River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Topics in Modal Analysis & Parameter Identification, Volume 9 Brandon J. Dilworth Timothy Marinone Michael Mains Proceedings of the 41st IMAC, A Conference and Exposition on Structural Dynamics 2023 River Publishers
Conference Proceedings of the Society for Experimental Mechanics Serie s Series Editor Kristin B. Zimmerman Society for Experimental Mechanics, Inc., Bethel, CT, USA
The Conference Proceedings of the Society for Experimental Mechanics Series presents early findings and case studies from a wide range of fundamental and applied work across the broad range of fields that comprise Experimental Mechanics. Series volumes follow the principle tracks or focus topics featured in each of the Society’s two annual conferences: IMAC, A Conference and Exposition on Structural Dynamics, and the Society’s Annual Conference & Exposition and will address critical areas of interest to researchers and design engineers working in all areas of Structural Dynamics, Solid Mechanics and Materials Research.
River Publishers Topics in Modal Analysis & Parameter Identification, Volume 9 Proceedings of the 41st IMAC, A Conference and Exposition on Structural Dynamics 2023 Brandon J. Dilworth • Timothy Marinone • Michael Mains Editors
Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-4380-047-7 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2024 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.
Prefac e Topics in Modal Analysis and Parameter Identification represents one of ten volumes of technical papers presented at the 41st IMAC, a Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, held February 13–16, 2023. The full proceedings also include volumes on Nonlinear Structures and Systems ; Dynamic s of Civil Structures ; Model Validation and Uncertainty Quantification ; Dynamic Substructures ; Special Topics in Structural Dynamics and Experimental Techniques ; Computer Vision and Laser Vibrometry ; Dynamic Environments Testing ; Sensor s and Instrumentation and Aircraft/Aerospace Testing Techniques ; an d Data Science in Engineering . Each collection presents early findings from experimental and computational investigations on an important area within structural dynamics. Topics in Modal Analysis and Parameter Identification represents papers on enabling technologies for dynamic systems and applications of modal analysis in specific application areas. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Lexington, MA, USA Brandon J. Dilworth San Diego, CA, USA Timothy Marinone Cincinnati, OH, USA Michael Mains v
Contents 1 Automated Operational Modal Analysis on a Full-Scale Wind Turbine Tower .................................. 1 Jens Kristian Mikkelsen, Esben Orlowitz, and Peter Møller Juhl 2 Combining Nontraditional Response Variables with Acceleration Data for Experimental Modal Analysis .. 9 P. M. Vinze, R. J. Allemang, A. W. Phillips, and R. N. Coppolino 3 A Somewhat Comprehensive Critique of Experimental Modal Analysis ......................................... 2 1 Robert N. Coppolino 4 OMA of a High-Rise TV Tower Using the Novel Poly-reference Complex Frequency Modal Identification Technique Formulated in Modal Model .............................................................. 4 1 Sandro D. R. Amador, Liga Gaile, Emmanouil Lydakis, and Rune Brincker 5 The New Poly-reference Complex Frequency Formulated in Modal Model (pCF-MM): A New Trend in Experimental Modal Analysis....................................................................................... 4 9 Sandro D. R. Amador and Rune Brincker 6 Mode Shape Identification Using Drive-by Monitoring: A Comparative Study.................................. 5 7 Kultigin Demirlioglu, Semih Gonen, and Emrah Erduran 7 Tips, Tricks, and Obscure Features for Modal Parameter Estimation............................................ 6 7 William Fladung and Kevin Napolitano 8 Modal Analysis Using a UAV-Deployable Wireless Sensor Network............................................... 7 5 Joud N. Satme, Ryan Yount, Jacob Vaught, Jason Smith, and Austin R. J. Downey 9 Vibration-Based Approach for Identifying Closely Spaced Modes in Space Frame Structures and Derivation of Member Axial Forces .............................................................................. 8 3 Mena Abdelnour and Volkmar Zabel 10 A Technique for Minimizing Robot-Induced Modal Excitations for On-Orbit Servicing, Assembly, and Manufacturing Structures......................................................................................... 8 9 Cory J. Rupp 11 Design Optimization of 3D Printed Chiral Metamaterials with Simultaneous High Stiffness and High Dampin g .................................................................................................................. 9 5 Wei-Chun Lu, Othman Oudghiri-Idrissi, Hrishikesh Danawe, and Serife Tol 12 Modal Analysis of a Coilable Composite Tape Spring Boom with Parabolic Cross Section .................... 9 9 Deven Mhadgut, Sheyda Davaria, Minzhen Du, Rob Engebretson, Gustavo Gargioni, Tyler Rhodes, and Jonathan Black 13 On the Behavior of Superimposed Orthogonal Structure-Borne Traveling Waves in Two-Dimensional Finite Surfaces ........................................................................................................... 10 7 William C. Rogers and Mohammad I. Albakri vii
viii Contents 14 Comparative Assessment of Force Estimation in MIMO Tests..................................................... 11 7 Odey Yousef, Fernando Moreu, and Arup Maji 15 Online Implementation of the Local Eigenvalue Modification Procedure for High-Rate Model Assimilation 12 1 Alexander B. Vereen, Emmanuel A. Ogunniyi, Austin R. J. Downey, Jacob Dodson, Adriane G. Moura, and Jason D. Bakos 16 Modal Correlation Is Required to Reduce Uncertainty in Shock Analysis and Testing......................... 12 9 Monty Kennedy and Jason Blough 17 Modal Analysis of a Time-Variable Ropeway System: Model Reduction and Vibration Instability Detection 13 3 Hugo Bécu, Claude-Henri Lamarque, and Alireza Ture Savadkoohi 18 Investigation of Rotating Structures’ Modal Response by Using DIC............................................. 14 5 Davide Mastrodicasa, Emilio Di Lorenzo, Bart Peeters, and Patrick Guillaume 19 Increasing Multi-Axis Testing Confidence Through Finite Element and Input Control Modeling ............ 15 1 Kaelyn Fenstermacher, Sarah Johnson, Aleck Tilbrook, Peter Fickenwirth, John Schultze, and Sandra Zimmerman 20 Vibration-Based Damage Detection of a Monopile Specimen Using Output-Only Environmental Models ... 16 3 Emmanouil Lydakis, Sandro D. R. Amador, Holger Koss, and Rune Brincker 21 Analysis of Traveling Wave Properties of Mechanical Metamaterial Structures: Simulation and Experiment 16 9 Hannes Fischer and Sebastian Tatzko 22 Data Sampling Frequency Impact on Automatic Operational Modal Analysis Application on Long-Span Bridges .................................................................................................. 17 3 Anno C. Dederichs and Ole Øiseth 23 Comparison of Two Possible Dynamic Models for Gear Dynamic Analysis ...................................... 18 3 Fabio Bruzzone and Carlo Rosso 24 Influence of Gearbox Flexibilities on Dynamic Overloads .......................................................... 19 3 Fabio Bruzzone and Carlo Rosso 25 Experimental Modal Analysis and Operational Deflection Shape Analysis of a Cantilever Plate in a Wind Tunnel with Finite Element Model Verification.......................................................... 20 3 David T. Will and Weidong Zhu
Chapter 1 Automated Operational Modal Analysis on a Full-Scale Wind Turbine Tower Jens Kristian Mikkelsen, Esben Orlowitz, and Peter Møller Juhl Abstrac t This chapter is concerned with the automated extraction of modal parameters (frequency and damping estimates) from accelerometer data measured on a full-scale wind turbine tower without its nacelle and blades installed. The proposed algorithm is based on an existing Operational Modal Analysis research software employing the Stochastic Subspace Identification algorithm for manual selection and extraction of modal parameters. The automatization of the algorithm is discussed in terms of the choices made and their consequences with respect to sensitivity and robustness. The algorithm is finally tested on a large experimental dataset consisting of 10 days of signals sampled at 25 Hz from two accelerometers mounted at the top of the tower in orthogonal directions. The automated algorithm is successful in time tracking the development of the first two modes with respect to frequency and damping despite the challenge posed by the fact that due to the high degree of symmetry in the setup the frequencies of the two modes are very similar. Keyword s Operational Modal Analysis · Modal parameter estimation · Automated method · Tracking of modes · Large structures 1.1 Introduction In an experimental modal analysis, also including Operational Modal Analysis (OMA), a frequently used step for the Modal Parameter Estimation (MPE) is the utilization of the so-called stabilization diagram or consistency diagram. The stabilization diagram is a user-interactive tool used to sort out real/physical modes of the structure under test from the unavoidable computational/noise modes that will be present in most MPE methods (e.g., Stochastic Subspace Identification (SSI)). The basic idea of the stabilization diagram is to track mode estimates as a function of model order allowed in the MPE method. The modes of a real physical system will be unaffected by the model order, and they are stable. Computational/noise modes, however, will change as the model order changes, and they will be unstable. Via the stabilization diagram, an experienced user manually selects an estimate of the modes being observable to him/her, and hence this is a subjective decision which depends on the users experience and training. In addition, it is a time-consuming process which is too cumbersome for tracking modal parameters of a structure over long periods of time. Tracking the modal parameters of a structure over time could be relevant for several reasons like model validation, structural health monitoring, etc. Therefore, an automation of the MPE avoiding the manual step of inspecting the stabilization diagram is desirable. The aim of this chapter is to track the modal parameters of the two first-order bending modes of a wind turbine tower over a time period of 10 days. To accomplish this, a method for automatic OMA is proposed that removes the need for a user-interactive stabilization diagram. The method is as such independent of the MPE method employed as long as results for a stabilization diagram are produced. For the present work, the Stochastic Subspace Identification (SSI) method has been chosen [1] as implemented in Ref. [2]. The proposed method is finally tested on 10 days experimental data set from a full-scale wind turbine tower. J. K. Mikkelsen · P. M. Juhl ( ) The Faculty of Engineering, University of Southern Denmark, Odense, Denmark e-mail: pmjuhl@sdu.dk; pmjuhl@sdu.dk E. Orlowitz Siemens Gamesa Renewable Energy, Brande, Denmark e-mail: Esben.Orlowitz@siemensgamesa.com © The Society for Experimental Mechanics, Inc. 2024 B. J. Dilworth et al. (eds.), Topics in Modal Analysis & Parameter Identification, Volume 9 , Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-34942-3_1 1
2 J. K. Mikkelsen et al. 1.2 Background 1.2.1 The Tower and Acceleration Data During the erection of a wind turbine prototype in Denmark, 1 the. 116 m tower (without nacelle and rotor) was standing alone for 10 days. A sketch of the tower dimensions is presented in Fig. 1.1a. In this period, two accelerometers were mounted orthogonally at the top of the tower and measured the acceleration at a sampling frequency of 25 Hz. Figure 1.1b shows the root mean square (RMS) values of the acceleration in the two directions in 30 minutes averages. The data are presented in its raw format in Ref. [3]. 1.2.2 COMSOL Model Tower Before developing the automation algorithm, a simplified version of the wind turbine tower was set up in COMSOL Multiphysics 5.6 to obtain simulated values for the expected eigenfrequencies of the two first-order bending modes. The simplified model is based on a technical drawing and is illustrated in Fig. 1.1a. The tower is fixed at the base and free at the top, mimicking a cantilever problem. An eigenfrequency study of the tower was computed [4] and the eigenfrequencies of the two first-order bending modes, . f1 , in both directions were found as.f1,x =f1,y =0.57123 Hz [5]. These two orthogonal modes are identical in frequency in this simulation since the tower has axial symmetry. In practice, there are imperfections in the tower and other effects leading to a separation of the eigenfrequencies of the modes. The shapes of the two bending modes are identical but in orthogonal direction. Because of this symmetry in the COMSOL model, it is expected that the two bending modes found in the data will be closely spaced in the frequency domain. Fig . 1. 1 (a ) Illustrated overview of the wind turbine tower showing the longitudinal sections. (b ) Averaged RMS acceleration data in both x- and y-direction. Each point shows the RMS average of the succeeding 30 minutes 1 Danish National Test Center Østerild https://wind.dtu.dk/Facilities/oesterild.
1 Automated Operational Modal Analysis on a Full-Scale Wind Turbine Tower 3 1.2.3 Manual OMA algorithm The input to the SSI algorithm employed [2] is a block of continuously sampled data from two accelerometers placed in the top of the tower in perpendicular directions. This allows for the identification of the two first-order modes of the tower only. Furthermore, the SSI algorithm requires user-set parameters such as the number of time lags and the maximum allowed model order, see, e.g., Ref. [6]. The SSI algorithm can result in a stabilization diagram. The stabilization diagram shows a graphical representation of the estimated modes by indicating the stable or unstable poles as a function of modal frequency and model order. Figure 1.2 shows a stabilization diagram of 30 minutes worth of data started at 40 hours into the time series. The range of the stabilization diagram has been truncated between.0.55H z an d.0.65 Hz to better visualize the two closely spaced modes. In an earlier work [3], some damping estimates were found based on manually picked stable poles around the two first-order modes. One of the primary issues in picking poles manually is the consistency from the user as well as the time consumption. The supplied data set is about 240 hours long, and picking a data block length of 30 minutes for analysis purposes effectively results in 480 different frequency stabilization diagrams to manually sort and store. The two frequencies that can be extracted from the green stable poles in the stabilization diagram are .f2 =0.59614 Hz and.f1 =0.59519Hz , which are quite close to the eigenfrequencies found in COMSOL. Furthermore, the two modes are very closely spaced as seen in Fig. 1.2, where the difference in the two chosen frequencies is .Δfmodes =0.00095Hz. (1.1 ) Thus, the frequency resolution needs to be finer than this resolution for the algorithm to properly discern the two modes. One of the main purposes of this work is to automate this process, so a person does not have to manually interact with the graphical interface of the stabilization diagram 480 times, as mentioned. Instead of picking the poles manually and subjectively by the visual aid of the stabilization diagram, this work aims at automating the procedure by utilizing the Modal Assurance Criterion (MAC) to pick the optimal pole and mode shape for each 30-minute interval of data. The MAC is used to quantitatively compare modal vectors of mode shapes. It essentially compares the likeliness between modal vectors by calculating the normalized dot product of two complex modal vectors [7]. The MAC value evaluates between 0 and 1 wherein an indication of likeliness between modal vectors is given with 0 being a low likeliness and 1 being a high likeliness. Fig . 1. 2 Stabilization diagram of 30 minutes of data starting 40 hours into the acceleration data. The underlying blue line is a Power Spectral Density (PSD) estimate to visually identify the physical poles, the green points are the stable poles, the red points are all unstable poles, the blue point is a stable frequency, and the pink points are stable frequency and damping estimate
4 J. K. Mikkelsen et al. 1.3 Analysis 1.3.1 Development of the Automated OMA As mentioned by the input for the SSI algorithm, the Automated OMA (AOMA) algorithm uses similar inputs but a few additions for automation, herein an optional frequency range where in which the first mode is expected to be found and lastly an Auto MAC (AMAC) threshold for finding the most consistent model order. The output parameters of the method are two poles and two modal vectors for the first two modes, respectively. This algorithm is shown in the flowchart in Fig. 1.3. The flowchart shows when certain inputs are used throughout the algorithm and what processes are performed. 1.3.2 Algorithm Review and Choice of Thresholds The list of poles and mode shapes from the SSI algorithm is extracted, circumventing the stabilization diagram graphical user interface, while the model order for each pole and the corresponding mode shapes are kept for later use. The complete list of poles is then converted to vectors of natural frequencies. Next, the two modes need to be isolated in a histogram. Since the first modes are already expected inside the frequency rang e o f .0.55 −0.6 Hz, a full histogram will not be created, only a truncated version. The width of histogram bins is determined by the frequency resolution which needs to be better than the difference between the two modes shown in Eq. (1.1) and is based on the frequency resolution which is dependent on the length of the signal as .Δfres = 1 T30minutes = 0.00056Hz. (1.2 ) The first modes of the wind turbine tower will most often occur around.0.59 Hz in practice [3], so the algorithm will pick the two maximum bins inside this area. This is seen in Fig. 1.4. As in the flowchart, if the difference between the two bin counts Fig . 1. 3 Flowchart of the developed automation algorithm that illustrates picking a set of poles and modal vectors for a 30-minute long time series
1 Automated Operational Modal Analysis on a Full-Scale Wind Turbine Tower 5 Fig . 1. 4 Histogram of the frequencies inside the area of interest. Time series start point is 40 hours into the data and the time series length is 30 minute s Fig . 1. 5 Model order of each extracted natural frequency of the two closely spaced modes. Here, the time instance of 40 hours into the signal is still used is larger than 50 in the case of this algorithm, the current time series is skipped as two modes cannot be discerned inside the frequency resolution. If two suspected physical modes can be discerned, the two bins of natural frequencies are isolated. Optimally, there will now be two bins from the histogram, each with 100 natural frequencies of similar values because of the maximum chosen model order of 100, but in practice because of noise and frequency resolution there will be fewer poles left. The optimal two poles and two modal vectors have to be chosen, a set from the bin with the lowest frequency and a set from the bin with the highest frequency. Each model order corresponding to a pole is plotted against the natural frequency to visualize the two closely spaced modes in frequency. This is shown in Fig. 1.5. To find the optimal pole and modal vector, an AMAC calculation is performed. In Fig. 1.5, each model order indicates a different pole and modal vector, and ideally all the blue points are the same mode shape and all the orange points are the same mode shape. Figure 1.6a shows each modal vector of the blue points in Fig. 1.5 compared to each other in the AMAC. This results in a contour plot where the color bar indicates the MAC value of each modal vector compared with the other modal vectors. The diagonal is unity since this is when a modal vector is compared to itself. To find the optimal modal vector—which determines the choice of the set of extracted poles and modal vectors—an acceptance threshold for the MAC value must be set where the number of model order’s corresponding modal vectors in Fig. 1.6a that fulfill the threshold is summed for each model order. The acceptance threshold is set at .MACthreshold ≥0.9. (1.3 ) The model order with the most modal vectors that comply with the acceptance threshold is the most consistent model order and determines the set which is picked by the algorithm. The consistency plot of the model orders in Fig. 1.6a is shown in Fig. 1.6b. For this particular example, the most consistent model order is found to be 26, which is consistent with 24 other modes at the threshold set in Eq. (1.3).
6 J. K. Mikkelsen et al. Fig . 1. 6 (a ) Contour plot where each modal vector from the blue points in Fig. 1.5 is compared to itself and all other modal vectors. The contour indicates the value of each MAC calculation of one model order’s corresponding modal vector with the next. The diagonal is thus unity as expected. (b ) Plot of consistent model orders. The MAC value threshold is set at . 0.9, the value where a model order is accepted. For each model order, the number of accepted MAC checks between the modal vectors above the threshold is summed. The figure shows how consistently the model orders adhere to the MAC threshold. Here, the time series of 40 hours into the signal is still used The MAC contour plot and the consistency check are to be performed for the orange points in Fig. 1.5 as well to get the second set of poles and modal vectors. Thus, the algorithm can continue to the next time increment where all the routines start over to pick the optimal poles and mode shapes. 1.3.3 Test of Thresholds The developed algorithm has several criteria and thresholds throughout that discards or includes certain data based on the specific case of the wind turbine tower and the chosen OMA method. These choices are fitted to this case and should be refitted to other use cases where different frequency ranges, frequency resolutions, or MAC thresholds could be optimal instead. The most important of these choices and their effect on the algorithm will now be discussed. In the case of this wind turbine tower, a threshold for the histogram of all extracted frequencies is chosen as to only include frequencies inside the area of .0.55H z to. 0.6 Hz as this is where the first modes are expected to be found in this specific data set. This can be changed based on other use cases. The frequency resolution is proportional to the reciprocal of the time series data length and is fixed in this case as the data length is constantly 30 minutes long. If the frequency resolution is worse than the difference in Eq. (1.1), the two modes cannot be separated which can lead to faulty damping estimates. However, the time series data length cannot be too long either since the SSI method assumes time-invariant data, and over longer time spans the recorded signal can become time dependent, a periodicity also observed in Fig. 1.1b. Thus, choosing between time dependency in the signal or greater frequency resolution is a fine balance. In Fig. 1.4, two closely spaced modes in frequency were observed. If a large difference between the two most prominent bins is observed, the algorithm concludes that it cannot identify two excited modes and thus skips the time series. Several factors can result in the algorithm not being able to identify two excited modes: the first factor is the frequency resolution since a better resolution can discern bins better and a second reason can be MATLAB’s histogram function that decides where the lines between histogram bins are drawn. Essentially, the algorithm cannot directly determine if a mode is excited in the data, but it can determine if two modes are insufficiently consistently present. The result of not being able the discern between the two modes can finally result in a wrong estimation of the damping if some frequencies are placed incorrectly in the histogram bins. The choice of a bin count difference of 50 being the threshold is based on the amount of extracted poles as well as the model order since a physical pole would ideally have 100 poles inside the bin if the maximum model order is chosen to be 100 as well. If the time series is shorter or the model order lower, one would have to change this bin difference threshold. The AMAC acceptance threshold for picking the modal vector most frequently like the other modal vectors is set at the threshold value in Eq. (1.3) but can be changed based on the specific use case. In Ref. [5], the value was not picked at random, and a sweep of values were analyzed to showcase the optimal value by changing the MAC threshold value from. 0.5 to 1. It was concluded that choosing a looser threshold value in the MAC essentially allows for the modal vectors to differ more from each other and that choosing a stricter threshold allows little change in the modal vectors.
1 Automated Operational Modal Analysis on a Full-Scale Wind Turbine Tower 7 Fig . 1. 7 Extracted natural frequencies of the first two modes as a function of time indicated by date. Each point indicates a 30-minute time series. Each vertical date line indicates midnight Fig . 1. 8 CMAC values of modal vectors compared to succeeding modal vectors for the low frequency mode (blue dots), their median value (orange dashed line), CMAC values of modal vectors compared to succeeding modal vectors for the high frequency mode (orange dots), and lastly, their median value (purple dashed line) 1.4 Results of the Algorithm The acceleration data seen in Fig. 1.1b in its raw format are used in the AOMA algorithm with the purpose of monitoring the change in the natural frequencies of the two closely spaced modes over time. The natural frequencies extracted for each 30 minutes of data over the 10 days of total data are shown as a function of time in Fig. 1.7. Here, a slight periodicity with respect to the time of day is observed, essentially showing an increase in the natural frequencies during midday and a decrease during the night. This is analyzed further in Ref. [5] but is not further relevant for this chapter. One of the main challenges for this method is to circumvent the risk of two closely spaced modes overlapping in frequency or swapping places, so identification of individual modes becomes difficult. To potentially identify this, a Cross MAC (CMAC) check between the modal vector of the extracted mode shape from the current time instance is compared to the modal vector of the extracted mode shape from the succeeding time instance. This is performed for both modes and is plotted in Fig. 1.8 as a function of time with dates indicating time on the horizontal axis. The median value of the CMAC values is plotted as two overlaying lines on the plot indicating a strong congruence between the modal vectors over time, meaning a potential swap in mode shapes is unlikely, but still inconclusive. 1.5 Conclusion The main purpose of the chapter was to develop an automation of the modal parameter estimation by operational modal analysis of a wind turbine tower. The automation is needed to avoid the time-consuming and subjective manual picking of modes from a stabilization diagram. Thus, the developed algorithm overall extracts two sets of poles and mode shapes for each time series serving as input to the program. The length of the time series and the sampling frequency can be altered by the user. The main use of the algorithm is monitoring of closely spaced modes over a lengthy period of time.
8 J. K. Mikkelsen et al. Thus, the use of the algorithm extends beyond an individual 30-minute measurement period. The primary way the algorithm has been developed is by analyzing the closely spaced natural frequencies as well as the similarity between modal vectors to determine the validity of two closely spaced modes being two different mode shapes. As a result of developing the algorithm, the measured acceleration data has been further analyzed after extraction, and the natural frequencies were plotted against the recorded time of day. The change in the mode shapes has also been monitored over time where a cross-modal assurance criterion check has been performed between two modes with 30-minute increments between them with the purpose of checking if the modes swap at any instances in time. This analysis was deemed inconclusive, but the median of the cross-modal assurance criterion values indicate good conformity between the extracted mode shapes over time. References 1. van Overschee, P., de Moor, B.: Subspace Identification For Linear Systems. Kluwer Academic Publishers, Dordrecht (1996) 2. Orlowitz, E.: Damping Estimation in Operational Modal Analysis. Ph.D. Thesis, 2015 3. Mikkelsen, J.K.: Damping estimation of a wind turbine. Technical report, University of Southern Denmark, Odense, 2022 4. COMSOL: Multiphysics cyclopedia: structural mechanics—eigenfrequency analysis (2018). https://www.comsol.com/multiphysics/ eigenfrequency-analysis?parent=structural-mechanics-0182-212. Accessed 6 April 2022 5. Mikkelsen, J.K.: Comparison of modal parameters of a wind turbine by operational modal analysis and a finite element model. Master’s Thesis, Odense, 2022 6. Orlowitz, E., Brandt, A.: Influence of noise in correlation function estimates for operational modal analysis. Topics Modal Anal. Testing 9 , 55–64 (2019) 7. Brandt, A.: Noise and Vibration Analysis: Signal Analysis and Experimental Procedures, 1st edn. Wiley, New York (2011)
Chapter 2 Combining Nontraditional Response Variables with Acceleration Data for Experimental Modal Analysis P. M. Vinze, R. J. Allemang, A. W. Phillips, and R. N. Coppolino Abstrac t Accelerometer data is the most commonly used data for experimental modal analysis of structures. Together with measuring applied force, it provides the basis for FRF estimation and subsequent modal parameter estimation and validation. As discussed in the paper by Dr. Coppolino (Experimental modal analysis using non-traditional response variables. In: IMAC Proceedings, 2021), there are situations where test analysis cross orthogonality is difficult to determine on inaccessible key regions of a test article. In that chapter, it is contended that it is in theory possible to augment data from accelerometers with data from other sensor sources at these key regions that have a proportionality to acceleration or displacement. This is important as strain and pressure have been shown to be useful measurements for modal analysis (Zienkiewicz et al., The finite element method: its basis and fundamentals, 6th edn. Butterworth-Heinemann, Oxford, p 563–584, 2005; Kranjc et al., J Sound Vib 332:6968, 2013; Kranjc et al., J Vib Control 22(2):371–381, 2016; Dos Santos et al., Strain-based experimental modal analysis: new concepts and practical aspects. In: Proceedings of ISMA. IEEE, Piscataway, p 2263– 2277, 2016; Dos Santos et al., An overview of experimental strain-based modal analysis methods. In: Proceedings of the international conference on noise and vibration engineering (ISMA), Leuven, p 2453–2468, 2014). But they have not been used in augmentation with acceleration. Two specific examples discussed are fluid pressure and strain. Experimentally, this presents several problems. For example, in the most simple structures it is expected to have maximum acceleration at locations of 0 strain and vice versa. This makes it difficult to relate the modal information contained in acceleration variable to the strain variable at the location of maximum acceleration. Given that the FRF information will have to be uniform in units, this is another cause of concern when combining pressure, strain, and acceleration. Use of strain, pressure, and acceleration data all together for modal analysis purposes would reduce the need to place accelerometers in locations that are difficult to access. This chapter aims to present experimental results of strain and pressure FRF-based modal analysis on a rectangular steel plate and attempts to propose ways to combine these variables in the modal parameter estimation process. Keyword s Experimental modal analysis · Strain · Pressure · Augmentation · Non traditional variables 2.1 Introduction In this research, two experiments were conducted on a rectangular steel place. The first experiment was an impact test with accelerometer and strain gauges as response measurements. The second experiment was another impact test with accelerometer, strain gauges, and microphones as the response measurements. Different ways of augmenting partial accelerometer data/results from partial accelerometer data were tried and MAC [2] values with full experimental data were checked. FRF synthesis from the modal vectors of the augmented data was also done and compared with measured acceleration FRF. Strain and pressure have been shown to be useful measurements for modal analysis [3–6] and [7]. But they have not been used in augmentation with acceleration. The aim of the MAC values and FRF synthesis comparisons was to show that augmenting accelerometer, pressure and strain data can give results that are close to results from accelerometer data. P. M. Vinze ( ) · R. J. Allemang · A. W. Phillips Department of Mechanical Engineering, College of Engineering and Sciences, University of Cincinnati, Cincinnati, OH, USA e-mail: vinzepm@mail.uc.edu R. N. Coppolino Measurement Analysis Corporation, Torrance, CA, USA © The Society for Experimental Mechanics, Inc. 2024 B. J. Dilworth et al. (eds.), Topics in Modal Analysis & Parameter Identification, Volume 9, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-031-34942-3_2 9
10 P. M. Vinze et al. In the paper by Coppolino [1], the case that was considered was that of a rod that is modeled as a MDOF mass spring system. One of the major changes from that model to a system like a rectangular steel plate is that the strain (for all the modes in the frequency range of interest) is no longer due to extension and compression but due to bending. Another difference that is important to point out is that strain cannot be modeled as the ratio of difference of displacement at two neighboring points of response measurements and the length. The reason this does not work is that the plate is a continuous system, and the experimental strain measured will be the local measurement at the given point. In this way if measurements are made on n points, there will be n strain measurements, whereas in the theoretical model there will be n− 1 strains. These two factors are a big deviation from the model discussed in the referred paper but the general idea that measurements like strained pressure should give similar modal results is what served as motivation to attempt this work 2.2 Experimental Setup The rectangular plate is rested on four rubber ball supports 6–8 inches inside of the four corners that approximate a free-free condition. The setup for the second experiment is shown in Fig. 2.1. For the first experiment, the setup was without the microphones and microphone stands. The strain gauges are placed next to the accelerometers and aluminum cubes were placed very close to provide a surface for X direction (along the long edge) impact (along the long edge on the plate). Figure 2.2 shows an image of the sensor placement. The numbers and circles etched on the plate are 160 equally spaced impact locations, a subset (40) of which have been used. Fig . 2. 1 Test setup Fig . 2. 2 Sensors on a plate
2 Combining Nontraditional Response Variables with Acceleration Data for Experimental Modal Analysis 11 2.3 Analysis A subset of the data gathered was used to get modal parameters. The idea was to work with several combinations of subsets to simulate a situation where different combinations of sensors on the structure could be worked with. This sieving of the full 9 × 40 FRF matrix was done in two ways: 1. Sieving by references only. This could be done before modal parameter estimation to create a new 3 × 40 FRF matrix that would contain one or two accelerometer reference points and the remaining reference points could be taken from the strain or pressure data for that reference points. When including pressure, it was important to scale the pressure in some way to a similar level of strain and acceleration. 2. Sieving by response as well as references. This means a subset of response points were chosen for acceleration references and another subset of response points for strain and in one case another subset of response points for pressure references. The choices are always made such that none of the 3 references or 40 response locations are completely missed out. Also some (3–5) response points are kept in common for all 3 reference locations. A composite modal vector is created by scaling the strain and pressure modal vectors based on the common response locations on the modal vectors and combining the scaled response at the locations that are not common between all three sensors. These two methods were followed in different combination of selections and checked with a full set of accelerometer FRF-based results. 2.4 Sieving by References Only The MACs were evaluated for strain and acoustic pressure-based modal vectors against the accelerometer-based modal vectors. The two MACs are shown in Figs. 2.3 and 2.4. This confirmed that pressure and strain data had the same modal information as acceleration data. Table 2.1 shows the modal frequencies and damping results for strain gauge and accelerometer-based modal analysis results and the percent difference between them. As can be seen the results agree expect for one damping result. Fig . 2. 3 Accelerometer-based versus strain-based modal vectors MACs
12 P. M. Vinze et al. Fig . 2. 4 Accelerometer-based versus microphone-based modal vectors MACs Table 2.1 Accelerometer-based versus strain-based modal frequencies and damping results S.No. Strain frequency (Hz) Strain damping (%) Accelerometer frequency (Hz) Accelerometer damping (%) % diff frequency % diff damping 1 40.2441 0.2488 40.255 0.2563 0.027077382 2.926258291 2 −40.2441 0.2488 −40.255 0.2563 0.027077382 2.926258291 3 43.0117 0.1258 43.0198 0.1401 0.018828539 10.206995 4 −43.0117 0.1258 −43.0198 0.1401 0.018828539 10.206995 5 92.931 0.0983 92.931 0.098 0 0.306122449 6 −92.931 0.0983 −92.931 0.098 0 0.306122449 7 103.679 0.029 103.6783 0.0284 0.000675165 2.112676056 8 −103.679 0.029 −103.6783 0.0284 0.000675165 2.112676056 9 115.5573 0.0274 115.5576 0.0275 0.000259611 0.363636364 10 −115.5573 0.0274 −115.5576 0.0275 0.000259611 0.363636364 11 138.3034 0.0331 138.2977 0.031 0.004121544 6.774193548 12 −138.3034 0.0331 −138.2977 0.031 0.004121544 6.774193548 13 172.9429 0.0206 172.9427 0.0199 0.000115645 3.51758794 14 −172.9429 0.0206 −172.9427 0.0199 0.000115645 3.51758794 15 200.731 0.0143 200.7539 0.0114 0.011407001 25.43859649 16 −200.731 0.0143 −200.7539 0.0114 0.011407001 25.43859649 17 244.1445 0.0264 244.1497 0.0246 0.002129841 7.317073171 18 −244.1445 0.0264 −244.1497 0.0246 0.002129841 7.317073171 19 277.0993 0.0257 277.0931 0.0265 0.002237515 3.018867925 20 −277.0933 0.0257 −277.0931 0.0265 7.21779E-05 3.018867925 21 297.2717 0.0353 297.2719 0.0351 6.72785E-05 0.56980057 22 −297.2717 0.0353 −297.2719 0.0351 6.72785E-05 0.56980057 23 300.9256 0.0148 300.9265 0.016 0.000299076 7.5 24 −300.9256 0.0148 −300.9265 0.016 0.000299076 7.5 25 308.9535 0.0356 308.9542 0.0351 0.000226571 1.424501425 26 −308.9535 0.0356 −308.9542 0.0351 0.000226571 1.424501425
2 Combining Nontraditional Response Variables with Acceleration Data for Experimental Modal Analysis 13 Fig . 2. 5 MAC between 2 accelerometer and 1 strain versus 3 accelerometer 2.5 Accelerometers and One Strain Gauge Modal vectors for data with two acceleration and one strain reference were compared with three acceleration modal vectors. The MAC comes out completely diagonal indicating that the modes are similar. MAC is shown in Fig. 2.5. 2.6 Accelerometer and Two Strain Gauges Modal vectors for data with one acceleration and two strain references were compared with three acceleration modal vectors. The MAC comes out completely diagonal indicating that the modes are similar. MAC is shown in Figs. 2.6 and 2.7 for the two different acceleration references. Another comparison that was done was to compare the density of pole estimates. There were differences in the density of the scatter plots, but it was different for different modes. There was not one set of data that gave better clusters of damping ratio values for all modes. This suggested that there was not any improvement in damping results from replacing accelerometers with strain gauges at some measurement locations. 2.7 Accelerometer and Microphones To combine accelerometer data with microphone data, it was important to consider the fact that there was a large difference in their magnitudes. The microphone data was observed to be around 6 orders of magnitude larger. An equation was set up such that modified pressure FRF data was expressed as . Hpm = R3∗ Hp+ R2/ω 2 + R1 where Hpm is the modified pressure data, Hp is measured pressure FRF, and R1, R2 and R3 are the terms determined in the least square sense by comparing Hpm to the displacement FRF on the same location. The average of all R3 values for all
14 P. M. Vinze et al. Fig . 2. 6 MAC between 1 accelerometer (location14) and 2 strain versus 3 accelerometer Fig . 2. 7 MAC between 1 accelerometer (location99) and 2 strain versus 3 accelerometer FRFs is calculated and used as the multiplier. The combining of pressure and displacement FRFs is only possible with the modified pressure because of the significant difference of scale of pressure and displacement data. Figures 2.8 and 2.9 show that the modal vectors obtained from combining pressure with acceleration measurements result in diagonal MAC values.
2 Combining Nontraditional Response Variables with Acceleration Data for Experimental Modal Analysis 15 Fig . 2. 8 MAC for 2 accelerometer 1 pressure modes versus 3 accelerometer modes Fig . 2. 9 MAC for 1 accelerometer 2 pressure modes versus 3 accelerometer modes
16 P. M. Vinze et al. 2.8 Sieving by References and Responses 2.8.1 Acceleration and Strain Two different modal parameter estimation processes were carried out on two different sets of sieved FRF data. Seventeen response locations with 3 acceleration references were selected and modal vectors were extracted for first 13 modes. Twentyeight response locations with 3 strain references were selected and modal vectors were extracted for first 13 modes. The 28 locations selected were such that there were exactly 5 locations in common with the 13 points selected for accelerometer data. Three out of these 5 locations on the two modal vectors (strain based and accelerometer based) were used to scale the strain modal vector upto the accelerometer modal vector. This was done by taking the point-by-point ratio of each of the 3 locations for the 13 modal vectors and averaging this value across the 3 ratio values. This yielded one multiplier for each strain based modal vector. Except for the 2 entries (corresponding to the 5th mode) in column 2, the ratio values are consistent across the three columns. The MAC between the composite vectors and modal vectors found from 40 × 3 accelerometer data also comes diagonal showing that the sets of vectors match. The crossMAC is shown in Fig. 2.10. Residuals were calculated for these modal vectors based on the Modal A values obtained from X-Modal and synthesized displacement FRFs were compared to measured FRFs. Some example comparisons are shown in Fig. 2.11 in magnitude and phase format. It can be seen that the FRFs compare well. Another variation tried was when doing the acceleration MPE, and one of the references were removed. The FRF data for this reference was then synthesized from the composite modal vector. The results obtained were also consistent with the measured acceleration FRF of the selected reference. 2.8.2 Acceleration, Strain, and Pressure Three different modal parameter estimation processes were carried out on three different sets of sieved FRF data. Seventeen responses with 3 acceleration references, 17 responses with 3 strain references, and 16 responses with 3 pressure references were taken such that 5 of the response locations were common. The acceleration response from these locations was used to scale the other two modal vectors to the acceleration modal vector magnitude. Similar to the data in Table 2.1, a ratio was evaluated between the common location modal vectors elements strain and acceleration and pressure and acceleration. The ratios were found to be generally consistent across 5 locations. This ratio was averaged and used to scale the strain and pressure-based modal vectors. A composite modal vector was then assembled using the three partial modal vectors. MAC was evaluated for the composite modal vector versus a full accelerometer-based modal vector. This MAC is shown in Fig. 2.12. Synthesized FRFs from these composite modal vectors were compared with measured FRFS and were observed to compare well as shown in Fig. 2.13. Synthesis of an absent reference has not been tried in this combined method yet. Fig. 2.10 MAC composite (acceleration and strain) modal vector versus normal modal vector
2 Combining Nontraditional Response Variables with Acceleration Data for Experimental Modal Analysis 17 Fig. 2.11 Experimental (solid orange) versus synthesized (dotted blue) FRFs from composite vector (acceleration and strain) 2.9 Conclusion Substituting acceleration reference with strain yielded close modal vector results as well as modal frequencies and damping. The difference in modal frequencies and damping is close to what is shown in Table 2.1 in all cases. There is variation in terms of how much of a scatter is observed in the damping estimates. But no consistent improvement due to adding strain data was observed across all modes. Substituting acceleration reference with acoustic pressure was not as simple. The pressure data had to be modified to bring it to a similar scale as the acceleration and strain data. Once that was done, the modal vector results did have a good MAC with full acceleration-based modal vectors. When data was filtered through references as well as responses, it was done post modal parameter estimation of both strain and acceleration reference subsets. The ratio of modal contributions at the common locations for strain and acceleration came
18 P. M. Vinze et al. Fig. 2.12 MAC composite (acceleration, strain and pressure) modal vector versus normal modal vector Fig. 2.13 Experimental (solid orange) versus synthesized (dotted blue) FRFs from composite vector (acceleration, strain and pressure)
2 Combining Nontraditional Response Variables with Acceleration Data for Experimental Modal Analysis 19 out very close to−1 and 1 except for the 5th mode. The composite modal vector MAC with accelerometer-based modal vector came out diagonal. FRF data synthesized based on the composite modal vector compared generally well with the measured FRF. Composite modal vector evaluated with pressure data included with acceleration and strain also had a good MAC with accelerometer-based modal vectors. FRF synthesized compared well with acceleration-based modal vector. The modal parameters obtained are the same or very close to what was obtained with the same amount of accelerometers, which brings up the question of whether replacing accelerometer data with other sensor data has yielded any improvements. It may be that due to its simplicity three accelerometers prove to be enough for the modal analysis of a rectangular plate. In that case, it might be useful to have a similar experiment of a structure that does not yield great results with three accelerometers. It will be worthwhile to also test a cylindrical container as originally discussed in the paper by Coppolino [1]. References 1. Coppolino, R.N.: Experimental modal analysis using non-traditional response variables. In: IMAC Proceedings. Springer, Cham (2021) 2. Allemang, R.J., Brown, D.L.: A correlation coefficient for modal vector analysis. In: Proceedings, International Modal Analysis Conference, pp. 110–116. Union College/Society for Experimental Mechanics/International Society for Optical Engineering Springer, Schenectady/Bethel/New York (1982) 3. Zienkiewicz, O.C., Taylor, R.L., Zhu, J.H.: The Finite Element Method: Its Basis and Fundamentals, 6th edn, pp. 563–584. ButterworthHeinemann, Oxford (2005) 4. Kranjc, T., Slavic, J., Boltezar, M.: The mass normalization of the displacement and strain mode shapes in a strain experimental modal analysis using the mass-change strategy. J. Sound Vib. 332, 6968 (2013) 5. Kranjc, T., Slavicˇ, J., Boltežar, M.: A comparison of strain and classic experimental modal analysis. J. Vib. Control. 22(2), 371–381 (2016) 6. Dos Santos, F.L.M., Peeters, B., Desmet, W., Góes, L.C.S.: Strain-based experimental modal analysis: new concepts and practical aspects. In: Proceedings of ISMA, pp. 2263–2277. IEEE, Piscataway (2016) 7. Dos Santos, F.L.M., Peeters, B., Lau, J., Desmet, W., Góes, L.C.S.: An overview of experimental strain-based modal analysis methods. In: Proceedings of the International Conference on Noise and Vibration Engineering (ISMA), pp. 2453–2468, Leuven (2014)
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