Data Consistency Assessment Function (DCAF)

Mechanical Systems and Signal Processing 141 (2020) 106688

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Data Consistency Assessment Function (DCAF) Yuanchang Chen a,⇑, Peter Avitabile a, Jacob Dodson b a b

Structural Dynamics and Acoustic Systems Laboratory, University of Massachusetts Lowell, One University Avenue, Lowell, MA 01854, United States Air Force Research Laboratory, Munitions Directorate, Fuzes Branch, Eglin Air Force Base, 306 W. Eglin Blvd., Bldg 432, Eglin AFB, FL 32542-5430, United States

a r t i c l e

i n f o

Article history: Received 20 November 2019 Received in revised form 18 January 2020 Accepted 27 January 2020

Keywords: Data consistency Expansion Mode shape Dynamic time response SEREP Polynomial expansion

a b s t r a c t A Data Consistency Assessment Function (DCAF) is developed to check the consistency of a measurement or set of measurements to all of the data in the entire data set. The inconsistent data can be precisely spotted and identified, which are recognized as the poorly measured data and can also be modified to be consistent with the rest of the data with an expansion process. The data can be either mode shapes, dynamic time response, frequency response functions or strain fields. Depending on the particular situation, three forms of the data expansion approach can be selected to implement the DCAF: System Equivalent Reduction Expansion Process (SEREP) with finite element model, SEREP with experimental mode shapes, and polynomial expansion. Some academic and industrial structures are used as examples to study the application of DCAF. Experimental data are used to validate the technique. Ó 2020 Elsevier Ltd. All rights reserved.

1. Introduction Many times, experimental modal data or time response data (displacement, acceleration or strain) is collected in order to understand the validity of a finite element model of a structure for structural dynamic applications. Frequency Response Functions (FRF) may be collected in order to identify the frequency, damping and mode shapes of the structure which are in turn used to compare to the finite element model frequencies and modes shapes obtained from an eigensolution from the mass and stiffness matrices developed to represent the structure. At times, the time response data is also collected and used for further validation of the model. When the time data is collected to form the FRFs, the data is typically evaluated by computing the coherence (or multiple coherence) function. However, this assessment is only applicable for the individual measurements and does not identify the relationship of all the measurements acquired relative to each other. Measurement issues can run a very wide range of possibilities for a measurement to become contaminated. A measurement may be contaminated to a small degree such as typically seen with background noise. But noise is not the only issue and the severity of contamination can increase to a point where a measurement may be totally erroneous, and its use in further analysis degrades the entire experimental model developed. Measurements can be corrupted with cable problems, saturation, clipping, along with many other issues. Also, an FRF measurement may even be mis-labeled to be a different point or the response transducer may be located at a different location or the orientation of the transducer may be wrong (such as in the case of triaxial accelerometers). Of course, in the process of developing an experimental modal model for instance, the data points are validated through the process of synthesizing the FRF used to extract parameters and compared to the original measurement. While this seems ⇑ Corresponding author. E-mail address: [email protected] (Y. Chen). https://doi.org/10.1016/j.ymssp.2020.106688 0888-3270/Ó 2020 Elsevier Ltd. All rights reserved.

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to validate the model, the simple fact is that it only validates the one particular measurement point and contains no information as to the consistency of that one data point to all the other data points that exist in the experimental data set. There could be a calibration error, mis-location of a transducer placement or mis-labeling of a transducer direction – none of which are identified by the synthesis and comparison with the original measurement. So the FRFs themselves are very separate, individual measurements and there is no tool to determine the consistency of all the data to the entire set of measurement points. However, while the Modal Assurance Criteria (MAC) [1] does lend some insight into the uniqueness of the vectors extracted, MAC is not a useful tool to check the general consistency of each individual measurement points relative to other measurement points. Other ‘‘MAC like” tools such as Frequency Response Assurance Criterion (FRAC) [2], Time Response Assurance Criterion (TRAC) [3], Coordinate Modal Assurance Criterion (CoMAC) [4] are also useful but again do not identify the relationship of a set of measurements to themselves and are only useful when comparing to other data sets. And while FRFs have multiple coherence and synthesis as tools to determine the individual measurement acceptance, raw measured time data (that creates the FRFs) has no tools to assist in determining the consistency of the data set. The work presented here attempts to provide a Data Consistency Assessment Function (DCAF) to be able to check the consistency of a measurement or set of measurements to all of the data in the entire data set. The basic approach is to remove one particular measurement from the complete set of data describing the structure (time response or modal shapes or strain) and then recreate that measurement from an expansion from the remaining set of data and then compare that data to the corresponding data point and determine a correlation coefficient. If the data point removed is actually consistently related to the data set then the correlation coefficient will approach 1.0; if the data is not consistently related then the correlation consistency will degrade depending on the severity of the mismatch. The DCAF is based on expansion of data which may take several forms depending on the particular situation a) A finite element model is available and then a System Equivalent Reduction Expansion Process (SEREP) expansion can be deployed using the finite element model mode shapes b) An experimental modal model is available and then a SEREP expansion can be deployed using the experimental mode shapes extracted c) No mode shapes are available and then a polynomial expansion can be deployed 2. Theory The development of the Data Consistency Assessment Function (DCAF) utilizes basic expansion equations either using a generalized inverse on the shape information [5] or using a polynomial based approach [6,7]. The basic equations for expansion are only summarized here with details in the referenced papers. The expansion approaches can be used to expand a sparse set of points to a larger set. For all model reduction and expansion algorithms [5,8–15], the relationship between the augmented set of DOFs and a reduced set of DOFs can be written as:



fXn g ¼

Xa Xna



¼ ½TfXa g

ð1Þ

where, subscript ‘n’ denotes the augmented configuration and ‘a’ denotes the sparse configuration. fX a g represents the displacement vector at ‘a’ degrees of freedom, and f Xn g represents the displacement vector at ‘n’ degrees of freedom. ½T  denotes the transformation matrix that can expand the displacement vector from the sparse configuration to the augmented configuration. Eq. (1) can also be used for the operating time response and can be written as.

½ERTOn  ¼ ½T½RTOa 

ð2Þ

where RTO indicates the real time operating data collected at the sparse set of ‘a’ degrees of freedom and ERTO indicates the expanded real time operating data collected at the full or augmented set of ‘n’ degrees of freedom (with the a and n subscripts the same as defined above). In this work, three ways are employed to form the transformation matrices. a) F-SEREP – shapes are expanded using the finite element mode shapes b) E-SEREP – shapes are expanded using experimentally measured mode shapes c) P-Expansion – shapes are expanded using the polynomial expansion process These are further explained next. a) SEREP expansion using FEA shapes (F-SEREP) The SEREP procedure produces reduced matrices for mass and stiffness that yield the exact frequencies and mode shapes as those obtained from the eigensolution of the full-size matrix. The physical system is mapped into modal vectors.

Y. Chen et al. / Mechanical Systems and Signal Processing 141 (2020) 106688

 fXn g ¼

Xa



Xna

 ¼ ½U n fpg ¼

Ua

3



Una

fpg

ð3Þ

where ½U n  is the analytical modal vector matrix containing ‘m’ columns, i.e. ‘m’ modes. ½U a  is ‘a’ rows submatrix of ½U n . fpg is the corresponding modal coordinate. The equation at ‘a’ degrees of freedom in Eq. (3) can be written as:

fXa g ¼ ½U a fpg

ð4Þ

Rewrite this equation for fpg

  T 1 ½U a T fXa g ¼ ½Ua g fXa g fpg ¼ ½U a  ½U a 

ð5Þ

½U a g is the generalized inverse matrix of ½U a . Substituting Eq. (5) into Eq. (3) gives.

fXn g ¼ ½U n ½U a g fXa g ¼ ½T fXa g

ð6Þ

Thus, the transformation matrix ½T in F-SEREP can be written as.

½T ¼ ½Un ½Ua g

ð7Þ

b) SEREP expansion using experimental mode shapes (E-SEREP) The transformation matrix ½T in E-SEREP is determined from the experimental mode shapes, instead of analytical mode shapes.

½T ¼ ½En ½Ea g

ð8Þ

where ½E denotes the experimental mode shapes extracted from the modal testing. c) Polynomial expansion (P-Expansion) In short, the transformation matrix ½T in P-Expansion can be written as the function of polynomial shape functions, which is mathematically shown as

½T ¼ f ð½Ca ; ½Cn Þ

ð9Þ

where ½C a  denotes the polynomial shape functions at sparse configuration and ½C n  denotes the polynomial shape functions at augmented configuration. The derivation of Eq. (9) is shown as follows. The displacement vector at sparse configuration can be decomposed into a series of orthogonal polynomial shape functions of the sparse configuration ½C a  with their polynomial coefficients fka g, and can be written as

fX a g ¼ ½Ca fka g

ð10Þ

½C a  is solely dependent on the coordinates of the sparse points. From Eq. (10), the polynomial coefficients of the sparse configuration fka g can be determined. g fka g ¼ ½Ca  fX a g

ð11Þ

where ½C a g is the generalized inverse matrix of ½C a . The coefficient of the augmented configuration fkn g and the coefficient of sparse configuration fka g differ by a scale. This scale is the function of ratio of point resolution between the augmented shape function ½C n  and the sparse shape function ½C a , so that fkn g can be calculated by scaling fka g.

n fkn g ¼f a fka g

ð12Þ

Similar with Eq. (10), displacement vector of the augmented configuration can be constructed by shape functions of the augmented configuration ½C n  and its corresponding coefficient fkn g, that is

fX n g ¼ ½Cn fkn g

ð13Þ

Just like ½C a  is determined by coordinates of sparse grid, ½C n  is determined by coordinates of the augmented grid. Substituting Eqs. (11) and (12) into Eq. (13) yields.

fXn g ¼ f

n ½Cn ½Ca g fXa g a

Thus, the transformation matrix ½T in P-Expansion can be written as.

ð14Þ

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½T ¼ f

n ½ Cn  ½ Ca  g a

ð15Þ

The derivation of P-Expansion is shown in details in Ref. [6]. In order to illustrate the DCAF process, a simple schematic illustrating a set of 25 points is used in Fig. 1 to identify the data consistency among all the measurements. Each of the three different expansion processes are shown in Fig. 1.

F-SEREP

E-SEREP

P-Expansion

SEREP using FEA

SEREP using Experimental Mode Shapes

Polynomial Expansion

Step 1: Mode Shapes at N-1 points 1st Mode 2nd Mode 3rd Mode

Expand from N-1 Point to N Point

Reconstructed Mode Shapes at N points 1st Mode 2nd Mode 3rd Mode

With Point 1 Removed

Step 2:

Correlation

Step 3:

Original Mode Shapes at N points 1st

Mode 2nd Mode 3rd Mode

Plot Correlation (MAC)

Step 4: Repeat Step 1 to Step 3 for All Points Mode Shapes at N-1 points 1st Mode 2nd Mode 3rd Mode

Step 5:

Integration

Absolute Difference from UNITY

With Point 1 Removed

With Point 2 Removed Data Consistency Assessment Function (DCAF)

With Point N Removed

Fig. 1. Process of DCAF for mode shape assessment.

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Step 1 to Step 4 Time Response at N-1 Points

Integration

Step 5: Absolute Difference from UNITY

With Point 1 Removed Time Response at N Points

With Point 2 Removed Data Consistency Assessment Function (DCAF)

With Point N Removed

Fig. 2. Process of DCAF for time response assessment.

The DCAF for mode shape expansion is shown in Fig. 1. The mode shape recovery is performed at each point for the complete set of points (25-points) one at a time. For example, point 1 is intentionally removed and the mode shape at all of the rest of the points (24-points) is expanded to the whole set of points (25-points). The recovered (expanded) mode shape at the complete set of points is correlated with the original mode shape at the corresponding points. Thus, a 25 by 1 MAC vector is produced in which each element correspondingly indicates the MAC value between the recovered mode shape and the original mode shape. The same process is performed for every point, one at a time, and in total 25 MAC vectors (25 by 1 vector) are obtained. These 25 MAC vectors are assembled to form one square MAC matrix (25 by 25 matrix). In the MAC matrix obtained, the right horizontal axis indicates the removed point configuration, i.e., the point that is removed or taken out. The left horizontal axis indicates the entire set of points. The vertical axis indicates the MAC value between the recovered mode shape and the original mode shape. No matter which removed point configuration is investigated, high correlations are shown at most points. To make any poor correlations in this MAC matrix easily discernable, the MAC matrix is subtracted from a matrix of the same size consisting of all ones which are a perfect correlation. The difference matrix is named DCAF. So the entire process is performed and the resulting DCAF matrix will be very close to zero if all of the data points are well correlated to each other. If a particular point is not consistently related to the other points then there will be a very consistent indication for that point over all data sets considered. This will be clearly shown in the example cases presented. The process of the DCAF for the time response assessment is shown in Fig. 2; this follows the same strategy as discussed above except that time data is evaluated in the expansion process. (In addition to time data, this process can also be extended for strain fields and is the subject of other papers currently under development). In this work, several structures are considered (a base-upright (BU) structure, a dryer cabinet base panel and a large-scale robot roadway structure) to study the application of the proposed Data Consistency Assessment Function (DCAF) using all of the various variations of the DCAF tool. These test cases are shown next.

3. Test cases evaluated The test cases developed in this paper utilized the Data Consistency Assessment Function (DCAF) with finite element model mode shapes as the expansion basis, with experimentally measured modes shapes as the expansion basis and with polynomial based expansion as the basis to show the DCAF in all the different scenarios possible. The three major cases identified are:  a base-upright (BU) structure,  a dryer cabinet base panel and  a large-scale robot roadway structure Each of these cases are presented next.

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3.1. Base-Upright (BU) The Base-Upright (BU) structure has been used for many correlation studies with a variety of measurement methodologies including accelerometers, lasers and optical techniques [16,17]. As shown in Fig. 3, this laboratory structure is comprised of two 3/4 in. (19 mm) thick aluminum plates attached together with two steel L-brackets. The base plate is

Test step up

Bandwidth 1024 Hz Resolution 0.25 Hz Spectral Line 4096

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Measured Mode Shapes 2nd Mode

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79.2 Hz

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548.1 Hz

617.2 Hz

1st

Mode

Fig. 3. Sample structure BU and its experimental mode shapes.

Fig. 4. Mode shape assessment on the BU using F-SEREP.

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24  24 in. (610  610 mm) and rigidly bolted to the tied to the ground at the four corners using concrete anchors, while the upright is 24  30 in. (610  762 mm). Because this structure is fairly well known, this structure was used for study for the DCAF technique. Both mode shapes and time response were considered for the BU.

3.1.1. Mode shape check Generally, the shapes are easier to identify outliers by visual inspection. However, for more complicated structures the 3D motion may not be so easy to decipher. The finite element mode shape data can be used for expansion with the DCAF approach to identify any outlier(s) that may exist in the mode shapes for the modes of the system. As shown in Fig. 3, a modal test is performed on the structure and the FRF at 25 points on the upright plate are measured using modal hammer excitation. The FRFs are then curvefitted to obtain the experimental mode shapes for the first eight out of plane modes. The 25 data points are shown in Fig. 3 with the corresponding lower order modes. Using the DCAF process, one point from the set of 25 points was removed and used to assess the data with the remaining 24 points. This process is continued for all 25 points to develop the DCAF matrix, as shown in Fig. 4.

Use Point 21 to Replace Point 1

(a)

Replace

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Use Point 21 to Replace Point 7 and Point 25 to Replace Point 4

Replace

20

24 25

Fig. 5. Mode shape assessment on the BU using F-SEREP when there is cable swapping.

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In the DCAF matrix, all of the values are very low indicating overall that there are no obvious outliers in the dataset. Thus, the measured mode shape on the BU is most likely consistent. This data set has proven to be fairly consistent from many other earlier studies too. In order to show the power of the DCAF, some of the data was intentionally contaminated by replacing one or more of the 25 points with the data collected at other points. For example, as shown in Fig. 5(a), the mode shape value at Point 1 for all modes is replaced by Point 21. The DCAF process is performed on the new set of data and the DCAF is plotted. As each of the data points is removed, all the separate DCAF assessments indicate that Point 1 does not appear to be consistently related with the rest of the 24 data points. Point 1 was the point that was intentionally contaminated and DCAF easily identifies that. Then, the same is done for Point 10, as

Measured Time Response at 25 points

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Point 20

24 25

Fig. 6. Measured time response on the BU.

Fig. 7. Time response assessment on the BU using F-SEREP.

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shown Fig. 5(b). Again, the DCAF clearly indicates that Point 10 is not consistently related with the other 24 points in the data set. Next two data points are intentionally contaminated as shown in Fig. 5(c). Using the DCAF, the data consistency is examined to identify the contaminated points. As shown in Fig. 5(c), the two contaminated points are identified to be inconsistent with the rest of the points. This further supports the validity of DCAF. Therefore, the DCAF can examine the consistency of the mode shape measured on the BU and the inconsistent data can be clearly identified. While shape information may be able to be identified visually, the DCAF clearly assists in the identification of data that is not consistently related to the rest of the data points. The dynamic time response data is much more difficult to determine if inconsistent data exists when looking at raw data and is evaluated next. 3.1.2. Time response check The 25 points sparse mesh used for mode shape assessment of the BU is also used for the time response assessment study, as shown in Fig. 6. The BU upright plate is subjected to hammer impact at the same corner with the modal testing. Accelerometer data was acquired from the 25 points, along with 5 additional, alternate points which are not part of the 25 points, as shown in Fig. 8. The 25 measured time response along with a zoomed-in view at Point 20 are shown in Fig. 6. The process of the DCAF for the time response expansion is shown in Fig. 2. Three separate expansion cases are studied with this data using

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Case A

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15 20

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Case B Case C Case D

Use Replacement Point c to Replace Point 2 Use Replacement Point c to Replace Point 10 Use Replacement Point e to Replace Point 3 Use Replacement Point c to Replace Point 17 Use Replacement Point e to Replace Point 8 Use Replacement Point c to Replace Point 22 Use Replacement Point e to Replace Point 4 Use Replacement Point b to Replace Point 6

Replacement Point

(A) Point 2 and Point 10 Contaminated

(B) Point 3 and Point 17 Contaminated

Point 10 and Point 2 Identified

Point 17 and Point 3 Identified

(C)

(D)

Point 8 and Point 22 Contaminated

Point 22 and Point 8 Identified

Point 4 and Point 6 Contaminated

Point 6 and Point 4 Identified

Fig. 8. Time response assessment on the BU using F-SEREP when there is cable swapping.

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a) Finite element model mode shapes for expansion (F-SEREP), b) Experimentally measured mode shapes for expansion (E-SEREP) and c) Polynomial expansion (P-Expansion). and the DCAF process is deployed for each expansion approach for the time response data. 3.1.2.1. SEREP expansion using analytical mode shapes of finite element model. In the DCAF shown in Fig. 7, the TRAC is over 0.9 at every point no matter what removed point configuration is selected. The DCAF indicates that the recovered time response is well correlated with the measured time response at all points. Thus, the set of 25 measured data on the BU is most likely consistent. In addition, the inconsistent point identification is further studied when two points are intentionally made to be inconsistent with the rest of the points. Two points of the whole set of points are replaced by completely different measurements made at different locations that are not part of the original 25 time data points when the original data was collected; these are from the 5 extra points collected as shown by the blue dots on the BU structure in Fig. 8. Using the DCAF, the data consistency is examined to identify the contaminated points. The DCAFs of four different two-points contaminated cases are shown in Fig. 8 as examples. As shown in Fig. 8, the two contaminated points are identified to be inconsistent with the rest

Fig. 9. Time response assessment on the BU using E-SEREP.

Fig. 10. Time response assessment on the BU using P-Expansion.

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of the points for each of the 4 different cases studied. This shows the power of the DCAF technique to identify erroneous or inconsistent points in a data set; clearly this would be very difficult to identify in the measured time response data. 3.1.2.2. SEREP expansion using experimental mode shapes. The previous case used finite element mode shapes for the expansion process (F-SEREP). In this case the experimentally measured mode shapes are used for expansion (E-SEREP). The DCAF is used to assess the data consistency of the same set of 25-points time response data. Fig. 9 produces similar results as that shown in Fig. 7 which is the companion expansion using the finite element mode shapes. Here the DCAF shows that the set of 25 measured data on the BU is most likely consistent. 3.1.2.3. Polynomial expansion. The previous two cases used finite element mode shapes for the expansion process (F-SEREP) and experimentally measured mode shapes for the expansion (E-SEREP). This case uses the polynomial approach for expansion for comparison. The DCAF is used to assess the data consistency of the same set of 25-points time response data and is shown in Fig. 10. This produces similar results as that shown in Figs. 7 and 9 which is the companion expansion. Here the DCAF shows that the set of 25 measured data on the BU is most likely consistent. In order to further show the power of DCAF, the P-Expansion is used to identify the intentionally contamination of data. Here, there are two points that are replaced by other points. The P-Expansion DCAF is used to assess the data consistency of

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Case B Case C Case D

Use Replacement Point c to Replace Point 2 Use Replacement Point c to Replace Point 10 Use Replacement Point e to Replace Point 3 Use Replacement Point c to Replace Point 17 Use Replacement Point e to Replace Point 8 Use Replacement Point c to Replace Point 22 Use Replacement Point e to Replace Point 4 Use Replacement Point b to Replace Point 6

Replacement Point

(A) Point 2 and Point 10 Contaminated

(B) Point 3 and Point 17 Contaminated

Point 10 and Point 2 Identified

Point 17 and Point 3 Identified

(C)

(D)

Point 8 and Point 22 Contaminated

Point 22 and Point 8 Identified

Point 4 and Point 6 Contaminated

Point 6 and Point 4 Identified

Fig. 11. Time response assessment on the BU using P-Expansion when there is cable swapping.

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the new data set. Fig. 11 shows 4 separate cases where two data points are contaminated and DCAF is able to identify that the points are inconsistently related to rest of the data points.

3.2. Dryer cabinet base panel In the previous section, using the DCAF, a set of time response data measured on the BU is examined and proven to be consistent; in addition, intentionally flawed measurements were used as replacements to good data to show that the erroneous data could be identified clearly. In this section, the DCAF is also used to examine the consistency of a set of measurements on a dryer cabinet base panel [18], shown in Fig. 12. The DIC optical measurement on the panel is performed to obtain the full field displacement time response using 3D DIC [19–23]. The test set up and test parameters setting is shown in Fig. 12. An impulse excitation is applied normal to one corner of the panel using an impact hammer having a soft hammer tip. The full field displacement time response of the speckled panel due to the impulse is measured as a reference. Only the displacement response at a portion of the panel is captured by DIC, shown by the illuminated area. 12,504 points are obtained in this effective measurement area.

(a)

Time Step Total Time Sampling Frequency Frequency Resolution Spectral Line

(b)

1/500 s 1.234 s 500 Hz 0.81 Hz 300

(c)

Fig. 12. (a) Dryer cabinet base panel. (b) Test configuration. (c) The effective area of the measurement.

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A subset of 25 points is selected from the effective measurement area, which is shown in Fig. 13. In this section, DCAF with two different expansion approaches are used to assess the consistency of this set of 25 time response. 3.2.1. SEREP expansion using analytical mode shapes of finite element model The F-SEREP DCAF is used to check the data consistency of this set of 25 points on the dryer panel, which is shown in Fig. 14. The DCAF shows that the majority of the data appears to be consistently related and there are not any obvious inconsistencies.

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Fig. 13. Measured time response on the dryer panel.

Fig. 14. Time response assessment on the dryer panel using F-SEREP.

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3.2.2. Polynomial expansion The P-Expansion DCAF is used to check the data consistency of this set of 25 points on the dryer panel, which is shown in Fig. 15. The DCAF shows that the majority of the data appears to be consistently related and there are not any obvious inconsistencies.

Fig. 15. Time response assessment on the dryer panel using P-Expansion.

Fig. 16. Test set up for roadway. (a) Roadway structure; (b) Deployment of accelerometers; (c) Data acquisition system; (d) Impact hammer.

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As shown in Figs. 14 and 15, the DCAF has low values which indicate that this set of data collected on the dryer panel is consistent. Therefore, the consistency of the data collected on the dryer panel can be identified using the DCAF. Here the data was well collected and there were no significant outliers in the time response data sets. In the next example there are some very obvious data points that are identified.

Impact

(a)

(b)

(c) Fig. 17. (a) Accelerometers and impact location; (b) Impact force; (c) Measured time response at 55 points.

Impact

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Fig. 18. A set of the 12-points base for data consistency assessment.

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(a)

Point 1 Point 2 Point 3 Point 4 Point 5 Point 6 Point 7 Point 8 Point 9 Point 10 Point 11 Point 12

(b) Fig. 19. Time response assessment on the 12-points base using P-Expansion. (a) DCAF; (b) Diagonal of DCAF.

TRAC 0.93 0.89 0.94 0.95 0.96 0.93 0.95 0.92 0.91 0.85 0.88 0.88

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Impact

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(a)

Point 1 Point 2 Point 3 Point 4 Point 5 Point 6 Point 7 Point 8 Point 9 Point 10 Point 11 Point 12 Point 13 Point 14 Point 15 Point 16 Point 17 Point 18

TRAC 0.92 0.87 0.92 0.93 0.94 0.92 0.92 0.92 0.88 0.78 0.83 0.82 0.82 0.94 0.80 0.61 0.74 0.83

(b) Fig. 21. Time response assessment on the 18-points base using P-Expansion. (a) DCAF; (b) Diagonal of DCAF.

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3.3. Large-scale robot roadway In this section, the DCAF is applied on a set of time response data measured on a large-scale robot roadway. The section of the roadway tested was approximately 10 feet by 20 feet (3 m by 6 m). There are ribs running across the structure to provide support. The structure and test set up including data acquisition system, deployment of accelerometers and hammer used for excitation are shown in Fig. 16. An impact time response testing has been performed under a random rapid impact. The time response is collected at an arrangement of 55 uniaxial accelerometers deployed on the upper surface of the plate, shown by the black dots in Fig. 17(a). The impact location is shown in Fig. 17(a) and the input force in both time and frequency domain are shown in Fig. 17(b). The measured 55 time data points are shown in Fig. 17(c). For this particular structure, there were many decking panels that were very loosely attached which caused measurement difficulties (rattles, nonlinear response, overload, saturation, etc.) During the testing performed, there were several measurements that were observed to have an overload and there was a concern as to the consistency of the data collected. The consistency of the measured data is examined by the DCAF. Because finite element model of the structure is not available, only the P-Expansion DCAF is performed. The DCAF is performed on a subset of 12-points, shown in Figs. 18 and 19. In Fig. 19(a), the DCAF does not show any significant data points that appear to be universally inconsistent. In addition, in Fig. 19(b), the TRAC table shows the diagonal of DCAF, and the overlay between the expanded response and the original response at the diagonal of DCAF is also shown, respectively. The TRAC values for all 12 points appear to be reasonable considering the complexity of the structure. The DCAF is then performed on a subset of 18-points, shown in Figs. 20 and 21. In Fig. 21(a), while the DCAF is not as clear as some of the previous cases, there is clearly a set of points related to Point 16 and Point 17 that appear to have some inconsistency with the rest of the data points. In addition, the TRAC for all 18 data points are reviewed in Fig. 21(b) and the TRAC associated with Point 16 and Point 17 are lower than the TRAC for the rest of the points. Clearly Point 16 and Point 17 are suspect. These points were removed from the data set and the results were improved. Because there is not ‘‘truth” model for this set of data, no additional analyses were performed. But clearly DCAF assisted in identifying data that does not appear consistent with the rest of the data in the data set. 4. Conclusion The Data Consistency Assessment Function (DCAF) was developed to assist in the identification of data points that do not appear to be consistently related to the entire data set. Existing tools such as coherence in a measurement or synthesis of measured data from a modal test are only useful for identifying the accuracy of the measurement to itself and has no relationship to the entire set of data. The DCAF helps to identify the consistency of data points to the entire set of data. The DCAF relies on an expansion process. Three different expansion processes are shown using finite element mode shapes or experimental modal shapes or polynomial expansion processes. Each of the three different expansions were shown to be useful approaches. Several different structures were used to show the application of the technique. In all cases studied and shown in this paper, DCAF was shown to be an effective tool to assess the consistency of data to the entire set. CRediT authorship contribution statement Yuanchang Chen: Software, Formal analysis, Investigation, Resources, Data curation, Writing - original draft, Visualization. Peter Avitabile: Conceptualization, Methodology, Validation, Writing - review & editing. Jacob Dodson: Supervision, Project administration, Funding acquisition. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements Some of the work presented herein was partially funded by Air Force Research Laboratory Award FA8651-16-2-0006 ‘‘Nonstationary System State Identification Using Complex Polynomial Representations”. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the United States Air Force. Distribution A. Approved for public release; distribution unlimited (96TW-2020-0002). The authors are grateful for the support obtained. References [1] R.J. Allemang, D.L. Brown, A correlation coefficient for modal vector analysis, in: Proceedings of the 1st International Modal Analysis Conference, 1982, pp. 110–116.

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