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\title{\Large\textbf{MINIMIZING SENSITIVITY OF BLADED DISKS TO MISTUNING}}


\author{\vspace{.25in}\\
\textbf{Joseph C. Slater \textnormal{and} Andrew J. Blair}\\
\\
 {\normalsize Department of Mechanical and Materials Engineering} \\
 {\normalsize Wright State University}\\
 {\normalsize Colonel Glenn Highway}\\
 {\normalsize Dayton, OH 45435}\\
}
\maketitle

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\begin{abstract}
High cycle fatigue due to vibration continues to be a leading cause
of blade failure in turbomachinery.  Designing for the alleviation
of high cycle fatigue is one of the most complex  problems
facing engineers.  The complexity of the problem has given
rise to many design propositions and many techniques for evaluating
their merit.  In this work, the Finite Element Method is used to
analyze minor hub design modifications intended to minimize the
propensity for mode localization in turbomachinery blades, thus
alleviating high cycle fatigue.
\end{abstract}


\subsubsection*{Nomenclature}
\begin{tabular}{lcl}
$[\mathbf{M}]$ & & mass matrix\\
$\{\mathbf{\Phi}\}$ & & normalized mode shapes \\
\\

$E$ & & Young's modulus \\
$M$ & & mass matrix\\
$R$ & & stress ratio \\
$VM$ & & Von Mises stress \\
\\

$\nu $ & & Poisson's Ratio \\
$\rho $ & & density \\
\\ 

$i$ & & mode shape \# \\
$m$ & & mistuned case \\
$t$ & & tuned case \\
\end{tabular}
\\ 

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\section{Investigation}

\indent In bladed disk assemblies, the disk acts as a coupling device between the
blades.  As the stiffness of the disk increases, blade coupling
decreases. It has been shown that weak interblade coupling leads to
high levels of mode localization when blades are 
mistuned\cite{pierre1,pierre2,wei1,ewins2,griffin1,kaza1,wei1}.  Mode localization also occurs in
bladed disks as a result of their symmetry.  Sets of axisymmetric
modes combine to form a basis set from which drastically localized
mode shapes can be generated.  Bladed disk assemblies are
traditionally designed to be symmetric for the sake of balance.  This leads
to two hypotheses to be investigated in this work:  1)  Decreasing
the stiffness of the disk by varying the geometry and/or material
composition may reduce mode localization due to mistuning, 2)
Destroying the symmetry of the disk, yet maintaining balance, may
reduce mode localization due to mistuning (The idea here being to 
split repeated modes such that they have unique natural frequencies).
  Each of these was
investigated separately, as well as combinations of the two.

A model of an eight bladed disk based on an experimental testbed  
 at Wright State University was constructed in {\small
\textsf{ANSYS}\textregistered} using eight noded ``brick"
elements (Figure (sorry, it's not in this doc)). The
model was designed to exhibit a propensity for mode localization
similar to that of turbomachinery bladed disk assemblies.  The bladed disk
model was adjusted to provide weak coupling between the blades
resulting in tightly packed sets of natural frequencies, eight
modes in each.  The blade deformation in the first set of modes is
predominately a first beam bending mode, in the second set of modes
it is predominately a first beam torsional mode, and in the third
set of modes it is predominately a second beam bending mode.

\begin{figure}
\begin{center}
\fbox{There once was a figure here}
\end{center}
\caption{\label{undeformed}Finite element model.}
\end{figure}

Three types of mistuning were chosen for the investigation.  The
first two were: one percent of the mass of one blade added to the
tip of one blade, and one percent blade mass removed from the tip
of a single blade.  This results in two cases of mistuning for the
symmetric models.  For non-symmetric models, however, each type
(addition and subtraction) must be investigated on each of four
different blades. This results in eight different cases of
mistuning for the non-symmetric models.  The preceding models,
representing minor damage cases to an individual blade or all but
one blade, are in agreement with accepted practice in prior 
studies\cite{pierre1,pierre2}.  The final type of mistuning cases is
random mistuning. Three random patterns were chosen such that the
mass added to the tip of each blade varied between plus or
minus one percent of the mass of one blade.

\subsection{Models}

\indent All models used in this study were variations of the symmetric,
constant stiffness system referred to as the baseline model (Figure
\ref{undeformed}). The dimensions of the
disk are as follows:  OD = 188 mm, ID = 87.5 mm, and thickness = 14
mm. The dimensions of each blade are: length
= 62 mm, width = 32 mm, and thickness = 1.5 mm. This model was constructed
entirely of eight noded brick elements ({\small
\textsf{ANSYS}\textregistered} element--- Solid 45), having three
degrees of freedom per node.  The material properties used were that of mild
steel:  Young's Modulus \(E= 200\) GPa, density $\rho= 7800
\mbox{ kg/m}^3$, and Poisson's Ratio $\nu = 0.3$. Lines were also included
in the geometry that allowed meshing to be performed in {\small
\textsf{ANSYS}\textregistered}. A proper mesh in this study had two
important properties. Yada yada.


\subsection{Mistuning}

\indent Three basic types of mistuning were chosen to best represent realistic challenges to the
robustness of a bladed disk design.  The first two were addition
and subtraction of mass to a single blade.  These were chosen to
represent foreign object damage (FOD) to either a single blade (subtraction) or
all but one blade (addition).  The final type of mistuning chosen
was the random addition and subtraction of mass from each blade,
representative of the small variance between blades due to
manufacturing variations or more distributed wear/damage.

Yada yada.

%\placedrawing[!ht]{blnumb.lp}{fig:bladenumbering}


Three random patterns of mistuning were also investigated.  As
mentioned before, the addition of one percent blade mass to each
blade was chosen as the tuned, or nominal case.  In determining the
random patterns to be used, the mass to be added to any location
was allowed to vary from zero to two percent of a single blade
mass.  The patterns
generated follow in Table \ref{random} and the blade numbering
scheme is shown in Figure (not in sample). 


\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
    & Random 1      &     Random 2  &    Random 3\\
\cline{2-4}
\raisebox{2ex}{Blade Number}  & $10^{-3}$ Kg   &    $10^{-3}$ Kg &    $10^{-3}$  Kg\\
\hline
\hline

1   &    0.0168  & 0.3343  & 0.3413\\
2    &   0.0260  & 0.2867  & 0.4431\\
3    &   0.2579  & 0.4529  & 0.3710\\
4    &   0.3267  & 0.4119  & 0.1278\\
5    &   0.0037  & 0.2565  & 0.0231\\
6    &   0.1866  & 0.0448  & 0.3583\\
7    &   0.0325  & 0.3183  & 0.1598\\
8    &   0.2032  & 0.2025  & 0.3080\\

\hline

\end{tabular}
\end{center}
\caption{\label{random} Random mistuning patterns used in the investigation.}
\end{table}




\section*{Acknowledgements}
Thanks to all those who've built LaTeX into what it is today.

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