Topics in Modal Analysis II, Volume 8

River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Topics in Modal Analysis II, Volume 8 Randall Allemang Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014 River Publishers

Conference Proceedings of the Society for Experimental Mechanics Series Series Editor TomProulx Society for Experimental Mechanics, Inc., Bethel, CT, USA

River Publishers Randall Allemang Editor Topics in Modal Analysis II, Volume 8 Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014

Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-896-5 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2014 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, or reproduction in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Preface v Topics in Modal Analysis II, Volume 8 represents one of the eight volumes of technical papers presented at the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, organized by the Society for Experimental Mechanics, and held in Orlando, Florida, February 3–6, 2014. The full proceedings also include volumes on Dynamics of Coupled Structures; Nonlinear Dynamics; Model Validation and Uncertainty Quantification; Dynamics of Civil Structures; Structural Health Monitoring; Special Topics in Structural Dynamics; and Topics in Modal Analysis I. Each collection presents early findings from experimental and computational investigations on an important area within structural dynamics. Topics in Modal Analysis II represents papers on enabling technologies for modal analysis measurements such as sensors and instrumentation and applications of modal analysis in specific application areas. Topics in this volume include: Finite element techniques Modal parameter identification Modal testing methods Shock and vibration The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Cincinnati,OH, USA Randall Allemang

Contents 1 Integrating Multiple Algorithms in Autonomous Modal Parameter Estimation................................. 1 R.J. Allemang and A.W. Phillips 2 Effects of Magneto-Mechanical Coupling on Structural Modal Parameters ..................................... 11 M. Kirschneck, D.J. Rixen, Henk Polinder, and Ron van Ostayen 3 Extraction of Modal Parameters of Micromachined Resonators in Higher Modes.............................. 19 AVSS Prasad, K.P. Venkatesh, Navakanta Bhat, and Rudra Pratap 4 Normalization of Experimental Modal Vectors to Remove Modal Vector Contamination...................... 29 A.W. Phillips and R.J. Allemang 5 Effective Use of Scanning Laser Doppler Vibrometers for Modal Testing........................................ 43 Ben Weekes and David Ewins 6 Precise Frequency Domain Algorithm of Half Spectrum and FRF................................................ 59 J.M. Liu, W.D. Zhu, M. Ying, S. Shen, and Y.F. Xu 7 Identification of a Time-Varying Beam Using Hilbert Vibration Decomposition ................................ 71 M. Bertha and J.C. Golinval 8 Recovery of Operational Deflection Shapes from Noise-Corrupted Measurement Data from CSLDV: Comparison Between Polynomial and Mode Filtering Approaches .............................. 83 P. Castellini, P. Chiariotti, E.P. Tomasini, M. Martarelli, D. Di Maio, B. Weekes, and D.J. Ewins 9 Exploiting Imaging Techniques to Overcome the Limits of Vibration Testing in High Excitation Level Conditions ......................................................................................................... 93 M. Martarelli, P. Castellini, P. Chiariotti, and E.P. Tomasini 10 An Experimental Modal Channel Reduction Procedure Using a Pareto Chart .................................. 101 William H. Semke, Kaci J. Lemler, and Milan Thapa 11 Unique Isolation Systems to Protect Equipment in Navy Shock Tests............................................. 111 Herb LeKuch, Kevork Kayayan, and Neil Donovan 12 Nonlinear High Fidelity Modeling of Impact Load Response in a Rod ........................................... 129 Yu Liu, Andrew J. Dick, Jacob Dodson, and Jason Foley 13 On the Role of Boundary Conditions in the Nonlinear Dynamic Response of Simple Structures.............. 135 Yu Liu and Andrew J. Dick 14 Evaluation of On-Line Algebraic Modal Parameter Identification Methods ..................................... 145 F. Beltrán-Carbajal, G. Silva-Navarro, and L.G. Trujillo-Franco 15 Ambient Vibration Test of Granville Street Bridge Before Bearing Replacement ............................... 153 Yavuz Kaya, Carlos Ventura, and Martin Turek vii

viii Contents 16 Vibration Testing and Analysis of A Monumental Stair ............................................................ 161 Mehdi Setareh and Xiaoyao Wang 17 Evaluation of Stop Bands in Periodic and Semi-Periodic Structures by Experimental and Numerical Approaches.................................................................................................. 171 P.G. Domadiya, E. Manconi, M. Vanali, L.V. Andersen, and A. Ricci 18 Operating Mode Shapes of Electronic Assemblies Under Shock Input ........................................... 179 Ryan D. Lowe, Jason R. Foley, David W. Geissler, and Jennifer A. Cordes 19 Comparison of Modal Parameters Extracted Using MIMO, SIMO, and Impact Hammer Tests on a Three-Bladed Wind Turbine...................................................................................... 185 Javad Baqersad, Peyman Poozesh, Christopher Niezrecki, and Peter Avitabile 20 Modal Test Results of a Ship Under Operational Conditions ...................................................... 199 Esben Orlowitz and Anders Brandt 21 Measuring Effective Mass of a Circuit Board........................................................................ 207 Randall L. Mayes and Daniel W. Linehan 22 Acoustic Cavity Modal Analysis for NVH Development of Road Machinery Cabins ........................... 219 Hongan Xu, Owen Dickinson, John Wang, and Hyunseok Kang 23 Strain-Based Dynamic Measurements and Modal Testing ......................................................... 233 Fábio Luis Marques dos Santos, Bart Peeters, Marco Menchicchi, Jenny Lau, Ludo Gielen, Wim Desmet, and Luiz Carlos Sandoval Góes 24 AIRBUS A350 XWB GVT: State-of-the-Art Techniques to Perform a Faster and Better GVT Campaign... 243 P. Lubrina, S. Giclais, C. Stephan, M. Boeswald, Y. Govers, and N. Botargues 25 Bayesian System Identification of MDOF Nonlinear Systems Using Highly Informative Training Data...... 257 P.L. Green 26 Finite Element Model Updating Using the Separable Shadow Hybrid Monte Carlo Technique ............... 267 I. Boulkaibet, L. Mthembu, T. Marwala, M.I. Friswell, and S. Adhikari 27 Bayesian System Identification of Dynamical Systems Using Reversible Jump Markov Chain Monte Carlo.............................................................................................................. 277 D. Tiboaca, P.L. Green, R.J. Barthorpe, and K. Worden 28 Assessment and Validation of Nonlinear Identification Techniques Using Simulated Numerical and Real Measured Data................................................................................................ 285 A. delli Carri and D.J. Ewins 29 Effects of Errors in Finite Element Models on Component Modal Tests.......................................... 299 Masayoshi Misawa and Hidenori Kawasoe 30 Estimating Frequency-Dependent Mechanical Properties of Materials........................................... 313 Jason R. Foley and Jacob C. Dodson 31 Flexible Dynamic Modeling of Turret Systems by Means of Craig-Bampton Method and Experimental Validation........................................................................................... 325 Fatih Altunel and Murat Aykan 32 Material Characterization of Gyroscope Isolator Using Modal Test Data........................................ 339 Özge Mencek and Murat Aykan 33 Loss Factors Estimation Using FEM in Statistical Energy Analysis............................................... 351 Takayuki Koizumi, Nobutaka Tsujiuchi, and Katsuyoshi Honsho 34 Investigation of Crossing and Veering Phenomena in an Isogeometric Analysis Framework .................. 361 Stefano Tornincasa, Elvio Bonisoli, Pierre Kerfriden, and Marco Brino 35 Influence of Fan Balancing in Vibration Reduction of a Braking Resistor ....................................... 377 F. Braghin, M. Portentoso, and E. Sabbioni

Contents ix 36 Vibrations of Discretely Layered Structures Using a Continuous Variation Model ............................. 385 Arnaldo J. Mazzei and Richard A. Scott 37 Next-Generation Random Vibration Tests............................................................................ 397 P.M. Daborn, C. Roberts, D.J. Ewins, and P.R. Ind 38 Optimal Phasing Combinations for Multiple Input Source Excitation............................................ 411 Kevin L. Napolitano and Nathanael C. Yoder

Chapter1 Integrating Multiple Algorithms in Autonomous Modal Parameter Estimation R.J. Allemang and A.W. Phillips Abstract Recent work with autonomous modal parameter estimation has shown great promise in the quality of the modal parameter estimation results when compared to results from experienced user interaction using traditional methods. Current research with the Common Statistical Subspace Autonomous Mode Identification (CSSAMI) procedure involves the integration of multiple modal parameter estimation algorithms into the autonomous procedure. The current work uses possible solutions from different traditional methods like Polyreference Time Domain (PTD), Eigensystem Realization Algorithm (ERA) and Polyreference Frequency Domain (PFD) that are combined in the autonomous procedure to yield one consistent set of modal parameter solutions. This final, consistent set of modal parameters is identifiable due to the combination of temporal information (the complex modal frequency) and the spatial information (the modal vectors) in a Z domain state vector of relatively high order (5–10). Since this Z domain state vector has the complex modal frequency and the modal vector as embedded content, sorting consistent estimates from hundreds or thousands of possible solutions is now relatively trivial based upon the use of a state vector involving spatial information. Keywords Autonomous • Modal parameter estimation • Pole weighted vector • State vector • Experimental structural dynamics Nomenclature Ni Number of inputs No Number of outputs NS Short dimension size NL Long dimension size N Number of vectors in cluster ¨i Discrete frequency (rad/s) [H(¨i)] FRF matrix (No Ni) r Mode number œr S domain polynomial root œr Complex modal frequency (rad/s) œr rCj!r r Modal damping !r Damped natural frequency zr Z domain polynomial root f rg Base vector (modal vector) f rg Pole weighted base vector (state vector) R.J. Allemang ( ) • A.W. Phillips Structural Dynamics Research Laboratory, Department of Mechanical and Materials Engineering, College of Engineering and Applied Science, University of Cincinnati, Cincinnati, OH 45221-0072, USA e-mail: Randall.Allemang@UC.EDU R. Allemang (ed.), Topics in Modal Analysis II, Volume 8: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-04774-4__1, © The Society for Experimental Mechanics, Inc. 2014 1

2 R.J. Allemang and A.W. Phillips 1.1 Introduction The desire to estimate modal parameters automatically, once a set or multiple sets of test data are acquired, has been a subject of great interest for more than 40 years. Even in the 1960s, when modal testing was limited to analog test methods, several researchers were exploring the idea of an automated test procedure for determining modal parameters [1–3]. Today, with the increased memory and compute power of current computers used to process test data, an automated or autonomous, modal parameter estimation procedure is entirely possible and is being evaluated by numerous researchers and users. Before proceeding with a discussion of how multiple modal parameter estimation algorithms can be combined into autonomous modal parameter estimation, some discussion of the current autonomous modal parameter estimation procedure is required. In general, autonomous modal parameter estimation refers to an automated procedure that is applied to a modal parameter estimation algorithm so that no user interaction is required once the process is initiated. This typically involves setting a number of parameters or thresholds that are used to guide the process in order to exclude solutions that are not acceptable to the user. When the procedure finishes, a set of modal parameters is identified that can then be reduced or expanded if necessary. The goal is that no further reduction, expansion or interaction with the process will be required. For the purposes of further discussion, the autonomous modal parameter estimation procedure is simply an efficient mechanism for sorting a very large number of solutions into a final set of solutions that satisfies a set of criteria and thresholds that are acceptable to the user. When multiple modal parameter estimation algorithms are combined into a single autonomous procedure, this yields more estimates of the modal parameters which contribute to a statistically more significant result. Currently, the user of autonomous modal parameter estimation is assumed to be very experienced and is using autonomous modal parameter estimation as a sophisticated tool to highlight the most likely solutions based upon statistics. The experienced user will realize that the final solutions may include unrealistic solutions or non-optimal solutions and further evaluation will be required. 1.2 Background In order to discuss the impact and use of multiple modal parameter estimation algorithms in autonomous modal parameter estimation, the importance of spatial information to the solution procedure is critical. Therefore, some background is needed to clarify terminology and methodology. This background has been provided in previous papers [4–7] and will only be highlighted here in terms of spatial information, modal parameter estimation and autonomous modal parameter estimation. 1.2.1 Spatial Information Spatial information, with respect to experimental modal parameter estimation, refers to the vector information and dimension associated with the inputs and outputs of the experimental test. Essentially, this represents the locations of the sensors in the experimental test. It is important to recognize that an experimental test should always include multiple inputs and outputs in order to clearly estimate different modal vectors and to resolve modal vectors when the complex natural frequencies are close, what is called repeated or pseudo-repeated roots. Since the data matrix, normally involving frequency response functions (FRF) or impulse response functions (IRF), is considered to be symmetric or reciprocal, the data matrix can be transposed, switching the effective meaning of the row and column index with respect to the physical inputs and outputs. ŒH.!i/ No Ni DŒH.!i/ T N i No (1.1) Since many modal parameter estimation algorithms are developed on the basis of either the number of inputs (Ni) or the number of outputs (No), assuming that one or the other is larger based upon test method, some nomenclature conventions are required for ease of further discussion. In terms of the modal parameter estimation algorithms, it is more important to recognize whether the algorithm develops the solution on the basis of the larger (NL) ofNi orNo, or the smaller (NS ) ofNi or No, dimension of the experimental data. For this reason, the terminology of long (larger of Ni or No) dimension or short (smaller of Ni orNo) dimension is easier to understand without confusion.

1 Integrating Multiple Algorithms in Autonomous Modal Parameter Estimation 3 Therefore, the nomenclature of the number of outputs (No) and number of inputs (Ni) has been replaced by the length of the long dimension of the data matrix (NL) and the length of the short dimension (NS) regardless of which dimension refers to the physical output or input. This means that the above reciprocity relationship can be restated as: ŒH.!i/ NL NS DŒH.!i/ T NS NL (1.2) Note that the reciprocity relationships embedded in Eqs. 1.1 and 1.2 are a function of the common degrees of freedom (DOFs) in the short and long dimensions. If there are no common DOFs, there are no reciprocity relationships and the data requirement for modern modal parameter estimation algorithms (multiple references) will not be met. Nevertheless, the importance of Eqs. 1.1 and 1.2 is that the dimensions of the FRF matrix can be transposed as needed to fit the requirement of specific modal parameter estimation algorithms. This impacts the size of the square matrix coefficients in the matrix coefficient, polynomal equation and the length of the associated modal (base) vector. 1.2.2 Autonomous Modal Parameter Estimation The interest in automatic modal parameter estimation methods has been documented in the literature since at least the mid 1960s when the primary modal method was the analog, force appropriation method [1–3]. Following that early work, there has been a continuing interest in autonomous methods that, in most cases, have been procedures that are formulated based upon a specific modal parameter estimation algorithm like the Eigensystem Realization Algorithm (ERA), the Polyreference Time Domain (PTD) algorithm or more recently the Polyreference Least Squares Complex Frequency (PLSCF) algorithm (which thebasis of the commercial version of the PLSCF, the PolyMAX ® method and the rational fraction polynomial algorithm with Z-domain generalized frequency (RFP-z)) [8]. A relatively complete list of autonomous and semi-autonomous procedures that have been reported can be found in a recent paper [4]. Each of these past procedures have shown some promise but have not yet been widely adopted. In many cases, the procedure focused on a single modal parameter estimation algorithm and did not develop a general procedure. Most of the past procedural methods focused on modal frequency (pole) density but depended on limited modal vector data to identify correlated solutions. Currently, due to increased computational speed and availability of memory, procedural methods can be developed that were beyond the computational scope of available hardware only a few years ago. These methods do not require any initial thresholding of the solution sets and rely upon correlation of the vector space of hundreds or thousands of potential solutions as the primary identification tool. The discussion in the following sections of the use of multiple modal parameter estimation algorithms in autonomous modal parameter estimation is based upon recent implementation and experience with an autonomous modal parameter estimation procedure referred to as the Common Statistical Subspace Autonomous Mode Identification (CSSAMI) method. The strategy of the CSSAMI autonomous method is to use a default set of parameters and thresholds to allow for all possible solutions from a given data set. This strategy allows for some poor estimates to be identified as well as the good estimates. The philosophy of this approach is that it is easier for the user to evaluate and eliminate poor estimates compared to trying to find additional solutions. The reader is directed to a series of previous papers in order to get an overview of the methodology and to view application results for several cases [4–7]. Note that much of the background of the CSSAMI method is based upon the Unified Matrix Polynomial Algorithm (UMPA) [8]. This means that this method can be applied to both low and high order methods with short or long dimension modal (base) vectors. This also means that most commercial algorithms could take advantage of this procedure. Note that high order, matrix coefficient polynomials normally have coefficient matrices of a dimension that is based upon the short dimension of the data matrix, NS. In these cases, it may be useful to solve for the complete, unscaled or scaled, modal vector of the large dimension, NL. This will extend the temporal-spatial information in the modal (base) vector so that the vector will be more sensitive to change. This characteristic is what gives the CSSAMI autonomous method a robust ability to distinguish between computational and structural modal parameters. 1.2.3 Pole Weighted Modal Vectors The key to estimating the modal parameters utilizing the CSSAMI autonomous procedure is formulating clusters of pole weighted modal vectors, or state vectors, from the estimates of modal parameters that are represented in a consistency diagram. These state vectors are formed from the modal vector estimates found as the consistency diagram is developed.

4 R.J. Allemang and A.W. Phillips −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 0 50 100 150 200 250 Real Imag State Vector (DOF) Fig. 1.1 Eighth order, pole-weighted vector (state vector) example When comparing modal (base) vectors, at either the short or the long dimension, a pole weighted vector can be constructed independent of the original algorithm used to estimate the poles and modal (base) vectors. For a given order k of the pole weighted vector, the modal (base) vector and the associated pole can be used to formulate the pole weighted vector as follows: f gr D 8 ˆ ˆ ˆ ˆ ˆ< ˆ ˆ ˆ ˆ ˆ: k r f gr : : : 2 r f gr 1 r f gr 0 r f gr 9 > > > > >= > > > > >;r f gr D 8 ˆ ˆ ˆ ˆ ˆ< ˆ ˆ ˆ ˆ ˆ: zk r f gr : : : z2 r f gr z1 r f gr z0 r f gr 9 > > > > >= > > > > >;r (1.3) While the above formulation (on the left) is possible, this form would be dominated by the high order terms if actual frequency units are utilized. Generalized frequency concepts (frequency normalization or Z domain mapping) are normally used to minimize this issue by using the Z domain form (zr ) of the complex modal frequency (œr) as shown above (on the right). The Z domain form of the complex natural frequency is developed as follows: zr De . r= max/ (1.4) zmr De m . r= max/ (1.5) In the above equations, max can be chosen as needed to cause the positive and negative roots to wrap around the unit circle in the Z domain without overlapping (aliasing). Normally, max is taken to be five percent larger than the largest frequency (absolute value of the complex frequency) identified in the roots of the matrix coefficient polynomial. Figures 1.1 and 1.2 are graphical representations of the pole weighted vector (state vector) defined in Eq. 1.3. In this example, the modal (base) vector (at the bottom of Fig. 1.1) is a real-valued normal mode that looks like one period of a

1 Integrating Multiple Algorithms in Autonomous Modal Parameter Estimation 5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Real Imag Fig. 1.2 Eighth order, pole-weighted vector (state vector) example—Top view sine wave. The successive higher orders, up to order eight, are shown in different colors moving up the vertical axis of this figure. The effect of scaling of the modal (base) vector by the higher powers of the Z domain frequency value causes the base vector to rotate in the real and imaginary space. Figure 1.2 shows the rotation affect clearly. Note that the choice of the order (k) of the pole weighted vector, therefore, just generates additional length and rotation in the pole weighted vector and gives varying sensitivity to comparisons between estimates. Futhermore, note that the choice of order (k) is independent of algorithm. State vectors are a natural part of the numerical formulation for all modal parameter estimation algorithms but this pole weighted vector (state vector), which looks similar, can be formed independently once the modal (base) vector is estimated and, thus, is not constrained by the algorithm. The choice of the order of the pole weighted vector (k) will depend upon the length of the modal (base) vector and is under continuing study at present. Since the magnitude of the Z domain frequency value is unity, there is no magnitude weighting involved. This rotation gives a method for a single vector to represent the modal (base) vector shape together with the complex-valued frequency. With respect to sorting and separating modal vectors that have similar shapes but different frequencies or similar frequencies but different modal vector shapes, this becomes a powerful parameter, together with modal vector correlation tools like the modal assurance criterion (MAC), for modal parameter estimation and for autonomous modal parameter estimation. 1.3 Multi-algorithm, Extended Consistency Diagrams Consistency diagrams, historically called stability diagrams, have almost always been utilized and developed for a specific modal parameter estimation algorithm. As such the numerical implementation can be different as a function of basis dimension (NS orNL), model order and/or subspace iteration. This would make it very hard to combine different algorithms into a single consistency diagram. However, every algorithm, at the point of the numerical implementation of the consistency diagram, has multiple sets of complex modal frequency and complex-valued modal vectors. The modal vectors may be of different length (NS or NL) as a function of algorithm. This potential mismatch in modal (base) vector length can be solved by restricting the long dimension to the DOFs of the short dimension or, more preferably, adding an extra step in the solution procedure to estimate the missing portion of the long dimension vectors, extending them from the short dimension DOFs to the long dimension DOFs. The latter approach is used in the following two figures as an example of extended consistency

6 R.J. Allemang and A.W. Phillips 0 500 1000 1500 2000 2500 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 Consistency Diagram Frequency (Hz) Model Iteration cluster pole & vector pole frequency conjugate non realistic 1/condition RFP−Z ERA PFD PTD Fig. 1.3 Extended consistency diagram—Conventional version diagrams based upon multiple modal parameter estimation algorithms. In these examples, the results from the individual algorithms are simply stacked into the extended consistency diagram with common sorting and evaluation settings. It should be noted that the order of the stacking of the different algorithms will affect the look of the consistency diagram but the CSSAMI autonomous procedure uses all of the estimated parameters and pays no attention to the sequential ordering and stability calculation involved in the consistency diagram. The data used for this, and all following examples in this paper, is FRF data taken from an impact test of a steel disc supported in a pseudo free-free boundary condition. The steel disc is approximately 2 cm. thick and 86 cm. in diameter with several small holes through the disc. The center area of the disc (diameter of approximately 25 cm.) has a thickness of approximately 6 cm. There are seven reference accelerometers and measured force inputs from an impact hammer are applied to thirty-six locations, including next to the seven reference accelerometers. The frequency resolution of the data is 5 Hz. While the disc is not as challenging as some industrial data situations that contain more noise or other complicating factors like small nonlinearities, the disc has a number of pseudo-repeated roots spaced well within the 5 Hz frequency resolution and a mix of close modes involving repeated and non-repeated roots which are very challenging. Based upon the construction of the disc, real-valued, normal modes can be expected and the inability to resolve these modes can be instructive relative to both modal parameter estimation algorithm and autonomous procedure performance. For the interested reader, a number of realistic examples are shown in other past papers including FRF data from an automotive structure and a bridge structure [4, 7]. Figure 1.3 is an example of using a conventional, sequential sorting procedure involving criteria for frequency, damping and modal vector consistency. Figure 1.4 is an example using a pole weighted vector (state vector) method of producing a similar consistency diagram. In this example, every estimate from every matrix coefficient polynomial solution from every algorithm is converted into a pole weighted vector of a specific order, in this case tenth order. Then, the consistency diagram is developed by using vector

1 Integrating Multiple Algorithms in Autonomous Modal Parameter Estimation 7 0 500 1000 1500 2000 2500 12 4356 78 10911 12 11 34 11 56 11 78 12 90 22 12 22 4322 56 22 78 32 0933 12 33 34 33 56 33 78 34 90 44 12 44 34 44 56 44 78 54 0955 12 55 34 55 56 55 78 56 90 66 12 66 34 66 56 66 78 76 0977 12 77 34 77 6577 Consistency Diagram Frequency (Hz) Model Iteration pweight = λ10 pwMAC≥0.99 pwMAC≥0.97 pwMAC≥0.94 pwMAC≥0.90 pwMAC≥0.85 pwMAC≥0.79 PTD ERA PFD RFP−Z Fig. 1.4 Extended consistency diagram—Pole weighted MAC version correlation methods (MAC) to identify consistency. A similar set of symbols, as those used in Fig. 1.3, are used to define increased levels of consistency as numerical solutions are added. Both methods work very well but the implementation of Fig. 1.4 is computationally easier and not subject to a frequency drift in the symbol path that can occur in the conventional implementation shown in Fig. 1.1. Note that the solid square symbols at the top of both consistency diagrams represent the solution found from the CSSAMI autonomous modal parameter estimation procedure applied to the information represented by each consistency diagram. Note that all of the above algorithms are using the same matrix polynomial equation normalization procedure which tends to yield clear consistency diagrams. Each consistency diagram can yield twice as many estimates of the desired modal parameters if both low and high matrix coefficient normalizations are utilized. This is also under current study. 1.4 Autonomous Modal Parameter Estimation with Extended Consistency Diagrams The CSSAMI autonomous procedure utilizes all solutions indicated by a symbol in the consistency diagram. If some symbols are not present, it means the user has decided not to view solutions identified by those symbols. This provides a way to remove solutions from the autonomous procedure that are clearly not reasonable. However, experience with the CSSAMI autonomous procedure has shown that some solutions that are often eliminated by users in an attempt to have a clear consistency diagram are often statistically consistent and useful. Figure 1.5 shows the solutions that are included in the autonomous procedure. The graphical representation on the left represents a MAC matrix involving the pole weighted vectors for all possible solutions from Fig. 1.3. The graphical

8 R.J. Allemang and A.W. Phillips 1000 800 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 600 400 200 0 -200 -400 -600 -800 -1000 -1000 -500 0 500 1000 Pole Number (±ωr) Ordered - Pole Weighted MAC Pole Number (±ωr) 1000 800 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 600 400 200 0 -200 -400 -600 -800 -1000 -1000 -500 0 500 1000 Pole Number (±ωr) Ordered - Pole Weighted MAC Pole Number (±ωr) Fig. 1.5 Pole weighted MAC of all consistency diagram solutions—Before and after threshold applied representation on the right represents the pole weighted vectors that remain after threshold and cluster size limitations are imposed. Each cluster that remains is evaluated, cluster by cluster, independently to estimate the best modal frequency and modal vector from that cluster. Note that both the positive frequency and negative frequency (complex conjugate) roots are included and identified separately as clusters. Figure 1.5 represents nearly 1,000 solution estimates spanning four different algorithms and 19 different solutions form each algorithm. Once the final set of modal parameters, along with their associated statistics, is obtained, quality can be assessed by many methods that have been used in the past. The most common example is to perform comparisons between the original measurements and measurements synthesized from the modal parameters. Another common example is to look at physical characteristics of the identified parameters such as reasonableness of frequency and damping values, normal mode characteristics in the modal vectors, and appropriate magnitude and phasing in the modal scaling. Other evaluations that may be helpful are unweighted and weighted modal assurance criterion (MAC) evaluation of the independence of the complete modal vector set, mean phase correlation (MPC) of each vector or any other method available. Naturally, since a significant number of pole weighted vectors are used in a cluster to identify the final modal parameters, traditional statistics involving mean and standard deviation are now available. 1.5 Summary and Future Work With the advent of more computationally powerful computers and sufficient memory, it has become practical to evaluate sets of solutions involving hundreds or thousands of modal parameter estimates and to extract the common information from those sets. If multiple modal parameter estimation algorithms can be combined into a single autonomous procedure, the statistics related to the common modal parameter estimation become even more meaningful. In most experimental cases studied so far, autonomous procedures give very acceptable results, in some cases superior results, in a fraction of the time required for an experienced user to get the same result. Future work will involve evaluating alternate numerical methods for combining algorithms into a single consistency diagram (equation normalization, order of the pole weighted vector, etc.) and as well as modal vector solution methods for identifying the best causal results (Do we get a normal mode when we expect a normal mode?). Numerical solution methods that identify both real-valued modal vectors (normal modes) and complex-valued modal vectors, when appropriate, would be truly autonomous. However, it is important to reiterate that the use of these autonomous procedures or wizard tools by users with limited experience is probably not yet appropriate. Such tools are most appropriately used by users with the experience to accurately judge the quality of the parameter solutions identified.

1 Integrating Multiple Algorithms in Autonomous Modal Parameter Estimation 9 Acknowledgements The author would like to acknowledge the collaboration and assistance from the graduate students and faculty of the Structural Dynamics Research Lab at the University of Cincinnati. In particular, the discussions and collaborations with Dr. David L. Brown have been instrumental in the progress made to this point. References 1. Hawkins FJ (1965) An automatic resonance testing technique for exciting normal modes of vibration of complex structures. In: Symposium IUTAM, “Progres Recents de la Mecanique des Vibrations Lineaires”, pp. 37–41 2. Hawkins FJ (1967) GRAMPA—An automatic technique for exciting the principal modes of vibration of complex structures. Royal Aircraft Establishment, RAE-TR-67-211 3. Taylor GA, Gaukroger DR, Skingle CW (1967) MAMA—A semi-automatic technique for exciting the principal modes of vibration of complex structures. Aeronautical Research Council, ARC-R/M-3590, 20pp 4. Allemang RJ, Brown DL, Phillips AW (2010) Survey of modal techniques applicable to autonomous/semi-autonomous parameter identification. In: Proceedings of international conference on noise and vibration engineering (ISMA), Katholieke Universiteit Leuven, Belgium, 42pp 5. Phillips AW, Allemang RJ, Brown DL (2011) Autonomous modal parameter estimation: methodology. In: Proceedings of international modal analysis conference (IMAC), 22pp 6. Allemang RJ, Phillips AW, Brown DL (2011) Autonomous modal parameter estimation: statistical considerations. In: Proceedings of international modal analysis conference (IMAC), 17pp 7. Brown DL, Allemang RJ, Phillips AW (2011) Autonomous modal parameter estimation: application examples. In: Proceedings of international modal analysis conference (IMAC), 26pp 8. Allemang RJ, Phillips AW (2004) The unified matrix polynomial approach to understanding modal parameter estimation: an update. In: Proceedings of international conference on noise and vibration engineering (ISMA)

Chapter2 Effects of Magneto-Mechanical Coupling on Structural Modal Parameters M. Kirschneck, D.J. Rixen, Henk Polinder, and Ron van Ostayen Abstract Structures that are exposed to a magnetic field experience magnetic forces. As these forces are geometry dependent they vary with the displacement of the structure that can result in an additional stiffness. Furthermore eddy currents induced by the movement of the structure can lead to an increased dissipation resulting in a higher damping value for the mechanical part of the system. This paper introduces calculation techniques for predicting these effects and validates them with measurements done on a simple set up in the lab. Keywords Modal parameter identification • Magneto-mechanical coupling • Monolithic eigenvalue problem 2.1 Introduction All ferro-magnetic objects, that are exposed to a magnetic field, experiences local forces. For an object at rest in a magnetic field these local forces cancel each other out and the net force on the object is zero. But when the magnetic field is such that the local forces do not balance each other out the object experiences a net force. In such a case the magnetic force has an effect on the mechanics of the system. At the same time the change of geometry due to movement will affect the magnetic field. These kind of systems are called two way magneto-mechanically coupled systems. In such as system the dynamical behavior of can be altered compared to its behavior without that coupling. This also has an impact on the modal parameters that the system displays under no coupling conditions. Certain configurations and geometries contribute to the impact of the effect of the magneto-mechanical coupling. In this paper such a system is introduced and it is shown how the change on modal parameters can be simulated and predicted. There has been extensive research on magneto-mechanical systems. In fact many transducer that transforms electric energy to mechanic energy or the other way around, i.e. electric machines, are magneto-mechanical coupled systems. Therefore the research on magneto-mechanical coupled systems began by the discovery of forces due to electric currents and their mentioning by Maxwell [7]. In light weight structures the opposing aims of making a structure as stiff as possible and as light as possible is commonly found. For these kind of structures that are exposed to magnetic fields, the exact knowledge of the dynamics of the structure might be crucial. The knowledge might allow to reduce the weight of the structure further. The same is true for electric machines that operate in places where weight reduction is essential. The rotors and stators of these machines are exposed to magnetic fields while being required to be as stiff and as light as possible. Applications can be found in electric cars or large off-shore direct-drive wind turbines. M. Kirschneck ( ) • R. van Ostayen Faculty of Mechanical, Maritime and Materials Engineering, Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands e-mail: m.kirschneck@tudelft.nl D.J. Rixen Faculty of Mechanical Engineering, Technical University of Munich, Boltzmannstr. 15, 85748 München, Germany H. Polinder Faculty of Electrical Engineering Mathematics and Computer Science, Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands R. Allemang (ed.), Topics in Modal Analysis II, Volume 8: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-04774-4__2, © The Society for Experimental Mechanics, Inc. 2014 11

12 M. Kirschneck et al. Recent research on dynamics of magneto-mechanical coupled system concentrated on one way coupled formulations [4, 8, 9]. The reason for this is that 3D magnetic calculations are expensive and are avoided unless absolutely necessary. Research on 3D two way coupled problems has been done but not applied to modal analysis [2]. 2.2 The Test Setup The test set up consists of a stator yoke, two permanent magnets and a flexible beam. Figure 2.1 shows a photo of the test set up and a 3D schematic of it. The coil that can be seen in the picture was not used for the experiments. The stator yoke is fixed to the table by clamps. The flexible beam is fixed to a table that can be moved. This construction allows to move the front part of the beam in and out of the air gap of the stator yoke. The two permanent magnets are located in the air gap and create the magnetic field that passively interacts with the structural dynamics (Table 2.1). Because neither the stator yoke nor the beam are slotted eddy currents are possible in the system and heat dissipation can occur. 2.2.1 Emerging Effects The beam is constructed in such a way that the first bending frequency in one direction is much lower than in the other directions. The bending mode shown in Fig. 2.1b will decrease the air gap length on one side of the beam while it is increased on the other side of the beam. This will change the magnetic field in the air gap. Due to fringe effects the magnetic flux density will rise on the side where the air gap length is reduced and diminish where the air gap lengthened. The resulting magnetic force that acts on the beam and pulls the beam in both air gaps towards the yoke will also change. Because the force does not depend linearly on the air gap length but is proportional to 1=l, where l is the air gap length, the forces will no longer even each other out and the beam will see a force pulling it in the same direction as the displacement. From a dynamical point of view this can be seen as an additional negative stiffness that is introduced into the system when the beam oscillates. As a result the oscillation frequency of the first bending mode will decrease. Additionally the time changing magnetic field will induce eddy currents in the stator yoke counter acting the change of the magnetic field. The result is that the peak of the magnetic field has a short delay compared to the peak of the displacement. permanent magnet beam yoke 1st Bending Mode Fig. 2.1 The test rig used for measurements Table 2.1 Specification of permanent magnets as documented by the suppliers (if documented) Property Value Height 2 mm Length 20 mm Width 10 mm Remanence flux density 1.32–1.37 T Coercity 860–995 kA m Relative permeability 1.056–1.26 Conductivity 5882–9090.9 S m

2 Effects of Magneto-Mechanical Coupling on Structural Modal Parameters 13 The force acting on the beam is therefore smaller while it moves away from the equilibrium position than when it moves towards it. This slows the oscillation down resulting in an increased damping. However the force is not linearly dependent on the velocity of the beam. This distorts the oscillation behavior of the beam slightly. 2.2.2 Mathematical Description of the System The system consists of two domains: the mechanical structural dynamics of the beam and the magnetic field. It is therefore necessary to use a coupled model of these two physics to describe it completely [6]. The mechanical system can be described as a second order system Rui C @ ij @xj Cfi;ext D0 (2.1) where denotes the stress tensor and fext the external force. The magnetic field can be described by the magnetic vector potential A r 1 r AD @A @t C 1 r Br (2.2) where represents the permeability of the material, Br the remanence flux density of the magnets and the conductivity of the material. Both parts of the system can store energy. Assuming a conservative system the energy between the mechanical system and the magnetic domain can be exchanged in both directions. The total energy in the system can therefore be calculated by W DWmech CWmag DWkin CWpot CWmag This can be seen as a potential energy for small displacements (the magnetic potential is not defined for all points in the domain due to singularities at corners. However as long as the integration path does not encircle such a singularity the energy is conservative.) As stated in [5] the change of the magnetic field energy can be described by dWmag Did Cfmagdu (2.3) where is the flux linkage, i the currents in any eventual coils, fmag the magnetic force and u the displacement. In this system however we can ignore the first term on the right hand side as there are no coils present. Extending this kind of analysis to the whole system it can be concluded that the only ways of energy entering or leaving the system is by means of external forces fext , coils, mechanical friction and ohmic losses. dW Dd PuC ieddy Cicoil d Cfext du (2.4) the above mentioned energy exchange by forcefmag dubecomes in this case an internal energy conversion from the magnetic domain to the mechanical domain and vis versa. It can be seen from (2.3) the magnetic force can be calculated using the principle of virtual work fmag D @Wmag @u (2.5) 2.2.3 Parameter Identification In order to determine the magnet properties of the steal used for beam and yoke impedance measurements were conducted. The permeability of metals depends on the manufacturing process. Therefore it is hard to predict this property beforehand. However, this property can be determined by measuring the impedance of a coil winded around the beam or the yoke. This property depends mainly on the conductivity of the material and the permeability. For structural steal that is used in this case the conductivity is roughly known. Therefore the impedance can be used to approximate the permeability. By simulating the same system in a 3D FEM program the permeability of the material can be estimated. Figure 2.2 shows the comparison between the measured values for the inductance and resistance and the calculated values for different permeabilities and conductivities of the iron material. The instrument used was lacking the capability to measure below a frequency of 20 Hz. It is presumed that due to the skin effect in the iron the inductance drops rapidly for some frequencies

14 M. Kirschneck et al. 0 100 200 300 400 500 600 700 800 900 1,000 0.2 0.4 0.6 0.8 1 Frequency [Hz] Resistance [Ω] Resistance of Coil measured Resistance of Coil simulated mr = 50 g = 10 7 S m Resistance of Coil simulated mr = 40 g = 10 7 S m Resistance of Coil simulated mr = 20 g = 8 . 106 S m Resistance of Coil simulated mr = 20 g = 10 7 S m 0 100 200 300 400 500 600 700 800 900 1,000 2 4 .10−4 Frequency [Hz] Inductance [H] Inductance of Coil measurement Inductance of Coil simulated mr = 50 g = 10 7 S m Inductance of Coil simulated mr = 40 g = 10 7 S m Inductance of Coil simulated mr = 20 g = 8 . 106 S m Inductance of Coil simulated m r = 20 g = 10 7 S m Fig. 2.2 Impedance measurements of the stator yoke below20Hz. Being able to measure this drop at very low frequencies would increase the impact of different permeability and conductivity values on the inductance leading to a more accurate determination of the material properties. At the frequencies measured it can be seen in Figs. 2.3 and 2.2 that the variation of the values has little effect on the inductance making a property identification difficult. It should also be noted that the devices used has a higher accuracy at higher frequencies. At 20 Hz the error is around 1% of the measured. Therefore it is more important to properly fit the measured data to the simulated one at higher frequencies (Table 2.2). 2.3 3D FEM Model Above it was discussed that the oscillation are not linear oscillation but slightly distorted. However, in the computer model the assumption is made that the coupled system oscillates linearly around the equilibrium position. In order to calculate the eigenparameters of this model a monolithic formulation is necessary as it has been done in [3], in [10] for electro-mechanical coupling and in [1] for piezo elements. A 3D analysis of the magnetic field is necessary because the change in magnetic field density can only be predicted accurately by taking the fringe effects around the edges of the magnets and the beam into account. A 2D model would neglect parts of the edges of the system and hence also part of the fringe effects. As a result the calculated change of magnetic force density due to the movement of the geometry that depends on the magnetic flux density would also be underestimated in a 2D model. This would lead to an underestimation of the effect of interest too. The FEM formulation for the uncoupled system can be looked up in literature [6, 15]. The derivation can be started from the energy of the system and by derivation with respect to the dofs qu the stiffness and mass matrices for the uncoupled system can be derived. The continuous functionA.x/ andu.x/ are approximated by the shape functionsN.x/ and the degrees of freedomq DŒquqA of the discrete system. uDN.x/ qu ADN.x/ qA

2 Effects of Magneto-Mechanical Coupling on Structural Modal Parameters 15 0 100 200 300 400 500 600 700 800 900 1,000 0.5 1 1.5 2 Frequency [Hz] Resistance [Ω] Resistance of Coilmeasured Resistance of Coilsimulated mr = 15 g = 10 7 S m Resistance of Coilsimulated mr = 75 g = 8 . 106 S m Resistance of Coilsimulated mr = 20 g = 10 7 S m 0 100 200 300 400 500 600 700 800 900 1,000 2 4 6 8 .10−4 Frequency [Hz] Inductance [H] Inductance of Coilmeasurement Inductance of Coilsimulated mr = 15 g = 10 7 S m Inductance of Coilsimulated mr = 75 g = 8 . 106 S m Inductance of Coilsimulated mr = 20 g = 10 7 S m Fig. 2.3 Impedance measurements of the beam Table 2.2 Chosen parameters for the models Property Value Remanence flux density 1:32 T Relative permeability 1:06 Conductivity 0:6 106 S m Relative permitivity 1 F m Properties of stator-yoke Conductivity 107 S m Relative permitivity 1F m Relative permeability 20 Properties of beam Conductivity 107 S m Relative permitivity 1 F m Relative permeability 20 Mass matrix Rayleig damping coefficient 40 Stiffness matrix Rayleig damping coefficient 6 10 6 Applying these approximations to the energies we can derive the matrices for the discrete system. Kuu D @2Wpot @q2 u D @2 @q2 u 1 2 Z uTCud (2.6) withCbeing the material stiffness matrix which is constant assuming a linear elastic material and a first order finite element. The Mass matrix can be calculated by taking the second derivative of the kinetic energy with respect to the acceleration of the displacement Mu D @2Wkin @Pq2 u D @2 @Pq2 u 1 2 Z PuT Pud (2.7)

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