performing the regression, the results may be more meaningful if the
predictor variables are normalized to their min-max range.
12
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. The
influence coefficients typically follow a Pareto Distribution.
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.
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
….
13
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.
14
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).
15
6. Appendices
Appendices contain information relevant to the execution of the project but would
interrupt the flow of the paper.
16