Data-Driven Approach for Identifying Mistuning in As-Manufactured Blisks

Small mistuning, namely, blade-to-blade variations in geometry and material properties of bladed disks, or blisks, can result is significantly greater vibration amplitudes and strain energy localization during operation when subject to a traveling-wave excitation [1,2]. Though nominally cyclic symmetric (i.e., tuned), as-manufactured blisks contain such unavoidable variations even before being used in operation. As such, many reduced-order models (ROMs) have been developed to solve the direct problem of predicting vibration amplitudes with known mistuning, among other known system parameters [3–6].

To utilize these ROMs effectively for accurate prediction of as-manufactured blisk dynamics, accurate mistuning identification (ID) (among other system parameters) for as-manufactured blisks is necessary. This is considered the inverse problem, where system parameters are estimated using known response data. This can be particularly challenging for blisks made as a single piece, as opposed to bladed disks with blades that can be removed for individual testing. For blisks specifically, many previous methods have been developed for mistuning ID [7–17]. However, many of these approaches require full-wheel (i.e., system-level) modal response information and/or measurements from across the entire blisk. Experimental modal response information can be difficult to obtain, particularly in frequency ranges of high modal density, which due to the small variations of mistuning and cyclicity of blisks are quite common [16]. In addition, the need for system-level response data can be a limitation if not all blades are accurately measured experimentally (if measured at all), especially for blisks with many sectors. For Refs. [15] and [18] specifically, geometric mistuning is considered using a coordinate measurement machine and blue-light fringe projector to record geometric measurements at many points across all blades, respectively. However, these methods do not directly account for variations in material properties when considering stiffness mistuning.

Other mistuning ID methods use blade isolation techniques such as mass detuning through added masses [18,19] or adding damping pads [20]. These methods can be intrusive and even infeasible due to geometric constraints. In addition, because blades cannot be completely decoupled, these methods can be subject to errors due to residual coupling, particularly if modal ping testing is used [20].

To address many of these challenges, a new data-driven sector-level approach using a neural network for mistuning ID is presented. As is commonly done for many ROMs and previous mistuning ID methods, mistuning is assumed to be deviations in cantilever-blade frequencies parameterized by a small variation in a blade's Young's modulus [1,3,4,12,21]. Distinct from these past methods, for the presented approach, no modal response information is required for identification. Instead, only a subset of forced responses within a sector from as little as a single engine order (EO) traveling-wave excitation, the excitation frequency, and forcing phase on a sector are needed to identify the mistuning of an individual sector. Hence, system-level response data is not required for identifying the mistuning of an individual sector. Because this approach is sector-level, the computational effort needed to create surrogate training data is significantly reduced compared to a full-wheel approach by substantially lowering the number of full-wheel simulations required. In addition, systematic procedures for excitation frequency selection as well as for noise mitigation while simultaneously removing effects of forcing amplitude are presented. This mistuning ID approach is validated using a high-fidelity finite element (FE) model with response data containing up to 10% relative and 5% absolute noise. Mistuning is shown to be accurately predicted separately using different EOs of excitation in a frequency region of high modal density.

First, a brief overview of feed-forward neural networks (FFNNs) is provided. Next, an overview of the FE blisk model to be used for data generation and validation is presented, along with relevant quantities required for mistuning ID. Lastly, the data-driven sector-level mistuning ID approach is presented. This will start with a general overview of the approach, followed by a procedure for excitation frequency selection, and ending with a procedure for response conditioning to help mitigate effects of noise on prediction accuracy, while simultaneously removing the effects of forcing amplitude.

Most generally, artificial neural networks are biologically inspired systems which aim to learn relationships between sets of inputs and outputs [22]. For an FFNN specifically, this consists of input values provided to an input layer, which are then passed through hidden layers connected via weights and biases, and eventually output from the output layer (i.e., a forward pass). This is illustrated using a simple FFNN as shown in Fig. 1.

where $Wi(L)(L−1)$ is row i of $W(L)(L−1), z(L−1)$ a column vector of all outputs from layer L – 1, $bi(L)$ the ith entry in $b(L)$⁠, and f_a the activation function. To calculate the final network outputs, input values are provided to the input layer, and data is successively passed through the hidden layers and eventually the output layer using Eq. (1). Typically, activation functions are nonlinear with common examples being the hyperbolic tangent, Rectified Linear Unit (ReLU), or the sigmoid functions [22]. For this study, the hyperbolic tangent fuction was found to be sufficient.

For a network to be trained, i.e., to fit the known output data for the provided inputs, weights and biases are updated. This is classified as supervised learning where known output values are available. A common method for updating weights and biases is backpropagation using a loss function. This loss function calculates the error between predictions and known outputs, with the goal to minimize the value of this loss function during training. To update weights and biases many methods have been developed, often using the gradient of the loss function with respect to the network weights and biases [23–25]. For this study, the scaled-conjugate gradient descent method [23] was found to be sufficient.

where N_s is the total number of predicted outputs across all training data, y_i is the predicted output, and $yi, actual$ is the known output corresponding to y_i [22].

where $|F|$ is the forcing amplitude and $j=−1$⁠.

The mistuning to be identified in this study is defined as small, random perturbations in the blade-alone stiffnesses varying across a nominally cyclic-symmetric blisk. This type of mistuning is parameterized as deviations in the cantilever blade-alone frequencies from a nominally tuned blade [3]. The deviation δⁱ in sector i is expressed as $δi=σni$⁠, where σ is a scalar that defines the mistuning magnitude, and nⁱ is a random value from a normal distribution with mean of 0 and a standard deviation of 1. This deviation is applied to the blade Young's modulus as $Eδi=(1+δi)Eo$⁠, where $Eδi$ and E_o are the mistuned and nominal Young's moduli, respectively. The application of this mistuning is visualized in Fig. 2, where the disk portion remains unchanged while the colored blades represent variations in their respective Young's modulus.

where $j=−1, Ko$ is the nominal tuned stiffness matrix, $ΔK$ is the added mistuning to the stiffness matrix, and $x̃$ and $f̃$ are vectors of the displacements and forcing in the frequency domain, respectively.

The fundamental idea of this approach is to use a single FFNN that takes as input response data from within any sector i, as well as the excitation frequency and phase of the forcing in that sector i, and provide as output the mistuning δⁱ in sector i. Generally, mistuning affects all sectors simultaneously. Several previous methods require blade isolation to decouple the identification [10,19,20]. For the method proposed here, we do not require blade isolation. Instead, we account for interactions among sectors. To incorporate these interactions, response data from the disk portion on the interfaces between a sector i and its neighboring sectors i + 1 and i – 1 are included. In addition, response data from the blade is included. For this work, blade-dominated first-bending modes of the blisk are considered, where the maximum blade responses occur at the blade tip. As discussed in Ref. [26], points with large responses on the blade are adequate for accurate mistuning ID. Only exterior out-of-plane responses along the axial direction are considered, which are most easily measured. Thus, possible response locations are shown are in Fig. 3, where $yi,j$ indicates a column vector of responses from sector i at excitation frequency j. Note that the real and imaginary components of selected responses are needed, and stored as individual entries such that $yi,j=[Re(yi,j)T, Im(yi,j)T]T$⁠. This data structure is used for the network input, which does not use complex-valued inputs directly. While Fig. 3 denotes some plausible response locations, only a small subset of responses from these plausible locations will be used.

Using a subset of responses from the locations in Fig. 3 and a parameter vector $pji$ containing the excitation frequency ω_j and the forcing phase $∠fi$ on sector i, the general neural network (NN) input and output structure is shown in Fig. 4. Because mistuning ID is an inverse reduced-order model (IROM) approach, we refer to the NN in this study as the IROM network. With this formulation, mistuning is identified for each sector independently at each excitation frequency of interest. Although the mistuning considered for this study is independent of frequency, by obtaining predictions across a frequency range a mean prediction can be calculated along with a standard deviation. This is addressed in further detail in Sec. 2.5.

It should be noted that only sector-level data is required for the IROM network. Here, we define a sector-level dataset as responses and parameters (i.e., forcing frequency and phase) associated with a single sector over an excitation frequency range. Thus, this approach has the significant advantage that for every simulation or experimental dataset for a blisk with N sectors, N sector-level data sets are generated, resulting in much fewer full-wheel simulations needed compared to a full-wheel approach. Even though identifying the mistuning for every sector of a blisk with N sectors requires N sector-level data sets (one sector-level dataset per sector), this is advantageous because even linear FE blisk models can be computationally expensive without using a ROM, particularly for industrial models.

To improve prediction accuracy, only excitation frequencies with large response amplitudes should be considered. This is primarily due to the noise-to-signal ratio being large for off-resonance frequency ranges. This implies the region around resonances where the blade response is largest needs to be selected for training and identification. Thus, a systematic approach to determine the excitation frequencies to use for training and identification is presented next.

First, for a given EO excitation, a frequency range containing the resonance frequency of a targeted mode is defined. For this study, an EO3 excitation is considered with a frequency range of 920 Hz to 1000 Hz. This frequency range covers the flat portion of first mode family (i.e., blade-dominated system modes) as shown in Fig. 5, where the targeted tuned first mode family nodal diameter (ND) 3 mode is indicated. Note that in a tuned blisk, only the ND3 mode is excited for EO3 excitation, however this is not the case for a mistuned blisk.

Next, the maximum response magnitude for a large responding blade tip DoF within this frequency range is found across all blades for all mistuning patterns to be used for training. Thus, for this FE blisk model with N = 18 sectors, this means finding 18 values for each mistuning pattern. Then, a threshold percentage value, denoted as α, is used to determine a fraction of these maximum values. Within the frequency range of interest, using the maximum value of the blade tip for a sector i (denoted $ymaxi$⁠) and the chosen α, the threshold response magnitude for that blade and mistuning pattern alone is $αymaxi$⁠. All excitation frequencies where sector i responds at or above $αymaxi$ are stored, and this process is repeated separately for all sectors and all mistuning patterns in the training dataset. Figure 6 shows an example noisy blade tip response in arbitrary units (denoted (a.u.)) with the maximum value, threshold, and stored frequencies indicated. Using a single value of α for all sectors across all mistuning patterns was found to be sufficient.

After repeating this process for all sectors and mistuning patterns in the training dataset, the number of instances that an excitation frequency results in a response for a blade tip at or above $αymaxi$ (i.e., was stored) is determined. That is, the number of times a frequency was stored when repeating this process for all sectors across all mistuning patterns is recorded. Lastly, a cutoff number of instances that an excitation frequency must have been stored in order for the responses at the frequency to be accepted and used for training and ID is chosen. An example of how many times a frequency was stored and which frequencies are accepted is shown in Fig. 7. Here, the cutoff threshold is shown, with only frequencies above this threshold accepted. It was found that the mean number of instances frequencies are stored provides a good initial guess for this cutoff. This frequency selection procedure was verified using a small set of responses to check that large responses relative to the entire frequency range occurred at accepted frequencies. In addition, it should be noted that the accepted frequencies are the same for all sectors across all mistuning patterns for training, and comprise the excitation frequencies used during ID for mistuning patterns not used for training, known as test cases.

This process for frequency selection serves two purposes. The first is to select the excitation frequencies around resonances by an automated method while accounting for the variety of blade responses that can occur due to mistuning, namely, finding frequencies that most often result in the largest responses in the considered frequency range across all blades and training mistuning patterns. The second is to reduce the impact of noise on frequency selection. This is accomplished by selecting excitation frequencies that not only yield large response magnitudes (above $αymaxi$ values), but also are stored more than a certain number of times across all blades and mistuning patterns. By rejecting excitation frequencies that (across all blades) often result in large noise-to-signal ratios (and hence results in worse mistuning ID) through this selection process, this data curation acts as a first step to mitigating noise effects through frequency selection.

An example of blade tip responses and accepted excitation frequencies using this selection method are shown in Fig. 8 for EO3 excitation with 10% relative and 5% absolute noise. These frequencies were selected by considering EO3 excited responses from 32 mistuning patterns all with $σ=0.01=1%$⁠, resulting in 576 sector-level data sets. Note that we do not enforce nor require mistuning patterns to have 0 mean.

To further mitigate the effects of noise while removing effects of forcing amplitude, a response conditioning technique is presented. As is often done for experimental data when noise is present, multiple trials are collected ideally using the same forcing conditions. Thus, assume that N_t separate trials of noisy response data are available, henceforward referred to as noisy data sets. This is simulated by computationally adding random relative absolute noise N_t separate times to noise-free responses.

Example conditioned blade tip responses from all sectors for accepted frequencies using respective $u1i$ vectors obtained with this conditioning procedure are plotted in Fig. 9. Conditioned responses using 1, 3, 5, and 10 noisy data sets with random 10% relative and 5% absolute noise are shown.

It can be seen in Fig. 9 that as more noisy data sets are available for sector i, the less noisy the approximation $u1i$ provides. In addition, an added benefit of this conditioning approach is that $u1i$ is normalized to have unit magnitude by definition of the SVD. Therefore, with this procedure effects of the forcing amplitude in the inputs to the network have been removed. This is significant because it removes the need for both known and uniform forcing amplitude across all blades and noisy data sets, with uniform forcing amplitude in particular being very difficult to achieve experimentally in bench or operating conditions. It should be noted, however, that if only a single noisy dataset is available, then $u1i$ is equivalent to normalizing $yi$ (which for this case is a column vector) so it has unit 2-norm, and no further noise mitigation is performed. Conversely, as $Nt→∞$⁠, the rank 1 approximation converges to the noise-free responses.

Lastly, the responses in $y¯i$ for sector i are parsed for each individual accepted excitation frequency, stored for a frequency j in the column vector $y¯i,j$⁠. Using responses in $y¯i,j$ and the corresponding forcing phase $∠fi$ and excitation frequency ω_j stored in $pji$⁠, an identified mistuning is obtained for all accepted frequencies, denoted $δi,j$⁠. Using all $δi,j$ values the mean identified mistuning across all accepted frequencies is the final identified value for δⁱ. This is repeated for all sectors of a blisk. This process is summarized in Fig. 10. In addition, a standard deviation for each mistuning prediction is calculated to indicate the spread in the identified mistuning for all sectors over the accepted frequency range.

First, mistuning ID results using IROM networks trained on and given only EO3 excitation response data are shown. Two cases of added relative and absolute noise are considered, and ID demonstrated for two point selection cases. Next, mistuning ID results from repeating this process using only EO9 excitation response data are shown. Once again, two cases of added relative and absolute noise are considered, and results are shown for the point selection case using fewer responses.

For an EO3 excitation, the ND3 mode of the first mode family shown in Fig. 5 is targeted. As shown in Figs. 7 and 8, the accepted excitation frequencies are between 939.25 Hz to 948.75 Hz (selected with 0.25 Hz resolution) out of the entire considered range of 920 Hz to 1000 Hz. A threshold percentage value of $α=0.35$ and cutoff number of 41 instances a frequency must have been stored (the mean number of instances over the considered frequency range) were used for frequency selection. A total of 32 mistuned full-wheel simulations using ANSYS Mechanical are performed for training data generation (i.e., 32 mistuning patterns) using $β=3.3×10−7 (rad/s)−1$⁠, which corresponds to $ζ≈9.95×10−4$⁠. Note that these mistuning patterns are not enforced to have 0 mean. Because this approach is sector-level, 32 full-wheel simulations for this FE model with N = 18 sectors generates 576 sector-level data sets over the accepted frequency range. For this amount of training data, a NN size of 4 hidden layers with 40 nodes each was used and found to be sufficient. 8-cores from two Intel® Xeon® E5-2609 v3 processors were used in parallel for network training, with the total training time taking approximately 3 to 5 min. All networks were trained using the MATLAB Neural Network Toolbox.

Two cases of DoF selections are considered as shown in Fig. 11. Both cases only use out-of-plane axial responses from the high and low interfaces and blade tip, with case 2 only differing from case 1 by removing a DoF from both the high and low interfaces. Note that case 2 is the minimum number of required points per sector to capture both motion in the blade and effects from neighboring sectors.

Using three noisy data sets for response conditioning and separately considering case 1 and case 2 responses, mistuning ID results for an example mistuning pattern not used for training (i.e., test case) are shown in Figs. 12 and 13. Error bars indicate the standard deviation of the identified mistuning values over the accepted frequencies. Two levels of added relative and absolute noise are considered, the maximum amount being 10% relative and 5% absolute noise as shown in Fig. 8. Note that for each panel within Figs. 12 and 13, a separate NN was trained using the specified amount of added noise and response data for cases 1 and 2.

As shown in Figs. 12 and 13, mistuning ID with this small subset of response from only EO3 excitation can be very accurate and is robust to even large amounts of noise. These plots show that increasing the noise does not significantly change the NN mean predictions, and has the effect of slightly increasing the standard deviation error bars.

where $δNNi$ and $δactuali$ are the identified and actual mistuning for sector i, respectively, and $max|δactual|$ is the maximum absolute value across all actual mistuning values from the mistuning pattern. Considering case 2 responses with 10% relative and 5% absolute noise, Fig. 14 shows the value of $E¯$ for four separate test mistuning patterns varying α.

As shown in Fig. 14, for $α≥0.35$ the mean percent errors change very little across all test cases. The larger errors at lower α values are caused by more data at lower noise-to-signal ratios being included in the training data. This is because lower α values result in more response data being above the required percentage value, thereby increasing the number of accepted frequencies, and vice versa for higher α. However, for $α≥0.35$ the accepted frequencies change very little due to the low damping of the system, resulting in the training data sets being very similar and therefore similar levels of mistuning ID accuracy. Of course, in the limit of $α→1$ only a very small subset of frequencies will be accepted and accuracy may decrease due to too few mistuning values used for averaging. Nevertheless, this illustrates that this method is robust to the choice of α, as long as it is sufficiently high to remove small responses while not decreasing the number of accepted frequencies too greatly.

General methods that can help further improve mistuning ID include:

1. Reduce noise in the data sets used for response conditioning

2. Increase the number N_t of noisy data sets for response conditioning because, as shown in Fig. 9, the noise-to-signal ratio of conditioned responses decreases as N_t increases

3. Provide response data from more locations because the ID can be improved when more data further captures effects from neighboring sectors or blade motion

4. Increase the amount of training data because mistuning is generated using random values from a normal distribution, and if not enough data at larger δⁱ values is provided, the accuracy of the ID for larger δⁱ can suffer

To illustrate that this data-driven mistuning ID approach is robust to the choice of EO excitation, mistuning ID is repeated using an EO9 excitation, targeting the first family ND9 mode. From Fig. 5, it is clear this mode lies in a frequency range of high modal density. The number of data sets and NN structure are the same as those used for mistuning ID using EO3 excitation response data.

Similar to EO3 excitation, the number of instances where a frequency was stored based on a blade tip DoF response magnitude versus excitation frequency is plotted for EO9 excitation in Fig. 15. Now, the accepted frequency range is 959.5 Hz to 973 Hz selected with 0.25 Hz resolution. A threshold percentage value of $α=0.35$ and cutoff number of instances of 110 were used for excitation frequency selection. Note that the same mistuning patterns used for training the NNs with EO3 excitation are now used with EO9 excitation.

Example blade tip responses for an EO9 excitation of a mistuned blisk are shown in Fig. 16 with 10% relative and 5% absolute noise added. The accepted frequency range found using the proposed methodology is indicated in Fig. 16. Due to high modal density in this frequency range, and because the blisk is mistuned, even using only EO9 excitation the range of frequencies for which blades have large response magnitudes is relatively wide.

Considering case 2 using three noisy data sets for response conditioning, mistuning ID of the same test case using EO9 excitation data with 5% relative and 3% absolute noise and 10% relative and 5% absolute noise, respectively, is shown in Fig. 17. Once again, ID using the minimal number of responses with significant noise remains accurate, with all actual δ values within 1 standard deviation of respective identified values. These results illustrate that data-driven mistuning ID can be performed accurately using single EO excitation response data for two separate EO excitations in a frequency range of high modal density. Again, note that case 2 illustrates the minimal number of responses per sector needed for this approach to capture both the blade and coupling effects from neighboring sectors.

A data-driven approach for identification of small mistuning in as-manufactured blisks was presented. This approach uses sector-level physical responses from a blisk subject to traveling-wave excitation, as well the forcing frequency and phase to identify the mistuning within each sector. In addition, a systematic procedure for excitation frequency selection as well as a procedure for noise mitigation while simultaneously removing forcing amplitude effects was presented. This approach was validated using computational surrogate data from a high-fidelity FE blisk model with N = 18 sectors, with significant relative and absolute noise added to response data.

A summary of the key advantages of the presented approach for mistuning ID are as follows:

1. Physical out-of-plane axial response data from a single EO excitation is used directly, without the need of any modal response information

2. Does not require blade isolation

3. Only sector-level response data is needed for IROM network training and identification. Thus, no system-level information is required, and N sector-level data sets are obtained per full-wheel simulation, significantly reducing the computational effort needed for training data generation

4. A minimal number of responses (i.e., case 2) can be used while still maintaining identification accuracy

5. Is agnostic to forcing and response amplitudes which may vary blade-to-blade in bench or operating conditions

6. Is robust to significantly noisy responses

7. Can be used in frequency ranges of high modal density

Possible future work includes extending this approach to other types of forcing and/or response data in particular at higher and less isolated mode families, as well as verifying the mistuning ID experimentally using response data from as-manufactured blisks. In addition, different IROM network architectures can be considered other than the FFNN used in this study, along with further investigation of the amount of training data needed for achieving both good accuracy and adequate network generalization to unseen test data.

• GUIde 6 Consortium.

The authors gratefully acknowledge the generous support offered by the GUIde 6 Consortium towards their investigation and research in this topic.

Symbols

M =

mass matrix

C =

damping matrix

K =

stiffness matrix

$Ko$ =

tuned stiffness matrix

$ΔK$ =

added mistuning to stiffness matrix

U =

matrix of left singular vectors

V =

matrix of right singular vectors

W =

neural network weight matrix

b =

neural network bias vector

f =

forcing vector

u =

left singular vector

v =

right singular vector

$x, x˙, ẍ$ =

displacement, velocity, and acceleration vectors

y =

out-of-plane axial displacement vector

z =

neural network general layer output vector

$E¯$ =

mean percent error

$Eo, Eδi$ =

nominal and mistuned Young's moduli

F =

forcing amplitude

f_i =

forcing on sector i

f_a =

activation function

N =

number of sectors

N_f =

number of frequencies

N_s =

number of network outputs across all training data

N_t =

number of trials taken of noisy response data

n =

random value from normal distribution

r =

matrix rank

EO =

engine order

ND =

nodal diameter

DoF =

degree of freedom

Σ =

diagonal matrix of singular values

α =

threshold percentage value

β =

proportional damping coefficient (stiffness only)

δ =

mistuning

ζ =

damping ratio

σ =

mistuning magnitude

ω =

excitation frequency