River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Topics in Modal Analysis & Testing, Volume 8 Brandon Dilworth Michael Mains Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020 River Publishers
Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics, Inc., Bethel, CT, USA
The Conference Proceedings of the Society for Experimental Mechanics Series presents early findings and case studies from a wide range of fundamental and applied work across the broad range of fields that comprise Experimental Mechanics. Series volumes follow the principle tracks or focus topics featured in each of the Society’s two annual conferences: IMAC, A Conference and Exposition on Structural Dynamics, and the Society’s Annual Conference & Exposition and will address critical areas of interest to researchers and design engineers working in all areas of Structural Dynamics, Solid Mechanics and Materials Research
River Publishers Topics in Modal Analysis & Testing, Volume 8 Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020 Brandon Dilworth • Michael Mains Editors
Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-4380-003-3 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2021 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 Topics in Modal Analysis & Testing represents one of eight volumes of technical papers presented at the 38th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, and held in Houston, Texas, February 10–13, 2020. The full proceedings also include volumes on Nonlinear Structures and Systems; Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; Dynamic Substructures; Special Topics in Structural Dynamics & Experimental Techniques; Rotating Machinery, Optical Methods & Scanning LDV Methods; and Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing. Each collection presents early findings from experimental and computational investigations on an important area within Structural Dynamics. Topics in Modal Analysis represents papers on enabling technologies for Modal Analysis measurements 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 Dilworth v
Contents 1 Modal Analysis on a Wind Turbine Blade Based on Wind Tunnel Experiments................................. 1 L. G. Trujillo-Franco, H. F. Abundis-Fong, R. Campos-Amezcua, and R. Gomez-Martinez 2 An Investigation of Vibrational Characteristics of Lap Joints Using Experimental and Analytical Methods................................................................................................................... 9 Thomas Roberts and Phillip J. Cornwell 3 Using Strain Gages as References to Calculate Free-Free Frequency Response Functions..................... 25 Kevin Napolitano and David Cloutier 4 A Principle for Obtaining Pragmatic Uncertainty Bounds on Modal Parameters............................... 31 Jonas G. Kjeld and Anders Brandt 5 On Partitioning of an SHM Problem and Parallels with Transfer Learning ..................................... 41 G. P. Tsialiamanis, D. J. Wagg, P. A. Gardner, N. Dervilis, and K. Worden 6 An Ontological Approach to Structural Health Monitoring........................................................ 51 G. P. Tsialiamanis, D. J. Wagg, I. Antoniadou, and K. Worden 7 Passive Aeroelastic Tailored Wing Modal Test Using the Fixed Base Correction Method ...................... 61 Natalie Spivey, Rachel Saltzman, Carol Wieseman, Kevin Napolitano, and Benjamin Smith 8 Nonlinear Normal Mode Estimation with Near-Resonant Steady State Inputs .................................. 85 Michael Kwarta and Matthew S. Allen 9 Automatic Modal Parameter Identification with Methods of Artificial Intelligence............................. 89 Maik Gollnick, Daniel Herfert, and Jan Heimann 10 A Single Step Modal Parameter Estimation Algorithm: Computing Residues from Numerator Matrix Coefficients of Rational Fractions ..................................................................................... 97 Nimish Pandiya, Christian Dindorf, and Wim Desmet 11 Improved Expansion Results Using Regularized Solutions......................................................... 107 Chris Beale, Ryan Schultz, and Deborah Fowler 12 Expansion of Coupled Structural-Acoustic Systems ................................................................ 131 Ryan Schultz, Dagny Beale, and Ryan Romeo 13 Expansion Methods Applied to Internal Acoustic Problems ....................................................... 141 Ryan Schultz and Dagny Joffre 14 Scaling an OMA Modal Model of a Wood Building Using OMAH and a Small Shaker ........................ 151 Osama Abdeljaber, Michael Dorn, and Anders Brandt 15 Quantitative Study on the Modal Parameters Estimated Using the PLSCF and the MITD Methods and an Automated Modal Analysis Algorithm....................................................................... 159 Silas Sverre Christensen, Stefano Manzoni, Marcello Vanali, Alfredo Cigada, and Anders Brandt vii
viii Contents 16 Empirical Models for the Health Monitoring of High-Rise Buildings: The Case of Palazzo Lombardia ..... 169 Marta Berardengo, Francescantonio Lucà, Stefano Manzoni, Marcello Vanali, and Daniele Acerbis 17 Towards Population-Based Structural Health Monitoring, Part II: Heterogeneous Populations and Structures as Graphs............................................................................................... 177 Julian Gosliga, Paul Gardner, Lawrence A. Bull, Nikolaos Dervilis, and Keith Worden 18 Developing a Correlation Criterion (SpaceMAC) for Repeated and Pseudo-repeated Modes.................. 189 Pranjal M. Vinze, Randall J. Allemang, and Allyn W. Phillips 19 Subsecond Model Updating for High-Rate Structural Health Monitoring ....................................... 201 Michael Carroll, Austin Downey, Jacob Dodson, Jonathan Hong, and James Scheppegrell 20 Phase Quadrature Backbone Curve for Nonlinear Modal Analysis of Nonconservative Systems.............. 207 Martin Volvert and Gaëtan Kerschen 21 Preliminary Results of Vibration Measurements on a Wind Turbine Test Bench................................ 211 Jesper Berntsen and Anders Brandt 22 Feasibility for Damage Identification in Offshore Wind Jacket Structures Through Numerical Modelling of Global Dynamics......................................................................................... 221 Mark Richmond, Ursula Smolka, and Athanasios Kolios 23 Use of Operational Modal Analysis to Identify Systems with Oscillatory Masses ............................... 227 Lasse Førde Thunbo, Niklas Carl Ørum-Nielsen, Tobias Friis, Sandro D. R. Amador, Evangelos Katsanos, and Rune Brincker 24 OMA-Based Modal Identification and Response Estimation of a Monopile Model Subjected to Wave Load............................................................................................................. 237 Jóhan Bech Húsgard, Frederik Alexander Hvelplund Uhre, Bruna Silva Nabuco, Renata Grabowsky, Sandro Amador, Evangelos Katsanos, Erik D. Christensen, and Rune Brincker 25 A Concept for the Estimation of Displacement Fields in Flexible Wind Turbine Structures ................... 247 Johannes Luthe, Andreas Schulze, János Zierath, Sven-Erik Rosenow and Christoph Woernle 26 Towards Population-Based Structural Health Monitoring, Part III: Graphs, Networks and Communities ........................................................................................................ 255 Julian Gosliga, Paul Gardner, Lawrence A. Bull, Nikolaos Dervilis, and Keith Worden 27 Utilization of Experimental Data in Elastic Multibody Simulation: Case Study on the Ampair 600 Turbine Blade ............................................................................................................ 269 Andreas Schulze, Johannes Luthe, János Zierath, and Christoph Woernle 28 On the Use of PVDF Sensors for Experimental Modal Analysis ................................................... 279 Tomaž Bregar, Blaž Starc, Gregor Cˇ epon and Miha Boltežar 29 Predicting Tool Wear Using Linear Response Surface Methodology and Gaussian Process Regression...... 283 Chandula T. Wickramarachchi, Timothy J. Rogers, Wayne Leahy, and Elizabeth J. Cross 30 Computer Aided Measurement Uncertainty Calculation by Modern DAQs for Raw Acceleration and Force Data in Modal Analysis..................................................................................... 287 David Kuntz, Thomas Petzsche, Martin Stierli, and William Zwolinski 31 Towards Population-Based Structural Health Monitoring, Part VII: EOV Fields – Environmental Mapping .................................................................................................................. 297 Weijiang Lin, Keith Worden, A. E. Maguire, and Elizabeth J. Cross 32 Investigation of Resistive Forces in Variable Recruitment Fluidic Artificial Muscle Bundles .................. 305 Jeong Yong Kim, Nicholas Mazzoleni, and Matthew Bryant 33 Numerical and Experimental Study on the Modal Characteristics of a Rotor Test Rig......................... 315 Verena Heuschneider, Florian Berghammer, and Manfred Hajek
Contents ix 34 A Comparison of Different Boundary Condition Correction Methods............................................ 323 Peter A. Kerrian 35 Deflection Shape Balancing: An Alternative to Modal Balancing.................................................. 337 W. Jason Morrell, B. Damiano, K. Hylton, and C. Jordan 36 Real-Time Theoretical and Experimental Dynamic Mode Shapes for Structural Analysis Using Augmented Reality ...................................................................................................... 351 Maimuna Hossain, John-Wesley Hanson, and Fernando Moreu 37 A Bottom-Up Approach to FE Model Updating of Industrial Structures......................................... 357 Daniel J. Alarcón, Fabian Keilpflug, and Peter Blaschke 38 Shaft Bending to Zero Nodal Diameter Disc Coupling Effects in Rotating Structures Due to Asymmetric Bearing Supports ...................................................................................... 379 G. Tuzzi, C. W. Schwingshackl, and J. S. Green 39 Dynamic Characterization of a Pop-Up Folding Flat Explorer Robot (PUFFER) for Planetary Exploration............................................................................................................... 383 John Bell, Laura Redmond, Kalind Carpenter, and Jean-Pierre de la Croix 40 A Framework for the Design of Rotating Multiple Tuned Mass Damper......................................... 393 Kévin Jaboviste, Emeline Sadoulet-Reboul, Olivier Sauvage, and Gaël Chevallier 41 Operational Modal Analysis of Wind Turbine Drivetrain with Focus on Damping Extraction................ 399 Pieter-Jan Daems, Cédric Peeters, Patrick Guillaume, and Jan Helsen 42 Investigation of Viscous and Friction Damping Mechanisms in a Cantilever Beam and Hanger System..... 407 Akhil Sharma, Aimee Frame, and Allyn W. Phillips 43 Detecting Nonsynchronous Heart Cells from Video – An Unsupervised Machine Learning Approach....... 415 William Anderson, Lauren Schneider, Li-Ming Richard Yeong, Kent Coombs, Pulak Nath, Jennifer Harris, David Mascareñas, and Bridget Martinez
Chapter 1 Modal Analysis on a Wind Turbine Blade Based on Wind Tunnel Experiments L. G. Trujillo-Franco, H. F. Abundis-Fong, R. Campos-Amezcua, and R. Gomez-Martinez Abstract This paper describes the evaluation of a time domain algebraic modal parameters identification methodology. This methodology is applied on a wind turbine blade. The natural frequencies and modal damping factors associated to the blade are estimated from measurements of velocities. A comparison with usual modal identification techniques is performed in order to evaluate and establish the main contributions of the proposed approach. The modal parameter identification algorithms are implemented to run (but not limited to) on a Matlab platform running in a PC using measurements obtained from a laser vibrometer and the corresponding data acquisition system. The results show the performance and parametric estimation of the proposed algebraic identification approach. Keywords Experimental modal analysis · Operational modal analysis · Wind tunnel experiments 1.1 Introduction The field of structural dynamics has a set of applications that is in a constant expansion due to the advances in mechanical design procedures and testing protocols and structural health monitoring schemes applied to engineering structures involved in the considerably diverse universe of mechatronic systems, for example the specific case of wind turbines that involve supporting structures and blades that are subjected to harmonic excitation product of their natural interaction with the air in normal or nominal operating conditions. In this context, modal analysis and modal testing are powerful technological tools with a solid theoretical and experimental background [1–4] widely applied to the validation of the mathematical models of the dynamic response of the mechanical systems. In both of the two main presentations of modal analysis: experimental modal analysis (EMA) and operational modal analysis (OMA) the mechanical design engineer have a reliable source of information about the dynamic nature of the system or constituting part of it, like the case of the blades, a vital part of a wind turbine for an energy generation system. In this work, we present experimental results of a modal analysis procedure performed on a wind turbine blade designed for small power applications. The test was performed in the two common formats: experimental modal analysis at laboratory conditions and operational modal analysis in real life-like operational conditions, the latter were simulated by wind tunnel experiments with turbulence generated by using a fixed pattern grid. We use velocity measurements in the analysis to determine the natural frequencies and modal damping factors. L. G. Trujillo-Franco ( ) Licenciatura en ingeniería mecánica automotriz, Universidad Politécnica de Pachuca, Zempoala, Hidalgo, México e-mail: luis.trujillo@upp.edu.mx H. F. Abundis-Fong División de Estudios de Posgrado e Investigación, Tecnológico Nacional de México/I.T. Pachuca, Pachuca de Soto, Hidalgo, Mexico e-mail: hugo.af@pachuca.tecnm.mx R. Campos-Amezcua Tecnológico Nacional de México/Centro Nacional de Investigación y Desarrollo Tecnológico, Interior Internado Palmira S/N, Col. Palmira, Cuernavaca 62490, Mexico e-mail: rafael.ca@cenidet.tecnm.mx R. Gomez-Martinez Instituto de Ingeniería, Universidad Nacional Autónoma de México, Mexico City, CDMX, Mexico e-mail: RGomezM@iingen.unam.mx © The Society for Experimental Mechanics, Inc. 2021 B. Dilworth (ed.), Topics in Modal Analysis & Testing, Volume 8, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47717-2_1 1
2 L. G. Trujillo-Franco et al. 1.2 Modal Analysis and Operational Calculus It is well known that the mathematical model for most of the engineering structures considered as flexible mechanical systems of ndegrees of freedom (DOF) under free vibration condition is given by the so called and continuously cited in the literature ordinary differential equation, in matrix form: M¨x+C˙x+Kx =0, x ∈Rn (1.1) here, x is the vector of displacements, and M, C and K are symmetric inertia, damping and stiffness n × n matrices, respectively. Those matrices do not obey or follow a known or fully established pattern, therefore it is common to make assumptions in order to guarantee the stability and physical coherence of the model. In general one can assume that (1.1) represents a real system when Kis positive definite and C≡0, and asymptotically stable when Cis positive definite (see, e.g., Inman [12]). On the other hand it is also well known that the coupled differential Eq. (1.1) can be transformed to modal coordinates qi, i =1, 2, · · · , n, as follows: ¨qi +2ζiωi ˙qi +ω 2 i qi =0 (1.2) x(t) =Ψq(t) (1.3) where ωi and ζi are the natural frequencies and damping ratios associated to the i-th vibration mode, respectively, and is the so-called n ×n modal matrix In notation of Mikusin´ski’s operational calculus [1, 5], the modal analysis representation or modal model (1.2) is described as: s 2 +2ζiωis +ω 2 i qi(s) =p0,i +p1,is (1.4) wherepo,i are constants depending on the system initial conditions at the time t0 ≥0. Using Eqs. (1.3) and (1.4), we obtain the expression for the physical displacements xi, in the form of a summation of independent single degree of freedom responses known as vibration modes: xi(s) = n j=1 ψij p0,j +p1,js s2 +2ζjωjs +ω 2 j (1.5) It is easy to prove that for each physical displacementxiwe have: pc(s) xi(s) =r0,i +r1,is +· · ·+r2n−2,is 2n−2 +r2n−1,is 2n−1 (1.6) where pc(s) is the characteristic polynomial of the mechanical system andri,j are constants which can be easily calculated by using the values of the system initial conditions as well as the modal matrix components ψij, and has the general polynomial form given by: pc(s) =s 2n +a2n−1s 2n−1 +· · ·+a1s +a0 (1.7) When using velocity measurements, instead of displacement measurements, one could simply multiply Eq. (1.5) bys and then describe the velocity output as: yi(s) = n j=1 ψij p0,js +p1,js 2 s2 +2ζjωjs +ω 2 j (1.8) The resulting output yi(s) =sxi(s) is a velocity output variable; described in a complex s domain. It is widely known that the roots of the characteristic polynomial (1.7) provide the damping factors and the damped natural frequencies, and hence the natural frequencies and damping ratios of the flexible structure. Here, we use a modal parameters identification approach to estimate the modal parameters of the mechanical system through the estimation of the positive coefficients ak
1 Modal Analysis on a Wind Turbine Blade Based on Wind Tunnel Experiments 3 of the system’s characteristic polynomial using only velocity measurements of some output variable, obtained from a wind tunnel experiment. 1.3 Output Only Modal Parameters Estimation We perform an algebraic identification approach to estimate the modal parameters of the mechanical system through the real-time estimation of the coefficients ak of the system’s characteristic polynomial as reported in [8, 9] using only velocity measurements of a single point of the blade. The application of the online algebraic identification scheme is performed using cumulative trapezoidal numerical integration with fixed sampling period of 4.16 μs. The specific algebraic identification scheme applied here is described on detail in [7] where is shown that by solving the algebraic Eq. (1.9) also detailed in [7–9] one obtains the parameter vector θ as: θ =A−1B= 1 Δ ⎡ ⎢⎢ ⎢⎢ ⎢⎣ Δ1 Δ2 . . . Δn−1 Δn ⎤ ⎥⎥ ⎥⎥ ⎥⎦ (1.9) Then, the algebraic identifiers to estimate the coefficients ak of the characteristic polynomial, avoiding singularities when the determinant Δ=det(A(t)) crosses by zero, are obtained with: ˆak = | Δk−1| |Δ| , k =1, 2, · · · , 2n−1 (1.10) Thus, one could implement the algebraic identifiers (1.10) using only any available acceleration, velocity or position measurements of any specific floor or degree of freedom. From the estimated coefficientsˆak, one can obtain the roots of the characteristic polynomial. Hence, the estimates of the natural frequencies ˆωni and damping ratios ˆζi are given by: ˆωni = ˆσ 2 i + ˆ ω 2 di , ˆζi =− ˆ σi ˆσ 2 i + ˆ ω 2 di (1.11) where ˆσi and ˆωdi are respectively estimates of the damping factors and damped natural frequencies of the mechanical system. The proposed algebraic identification scheme is shown in Fig. 1.1, where the block diagram shows the general structure of the proposed approach. Notice that the data acquisition system samples the velocity at only one specific point or test location of the wind turbine blade (in the horizontal axis direction at a precise fixed sample rate of 2.4Khz), and then, those samples are sent to a standard PC running under Windows 10 ® to finally perform the time domain identification scheme. 1.4 Wind Turbine Blade In the present work, we report the results of a vibrations analysis of a wind turbine blade made of a composite material (glass fiber). First, we performed a traditional modal testing based on impact hammer response analysis assuming free clamped boundary conditions as shown in Fig. 1.2a. The impact hammer testing was performed by acquiring velocity measurements using a Polytec ® portable laser vibrometer model PVD-100 at 2.4ksps at one fixed reference point and 9 different locations of excitation according to Maxwell’s reciprocity principle [2, 3]. The complete set of measurements, corresponding to 9 different points of excitation is shown in Fig. 1.3, where Fig. 1.3a shows the free decay response, in the time domain, whereas Fig. 1.3b shows the resulting frequency response functions (FRF) of each point. Finally, the estimation, by applying the classic peak picking technique, of the first 7 natural frequencies and modal damping factors are reported in Table 1.1.
4 L. G. Trujillo-Franco et al. Fig. 1.1 Flowchart of the proposed modal parameters estimation scheme Fig. 1.2 Wind tunnel set up for modal testing of the wind turbine blade: (a) Free clamped boundary conditions and (b) Power-distance mount and focus stabilization of the portable laser vibrometer Fig. 1.3 Impact response of the blade; (a) Free vibration decay in the time domain (b) FRF at different excitation points The detailed FRF corresponding to the point number 7 is shown in Fig. 1.4 where the first six resonances are marked in red. By a brief examination of the experimental FRF, we can assume the dynamic behavior of the wind turbine blade in the specific bandwidth of 300 Hz is dominantly linear and lightly damped considering the velocity range of 0.6 m/s. Even though
1 Modal Analysis on a Wind Turbine Blade Based on Wind Tunnel Experiments 5 Table 1.1 Impact hammer test modal parameters identification Mode Natural frequency [Hz] Modal damping % 1 9.15 0.05 2 32.6 0.19 3 75.8 0.09 4 141.0 0.26 5 175.8 0.06 6 214.24 0.05 7 275.03 0.30 Fig. 1.4 Experimental FRF of point 7 corresponding to excitation point 2 Fig. 1.5 Wind turbine experiments set up; (a) flowchart and (b) fixed pattern (passive) turbulence generator grid the construction material of the blade is a composite material (glass fiber) the performed modal testing does not evidence the presence of non-linearity or distortion in the experimental FRF. 1.5 Wind Tunnel Experiment It is well known that in normal operation conditions, the blades, as the main eolic energy conversion elements of the wind turbines (including small power systems), are subjected to harmonic excitations. In the ideal case of laminar flow (not common in real environmental conditions) the main stresses are static; nevertheless, the fatigue issues are considerably common in real life applications. In order to perform a realistic operational conditions test, we introduce a fixed pattern (square panel) grid in the wind tunnel (Fig. 1.5), with the purpose of generating turbulence in the laminar-like flux wind produced by the turbine of the wind tunnel that produces the controlled and variable speed wind excitation as shown in Fig. 1.5.
6 L. G. Trujillo-Franco et al. Fig. 1.6 Dynamic response of the wind turbine blade under turbulent wind excitation. In (a) time domain velocity signals at different wind speeds, and (b) FRF a corresponding to different wind speeds Fig. 1.7 FRF at wind speed 18 m/s In order to evaluate the identification scheme using the same boundary conditions with operational or turbulent wind excitation, the wind turbine blade was subjected to 8 different wind speed references in the interval [2–9] m/s. The time domain responses corresponding to the different wind speeds referenced to the 15th point or response location are those shown in Fig. 1.6a. Naturally, the amplitude of the velocity signal increases according to the wind speed, however, the frequency content of the signal is (in terms of harmonic content) the same and it shows the inherent dynamic behavior of the blade [1] as it is depicted in Fig. 1.6b. The FRF corresponding to the wind speed of 18 m/s is reported with detail in Fig. 1.7, where the resonances are marked with red and only the first seven bending modes are analyzed. 1.6 Application of the Time Domain Identification Scheme The proposed time domain modal parameters estimation scheme has been reported and detailed in several previous works [6, 7–12] where the online methodology of this approach has been evaluated. Here, we use the same methodology in its off-line configuration. The same algebraic problem expressed on (1.9) is solved in a post-processing context. Roughly speaking, we take a buffer of 2400 samples as shown in the flowchart of Fig. 1.1, and then, we apply the algebraic identification scheme (1.11) to the time domain array of experimental values. The results of the application of the time domain identification scheme are reported in Table 1.2. where a comparison with those obtained with the impact hammer testing is shown.
1 Modal Analysis on a Wind Turbine Blade Based on Wind Tunnel Experiments 7 Table 1.2 Wind tunnel test modal parameters identification Natural frequency [Hz] Modal damping % Mode Peak picking Time domain algebraic Peak picking Time domain algebraic 1 9.04 8.82 0.027 0.032 2 32.02 32.0 0.17 0.23 3 74.45 75.2 0.08 0.07 4 138.21 137.8 0.24 0.26 5 170.82 169.6 0.06 0.08 6 212.66 211.56 0.03 0.04 7 275.56 273.2 0.3 0.37 1.7 Conclusion An algebraic and time domain identification approach for the estimation of the natural frequencies and damping ratios of a lumped parameters vibrating mechanical system is presented, specifically, a wind turbine blade, subjected to turbulent wind excitation in a wind tunnel environment. The values of the coefficients of the characteristic polynomial of the mechanical system are firstly estimated from a data buffer, and then the modal parameters are obtained. In the design process, we have considered that measurements of only one velocity output, or measurement point is available for the implementation. It is also important to consider that one could easily extend the results for situations where acceleration or position measurements are available. The algebraic modal parameter identification was tested for a lumped parameters N-DOF mechanical system excited by a turbulent wind in wind tunnel conditions. In general, the experimental results show a satisfactory performance of the proposed identification. Acknowledgements We appreciate the support of the wind tunnel facility’s technical staff of Instituto de Ingeniería of Universidad Nacional Autónoma de México. We express our gratitude to R. Sanchez, I.M. Arenas, O. Rosales and M.A. Mendoza for their special attention and disposition to collaborate in this project. References 1. Brincker, R., Ventura, C.E.: Introduction to operational modal analysis. Wiley (2015) 2. Heylen, W., Lammens, S., Sas, P.: Modal analysis, theory and testing. Katholieke Universiteit Leuven, Belgium (2003) 3. Soderstrom, T., Stoica, P.: System identification. Prentice-Hall, New York (1989) 4. Chopra, A.K.: Dynamics of structures theory and applications to earthquake engineering. Prentice-Hall (1995) 5. Mikusin´ski, J.: Operational calculus, vol. 1, 2nd edn. PWN & Pergamon, Warsaw (1983) 6. Fliess, M., Sira-Ramírez, H.: An algebraic framework for linear identification. ESAIM: Control, Optimization and Calculus of Variations. 9, 151–168 (1996) 7. Beltran-Carbajal, F., Silva-Navarro, G., Trujillo-Franco, L.G.: Evaluation of on-line algebraic modal parameter identification methods. In: Proceedings of the 32nd international modal analysis conference (IMAC XXXII), vol. 8, pp. 145–152 (2014) 8. Beltran-Carbajal, F., Silva-Navarro, G., Chávez-Conde, E.: Design of active vibration absorbers using on line estimation of parameters and signals. In: Beltrán-Carbajal, F. (ed.) Vibration analysis and control. New trends and developments. InTech, Croatia (2011) 9. Beltran-Carbajal, F., Silva-Navarro, G.: Algebraic parameter identification of multi-degree-of-freedom vibrating mechanical systems. In: Proceedings of the 20th international congress on sound and vibration (ICSV20), Bangkok, Thailand, pp. 1–8 (2013) 10. Beltran-Carbajal, F., Silva-Navarro, G.: Adaptive-like vibration control in mechanical systems with unknown parameters and signals. Asian J. Control. (2013). https://doi.org/10.1002/asjc.727, to appear, 2013 11. Beltran-Carbajal, F., Silva-Navarro, G., Arias-Montiel, M.: Active unbalance control of rotor systems using on-line algebraic identification methods. Asian J. Control. (2013). https://doi.org/10.1002/asjc.744, to appear, 2013 12. Inman, D.J.: Vibration with control. Wiley, New York (2006)
Chapter 2 An Investigation of Vibrational Characteristics of Lap Joints Using Experimental and Analytical Methods Thomas Roberts and Phillip J. Cornwell Abstract Many structures are assembled with components that are joined together with connections such as lap joints, making it important to understand how to effectively model and identify damage, such as loosening bolts, in these connections. The structures examined in this work were two bars joined by simple lap joints – a solid structure, a welded structure, and a bolted structure. Experimental modal analysis and finite element models were used to determine the natural frequencies, damping ratios, and mode shapes for each of the different structure configurations. The first goal of this work was to determine if changes in the natural frequencies and damping ratios were large enough to distinguish between different types of structures and between experimental and analytical models. Although differences were present, results showed that natural frequencies and damping ratios are not extremely reliable metrics for determining the differences in these structures. Damage in the bolted structure was investigated by loosening or removing a bolt. Bonded contact regions were implemented within the bolted structure FE models to simulate the effects of a loosened connection while maintaining linearity for modal analysis. The end-goal in this aspect of the research was to ascertain whether a fractional strain energy method via mode shape curvature could be used to determine the location and intensity of damage in a structure. For convenience, a MATLAB GUI was developed to implement this technique. The strain energy method was unsuccessful in identifying differences between structures or damage within the bolted structure, for the differences in mode shape curvatures was not significant enough. Results from the finite element model, however, exhibited significant enough differences to distinguish the bolted structure from the solid and welded structures as well as to detect several different simulated forms of damage. Keywords Bolted joints · Welded joints · Structural health monitoring · Strain energy · Modal analysis 2.1 Introduction This work involved experimental modal analysis, finite element models, and the structural health monitoring technique called the fractional strain energy method. The structures examined in this study have two different types of lap joints – welded and bolted. These types of joints are very common in real structures, and it is important to understand how the joints change the dynamic characteristics of the structure. McCarthy et al. emphasized the importance of analyzing mechanical joints, as they are likely to be the weakest points within a structure [1]. Mechanical joints introduce factors such as bolt bending, bolt pre-loading, and non-linear stress/strain relationships, which are very hard to model analytically. Previous research on the vibrational characteristics of lap joints, particularly bolted ones, points out that it can be difficult to distinguish different levels of bolt pretension based on natural frequencies alone [2, 3]. For that reason, the experimental aspects of this work are focused mostly on detecting more severe cases of damage, such as a missing bolt. Sun et al. found that there is some correlation between a tighter bolted lap joint and a lower resulting damping ratio [2]. Other comparisons between different types of lap joint structures, such as welded and bolted, are verified in this work. Zaman et al. studied the differences between a welded lap joint structure and bolted lap joint structure. The results from that work show that even when mass properties are similar, welded lap joints tend to be stiffer than bolted lap joints [4]. Another common method for analyzing the vibrational characteristics of lap joints is utilizing guided waves. This work aims to take a simpler approach to the problem, but Du et al. and Kedra et al. describe the guided wave analysis process in detail [5, 6]. T. Roberts ( ) · P. J. Cornwell Department of Mechanical Engineering, Rose-Hulman Institute of Technology, Terre Haute, IN, USA e-mail: Thomas.Roberts@Utah.edu; robertt1@rose-hulman.edu; cornwell@rose-hulman.edu © The Society for Experimental Mechanics, Inc. 2021 B. Dilworth (ed.), Topics in Modal Analysis & Testing, Volume 8, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47717-2_2 9
10 T. Roberts and P. J. Cornwell Previous research has shown multiple different ways to represent structures with joints in a finite element model. Many studies have attempted to analyze these factors using a two-dimensional model [1], as the computation cost of a threedimensional model is much greater. If a three-dimensional model is used, Kim et al. offers multiple suggestions for accurately modeling a bolted joint. Pretension can be modeled rather simply as a thermal deformation or an external strain in the bolt. The most simplistic representation for the fastener is a solid model as opposed to an assembly of bolts, washers, and nuts [7]. Using a solid representation of the fasteners allows for more simple external geometries and thus a smoother, more accurate mesh using brick elements [7]. More complex fastener models, such as assemblies of nuts, bolts, and washers with contact conditions, can be used to produce a model that more closely resembles the real world. The main disadvantages of more complex models are the uncertainty in contact parameters and the increased simulation time. However, the increased complexities of this type of model may allow for a more accurate representation of energy dissipation within the bolted joint. Unfortunately, modeling techniques that use pre-stress of any kind or contact parameters that are anything but bonded cannot be used in modal analysis. Modal analysis studies require that all properties of the model be linear. If more complex modeling techniques are desired, then time-dependent solvers can be used to determine the dynamic response from the FE model. If the computational capability is available, some researchers prefer to use nonlinear FE solvers to analyze problems like this. For example, Sun et al. used penalty stiffness contact in the entirety of a bolted lap joint model and solved for the frequency response with a nonlinear solver [2]. Kim et al. proposed a method for simulating bolt pretension that requires only boned contact regions in the model. A cylindrical-shaped region around the bolt hole was used for applying bonded contact constraints between the two plates in the joint [7]. Additionally, the solid bolt model was bonded to those same contact regions. Many methods have been proposed to model welded joints using finite elements. The most practical method of those researched breaks up the model into three zones: base metal, weld metal, and heat-affected metal. Each of these zones is a solid model, and the zones are assembled together and connected with contact parameters within the finite element program [8]. This method allows for the inside of the joint (where the weld has not penetrated) to be modeled without contact, similar to the actual specimen. Additionally, pre-stresses due to the intense heat of the welding process can be applied to specific regions of the model; since the heat-affected zone is a separate body, it can be isolated in order to apply the heat pre-stress. Lastly, the amount of filler material left as a product of the welding process can be controlled very easily using this method of assembling the model. Another objective of this work was to utilize structural health monitoring techniques to compare structures with and without damage. The chosen method was a fractional strain energy method, proposed by Stubbs for beam-like structures [9] and extended by Cornwell [10] for plate-like structures. The fractional strain energy method uses the curvature of mode shapes to calculate a damage index. Experimental mode shapes and FE mode shapes were used in the method for detecting differences between the solid, welded and bolted structures. Additionally, the fractional strain energy method was used to detect differences between an undamaged bolted structure and a damaged one with a missing bolt. This comparison was again made with both experimental and analytical mode shape information. 2.2 Experimental Methods Roving hammer tests were performed on three structures – one solid, one welded, and one bolted, as shown in Fig. 2.1. Each structure was constructed from 6061 Aluminum bar stock. The solid structure was machined from a solid piece of ½” ×2 stock. The bolted and welded structures were constructed from two pieces of ¼” ×2 stock. Welded joints contain a filler material that was assumed to be 4043 Aluminum, and the bolts in the bolted structure were medium carbon alloy steel. The first method used for characterizing the modal properties of the lap joint structures was a roving hammer modal test. Each structure was tested at 45 different points in order to create a grid with nodes spaced one-inch by one-inch apart. Mode shapes, natural frequencies, and damping ratios were computed for each of the three lap joint structures. In addition, the bolted structure was tested with one of the four bolts removed from the assembly. In the context of structural health monitoring (SHM), a structure with a missing bolt would be considered damaged when being compared to a structure with all four bolts adequately tightened to 25 inch-lbs. The bolted structure was only tested with all bolts fully tightened or one bolt completely removed. The experimental modal tests were mapped to a planar geometry with one-inch by one-inch node spacing. The geometry used for the modal test had 45 nodes, and 72 tracelines. For the fractional strain energy method, a total of 28 four-node shells and three separate fifteen-node beams were defined. Figure 2.2 shows the geometry used for all experimental modal tests. The planar geometry of the structures was set to the x- and y-planes and the response degree-of-freedom was in the positive
2 An Investigation of Vibrational Characteristics of Lap Joints Using Experimental and Analytical Methods 11 Fig. 2.1 Solid, welded, and bolted lap joint structures (front to rear, respectively) used for experimental analysis 30 29 45 15 28 44 15 27 43 14 26 42 13 25 41 12 24 40 10 11 23 39 10 22 38 9 x 21 37 8 20 36 7 19 35 5 6 18 34 5 -1 17 33 4 0 16 32 3 1 1 z 31 y 2 0 0 1 -1 Fig. 2.2 Geometry used for mapping experimental modal data to the mode shapes of the structure z-axis for detecting in-plane (I.P) modes. In order to observe out-of-plane modes (O.P.), the response degree of freedom was changed to the negative y-axis. Each structure was tested three times in a randomized order. 2.3 Analytical Methods Finite element models of each of the three lap joint structures were used for modal analysis studies, and the results from the modal studies were compared to the results from the experimental modal tests. Each model was configured with the same meshing parameters and appropriate material properties. Table 2.1 summarizes the material properties used to configure the
12 T. Roberts and P. J. Cornwell Table 2.1 Material properties used for the three lap joint FE models Material Density [g/cm3] Young’s modulus [GPa] Poisson’s ratio AL6061 2.65 66.0 0.33 AL4043 2.68 75.0 0.33 Steel alloy 7.70 205 0.29 Bar sections Weld bead Fig. 2.3 Geometry for the welded structure FE model consisting of two bar sections and a weld bead three different finite element models. Just as in the experimental modal tests, the boundary conditions for each FE model were assumed to be free-free. 2.3.1 Solid Structure FE Model The solid structure was a rather simple model. The entire model was assigned AL 6061 material properties, and a global mesh sizing of two millimeters was applied to the whole model in order to produce at least three elements through the thickness of the model. No pre-stress or contact parameters were used in configuring the solid structure model. 2.3.2 Welded Structure FE Model The welded structure model was composed of three parts – two bar sections and one weld bead. The two identical bar sections were generated as pieces of flat bar that would fit together to form the final shape of the structure. The bar sections had a small portion of material removed in order for the weld bead to be assembled into the model. The weld bead was created to have a radius of half the thickness of the bar material. During the welding process, the filler material bonds the bars together. To model this characteristic of the welded joint, AL 6061 material was removed and replaced with the AL 4043 weld bead. The pieces used for this model are shown in Fig. 2.3. The three-piece model was assembled using bonded contact conditions around the weld bead. Only the mating faces between the weld bead and the bar sections were bonded together; the two bar sections had no contact conditions between them. This contact condition most closely resembles the actual structure, as the only connection between the bar sections is where the weld bead has penetrated. 2.3.3 Bolted Structure FE Model The bolted structure model was constructed of two bar sections with matching holes and a bolt for each set of holes (four bolts in all). The bar sections and bolts were assigned AL 6061 and Steel Alloy material properties, respectively. The geometry used for the bolted structure model is shown in Fig. 2.4. The bolts were modeled as an assembly of a bolt, nut, and two washers. Creating a solid geometry of the bolt assembly allowed for a much simpler model as the contact between components and the interactions of bolt/nut threads were not a concern for this analysis. Since the FE models were to be used in a modal study, all contact parameters were bonded, and no pretension was included in the model.
2 An Investigation of Vibrational Characteristics of Lap Joints Using Experimental and Analytical Methods 13 Bar sections Solid bolt models Fig. 2.4 Geometry for the bolted structure FE model consisting of two bar sections and four solid bolt models Contact regions Fig. 2.5 Contact regions used to simulate pretension in the bolted structure FE model In order to allow for the approximation of a loosened connection in the modal study, the bars were bonded to the bolts using a circular region around each bolt hole, as shown in Fig. 2.5. A similar method was successfully employed by Liao et al. [11]. These circular regions had shared topology with their respective bar sections, and contact was assigned between the bars and between the bars and the bolt head/nuts. Different levels of pretension were approximated by changing the size of the circular contact regions around the bolt heads. Multiple trials of the bolted structure FE model were solved. First, the sizing of the contact region around each bolt was adjusted to model the effects of a loosened connection in the model. In general, a smaller contact region around each bolt resulted in a model that was less stiff, thus simulating the effects of loosening a bolt in the model while mass properties remain constant. Next, one bolt was completely removed from the model along with its contact parameters. In this type of model, the mass properties and the component contact are different from the undamaged structure. SHM methods were also used to compare the differences between complete (healthy) structures and structures with missing components or differing size contact regions (damaged). 2.3.4 Fractional Strain Energy Method Changes in natural frequencies and damping ratios do not often provide sufficient information to determine the presence of damage in a structure. Even if natural frequencies and damping ratios are sensitive to damage, they cannot be used to locate the damage. In this work, a fractional strain energy method was implemented to compare experimental data and analytical results. The fractional strain energy method (SEM) was implemented for generic planar structures, such as a beam or plate, that have only a single degree-of-freedom of modal response. The fractional strain energy for a beam element for a particular mode shape, ψi(x), is Ui = 1 2 l 0 EI ∂ 2 ψi ∂x2 2 dx. (2.1)
14 T. Roberts and P. J. Cornwell and the fractional strain energy for a plate element for a particular mode shape, ψi(x, y), is Ui = D 2 b 0 a 0 ∂ 2 ψi ∂x2 2 + ∂ 2 ψi ∂y2 2 +2ν ∂ 2 ψi ∂x2 ∂ 2 ψi ∂y2 +2(1−ν) ∂ 2 ψi ∂x∂y 2 dxdy. (2.2) For this analysis, thex-direction is the long dimension of the lap joint structures, and they-direction is the short dimension. Refer to Fig. 2.2 for a visualization. The plates or beams are then subdivided and the strain energy for each sub-region within themodel, Uijk, can be calculated by integrating over the area of each sub-region. The fractional strain energy at location jk is defined to be fijk = Uijk Ui (2.3) The damage index, βjk, for each subdivision of the plate is then defined as βjk = m i−1 f∗ijk m i−1 fijk , (2.4) where f∗ ijk is the fractional strain energy for the damaged structure, and mis the number of measured modes. The fractional strain energy damage index is lastly normalized by its mean, βk, and standard deviation, σk, to create the final metric used to compare healthy and damaged structures: Zk = βk −βk σk . (2.5) Further detail about the development of the fractional strain energy method can be found in [10]. 2.4 User Interface Development The algorithm for the fractional strain energy method described in the previous section was originally part of the MATLAB programDIAMOMD[6, 12]. DIAMOND(Damage Identification and Modal Analysis of Data) is a program created at Los Alamos National Laboratory under the auspices of the US Department of Energy. The strain energy algorithms for both beam and shell element computations were extracted fromDIAMONDand modified to run independently using simpler forms of data input. The splash page of the new strain energy user interface (UI) is shown in Fig. 2.6. Fig. 2.6 Main window of the strain energy method user interface. From here, the user can access the UI’s three main functions
2 An Investigation of Vibrational Characteristics of Lap Joints Using Experimental and Analytical Methods 15 Fig. 2.7 SEM with Shells window. Here the user inputs geometry, undamaged and damaged modes, and selects which modes to use in the computation The window shown in Fig. 2.6 is where the user accesses the UI’s three main functions – SEM with Beams, SEM with Shells, and Convert FE Data to Modes. The two strain energy methods work in similar ways and require the same inputs. Once given the correct input, each function executes its respective method and produces damage identification results. Figure 2.7 shows the window for executing the SEM with Shells function (the window for the SEM with Beams functions is visually identical with the exception of the figure’s title and resulting plot format). To utilize the SEM functions within the UI, the user must input a valid geometry file, an undamaged modes file, and a damaged modes file. The UI requires these files to be in MATLAB data format. Geometry is specified by Nodes, Tracelines, Beams, Quads, and Shells. Files for undamaged and damaged modes can be either direct modal analysis output from DIAMOND, or the user can generate their own mode shape files from other modal analysis software packages. The SEM UI only requires a MATLAB Structure data type of the nodal displacements that pertain to the geometry and a confirmation that the response degree-of-freedom is in the positive z-axis. Some notable assumptions within the SEM UI are that it requires planar geometries with the response degree-of-freedom being in the positive z-axis. The SEM with Beams function requires that the beams used in the computation be parallel in either the x- or y-axis, but beams need not be the same length in this function. The third function within the SEM UI is mapping FE mode shapes to geometry that can be used within the strain energy methods. The Convert FE Data to Modes window, shown in Fig. 2.8, has two capabilities – mapping FE mode shapes to an existing geometry and mapping FE mode shapes to a new geometry. The user inputs a reference geometry (the geometry in Fig. 2.8 is the same as shown in Fig. 2.2) MATLAB file and a Microsoft Excel file containing the FE mode shapes that are defined by paths in the geometry. The format required for the FE Displacement Excel File can be found in the open-source example data that accompanies the UI. The UI then configures itself to map the Current Path to the nodes that the user inputs in the Nodes for Path dialog box (node numbers separated by commas). This window in the UI relies on similar geometry assumptions as previously stated with some extra constraints. The Convert FE Data to Modes functions require that paths be parallel in either the x- or y-axis and the same length, and it is advantageous to list the paths in ascending order with the most negative x- or y-direction path first. ANSYS Workbench 19.1 was used in this work to create FE models and do the modal analysis. ANSYS allows the user to create paths along edges in a geometry, and this functionality allows the user to extract nodal displacements to be used in the SEM UI. Figure 2.9 shows a screen shot of a path and its corresponding data from within an ANSYS modal analysis. The second functionality of the Convert FE Data to Modes window (which is still in development) is mapping FE displacements to a new geometry. For this function, the user only inputs the FE Displacements Excel File and the number of nodes to discretize each path into. In this function it is critical that all paths be parallel, the same length, and listed in the correct order. Once given the proper information, the UI generates the mode shape data and the geometry file to be used
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