System Identification and Modal Extraction from … Conference... · System Identification and...

American Institute of Aeronautics and Astronautics

Presentation approved for public release: 412 TW-PA-13170

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System Identification and Modal Extraction from Response Data

P. Chase Schulze,* Peter M. Thompson, Ph.D.,† Brian P. Danowsky,‡ and Dongchan Lee, Ph.D.§ Systems Technology, Inc., Hawthorne, CA, 90250

and

Chuck Harris, Ph.D.** Air Force Test Center, Edwards AFB, CA, 93524

Aeroelastic stability or flutter testing is a critical part of the development of any new aircraft system, or configuration expansion. Each new aircraft and configuration needs to be assessed to ensure that it is free of any excessive and possibly destructive aeroelastic interactions. Such information could be determined from an appropriate model, but this information and insight is only as good as the model it was derived from. As such, there is a need to have a toolset that will identify necessary aeroelastic modal frequencies, damping ratios and mode shapes directly from flight test data and also compare this information against a known model. An additional complicating factor is that the driving excitation inputs to the aircraft are not always completely known, hence the identification techniques need to be capable of operating using only output sensor data. To meet these challenges, three identification methods that leverage both time and frequency domains are presented here and a variety of comparison metrics are presented to evaluate the relative accuracy of the identification approaches. Additionally, a real-time demonstration of many of these techniques was also conducted to validate the capabilities of the identification methods and comparison metrics.

I. Introduction he modal extraction problem has three primary parts: identification of modal frequencies, identification of modal damping ratios and identification of the aeroelastic mode shapes. This particular problem, which includes

the comparison of these identified mode shapes against a known model, adds an additional complication; how should a complex mode shape be represented and compared against another and what qualifies these shapes as matching? To add to this challenge, these identifications must occur in near real-time during a flight test so that the results can be presented in a clear manner to the controls room engineer.

Three unique identification methods are presented here to meet these requirements, two based in the frequency domain and the third in the time domain. Each method is included to serve as validation for the others, particularly when comparing the frequency and time domain identification results against one another. Since each method approaches the problem from a completely different manner, a greater amount of confidence can be placed in their results when they match well with one another. All of the presented methods can identify the required modal frequencies and damping ratios, but only the time domain and one of the frequency domain methods can identify aeroelastic mode shapes as well.

* Staff Engineer, Research, AIAA Member. † Chief Scientist, AIAA Senior Member. ‡ Principal Research Engineer, AIAA Senior Member. § Principal Specialist. ** Technical Expert Structures Brach.

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Each of the identification methods requires only the output sensor data and no knowledge of the input signals, whether they are known or unknown, is required for identification. This is particularly important for those instances where the exact input or excitation signal is unknown (e.g., atmospheric turbulence). Even when the input signal is known, the intended and actual excitation can still exhibit some discrepancies. These three methods are detailed below: the Curve-fitting Frequency Domain Decomposition (CFDD, frequency domain), Bendat and Piersol’s Half-Power Bandwidth (B&P, frequency domain) and the Generalized Realization Algorithm (GRA, time domain). CFDD and GRA are capable of estimating and identifying the mode shapes in addition to the modal frequencies and damping ratios.

Comparison of these identified mode shapes is facilitated primarily through the use of three different mode shape comparison diagrams: the modal elements diagram [1], the Taylor diagram [2, 3] and by directly overlaying either the estimated real mode shapes or by visualizing the fully complex mode shape animations.

Having addressed both the identification of the necessary modal properties and the display and comparison of the identified and modeled mode shapes, these analysis methods and metrics were applied in a real-time environment. Using a high order aeroelastic F/A-18C model and a fixed based piloted simulator, a simulated discrete tracking task was flown and its data analyzed in real-time using the CFDD and B&P identification methods and applying the above metrics. The comparative metrics previously described, in addition to time-varying frequency and damping plots were combined into a demonstration interface. This demonstration highlights the ability of the CFDD and B&P identification routines to accurately and rapidly identify the necessary information from real-time flight data and display said information in a meaningful and easy to understand manner.

II. Identification Methods For the demonstration described herein, only the CFDD and B&P methods were used as the identification

algorithms. The displayed results used the CFDD results only. Each of these three methods will be a part of the final toolset in order to serve as verification of one another, but due to time constraints only the frequency domain methods were implemented here. An example is provided for the GRA method and includes results from the CFDD and B&P identifications of the same data.

A. CFDD The classical approach used to identify modal characteristics of a vibrating system has been based on picking the

peaks in the power-spectral density diagram [4]. For well separated modes, the technique can give a reasonable estimation of natural frequencies and mode shapes but when the modes are closely spaced, the estimation of the mode shapes and natural frequencies can become heavily biased [5]. Also as Bendat and Piersol pointed out, the autospectra of the output signals do not provide sufficient information to estimate accurate normal modes of vibration [4]. The cross-spectra magnitude and phase along with coherence are also equally important to give an accurate estimation of the normal modes.

Figure 1: ID Tail Example


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Figure 2 present the autospectral density plots of the outputs measured at the tip of an aeroelastic model of a vertical tail and rudder, pictured in Figure 1. This model is based on a linear finite element model composed of quadrilateral plate elements [6] and a linear aerodynamic model based on the unsteady doublet lattice method [7,8]. These outputs were simulated using the full order aeroelastic model. As shown in Figure 2 all three modal frequencies are identified in all of the autospectral plots.

a) wing tip out-of-plane (z) translation

b) wing tip chordwise (x) rotation

c) wing tip spanwise (y) rotation

d) rudder deflection

Figure 2: Autospectrums

Picking peaks from the autospectral plots of the outputs does not provide enough information. This point can be further illustrated with the cross-spectral density plots shown in Figure 3a. The cross-spectral density plot between the z translation and the y rotation signals reveals that all three dominant peaks at the modal frequencies do not reflect the mode shapes. This is true because the z translation and the y rotation are virtually in sync without significant phase difference. As shown in Figure 3b, the cross-spectral density plot between the heave and the torsion modes reveals that the three peaks are truly the dominant modes since the phase angles show either ±180° or zero degree at each dominant peak. For the cross-spectra density plot between the z translation and the rudder deflection shown in Figure 3c only the first two peaks are the dominant modes since the last peak does not show any change in phase angle.

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a) z translation and y rotation

b) z translation and x rotation

c) translation and rudder deflection

Figure 3: Cross-spectra

The importance of the cross-spectra densities between each measured output is clearly indicated in the above study. However, using classical frequency domain analysis for picking dominant peaks imposes a great deal of practical difficulties. All cross-spectral density plots need to be investigated but it is not practical when there are many measurements to deal with. To remedy this issue, Brincker et al. suggested the use of the Singular Value Decomposition (SVD) of the power spectral-density matrix [5, 9]. In this approach, the power spectral-density matrix is first built from all measured outputs containing diagonal elements with the autospectral density of each output and off-diagonal elements with the cross-spectral densities of individual outputs. The power spectral-density matrix at each frequency of interest is then decomposed by taking the SVD of the matrix. The Frequency Domain Decomposition (FDD) theory states that the first singular vector corresponding to the first singular value is an estimation of the mode shape corresponding to a single degree of freedom system [5, 9]. This technique was applied to the measured outputs (various measurements on the tail) simulated with the full order aeroservoelastic model and shown in Figure 4. The singular values plot shows three well separated dominant peaks which correspond to individual single degree of freedom dominant modes. It should be noted that the measured outputs are not properly normalized so that the singular value plots shows two distinctive groups. These spectra take all measurements into account.

With the identified autospectral density function, the natural frequencies and the damping ratios can easily be estimated. Two different techniques are available to estimate the natural frequencies and damping ratios. The first approach is based on the inverse Fourier transformation of the identified autospectral density function. In this technique, the singular vector at each peak, which is an estimate of the mode shape of a single degree of freedom system, is compared with other singular vectors within a frequency range around the peak. This is called the Modal Assurance Criteria (MAC) test and if the predetermined MAC value is not satisfied the corresponding autospectral

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density function at the frequency is set to zero. The MAC is a standard metric widely used to evaluate the correlation between two modal vectors, generally experimental vs. modeled mode shapes [10]. This procedure is repeated for each identified peak. For each peak, the reconstructed autospectral density function is converted back to the time domain with the inverse Fourier Transformation. The natural frequency and the damping ratio are found by estimating crossing times and logarithmic decrement in the time domain. This technique shows promising results [5, 9] but imposes a great deal of overhead in memory since all the singular vectors need to be saved for each frequency point for the MAC test later.

A newly developed technique called the Enhanced Frequency Domain Decomposition (EFDD) remedies this extra overhead by fitting each identified peaks with an assumed form of a frequency response function [11].

The FDD and EFDD techniques show promise for the problem of identifying an unknown system with output-only measurements. The techniques investigated here can be combined to develop a technique with robustness and accuracy. In most system identification problems the difficult part is to estimate unknown damping ratios and the natural frequencies since those estimates are closely related to the poles of vibrating systems. The inaccurate estimation of the poles usually leads to an unstable simulation when used with the output-error or least squares error based system identification schemes. With the FDD/EFDD techniques, the estimation of unknown poles can be easily carried out, reducing unknown system parameters dramatically and as a result robust and stable system identification.

Figure 4: Singular Values of the Decomposed Spectral Density Matrix

The CFDD technique, derived from the EFDD technique, is a method to directly identify the natural frequencies and the damping ratios from the Single Degrees of Freedom (SDOF) spectrum of measured outputs. As the name indicates, this method fits each SDOF spectrum with a 2nd order transfer function and the roots of the denominator of the identified transfer function provides the estimation of the natural frequencies and the damping ratios of the individual SDOF vibration modes. In theory, the transfer function of a power spectral density function can be factorized into two parts such as, ( ) ( ) ( )S s H s H s= − , where s jω= and the magnitude of ( )S s is the power spectral density function itself. In the current study, the transfer function ( )H s is the individual SDOF vibration modes with 2nd order denominator polynomials. Since ( ) ( ) ( )S s H s H s= − ,the fitted transfer function becomes fourth order. As a result, the identified 4th order transfer functions show the symmetry about the imaginary axis in their poles locations (same frequency and same damping magnitude with both positive and negative values). The 2nd order pole with positive damping (stable) is then used since the time series data does not indicate diverging behavior.

B. B&P To serve as a further check on the identified frequencies and damping ratios, a secondary frequency domain

identification method was also used; Bendat and Piersol’s half-power bandwidth. This method uses the same peak frequency and power spectral density information determined for the above CFDD method as a starting point. Unlike the CFDD method it does not provide identified mode shapes and as such its use here is to serve as a check on the CFDD and GRA (to be described) results. Bendat and Piersol’s method for estimating structural damping and modal frequencies [4] is based on the half-power point bandwidth of gain factor around the peak frequency rf given by,

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1 where ( ) ( ) ( )2r rB f f H f H f H f= − = = (1)

The equivalent damping ratio for the mode can be estimated as

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As rules of thumb, the damping ratio estimation through Eq. (2) needs to satisfy four conditions:

1. The autospectrum of the excitation must be reasonably uniform over the frequency of the normal mode. 2. The modal damping must be small (ζ ≤ 0.05). 3. The analysis resolution bandwidth must be much smaller than the half-power point bandwidth of the mode. 4. The mode must not overlap heavily with neighboring modes. Bendat and Piersol’s method works well

when there are well spaced sharp peaks in the spectrum.

C. GRA The GRA [12, 13] method is a time domain subspace-based identification approach that utilizes the output time

series information to estimate the A and C state and output matrices by solving a linear least squares problem. Recent innovations by Miller and de Callafon make use of the input time history data when available to setup and use the linear least squares problem to estimate the B, and D input and mixing matrices respectively and the initial condition. The complex eigenvalues and eigenvectors of the A matrix provide the estimation of the modal properties and shapes. The GRA method was used to identify the same finite element tail model as was used in the CFDD description, only using a different output set, as indicated in Figure 5. The z translation sensors at each of these nodes provided the data for the identification results tabulated in Table 1. The frequency and damping values closely match the model well, particularly considering the respective magnitudes of the frequency and damping values.

Figure 5: STIFEM Sensor Locations.

Table 1: Frequency and damping comparison – STIFEM.

Analytical model CFDD Bendat & Piersol GRA

Mode Frequency (rad/s) Damping Frequency

(rad/s) Damping Frequency (rad/s) Damping Frequency

(rad/s) Damping

First bending 114 0.0438 114.08 0.0357 113.47 0.0365 116 0.0194

First torsion 307 0.0243 307.24 0.0198 306.65 0.0191 298 0.0203

Second bending 376 0.000246 375.84 0.0097 377.17 0.0093 375 0.0034


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III. Modal Analysis Visualization Tools The most direct means of comparing two mode shapes would be to overlay the shapes as seen in Figure 6.

Comparing two mode shapes directly allows for a quick qualitative evaluation and can be done for the simple mode shape in Figure 6 or for the more complicated shape seen in Figure 7. This method of evaluation can only show that the shapes are similar, or representative of the same mode, but cannot convey how accurately one shape matches the other in any quantitative sense. This is not to say that this information is useless, it can be quite valuable, particularly for an animated shape where the complex motion of the shape can be properly visualized. To address this deficiency, a spectrum of comparative figures was proposed and evaluated of which two are described in more detail here: modal element comparisons and the Taylor diagram.

In general, the aeroelastic mode shapes under consideration here are complex-valued, indicating that not all points are at their peak deflection values at the same time and hence the shape cannot be truly represented with a single image. To avoid this issue, synchronous mode approximation was applied to the reference mode shapes. This method rotates the complex vector on the complex plane in order to minimize the imaginary portion (maximizing the real portion) while preserving the magnitude of the vector [14]. Once this was done the real portions of the mode shape, now the dominant terms in the complex vector, were plotted. When comparing a mode shape against this reference shape, the phase of the new shape was adjusted to match that of the reference shape so that an accurate comparison could be made between the two shapes. This adjustment was utilized for all comparison figures and metrics except the mode shape animations.

Figure 6: Sample Overlay of Mode Shapes

Figure 7: Sample #2 of Mode Shape Overlay

A. Modal Element Comparison The mode shape elements method [1] works by directly comparing the elements of the eigenvector between the

identified and model mode shapes. The model elements are plotted along the x-axis while the identified mode shape elements are plotted along the y-axis. If the model and identified mode shapes were to match perfectly, all of the data would fall along the y = x dashed line as in Figure 8. If the data does not fall along the line with slope 1, it generally has one of three other patterns: 1) a linear line that does not have a slope of 1, 2) a linear line with some scatter, or 3) poorly or no defined linear line with an excessive amount of scatter. The first pattern is likely due to a scaling error of either shape while the second points to some inaccuracy in either the model or estimated mode shape. If the data has more than a marginal amount of scatter, there could either be some larger inaccuracies in either shape or the two shapes being compared are not related to the same mode. In addition to this direct graphical comparison of the mode shape elements, the MAC value is also provided for each mode in the legend of the figure. The equation for this is given below in (3) where φID and φm represent the identified and model mode shape eigenvectors respectively and T represents the transpose operation.

( )( )IDTIDm

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In Figure 8 the lowest of the displayed MAC values also corresponds to the set of points that is least linear, reflecting the intended trend and type of information to be presented.


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Figure 8: Example Mode Shape Element Diagrams

B. Taylor Diagram The Taylor diagram offers a way to examine statistical measures that describe how well two patterns match one

another [2, 3]. Originally conceived to evaluate and monitor climate model performance, its ability to present multiple statistical measures of pattern matching makes it a possible candidate for comparing mode shapes. The three measures used, correlation coefficient, root-mean-square (RMS) difference, and the standard deviations define a single point in the Taylor diagram. These particular statistical measures are appropriate given the ability of the correlation coefficient and RMS to quantify pattern similarity and quantify differences in two fields respectively. The correlation coefficient (R), center patterned RMS ( E′ ) and standard deviations (σ ) are defined as given below:

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rf

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where the identified and model data are defined as f and r respectively and their averages are denoted f and r . The relation between the standard deviations, RMS and correlation coefficient is given geometrically in Figure

9a [2] and a sample Taylor diagram is shown in Figure 9b.

a) Taylor Diagram Statisical Relations

b) Sample Taylor Diagram

Figure 9: Taylor Diagram Statistical Relations [2].

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The various statistical metrics have all been normalized by the model’s standard deviation ( rσ ). The standard deviation of the model, shown as a red circle, is plotted on the x-axis at a value of 1. Radii of constant RMS values are plotted as the dashed green lines centered on the red circle and lines of constant correlation coefficient radiate out from the origin. Qualitatively, the closer any single point is to the model normalized standard deviation (red circle) the better the model and identified mode shape match.

C. Mode Shape Animation and Overlay The final diagram used in this demonstration was a mode shape overlay. The identified and model mode shapes

were directly overlaid with one another to compare their relative shapes. The dotted blue lines are provided to show the un-deformed shape of the wing.

For this demonstration the mode shapes were not animated but will be for future iterations of the toolset. The shapes will be animated using their fully complex representation. This animation will be performed using the complex representation of a wave applied separately to the real and imaginary portions of the mode shape:

( ) ( )( ) ( ) ( )( )Re Re Im Imi t i te eω ωϕ ϕ+

The mode shape is defined as ϕ, the frequency of the motion as ω, and t was the time vector defining the duration of the animation. For ease of visualization, ω was defined at a fixed value for all mode shapes as opposed to using the associated frequency for each mode shape. The synchronous mode adjustment used for the other mode shape comparison methods is not required for this diagram as it will show the true complex nature of the mode shape. The phase alignment of each of the modes is still required though, so that the motion can be synchronized between the identified and modeled mode shape.

Figure 10: Mode Shape Animation and Overlay

IV. Real-Time Demonstration

A. F/A-18C Aeroelastic Model For this demonstration, a linear time invariant (LTI) F/A-18C model was utilized. This model is based on a

CFD/CSD-based reduced order model (ROM) of the F/A-18C aircraft in steady level flight at Mach 0.7, 10,000 ft. MSL. Model details (structural modes, etc.) are documented in Refs. [15] and [16].

The ROM was based on a constrained structural model without free rigid body degrees of freedom. This model was combined with a rigid body model of the same aircraft operating at the same flight condition. The original model was outfitted to include many sensor outputs measuring both translational and angular displacements, velocities and accelerations at various locations throughout the aircraft structure.


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Figure 11: Combined F/A-18C Flexible and Rigid Body Model

The model is outfitted with four control inputs described in Table 2 below. The control surfaces commands are equal and opposite moments acting along the respective hinge line. These moments are linearly related to control surface deflection angles as described in Table 3.

Table 2: Control inputs.

Input Name Description Units stabs down collective stabilators (positive

down deflection) in-lb

differential aileron (roll left) differential ailerons (positive right down, left up)

in-lb

rudders left collective rudders (positive left) in-lb thrust engine thrust force lb

Table 3: Control input conversions.

Input Name Conversion factor for radian input

Conversion factor for degree input

stabs down 4.3373E6 in-lb/rad 7.57E4 in-lb/deg differential aileron (roll left) 4.1066E5 in-lb/rad 7.1674E3 in-lb/deg

rudders left 3.2284E8 in-lb/rad 5.6346E6 in-lb/deg The sensor nodes have been distributed throughout the aircraft structure and are shown in Figure 12a and b. This

model has 91 defined sensor nodes. Each node coordinate frame is pinned to the structure so that its measurement is representative of a mounted sensor. The orientation of all node coordinate frames are aligned with the inertial frame at equilibrium (x – forward, y – right, z – down). At each sensor location, the model measures 18 quantities, providing for a total of 1638 possible output signals. Of these, the z-acceleration signals at 17 different node locations on the right wing and missile were retained for use with the CFDD and B&P identification methods. The location of these signals and resulting shape are given in Figure 12c. Each intersection in the enlarged portion of the wing represents a sensor location.

F/A-18C Flex

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a) View from Behind the Right Wing

b) Top View

c) Enlarged View of Retained Nodes

Figure 12: F/A-18C Model Sensor Node Locations

B. Simulator The STI fixed-base pilot-in-the-loop flight simulator was utilized. The key elements of the simulator including

the pilot are identified in Figure 13. The McFadden feel system is a key component in that it provides proprioceptive force feedback cues that enhance the fidelity of the simulation. Features of the PC-based simulation and projected display are described below.

Figure 13: Pilot-in-the-Loop Simulator Elements

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The simulator shown in Figure 14 is a win32 console application designed to interface with Matlab™ for data input and output. It is capable of simultaneously simulating the time response to arbitrary input of a user defined Simulink© model, which represents aircraft dynamics.

A user defined Simulink© model can contain arbitrary complexity which can include (but is not limited to): Linear airframe equations of motion defined by state space models; Nonlinear airframe equations of motion; Multiple aircraftmodels; Nonlinear software rate limits and actuator models; Data recording of unlimited output states; Vehicle dynamics that update at up to 1000 Hz and graphics that update at a minimum of 50 Hz; and Texture-mapped PC graphics with a superimposed head-up display that supports pitch and roll axis tracking tasks.

Figure 14: STI Simulator with McFadden Control Loader and Projected Display

During a run, the pilot is presented a HUD superimposed on a suitable 3-D environment (see Figure 14). At the end of a run, the input vectors and output time responses of all simulated systems are saved to a Matlab™ file for future analysis. The simulator uses tracking bars on the HUD to support sum-of-sines and step and ramp discrete tracking tasks. These tasks provide the continuous tracking environment associated with operational tasks such as close formation flying and aerial refueling

C. Interface and Data Acquisition The demonstration interface and figures were created in the Matlab© environment. The necessary data was

processed using the identification algorithms within this Matlab© environment as well. This interface was intended only to be a visual mechanism by which the near real-time capabilities of the identification methods could be shown, many of its intended features and advanced functionalities have not been included.

A data file containing the necessary sensor data from the simulator was saved every 3 seconds and contained the previous 15 seconds worth of output data. The most recently saved data file was loaded into the Matlab© workspace and processed using the CFDD and B&P identification algorithms. The interface is then updated with the relevant information and the next set of data read in before repeating the process. Note that this interface contains all of the previously described mode shape comparison figures with the exception of the error diagrams; these were omitted for readability and space considerations but the necessary information to generate these is already available. The time varying frequency and damping plots use the last time of the file being analyzed, so if the file contained data for the 15-30 second interval, the data point would appear at 30 seconds. If no identification could be made due to inadequate excitation of any of the modes, no new information would appear and the next set of data would be loaded in and processed. The diagram in the lower right of the interface presents a time history of the maximum SVD values (shown in blue), the identified peaks and the transfer functions used to fit them (shown in red). When multiple sets of data have been processed, this image will show the time variation of this information, with lighter colored lines representing identifications further back in time.

D. Demonstration Results For this demonstration, the pilot flew a discrete pitch and roll tracking task, requiring inputs in both the

longitudinal and lateral axis. The duration of this task was ~125 seconds and the profile for each axis is given in Figure 15.


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Figure 15: Discrete Tracking Demonstration Task

This demonstration began with the launching and initialization of STI’s simulator. As a part of this initialization, a Matlab™ environment and workspace was initiated and the identification interface seen in Figure 16was also created, only without any data processed. The simulator environment and visuals were started and the pilot began to fly the tracking task shown in Figure 15. The Start Analysis button in the identification interface was then selected to prompt the interface to begin searching for data and performing its identification using the CFDD and B&P methods. Once the appropriate amount of data had been gathered, the simulator generated the necessary data file and the interface began to process this data. After the successful identification of the necessary modal properties the figure automatically updated to look similar to that seen in Figure 16.

For this demonstration, the interface is comparing the lowest frequency identified mode with the known first bending mode at ~31 rad/s with a damping ratio of ~0.02. The correct identification of this mode is highlighted in the comparison figures and metrics shown in Figure 16. The modal element comparison results have the desired slope of 1and the MAC value is very near 1 as well. The statistical measures in the Taylor diagram also reflect an identified mode that matches the model very well. The standard deviation of the identified mode shape is also very nearly 1, matching the standard deviation of the model. The RMS value has also been minimized and the correlation coefficient is essentially 1. In addition, there is no distinguishable difference in the mode shape overlay. Any discrepancies in the mode shapes would be clearer in an animated version of this figure, a planned feature of the future iteration of this interface. All of these figures and values indicate that the identified mode matches the comparative mode from the model.

The identification made prior to that shown in the interface in Figure 16 incorrectly identified a false mode at ~15 rad/s as can be seen in the frequency time history diagram. This occurred only because the algorithm found and fit a peak in the SVD values. As this interface was to demonstrate the real-time process only, the necessary filtering and comparison algorithms were not in place to prevent this from occurring. Such an occurrence will not be an issue in the final iteration of this toolset.


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Figure 16: Demonstration Interface

V. Conclusions In this program, two identification techniques were demonstrated to be effective at correctly identifying the aeroelastic modal frequencies, damping ratios and mode shapes in a real-time environment while a third has been discussed for later inclusion into the toolset. By incorporating techniques from both the time and frequency domains greater confidence can be placed in their accuracy and validity, particularly for cases where a comparative model was unavailable. The accuracy of the identified mode shapes was examined using a range of analysis metrics and figures, allowing for a quick and clear indication of the accuracy of the identifications or by extension, the accuracy of the relevant model. These methods and metrics were also successfully demonstrated in a real-time simulated flight test using sensor data generated by STI’s fixed based simulator.

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