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Section A.1 in Appendix A. It should be noted, that despite using singular value
decomposition instead of principal component analysis, the aforementioned equation
remains valid; the  and x matrices are simply replaced by the  and V matrices,
respectively. Now, accurately constructed plots of the first three mode vectors allow for
the visualization of the most significant amounts of deviation between the asmanufactured blades and the design intent.
A.3 Principal Component Analysis Code Validation
The need for the validation of the code developed at Virginia Tech, which
performs principal component analysis on two-dimensional compressor blade data, was
twofold. First, it was critical to ensure that implementing the singular value
decomposition of the alternative form of the data matrix, opposed to the eigenvalue
decomposition of the data matrix’s covariance matrix, would not have a deleterious effect
on the final results; secondly, code validation was needed to ensure that the code’s
preprocessing routine would not impart any undesired alteration to the original data set.
Two simple test cases, based on altering only the chord length of the rotor’s hub
section, were employed to carry out the code validation; these test cases are outlined
below and consist of using a data set with zero variance and a randomly distributed data
set.
A.3.1 Zero Variance Data Set
The zero variance data set test case was the simpler of the two test cases
employed, and was intended to primarily test the preprocessing routine of the Virginia
Tech code. By implementing principal component analysis on a zero variance data set, it
ensured that the deviations returned from the principal component analysis would be
equivalent to the mean deviation between the design intent and altered geometry, as long
as the preprocessing routine did not inaccurately reconstruct the data set. Creation of the
zero variance data set required identical deviations throughout the data set, and obtaining
such a data set was accomplished by replicating a single altered geometry 96 times.
Construction of the single altered geometry was carried out by using the Rolls Royce
blade profile generating code BD76 to scale only the chord length of the hub design
intent profile in Section 2.2 by 19.4%.
Implementation of the Virginia Tech code, with the aforementioned zero variance
data set, revealed that the preprocessing routine did not impart any undesired alterations
to the data set during reconstruction. Evidence of this fact was seen in two of the
principal component analysis outputs. First, mode one represented all of the deviation in
the data set, which should occur when the scatter of the data is in a single direction; in
this case, the direction in which the x-y coordinate pairs were altered to change the chord
length of the altered geometry. And secondly, the chord length for mode one was 19.4%
greater than that of the design intent, which is equivalent to the mean deviation between
the design intent and altered geometry.
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