Model Validation and Uncertainty Quantification, Volume 3

River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Model Validation and Uncertainty Quantification, Volume 3 Zhu Mao Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020 River Publishers

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 Model Validation and Uncertainty Quantification, Volume 3 Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020 Zhu Mao Editor

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Preface Model Validation and Uncertainty Quantification represents one of eight volumes of technical papers presented at the 38th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, and held in Houston, Texas, February 10–13, 2020. The full proceedings also include volumes on Nonlinear Structures and Systems; Dynamics of Civil Structures; Dynamic Substructures; Special Topics in Structural Dynamics & Experimental Techniques; Rotating Machinery, Optical Methods & Scanning LDV Methods; Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing; and Topics in Modal Analysis &Testing. Each collection presents early findings from experimental and computational investigations on an important area within Structural Dynamics. Model Validation and Uncertainty Quantification (MVUQ) is one of these areas. Modeling and simulation are routinely implemented to predict the behavior of complex dynamical systems. These tools powerfully unite theoretical foundations, numerical models, and experimental data which include associated uncertainties and errors. The field of MVUQ research entails the development of methods and metrics to test model prediction accuracy and robustness while considering all relevant sources of uncertainties and errors through systematic comparisons against experimental observations. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Lowell, MA, USA ZhuMao v

Contents 1 Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method ......................... 1 Daniel C. Kammer, Paul Blelloch, and Joel Sills 2 Bayesian Uncertainty Quantification in the Development of a New Vibration Absorber Technology......... 19 Nicolas Brötz and Peter F. Pelz 3 Comparison of Complexity Measures for Structural Health Monitoring ........................................ 27 Hannah Donajkowski, Salma Leyasi, Gregory Mellos, Chuck R. Farrar, Alex Scheinker, Jin-Song Pei, and Nicholas A. J. Lieven 4 Selection of an Adequate Model of a Piezo-Elastic Support for Structural Control in a Beam Truss Structure.................................................................................................................. 41 Jonathan Lenz, Maximilian Schäffner, Roland Platz, and Tobias Melz 5 Impact Load Identification for the DROPBEAR Setup Using a Finite Input Covariance (FIC) Estimator ................................................................................................................. 51 Peter Lander, Yang Wang, and Jacob Dodson 6 Real-Time Digital Twin Updating Strategy Based on Structural Health Monitoring Systems ................. 55 Yi-Chen Zhu, David Wagg, Elizabeth Cross, and Robert Barthorpe 7 On the Fusion of Test and Analysis.................................................................................... 65 Ibrahim A. Sever 8 Design of an Actuation Controller for Physical Substructures in Stochastic Real-Time Hybrid Simulations ............................................................................................................... 69 Nikolaos Tsokanas and B. Stojadinovic 9 Output-Only Nonlinear Finite Element Model Updating Using Autoregressive Process........................ 83 Juan Castiglione, Rodrigo Astroza, Saeed Eftekhar Azam, and Daniel Linzell 10 Axle Box Accelerometer Signal Identification and Modelling...................................................... 87 Cyprien A. Hoelzl, Luis David Avendano Valencia, Vasilis K. Dertimanis, Eleni N. Chatzi, and Marcel Zurkirchen 11 Kalman-Based Virtual Sensing for Improvement of Service Response Replication in Environmental Tests.................................................................................................. 93 Silvia Vettori, Emilio Di Lorenzo, Roberta Cumbo, Umberto Musella, Tommaso Tamarozzi, Bart Peeters, and Eleni Chatzi 12 Virtual Sensing of Wheel Position in Ground-Steering Systems for Aircraft Using Digital Twins............. 107 Mattia Dal Borgo, Stephen J. Elliott, Maryam Ghandchi Tehrani, and Ian M. Stothers 13 Assessing Model Form Uncertainty in Fracture Models Using Digital Image Correlation ..................... 119 Robin Callens, Matthias Faes, and David Moens vii

viii Contents 14 Identification of Lack of Knowledge Using Analytical Redundancy Applied to Structural Dynamic Systems ........................................................................................................ 131 Jakob Hartig, Florian Hoppe, Daniel Martin, Georg Staudter, Tugrul Öztürk, Reiner Anderl, Peter Groche, Peter F. Pelz, and Matthias Weigold 15 A Structural Fatigue Monitoring Concept for Wind Turbines by Means of Digital Twins ..................... 139 János Zierath, Sven-Erik Rosenow, Johannes Luthe, Andreas Schulze, Christiane Saalbach, Manuela Sander, and Christoph Woernle 16 Damage Identification of Structures Through Machine Learning Techniques with Updated Finite Element Models and Experimental Validations...................................................................... 143 Panagiotis Seventekidis, Dimitrios Giagopoulos, Alexandros Arailopoulos, and Olga Markogiannaki 17 Modal Analyses and Meta-Models for Fatigue Assessment of Automotive Steel Wheels........................ 155 S. Venturini, E. Bonisoli, C. Rosso, D. Rovarino, and M. Velardocchia 18 Towards the Development of a Digital Twin for Structural Dynamics Applications............................. 165 Paul Gardner, Mattia Dal Borgo, Valentina Ruffini, Yichen Zhu, and Aidan Hughes 19 An Improved Optimal Sensor Placement Strategy for Kalman-Based Multiple-Input Estimation............ 181 Lorenzo Mazzanti, Roberta Cumbo, Wim Desmet, Frank Naets, and Tommaso Tamarozzi 20 Towards Population-Based Structural Health Monitoring, Part IV: Heterogeneous Populations, Transfer and Mapping .................................................................................................. 187 Paul Gardner, Lawerence A. Bull, Julian Gosliga, Nikolaos Dervilis, and Keith Worden 21 Feasibility Study of Using Low-Cost Measurement Devices for System Identification Using Bayesian Approaches ............................................................................................................... 201 Alejandro Duarte and Albert R. Ortiz 22 Kernelised Bayesian Transfer Learning for Population-Based Structural Health Monitoring................. 209 Paul Gardner, Lawrence A. Bull, Nikolaos Dervilis, and Keith Worden 23 Predicting System Response at Unmeasured Locations Using a Laboratory Pre-Test .......................... 217 Randy Mayes, Luke Ankers, and Phil Daborn 24 Robust Estimation of Truncation-Induced Numerical Uncertainty ............................................... 223 François Hemez 25 Fatigue Crack Growth Diagnosis and Prognosis for Damage-Adaptive Operation of Mechanical Systems ................................................................................................................... 233 Pranav M. Karve, Yulin Guo, Berkcan Kapusuzoglu, Sankaran Mahadevan, and Mulugeta A. Haile 26 An Evolutionary Approach to Learning Neural Networks for Structural Health Monitoring ................. 237 Tharuka Devendra, Nikolaos Dervilis, Keith Worden, George Tsialiamanis, Elizabeth J. Cross, and Timothy J. Rogers 27 Bayesian Solutions to State-Space Structural Identification........................................................ 247 Timothy J. Rogers, Keith Worden, and Elizabeth J. Cross 28 Analyzing Propagation of Model Form Uncertainty for Different Suspension Strut Models................... 255 Robert Feldmann, Maximilian Schäffner, Christopher M. Gehb, Roland Platz, and Tobias Melz 29 Determining Interdependencies and Causation of Vibration in Aero Engines Using Multiscale Cross-Correlation Analysis............................................................................................. 265 Manu Krishnan, Ibrahim A. Sever, and Pablo A. Tarazaga 30 Dynamic Data Driven Modeling of Aero Engine Response ......................................................... 273 Manu Krishnan, Serkan Gugercin, Ibrahim Sever, and Pablo Tarazaga 31 Nonlinear Model Updating Using Recursive and Batch Bayesian Methods ...................................... 279 Mingming Song, Rodrigo Astroza, Hamed Ebrahimian, Babak Moaveni, and Costas Papadimitriou

Contents ix 32 Towards Population-Based Structural Health Monitoring, Part I: Homogeneous Populations and Forms... 287 Lawerence A. Bull, Paul A. Gardner, Julian Gosliga, Nikolaos Dervilis, Evangelos Papatheou, Andrew E. Maguire, Carles Campos, Timothy J. Rogers, Elizabeth J. Cross, and Keith Worden 33 A Detailed Assessment of Model Form Uncertainty in a Load-Carrying Truss Structure...................... 303 Robert Feldmann, Christopher M. Gehb, Maximilian Schäffner, Alexander Matei, Jonathan Lenz, Sebastian Kersting, and Moritz Weber 34 Recursive Nonlinear Identification of a Negative Stiffness Device for Seismic Protection of Structures with Geometric and Material Nonlinearities ......................................................................... 315 Kalil Erazo and Satish Nagarajaiah 35 Adequate Mathematical Beam-Column Model for Active Buckling Control in a Tetrahedron Truss Structure.................................................................................................................. 323 Maximilian Schaeffner, Roland Platz, and Tobias Melz 36 Site Characterization Through Hierarchical Bayesian Model Updating Using Dispersion and H/V Data.... 333 Mehdi M. Akhlaghi, Mingming Song, Marshall Pontrelli, Babak Moaveni, and Laurie G. Baise 37 BAYESIAN Inference Based Parameter Calibration of a Mechanical Load-Bearing Structure’s Mathematical Model .................................................................................................... 337 Christopher M. Gehb, Roland Platz and Tobias Melz 38 Uncertainty Propagation in a Hybrid Data-Driven and Physics-Based Submodeling Method for Refined Response Estimation .......................................................................................... 349 Bhavana Valeti and Shamim N. Pakzad 39 Adaptive Process and Measurement Noise Identification for Recursive Bayesian Estimation.................. 361 Konstantinos E. Tatsis, Vasilis K. Dertimanis, and Eleni N. Chatzi 40 Effective Learning of Post-Seismic Building Damage with Sparse Observations ................................ 365 Mohamadreza Sheibani and Ge Ou 41 Efficient Bayesian Inference of Miter Gates Using High-Fidelity Models......................................... 375 Manuel A. Vega, Mukesh K. Ramancha, Joel P. Conte, and Michael D. Todd 42 Two-Stage Hierarchical Bayesian Framework for Finite Element Model Updating............................. 383 Xinyu Jia, Omid Sedehi, Costas Papadimitriou, Lambros Katafygiotis, and Babak Moaveni 43 Bayesian Nonlinear Finite Element Model Updating of a Full-Scale Bridge-Column Using Sequential Monte Carlo.............................................................................................................. 389 Mukesh K. Ramancha, Rodrigo Astroza, Joel P. Conte, Jose I. Restrepo, and Michael D. Todd 44 Optimal Input Locations for Stiffness Parameter Identification................................................... 399 Debasish Jana, Dhiraj Ghosh, Suparno Mukhopadhyay, and Samit Ray-Chaudhuri 45 Modal Identification and Damage Detection of Railway Bridges Using Time-Varying Modes Identified from Train Induced Vibrations ............................................................................ 405 Ashish Pal, Astha Gaur, and Suparno Mukhopadhyay 46 Test-Analysis Modal Correlation of Rocket Engine Structures in Liquid Hydrogen – Phase II................ 413 Andrew M. Brown and Jennifer L. DeLessio 47 An Output-Only Bayesian Identification Approach for Nonlinear Structural and Mechanical Systems...... 431 Satish Nagarajaiah and Kalil Erazo

Chapter 1 Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method Daniel C. Kammer, Paul Blelloch, and Joel Sills Abstract Time-domain coupled loads analysis (CLA) is used to determine the response of a launch vehicle and payload system to transient forces, such as liftoff, engine ignitions and shutdowns, jettison events, and atmospheric flight loads, such as buffet. CLA, using Hurty/Craig-Bampton (HCB) component models, is the accepted method for the establishment of design-level loads for launch systems. However, uncertainty in the component models flows into uncertainty in predicted system results. Uncertainty in the structural responses during launch is a significant concern because small variations in launch vehicle and payload mode shapes and their interactions can result in significant variations in system loads. Uncertainty quantification (UQ) is used to determine statistical bounds on prediction accuracy based on model uncertainty. In this paper uncertainty is treated at the HCB component-model level. In an effort to account for model uncertainties and statistically bound their effect on CLA predictions, this work combines CLA with UQ in a process termed variational coupled loads analysis (VCLA). The modeling of uncertainty using a parametric approach, in which input parameters are represented by random variables, is common, but its major drawback is the resulting uncertainty is limited to the form of the nominal model. Uncertainty in model form is one of the biggest contributors to uncertainty in complex built-up structures. Modelform uncertainty can be represented using a nonparametric approach based on random matrix theory (RMT). In this work, UQ is performed using the hybrid parametric variation (HPV) method, which combines parametric with nonparametric uncertainty at the HCB component model level. The HPV method requires the selection of dispersion values for the HCB fixed-interface (FI) eigenvalues, and the HCB mass and stiffness matrices. The dispersions are based upon component testanalysis modal correlation results. During VCLA, random component models are assembled into an ensemble of random systems using a Monte Carlo (MC) approach. CLA is applied to each of the ensemble members to produce an ensemble of system-level responses for statistical analysis. The proposed methodology is demonstrated through its application to a buffet loads analysis of NASA’s Space Launch System (SLS) during the transonic regime 50 s after liftoff. Core stage (CS) section shears and moments are recovered, and statistics are computed. Keywords Uncertainty quantification · Hurty/Craig-Bampton · Random matrix · Model form · Coupled loads analysis Acronyms CLA coupled loads analysis CS core stage DCGM diagonal cross-generalized mass metric DOF degrees of freedom FEM finite element model FI fixed-interface HCB Hurty/Craig-Bampton HPV hybrid parametric variation D. C. Kammer ( ) · P. Blelloch ATA Engineering, Inc., San Diego, CA, USA e-mail: daniel.kammer@wisc.edu; paul.blelloch@ata-e.com J. Sills NASA Johnson Space Center, Houston, TX, USA e-mail: joel.w.sills@nasa.gov © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_1 1

2 D. C. Kammer et al. ICPS interim cryogenic propulsion stage ISPE integrated spacecraft payload element LSRB left solid rocket booster LVSA launch vehicle stage adapter MC Monte Carlo ME maximum entropy MEM modal effective mass MUF model uncertainty factor MPCV Multi-Purpose Crew Vehicle MSA MPCV stage adapter NPV nonparametric variation OTM output transformation matrix PDF probability distribution function RINU replaceable inertial navigation unit RMS root mean square RMT random matrix theory RSRB right solid rocket booster SLS Space Launch System UQ uncertainty quantification VCLA variational coupled loads analysis 1.1 Introduction Uncertainty in structural responses during launch is a significant concern in the development of spacecraft and launch vehicles. Small variations in launch vehicle and payload mode shapes and their interactions can result in significant variations in system loads. Time-domain coupled loads analysis (CLA) is used to determine the response of a launch vehicle and payload system to transient forces, such as liftoff, engine ignitions and shutdowns, jettison events, and atmospheric flight loads. Atmospheric flight loads include static-aeroelastic, turbulence/gust, and buffet loads. CLA, using Hurty/CraigBampton (HCB) [1] component models, is the accepted method for the establishment of design-level loads for launch systems, but it is an expensive and time-consuming process that is performed a limited number of times using a sequence of increasingly more accurate system models. Ideally, a modal test of the integrated system is performed, and the resulting test-correlated system model is used in the final CLA. However, in many cases, an integrated-system test is not practical, so the accuracy of the system model is dependent on the accuracy of component models that are correlated and updated using component test results. Unfortunately, component test configurations are rarely, if ever, the same as component flight configurations, resulting in uncertainty in the component and, thus, system models. In addition, as a program advances, inevitable increasing cost and time constraints begin to limit the number of component tests that are actually performed before launch, which, again, increases the amount of uncertainty in the models and CLA results. In an effort to account for model uncertainties and their effect on CLA predictions, CLA sensitivity analyses could be performed, but due to the high cost of multiple CLAs, this has been impractical and typically not done. Another standard practice is to apply model uncertainty factors (MUF) to the predicted CLA load results. This is simple, but the MUF approach is not based on test results of the specific system it is being applied to, and such a blanket approach may be needlessly conservative in some instances, but unconservative in others. Uncertainty quantification (UQ) is used to determine statistical bounds on prediction accuracy based on model uncertainty. In the case of NASA’s Space Launch System (SLS), uncertainty is modeled at the HCB component-model level because it has the most direct link to modal test results. This work combines CLA with UQ in a process termed variational coupled loads analysis (VCLA) [2]. In the past, VCLA has not been feasible due to computational expense. But due to advances in computational speed, software, and mathematical approaches, VCLA is beginning to be practical, provided that care is taken in how the variations are modeled. The modeling of uncertainty using a parametric approach, in which finite element model (FEM) input parameters are represented by random variables, is common. The major drawback is that the resulting uncertainty is limited to the form of the nominal model. Uncertainty in model form is one of the biggest contributors to uncertainty in complex built-up structures. One way to treat model-form uncertainty is to use a nonparametric approach based on random matrix theory (RMT). In this work, UQ is performed using the hybrid parametric variation (HPV) [3] method, which combines parametric with nonparametric uncertainty at the HCB component-model level. The HPV method requires the selection of dispersion values for the HCB fixed-interface eigenvalues, and the HCB mass and stiffness matrices. The

1 Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method 3 dispersions are based upon component test-analysis modal correlation results. During VCLA, random component models are assembled into an ensemble of random systems using a Monte Carlo (MC) approach. CLA is applied to each of the systems to produce an ensemble of system-level responses for statistical analysis. The proposed methodology is demonstrated through its application to a buffet loads analysis of the SLS during the transonic regime at 50 s after liftoff. Core stage (CS) section shears and moments are recovered, and statistics, such as P99/90 non-exceedance values, are computed. 1.2 Theory The SLS consists of components that are assembled into the launch vehicle. In order to predict system performance, FEMs of the components are developed, reduced to HCB representations, and assembled to represent different phases of flight. There is always uncertainty in every model, which flows into uncertainty in predicted system results. For the SLS, it is natural to treat the model uncertainty at the HCB component-model level. The HCB component-model displacement vector is given by uHCB = x T t q T T, where xt is the vector of physical displacements at the component interface, and q is the vector of generalized coordinates associated with the component fixed-interface (FI) modes. Given the assumption that the FI modes are mass normalized, the corresponding HCB mass and stiffness matrices have the form MHCB = MS Mtq MT tq I KHCB = KS 0 0 λ (1.1) in which MS and KS are the component physical mass and stiffness matrices statically reduced to the interface, Mtq is the mass coupling between the interface and the fixed-interface modes, I is an identity matrix, and λis a diagonal matrix of the FI mode eigenvalues. Details of the HCB component-model derivation can be found in reference [1]. In this work, uncertainty in the component HCB representations is quantified using the HPV approach, which combines parametric with nonparametric uncertainty. Purely parametric uncertainty approaches are the most common in the structural dynamics community. Component parameters that are inputs to the FEM, such as Young’s modulus, mass density, geometric properties, etc., are modeled as random variables. Parametric uncertainty can be propagated into the system response using a method such as stochastic finite element analysis [4]. The advantage of the parametric approach is that each random set of model parameters represents a specific random FEM. However, there are disadvantages associated with the parametric method: it can be very time consuming, there are an infinite number of ways to parameterize the model, and the selected parameter probability distributions are generally not available. The most significant drawback is that the uncertainty represented is limited to the form of the nominal FEM. It is known that most errors in a FEM stem from modeling assumptions or model-form errors, not parametric errors. Therefore, in practice, the parameter changes are merely surrogates for the actual model errors. In the case of HPV, the HCB components are parameterized in terms of the FI eigenvalues, not the inputs to the original FEM. While there is not a simple direct connection between the random FI eigenvalues and a random component FEM, there is a direct connection to the corresponding random HCB component. Uncertainty in model form is likely the largest contributor to uncertainty in complex built-up structures, as it cannot be directly represented by model parameters and thus cannot be included in a parametric approach. Familiar examples include unmodeled nonlinearities, errors in component joint models, etc. Instead, model-form uncertainty can be modeled using random matrix theory (RMT), where a probability distribution is developed for the matrix ensemble of interest. RMT was introduced and developed in mathematical statistics by Wishart [5], and more recently, Soize [6, 7] developed a nonparametric variation (NPV) approach to represent model-form uncertainty in structural dynamics applications. Soize’s approach was extended by Adhikari [8, 9] using Wishart distributions to model random structural mass, damping, and stiffness matrices. The nonparametric matrix-based approach to representing structural uncertainty has been used extensively in aeronautics and aerospace engineering applications [10, 11, 12]. Soize [7] employed the maximum entropy (ME) principle to derive the positive and positive-semidefinite ensembles SE+ and SE+0 that follow a matrix variate gamma distribution and are capable of representing random structural matrices. This means that the matrices in the ensembles are real and symmetric and possess the appropriate sign definiteness to represent structural mass, stiffness, or damping matrices. As the dimension of the random matrixnincreases, the matrix variate gamma distribution converges to a matrix variate Wishart distribution. In structural dynamics applications, the matrix dimensions are usually sufficient to give a negligible difference between the two distributions. In letting ensemble member random matrix Gbe any of the random mass, stiffness, or damping matrices, it is assumed in this work that Gfollows a matrix variate Wishart distribution, G~Wn(p, ). In general, a Wishart distribution with parameters p and can be thought of as the sum of the outer product of p independent random vectors Xi all having a multivariate normal distribution with zero mean and

4 D. C. Kammer et al. covariance matrix . Parameter pis sometimes called the shape parameter. The random matrix Gcan be written as G= p i=1 XiXT i Xi ∼Nn (0, ) (1.2) where the expected value is given by E(G) =G=p (1.3) The dispersion or normalized standard deviation of the random matrixGis defined by the relation δ 2 G = E G−G 2 F E G 2 F (1.4) inwhich ∗ 2 F is the Frobenius norm squared, or trace(∗ T ∗). It can be shown that Eq. (1.4) reduces to the expression δ 2 G = 1 p ⎡ ⎣ 1+ tr G 2 tr G T G ⎤ ⎦= 1 p [1+γG] (1.5) where γG = tr G 2 tr G T G can be thought of as a measure of the magnitude of the matrix. The uncertainty in the random matrix Gis dictated by the shape parameter p, the number of inner products in Eq. (1.2). The larger the value of p, the smaller the dispersion δG. There may be instances when it is desirable to have the same amount of uncertainty in two or more substructures. Suppose G1 and G2 represent structural matrices, such as stiffness, from two different system components. In order to have equivalent uncertainty in the two matrices, the shape parameter p must be the same for both ensembles. However, Eq. (1.5) shows that even if p1 =p2 =p, the dispersion values are not the same in general, δ 2 G1 = δ 2 G2 , unless γ1 =γ2. A more useful definition of matrix dispersion is the normalized dispersion δGn = δG1 √1+γ1 = δG2 √1+γ2 = 1 √p (1.6) which is independent of the matrix magnitude γG. It is important to realize, however, that just because two components have the same normalized dispersion, this does not mean that they will have the same uncertainty relative to a specific metric used to quantify component uncertainty, such as modal parameter uncertainty. Where the normalized dispersion is small, the modal statistics of the two components tend to be close, but as the normalized dispersion increases, the modal statistics for the two components become more disparate due to nonlinearity. Therefore, while Eq. (1.6) can be used to initially set a component matrix dispersion, it should ultimately be checked and adjusted based on the specific metric being used to quantify component uncertainty. Adhikari [8] referred to the random matrix method developed by Soize [6, 7] as Method 1. The Wishart parameters are selected as p and =Go/p where Go is the nominal value of G. The mean of the distribution is given by Eq. (1.3) as G=p =p(Go/p) =Go. Therefore, Method 1 preserves the nominal matrix as the mean of the ensemble. In general, the nominal matrix can be decomposed in the form Go =LL T (1.7) In the case of a positive definite matrix, this would just be the Cholesky decomposition. Let (n×p)matrixXbe givenby X= x1 x2 · · · xp (1.8) inwhich xi is an (n×1) column vector containing standard random normal variables such that xi~Nn(0, In). Note that p≥n must be satisfied in order for Gto be full rank. An ensemble member G~Wn(p, Go/p) can then be generated for MC analysis using the expression G= 1 p LXXTLT (1.9)

1 Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method 5 It has been found that ensembles of random component mass matrices are best represented using Method 1. Adhikari [8] noted that Method 1 does not maintain the inverse of the mean matrix as the mean of the inverse; that is E G−1 =[E(G)]−1 =G−1 (1.10) In some cases, the two can be vastly different, which is clearly not physically realistic. Instead, he proposed Method 3, in which the Wishart parameters are selected as p and =Go/θ where θ = 1 δ 2 G [1+γG] −(n+1) (1.11) An ensemble member G~Wn(p, Go/θ) can then be generated using the relation G= 1 θ LXXTLT (1.12) In this case, the inverse of the mean matrix is preserved as the mean of the ensemble inverses, where the mean matrix is now given by G=p =p(Go/θ) = p θ Go (1.13) In Method 3, the dispersion defined in Eq. (1.4) is now calculated with respect to the mean given in Eq. (1.13), while Eqs. (1.5) and (1.6) still hold. It has been determined that ensembles of random component stiffness matrices are best represented using Method 3. Therefore, the nonparametric portion of the HPV method is based on a Method 1 randomization of the component mass matrix and a Method 3 randomization of the component stiffness matrix. In this approach, the random component mass and stiffness matrices are assumed to be independent. Note that the Wishart matrix uncertainty model results in uncertainty both in mode shapes and in frequencies. However, MC simulation and analysis has shown that component mode shapes tend to be sensitive to the nonparametric matrix randomization provided by Methods 1 and 3, but the corresponding modal frequencies tend to be relatively insensitive. Therefore, a parametric component of uncertainty was added to the HPV approach in which the eigenvalues of the FI modes in the component HCB representation are also assumed to be random variables. ME is also used to derive the probability distribution function (PDF) for the HCB FI eigenvalues. The ME principle produces a PDF based solely on the available information, and in this case, there are two pieces of information available: First, the ith random eigenvalue must be strictly positive. Second, the expected value of the random eigenvalue is given by the nominal value. The application of ME yields a gamma distribution, λri~G(ki, θi), where the shape parameter ki and the scale parameter θi are given by ki = δ−2 i and θi =λiδ 2 i , inwhich δi is the corresponding coefficient of variation, or dispersion [13]. The FI eigenvalues are then independent random parameters within the HCB component stiffness matrix. The validity and impact of this assumption can be determined by considering only the FI component of the HCB substructure representation. The corresponding equation of motion in physical coordinates is given by Moo ¨uo +Koouo =Fo (1.14) while the equation of motion in FI modal coordinates is I ¨q +λq =φTFo (1.15) where the modal stiffness λis a diagonal matrix of FI modal eigenvalues from Eq. (1.1). The modal stiffness is given by λ =φTKooφ (1.16) The physical stiffness can be recovered by pre-multiplying Eq. (1.16) byMooφand post-multiplying byφ TMoo producing Mooφφ TKooφφ TMoo =Mooφλφ TMoo (1.17) If all FI modes are retained, thenφφ TMoo =I. In considering only the ith mode, Eq. (1.17) becomes Kooi =P T i KooPi =Mooφiλiφ T i Moo (1.18)

6 D. C. Kammer et al. where Pi = φiφ T i Moo is an oblique projector [14] onto the column space spanned by the ith FI mode and Kooi is the contribution of the ith FI mode to the FI physical stiffness matrix. The physical stiffness can then be expressed as Koo = no i=1 Kooi = no i=1 λi Mooφiφ T i Moo (1.19) Therefore, the randomized physical stiffness matrix that corresponds to the randomized FI eigenvalues λri is given by Koor = no i=1 αiKooi = no i=1 αiλi Mooφiφ T i Moo = n i=1 λri Mooφiφ T i Moo (1.20) whereαi is a random variable selected from a gamma distribution with a mean value of 1.0 andno is the number of FI modes. Equation (1.20) indicates that the parameterization of the FI stiffness in modal space using the eigenvalues λi corresponds to the parameterization of the FI stiffness in physical space using the matrices Kooi. The randomized stiffness parameters Koori =αiKooi are independent of one another, meaning that changing the ith stiffness parameter has no impact on any of the other stiffness parameters. In summary, the FI eigenvalue parameterization used in the HPV method is equivalent to a parameterization within physical FI space where the stiffness components for the individual modes are scaled byno independent random parameters. The resulting sum produces a random physical stiffness matrix that preserves the nominal FI modes, but the generalized stiffness or eigenvalues vary. While the HPV FI eigenvalue parameterization preserves the model form of the HCB representation stiffness matrix, it does not preserve the model form of the physical FI stiffness matrix. Therefore, the variation of the HCB FI eigenvalues is a purely parametric variation with respect to the HCB representation, but it is not a purely parametric variation with respect to the physical stiffness partition Koo, meaning that it also results in modelform uncertainty. Also, if the corresponding random HCB stiffness matrix in its entirety is back transformed into physical coordinates, then not only is the Koo partition randomized as discussed above, but the Kaa and Kao partitions, corresponding to the interface DOF, are also randomized such that the sign-definiteness and the rigid body modes are preserved. The fact that this parameterization affects the FI eigenvalues but not the FI mode shapes allows it to be paired with the NPV method to produce a maximal impact on the HCB frequencies and a minimal impact on the HCB mode shapes. Therefore, the HPV approach provides the capability to almost independently adjust the uncertainty in the component frequencies and mode shapes, by adjusting the dispersion of the FI eigenvalues vs. the dispersions of the HCB mass and stiffness matrices. It is important to emphasize that this parameterization is not equivalent to the usual model input parameterization of the FEM. In contrast, if all of the random FI eigenvalues are varied in unison, such they are no longer independent, but perfectly correlated, then the random FI partition of the stiffness matrix in physical space is given by Koor = n i=1 αiKooi =α n i=1 Kooi =αKoo (1.21) which is just a random scaling of the nominal stiffness. This randomization of the FI eigenvalues then preserves the model form of the FI partition of the physical stiffness matrix as well as the model form of the entire HCB stiffness matrix. As in the previous case, if the corresponding random HCB stiffness matrix is transformed back into physical coordinates, then the FI partition of the physical stiffness matrix is a scaled version of the nominal FI partition with the random scale factor α. The model form of the other random physical stiffness partitions involving the interface is not necessarily maintained. During each iteration in an MC analysis, a random draw of HCB FI eigenvalues is selected to generate a random HCB component stiffness matrix as described. The mean of this ensemble would just be the nominal HCB stiffness matrix. However, for each iteration, the parametrically randomized HCB stiffness is treated as the nominal matrix, and Method 3 is applied to provide model-form uncertainty on top of the FI eigenvalue parametric uncertainty. This is analogous to the approach proposed by Capiez-Lernout [10] for separating parametric and nonparametric uncertainty. Component frequency uncertainty can be based on component test-analysis frequency correlation, and the nonparametric mass and stiffness dispersion can be based on the corresponding orthogonality and cross-orthogonality results. The component slosh modes are likely to have much less uncertainty associated with them. In addition, there may be times when the component mass must be randomized while the rigid body mass is preserved. A special methodology has been developed within the HPV method to preserve the component slosh modes and rigid body mass when desired; the same methodology can be applied to any subset of the component modes. When the nominal matrix is positive semidefinite, such as in the case of an SLS flight component that has rigid body modes and a positive semidefinite stiffness matrix, special steps must be taken to decompose the nominal HCB stiffness matrix for subsequent randomization using Method 3 and Eq. (1.12). As in the case of component mass randomization, the slosh mode stiffness must be preserved. In addition, the rigid body stiffness must be preserved as the null matrix. Details of the methodology’s implementation within the HPV framework are given in reference [3].

1 Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method 7 1.3 Selection of Component Eigenvalue and Stiffness Matrix Dispersion Values The HPV approach for modeling component uncertainty requires the selection of dispersion values for the HCB component FI eigenvalues, mass matrix, and stiffness matrix. Ideally, these dispersion values are selected for each component based on component modal test results. This is because test-analysis modal correlation metrics are used to determine the dispersions. Test-analysis frequency error is used to identify the HCB FI eigenvalue uncertainties, but one of the biggest challenges in the propagation of component test-analysis frequency error into uncertainty in the HCB flight configuration FI modes is that the component test configuration and the component flight configuration boundary conditions and/or hardware are almost never the same. Because of this, it is difficult to match test configuration modes with flight configuration FI modes. The boundary condition mismatch can be alleviated using a mixed-boundary approach. In general, the HCB flight configuration FI modes will be over-constrained when compared to the test configuration modes. Therefore, the HCB stiffness matrix in Eq. (1.1) can be written as KHCB = KS 0 0 λ = ⎡ ⎣ Kcc Kcb 0 Kbc Kbb 0 0 0 λ ⎤ ⎦ (1.22) where the HCB flight configuration set of boundary degrees of freedom (DOF) have been divided into two subsets: the c-set contains all DOF that are free in the component test configuration, and the b-set contains the DOF that are constrained in the component test configuration. When the HCB flight configuration is constrained at the test configuration interface DOF (b-set), it produces the mass and stiffness matrices MC = Mcc Mcq Mqc Mqq KC = Kcc 0 0 λ (1.23) with corresponding eigenvalues λC and mass normalized eigenvectors φC = φ T cc φ T cq T . These eigenvalues and eigenvectors are consistent with the boundary conditions of the test configuration modes used in the component test-analysis correlation. Error or uncertainty in the analytical test configuration eigenvalues can be much more easily mapped onto uncertainty λC in the eigenvalues of the system in Eq. (1.23). The HCB representation of the component using λC and φC as FI modal properties has the stiffness matrix and corresponding displacement vector given by KB = KSb 0 0 λC uB = x T b qT C T (1.24) where KSb is KS statically reduced to the b-set, xb is the physical displacement of the b-set, andqC are the modal coordinates of the FI modes with the c-set free. The transformation between displacement vector uB and the original HCB displacement vector uHCB is given by uHCB = ⎧ ⎨ ⎩ xc xb q ⎫ ⎬ ⎭ = ⎡ ⎣ ψ φcc I 0 0 φcq ⎤ ⎦ xb qC =TuB (1.25) The relation between KB and KHCB is then KB =T TKHCBT (1.26) The test configuration HCB FI eigenvalues λC can be randomized (λCr) based upon the component test-analysis correlation results, and the uncertainty can be propagated into the random flight configuration HCB component stiffness (KHCBr) using the expression KHCBr =T−TKBrT−1 =T−T KSb 0 0 λCr T−1 (1.27)

8 D. C. Kammer et al. Details of the procedure used to assign the uncertainty to the eigenvalues λC are discussed in the following section. Reference [3] details a procedure for identifying the HCB mass matrix dispersion based upon the component test mode selforthogonality matrix. Alternatively, the mass matrix dispersion could be based on engineering judgment, past experience, and historical results. In this work, it is assumed that the mass representation of the components is accurate, so the mass matrix is not dispersed. Once the eigenvalue dispersions have been identified, test-analysis cross-orthogonality is used to identify the dispersion of the component stiffness matrix. In this case, the root mean square (RMS) diagonal value of the component test-analysis cross-orthogonality matrix, referred to as the diagonal cross-generalized mass (DCGM), is used as the metric. For this, a series of MC analyses is performed in which the HCB stiffness matrix dispersion value is swept over a range, and the most probable value of DCGM is computed for each MC analysis. The goal is to select the stiffness dispersion value that gives the most probable DCGM value that is equal to the test result. Details of the identification procedure are discussed in the following section and in reference [3]. 1.4 Nominal ICPS/LVSA Based on ISPE Configuration 3 Dispersion values for the combined nominal interim cryogenic propulsion stage (ICPS)/launch vehicle stage adapter (LVSA) HCB components are based on the integrated spacecraft payload element (ISPE) configuration 3 modal test-analysis correlation results. The FEM representation of ISPE configuration 3 is shown in Fig. 1.1. There are 11 FEM target modes matched to 11 of 19 test modes. Only these target modes are considered in this analysis, because the other eight modes are dominated by the MPCV stage adapter (MSA)/Multi-Purpose Crew Vehicle (MPCV) simulator, which is not part of the ICPS/LVSA component. The test-analysis frequency correlation results are listed in Table 1.1. The nominal model accurately predicts the first bending test mode frequencies. This is consistent with the static test results, which showed good agreement between the nominal model and the test results for overall bending and axial stiffness. Only one second-order bending test mode was identified in the test data, and no axial modes were identified. The nominal model does a poor job of predicting the second-order bending test mode frequency, and the LVSA shell test mode frequencies are even less accurately predicted. Note that the frequency error is calculated relative to the nominal FEM frequency rather than the test; this is done because the uncertainty analysis is performed relative to the FEM HCB representation. As mentioned, the SLS HCB flight components and the test components do not match in most cases. In the case of the ICPS/LVSA HCB flight component, there is no MSA nor MPCV simulator. In addition, the ISPE configuration 3 is not tested in the HCB flight configuration. At the time of this writing, the mixed-boundary approach discussed above has not yet been applied to the ICPS/LVSA HCB flight component. Instead, the test configuration modal frequency uncertainties are mapped directly onto the ICPS/LVSA HCB flight component FI modes. It is assumed that the component test correlation results can be used as an indicator of what level of uncertainty is expected in the corresponding HCB component model. Fig. 1.1 ISPE configuration 3 FEM representation

1 Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method 9 The first element of uncertainty to be specified is the dispersion of the 33 non-slosh FI modal frequencies for the HCB flight component. These modes have the base of the LVSA and the top of the ICPS at the MSA interface constrained, whereas during the test, the base of the LVSA is attached to the core simulator, and the top of the ICPS is attached to the MSA/MPCV simulator. Test-analysis frequency error is mapped onto the FI modes using modal effective mass (MEM). The nominal FEM ISPE configuration 3 MEM is dominated by the fundamental bending and axial modes, and to a lesser extent, the second-order bending modes. The LVSA shell modes have little or no MEM. Based on MEM, the HCB FI modes are placed in three different bins of frequency uncertainty. Bin 1, associated with the FEM configuration 3 first bending pair, is assigned a frequency dispersion of 1.39%, corresponding to the RMS error in the prediction of the first bending configuration 3 test mode pair frequencies. The modal frequencies are assumed to follow the gamma distribution described previously. Bin 2 has a frequency dispersion of 9.01%, corresponding to the test-analysis frequency error of the second-order bending configuration 3 mode. FI modes that have little or no MEM, analogous to the LVSA shell test modes, are assigned to bin 3 with a frequency dispersion of 10.91%, corresponding to the RMS frequency error in the configuration 3 LVSA shell modes. Once the FI eigenvalue uncertainty is applied, the dispersion of the stiffness matrix is determined by computation of the DCGM metric based on cross-orthogonality, which is the RMS value of the diagonal after modes are matched and resorted accordingly. Based on the nominal ISPE configuration 3 cross-orthogonality matrix for the 11 target modes, the dispersion metric has the value DCGM=90.44. The goal is to select a stiffness dispersion value for the MC analysis such that the test-based metric value is the most probable. The MC analysis is based on the first 37 nominal HCB elastic non-slosh modes and 3000 ensemble members. The selected FI eigenvalue uncertainties are applied, and then a series of MC analyses are performed with stiffness dispersion swept over a range of values. A stiffness dispersion of 7%, which corresponds to a normalized dispersion of 0.85%, produces the most probable DCGM value that agrees with the test value. 1.5 Updated ICPS/LVSA Based on ISPE Configuration 3 A posttest correlation was performed to update the ISPE configuration 3 FEM; thus, dispersion values for the updated ICPS/LVSA HCB component are based on the updated ISPE configuration 3 FEM. Like the nominal model analysis, there are 11 FEM target modes matched to 11 of 19 test modes. The test-analysis frequency correlation results are also listed in Table 1.1. Note that the frequency error for the LVSA shell modes and the second-order bending modes have been dramatically reduced. There are now 243 HCB FI non-slosh modes in the updated ICPS/LVSA fifty-seconds-of-ascent HCB component. The FI MEM is calculated and the same three-bin uncertainty assignment strategy is applied. Bin 1 is assigned a frequency dispersion of 2.04%, corresponding to the RMS error in the prediction of the first bending test mode pair. Bin 2 is assigned a frequency dispersion of 4.83%, corresponding to the test-analysis frequency error of the second-order bending test mode. As in the nominal model analysis, the remaining FI modes have little or no MEM, analogous to the LVSA shell test modes. Therefore, these FI modes are assigned to bin 3 with a frequency dispersion of 1.97%, corresponding to the RMS frequency error in the configuration 3 LVSA shell modes. Based on the updated ISPE configuration 3 cross-orthogonality matrix for the 11 target modes, the dispersion metric has the test value DCGM=95.43. The selected FI eigenvalue uncertainties were applied, and then a series of MC analyses were performed with stiffness dispersion swept over a range of values. In this case, the MC analysis was based on the first 35 Table 1.1 Test-analysis frequency error for ISPE configuration 3 nominal model Testmode Pretest mode Error (%) Posttest mode Error (%) Description 1 6 −0.16 6 −0.89 First bending 2 5 −1.94 5 −2.70 First bending 3 9 10.13 7 1.43 LVSA shell ND 5 4 10 9.95 8 1.23 LVSA shell ND 5 5 11 7.21 9 3.96 LVSA shell ND 4 6 12 6.97 10 3.50 LVSA shell ND 4 9 14 12.90 14 −0.23 LVSA shell ND 6 10 13 12.60 13 −0.57 LVSA shell ND 6 14 24 15.29 20 −0.01 LVSA shell ND 7 15 23 14.98 19 −0.33 LVSA shell ND 7 19 22 −8.63 24 −4.72 Second bending

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