Dynamics of Coupled Structures, Volume 4

River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Dynamics of Coupled Structures, Volume 4 Matt Allen Randall L. Mayes Daniel Rixen Proceedings of the 34th IMAC, A Conference and Exposition on Structural Dynamics 2016 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

River Publishers Matt Allen • Randall L. Mayes • Daniel Rixen Editors Dynamics of Coupled Structures, Volume 4 Proceedings of the 34th IMAC, A Conference and Exposition on Structural Dynamics 2016

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Preface Dynamics of Coupled Structures represents one of ten volumes of technical papers presented at the 34th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics and held in Orlando, Florida, January 25–28, 2016. The full proceedings also include volumes on nonlinear dynamics; dynamics of civil structures; model validation and uncertainty quantification; sensors and instrumentation; special topics in structural dynamics; structural health monitoring, damage detection, and mechatronics; rotating machinery, hybrid test methods, vibro-acoustics, and laser vibrometry; and shock and vibration, aircraft/aerospace, energy harvesting, acoustics and optics, and topics in modal analysis and testing. Each collection presents early findings from experimental and computational investigations on an important area within structural dynamics. Coupled structures or substructuring is one of these areas. Substructuring is a general paradigm in engineering dynamics where a complicated system is analyzed by considering the dynamic interactions between subcomponents. In numerical simulations, substructuring allows one to reduce the complexity of parts of the system in order to construct a computationally efficient model of the assembled system. A subcomponent model can also be derived experimentally, allowing one to predict the dynamic behavior of an assembly by combining experimentally and/or analytically derived models. This can be advantageous for subcomponents that are expensive or difficult to model analytically. Substructuring can also be used to couple numerical simulation with real-time testing of components. Such approaches are known as hardware-in-the-loop or hybrid testing. Whether experimental or numerical, all substructuring approaches have a common basis, namely, the equilibrium of the substructures under the action of the applied and interface forces and the compatibility of displacements at the interfaces of the subcomponents. Experimental substructuring requires special care in the way the measurements are obtained and processed in order to assure that measurement inaccuracies and noise do not invalidate the results. In numerical approaches, the fundamental quest is the efficient computation of reduced order models describing the substructure’s dynamic motion. For hardware-in-the-loop applications, difficulties include the fast computation of the numerical components and the proper sensing and actuation of the hardware component. Recent advances in experimental techniques, sensor/actuator technologies, novel numerical methods, and parallel computing have rekindled interest in substructuring in recent years leading to new insights and improved experimental and analytical techniques. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Madison, WI, USA Matt Allen Albuquerque, NM, USA Randall L. Mayes Garching, Bayern, Germany Daniel Rixen v

Contents 1 Verification of Experimental Component Mode Synthesis in the Sierra Analysis Framework................. 1 Brian C. Owens and Randall L. Mayes 2 Multi-DoF Interface Synchronization of Real-Time-Hybrid-Tests Using a Recursive-Least-Squares Adaption Law: A Numerical Evaluation ............................................... 7 Andreas Bartl, Johannes Mayet, Morteza Karamooz Mahdiabadi, and Daniel J. Rixen 3 Controls Based Hybrid Sub-Structuring Approach to Transfer Path Analysis .................................. 15 Joseph A. Franco, Rui M. Botelho, and Richard E. Christenson 4 Force Identification Based on Subspace Identification Algorithms and Homotopy Method.................... 25 Zhenguo Zhang, Xiuchang Huang, Zhiyi Zhang, and Hongxing Hua 5 Response DOF Selection for Mapping Experimental Normal Modes-2016 Update ............................. 33 Robert N. Coppolino 6 Experimental Modal Substructuring with Nonlinear Modal Iwan Models to Capture Nonlinear Subcomponent Damping................................................................................................ 47 Matthew S. Allen, Daniel Roettgen, Daniel Kammer, and Randy Mayes 7 A Modal Model to Simulate Typical Structural Dynamic Nonlinearity........................................... 57 Randall L. Mayes, Benjamin R. Pacini, and Daniel R. Roettgen 8 Optimal Replacement of Coupling DoFs in Substructure Decoupling ............................................ 77 Walter D’Ambrogio and Annalisa Fregolent 9 State-Space Substructuring with Transmission Simulator ......................................................... 91 Maren Scheel and Anders T. Johansson 10 Applying the Transmission Simulator Techniques to the Ampair 600 Wind Turbine Testbed ................. 105 Johann Gross, Benjamin Seeger, Simon Peter, and Pascal Reuss 11 Effect of Interface Substitute When Applying Frequency Based Substructuring to the Ampair 600 Wind Turbine Rotor Assembly.................................................................................... 117 Morteza Karamooz Mahdiabadi, Andreas Bartl, and Daniel J. Rixen 12 Improving Floor Vibration Performance Using Interstitial Columns ............................................. 123 Michael J. Wesolowsky, J. Shayne Love, Todd A. Busch, Fernando J. Tallavo, and John C. Swallow 13 Probabilistic Model Updating of Controller Models for Groups of People in a Standing Position............. 131 Albert R. Ortiz and Juan M. Caicedo 14 Fundamental Frequency of Lightweight Cold-Formed Steel Floor Systems ..................................... 137 S. Zhang and L. Xu 15 Fundamental Studies of AVC with Actuator Dynamics ............................................................. 147 E.J. Hudson, P. Reynolds, and D.S. Nyawako vii

viii Contents 16 Mitigating Existing Floor Vibration Issues in a School Renovation ............................................... 155 Linda M. Hanagan 17 Vibration Serviceability Assessment of an In-Service Pedestrian Bridge Under Human-Induced Excitations ........................................................................................... 163 Amir Gheitasi, Salman Usmani, Mohamad Alipour, Osman E. Ozbulut, and Devin K. Harris 18 Numerical and Experimental Studies on Scale Models of Lightweight Building Structures.................................................................................... 173 Ola Flodén, Kent Persson, and Göran Sandberg 19 A Wavelet-Based Approach for Generating Individual Jumping Loads .......................................... 181 Guo Li and Jun Chen 20 A Numerical Round Robin for the Prediction of the Dynamics of Jointed Structures .......................... 195 J. Gross, J. Armand, R.M. Lacayo, P. Reuss, L. Salles, C.W. Schwingshackl, M.R.W. Brake, and R. J. Kuether 21 A Method to Capture Macroslip at Bolted Interfaces ............................................................... 213 Ronald N. Hopkins and Lili A.A. Heitman 22 A Reduced Iwan Model that Includes Pinning for Bolted Joint Mechanics ...................................... 231 M.R.W. Brake 23 Nonlinear Vibration Phenomena in Aero-Engine Measurements.................................................. 241 Ibrahim A. Sever 24 Instantaneous Frequency and Damping from Transient Ring-Down Data ....................................... 253 Robert J. Kuether and Matthew R.W. Brake 25 Explicit Modelling of Microslip Behaviour in Dry Friction Contact .............................................. 265 C.W. Schwingshackl and A. Natoli 26 Modal Testing Through Forced Sine Vibrations of a Timber Footbridge......................................... 273 Giacomo Bernagozzi, Luca Landi, and Pier Paolo Diotallevi 27 Damping Characteristics of a Footbridge: Mysteries and Truths.................................................. 283 Reto Cantieni, Anela Bajric´, and Rune Brincker 28 A Critical Analysis of Simplified Procedures for Footbridges’ Serviceability Assessment ...................... 293 Federica Tubino and Giuseppe Piccardo 29 Human-Induced Vibrations of Footbridges: The Effect of Vertical Human-Structure Interaction............ 299 Katrien Van Nimmen, Geert Lombaert, Guido De Roeck, and Peter Van den Broeck 30 Nonlinear Time-Varying Dynamic Analysis of a Multi-Mesh Spur Gear Train.................................. 309 Siar Deniz Yavuz, Zihni Burcay Saribay, and Ender Cigeroglu 31 Energy Dissipation of a System with Foam to Metal Interfaces.................................................... 323 Laura D. Jacobs, Robert J. Kuether, and John H. Hofer 32 Nonlinear System Identification of Mechanical Interfaces Based on Wave Scattering .......................... 333 Keegan J. Moore, Mehmet Kurt, Melih Eriten, D. Michael McFarland, Lawrence A. Bergman, and Alexander F. Vakakis 33 Studies of a Geometrical Nonlinear Friction Damped System Using NNMs...................................... 341 Martin Jerschl, Dominik Süß, and Kai Willner 34 Scale-Dependent Modeling of Joint Behavior ........................................................................ 349 Kai Willner 35 Robust Occupant Detection Through Step-Induced Floor Vibration by Incorporating Structural Characteristics ........................................................................................................... 357 Mike Lam, Mostafa Mirshekari, Shijia Pan, Pei Zhang, and Hae Young Noh

Contents ix 36 Assessment of Large Error Time-Differences for Localization in a Plate Simulation ........................... 369 Americo G. Woolard, Austin A. Phoenix, and Pablo A. Tarazaga 37 Gender Classification Using Under Floor Vibration Measurements............................................... 377 Dustin Bales, Pablo Tarazaga, Mary Kasarda, and Dhruv Batra 38 Human-Structure Interaction and Implications ..................................................................... 385 Lars Pedersen 39 Study of Human-Structure Dynamic Interactions................................................................... 391 Mehdi Setareh and Shiqi Gan 40 Characterisation of Transient Actions Induced by Spectators on Sport Stadia.................................. 401 A. Quattrone, M. Bocian, V. Racic, J.M.W. Brownjohn, E.J. Hudson, D. Hester, and J. Davies 41 Recent Issues on Stadium Monitoring and Serviceability: A Review.............................................. 411 Ozan Celik, Ngoan Tien Do, Osama Abdeljaber, Mustafa Gul, Onur Avci, and F. Necati Catbas 42 Characterising Randomness in Human Actions on Civil Engineering Structures ............................... 417 S. Živanovic´, M.G. McDonald, and H.V. Dang 43 Optimal Restraint Conditions for the SID-IIs Dummy with Different Objective Functions.................... 425 Yibing Shi, Jianping Wu, and Guy S. Nusholtz 44 A Comparison of Common Model Updating Approaches .......................................................... 439 D. Xu, M. Karamooz Mahdiabadi, A. Bartl, and D.J. Rixen 45 Experimental Coupling and Decoupling of Engineering Structures Using Frequency-Based Substructuring .............................................................................. 447 S. Manzato, C. Napoli, G. Coppotelli, A. Fregolent, W. D’Ambrogio, and B. Peeters 46 New FRF Based Methods for Substructure Decoupling ............................................................ 463 Taner Kalaycıog˘lu and H. Nevzat Özgüven 47 Experimental Determination of Frictional Interface Models....................................................... 473 Matthew S. Bonney, Brett A. Robertson, Marc Mignolet, Fabian Schempp, and Matthew R. Brake 48 Effects of Experimental Methods on the Measurements of a Nonlinear Structure .............................. 491 S. Catalfamo, S.A. Smith, F. Morlock, M.R.W. Brake, P. Reuß, C.W. Schwingshackl, and W.D. Zhu 49 Stress Waves Propagating Through Bolted Joints ................................................................... 501 R.C. Flicek, K.J. Moore, G.M. Castelluccio, M.R.W. Brake, T. Truster, and C.I. Hammetter 50 A Comparison of Reduced Order Modeling Techniques Used in Dynamic Substructuring .................... 511 Daniel Roettgen, Benjamin Seeger, Wei Che Tai, Seunghun Baek, Tilán Dossogne, Matthew Allen, Robert Kuether, Matthew R.W. Brake, and Randall Mayes

Chapter 1 Verification of Experimental Component Mode Synthesis in the Sierra Analysis Framework Brian C. Owens and Randall L. Mayes Abstract Experimental component mode synthesis (CMS) seeks to measure the fundamental modes of vibration of a substructure and develop a structural dynamics model of an as-built structural component through modal testing. Experimental CMS has the potential to circumvent laborious and costly substructure model development and calibration in lieu of a structural dynamics model obtained directly from experimental measurements. Previous efforts of interfacing an experimental CMS model with a production finite element code proved cumbersome. Recently an improved “Craig-Mayes” approach casts an experimental CMS model in the familiar Craig-Bampton form. This form is easily understood by analysts and more readily interfaced with non-trivial, discrete finite element models. The approach/work-flow for interfacing an experimental Craig-Mayes CMS model with the Sierra analysis framework is discussed and the procedure is demonstrated on a verification problem. Keywords Component mode synthesis • Craig-Bampton • Substructure • Sierra • Finite elements Nomenclature CMS Component mode synthesis FE Finite element(s) M Mass matrix TS Transmission simulator 1.1 Introduction The concept of experimental component mode synthesis (CMS) seeks to measure the fundamental modes of vibration of a substructure and develop a structural dynamics model of a component or subsystem which may be inserted into an analytical model of a higher-level system. Strengths of experimental CMS allow for one to circumvent laborious and costly substructure model development and calibration in lieu of a structural dynamics model obtained directly from experimental measurements. Furthermore, experimental CMS allows for a better modelling capability of as-built structural components. This work will interface an experimentally derived “Craig-Bampton” like substructure with a discrete finite element model within Sandia National Laboratories Sierra analysis framework [1, 2]. A previously developed transmission simulation approach is employed to match interface locations with a discrete system level finite element model. Previous efforts will be discussed and strengths of the current approach in streamlining the use of experimentally derived substructures will be highlighted. This approach will be discussed and verification exercises will be presented. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy National Nuclear Security Administration under Contract DE-AC04-94AL85000. B.C. Owens ( ) • R.L. Mayes Sandia National Laboratories, P.O. Box 5800 – MS0346, Albuquerque, NM 87185, USA e-mail: bcowens@sandia.gov; rlmayes@sandia.gov © The Society for Experimental Mechanics, Inc. 2016 M. Allen et al. (eds.), Dynamics of Coupled Structures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-29763-7_1 1

2 B.C. Owens and R.L. Mayes 1.2 Craig-Mayes Experimental Sub-Structuring Method The Craig-Mayes experimental dynamic sub-structuring method improves upon previous experimental sub-structuring methods by representing the substructure system matrices (mass, stiffness, and damping) in a Craig-Bampton [3] like form. This form contains structural matrices with generalized/modal and interface degrees of freedom. This approach uses a transmission simulator to model the interface of the sub-structure to the remainder of the system. The transmission simulator approach requires an accurate discrete finite element model of the transmission simulator/fixture to accurately recover the interface degrees of freedom in an experimental substructure. The Craig-Mayes experimental sub-structuring method and transmission simulator approach are discussed in references [4, 5]. 1.3 Interface of Experimental CMS Model to Sierra Previous efforts of coupling experimental CMS models within the Sierra framework employed multi-point constraints and a non-Craig-Mayes CMS representation. This approach proved overly cumbersome for all but the simplest model configurations, and was prone to numerical conditioning issues. The Craig-Mayes format provides a readily realizable interface to a high fidelity structural dynamics model with an interface similar to CMS or “super element” model derived purely from analytical methods. The experimental CMS model is typically provided by experimentalists as a collection of Craig-Bampton like mass, stiffness, and damping matrices [4]. Note that the damping matrix is not required to define a baseline experimental CMS model, but is readily available from experimental measurements. The Craig-Bampton like matrices are p p in dimension, such that pDmCn. Here, mis the number of modes retained in the CMS reduction, and n is the number of interface degrees of freedom in the CMS model. The required form of these equations is shown in Eq. (1.1). Note that the form of the mass matrix is shown, but identical forms are required for the stiffness and damping matrices. These matrices are symmetrical in nature. OMdefines couplings between the generalized (modal) degrees of freedom in the CMS model (this matrix should be diagonal in nature), Mdefines couplings between the interface degrees of freedom, and eMdefines the couplings between generalized and interface degrees of freedom. MCMS D" OMm m eMm n eM T m n Mn n # (1.1) An r 3 coordinate array is also required that defines the coordinates of the r interface points. In addition to interface point coordinates an n 1 map array is required that specifies the “local” degrees of freedom of the interface degrees of freedom. The order of this array should be consistent with the ordering of the coordinate array. In summary, the following data is required to accompany an experimentally derived CMS model: • Number of modes retained in the CMS reduction (m) • Number of interface degrees of freedom (n) • Interface point coordinate array • CMS mass matrix • CMS stiffness matrix • Interface degree of freedom map array • CMS damping matrix (optional) This information can be provided to the MATLAB based “CMS Toolkit” to create a Sierra super element of the experimental CMS model. The CMS toolkit creates two files. First, an Exodus finite element mesh is created defining the geometry of the super element. This includes the coordinates of the interface nodes for the n-node super element. Next, the formulation of this super element (mass, stiffness, and damping) are characterized in a NetCDF binary file. The super element Exodus file is inserted into a discrete finite element model using GJOIN [6] or a similar mesh joining utility. From here, the super element NetCDF file is referenced in a Sierra input deck and subsequent modal, vibration, or transient analysis accounts for the coupling between the discrete finite element model and the experimentally derived Craig-Mayes substructure. This workflow is depicted in Fig. 1.1.

1 Verification of Experimental Component Mode Synthesis in the Sierra Analysis Framework 3 Fig. 1.1 Workflow for interfacing an experimentally derived CMS model in Sierra analysis 1.4 Demonstration This section presents a demonstration of the aforementioned process for interfacing an experimentally derived Craig-Mayes substructure to a discrete finite element modal for Sierra-SD analysis. First the model/test configuration is described followed by results of the exercise. 1.4.1 Configuration A 2-D simple beam configuration documented in reference [4] was considered for a proof of concept analysis for interfacing an experimentally derived CMS substructure model with Sierra-SD analysis. The configuration is shown in Fig. 1.2. Two beams are connected together over a specified region of overlap. The left beam is to be modeled by finite elements, whereas the dynamics of the right beam are measured experimentally and an experimental CMS model is derived. This is done using a “transmission simulator” shown as “TS Beam” in Fig. 1.2. Details of the transmission simulator are elaborated on in references [4, 5]. The transmission simulator essentially allows one to generate interface degree of freedom responses at discrete locations from those measured from a modal test. This is a very convenient means for interfacing an experimentally derived CMS model to discrete points of a finite element model. Note that there are five nodes in the overlap between the left finite element beam and the right experimental CMS beam. Thus, the transmission simulator approach was used to derive an experimental CMS model with five interface points (coincident with the finite element nodes). Each interface point had 3 degrees of freedom (axial translation, bending translation, and rotation). Therefore, a total of 15 interface degrees of freedom exist in the model. Three modes were retained in the CMS reduction. This resulted in CMS mass, stiffness, and damping matrices that had dimension of 18 18. 1.4.2 Results The Craig-Mayes substructure model of the beam was interfaced to the discrete “FE Beam” model in Sierra-SD described in Sect. 1.2. Results show good agreement between the “truth model” described in reference [4] and the Sierra-SD implementation. Table 1.1 presents a comparison of modal frequencies. The first five modes have 1 % error or less and

4 B.C. Owens and R.L. Mayes FE Beam Final System Beam TS Beam Experimental Beam Fig. 1.2 2D beam configuration [4] Table 1.1 Comparison of Sierra-SD sub-structured modal frequency vs. truth frequency Truth frequency (Hz) [4] Sierra-SD sub-structured frequency (Hz) Error (%) 212.0 210.4 0.7 574.6 568.6 1.0 1121.0 1132.0 1.0 1867.3 1863.9 0.2 2750.2 2767.6 0.6 3341.7 3383.9 1.3 3949.6 4003.2 1.4 5115.9 5105.0 0.2 5965.5 5945.8 0.3 Fig. 1.3 Comparison of “truth” and Sierra-SD sub-structured lower bending mode shapes (solidlineDtruth model, circleDsub-structured model) the 6th–9th modes have at most 1.4 % error. Bending mode shape comparisons of a discrete finite element model of the entire system and those of the discrete “FE beam” coupled with the Craig-Mayes experimental beam are shown in Figs. 1.3 and 1.4. Solid lines represent the finite element results of the complete system while markers represent the mode shape of the discrete left beam coupled with the Craig-Mayes right beam. Overall, good agreement is seen between the “truth” FEM mode shapes and those of the discrete finite element model of the left beam coupled to the Craig-Mayes substructure of the right beam. Some differences are apparent for the 4th–7th bending mode shapes in the vicinity of the interface to the Craig-Mayes substructure. This may be due to some artifacts of the transmission simulator approach providing an increased stiffening effect at this location.

1 Verification of Experimental Component Mode Synthesis in the Sierra Analysis Framework 5 Fig. 1.4 Comparison of “truth” and Sierra-SD sub-structured higher bending mode shapes (solid lineDtruth model, circleDsub-structured model) 1.5 Conclusions This paper has presented the motivation for using experimentally derived substructures within the Sierra analysis framework. The Craig-Mayes sub-structuring approach allows for a straightforward interface of an experimental CMS model with a discrete finite element model by using a representation similar to the Craig-Bampton CMS approach. This allows for the experimental CMS model to be treated virtually the same way as a numerically derived Craig-Bampton “super element”, although the experimental model may be prone to some numerical conditioning issues as a result of flaws in measurement data and mathematical operations being performed on that data. The work-flow of interfacing a Craig-Mayes model with the Sierra analysis framework was discussed and the process was demonstrated successfully on a proof-of-concept application. Future work will consider more complicated substructures. This may include sub-structures generated from actual experimental data or substructures derived from “virtual” modal testing with the Sierra-SD analysis software. The concept of virtual modal testing allows for more idealized accelerometer data to be considered within the general process of an experimental sub-structuring method while allowing control of the imperfections in the test data through the introduction of measurement noise or other flaws. References 1. Reese, G., Bhardwaj, M., Walsh, T.: Sierra Structural Dynamics-Theory Manual. Sandia National Laboratories, Albuquerque (2014) 2. Sierra Structural Dynamics Development Team: Sierra Structural Dynamics-User’s Notes. Sandia National Laboratories, Albuquerque (2014) 3. Craig, R.R., Bampton, M.C.: Coupling of substructures for dynamic analysis. AIAA J. 6(7), 1313–1319 (1968) 4. Mayes, R.L.: A Craig-Bampton experimental dynamic substructure using the transmission simulator method. In: Proceedings of the 32nd International Modal Analysis Conference, Orlando, February 2015 5. Allen, M.S., Kammer, D.C., Mayes, R.L.: Experimental based substructuring using a Craig-Bampton transmission simulator model. In: Proceedings of the 32nd International Modal Analysis Conference, Orlando, February 2014 6. Sjaardema, G.D.: GJOIN: a program for merging two or more GENESIS databases, SAND92-2290 (1992)

Chapter 2 Multi-DoF Interface Synchronization of Real-Time-Hybrid-Tests Using a Recursive-Least-Squares Adaption Law: A Numerical Evaluation Andreas Bartl, Johannes Mayet, Morteza Karamooz Mahdiabadi, and Daniel J. Rixen Abstract Cyber Physical Testing or Real Time Hybrid Testing is a Hardware-In-The-Loop approach allowing for tests of structural components of complex machines with realistic boundary conditions by coupling virtual components. The need to actuate the physical interface makes the tests on structural systems challenging. In order to deal with stability and accuracy issues, we propose the use of an Adaptive Feed-Forward Cancellation approach with a Recursive Least Squares (RLS) adaption law for interface synchronization of harmonically excited systems. The interface forces are generated from multiple harmonic components of the excitation force. A RLS adaption law sets the amplitudes and phases of the harmonic interface force components and minimizes the interface gap. One major practical advantage of using a RLS adaption law is that only one forgetting factor has to be chosen compared to other adaption algorithms with various tuning parameters. As a consequence, it is possible to test systems with multiple interface DoF. In order to illustrate the performance and robustness of the proposed testing algorithm, the contribution includes a numerical investigation on a lumped mass system. Keywords Hybrid testing • Hardware-in-the-loop • Real-time substructuring • Interface synchronization • Recursive least squares 2.1 Introduction Real Time Hybrid Testing, Cyber Physical Testing or Hardware-in-the-Loop for structural systems is a testing approach connecting experimental test rigs (experimental component) with simulation models (virtual component) in a real time test (see Figs. 2.1 and 2.2). In contrast of testing the experimental component by applying fictitious load cases, realistic boundary conditions are provided in these test procedures. The approach is always valuable were neither full experimental tests nor full simulations are applicable. Real Time Hybrid Testing was applied in engineering of earthquake save civil structures in [3, 12, 17]. A testing example on a full wind turbine nacelle is presented in [6] and an automotive application is given in [18]. The objective of the interface synchronization control in Real Time Hybrid Testing (RTHT) is to satisfy equilibrium and compatibility constraints within the desired frequency range. Consider for example structural applications with commonly low damping of the overall system. Controlling a system with poles close to the imaginary axis can cause instability of the real time test due to small control errors and inaccuracies in measurement or actuation. The problem of interface synchronization is closely linked to actual compensation methods. The performance of actuator compensation methods is compared in [5]. The authors of [15, 21] present frameworks for the development of RTHT controllers. A Linear-Quadratic-Regulator controller framework is presented in [22]. In the contribution [7] the Real Time Hybrid Testing problem is analyzed with conventional control theory. As in many applications the dynamics of the experimental substructures are unknown, Model Reference Adaptive Control (MRAC) is proposed as a control strategy in [19, 23]. A widely used approach is based on polynomial forward prediction used in [8, 12] for compensation of the actuator dynamics. The authors of [24] extend this approach by gain and phase estimation. More recently neuronal network feedforward compensation for the use in Real Time Hybrid Testing were proposed in [16]. Model Predictive Control is proposed as a control strategy for RTHT in [20]. In [1] we presented a adaptive feedforward algorithm with a harmonic regressor (see e.g. [2, 4, 10]) applied to RTHT. The adaption is based on a gradient algorithm. This approach is closely linked to fxLMS Algorithm as presented in [14]. However, in case of multiple DoF interfaces the choice of the adaption gain matrix, which defines the stability of the algorithm, is getting impractical. The entries of the adaption gain matrix can vary within several orders of magnitude and wrong choices may cause instability of the test. Therefore, we propose in this contribution an adaptive feedforward filter with harmonic A. Bartl ( ) • J. Mayet • M.K. Mahdiabadi • D.J. Rixen Technical University of Munich, Boltzmannstraße 15, D-85748 Garching, Germany e-mail: andreas.bartl@tum.de © The Society for Experimental Mechanics, Inc. 2016 M. Allen et al. (eds.), Dynamics of Coupled Structures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-29763-7_2 7

8 A. Bartl et al. Overall System Virtual Component (Numerical Model) Experimental Component Fig. 2.1 The overall system is split into a virtual and an experimental component Fig. 2.2 The test rig is coupled with sensors and actuators to the virtual component running on a real time computer regressor based on a Recursive Least Squares (RLS) adaption law with only a single tuning parameter. The fact that the user has only to choose one tuning parameter makes the suggested approach applicable to carry out various tests on systems with a multi DoF interface. 2.2 Hybrid Testing Problem Formulation The objective of the RTHT control is to satisfy compatibility (Eq. (2.1)) and equilibrium (Eq. (2.2)) constraints between virtual and experimental components. The Boolean matrices GV andGE are selecting the interface forces and displacements (see [9] for details). yb,V and yb,E are the interface displacement vectors. The interface gap is denoted as e. fb,V and fb,E are the interface force vectors GVu GEuDyb,V yb,E De D0 (2.1) GVfb,V CGEfb,E D0 (2.2) In principal two distinctive ways for setting up an control scheme do exist. One possibility is to define the interface displacements as compatible and controlling the interface forces in order to achieve equilibrium. In contrast one can define the interface forces as forces with equal magnitude and opposite sign, controlling the interface gap. In this contribution, we use the latter one, which is comparable to the to the dual formulation in substructuring (see [9] for details). In practice, this foregoing is absolutely meaningful since one will end up with forces as controller setpoint rather than gaps which would necessarily require inner-loop actuator control algorithms. The applied forces are subsequently measured and applied to the virtual subcomponent with opposite sign. The dynamics of both components are given by Eq. (2.3). MV 0 0 ME RuV RuE C DV 0 0 DE PuV PuE C KV 0 0 KE uV uE C GT V GT E D fV fE (2.3) The objective of an interface synchronization controller will be to apply such that the interface gap e is closed. The assumptions of the control strategy are an harmonic excitation and steady-state system behaviour. The block diagram of the overall control system is given in Fig. 2.3. Before deriving the algorithm the hybrid testing problem is reformulated such that it can be used for an adaptive feedforward compensator in this section. The corresponding state space formulations are given in Eqs. (2.4) and (2.5).

2 Multi-DoF Interface Synchronization of Real-Time-Hybrid-Tests Using: : : 9 Fig. 2.3 The block diagram shows the hybrid test with adaptive feedforward compensation. Harmonic excitation on both the experimental and virtual component are possible. The actuator is exciting the experimental and contrariwise the virtual component in the present of real and virtual harmonic excitations. The controller adapts phase and gain of the harmonic inputs to the actuator such that the interface gap e is closed. (adapted from[1]) PxV D 0 I M 1 V KV M 1 V DV „ ƒ‚ … AV xV C 0 M 1 V GT V „ ƒ‚ … BV C 0 M 1 V „ƒ‚… EV fV yV DGVuV D GV 0 „ƒ‚… CV xV (2.4) PxE D 0 I M 1 E KE M 1 E DE „ ƒ‚ … AE xE 0 M 1 E GT E „ ƒ‚ … BE C 0 M 1 E „ƒ‚… EE fE yE DGEuE D GE 0 „ƒ‚… CE xE (2.5) The interface responses can be written as yV D Z t t0 CVe AV.t /BV d „ ƒ‚ … contribution of interface excitation with transfer function HV.j !/ C Z t t0 CVe AV.t /EVfVd „ ƒ‚ … contribution of external excitation C CVe AV.t /x V.t0/ „ ƒ‚ … contribution of initial conditions (2.6) yE D Z t t0 CEe AE.t /BE d „ ƒ‚ … contribution of interface excitation with transfer function HE.j !/ CZ t t0 CEe AE.t /EEfEd „ ƒ‚ … contribution of excitation C CEe AE.t /x E.t0/ „ ƒ‚ … contribution of initial conditions (2.7) Assuming harmonic excitations and steady state behavior, the contribution of initial conditions are neglected. The interface forces can be expressed as a combination of harmonic functions: D mX iD1 Wi.t/ i Wi.t/ D Inn cos.˛i/ Inn sin.˛i/ with Wi 2Rn 2n (2.8)

10 A. Bartl et al. In Eq. (2.8) the regressor matrixWi.t/ contains the amplitudes for the cosine and sine part of the interface forces and thus the phase angle˛i DR t 0 !i.t/dt and frequency!i.t/, which are allowed to vary slowly. The important parameter vector i defines the phases and amplitudes of the interface forces. Since we assume steady state behavior yV and yE can now be rewritten as yV D mX iD1 Wi.t/PV,i i „ ƒ‚ … influence interfaceforces C Wi.t/ V,i „ ƒ‚ … influence external excitation.disturbance/ DW.t/PV C mX iD1 Wi.t/ V,i (2.9) yE D mX iD1 Wi.t/PE,i i „ ƒ‚ … influence interface forces C Wi.t/ E,i „ ƒ‚ … influence external excitation.disturbance/ DW.t/PE C mX iD1 Wi.t/ E,i, (2.10) where the matrices PV,i and PE,i are created using transfer function HV.j!1/ and HE.j!1/ respectively (see Fig. 2.3 and Eqs. (2.11) and (2.12)). These matrices basically apply a phase shift and gain to the parameter vector . The vectors V,i and E,i define phase and amplitude of the contributions of the external forces to the interface displacements. PV,i D Re.HV.j!i// Im.HV.j!i// Im.HV.j!i// Re.HV.j!i// D PR,V,i PI,V,i PI,V,i PR,V,i with PV,i 2R2n 2n (2.11) PE,i D Re.HE.j!i// Im.HE.j!i// Im.HE.j!i// Re.HE.j!i// D PR,E,i PI,E,i PI,E,i PR,E,i with PE,i 2R2n 2n (2.12) 2.3 Adaptive Feedforward Algorithm In order to couple virtual and experimental components, the parameter vector has to be chosen such that the interface gap e is closed. In order to adapt online, the use of a recursive least squares algorithm (see e.g. [11, 13]) is proposed, which minimizes the integral cost functional J defined in Eq. (2.13). Note that for the derivation of the adaption law, the above mentioned functions are used in their time discretized form. Here we use brackets to indicate a specific time instance. The cost functional includes a forgetting factor 2 Œ0, 1 , which enables a decreasing weighting of old values of eTŒi eŒi at ith time instances. The phase and gain matrices PE and PV as well as PA, which characterizes the actuator dynamics, are combined toP. JŒk D kX iD0 k ieT Œi eŒi with eŒi DyEŒi yVŒi DWŒi .PE PV/PA „ ƒ‚ … P Œi CWŒi . EŒi VŒi „ ƒ‚ … Œi / (2.13) Starting point for deriving the adaption law for the hybrid testing problem is the solution of the least squares problem, which is then rearranged as recursive algorithm: 0 D @JŒk @ Œk D kX iD0 2 k i PTWŒi TWŒi P Œk CPTWŒi TWŒi Œi (2.14) Œk D kX iD0 k iPTWŒi TWŒi P! 1 „ ƒ‚ … RŒk kX iD0 k iPTWŒi TWŒi Œi ! (2.15)

2 Multi-DoF Interface Synchronization of Real-Time-Hybrid-Tests Using: : : 11 The solution of the least squares problem for the next time step k C1 is arranged as follows: Œk C1 D RŒkC1 ‚ …„ ƒ kX iD0 kC1 iPTWŒi TWŒi PCPTWŒk C1 TWŒk C1 P! 1 · kX iD0 kC1 iPTWŒi TWŒi Œi PTWŒk C1 TWŒk C1 Œk C1 ! Applying the Woodbury matrix identity allows to replace the inverse of the regressor matrix: Œk C1 D RŒkC1 ‚ …„ ƒ 1 RŒk 1 RŒk PTWŒk C1 T I C 1 WŒk C1 PRŒk PTWŒk C1 T 1 WŒk C1 PRŒk !· kX iD0 k iPTWŒi TWŒi Œi PTWŒk C1 TWŒk C1 Œk C1 ! Further simplification of the equations finally yields the recursive least squares adaption law: Œk C1 D 1 .RŒk PTWŒk C1 T /.I C 1 WŒk C1 PRŒk PTWT Œk C1 / 1 (2.16) Œk C1 D Œk C Œk C1 .WŒk C1 P Œk CWŒk C1 Œk C1 / „ ƒ‚ … e0ŒkC1 (2.17) RŒk C1 D 1 .RŒk Œk C1 WŒk C1 PRŒk / (2.18) Note that e0 is the a-priori gap, which can be measured, whereas e is the a-posteriori interface gap, which is used in the cost functional J. The RLS algorithm allows the practical application of adaptive feedforward compensation in Real Time Hybrid Testing with multiple DoF interfaces as a single forgetting factor has to be chosen. Note that the phase and gain matrix P characterizing plant dynamics are used in the adaption law. Pcan be identified prior to the adaption process by exciting each actuation DoF separately or with uncorrelated noise. 2.4 Numerical Case Study The algorithm is applied to a simple lumped mass problem with a two DoF interface. The arrangement of the masses is illustrated in Fig. 2.4. The mass and stiffness parameters are given in Table 2.1. Proportional damping with a stiffness proportional coefficient ˛ D0.01and a mass proportional coefficient ˇ D0.001is used, which confers a modal damping of 0.5% to the submodels. The models and the interface synchronization control were implemented in Matlabo˝ Simulinko˝. The excitation forcefV,ext DP 4 iD1 D Ai sin!it was applied on mass 1. The excitation frequencies were!1 D20 1 rad , !2 D30 1 rad , !3 D50 1 rad and!4 D60 1 rad . The amplitudes A1 D4N, A2 D10N, A3 D10NandA4 D20N. The forgetting factor for the RLS algorithm was chosen as D0.99 The identification was running for 10s with an excitation of 5s on each actuation DoF. The adaption with the RLS algorithm starts at t D10s. Figures 2.5 and 2.6 show the interface synchronization for both interface DoF. In the investigated case the algorithm adapts within 2s and is then accurately ensuring compatibility. The adaption time is depending on the properties of the coupled components. Figure 2.7 shows the comparison of the displacement of mass 4 with the reference overall system. After the adaption process the reference system is simulated accurately. In all of our numerical studies the algorithm was found to be very robust. As indicated by Fig. 2.8 noisy force and displacement signals have little impact on adaption time and stability issues in this numerical case study which indicates a good feasibility for practical implementation.

12 A. Bartl et al. Fig. 2.4 Arrangement of the lumped mass system used for numerical studies Table 2.1 System parameters used in the numerical case study Virtual component (V) Stiffness (N/m) Mass (kg) kV,1 25,000,000 mV,1 10 kV,12 10,000,000 mV,2 3 kV,13 10,000,000 mV,3 3 kV,24 10,000,000 mV,4 3 kV,35 10,000,000 mV,5 3 kV,45 10,000,000 mV,6 2 kV,58 500,000 mV,7 2 kV,46 20,000,000 mV,8 4 kV,67 20,000,000 Test specimen (EXP) Stiffness (N/m) Mass (kg) kEXP,13 2,500,000 mEXP,1 2 kEXP,23 2,000,000 mEXP,2 4 kEXP,34 10,000,000 mEXP,3 8 kEXP,4 10,000,000 mEXP,4 5 2.5 Conclusion In this paper we propose an adaptive feedforward technique with harmonic regressor for interface synchronization in Real Time Hybrid Testing. The approach makes use of the assumption of harmonic excitation and steady state. It addresses stability and accuracy issues in cases where the simulated overall system is a structural system with low damping. Multiple DoF interfaces are necessary in many applications. As the choice of adaption gain parameters is getting a complex task for tests with multiple DoF interfaces, we propose the use of a recursive least square algorithm for the adaption of the harmonic parameters with only a single parameter for the controller design. Future work will include the experimental validation on a test rig with a multiple DoF interface as well as the comparison with other interface synchronization techniques for Real Time Hybrid Testing.

2 Multi-DoF Interface Synchronization of Real-Time-Hybrid-Tests Using: : : 13 Fig. 2.5 Interface synchronization for the first interface DoF: the left hand figure shows the adaption process, the right hand figure shows the synchronization in the adapted state Fig. 2.6 Interface synchronization for the second interface DoF: the left hand figure shows the adaption process, the right hand figure shows the synchronization in the adapted state Fig. 2.7 Displacement of mass 4during the adaption process compared with the reference overall system

14 A. Bartl et al. Fig. 2.8 Interface synchronization for the first interface DoF with added noise on force and displacement signals: the left hand figure shows the adaption process, the right hand figure the synchronization in the adapted state References 1. Bartl, A., Mayet, J., Rixen, D.J.: Adaptive feedforward compensation for real time hybrid testing with harmonic exciation. In: Proceedings of the 11th International Conference on Engineering Vibration, Sept 2015 2. Bayard, D.S.: A general theory of linear time-invariant adaptive feedforward systems with harmonic regressors. IEEE Trans. Autom. Control 45(11), 1983–1996 (2000) 3. Blakeborough, A., Darby, A., Williams, M.: The development substructure of real-time testing. Philos. Trans. Math. Phys. Eng. Sci. 359(1786), 1869–1891 (2001) 4. Bodson, M., Sacks, A., Khosla, P.: Harmonic generation in adaptive feedforward cancellation schemes. IEEE Trans. Autom. Control 39(9), 1939–1944 (1994) 5. Bonnet, P.A.: The development of multi-axis real-time substructure testing. Ph.D. thesis (2006) 6. Bosse, D., Radner, D., Schelenz, R., Jacobs, G.: Analysis and application of hardware in the loop wind loads for full scale Nacelle ground testing. DEWI Mag. 43, 65–70 (2013) 7. Botelho, R.M., Christenson, R.E.: Mathematical equivalence between dynamic substructuring and feedback control theory. In: Proceedings of the 33rd IMAC (2015) 8. Darby, A.P., Williams, M.S., Blakeborough, A.: Stability and delay compensation for real-time substructure testing. J. Eng. Mech. 128, 1276– 1284 (2002) 9. De Klerk, D., Rixen, D.J., Voormeeren, S.N.: General framework for dynamic substructuring: history, review and classification of techniques. AIAAJ. 46(5), 1169–1181 (2008) 10. Glover, J.: Adaptive noise canceling applied to sinusoidal interferences. IEEE Trans. Acoust. Speech Signal Process. 25, 484–491 (1977) 11. Haykin, S.: Adaptive Filter Theory. Upper Saddle River, Prentice Hall (2010) 12. Horiuchi, T., Inoue, M., Konno, T., Namita, Y.: Real-time hybrid experimental system with actuator delay compensation and its application to a piping system with energy absorber. Earthq. Eng. Struct. Dyn. 28(10), 1121–1141 (1999) 13. Ioannou, P., Sun, J.: Robust Adaptive Control. Dover Publications, New York (2013) 14. Jungblut, T., Wolter, S., Matthias, M., Hanselka, H.: Using numerical models to complement experimental setups by means of active control of mobility. Appl. Mech. Mater. 70, 357–362 (2011) 15. Li, G.: A generic dynamically substructured system framework and its dual counterparts, pp. 10101–10106 (2014) 16. Li, G., Na, J., Stoten, D.P., Ren, X.: Adaptive neural network feedforward control for dynamically substructured systems. IEEE Trans. Control Syst. Technol. 22(3), 944–954 (2014) 17. Nakashima, M., Kato, H., Takaoka, E.: Development of real-time pseudo dynamic testing. Earthq. Eng. Struct. Dyn. 21, 79–92 (1992) 18. Plummer, A.: Model-in-the-loop testing. Proc. Inst. Mech. Eng. I: J. Syst. Control Eng. 220, 183–199 (2006) 19. Stoten, D., Hyde, R.: Adaptive control of dynamically substructured systems: the single-input single-output case. Proc. Inst. Mech. Eng. I: J. Syst. Control Eng. 220(2), 63–79 (2006) 20. Stoten, D., Li, G., Tu, J.: Model predictive control of dynamically substructured systems with application to a servohydraulically actuated mechanical plant. IET Control Theory Appl. 4(2), 253–264 (2010) 21. Stoten, D., Tu, J., Li, G.: Synthesis and control of generalized dynamically substructured systems. Syst. Control Eng. 223, 371–392 (2010) 22. Tu, J.: Development of numerical-substructure-based and output-based substructuring controllers. In: Structural Control and Health Monitoring, June 2012, pp. 918–936. Wiley, New York (2013) 23. Wagg, D.J., Stoten, D.P.: Substructuring of dynamical systems via the adaptive minimal control synthesis algorithm. Earthq. Eng. Struct. Dyn. 30(6), 865–877 (2001) 24. Wallace, M.I., Wagg, D.J., Neild, S.a.: An adaptive polynomial based forward prediction algorithm for multi-actuator real-time dynamic substructuring. Proc. R. Soc. A Math. Phys. Eng. Sci. 461(2064), 3807–3826 (2005)

Chapter 3 Controls Based Hybrid Sub-Structuring Approach to Transfer Path Analysis Joseph A. Franco, Rui M. Botelho, and Richard E. Christenson Abstract In the design of mechanical systems, there are constraints imposed on the vibration of mechanical equipment to limit the vibration transmission into its support structure. To accurately predict the coupled system response, it is important to capture the coupled interaction of the two portions, i.e., the mechanical equipment and the support structure, of the mechanical system. Typically during a design, the analysis of the full mechanical system is not possible because a large part of the system may be non-existent. Existing methods known as Transfer Path Analysis and Frequency Based Substructuring are techniques for predicting the coupled response of vibrating mechanical systems. In this paper, a control based hybrid substructuring approach to Transfer Path Analysis is proposed. By recognizing the similarities between feedback control and dynamic substructuring, this paper demonstrates that this approach can accurately predict the coupled dynamic system response of multiple substructured systems including operating mechanical equipment with a complex vibration source. The main advantage of this method is that it uses blocked force measurements in the form of a power spectral density matrix measured uncoupled from the rest of the system. This substructuring method is demonstrated using a simplified case study comprised of a two-stage vibration isolation system and excited by operating mechanical equipment. Keywords Transfer path analysis • Frequency based substructuring • Hybrid substructuring • Feedback control • System level vibration analysis 3.1 Introduction Vibration of mechanical equipment can result in fatigue, detection, and/or environmental concerns for a structural system. A critical aspect of the design of systems that include mechanical equipment is quantifying the level of transmitted vibration energy through the supporting structure. The system design typically consists of strict constraints imposed on the vibration transmission of the mechanical equipment through the support structure. During the design phase of a system, the mechanical equipment is pre-existing either from previous designs or they are commercially available components purchased from a vendor. The support structure is typically non-existent and is designed and optimized using Computer Aided Design (CAD) software. This makes testing of the full mechanical system, impossible. For these reasons, the analysis of the mechanical system normally requires the combination of multiple quantifications of dynamics of various substructures of the mechanical system. Existing methods known as Transfer Path Analysis (TPA) are frequency response functions (FRF) based techniques that describe the dynamics of the mechanical system by the multiplication of the FRFs of the system substructures. This method can also be used to combine theoretical (FEM) models and experimental measurements of system substructures. Some of these methods were developed by Plunt [1, 2] for the automotive industry and Darby [3] for the marine industry. However, the disadvantage of these TPA methods is that they do not always consider the dynamic coupling between the receiving and exciting substructures. This limitation becomes critical at low frequencies due to the interaction between the modes of the individual substructures. Variations of TPA methods are known as Frequency Based Substructuring (FBS) methods which allow for the calculation of the entire mechanical system dynamic response based on the FRFs of the system substructures using various methods. Primary developments of FBS methods are Crowley et al. [4], Jetmundsen et al. [5], Imregun and Robb [6] and later on Gordis [7] and de Klerk [8]. Generally, this work demonstrated a wide variety of methods to couple substructures based on J.A. Franco ( ) • R.M. Botelho • R.E. Christenson Department of Civil and Environmental Engineering, University of Connecticut, 261 Glenbrook Road Unit 3037, Storrs, CT 06269-3037, USA e-mail: joseph.franco@uconn.edu © The Society for Experimental Mechanics, Inc. 2016 M. Allen et al. (eds.), Dynamics of Coupled Structures, Volume 4, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-29763-7_3 15

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