# Reduced-Order Model Development for Airfoil Forced Response

Authors: Jeffrey M. Brown; Ramana V. Grandhi
Journal: International Journal of Rotating Machinery
 (2008)

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## Abstract

Two new reduced-order models are developed to accurately and rapidly predict geometry
deviation effects on airfoil forced response. Both models have significant application to improved
mistuning analysis. The first developed model integrates a principal component analysis
approach to reduce the number of defining geometric parameters, semianalytic eigensensitivity
analysis, and first-order Taylor series approximation to allow rapid as-measured airfoil
response analysis. A second developed model extends this approach and quantifies both random
and bias errors between the reduced and full models. Adjusting for the bias significantly
improves reduced-order model accuracy. The error model is developed from a regression analysis
of the relationship between airfoil geometry parameters and reduced-order model error,
leading to physics-based error quantification. Both models are demonstrated on an advanced
fan airfoil's frequency, modal force, and forced response.


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## Body

## 1. Introduction

Effective
airfoil dynamic response analysis ensures rotor reliability and requires
prediction of resonance avoidance margin, forced response, and mistuning.
Standard practices predict dynamic response using finite element models (FEMs)
of design intent geometries. While sufficient for some cases, this standard
approach does not explicitly consider airfoil structural response variations
caused by random manufacturing deviations from design intent geometries.
Because hundreds or thousands of these simulations would be required to assess
effects from random variations, a new more efficient airfoil modal and a forced
response prediction process are required.Existing literature contains a significant body of
work developing efficient mistuned rotor forced response prediction using
reduced-order models (ROMs) [1–6]. These efforts have shown strong amplification of
rotor forced response caused by small perturbations in blade-to-blade
frequency. While effective, these prior models are limited in two significant
ways. First, they assume that airfoil frequencies vary, but airfoil mode shapes
remain nominal. This assumption enables computational efficiencies but
geometric deviations clearly alter blade-to-blade mode shapes, thus altering
each blade's modal force and impacting mistuned response. Their second
limitation is the required experimentally obtained blade-to-blade frequency
variation input. Such empirical measures are subject to error, particularly for
integrally bladed rotors (IBRs) and the known challenge to isolate their
individual airfoil frequencies from the rotor system response. These
experimental results also are not connected to airfoil geometric parameters
that can be controlled in the design process for acceptable frequency scatter
manufacturing. Because of these limitations, physics-based ROMs of airfoil
modal and forced response that explicitly account for geometric deviations are
needed as they provide accurate input to existing
mistuning models and include frequency scatter in design. Further, since
existing mistuning prediction methods do not consider mode shape variation, a
ROM is needed to show the significance of mode shape variation on forced
response and leads to future improved mistuning analysis tools.ROM development begins with an approach to create a
reduced set of geometry parameters defining manufacturing variation. Previous
efforts in reduced-order airfoil geometry modeling include Garzon and
Darmofal's use of principal component analysis (PCA) [7]. The PCA approach is a
common statistical method that creates a reduced basis space through an
eigenanalysis of the covariance between parameter deviations [8]. The research's results
demonstrate the effectiveness of the technique for turbomachinery applications.
An alternate approach was demonstrated by Capiez-Lernout et al. in their
development of a technique characterizing manufacturing tolerances for mistuned
bladed disk with a dispersion parameter [9]. This ad-hoc estimation of the geometry effects on
response does not directly depend on measured geometry but does have
computational advantages. Because the PCA approach is directly related to
measured deviations, it is applied in this ROM development.With a reduced geometry model determined, development
of a reduced-order response method remains. Taylor series approximations are an
attractive method assuming that the required sensitivities can be efficiently
calculated. Methods to rapidly predict sensitivities of modal response,
eigensensitivities, have been developed by Fox and Kapoor for unique
eigenproblems and Friswell for cyclic symmetry problems [10, 11]. These equations are
semianalytic and allow sensitivity calculation from a single FEM solution with
efficiency improvements described later. Such approaches have been widely used
in optimization applications, but not for airfoil modal response approximation
over the range of manufacturing deviations considered in this effort.With these existing tools, the first of two airfoil
response ROMs, the standard ROM, is developed. First, PCA is used to create a
reduced basis set of the manufacturing deviations. Eigensensitivities are then
efficiently calculated semianalytically with respect to this new basis, and
these are used in a first-order Taylor series modal response approximation.
These approximate modal quantities are then used in a modal domain forced
response analysis. When combined, the integrated approaches lead to an
exceptionally efficient and accurate model.Though accurate, as with all
approximations, there is error. Model errors have been widely recognized as
critical to the design and analysis  process,  and
the need for their accounting has been outlined in several professional
editorial policy papers [12, 13]. In this research, a second ROM is developed, an
error-quantified ROM, that captures the error developed in the model reduction
process. This a posterior error model requires a linear regression model of the
errors obtained between a limited number of full model and standard ROM
comparisons. Results from the model are able to reduce the standard ROM error
and quantify the approximate model uncertainty.It is noted that these models do not account for the
impact of geometric deviations on unsteady aerodynamic loading. While this may
be an important factor in the prediction of forced response, the development of
a reduced-order model for aerodynamics is an ongoing challenge not considered
in this research.The following sections develop the two ROMs. How
measured airfoil deviations are reduced to a practical number of parameters
with PCA is defined in Section2. Section 3 develops the standard reduced-order
modal and forced response models, and Section 4 introduces the error-quantified
ROM. These sections are followed by results from a real-world component that
show the significance of geometric deviations from design intent and
demonstrate the accuracy of both developed models. The application of these new
tools provides improved input to existing mistuning prediction models that show
the effect of geometrically induced mode shape variation on forced response and
create a model that can be used for future mistuning tool developments.


## 2. Reduced-Order Airfoil Geometry Model

In the past, deviations from design intent have been
checked with shaped tools and manual gages. Such devices are pass-fail tools
providing no quantitative response information back to the engineer. Because of
the rotor response sensitivity to geometric variations, new measurement
techniques are desired. One approach uses coordinate measurement machines
(CMMs) that collect data through a geometry traversing probe that obtains
spatial data points at regular intervals. Each measured airfoil may provide
thousands of measured data points. Assessing the sensitivity of each of these
locations to perturbation would require significant computational resources,
hence the need for a reduced-order geometry model retaining a limited set of
parameters quantifying geometry deviations. PCA is attractive given its ease of
implementation and the creation of minimum, a set of retained basis vectors to
represent correlated geometry variations.PCA is implemented by storingn measured
three-dimensional coordinate data points in vector x∈ℝ3n. A set of p measured
airfoils results in matrix, X∈ℝ3n,p. Since we are interested in variations from the
average blade, the mean value of each row is subtracted from each member of the
row to give a matrix of measured deviations, ΔX, where each element is (1)Δxi,j=xi,j−x¯i,i=1,2,…,3n;j=1,2,…,p, where x¯i is the average
of the ith row. It is important
to note that the average, x¯i, is not necessarily the original design intent. Also,
subtracting the row mean from each element makes the expected value of each row
zero. The first-order covariance matrix of ΔX defines the
statistical relationship between a measurement point deviation and all other
points, and its eigensolution leads to eigenvectors that can be used to form a
new subspace optimally representing variation. This is written in standard
eigenproblem form: (2)Cov(ΔX)Ψ=ΨD, where D and Ψ are the
eigenvalue and eigenvector matrices, respectively. The eigenvectors are the
principal components modes of the measured data, and the eigenvalues are the
principal component variances that indicate the data variance each principal
component captures. Based on these eigenvalues, graphical and statistical
methods can be used to retain a limited set of basis vectors. Also of
importance, the principal components are orthogonal, and therefore,
uncorrelated statistically. The PCA transforms a large set of correlated
parameters into a small set of uncorrelated parameters.Transformation of the measured deviations,ΔX, to the principal component basis requires the linear
operation: (3)Z=ΨT[ΔX], where the eigenfunction matrix
is multiplied by the deviation matrix to give the z-score matrix, Z∈ℝm,p with m the number of
retained principal component modes. These scores are effectively regression
coefficients for the new principal component basis and define the participation
of each PCA mode in each measured geometry. The above algorithms, (1)–(3),
are the covariance method of PCA, and the columns
of Z represent the
Karhunen-Loeve transformation.How these z-scores
and principal component modes are integrated into a reduced-order forced
response model is described in the following section.


## 3. Standard Blade-Alone Forced Response Reduced-Order Model

The approach used in this ROM development is Taylor
series approximation using first-order sensitivities. Sensitivity calculations
can be computationally expensive when calculated numerically via finite
difference methods that require an FEM evaluations for each design parameter.
This work proposes to use semianalytic methods that replace the costly
calculations of the process, that is, decomposing the stiffness and mass matrix
and solving the matrix eigenvalue problem.Combining the semianalytic sensitivity methods defined
in [14] with the
reduced-order geometry model results from Section 2 leads to the following
principal component mode response sensitivities: (4)∂λi∂ψj=ϕiT(∂K∂ψj−λi∂M∂ψj)ϕi,∂ϕi∂ψj=∑g=1rcigϕg,  where i identifies the
vibration mode number, j is the
principal component mode number, and r is the total number
of retained vibration modes. The constant terms are calculated: (5)cig=ϕgT(∂K/∂ψj−λg(∂M/∂ψj))ϕi(λg−λi),cii=−12ϕiT∂M∂ψjϕi, where λi and ϕi are eigenvalue
and mass-normalized eigenvector, K and M are mass and
stiffness matrices, and ψj are retained
principal component modes. The stiffness and mass matrix derivatives are
numerically computed through nominal and perturbed finite element models.
Forced response sensitivity can also be calculated directly but is not done so
here because of the need to explicitly retain frequency and mode shape
sensitivities for use as input to mistuning and modal force predictions.Prediction of modal stress sensitivity requires the
derivative of the strain-displacement equation. Differentiating this equation
with respect to thejth principal
component mode gives (6)∂σi∂ψj=DB∂ϕi∂ψj+D∂B∂ψjϕi, where D is the
elasticity matrix, B is the
strain-displacement matrix, and σi is the stress
vector of the ith vibration mode.
As with the mass and stiffness matrices, the sensitivity of the
strain-displacement matrix is calculated numerically.Once the sensitivities have been computed, the standard
ROM eigenvalues and eigenvectors are computed with a first-order Taylor series
expansion. The approximations are(7)λ˜i=λi0+∑j=1m∂λi∂ψjdψj,ϕ˜i=ϕi0+∑j=1m∂ϕi∂ψjdψj, where λi0 and ϕi0 are the average
eigenvalue and eigenvector results and the tilde symbol annotates an
approximation. The increment dψj is the jth z-score value
for a given measured airfoil. A first-order approximation was chosen over
higher-order methods because of its simplicity and its accurate performance in
the demonstration problem. Further work exploring the use of higher-order
methods does have merit of should a situation be found where the current
approach has unacceptable accuracy.The forced response ROM is based on modal domain
transformation of the equation of motion using the approximate values for (7) while assuming harmonic forcing and motion: (8)(−ωf2+i2λ˜iωfζi+λ˜i2)α˜i=ϕ˜iTf,where ωf is the forcing
frequency, ζi is the modal
damping, ϕ˜iTf is the
approximate modal force, and α˜i is the
approximate modal coordinates: (9)α˜i=ϕ˜iTf(λ˜i2−ωf2)+i(2λ˜iωfζi), which gives the participation of
the ith approximate
mode. The approximate displacement vector, u˜, is computed in the approximate modal
domain: (10)u˜=Φ˜α˜, where the algebra represents
modal summation.While the ROM presented in this section does reduce
the costly modal analysis procedures from expensive matrix computations to
simple arithmetic, there is an error introduced in the approximation. The next
section describes how to improve this model using the developed
error-quantification technique.


## 4. Error Quantified Reduced-Order Model

Models in general have an unquantified error between
their result and the true value. Accounting for this error and providing an
error bound on the result ensure proper model application. This section
develops an approach to quantify the error between the eigensensitivity-based
approximate models developed in Section3 and full FEM solutions. This
quantification includes analysis for reducible errors related consistently to
design parameter variations, that is, bias and random errors that are
irreducible without modifying the model form. This error quantification
approach is used to reduce error instead of pursuing higher-order approximation
methods to avoid the complexity and to develop the error quantifying approach
that is applicable to even these more advanced approximations.The developed error model is an a posterior model that
requires comparison of a limited number,k, of full solutions to the standard ROM. These models
are referred to as training models that provide the error analysis data. The
error is quantified as the discrepancy vector between standard ROM and full
model results: (11)δk=f(zk)−f˜(zk),k=1,2,…,p, where the functions represent
the simulation of a modal response at the kth vector of
z-scores defining an airfoil, and the tilde represents the standard ROM
approximation. The analysis of the relationship between the vector δ and the
components of zk determines the
existence of a physical relationship between principal component mode magnitude
and discrepancy. A discrepancy model as a function of z-scores is constructed
from the regression analysis: (12)δ=Fβ+ε, where F is a matrix of
defined regression functions, β is a vector of
unknown regression coefficients, and ε is a normally
distributed zero mean error term. In the error-quantified ROM, the regression
functions are components of the Z matrix that
defines the airfoil geometries. As an example, the matrix form for a regression
model that includes a constant and all linear terms is (13)F=[1Z1,1Z1,2…Z1,p1Z2,1Z2,2⋮⋱1Zm,p], where the first column is the
regression coefficient for the constant model term b0. Should the discrepancy data show nonlinear
characteristics, additional nonlinear regression terms can be added. The
regression coefficients are determined so that the error between the regression
model and the data is minimized through solution of the linear least-squares
problem: (14)(FTF)−1FTδ=[b0b1⋮bm], where b values are the
most likely estimates of the regression coefficient vector β. The error term, ε, should be uncorrelated, normally distributed with
zero mean and constant variance for statistical modeling assumptions to be met
that allow confidence interval prediction. The linear model developed from (12)
is added to the approximate model developed in Section 3 to develop the
error-quantified ROM for the pth airfoil: (15)λ˜i=λi0+∑j=1n∂λi∂ψjdψj+(zp*)Tβ+εp,ϕ˜i=ϕi0+∑j=1n∂ϕi∂ψjdψj+(zp*)Tβ+εp, where zp* is the vector
of z-scores for the pth airfoil with
the addition of a leading value of one to account for the constant b0 term. Because
the error terms are modeling error as the difference between full FEM and
standard ROM, the addition of these terms reduces the error. Predictor
variables, members of β, are only included in the model if they show
statistical significance to the error. An advantage of this process is that the
PCA produces an orthogonal set of predictor variables which simplifies
determination of the parameter significance. These error-quantified modal
quantities are then used in an error-quantified forced response solution using
the modal domain approach from the previous section.


## 5. Numerical Results

The sensitivity of blade-alone modal and forced
response to geometric deviations from design intent and the effectiveness of
both the standard ROM and error-quantified ROM is shown on an advanced
sixteen-bladed low-aspect ratio IBR. This IBR, the advanced damping low-aspect
ratio fan (ADLARF), has been rigorously studied under the GUIde consortium, a
joint government, university, and industrial program to fund research in
turbomachinery forced response [15–17]. Because airfoil geometry
measurements are not available for this rotor, measured deviations from a
related industrial IBR fan stage are used. Full FEMs of the as-measured models
of the sixteen airfoils are used to assess sensitivity to variations from
design intent, create the error model training data, and quantify the accuracy
of the two developed ROMs. While these ROMs do not directly provide mistuning
results, they provide the required data for previously referenced mistuning
models that account for structural coupling.Modal calculations were made with a blade-alone finite
element fan blade model. The blade approximately spans 12 inches with a 9-inch
chord length. The model contains linear hexahedral elements with an element
edge length on 0.25 inches, resulting in 7722 degrees-of-freedom, uses common
Ti 6Al-4V material properties, and all degrees of freedom are fixed at the
blade root. This is a high quality, but not fully converged model, that was
used to reduce computational requirements during the development process. A
more rigorous converged mesh analysis was conducted to ensure that the
unconverged model does not impact any of the research conclusions. The
converged model had nearly 50 000 degrees-of-freedom, and frequency variation
results for the as-measured models, for the twentieth and most complex mode,
were within a maximum of 0.02 percent between the investigated and converged
models. This demonstrated that the geometric deviations had nearly the same
percentage effect on response variation regardless of mesh density.Results are obtained from the first twenty modes,
covering responses from first flex at 360 Hz to approximately 7000 Hz. Figure1
shows the mode shapes for the fifteenth and sixteenth modes, which are
discussed in greater detail in the proceeding sections. The models were created
parametrically with the coordinate measurement machine data points used as the
parameters. With this model, airfoil geometry variations were automatically generated
through a script file, and mesh topology remained consistent with each model.
Post processing was also conducted through scripting to ensure error-free
result tabulation.Mode fifteen and mode sixteen.
(a)(b)


### 5.1. Reduced-Order Geometry Model Results

An available set of compressor airfoil measured
geometry deviations and the ADLARF nominal geometry provided representative
as-measured geometry. Figure2 shows one measured geometry deviation profile
representative of the remaining airfoils, both as a blade surface contour plot
and a three-dimensional surface plot. Correlation between surface deviation
across the blade is evident and shows that a reduced-order geometry model
should account for spatial correlation. The measurement also shows negative
deviations near the tip and positive deviations near the base. Such a pattern
could be developed from variations in the vertical alignment of the part during
manufacture. The probability distribution of the set of all measured deviations
which is non-Gaussian has a mean value of nearly zero, a standard deviation of
0.003 inches, a minimum of −0.015 inches, and maximum of 0.011 inches.Airfoil surface deviation (Blade 1).
(a)(b)PCA of the sixteen measured blades generates fifteen
principal components,Ψ∈R3n,15. Figure 3 shows the percentage total variance of each
principal component mode, and it is shown that the first fifteen modes capture
all measured deviations. As expected, the modes are ordered by decreasing
variance modeling. Because fifteen features fully describe the blade geometry
deviations, there is a significant computational cost reduction associated with
the Taylor series approximations. If PCA had not reduced the geometry deviation
degrees-of-freedom to fifteen, one sensitivity calculation would be needed for
each FEM surface node degree-of freedom, requiring nearly 2700 simulations.Figure 3
Total variance explained by principal components.


### 5.2. Standard and Error-Quantified Reduced-Order Model Results

The ROMs developed in Sections2 and 3 are
demonstrated on the as-measured rotor. Each subsection first includes results
showing the full FEM predicted response variations of the sixteen as-measured
airfoils. These results justify the need to account for geometrically induced
variations. The subsections then continue to show the ROM's accuracy in
predicting blade-to-blade variations for a selected critical mode
that provides the training data used to determine the model bias
and random error and the two ROM's maximum errors over the first twenty
modes.


#### 5.2.1. Frequency Results

The IBR frequency variation predicted from the sixteen
as-measured airfoils for the first twenty modes, normalized by the average
frequency, is shown in the Figure4 box-and-whisker plot. A box-and-whisker
plot displays the four quartiles of data for each data set, displaying the
median as the horizontal dash bisecting the rectangular box into the first
upper- and lower-data quartiles. The dashed vertical lines attached to these
boxes show the upper- and lower-second quartiles. Addition symbols are for
outlier data. Results show the largest frequency variation interval covering
greater than ±2% of the average
value for sixteenth mode with the mean range of variation for all modes nearly ±1%. It is seen that the normalized frequency deviation
does not appear to significantly increase with the increasing mode number
because of the normalization. The absolute variation in frequency does increase
with mode number. Further analysis of the coefficient of variation, the data
mean divided by its standard deviation, does show an increasing trend in
normalized variation.Figure 4
Airfoil frequency variation.While these are small deviations, they are in a range
shown to lead to maximum mistuning amplifications. The close proximity of
blade-to-blade frequencies causes multiple mode excitations at a single forcing
frequency and summation of modal energy. Mistuning response will be highly
sensitive to the exact pattern, so accurate prediction of each blade frequency
is required. The predicted frequency variations can provide the necessary input
to existing mistuned forced response ROMs and avoids experimental frequency
measurement. Explicit geometric modeling also physically links design
parameters to the frequency variations that lead to mistuned amplification.
Understanding gained through these ROMs can lead to design changes or
manufacturing process controls that will lead to improved IBR reliability.Figure5 plots the comparison between the predicted
blade-to-blade IBR frequencies from the full finite element model, standard
ROM, and the error-quantified ROM. Results for the sixteenth mode are shown
because it had the largest frequency scatter in the first twenty modes, as seen
in Figure 4, and also have the largest error between full model and standard
ROM predictions as shown in Figure 7. Even though Figure 5 shows the ROMs at
their worst, it is seen that the standard ROM does an admirable job predicting
blade-to-blade frequency deviations and captures the blade-to-blade trend in
frequency deviation. Because mistuning is highly sensitive to frequency
magnitudes, a reduced error model is still desirable. Results show that the error-quantified
ROM greatly improved accuracy. The airfoils with the greatest error from the
standard ROM, three, four, eight, ten, fifteen, and sixteen show a marked
improvement with the error-quantified ROM.Figure 5
Frequency prediction comparison (Mode 16).The input for the bias and random error terms of the
error-quantified ROM was constructed from the a posterior discrepancy analysis
between full FEM and standard ROM. Figure6 shows the errors for the sixteenth
mode plotted against the z-scores of the retained principal component modes.
There is a clear linear correlation between the residual value and the z-score
magnitude of the twelfth principal component mode, while all other modes appear
randomly distributed. This linear relationship was seen for all twenty modes.
Because there is a predictable trend between the residual and twelfth principal
component mode magnitude, the error quantified reduced-order model from (15) will account for model bias and improve accuracy. Each mode has its own
regression coefficients based on that mode's data. The remaining error not
accounted for as bias can be included in the random error parameter of the
error-quantified ROM.Figure 6
Correlation of PC mode parameters and frequency residual (Mode 16).Figure 7
Maximum error calculation for airfoil frequency.It is noted that standard regression modeling practice
avoids validating the model with the data used to create the regression model.
This practice was not followed for the results shown. This is acceptable
because the strong linear relationship in the data shows that the result is
more than just a random phenomenon. Nonetheless, a set of 50 randomly generated
airfoil geometries based on the statistics from the PCA analysis was analyzed
with the full FEM and ROMs. These results did not change any of the conclusions
based on the sixteen as-measured airfoils and show that the error-quantified
model is applicable to the larger domain of random airfoils.Figure7 shows the maximum percent error between the
two ROMs and the full FEM for all twenty vibration modes. This maximum error is
obtained for each mode by computing the absolute difference between each ROM
and full model, dividing by the full model value for each of the sixteen
airfoils and plotting only the maximum of these sixteen error results. This is
again a worst case looks at the models, and the average blade error of the ROMs
is significantly lower as can be seen in Figure 5. Figure 7 shows that while
the standard ROM had a maximum percent error below 0.5% for the first 10 modes
and below 1% for all but one of the remaining modes, the error-quantified ROM
predicted much improved results nearly identical to the full model. Error
percentages from the error-quantified ROM for the first twenty modes are below
0.1% error. The figure shows that the error-quantified ROM reduced the maximum
percent error for all the first twenty modes by well over 75%. While the
standard ROMs accuracy may be considered sufficient, the high sensitivity of
mistuned response to variations on the order of its error indicates that the
error-quantified model may be more appropriate.


#### 5.2.2. Modal Force Results

While frequency deviation has been a subject of study
because of its relevance to frequency-based mistuning ROMs, mode shape
deviation has received limited investigation. In this section, the mode shape
variations are not shown directly, instead the useful heuristic of modal force
deviations is shown because of its role in forced response prediction. Modal
force, the inner product of the mode shape and loading vectors, is the quantity
on the right-hand side of the modal equation of motion, and its variation has a1:1 correspondence to forced displacement variation. Nominal unsteady loading
predictions for a defined harmonic from a related IBR were used in the modal
force calculation.A modal force variation box-and-whisker plot for the
as-measured IBR is shown in Figure8. It is evident that these deviations are
much larger than the frequency scatter. The first significant variation in
modal force is shown at the fifth mode with an upper bound 29% larger than the
average value. Considering the set of the first twenty modes, several modes are
observed with upper bounds near 20%, with mode fifteen notable for a 55% upper
bound. These significant variations in modal force directly impact variation in
airfoil forced response, and these are not explicitly accounted for in current
design or mistuning analysis practices. These variations are in addition to
those blade-to-blade stress variations caused by mistuning, that should be
accounted for to reliably predict forced response variations, and demonstrate
the need for a eigenvector response ROM that accounts for geometry variation.Figure 8
Airfoil modal force variation.Figure9 plots the blade-to-blade comparison between the full FEM, the standard, and error-quantified ROMs for the fifteenth mode
modal force prediction. The fifteenth mode was selected because the as-measured
airfoil results for this mode had the largest modal force variation, shown in
Figure 8, and also has the largest error between standard ROM and full models
as seen in Figure 11. Again, this shows the two ROMs at their worst. As seen in
Figure 9, the standard ROM accurately captures the trend of modal force
variation. The error-quantified ROM improves the approximation for nearly all
airfoils, in particular the third, fourth, eighth, tenth, fifteenth, and
sixteenth airfoil. These airfoils are the same that were shown to have the
largest frequency error, also had the largest modal force error, and were also
effectively accounted for by the error-quantified ROM.Figure 9
Modal force prediction comparison (Mode 15).The discrepancy
calculated between the standard ROM and full FEM modal force solutions of the
as-measured IBR is plotted with respect to the
airfoil z-scores in Figure10. Again as with the frequency results, principal
component mode twelve shows a linear relationship between the error and airfoil
z-score value. When compared to the frequency residuals of Figure 6, the linear
relationship is still obvious but with more random variation.Figure 10
Correlation of PC mode parameters and modal force residual (Mode 15).Figure 11
Maximum error calculation for airfoil modal force.Figure11 plots the maximum error between the full FEM and ROMs for the first twenty modes. This maximum error is obtained for each
mode by computing the absolute difference between standard ROM and full model,
dividing by the full model value for each of the sixteen airfoils and plotting
only the maximum of the sixteen errors. It is initially observed that the errors
are significantly larger than for frequency, but this is not unexpected as the
variations in modal force are significantly larger. For the first twenty modes,
the standard ROM performs adequately with more than half the modes below 5% and
most remaining modes below errors below 10% with the exception of the
thirteenth, fifteenth, and nineteenth modes. The error-quantified ROM reduces
the error for these modes in addition to significant error reduction for the
fifth mode. The error-quantified ROM reduces the error for these modes by over
50%. In general, it is seen that the error-quantified model is providing a
reduced benefit to the ROM when compared to the frequency results, but still
enables more accurate modal force prediction for the first twenty modes within
6% of full model results. The remaining error can be accounted for with the
error-quantified ROM random error term.


#### 5.2.3. Forced Response Results

While the variation in modal force is a significant
contributor to forced response, it only accounts for variations in mode shape
displacements. Variations in modal stress and frequency will also impact forced
response variation and this section's results account for those effects. The
maximum forced stress variation box-and-whisker plot is shown in Figure12.
Considering the set of the first twenty modes, several modes are observed with
upper bounds near 20%, with mode fifteen notable for a 68% upper bound.Figure 12
Airfoil forced stress variation.Comparison of these plots to
the modal force variation results of Figure8 shows similar magnitudes of
variation for each mode, but closer inspection shows differences on many modes.
Mode nine's upper bound on airfoil forced response is 38% greater than average
airfoil, while its modal force upper bound was 19%. Mode fifteen shows a 68%
increase in upper bound stress while the modal force upper bound variation for
the mode was 55%. These 19% and 13% increases in
stress upper bounds are caused by
variations in the maximum modal stress caused by geometric deviations. This
demonstrates the importance of accounting for modal stress variations in the
developed ROMs.The maximum forced stress blade-to-blade prediction is
compared between the standard ROM, error-quantified reduced-order ROM, and full
model in Figure13. The fifteenth mode was selected because the as-measured
results for this mode, shown in Figure 8, had the largest forced response
scatter and also has the largest error between full models and standard ROM,
shown in Figure 14. Figure 13 shows that the standard ROM does a good job
representing the full model results and accurately captures the blade-to-blade
trend in forced stress values. The figure also shows that the error-quantified ROM
improves the approximation for all airfoils, in particular the third, fourth,
eighth, tenth, fifteenth, and sixteenth airfoil.Figure 13
Forced stress prediction comparison (Mode 15).Figure 14
Maximum error calculation for airfoil forced stress.The a posterior training data used to create the error
quantified reduced-order model is not shown but is almost identical to Figure10.Figure14 plots the maximum error between the ROMs and
the full models for the first twenty modes. For these modes, the standard ROM
does well with most errors below 5%, with the exception of modes five, fifteen,
and sixteen. The error-quantified ROM reduces the error for many of the larger
errors in this range, particularly modes five, fifteen, and sixteen where error
is reduced by nearly 50%. In general, it is seen that the error-quantified
model is not as effective in correcting for bias as it was for frequency but
still enables forced response force prediction for the first twenty modes
within 5% of full model results. The remaining random error can be accounted
for with the random error term.


### 5.3. Explanation of Error Root Cause

It is necessary to consider why a single principal
component mode has shown such a strong linear relationship to error. While a
precise explanation is not possible, there are several factors which lead to a
supported argument. First, the linear approximations of frequency by the
standard first-order sensitivity ROM work well, and further analysis
demonstrated that the frequency deviations over the range considered and for a
single PCA mode are practically linear. It is then also significant that the
relationship between frequency error and the twelfth PCA mode participation is
likewise practically linear. Therefore, there is a clear evidence that the
sensitivity of the twelfth PCA mode has been inaccurately calculated because
the magnitude of the PCA mode is still enabling an improved prediction. Because
an automated process was used to generate input files and process results, it
is not likely that a user-based processing error was the cause of this error.
It is then likely that the error resides in the numerical accuracy of the
finite element solution software used in this effort. The magnitude of the
twelfth mode was smaller relative to the other modes, and it is possible it
fell below a limit that the software was able to accurately consider
independently from the other variations. While this error could be addressed
with FE software improvements, the error-quantified model shows its benefit by
both identifying the error, what parameter is causing it, and also correcting
the model. The model also retains its value of accounting for random errors,
which is significant when conducting a probabilistic analysis or developing
predictive confidence bounds.


## 5.1. Reduced-Order Geometry Model Results

An available set of compressor airfoil measured
geometry deviations and the ADLARF nominal geometry provided representative
as-measured geometry. Figure2 shows one measured geometry deviation profile
representative of the remaining airfoils, both as a blade surface contour plot
and a three-dimensional surface plot. Correlation between surface deviation
across the blade is evident and shows that a reduced-order geometry model
should account for spatial correlation. The measurement also shows negative
deviations near the tip and positive deviations near the base. Such a pattern
could be developed from variations in the vertical alignment of the part during
manufacture. The probability distribution of the set of all measured deviations
which is non-Gaussian has a mean value of nearly zero, a standard deviation of
0.003 inches, a minimum of −0.015 inches, and maximum of 0.011 inches.Airfoil surface deviation (Blade 1).
(a)(b)PCA of the sixteen measured blades generates fifteen
principal components,Ψ∈R3n,15. Figure 3 shows the percentage total variance of each
principal component mode, and it is shown that the first fifteen modes capture
all measured deviations. As expected, the modes are ordered by decreasing
variance modeling. Because fifteen features fully describe the blade geometry
deviations, there is a significant computational cost reduction associated with
the Taylor series approximations. If PCA had not reduced the geometry deviation
degrees-of-freedom to fifteen, one sensitivity calculation would be needed for
each FEM surface node degree-of freedom, requiring nearly 2700 simulations.Figure 3
Total variance explained by principal components.


## 5.2. Standard and Error-Quantified Reduced-Order Model Results

The ROMs developed in Sections2 and 3 are
demonstrated on the as-measured rotor. Each subsection first includes results
showing the full FEM predicted response variations of the sixteen as-measured
airfoils. These results justify the need to account for geometrically induced
variations. The subsections then continue to show the ROM's accuracy in
predicting blade-to-blade variations for a selected critical mode
that provides the training data used to determine the model bias
and random error and the two ROM's maximum errors over the first twenty
modes.


### 5.2.1. Frequency Results

The IBR frequency variation predicted from the sixteen
as-measured airfoils for the first twenty modes, normalized by the average
frequency, is shown in the Figure4 box-and-whisker plot. A box-and-whisker
plot displays the four quartiles of data for each data set, displaying the
median as the horizontal dash bisecting the rectangular box into the first
upper- and lower-data quartiles. The dashed vertical lines attached to these
boxes show the upper- and lower-second quartiles. Addition symbols are for
outlier data. Results show the largest frequency variation interval covering
greater than ±2% of the average
value for sixteenth mode with the mean range of variation for all modes nearly ±1%. It is seen that the normalized frequency deviation
does not appear to significantly increase with the increasing mode number
because of the normalization. The absolute variation in frequency does increase
with mode number. Further analysis of the coefficient of variation, the data
mean divided by its standard deviation, does show an increasing trend in
normalized variation.Figure 4
Airfoil frequency variation.While these are small deviations, they are in a range
shown to lead to maximum mistuning amplifications. The close proximity of
blade-to-blade frequencies causes multiple mode excitations at a single forcing
frequency and summation of modal energy. Mistuning response will be highly
sensitive to the exact pattern, so accurate prediction of each blade frequency
is required. The predicted frequency variations can provide the necessary input
to existing mistuned forced response ROMs and avoids experimental frequency
measurement. Explicit geometric modeling also physically links design
parameters to the frequency variations that lead to mistuned amplification.
Understanding gained through these ROMs can lead to design changes or
manufacturing process controls that will lead to improved IBR reliability.Figure5 plots the comparison between the predicted
blade-to-blade IBR frequencies from the full finite element model, standard
ROM, and the error-quantified ROM. Results for the sixteenth mode are shown
because it had the largest frequency scatter in the first twenty modes, as seen
in Figure 4, and also have the largest error between full model and standard
ROM predictions as shown in Figure 7. Even though Figure 5 shows the ROMs at
their worst, it is seen that the standard ROM does an admirable job predicting
blade-to-blade frequency deviations and captures the blade-to-blade trend in
frequency deviation. Because mistuning is highly sensitive to frequency
magnitudes, a reduced error model is still desirable. Results show that the error-quantified
ROM greatly improved accuracy. The airfoils with the greatest error from the
standard ROM, three, four, eight, ten, fifteen, and sixteen show a marked
improvement with the error-quantified ROM.Figure 5
Frequency prediction comparison (Mode 16).The input for the bias and random error terms of the
error-quantified ROM was constructed from the a posterior discrepancy analysis
between full FEM and standard ROM. Figure6 shows the errors for the sixteenth
mode plotted against the z-scores of the retained principal component modes.
There is a clear linear correlation between the residual value and the z-score
magnitude of the twelfth principal component mode, while all other modes appear
randomly distributed. This linear relationship was seen for all twenty modes.
Because there is a predictable trend between the residual and twelfth principal
component mode magnitude, the error quantified reduced-order model from (15) will account for model bias and improve accuracy. Each mode has its own
regression coefficients based on that mode's data. The remaining error not
accounted for as bias can be included in the random error parameter of the
error-quantified ROM.Figure 6
Correlation of PC mode parameters and frequency residual (Mode 16).Figure 7
Maximum error calculation for airfoil frequency.It is noted that standard regression modeling practice
avoids validating the model with the data used to create the regression model.
This practice was not followed for the results shown. This is acceptable
because the strong linear relationship in the data shows that the result is
more than just a random phenomenon. Nonetheless, a set of 50 randomly generated
airfoil geometries based on the statistics from the PCA analysis was analyzed
with the full FEM and ROMs. These results did not change any of the conclusions
based on the sixteen as-measured airfoils and show that the error-quantified
model is applicable to the larger domain of random airfoils.Figure7 shows the maximum percent error between the
two ROMs and the full FEM for all twenty vibration modes. This maximum error is
obtained for each mode by computing the absolute difference between each ROM
and full model, dividing by the full model value for each of the sixteen
airfoils and plotting only the maximum of these sixteen error results. This is
again a worst case looks at the models, and the average blade error of the ROMs
is significantly lower as can be seen in Figure 5. Figure 7 shows that while
the standard ROM had a maximum percent error below 0.5% for the first 10 modes
and below 1% for all but one of the remaining modes, the error-quantified ROM
predicted much improved results nearly identical to the full model. Error
percentages from the error-quantified ROM for the first twenty modes are below
0.1% error. The figure shows that the error-quantified ROM reduced the maximum
percent error for all the first twenty modes by well over 75%. While the
standard ROMs accuracy may be considered sufficient, the high sensitivity of
mistuned response to variations on the order of its error indicates that the
error-quantified model may be more appropriate.


### 5.2.2. Modal Force Results

While frequency deviation has been a subject of study
because of its relevance to frequency-based mistuning ROMs, mode shape
deviation has received limited investigation. In this section, the mode shape
variations are not shown directly, instead the useful heuristic of modal force
deviations is shown because of its role in forced response prediction. Modal
force, the inner product of the mode shape and loading vectors, is the quantity
on the right-hand side of the modal equation of motion, and its variation has a1:1 correspondence to forced displacement variation. Nominal unsteady loading
predictions for a defined harmonic from a related IBR were used in the modal
force calculation.A modal force variation box-and-whisker plot for the
as-measured IBR is shown in Figure8. It is evident that these deviations are
much larger than the frequency scatter. The first significant variation in
modal force is shown at the fifth mode with an upper bound 29% larger than the
average value. Considering the set of the first twenty modes, several modes are
observed with upper bounds near 20%, with mode fifteen notable for a 55% upper
bound. These significant variations in modal force directly impact variation in
airfoil forced response, and these are not explicitly accounted for in current
design or mistuning analysis practices. These variations are in addition to
those blade-to-blade stress variations caused by mistuning, that should be
accounted for to reliably predict forced response variations, and demonstrate
the need for a eigenvector response ROM that accounts for geometry variation.Figure 8
Airfoil modal force variation.Figure9 plots the blade-to-blade comparison between the full FEM, the standard, and error-quantified ROMs for the fifteenth mode
modal force prediction. The fifteenth mode was selected because the as-measured
airfoil results for this mode had the largest modal force variation, shown in
Figure 8, and also has the largest error between standard ROM and full models
as seen in Figure 11. Again, this shows the two ROMs at their worst. As seen in
Figure 9, the standard ROM accurately captures the trend of modal force
variation. The error-quantified ROM improves the approximation for nearly all
airfoils, in particular the third, fourth, eighth, tenth, fifteenth, and
sixteenth airfoil. These airfoils are the same that were shown to have the
largest frequency error, also had the largest modal force error, and were also
effectively accounted for by the error-quantified ROM.Figure 9
Modal force prediction comparison (Mode 15).The discrepancy
calculated between the standard ROM and full FEM modal force solutions of the
as-measured IBR is plotted with respect to the
airfoil z-scores in Figure10. Again as with the frequency results, principal
component mode twelve shows a linear relationship between the error and airfoil
z-score value. When compared to the frequency residuals of Figure 6, the linear
relationship is still obvious but with more random variation.Figure 10
Correlation of PC mode parameters and modal force residual (Mode 15).Figure 11
Maximum error calculation for airfoil modal force.Figure11 plots the maximum error between the full FEM and ROMs for the first twenty modes. This maximum error is obtained for each
mode by computing the absolute difference between standard ROM and full model,
dividing by the full model value for each of the sixteen airfoils and plotting
only the maximum of the sixteen errors. It is initially observed that the errors
are significantly larger than for frequency, but this is not unexpected as the
variations in modal force are significantly larger. For the first twenty modes,
the standard ROM performs adequately with more than half the modes below 5% and
most remaining modes below errors below 10% with the exception of the
thirteenth, fifteenth, and nineteenth modes. The error-quantified ROM reduces
the error for these modes in addition to significant error reduction for the
fifth mode. The error-quantified ROM reduces the error for these modes by over
50%. In general, it is seen that the error-quantified model is providing a
reduced benefit to the ROM when compared to the frequency results, but still
enables more accurate modal force prediction for the first twenty modes within
6% of full model results. The remaining error can be accounted for with the
error-quantified ROM random error term.


### 5.2.3. Forced Response Results

While the variation in modal force is a significant
contributor to forced response, it only accounts for variations in mode shape
displacements. Variations in modal stress and frequency will also impact forced
response variation and this section's results account for those effects. The
maximum forced stress variation box-and-whisker plot is shown in Figure12.
Considering the set of the first twenty modes, several modes are observed with
upper bounds near 20%, with mode fifteen notable for a 68% upper bound.Figure 12
Airfoil forced stress variation.Comparison of these plots to
the modal force variation results of Figure8 shows similar magnitudes of
variation for each mode, but closer inspection shows differences on many modes.
Mode nine's upper bound on airfoil forced response is 38% greater than average
airfoil, while its modal force upper bound was 19%. Mode fifteen shows a 68%
increase in upper bound stress while the modal force upper bound variation for
the mode was 55%. These 19% and 13% increases in
stress upper bounds are caused by
variations in the maximum modal stress caused by geometric deviations. This
demonstrates the importance of accounting for modal stress variations in the
developed ROMs.The maximum forced stress blade-to-blade prediction is
compared between the standard ROM, error-quantified reduced-order ROM, and full
model in Figure13. The fifteenth mode was selected because the as-measured
results for this mode, shown in Figure 8, had the largest forced response
scatter and also has the largest error between full models and standard ROM,
shown in Figure 14. Figure 13 shows that the standard ROM does a good job
representing the full model results and accurately captures the blade-to-blade
trend in forced stress values. The figure also shows that the error-quantified ROM
improves the approximation for all airfoils, in particular the third, fourth,
eighth, tenth, fifteenth, and sixteenth airfoil.Figure 13
Forced stress prediction comparison (Mode 15).Figure 14
Maximum error calculation for airfoil forced stress.The a posterior training data used to create the error
quantified reduced-order model is not shown but is almost identical to Figure10.Figure14 plots the maximum error between the ROMs and
the full models for the first twenty modes. For these modes, the standard ROM
does well with most errors below 5%, with the exception of modes five, fifteen,
and sixteen. The error-quantified ROM reduces the error for many of the larger
errors in this range, particularly modes five, fifteen, and sixteen where error
is reduced by nearly 50%. In general, it is seen that the error-quantified
model is not as effective in correcting for bias as it was for frequency but
still enables forced response force prediction for the first twenty modes
within 5% of full model results. The remaining random error can be accounted
for with the random error term.


## 5.2.1. Frequency Results

The IBR frequency variation predicted from the sixteen
as-measured airfoils for the first twenty modes, normalized by the average
frequency, is shown in the Figure4 box-and-whisker plot. A box-and-whisker
plot displays the four quartiles of data for each data set, displaying the
median as the horizontal dash bisecting the rectangular box into the first
upper- and lower-data quartiles. The dashed vertical lines attached to these
boxes show the upper- and lower-second quartiles. Addition symbols are for
outlier data. Results show the largest frequency variation interval covering
greater than ±2% of the average
value for sixteenth mode with the mean range of variation for all modes nearly ±1%. It is seen that the normalized frequency deviation
does not appear to significantly increase with the increasing mode number
because of the normalization. The absolute variation in frequency does increase
with mode number. Further analysis of the coefficient of variation, the data
mean divided by its standard deviation, does show an increasing trend in
normalized variation.Figure 4
Airfoil frequency variation.While these are small deviations, they are in a range
shown to lead to maximum mistuning amplifications. The close proximity of
blade-to-blade frequencies causes multiple mode excitations at a single forcing
frequency and summation of modal energy. Mistuning response will be highly
sensitive to the exact pattern, so accurate prediction of each blade frequency
is required. The predicted frequency variations can provide the necessary input
to existing mistuned forced response ROMs and avoids experimental frequency
measurement. Explicit geometric modeling also physically links design
parameters to the frequency variations that lead to mistuned amplification.
Understanding gained through these ROMs can lead to design changes or
manufacturing process controls that will lead to improved IBR reliability.Figure5 plots the comparison between the predicted
blade-to-blade IBR frequencies from the full finite element model, standard
ROM, and the error-quantified ROM. Results for the sixteenth mode are shown
because it had the largest frequency scatter in the first twenty modes, as seen
in Figure 4, and also have the largest error between full model and standard
ROM predictions as shown in Figure 7. Even though Figure 5 shows the ROMs at
their worst, it is seen that the standard ROM does an admirable job predicting
blade-to-blade frequency deviations and captures the blade-to-blade trend in
frequency deviation. Because mistuning is highly sensitive to frequency
magnitudes, a reduced error model is still desirable. Results show that the error-quantified
ROM greatly improved accuracy. The airfoils with the greatest error from the
standard ROM, three, four, eight, ten, fifteen, and sixteen show a marked
improvement with the error-quantified ROM.Figure 5
Frequency prediction comparison (Mode 16).The input for the bias and random error terms of the
error-quantified ROM was constructed from the a posterior discrepancy analysis
between full FEM and standard ROM. Figure6 shows the errors for the sixteenth
mode plotted against the z-scores of the retained principal component modes.
There is a clear linear correlation between the residual value and the z-score
magnitude of the twelfth principal component mode, while all other modes appear
randomly distributed. This linear relationship was seen for all twenty modes.
Because there is a predictable trend between the residual and twelfth principal
component mode magnitude, the error quantified reduced-order model from (15) will account for model bias and improve accuracy. Each mode has its own
regression coefficients based on that mode's data. The remaining error not
accounted for as bias can be included in the random error parameter of the
error-quantified ROM.Figure 6
Correlation of PC mode parameters and frequency residual (Mode 16).Figure 7
Maximum error calculation for airfoil frequency.It is noted that standard regression modeling practice
avoids validating the model with the data used to create the regression model.
This practice was not followed for the results shown. This is acceptable
because the strong linear relationship in the data shows that the result is
more than just a random phenomenon. Nonetheless, a set of 50 randomly generated
airfoil geometries based on the statistics from the PCA analysis was analyzed
with the full FEM and ROMs. These results did not change any of the conclusions
based on the sixteen as-measured airfoils and show that the error-quantified
model is applicable to the larger domain of random airfoils.Figure7 shows the maximum percent error between the
two ROMs and the full FEM for all twenty vibration modes. This maximum error is
obtained for each mode by computing the absolute difference between each ROM
and full model, dividing by the full model value for each of the sixteen
airfoils and plotting only the maximum of these sixteen error results. This is
again a worst case looks at the models, and the average blade error of the ROMs
is significantly lower as can be seen in Figure 5. Figure 7 shows that while
the standard ROM had a maximum percent error below 0.5% for the first 10 modes
and below 1% for all but one of the remaining modes, the error-quantified ROM
predicted much improved results nearly identical to the full model. Error
percentages from the error-quantified ROM for the first twenty modes are below
0.1% error. The figure shows that the error-quantified ROM reduced the maximum
percent error for all the first twenty modes by well over 75%. While the
standard ROMs accuracy may be considered sufficient, the high sensitivity of
mistuned response to variations on the order of its error indicates that the
error-quantified model may be more appropriate.


## 5.2.2. Modal Force Results

While frequency deviation has been a subject of study
because of its relevance to frequency-based mistuning ROMs, mode shape
deviation has received limited investigation. In this section, the mode shape
variations are not shown directly, instead the useful heuristic of modal force
deviations is shown because of its role in forced response prediction. Modal
force, the inner product of the mode shape and loading vectors, is the quantity
on the right-hand side of the modal equation of motion, and its variation has a1:1 correspondence to forced displacement variation. Nominal unsteady loading
predictions for a defined harmonic from a related IBR were used in the modal
force calculation.A modal force variation box-and-whisker plot for the
as-measured IBR is shown in Figure8. It is evident that these deviations are
much larger than the frequency scatter. The first significant variation in
modal force is shown at the fifth mode with an upper bound 29% larger than the
average value. Considering the set of the first twenty modes, several modes are
observed with upper bounds near 20%, with mode fifteen notable for a 55% upper
bound. These significant variations in modal force directly impact variation in
airfoil forced response, and these are not explicitly accounted for in current
design or mistuning analysis practices. These variations are in addition to
those blade-to-blade stress variations caused by mistuning, that should be
accounted for to reliably predict forced response variations, and demonstrate
the need for a eigenvector response ROM that accounts for geometry variation.Figure 8
Airfoil modal force variation.Figure9 plots the blade-to-blade comparison between the full FEM, the standard, and error-quantified ROMs for the fifteenth mode
modal force prediction. The fifteenth mode was selected because the as-measured
airfoil results for this mode had the largest modal force variation, shown in
Figure 8, and also has the largest error between standard ROM and full models
as seen in Figure 11. Again, this shows the two ROMs at their worst. As seen in
Figure 9, the standard ROM accurately captures the trend of modal force
variation. The error-quantified ROM improves the approximation for nearly all
airfoils, in particular the third, fourth, eighth, tenth, fifteenth, and
sixteenth airfoil. These airfoils are the same that were shown to have the
largest frequency error, also had the largest modal force error, and were also
effectively accounted for by the error-quantified ROM.Figure 9
Modal force prediction comparison (Mode 15).The discrepancy
calculated between the standard ROM and full FEM modal force solutions of the
as-measured IBR is plotted with respect to the
airfoil z-scores in Figure10. Again as with the frequency results, principal
component mode twelve shows a linear relationship between the error and airfoil
z-score value. When compared to the frequency residuals of Figure 6, the linear
relationship is still obvious but with more random variation.Figure 10
Correlation of PC mode parameters and modal force residual (Mode 15).Figure 11
Maximum error calculation for airfoil modal force.Figure11 plots the maximum error between the full FEM and ROMs for the first twenty modes. This maximum error is obtained for each
mode by computing the absolute difference between standard ROM and full model,
dividing by the full model value for each of the sixteen airfoils and plotting
only the maximum of the sixteen errors. It is initially observed that the errors
are significantly larger than for frequency, but this is not unexpected as the
variations in modal force are significantly larger. For the first twenty modes,
the standard ROM performs adequately with more than half the modes below 5% and
most remaining modes below errors below 10% with the exception of the
thirteenth, fifteenth, and nineteenth modes. The error-quantified ROM reduces
the error for these modes in addition to significant error reduction for the
fifth mode. The error-quantified ROM reduces the error for these modes by over
50%. In general, it is seen that the error-quantified model is providing a
reduced benefit to the ROM when compared to the frequency results, but still
enables more accurate modal force prediction for the first twenty modes within
6% of full model results. The remaining error can be accounted for with the
error-quantified ROM random error term.


## 5.2.3. Forced Response Results

While the variation in modal force is a significant
contributor to forced response, it only accounts for variations in mode shape
displacements. Variations in modal stress and frequency will also impact forced
response variation and this section's results account for those effects. The
maximum forced stress variation box-and-whisker plot is shown in Figure12.
Considering the set of the first twenty modes, several modes are observed with
upper bounds near 20%, with mode fifteen notable for a 68% upper bound.Figure 12
Airfoil forced stress variation.Comparison of these plots to
the modal force variation results of Figure8 shows similar magnitudes of
variation for each mode, but closer inspection shows differences on many modes.
Mode nine's upper bound on airfoil forced response is 38% greater than average
airfoil, while its modal force upper bound was 19%. Mode fifteen shows a 68%
increase in upper bound stress while the modal force upper bound variation for
the mode was 55%. These 19% and 13% increases in
stress upper bounds are caused by
variations in the maximum modal stress caused by geometric deviations. This
demonstrates the importance of accounting for modal stress variations in the
developed ROMs.The maximum forced stress blade-to-blade prediction is
compared between the standard ROM, error-quantified reduced-order ROM, and full
model in Figure13. The fifteenth mode was selected because the as-measured
results for this mode, shown in Figure 8, had the largest forced response
scatter and also has the largest error between full models and standard ROM,
shown in Figure 14. Figure 13 shows that the standard ROM does a good job
representing the full model results and accurately captures the blade-to-blade
trend in forced stress values. The figure also shows that the error-quantified ROM
improves the approximation for all airfoils, in particular the third, fourth,
eighth, tenth, fifteenth, and sixteenth airfoil.Figure 13
Forced stress prediction comparison (Mode 15).Figure 14
Maximum error calculation for airfoil forced stress.The a posterior training data used to create the error
quantified reduced-order model is not shown but is almost identical to Figure10.Figure14 plots the maximum error between the ROMs and
the full models for the first twenty modes. For these modes, the standard ROM
does well with most errors below 5%, with the exception of modes five, fifteen,
and sixteen. The error-quantified ROM reduces the error for many of the larger
errors in this range, particularly modes five, fifteen, and sixteen where error
is reduced by nearly 50%. In general, it is seen that the error-quantified
model is not as effective in correcting for bias as it was for frequency but
still enables forced response force prediction for the first twenty modes
within 5% of full model results. The remaining random error can be accounted
for with the random error term.


## 5.3. Explanation of Error Root Cause

It is necessary to consider why a single principal
component mode has shown such a strong linear relationship to error. While a
precise explanation is not possible, there are several factors which lead to a
supported argument. First, the linear approximations of frequency by the
standard first-order sensitivity ROM work well, and further analysis
demonstrated that the frequency deviations over the range considered and for a
single PCA mode are practically linear. It is then also significant that the
relationship between frequency error and the twelfth PCA mode participation is
likewise practically linear. Therefore, there is a clear evidence that the
sensitivity of the twelfth PCA mode has been inaccurately calculated because
the magnitude of the PCA mode is still enabling an improved prediction. Because
an automated process was used to generate input files and process results, it
is not likely that a user-based processing error was the cause of this error.
It is then likely that the error resides in the numerical accuracy of the
finite element solution software used in this effort. The magnitude of the
twelfth mode was smaller relative to the other modes, and it is possible it
fell below a limit that the software was able to accurately consider
independently from the other variations. While this error could be addressed
with FE software improvements, the error-quantified model shows its benefit by
both identifying the error, what parameter is causing it, and also correcting
the model. The model also retains its value of accounting for random errors,
which is significant when conducting a probabilistic analysis or developing
predictive confidence bounds.


## 6. Conclusion

This effort demonstrated the impact of geometry
deviations from design intent on the modal and forced response behavior of
airfoils. Significant variations were shown on frequency and these variations
are significant to mistuned rotor response. The ability to predict these
efficiently with the developed ROMs can significantly improve current mistuning
analysis and design procedures. It was shown that mode shape variations were
more sensitive than frequency variations and these led to large variations in
forced response. These variations are not currently accounted for in design,
but the developed ROMs begin the process to do so. The ROMs are based on PCA
reduction in geometry parameters and an eigensensitivity-based approximation to
reduced response solution times. The error between this model and full models
was quantified, and a linear regression model was demonstrated to quantify
which parameter was contributing to error. Knowledge of this relationship led
to an improvement in the model accuracy. The error was likely caused by
limitations in the numerical accuracies of the FE software and this was
corrected by the error-quantified ROM.



---

*Source: 387828-2008-04-21.xml*