Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing, Volume 7

River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Chad Walber Matthew Stefanski Steve Seidlitz Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics 2021 River Publishers Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing, Volume 7

Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics, Inc., Bethel, CT, USA

The Conference Proceedings of the Society for Experimental Mechanics Series presents early findings and case studies from a wide range of fundamental and applied work across the broad range of fields that comprise Experimental Mechanics. Series volumes follow the principle tracks or focus topics featured in each of the Society’s two annual conferences: IMAC, A Conference and Exposition on Structural Dynamics, and the Society’s Annual Conference & Exposition and will address critical areas of interest to researchers and design engineers working in all areas of Structural Dynamics, Solid Mechanics and Materials Research

River Publishers Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing, Volume 7 Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics 2021 Chad Walber • Matthew Stefanski • Steve Seidlitz Editors

Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-4380-017-0 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, or reproduction in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Preface Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting &Dynamic Environments Testing represents one of nine volumes of technical papers presented at the 39th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, and held between February 8 and 11, 2021. The full proceedings also include volumes on nonlinear structures and systems; dynamic substructures; model validation and uncertainty quantification; dynamic substructures; special topics in structural dynamics and experimental techniques; rotating machinery, optical methods, and scanning LDV methods; topics in modal analysis and parameter identification; and data science in engineering. Each collection presents early findings from experimental and computational investigations on an important area within sensors and instrumentation and other structural dynamics areas. Topics represent papers on calibration, smart sensors, practical issues improving energy harvesting measurements, shock calibration and shock environment synthesis, and applications for aircraft/aerospace structures. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Depew, NY, USA Chad Walber Dayton, OH, USA Matthew Stefanski Minneapolis, MN, USA Steve Seidlitz v

Contents 1 Exploring Iterative Optimization Methods to Develop a MIMO Control Input ....................... 1 J. Justin Wilbanks, Ryan A. Schultz, and Brian C. Owens 2 All-Electric X-Plane, X-57 Mod II Ground Vibration Test ............................................... 19 Natalie Spivey, Samson Truong, and Roger Truax 3 Mechanical Environment Test Specifications Derived from Equivalent Energy in Fixed Base Modes, with Frequency Shifts from Unit-to-Unit Variability............................................. 41 Troy J. Skousen and Randy L. Mayes 4 Investigation of Transmission Simulator-Based Response Reconstruction Accuracy ................. 65 Matthew J. Tuman, Christopher A. Schumann, Matthew S. Allen, Washington J. Delima, and Eric Dodgen 5 A Proposed Standard Random Vibration Environment for BARC and the Boundary Condition Challenge........................................................................................... 77 Ryan Schultz, Tyler Schoenherr, and Brian Owens 6 Assessment of Metrics Between Acceleration Power Spectral Density Metrics and Failure Criteria 85 Dagny Beale, William Larsen, and Peter Coffin 7 Using Parameterized Optimization to Model a Slip Table................................................ 103 Julie Pham and Tyler F. Schoenherr 8 Using Modal Projection Error to Evaluate ................................................................. 111 Tyler F. Schoenherr and Jelena Paripovic 9 WaveHit: The First Smart Impulse Hammer for Fully Automatic Impact Testing.................... 139 Daniel Herfert and Andreas Lemke 10 Aeroelastic Analysis Using Ground Vibration Test Modes ............................................... 147 David Cloutier and Eric Parker-Martin 11 Localizing Perturbed Objects in a Room with Reflective Boundaries Using Dispersed Acoustic Measurements....................................................................................... 161 Michael J. Gassen, Ian C. Marts, Mitchell J. Roberts, Brian M. West, and Jeffery D. Tippmann 12 Application of Smartphones in Pavement Deterioration Identification Using Artificial Neural Network......................................................................................................... 167 A. Moghadam and R. Sarlo 13 Impacts of Test Fixture Connections of the BARC Structure on Its Dynamical Responses........... 175 K. Jankowski, H. Sedillo, A. Takeshita, J. Barba, A. Bouma, and A. Abdelkefi 14 Experimental and Computational Investigations on Fixture Interference for BARC Systems ....... 179 A. Takeshita, H. Sedillo, K. Jankowski, J. Barba, A. Bouma, and A. Abdelkefi vii

viii Contents 15 Aeroelastic Test of the Nixus FBW Sailplane............................................................... 183 Paulo Iscold and William Fladung 16 Operational Modal Analysis of the Space Launch System Mobile Launcher on the Crawler Transporter ISVV-010 Rollout ............................................................................... 199 James C. Akers and Joel W. Sills 17 Structural Damage Detection in Civil Engineering with Machine Learning: Current State of theArt ....................................................................................................... 223 Onur Avci, Osama Abdeljaber, and Serkan Kiranyaz 18 Nonlinear Analysis and Characterization of Piezoaeroelastic Energy Harvesters with Discontinuous Nonlinearities ........................................................................... 231 Adam Bouma, Erik Le, Rui Vasconcellos, and Abdessattar Abdelkefi 19 Basic Vibration Analysis in a Laboratory Classroom Using Virtual Instruments ..................... 235 William H. Semke 20 Model Class Selection and Model Parameter Identification on Piezoelectric Energy Harvesters.... 245 Alejandro Poblete and Rafael O. Ruiz

Chapter 1 Exploring Iterative Optimization Methods to Develop a MIMO Control Input J. Justin Wilbanks, Ryan A. Schultz, and Brian C. Owens Abstract In general, existing methods to develop an effective input for multiple-input/multiple-output (MIMO) control do not offer flexibility to account for limitations in experimental test setups or tailor the control to specific test objectives. The work presented in this paper introduces a method to leverage global optimization approaches to define a MIMO control input to match a data set representing field data. This contrasts with traditional MIMO input estimation methods which rely on direct inverse methods. Efficacy of the iterative optimization method depends on the objective function and optimization method used as well as the definition of the format of the input cross-power spectral density (CPSD) matrix for the optimization routine. Various objective functions are explored in this work through sampling as well as implementation within the iterative optimization process and their impact on the resulting output CPSD. Performance of iterative optimization is assessed against the traditional, direct pseudoinverse method of obtaining the input CPSD as well as the buzz method and weighted least squares (LS). Constraints can be used within the optimization process to control the magnitude and other aspects of the input CPSD, which allows for shaker limitations to be accounted for, among other considerations. Iterative optimization can provide the best input CPSD possible for a test setup while accounting for any shortcomings the setup may have, including force and voltage constraints, which is not possible with traditional methods. Keywords MIMO control · Global optimization methods · Vibration testing · Dynamic modeling · Multi-axis testing 1.1 Introduction Multiple-input/multiple-output (MIMO) control methodologies are becoming more common in the structural dynamics community alongside multi-axis vibration testing due to the benefits that multi-axis tests have compared to traditional single-axis vibration testing [1]. Free-free MIMO vibration testing provides effective reproduction of the aerodynamic operating environments of vehicles unlike single-axis testing that creates representations that often differ significantly from the operating environments [1, 2]. Daborn et al. showed single-axis testing can alter the dynamics of the test article due to attachment to a shaker and can lead to poor recreation of stress and strain patterns in the test article arising from relative phase deviations in response locations that causes unrealistic failure modes [1]. Various papers have shown the successful recreation of aerodynamic environments for flight vehicles with multi-axis testing [3–5]. Traditional methods in acquiring the input cross-power spectral density (CPSD) to recreate a set of field data do not offer substantial flexibility in terms of constraining the voltage or force required at the shakers used in the test. Using iterative optimization allows for various constraints to be applied to the developed input while limiting the impact on the dB error or other objective function. The following sections of the paper introduce the MIMO control problem for iterative optimization, which is used to observe the impact of the optimization process, objective function, and input constraints on the efficacy of the input derived through optimization relative to traditional methods. Sandia National Laboratories is a multimission laboratory managed and operated by the National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the US Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. J. J. Wilbanks ( ) · R. A. Schultz · B. C. Owens Sandia National Laboratories, Albuquerque, NM, USA e-mail: jjwilba@sandia.gov © The Society for Experimental Mechanics, Inc. 2022 C. Walber et al. (eds.), Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing, Volume 7, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-75988-9_1 1

2 J. J. Wilbanks et al. 1.2 Definition of MIMO Control Problem for Iterative Optimization 1.2.1 Design Variable Choice to Account for Input and Output CPSD Constraints To set up the MIMO control problem for iterative optimization, a set of design variables must be chosen to ensure that the resulting input CPSD, Sxx, is positive definite. The forward MIMO control problem can be defined as: Syy =HyxSxxHyx H (1.1) where Sxx denotes the input CPSD that must be positive definite, Syy the output CPSD, Hyx the transfer function matrix, and ( )H the conjugate transpose. To ensure Sxx is positive definite, the design variables for the iterative optimization process are entries in an upper triangular matrix, Lmat, that is used to create the input CPSD matrix using a Cholesky factorization: Sxx =L H matLmat (1.2) where Lmat is stored using a vector form within the optimization process: [Lmat] = ⎡ ⎢⎢ ⎢⎣ L1,1 L1,2 . . . L1,n 0 L2,2 . . . L2,n . . . 0 . . . 0 . . . 0 . . . Ln,n ⎤ ⎥⎥ ⎥⎦ →{Lvec}= ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ L1,1 L1,2 L2,2 . . . L1,n L2,n . . . Ln,n ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (1.3) where Li, p denotes the element of the Lmat matrix in the i th row and pth column and n the number of input locations for the MIMO control problem. Optimization with design variables chosen to be entries in the Lmat matrix enforces positive definiteness Sxx, which is not the case if optimization is performed with auto-power spectral densities (APSDs) and crossterms of Sxx as design variables or APSDs, coherence, and phase. The design variable choice can be used alongside various optimization algorithms and objective functions. 1.2.2 Optimization Workflow Optimization is performed at discrete frequency lines for a set of field data using the elements of Lvec, dv as the design variables, where Lvec, dv is defined as: Lvec,dv = Lvec,Re Lvec,Im (1.4) where Lvec, Re and Lvec, Im contain the elements of the triangular matrices Lmat, Reand Lmat, Im used to form the real and imaginary portions of Sxx with the Cholesky factorization introduced in Eq. (1.2), respectively. The Cholesky factorization of the real and imaginary portions of Sxx is: Sxx,Re =Lmat,Re HLmat,Re Sxx,Im =Lmat,Im HLmat,Im Sxx =Sxx,Re +j ∗Sxx,Im (1.5)

1 Exploring Iterative Optimization Methods to Develop a MIMO Control Input 3 where Sxx, Re denotes the real portion of Sxx, Sxx, Im the imaginary portion of Sxx, Lmat, Re the upper triangular matrix for the Cholesky factorization of Sxx, Re, andLmat, Imthe upper triangular matrix for the Cholesky factorization of Sxx, Im. Lmat, Re is formed directly fromLvec, Re. Lmat, Imis found using the following expressions to maintain the positive definiteness of Sxx, Im: Lmat,Im = ⎡ ⎢⎢ ⎢⎢ ⎣ 0 Lmat,Im1,2 Lmat,Im1,3 . . . Lmat,Im1,2 0 0 Lmat,Im2,3 . . . . . . . . . . . . 0 . . . Lmat,Imn−1,2 0 0 0 0 0 ⎤ ⎥⎥ ⎥⎥ ⎦ + ⎡ ⎢⎢ ⎢⎣ 1 0 . . . 0 0 1 . . . 0 . . . 0 . . . 0 . . . 0 . . . 1 ⎤ ⎥⎥ ⎥⎦ (1.6) where Lvec,Im = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ Lmat,Im1,2 Lmat,Im1,3 Lmat,Im2,3 . . . Lmat,Im1,2 . . . Lmat,Imn−1,2 ⎫ ⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎭ (1.7) andLmat,Imi,p denotes one of the upper triangular terms in the i th rowandpth columnof Lmat, Im taken directly fromLvec, Im, shown in Eq. (1.7). Once Sxx, Im is formed using Eq. (1.5), the diagonal entries resulting from the addition of the identity matrix in the definition of Lmat, Im are removed. The complete optimization process can be broken down as follows: 1 Make an initial guess for the complete Lmatrix in the form of Lvec, dv 2 Extract real and imaginary parts from the current guess vector, Lvec, dv 3 Form triangular matrices, Lmat, Re and Lmat, Im, from current Lvec, dv 4 FormSxx, Re and Sxx, Im fromLmat, Re andLmat, Im using Eq. (1.5) 5 Form complete input matrixSxx as sumof Sxx, Re and Sxx, Im 6 Solve forward MIMO control problem, defined with Eq. (1.1), to get response from this optimization iteration 7 Compute one of several scalar objective functions to compare current response to the field data where the objective functions used in this paper are explored in Sect. 1.3. In general, the Sxx formed using the method outlined above will be Hermitian positive definite, but numerically it could become slightly nonpositive definite. Therefore, a small correction can be applied to ensure the resultingSxx is Hermitian positive definite: Sxx =Sxx +Sxx H−SxxD SxxD = Sxx 0 if i =p otherwise (1.8) whereSxxDdenotes the matrix of just the diagonal elements of Sxx, i the row index of a matrix element, andpthe column index of a matrix element. As an additional guarantee of positive definiteness, the eigenvalues of Sxx are adjusted to numerically enforce the constraint, if needed. The process outlined in Steps 1–7 above are repeated until an acceptable solution is obtained based on the parameters used with the optimization algorithms implemented with DAKOTA, an optimization software created at the Sandia National Laboratories, for each frequency step within the frequency range of interest [6].

4 J. J. Wilbanks et al. 1.2.3 Example Problem Definition The example problem used in this paper is the payload positioned at the top of the rocket assembly, modeled as a beam, as shown in blue in Fig. 1.1. Only transverse vibration is considered with two degrees of freedom, transverse displacement and rotation, for each node of the beam model. The payload is modeled as free-free with 25 elements in total and analyzed for the chosen frequency range from 20 to 400 Hz in 1 Hz increments. Elements are defined to be solid aluminum. Modal damping is defined at 2% of critical for the system. Input for the system is applied at two locations, 50.8 cm and 533.4 cm from the bottom of the payload that attaches to the rocket assembly, with the output measured at four locations along the payload, 127 cm, 177.8 cm, 406.4 cm, and 508 cm from the payload bottom. The field environment defined for the input is a random force power spectral density (PSD) with a level of 198.0 N2/Hz and variance of 784.3 N4/Hz2 down the length of the payload. Figure 1.2 shows the field input applied to the node at 50.8 cm along the payload length. The inputs from the field environment are applied to the frequency response functions (FRFs) of the system to get the response of the field environment, SyyField. The optimization will then determine a set of inputs to best make the payload respond as it did to the field environment forces. 1.3 Exploration of Objective Functions with Sampling Latin hypercube sampling (LHS) is completed within the DAKOTA framework for the example problem to observe the sensitivity of the objective functions (OFs) to the elements of Lvec, dv with a near random sampling [6]. There is not a well-accepted OF for the optimization problem presented in this paper. Therefore, three OFs are explored in this paper to determine the best function to use in order to acquire an output APSD sum that closely follows the field value. The OFs used are output CPSD dB error, output CPSD mean squared error, and output APSD sum error. The output CPSD dB error is calculated as: Fig. 1.1 Example beam model representing rocket with payload attached for MIMO input development with iterative optimization (Note: Nodes shown are only for example to show modeling methodology)

1 Exploring Iterative Optimization Methods to Develop a MIMO Control Input 5 Fig. 1.2 Random force PSD applied at Input Node 1 (50.8 cm) along the payload OFdB = ErrordB,UT ∗W 1 (1.9) ErrordB(m,n) = 10log10 SyyOptimized,Re (i,p) SyyField,Re (i,p) + 10log10 SyyOptimized,Im (i,p) SyyField,Im (i,p) (1.10) where SyyOptimized,Re denotes the real component of the optimized output CPSD, SyyOptimized,Im the imaginary component of the optimized output CPSD, SyyField,Re the real component of the field output CPSD, andSyyField,Imthe imaginary component of the field output CPSD. In Eqs. (1.9) and (1.10), ErrordB denotes a matrix containing the dB error for each element of SyyOptimized, ErrordB, UT is the upper triangular terms of ErrordB to prevent cross-terms from being counted twice, and Wis a weighting matrix to apply the cross-term weighting designated for the optimization process. i and p signify the row and column index of an element of the matrices in Eq. (1.10), respectively. The output CPSD mean squared error (MSE) objective function is defined as: OFMSE = sum SyyOptimized −SyyField 2 numel SyyOptimized (1.11) and the output APSD sum error objective function is OFSum = sum GyyOptimized −sum GyyField (1.12) where GyyOptimized denotes the optimized output APSD and GyyField the field output APSD. Figure 1.3 provides the objective function surfaces for the objective functions presented in Eqs. (1.9), (1.10), (1.11), and (1.12) at the 400 Hz frequency line and a cross-term weight of 0.25 used for the dB error calculations. Using dB error creates a surface with a larger range of values and maximum gradients, by a factor of at least 300%, at the frequency line shown. Qualitatively, the surface is more complex with a smaller number of points close to the minimum objective function, improving the resulting APSD sum calculation as well as the noise observed in the output CPSD from the determined input, which is discussed further in Sect. 1.5. For the two input locations considered, Lvec, dv contains a three-term vector

6 J. J. Wilbanks et al. Fig. 1.3 Objective function surfaces created with LHS for objective functions based on (a) output CPSD dB error, (b) output CPSD mean squared error, and (c) output APSD sum error representing Lvec, Re and a single value representing Lvec, Im. The surfaces shown in this section vary the first two terms of Lvec, dv. The dB error objective function maintains similar shapes as the frequency is varied through the required range. Fig. 1.4 compares the dB error objective function surface created when sampling over two of the four components of Lvec, dv. The range of values for the objective function varies by less than 10%, and the maximum gradients remain on the same order of magnitude on both frequency lines, 150 Hz in Fig. 1.4a and 400 Hz in Fig. 1.4b. The impact of the cross-term weight on the surface created by sampling the dB error objective function with LHS is shown in Fig. 1.5. As the cross-term weight is decreased, the range of objective function values decreases by 95%, while the maximum gradients observed as the first and second components of Lvec, dv vary and remain within 25% of the values observed with a unity cross-term weight. 1.4 Applying Iterative Optimization to MIMO Control Problem 1.4.1 Impact of Optimization Algorithm on Output CPSD DAKOTA contains various optimization algorithms that can be leveraged to solve the iterative optimization problem for a MIMO control input [6]. The DAKOTA *.inp file is used to define the solver and related parameters and to call a MATLAB

1 Exploring Iterative Optimization Methods to Develop a MIMO Control Input 7 Fig. 1.4 Output CPSD dB error objective function surfaces with a fixed cross-term weight of 0.25 at frequency points (a) 150 Hz and (b) 400Hz Fig. 1.5 Output CPSD dB error objective function surfaces with fixed cross-term weights of (a) 0.00, (b) 0.25, (c) 0.50, (d) 0.75, and (e) 1.00 wrapper that is used to complete the calculations outlined in Sect. 1.2. A simple visual basic file is used to iteratively call DAKOTA to complete the optimization process at each frequency line throughout the frequency range of interest. Solvers implemented in the example problem introduced are: • SOGA: Single-objective genetic algorithm. • COLINY EA: Evolutionary algorithm contained in the Sandia Colin Optimization Library (SCOLIB) collection of nongradient-based optimizers that support the Common Optimization Library Interface (COLIN). • COLINY PS: Derivative-free pattern search included as part of SCOLIB collection of nongradient-based optimizers. • CONMIN FRCG: Gradient-based optimization approach. where [6] provides further details on development and implementation of these and other solvers. Table 1.1 provides a comparison at a single frequency line at 400 Hz to observe the performance of the algorithms in the high-frequency region

8 J. J. Wilbanks et al. Table 1.1 Comparison of objective function reduction obtained with each of the solution algorithms Algorithm Frequency [Hz] Cross-term weight OF reduction [%] SOGA 400 0.25 92% COLINYEA 400 0.25 97% COLINYPS 400 0.25 75% CONMIN FRCG 400 0.25 75% Fig. 1.6 Impact of optimization algorithm on solution obtained for MIMO control input with dB error objective function and a fixed cross-term weight where large deviations and noise are observed. Figure 1.6 shows the impact of implementing each of these solvers on the payload example and the resulting APSD sum. The objective function used in all the cases shown in Fig. 1.6 and Table 1.1 is dB error, shown in Eqs. (1.9) and (1.10), with a fixed cross-term weight of 0.25. In the optimization runs presented in the following sections, the values of the real components of Lvec, dv are varied from 1.0e-05 to 5, and the imaginary components are varied from−100 to 100. The initial guess for Lvec, dv is defined so that all real components have a value of 0.05 and the imaginary component is 0. With the complex surface of the dB error objective function, the two evolutionary algorithms, SOGA and COLINY EA, outperform the gradient-based CONMIN FRCG and pattern search method, COLINY PS, with reductions of the objective function from its initial value being at least 20% more these evolutionary algorithms. The APSD sum obtained using the evolutionary algorithms more closely follows that of the field or APSD sum obtained with standard least squares (LS), which can be observed in Fig. 1.6. Standard LS is the direct pseudoinverse method of obtaining the input CPSD. Using the COLINY EA, the solution technique improved objective function reduction in Table 1.1 and the APSD sum tracking performance presented in Fig. 1.6 by at least 5% when comparing it to the other evolutionary algorithm, SOGA. When comparing the results of COLINY EA to CONMIN FRCG, the objective function reduction is improved by more than 20%, and the average percent difference for the APSD sum calculation with the field is reduced by over 10%. Figure 1.7 provides the objective function value obtained with the various solvers as a function of frequency. The COLINY EA solver maintains the best performance throughout the complete frequency range observed. COLINY EA results in a reduction of the mean objective function value of 63% and 29% compared to SOGA and CONMIN FRCG, respectively. Due to its increased performance, COLINY EA was chosen as the default solver to use in the development of a MIMO control input through iterative optimization. In the comparisons presented in Figs. 1.6 and 1.7, the solver options were chosen in order to result in the same order of magnitude of computation time, around 1 minute of computation time for one frequency line. In the evolutionary algorithms, COLINY EA and SOGA, the same population size and maximum function evaluation parameter were used, 100 and 2000, respectively. A 10 Hz frequency step was used in all cases.

1 Exploring Iterative Optimization Methods to Develop a MIMO Control Input 9 Fig. 1.7 Impact of optimization algorithm on objective function value as a function of frequency Fig. 1.8 Impact of cross-term weight on solution obtained for MIMO control input with dB error objective function 1.4.2 Impact of Iterations and Other Optimization Parameters Another parameter that can be varied with the dB error objective function, provided in Eqs. (1.9) and (1.10), is the cross-term weight applied with matrixW. Figure 1.8 shows the impact of the cross-term weight definition on tracking of the field APSD sum. The cross-term weight can either be fixed at a constant value throughout frequency or used as an additional design variable in the optimization process. Maximum and minimum percent errors stay similar as the cross-term weight is varied, but the mean percent error decreases as the cross-term weight increases, a total reduction of 18% when going from a weight of 0 to 1. The best case in reducing the error in the optimized APSD sum compared to the field is to use a variable weight that can be optimized at each frequency step. The objective function values for each cross-term weight as a function of frequency can be observed in Fig. 1.9. Values of the objective function are directly dependent on the weighting value used; therefore, the lower the cross-term weight applied, the lower the maximum and mean objective function values. As the cross-term weight is increased from 0 to 1, both the maximum and mean objective function values throughout the frequency range increase to over 350% of their original value.

10 J. J. Wilbanks et al. Fig. 1.9 Impact of cross-term weight on dB error objective function value as a function of frequency Fig. 1.10 APSD sum obtained with objective functions based on dB error with variable cross-term weights and mean squared error using a fine frequency step Using the cross-term weight as a design variable alongside the elements of Lvec, dv allows for the optimization process to choose the cross-term weight that allows for the lowest objective function value. Using a variable cross-term weight decreases the mean objective function value by 37% compared to a fixed cross-term weight of 0. A fine frequency step results in smoother resulting output APSD sums, but can suffer from some slight noise based on the optimization parameters used as well as the range provided for DAKOTA to search in for each of the elements of Lvec, dv. Figure 1.10 compares the resulting APSD sum using the dB error objective function with variable cross-term weight (dashed light blue line) and the mean squared error objective function (solid green line) to the field and APSD sum acquired with the standard least squares (LS) method with a frequency step of 1 Hz. The dB error and MSE objective functions behave similar in terms of the mean percent difference between the resulting APSD sum and the field value at 13.9% and 12.7%, respectively. These values are slightly higher than the average difference of 4.29% observed when generating the MIMO control input with the standard LS method. The results of the iterative optimization, in terms of noise and error, could be improved through modification of the initial guesses at each frequency step based on previous solutions or refinement of the various optimization parameters.

1 Exploring Iterative Optimization Methods to Develop a MIMO Control Input 11 Fig. 1.11 Cross-term weight as a function of frequency chosen through optimization with DAKOTA for objective function based on dB error with variable cross-term weights Figure 1.11 provides the resulting cross-term weight history when allowing DAKOTA and the COLINY EA solver to choose the weight at each frequency step as a design variable. In the analysis, the cross-term weight could vary from 1.00e05 to 1, with an initial value of 0.25. The mean cross-term weight chosen in Fig. 1.11 is 0.0078 with the maximum value chosen equal to 0.4047. Above a frequency of 300 Hz, the cross-term weights tend to be larger with the maximum crossweight occurring at 337 Hz. 1.5 Comparing Iterative Optimization to Other Methods for MIMO Control Input Development Figure 1.12 compares the output APSD sum obtained with iterative optimization with the dB error objective function and a variable cross-term weight to several more traditional methods of generating the MIMO control input. These alternative methods include standard LS, weighted LS, and the buzz method. The buzz method is a direct pseudoinverse method where the target response CPSD cross-terms are replaced with the coherence and phase derived from the FRFs of the lab system. In terms of the mean percent difference, the best traditional method is a weighted LS method with a mean percent difference of 2.6%, which is 38.5% less than the value observed with the standard LS method. Using the dB error objective function, the mean percent difference is 13.9%. Additionally, the dB error objective function tends to follow the buzz method more closely than the other methods, especially at lower frequencies, with the mean percent difference between the optimization process and buzz method equal to 14.1%. Figures 1.13 and 1.14 compare the traditional methods to the other objective functions presented in Sect. 1.3, mean squared error and APSD sum, respectively. Using mean squared error as an objective function results in an output APSD sum that closely follows the solution provided with standard and weighted LS. The mean percent difference between the field and the output APSD sum obtained with the mean squared error objective function is 12.7%. Comparing the result obtained with optimization and the MSE objective function to the methods based on LS, the mean percent difference is 10.1% and 11.01% for the standard and weighted LS methods, respectively. Observing Fig. 1.14, an objective function based on APSD sum error has inherently more noise compared to the other two objective functions using the same optimization parameters and design space. Additionally, the mean percent difference between the optimization methods with the output APSD sum to the field value is increased to 63.1%, which is over 45% more than the other two objective functions.

12 J. J. Wilbanks et al. Fig. 1.12 Comparing APSD sum obtained with objective functions based on dB error with variable weight to traditional methods of obtaining MIMO control input Fig. 1.13 Comparing APSD sum obtained with objective functions based on mean squared error to traditional methods of obtaining MIMO control input Based on Figs. 1.12, 1.13, and 1.14, iterative optimization with either the dB error objective function with variable weight or MSE results in performance that is comparable to traditional, direct pseudoinverse methods. As stated previously, the error observed in the iterative optimization cases could be improved by increasing the design space of the optimization problem, improving initial guesses through previous frequency step solutions, or altering the optimization parameters. Additionally, constraints can be applied to control the input magnitude, shaker voltage, or other parameter during the optimization process, which is introduced in the following section.

1 Exploring Iterative Optimization Methods to Develop a MIMO Control Input 13 Fig. 1.14 Comparing APSD sum obtained with objective functions based on absolute APSD sum error to traditional methods of obtaining MIMO control input 1.6 Accounting for Experimental Limitations in Optimization Application 1.6.1 Directly Constraining MIMO Control Input Constraints can be applied within the optimization process using inequality constraints. One of the most straightforward constraints to apply is the magnitude of the input CPSD, where magnitude is defined with the 2-norm. This could be done to limit the input based on the maximum force of the shaker. Figure 1.15 provides the magnitude of the input CPSD defined through optimization as a function of frequency with varying constraints. Restricting the input CPSD magnitude to 1187.2 N2/Hz decreases the peak and mean values of the input CPSD magnitude by 65% and 24%, respectively. Further limiting the input CPSD magnitude to 494.7 N2/Hz increases the reduction of the peak and mean values to 75.7% and 33.7%, respectively. These reductions come at the cost of increasing the objective function values obtained through optimization by 11%, which can be observed in Fig. 1.16. The impact on the mean percent difference between the output APSD sum and the field is less than 10% while maintaining the input CPSD magnitude below the specified values. 1.6.2 Accounting for Shaker Limitations with Electromechanical Model The lumped parameter electromechanical model shown in Fig. 1.17 combines the mechanical system in Fig. 1.17a with the circuit in Fig. 1.17b and is represented using the following differential equation: [MShaker] ⎧ ⎪⎨ ⎪⎩ ¨x1 ¨x2 ¨x3 ¨I ⎫ ⎪⎬ ⎪⎭ +[CShaker] ⎧ ⎪⎨ ⎪⎩ ˙x1 ˙x2 ˙x3 ˙I ⎫ ⎪⎬ ⎪⎭ +[KShaker] ⎧ ⎪⎨ ⎪⎩ x1 x2 x3 I ⎫ ⎪⎬ ⎪⎭ = ⎧ ⎪⎨ ⎪⎩ F1 F2 F3 e ⎫ ⎪⎬ ⎪⎭ (1.13)

14 J. J. Wilbanks et al. Fig. 1.15 Constrained input CPSD magnitude with various limits during optimization with objective function based on dB error with a fixed cross-term weight of 0.75 Fig. 1.16 Impact of constraining the input CPSD magnitude during optimization on objective function based on dB error with a fixed cross-term weight of 0.75 where the mass (MShaker), damping (CShaker), and stiffness (KShaker) matrices are defined as: [MShaker] = ⎡ ⎢⎢ ⎣ m1 0 0 0 0 m2 0 0 0 0 m3 0 0 0 0 0 ⎤ ⎥⎥ ⎦ (1.14) [CShaker] = ⎡ ⎢⎢ ⎣ (c12 +c13) −c12 −c13 0 −c12 c12 0 0 −c13 0 c13 0 fEMF −fEMF 0 L ⎤ ⎥⎥ ⎦ (1.15)

1 Exploring Iterative Optimization Methods to Develop a MIMO Control Input 15 Fig. 1.17 (a) Mechanical components and (b) electrical circuit of lumped parameter electromechanical shaker model presented and calibrated in [7] Table 1.2 Shaker parameters used in input force and voltage constraints Parameter Value Units m1 0.44 kg m2 15 kg m3 2.6 kg k12 1.10E+04 N/m k13 6.84E+06 N/m c12 9.6 N/(m/s) c13 0.1 N/(m/s) R 4 Ohm L 5.00E-04 +6.00E-04*j Henry fForce 36 N/A [KShaker] = ⎡ ⎢⎢ ⎣ (k12 +k13) −k12 −k13 −fForce −k12 k12 0 fForce −k13 0 k13 0 0 0 0 R ⎤ ⎥⎥ ⎦ (1.16) In Eq. (1.13), ˙( )and ¨( ) denote the first and second time derivatives, respectively. The connection of the electric subsystem to the mechanical subsystem is the coil force, FCoil, which is defined as the product of the force factor (fForce) and the current in the circuit provided in Fig. 1.17b. This connection also creates a voltage drop in the form of eEMF, which is defined as fEMF (˙x1 − ˙x2), where fEMF denotes a constant accounting for the velocity dependence of eEMF. The magnitude of fEMF is defined to be equal to fForce in the model tuning conducted in [7] with units of Vs/m. k12 and k13 are the stiffness of the springs connecting Mass 1 (m1) toMass 2 (m2) and Mass 3 (m3), respectively. c12 and c13 represent the damping between m1 and m2 and m1 and m3, respectively. L and Rare the impedance and resistance in the circuit in Fig. 1.17b, respectively. x1, x2, and x3 in Eq. (13) denote the displacements of Mass 1, Mass 2, and Mass 3, respectively. I ande in Eq. (1.13) are the current and voltage input in the circuit shown in Fig. 1.17b, respectively. Table 1.2 provides the parameter definitions used in Eqs. (1.13), (1.14), (1.15), and (1.16) [7, 8]. F1, F2, F3 are defined to be zero since there are no external forces in this case [7]. The complete transfer function matrix resulting from the system outlined in Eqs. (1.13), (1.14), (1.15), and (1.16) is: [HShaker] = −ω 2 [MShaker] +jω[CShaker] +[KShaker] −1 (1.17)

16 J. J. Wilbanks et al. Fig. 1.18 Constrained shaker voltage with various limits during optimization with objective function based on dB error with a fixed cross-term weight of 0.75 Fig. 1.19 Impact of constraining the shaker voltage during optimization on objective function based on dB error with a fixed cross-term weight of 0.75 Components of this matrix can be used to define the transfer functions needed to relate the shaker voltage and force to the input required from the shakers. The transfer function relating the voltage in the circuit shown in Fig. 1.17b to the force applied to the test article can be expressed as [7]: HFStinger,e =k13 Hx3,e −Hx1,e (1.18) which can be used along with components of HShaker to approximate the shaker voltage required to impart the attached location of the test article with the correct force. Using the transfer functions relating the stinger force of the shaker to the voltage, constraints can be defined to limit the shaker voltage in the optimization process. Figure 1.18 plots the voltage spectral density for both shakers under varying constraints with the corresponding objective function values as a function of frequency provided in Fig. 1.19. Inequality constraints limiting the voltage spectral density below 30 V2/Hzand7V2/Hz are successfully applied in Fig. 1.18. Restricting the voltage increases the mean objective function value by 12% and 17% compared to the unconstrained case for the low and

1 Exploring Iterative Optimization Methods to Develop a MIMO Control Input 17 high constraints, respectively. The high constraint (30 V2/Hz) increases the mean percent difference in the output APSD sum from the field value by 24.1%, and the low constraint (7 V2/Hz) reduces the mean percent difference by 26.2%. Constraints on the shaker voltage can have a larger impact on the ability of the optimization process to generate effective inputs for matching field measurements compared to limits applied to the input CSPD magnitude. However, implementing these constraints in the optimization process allows for solutions to be obtained that mitigate the increase in error of the output APSDs, and in some instances can improve the results if the original solution overestimated the response. In each case, the ability of applying various shaker constraints while maintaining an effective MIMO control input provides additional flexibility to the test engineer without undue burden in the design process. 1.7 Conclusions and Future Work Iterative optimization is an effective alternative to traditional methods in obtaining the input CPSD to best match response of a lab test article to a set of field response data. Using an objective function based on dB error or mean squared error results in mean percent errors in the APSD sum calculated with the developed input that are comparable to several traditional methods, roughly within 10% of the error seen in the traditional methods. The results obtained through the iterative optimization process are dependent on the parameters chosen for the optimization algorithm as well as the initial guess for the solution. Objective function minimization and tracking performance of the solution with the field output APSDs can be improved by increasing the initial population size alongside the allowable function evaluations as well as informing the initial guess for each frequency step based on previous solutions. Additionally, using iterative optimization to acquire the input CSPD for a field environment allows for constraints to be applied alongside the process to maintain the input CPSD magnitude or shaker voltages below a predefined level allowing for a test engineer to account for limitations in test setups and test equipment, like shakers and amplifiers. These limitations can be accounted for while maintaining efficacy by mitigating the impact to the error metric or tracking performance of the output APSDs. The additional flexibility provided by these iterative optimization techniques comes at a cost. The computation time required to determine a solution using an iterative optimization technique is much greater than the time required for direct pseudoinverse solutions. Future work can focus on improving the quality of the initial guesses at each frequency step based on previous solutions, exploring methods to decrease dependency of the method on the forward MIMO control problem, and implementing the method on complex systems with many input points. References 1. Daborn, P.M., Ind, P.R., Ewins, D.J.: Enhanced ground-based vibration testing for aerodynamic environments. Mech. Syst. Signal Process. 49(1–2), 165–180 (2014) 2. Whiteman, W., Berman, M.: Inadequacies in uniaxial stress screen vibration testing. J. IEST. 44(4), 20–23 (2001) 3. Roberts, C., Ewins, D.J.: Multi-axis vibration testing of an aerodynamically excited structure. J. Vib. Control. 24(2), 427–437 (2018) 4. Ross, M., Jacobs, L.D., Tipton, G., Nelson, G., Cross, K., Hunter, N., Harvie, J.: 6-DOF shaker test input derivation from field test. In: Shock & Vibration, Aircraft/Aerospace, Energy Harvesting, Acoustics & Optics, Volume 9, pp. 11–22. Springer, Cham (2017) 5. Daborn, P.M.: Scaling up of the impedance-matched multi-axis test (IMMAT) technique. In: Shock & Vibration, Aircraft/Aerospace, Energy Harvesting, Acoustics & Optics, Volume 9, pp. 1–10. Springer, Cham (2017) 6. Adams, B., et al.: Dakota 6.10 Reference Manual Documentation. Retrieved April 14, 2019 from https://dakota.sandia.gov/content/latestreference-manual (2019) 7. Schultz, R.: Calibration of shaker electro-mechanical models. In: Special Topics in Structural Dynamics & Experimental Techniques, Volume 5, pp. 133–144. Springer, Cham (2019) 8. Tiwari, N., Puri, A., Saraswat, A.: Lumped parameter modeling and methodology for extraction of model parameters for an electrodynamic shaker. J. Low Freq. Noise Vib. Active Control. 36(2), 99–115 (2017)

Chapter 2 All-Electric X-Plane, X-57 Mod II Ground Vibration Test Natalie Spivey, Samson Truong, and Roger Truax Abstract As part of the National Aeronautics and Space Administration New Aviation Horizons initiative to demonstrate and validate future high-impact concepts and technologies, the X-57 Maxwell airplane – the first all-electric X-plane – was conceived to advance research in the area of electric propulsion to show the feasibility of minimizing fuel use, reducing emissions, and lowering noise during flight. Through several configuration modifications to the X-57 airplane, validation of electrical-powered flight with increasing efficiency between each modification when compared to the baseline original airplane is anticipated. In the case of the X-57 Modification II airplane, a ground vibration test was needed to identify the airplane structural modes and use them to update and validate the finite element model. To determine the airworthiness of the airplane, the updated finite element model will be utilized to investigate both classical and whirl flutters. The X-57 Modification II ground vibration test was performed by the National Aeronautics and Space Administration Armstrong Flight Research Center Flight Loads Laboratory. This paper will highlight the testing performed to acquire the modal data as well as the results. Keywords Electric · Ground vibration test · Modal · X-57 airplane Nomenclature A/C Aircraft accel Accelerometer AFRC Armstrong Flight Research Center ail Aileron anti-sym Anti-symmetric AW1T Anti-symmetric wing first torsion AW1B Anti-symmetric wing first bending AW2B Anti-symmetric wing second bending BCM Battery control module CAD Computer-aided design DOF Degrees of freedom F1LB Fuselage first lateral bending F1VB Fuselage first vertical bending F/A Fore/aft FEM Finite element model FLL Flight Loads Laboratory GVT Ground vibration test HL High-lift lat Lateral long Longitudinal MLG Main landing gear N. Spivey ( ) · S. Truong · R. Truax Armstrong Flight Research Center, National Aeronautics and Space Administration, Edwards, CA, USA e-mail: natalie.d.spivey@nasa.gov © The Society for Experimental Mechanics, Inc. 2022 C. Walber et al. (eds.), Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing, Volume 7, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-75988-9_2 19

20 N. Spivey et al. Mod Modification NASA National Aeronautics and Space Administration OOP Out of phase rotn Rotation SW1B Symmetric wing first bending SW1T Symmetric wing first torsion SW2B Symmetric wing second bending SWFA Symmetric wing fore/aft sym Symmetric TE Trailing edge vert Vertical 2.1 Introduction The X-57 Maxwell will incorporate a distributed electrical propulsion system and is derived from a modified baseline Italian general aviation fleet airplane called the Tecnam P2006T (Costruzioni Aeronautiche Tecnam, Capua, Italy) [1, 2]. The X-57 airplane is being redesigned over several configuration modifications in order to evaluate the performance to the original baseline airplane, as shown in Fig. 2.1. Modification (Mod) I is the baseline Tecnam P2006T airplane with two mid-wing gas-powered engines; Mod II is Mod I redesigned with two electric motors powered by traction batteries (see Fig. 2.2). Mod IV will incorporate a new wing design with the two electric cruise motors moved to the wingtips and 12 high-lift (HL) electric motors, mounted and distributed along the wing leading edge; the previous Mod III configuration and objectives have been merged with the Mod IV configuration. The goal of the X-57 airplane project is to demonstrate a 500 percent increase in high-speed cruise efficiency, zero in-flight carbon emissions, and a 15-dB reduction in noise levels. The National Aeronautics and Space Administration (NASA) Armstrong Flight Research Center (AFRC) Flight Loads Laboratory (FLL) conducted a ground vibration test (GVT) on the X-57 Mod II airplane, as shown in Fig. 2.3, in order to gather modal data of a near flight-ready configuration to correlate and validate the Mod II airplane beam finite element Fig. 2.1 X-57 airplane project configuration modifications

2 All-Electric X-Plane, X-57 Mod II Ground Vibration Test 21 Fig. 2.2 X-57 Mod II airplane, artist illustration Fig. 2.3 X-57 Mod II airplane ground vibration test setup on soft supports to simulate a free-free boundary condition model (FEM) to the airplane GVT modes [3]. After correlating the FEM to match the GVT data, the updated FEM will be used in both the final classical and whirl flutter analyses that will be used for evaluating aeroelastic airworthiness for the X-57 Mod II airplane flights. Two separate airplane boundary conditions were conducted. One setup was with the X-57 Mod II airplane on a soft-support system to simulate the free-flight boundary conditions. The second boundary condition tested was with the airplane on-tires to characterize the on-ground modes for future airplane ground motor testing safety clearance and potential follow-on of ground GVTs when the airplane will be taxiing on the runway. Multiple test configurations were conducted during the X-57 Mod II GVT, each configuration with a different target airplane structural mode of interest; these test configurations dictated which locking mechanisms were to be used in the cockpit and on the control surfaces. Locking

22 N. Spivey et al. Table 2.1 Primary objectives for the X-57 Mod II airplane ground vibration test devices are commonly used to constrain moving components during GVTs. A total of 191 test runs were performed for this Mod IIGVT. 2.2 Test Objectives The purpose of the X-57 Mod II airplane GVT was to gather the modal frequencies and mode shapes of the airplane in a flight-ready configuration. The GVT will be used to correlate and validate the Mod II airplane FEM to that of the airplane GVT elastic modes. The resultant updated FEM will be utilized for flutter analysis predictions, which are critical for flight flutter envelope expansion [4]. The primary test objective and the main structural modes of interest during the GVT involved modes that are part of the predicted airplane flutter mechanism (see Table 2.1). For classical flutter, the flutter mechanism was a concern because of the coupling of the horizontal stabilator rotation and the fuselage first vertical bending (F1VB) modes. For whirl flutter, the flutter mechanism and modes of concern were the motor assembly lateral bending and vertical bending [5, 6]. In addition to the airplane elastic flutter mechanism modes, the airplane rigid-body modes were also part of the primary test objective (see Table 2.1). The newly designed X-57 soft-support system needed to provide adequate frequency separation of the airplane elastic and rigid modes to avoid potential coupling. Frequency separation would characterize the effectiveness of the soft supports to simulate the airplane in a free-flight environment, which greatly simplifies the post-test FEM correlation to the GVT results. The remainder of the airplane elastic modes were either secondary or tertiary objectives and were not expected to contribute to the flutter mechanism. The secondary objective modes involved the fuselage lateral bending and torsion modes, wing modes, and landing gear modes. The tertiary modes were higher-order wing modes and control surface modes. Identification of the secondary and tertiary modal frequencies and mode shapes would help assess the response of the airplane compared to the FEM predictions. Two boundary conditions were evaluated to meet the objectives for the X-57 Mod II GVT. The first boundary condition setup had the airplane suspended on a soft-support system to simulate a free flight; the second boundary condition setup had the airplane on-tires resting on the hangar floor. The free-free setup utilized the newly designed X-57 soft-support system to simulate the airplane in free flight with essentially no boundary conditions interfacing or touching the airplane. This setup allowed an apples-to-apples comparison of the GVT data to the FEM free-free modal analysis, making the FEM updating and correlation process easier post-test. The updated FEM will be utilized in the flutter analysis to clear the Mod II airplane for flight. The flutter flight-testing team will also use the GVT results in the control room to assist in assessing the airplane modes during flight flutter envelope expansion clearance. The on-tires GVT configuration was intended to gather airplane modal data that would be utilized for several different reasons. The first reason was to obtain the baseline characterization of the motor assembly modes for safety clearance of near-term ground motor testing before flight. The second reason was to baseline the airplane on-tires modes in order to avoid the need to repeat a more complicated airplane GVT on soft supports (if needed) – in case potential hardware changes before flight-testing raised flutter concerns that might not be alleviated or well predicted by FEM adjustments alone. The third reason was to gather modal data that would be comparable to low-speed airplane taxi testing as a buildup to higher-speed taxi testing and flight-testing. 2.3 Test Description The following sections describe the X-57 Mod II test article, the finite element model, and the details of the GVT. The X-57 Mod II airplane GVT was conducted from November to December 2019 at NASA AFRC.

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