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
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CONTENTS
CONTENTS ..................................................................................................................... iii
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
ACKNOWLEDGMENT .................................................................................................. vi
ABSTRACT .................................................................................................................... vii
1. INTRODUCTION / BACKGROUND ........................................................................ 1
iii
LIST OF TABLES
iv
LIST OF FIGURES
v
ACKNOWLEDGMENT
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
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ABSTRACT
Type the text of your abstract here.
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1. INTRODUCTION / BACKGROUND
This project deals with prediction of the natural frequencies of turbine blades. It is
important to predict turbine blade natural frequency to prevent resonance during an
operating condition. Turbine blades experience unsteady forcing functions constantly
and 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].
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 such factors can
be: number of upstream vanes (or 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.
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 (like idle, climb, or cruise). 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 conditions.
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 excitation factors. 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.
Usually the lines are not perfectly horizontal because they vary depending on engine
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speed and temperature. For turbine blades, the 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.
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.
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 geometric 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. For higher degree design spaces, it is not immediately
obvious how to explore the boundaries and interior regions of this space.
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 were 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
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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
was 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.
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. [3]
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.
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2. METHODOLOGY
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.
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.
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 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 end-wall so the losses are eliminated. The shroud
can also be used as a vibratory friction damper. Each shroud can be made to interlock
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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.
In order to generate random sets of parameters for the solid models, matlab will be
used to create multivariate normal distributions with no covariance. Matlab implements
this using the function mvtrnd(). I will define a maximum and minimum value for each
parameter I want to vary and transform the normalized (-1 to 1) variables into dimensional variables between the minimum and maximum for each geometric variable.
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