River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Topics in Modal Analysis, Volume 7 Randall Allemang James De Clerck Christopher Niezrecki Alfred Wicks Proceedings of the 31st IMAC, A Conference on Structural Dynamics, 2013 River Publishers
Conference Proceedings of the Society for Experimental Mechanics Series Series Editor TomProulx Society for Experimental Mechanics, Inc., Bethel, CT, USA
River Publishers Randall Allemang • James De Clerck • Christopher Niezrecki • Alfred Wicks Editors Topics in Modal Analysis, Volume 7 Proceedings of the 31st IMAC, A Conference on Structural Dynamics, 2013
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Preface Topics in Modal Analysis, Volume 7: Proceedings of the 31st IMAC, A Conference on Structural Dynamics, 2013 represents one of seven volumes of technical papers presented at the 31st IMAC, A Conference and Exposition on Structural Dynamics, 2013 organized by the Society for Experimental Mechanics, and held in Garden Grove, California February 11–14, 2013. The full proceedings also include volumes on Nonlinear Dynamics; Experimental Dynamics Substructuring; Dynamics of Bridges; Dynamics of Civil Structures; Model Validation and Uncertainty Quantification; and, Special Topics in Structural Dynamics. Each collection presents early findings from experimental and computational investigations on an important area within Structural Dynamics. Modal Analysis continues to be the core around which IMAC is built. The topics presented in this volume represent the state of the art in Modal Analysis as well as the latest advances in this area. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Cincinnati, OH, USA Randall Allemang Houghton, MI, USA James De Clerck Lowell, MI, USA Christopher Niezrecki Blacksburg, VA, USA Alfred Wicks v
Contents 1 Damage Detection Using Flexibility Proportional Coordinate Modal Assurance Criterion.................... 1 Luciana Balsamo, Suparno Mukhopadhyay, Raimondo Betti, and Hilmi Lus 2 Automated Selection of Damage Detection Features by Genetic Programming.................................. 9 Dustin Harvey and Michael Todd 3 Optimal Selection of Artificial Boundary Conditions for Model Update and Damage Detection – Part 1: Theory ............................................................................................. 17 Joshua H. Gordis and L.T. Konstantinos Papagiannakis 4 Optimal Selection of Artificial Boundary Conditions for Model Update and Damage Detection – Part 2: Experiment ........................................................................................ 37 Joshua H. Gordis 5 Detection of Mass Change on a Glass Plate........................................................................... 61 Jannick B. Hansen, Rune Brincker, and Manuel L. Aenlle 6 Vibro-Acoustic Research on a Full-Scale Aircraft Structure....................................................... 67 Christian Koehne, Delf Sachau, and Mirko Schaedlich 7 Control of Dynamic Mass as Boundary Condition for Testing Substructures.................................... 77 Manuel Baschke, Michael Krepl, and Delf Sachau 8 Multi-body-Simulation of a Self Adaptive Torsional Vibration Absorber ........................................ 85 Delf Sachau and Jonas Hanselka 9 Combined Optimization of Actuator/Sensor Positions and Weighting Matrices for an Active Noise Reduction System................................................................................................. 93 Jan Foht and Delf Sachau 10 SSDI Technique Evolution to Improve Attenuation Performances with Random Disturbances ............... 99 M. Berardengo, S. Manzoni, M. Redaelli, and M. Vanali 11 Geometrically Nonlinear Dynamic Analysis of Piezoelectric Integrated Thin-Walled Smart Structures ..... 107 S.Q. Zhang and R. Schmidt 12 Linear/Nonlinear Reduced-Order Substructuring for Uncertainty Quantification and Predictive Accuracy Assessment.................................................................................................... 117 Timothy Hasselman, George Lloyd, and Ryan Schnalzer 13 Damage Detection in an Energy Flow Model Including Parameter Uncertainty................................. 131 Marcela Rodrigues Machado and Jose Maria Campos Dos Santos 14 A Coupled Approach for Structural Damage Detection with Incomplete Measurements....................... 141 George James, Tim Cao, Mo Kaouk, and David Zimmerman 15 Efficient and Robust Solution of Inverse Structural Dynamic Problems for Vibration Health Monitoring .. 155 KengC. Yap vii
viii Contents 16 Finite Element-Based Damage Detection Using Expanded Ritz Vector Residuals ............................... 167 Stuart G. Taylor, George Khoury, Michael D. Todd, and David C. Zimmerman 17 Proportional Damping from Experimental Data .................................................................... 179 Brian Schwarz and Mark Richardson 18 Superior Damping of Hybrid Carbon Fiber Composites Grafted by ZnO Nanorods ........................... 187 A. Alipour Skandani, N. Masghouni, and M. Al-Haik 19 Advanced Identification Techniques for Operational Wind Turbine Data...................................................................................................... 195 Simone Manzato, Jonathan R. White, Bruce LeBlanc, Bart Peeters, and Karl Janssens 20 Tracking and Removing Modulated Harmonic Components with Spectral Kurtosis and Kalman Filters ... 211 Jean-Luc Dion, Cyrille Stephan, Gae¨l Chevallier, and Hugo Festjens 21 Vibration Reduction of Brush Cutter ................................................................................. 225 Nobuyuki Okubo, Hiroyuki Nakagawa, Kohei Furuya, and Takeshi Toi 22 Design of a Test Setup for Measuring Dynamic Stiffness of Vibration Isolators ................................. 235 Canan Uz, Gokhan O. Ozgen, and Ender Cigeroglu 23 An Impact Excitation System for Repeatable, High-Bandwidth Modal Testing of Miniature Structures..... 249 Bekir Bediz, Emrullah Korkmaz, and O. Burak Ozdoganlar 24 Replicating Aerodynamic Excitation in the Laboratory ............................................................ 259 P.M. Daborn, P.R. Ind, and D.J. Ewins 25 A Systematic Approach to Modal Testing of Nonlinear Structures................................................ 273 A. delli Carri and D.J. Ewins 26 Fiber Optics Sensing of Stressing and Fracture in Cylindrical Structures ............................................................................................... 287 Shen-en Chen, Benjamin Smith, and Peng Wang 27 Real-Time Damage Identification in Nonlinear Smart Structures Using Hyperchaotic Excitation and Stochastic Estimation .............................................................................................. 295 Shahab Torkamani, Eric A. Butcher, and Michael D. Todd 28 Damage Detection Based on Electromechanical Impedance Principle and Principal Components............ 307 Mario Anderson de Oliveira, Jozue Vieira Filho, Vicente Lopes Jr., and Daniel J. Inman 29 Enhanced Modal Wavelet Analysis for Damage Detection in Beams .............................................. 317 Mario Algaba, Mario Sol´ıs, and Pedro Galv´ın 30 Linear Projection Techniques in Damage Detection Under a Changing Environment .......................... 325 Salma Mozaffari Kojidi, Michael Do¨hler, Dionisio Bernal, and Yang Liu 31 Modal Reduction Based on Accurate Input-Output Relation Preservation ...................................... 333 M. Khorsand Vakilzadeh, S. Rahrovani, and T. Abrahamsson 32 Fast Precise Algorithm of Computing FRF by Considering Initial Response .................................... 343 J.M. Liu, W.D. Zhu, M.Ying, and S. Shen 33 Development of Full Space System Model Modes from Expansion of Reduced Order Component Modal Information ...................................................................................................... 353 Christopher Nonis, Louis Thibault, Timothy Marinone, and Peter Avitabile 34 Damage Localization from the Image of Changes in Flexibility ................................................... 369 Dionisio Bernal 35 Spectral Element Method for Cable Harnessed Structure.......................................................... 377 Jiduck Choi and Daniel J. Inman
Contents ix 36 Analytic Formula Derivation for a Rolling Tire with a Ring Model ............................................... 389 Jongsuh Lee, Peter Kindt, Bert Pluymers, Paul Sas, and Semyung Wang 37 Nonlinear Identification of the Viscous Damping of the Resistor for Nuclear Plants............................ 397 Giancarlo Galli, Francesco Braghin, and Edoardo Sabbioni 38 Effect of Spin Speed on Stability Lobes in High Speed Machining ................................................ 407 Hasan Yılmaz and Ender Cigeroglu 39 Chatter Reduction in Turning by Using Piezoelectric Shunt Circuits ............................................. 415 Ufuk Yigit, Ender Cigeroglu, and Erhan Budak 40 Damage Quantification from the Column Space of Flexibility Changes .......................................... 421 Dionisio Bernal 41 State Estimate of Wind Turbine Blades Using Geometrically Exact Beam Theory.............................. 427 Stuart G. Taylor, Darby J. Luscher, and Michael D. Todd 42 Damage Index Matrix: A Novel Damage Identification Method Using Hilbert-Huang Transformation...... 439 Ali Zarafshan and Farhad Ansari 43 An Approach to the Moving Load Problem for Multiple Cracked Beam......................................... 451 N.T. Khiem, T.H. Tran, and N.V. Quang 44 Detection of Structural Damage Through Nonlinear Identification by Using Modal Testing .................. 461 Murat Aykan and H. Nevzat O¨ zgu¨ven 45 Vibration Fatigue Analysis of a Cantilever Beam Using Different Fatigue Theories ............................ 471 Yusuf Eldogˇan and Ender Cigeroglu 46 Automated Modal Analysis Based on Statistical Evaluation of Frequency Responses .......................... 479 Vahid Yaghoubi and Thomas Abrahamsson 47 The Modal Observability Correlation as a Modal Correlation Metric............................................ 487 Vahid Yaghoubi and Thomas Abrahamsson 48 A Modal Test Method Based on Vibro-acoustical Reciprocity ..................................................... 495 W.D. Zhu, J.M. Liu, Y.F. Xu, and H.Q. Ying 49 Reactionless Test to Identify Dynamic Young’s Modulus and Damping of Isotropic Plastic Materials........ 511 Peter Blaschke and Torsten Schneider 50 Real-Time Modal Analysis of Shell-Shaped Objects Using High-Frame-Rate Structured-Light-Based Vision......................................................................................... 517 Hua Yang, Qingyi Gu, Tadayoshi Aoyama, Takeshi Takaki, and Idaku Ishii 51 Field and Numerical Testing of the BWE SchRs4600.50 Dynamic Behavior ..................................... 525 Damian Pietrusiak, Przemysław Moczko, and Jerzy Czmochowski 52 Modal Analysis of Rotating Carbon Nanotube Infused Composite Beams ....................................... 533 C. DeValve, N. Ameri, P. Tarazaga, and R. Pitchumani 53 Modal Analysis and Dynamic Monitoring of a Concentrating Solar Heliostat................................... 543 Adam Moya, Clifford Ho, Jeremy Sment, Todd Griffith, and Joshua Christian 54 Identification of Stability Cutting Parameters Using Laser Doppler Vibrometry ............................... 553 D. Olvera, A. El´ıas-Zu´n˜iga, M. Pineda, E. Macias, O. Mart´ınez, L.N. Lo´pez de Lacalle, and C. Rodr´ıguez 55 System Identification Using Kalman Filters.......................................................................... 561 F. Abid, G. Chevallier, J.L. Blanchard, J.L. Dion, and N. Dauchez 56 Identification of Time-Varying Nonlinear Systems Using Differential Evolution Algorithm................... 575 Nevena Perisic, Peter L Green, Keith Worden, and Poul Henning Kirkegaard
x Contents 57 Experimental Verification and Improvement of Dynamic Characterization Method for Structural Joints... 585 S¸erife Tol and H. Nevzat O¨ zgu¨ven 58 Transfer Functions to Measure Translational and Rotational Velocities with Continuous-Scan Laser Doppler Vibrometry ............................................................................................. 597 Shifei Yang and Matthew S. Allen 59 Empirical Slow-Flow Identification for Structural Health Monitoring and Damage Detection................ 617 Young S. Lee, D. Michael McFarland, Lawrence A. Bergman, and Alexander F. Vakakis 60 Continuous Scanning for Acoustic Field Characterization......................................................... 625 Carlos E. Garcia, Sriram Malladi, and Pablo A. Tarazaga 61 Operating Deflection Shapes of a Violin String via High Speed/High Resolution Videography................ 637 Chuck Van Karsen, Troy Bouman, and Geoff Gwaltney 62 Automated Measurement Grid Generation for Scanning Laser Doppler Vibrometers ......................... 645 L. Pesaresi and C.W. Schwingshackl 63 Mode Filtering of Continuous Scanning Laser Doppler Vibration Data.......................................... 655 P. Castellini, P. Chiariotti, and M. Martarelli 64 The Characterization of the Time Delay Problem in Hardware in the Loop System Applications ............ 661 C.A.G. Carrillo, J.V. Ferreira, and P.S. Meirelles 65 Optimal Placement of Piezoelectric Patches on a Cylindrical Shell for Active Vibration Control ............. 673 Caner Gencoglu and H. Nevzat O¨ zgu¨ven 66 Adaptive Feedback Linearisation and Control of a Flexible Aircraft Wing ...................................... 683 S. Jiffri, J.E. Mottershead, and J.E. Cooper 67 Limit Cycle Assignment in Nonlinear Aeroelastic Systems Using Describing Functions and the Receptance Method............................................................................................ 701 Xiaojun Wei and John E. Mottershead 68 Investigation of an Active Structural Acoustic Control System on a Complex 3D Structure ................... 715 S. Kulah, U. Aridogan, and I. Basdogan 69 Development of a Stabilized Pan/Tilt Platform and the State of the Art .......................................... 723 M. Burcak Ozkok and Ali Osman Boyacı 70 Dynamic Equations for an Anisotropic Cylindrical Shell ........................................................... 731 Reza Okhovat and Anders Bostro¨m 71 Expansion of Nonlinear System Response Using Linear Transformation Matrices from Reduced Component Model Representations ................................................................................... 743 Tim Marinone, Louis Thibault, and Peter Avitabile 72 Explicit Construction of Rods and Beams with Given Natural Frequencies...................................... 771 A.Morassi 73 A Metric for Modal Truncation in Model Reduction Problems Part 1: Performance and Error Analysis.... 781 Sadegh Rahrovani, Majid Khorsand Vakilzadeh, and Thomas Abrahamsson 74 A Metric for Modal Truncation in Model Reduction Problems Part 2: Extension to Systems with High-Dimensional Input Space ........................................................................................ 789 Sadegh Rahrovani, Majid Khorsand Vakilzadeh, and Thomas Abrahamsson 75 On Gramian-Based Techniques for Minimal Realization of Large-Scale Mechanical Systems ................ 797 Sadegh Rahrovani, Majid Khorsand Vakilzadeh, and Thomas Abrahamsson
Chapter1 Damage Detection Using Flexibility Proportional Coordinate Modal Assurance Criterion Luciana Balsamo, Suparno Mukhopadhyay, Raimondo Betti, and Hilmi Lus Abstract In the recent years, vibration-based identification techniques have attracted the attention of the civil engineering community, as these methods can be naturally incorporated into automated continuous structural health monitoring procedures. It is a generally accepted approach to model the damage and deterioration of a structural element through stiffness reduction. For this reason, a feature tailored so as to be well correlated to the expected differences between the undamaged and damaged flexibility matrices, such as the recently proposed Flexibility Proportional Coordinate Modal Assurance Criterion (FPCOMAC), is ideally suited to be exploited as damage sensitive feature. We present a statistical pattern recognition based damage detection method that employs FPCOMAC as damage sensitive feature. The proposed methodology is executed according to the training and testing phases typical of the pattern recognition framework. Particular effort is devoted to test the ability of the method to correctly identify the damage when response time histories used in the training are measured in different environmental conditions. The formulation is derived considering a shear-type structural system. Results obtained by considering a 7 DOFs shear-type system prove the efficiency of the method in detecting and locating the damage, irrespective of damage severity and environmental effects, under the conditions that the damage amount is greater than the structural variations caused by the external factors and the amount of data is reasonably large. Keywords Statistical pattern recognition • Structural damage detection • FPCOMAC 1.1 Introduction The task of ascertaining the health conditions of a structure by analyzing its vibration response is commonly termed as vibration based structural health monitoring. The most common approach to the problem of structural health monitoring is to identify the modal characteristics of the system under known conditions; the assessment of damage occurrence is then pursued by comparing the reference set of modal characteristics against a new set of parameters identified from new instances of the structural vibration response. In the last three decades, these were the methods most dominantly used and, then, the available algorithms are particularly efficient and able not only to detect the damage, but also to locate it as well as to indicate the extent of the detected structural alterations. Nonetheless, to accomplish such a high level of damage detection, the model based methods necessitate of very high quality time histories recorded from a vast number of sensors located at crucial points in the structure. In fact, the performance of the system identification algorithm is intimately correlated to the quality of the available response time histories. Moreover, the system identification algorithm itself may be based on assumptions (e.g. linearity of the system, stationarity of the response) that make it unsuited for certain kind of scenarios. L. Balsamo ( ) • S. Mukhopadhyay Ph.D. Candidate, Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027, USA e-mail: lb2591@columbia.edu; sm3315@columbia.edu R. Betti Professor, Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, NY 10027, USA e-mail: betti@civil.columbia.edu H. Lus Associate Professor, Department of Civil Engineering, Bogazici University, 34342 Bebek, Istanbul, Turkey e-mail: hilmilus@boun.edu.tr R. Allemang et al. (eds.), Topics in Modal Analysis, Volume 7: Proceedings of the 31st IMAC, A Conference on Structural Dynamics, 2013, Conference Proceedings of the Society for Experimental Mechanics Series 45, DOI 10.1007/978-1-4614-6585-0 1, © The Society for Experimental Mechanics 2014 1
2 L. Balsamo et al. On the other hand, over the last 10 years, the discipline of statistical pattern recognition has attracted the interest of a vast community within civil engineering. Its most appealing characteristic is that of not requiring a physical model of the system to be identified. Instead, the structure in analysis is represented by the probability distribution model of some damage sensitive features extracted from the vibration measurements, which are not directly correlated to the system structural characteristics: for instance, very popular damage sensitive features for statistical pattern recognition are the Auto-Regressive (AR) or Auto-Regressive with eXogeneous input (ARX) model coefficients (see [1, 2]). The main drawback of this approach is that of requiring a vast number of time histories to be employed while the beneficial effects are results that are not strongly corrupted by noise and input effects, as the large number of data employed filter out noise and external factors effects. The two approaches, i.e. model and non-model based techniques, are perceived as mutually exclusive; nevertheless, both methods have advantages and disadvantages, and should be then applied in combination. For this reason, in this paper a technique is proposed that makes use of a combination of the two. The deviation from 1 of the recently introduced Flexibility Proportional Coordinate Modal Assurance Criterion (FPCOMAC) is used as damage sensitive feature. FPCOMAC [3] is defined by means of the mode shape matrices of the healthy and possibly damaged systems. Therefore, a system identification algorithm is needed to derive these features from the system time histories. The damage detection method is developed according to the statistical pattern recognition paradigm: during training, a large number of feature vectors are extracted from the time histories of the healthy structure. Small perturbations around the mean value of stiffness and mass properties are considered, in order to represent external factors effects, such as environmental effects. The necessity of approaching the problem within the probabilistic framework is indeed required due to the presence of variations conveyed by environmental or operational effects. During the training operation, a threshold is identified which is able to label as undamaged all the observations of the feature falling within a given confidence level region. During testing, the features extracted from samples recorded on the system under unknown conditions are compared against the trained threshold to assess damage occurrence. The necessary derivations in the formulation of the damage detection algorithm are obtained considering shear-type models of the structural systems. Through numerical simulations, it is shown that the proposed algorithm is capable of locating and estimating damage extent reliably well. 1.2 Damage Sensitive Feature Flexibility Proportional Coordinate Assurance Criterion (FPCOMAC or FPCOMACk/ was recently proposed in an effort to introduce a criterion, whose deviation from 1 gives a measure of the error in the .k;k/-th element of the flexibility matrix estimated by employing mode shapes identified from the system response time histories, where with error, we refer to the change due to damage in the estimated flexibility matrix. Let us assume real valued mode shapes are evaluated at each degree of freedom (DOF) of a given N-DOF system: let i 2 RN 1 and i 2 RN 1 be mode shapes for the i-th mode of vibration of the system in the healthy and damaged states, respectively. Furthermore, let us denote with 2RN 1 the set of natural frequencies of the healthy structure, and with 2RN 1 the set of natural frequencies of the possibly damaged system. Under the assumptions that damage may be modeled thorough stiffness reduction, without any mass alteration, and that the mode shapes are mass normalized, the flexibility matrices of the structure under undamaged and damaged conditions may be given as in Eqs. (1.1) and (1.2), respectively F D N X iD1 i T i i (1.1) PD N X iD1 i T i i (1.2) FPCOMAC is derived in such a way that its deviation from 1 attempts to give a measure of the difference between the values of the .k;k/-th element of the healthy (F) and damaged (P) system flexibility matrices. The deviation from 1 of the FPCOMAC is denoted as eFPCOMAC, which is the criterion actually employed as damage sensitive feature in this work, and whose analytical expression is given in Eq. (1.3), for the k-thDOF: eFPCOMACk D1 N PiD0 k i k 2 k i k 2 2 i;k i N PiD1 2 i;k i (1.3)
1 Damage Detection Using Flexibility Proportional Coordinate Modal Assurance Criterion 3 A value of eFPCOMAC less than 0 would be indicative of damage, if the representations of the modal characteristics of the reference structure were unique. Nonetheless, when taking into account environmental and operational effects, the modal characteristics of the reference system vary, making it necessary the definition of a new threshold to assess damage occurrence. 1.2.1 Mode Shape Normalization A clarification is necessary regarding the mode shapes employed for the damage detection algorithm. In fact, the mode shapes identified through an output-only system identification algorithm, like Enhanced Canonical Correlation Analysis [4], diagonalize the mass matrix, but are not proportional to mass normalized mode shapes by a single scalar multiplicative constant, whereas the FPCOMAC feature is derived by assuming the mode shapes of the reference system as mass normalized, or at least proportional to the mass normalized ones by a constant. In the present section a normalization procedure valid for systems that can be modeled through a diagonal mass matrix is proposed. Let us then denote with ˚the identified N N mode shape matrix, and with MtheN N diagonal mass matrix so that: ˚TM˚Ddiagfd1; :::;dNg (1.4) while the condition we want to achieve is given in Eq. (1.5): O˚ TMO ˚D˛ I (1.5) where I is theN N identity matrix, and’is a constant multiplicative factor. To pursue the outcome outlined in Eq. (1.5), it is necessary to find a scaling matrixˇsuch that: O˚D˚ ˇ (1.6) where ˇ Ddiagfˇ1; : : :;ˇNg is a scalar diagonal matrix. By plugging Eq. (1.6) into Eq. (1.5), and setting ˛ Dˇ2 1=d1, we obtain the following expression for the inverse of the mass matrixM: M 1 D d1 ˇ2 1 ˚ˇ2˚T Dd1˚ ˚T (1.7) where the elements of the diagonal matrix D diagf1;ˇ2 2=ˇ2 1;ˇ2 3=ˇ2 1; : : :;ˇ2 N=ˇ2 1g D diagf1; 2 2; 2 3; : : :; 2 Ng may be retrieved by solving the system of Eq. (1.8) obtained by taking into account the properties of M, i.e. that the matrix is diagonal and then symmetric: N X iD2 2 i k;i l;i D k;1 l;1 8k ¤l, andk; l 2Œ1; :::;N (1.8) where i;jrepresents the (i,j)-th element of the mode shape matrix ˚. The identified mode shape matrix can then be scaled through : Q˚D˚ 1=2 D 1 ˇ1 O˚ (1.9) This scaling produces a set of normalized mode shapes that differ from the mass normalized ones by one single multiplicative factor d1, as shown in Eq. (1.10): Q˚ TMQ ˚D 1 ˇ2 1 O˚ TMO ˚D ˛ ˇ2 1 I D 1 d1 I (1.10)
4 L. Balsamo et al. 1.2.2 Damage Location The flexibility matrix of anN DOFs shear-type system can be expressed as: F D 2 6 6 6 6 6 4 f1 f1 f1 f1 f1 Cf2 f1 Cf2 : : : : : : : : : : : : f1 f1 Cf2 N PiD1 fi 3 7 7 7 7 7 5 (1.11) where fi is the inter-story flexibility between the .i 1/-th and i-th DOFs. Herein, structural damage is modeled by increasing the inter-story flexibility by a factor i greater than 1. It can be proved that the difference between two consecutive eFPCOAMCk coefficients serves as damage locator. In fact, eFPCOMACk is proportional to the ratio between the difference of the .k;k/-th elements of the undamaged and damaged flexibility matrices, and the .k;k/-th element of the undamaged flexibility matrix, as clarified by Eq. (1.12): eFPCOMACk / Fk;k Pk;k Fk;k (1.12) Then, Eq. (1.12) may be rewritten in terms of the damage factor i , as shown in Eq. (1.13): eFPCOMACk / k PiD1 .1 i /fi k PiD1 fi (1.13) It is worth noting that the parameter i is equal to 1 when no damage occurred between the .i 1/-th andi-th DOFs, while takes on values greater than 1 if the inter-story between.i 1/-thandi-th DOFs is damaged. Then, if the structure is healthy, eFPCOMACk is equal to zero, while if damage has occurred between.k 1/-th andk-th DOFs, eFPCOMACk will showa value lesser than that of eFPCOMACk 1. In fact, by using Eq. (1.13), through simple algebraic operations, it is possible to obtain the following relation between the difference of two consecutive eFPCOMACk values: eFPCOMACk eFPXOMACk 1 / k 1 X jD1 . j k/fj (1.14) It is then easy to verify that a value of eFPCOMACk smaller than the one of eFPCOMACk 1 is indicative of damage occurrence between the .k 1/-thandk-th floors: eFPCOMACk <eFPCOMACk 1 ) k > k 1 PjD1 jfj k 1 PjD1 fj (1.15) Indeed, by noting that the right-hand-side of the inequality in Eq. (1.15) may take only values greater or equal to 1, it is apparent that k can take only on values strictly greater than 1, i.e. indicative of an increase in flexibility, if eFPCOMACk is lesser than eFPCOMACk 1.
1 Damage Detection Using Flexibility Proportional Coordinate Modal Assurance Criterion 5 1.3 Damage Detection Algorithm Up to this point, it has been shown that eFPCOMAC has a natural threshold at 0, so that if a deterministic model of the reference system was available, the damage detection routine could be performed avoiding any further processing. Nonetheless, external factors such as environmental or operational effects may cause significant changes in the structural characteristics that could be misinterpreted as manifestation of damage. For this reason, it is important to identify a range of values for each of the N eFPCOMACk’s within which the monitored structure may be considered healthy. In other words, it is necessary to determine a threshold for the damage sensitive feature which is able to distinguish between changes due to external effects and deviations of the vibration response due to damage. As aforementioned, eFPCOMAC is ideally suited as damage detector for shear-type systems. For this reason, the algorithm proposed in the following paragraphs is intended for application on this kind of structures. 1.3.1 Training Let us assume that nsets of time histories are recorded on the structure in its reference, usually undamaged, configuration. In this paper, with set of time histories, we refer to the ensemble of time histories measured during a single monitoring session through all the available sensors, M, placed on the system in analysis. Here, it is important to note that if M <N, amode shape expansion approach like the one proposed by Mukhopadhyay in [5] must be applied to get the full mode shape matrix. For each of the n sets of time histories the following operations take place, where the superscript .i/ indicates a quantity extracted from the i-th set of time histories: 1. Identification of the mode shape matrix .i/ 2RN N; 2. If an output-only system identification procedure is applied, a normalization of .i/ following the procedure described in Sect. 1.2.1 must be performed; 3. Identification of the modal frequencies vector .i/ 2RN 1; 4. Evaluation of the threshold for eFPCOMACk executing the following pseudo-code: for i D1 tondo Define RD .i/I Define r D .i/I for k D1to ndo Define TD .k/I Define t D .k/I for j D1 toN do Evaluate eFPCOMACj according to equation.17/ W eFPCOMAC.i 1/ nCk j D1 N PiD0 kRi k 2 kTi k 2 T2 i;k ti N PiD1 R2 i;k ri (1.16) end for j end for k end for i 5. Sorting of the n2 eFPCOMAC.i 1/ nCk j in ascending order and pick the element whose value is exceeded by the 95% of the samples as threshold for the j-th eFPCOMAC.
6 L. Balsamo et al. 1.3.2 Testing Herein, it is assumed that the number of data-sets available at testing is Qn < n. The testing is performed according to the following operations, where the superscript .k/ refers to the k-th of the Qn vibration response time histories measured on the possibly damaged system: 1. Identification of the modal characteristics of the system in unknown conditions: • .k/ 2RN N, matrix of mode shapes; • .k/ 2RN 1, vector of modal frequencies; In this case no normalization procedure is required, as the ‘non-proportionality’ of the identified mode shapes to mass normalized mode shapes is taken into account by the norm ratio in the eFPCOMAC definition. 2. Evaluation of Qn neFPCOMACj executing the following pseudo-code: for i D1tondo Define RD .i/I Define r D .i/I for k D1to Qndo ı Define TD .k/I ı Define t D .k/I for j D1toN do Evaluate eFPCOMAC TESTj according to equation.18/ W eFPCOMAC TEXT.i 1/ nCk j D1 N PiD0 kRi k 2 kTi k 2 T2 i;k ti N PiD1 R2 i;k ri (1.17) end for j end for k end for i 3. Perform a double check on the retrieved parameters according to the following criteria: for s D1to n Qndo for j D1toN do if .eFPCOMAC TEST .i/ j <Thresholdj/ &.eFPCOMAC TEST.i/ j < eFPCOMAC TEST .i/ j 1/ countj Dcountj C1 end for j end for k end for i 4. Damage is located between the .j 1/-th and j-th DOFs if countj has a value greater than 60%n Qn. The 60%n Qn threshold value is the result of an optimization problem aimed to minimize the damage location error rate.
1 Damage Detection Using Flexibility Proportional Coordinate Modal Assurance Criterion 7 Table 1.1 Results of the algorithm validation Type I and II error rates of the proposed algorithm Damage scenario First inter-story Second inter-story Third inter-story Fourth inter-story Fifth inter-story Sixth inter-story Seventh inter-story DS0 0%(Type I) 0%(Type I) 0%(Type I) 0%(Type I) 0%(Type I) 0%(Type I) 0%(Type I) DS1 1% (Type II) 0%(Type I) 0%(Type I) 0%(Type I) 2%(Type I) 0%(Type I) 1%(Type I) DS2 0%(Type I) 0%(Type I) 0% (Type II) 0%(Type I) 2%(Type I) 1%(Type I) 1%(Type I) DS3 0%(Type I) 0%(Type I) 0%(Type I) 0%(Type I) 8% (Type II) 0%(Type I) 1%(Type I) DS4 0%(Type I) 0%(Type I) 0%(Type I) 0%(Type I) 0% (Type II) 0%(Type I) 1%(Type I) DS5 0%(Type I) 0%(Type I) 0% (Type II) 0%(Type I) 6% (Type II) 0%(Type I) 1%(Type I) DS6 0%(Type I) 0%(Type I) 0% (Type II) 0%(Type I) 0% (Type II) 0%(Type I) 0%(Type I) 1.4 Results To test the ability of the proposed algorithm, we consider a 7 DOFs shear-type system whose response is simulated using the MATLAB Control System Toolbox Kit. For the training, 100 data-sets are simulated, i.e. n is equal to 100. Each data-set, consisting of the ensemble of seven acceleration response time histories recorded at each mass, is obtained by exciting the system at each DOF via white noise 1 min long time histories. Then, for each DOF, the overall training data are 100 min long. At each realization, the values of the i-th inter-story stiffness ki are randomly picked within the range Œ5:7 10 6, 6:3 106 N=m, the values of the i-th mass mi are randomly picked within the rangeŒ2:3 10 3;2:6 103 kg, while Rayleigh Damping model is employed to account for the damping effects. Testing is performed by using 10 data-sets, i.e. Qnis 10, of simulated acceleration response time histories recorded at each DOF of the system, creating a total of 10 min data recording for each DOF. The stochastic subspace identification algorithm ECCA is exploited for the identification of mode shapes and modal frequencies. Seven damage scenarios are considered: 1. No damage (from here on referred to as DS0); 2. Ten percent decrease of the inter-story stiffness between ground and the first floor (DS1); 3. Ten percent decrease of the inter-story stiffness between second and third floors (DS2); 4. Ten percent decrease of the inter-story stiffness between fourth and fifth floors (DS3); 5. Twenty percent decrease of the inter-story stiffness between fourth and fifth floors (DS4); 6. Twenty percent decrease of the inter-story stiffness between second and third floors, and between fourth and fifth floors (DS5); 7. Twenty percent decrease of the inter-story stiffness between second and third floors, and between fourth and fifth floors (DS6); For each of the 7 damage scenarios, 100 experiments are run and the results of the simulations are presented in Table 1.1 in terms of error rates. The problem of damage detection may be restated according to the binary decision theory syntax: given the null hypothesis .H0/ that the system is in its healthy condition, the task of damage detection is that of assessing whether the null hypothesis can be accepted or rejected. Rejecting the null hypothesis, when instead it should be accepted, is known as Type I error whereas accepting the null hypothesis, when instead it should be rejected, is defined as Type II error. Let us now dwell on the damage location problem, and let us consider the problem of assessing whether the area between the (i 1)-th and i-th DOFs, here onwards referred to as i-th inter-story, is damaged. If the i-th inter-story is healthy but it is declared damaged, a Type I error is made, whereas if the i-th inter-story is damaged but it is declared healthy, a Type II error is made. This is the criterion to consider when reading Table 1.1. For example, Damage scenario 1 represents the condition when damage occurs at the first inter-story; then, a Type II error occurs if the first inter-story is declared healthy, while a Type I error is observed if the other inter-stories are declared damaged. In particular, for damage scenario 1, the Type II error rate is equal to 1%, while Type I error rate is equal to 3%. The results prove the robustness of the algorithm both in detecting and locating the damage, as both Types I and II error rates are satisfactorily low. Nonetheless, it is worth to make some considerations on the outcomes of damage scenarios DS3 and DS5, both representing the case where damage occurred at the fifth floor, since they are characterized by the poorest, albeit still acceptable, quality. These results are due to the combination of two aspects. Firstly, the characteristics of the FPCOMAC are such that a low damage at higher inter-stories is difficult to detect, as it is evident by analyzing the analytical definition of eFPCOMACk given in Eq. (1.13). In particular, the value of eFPCOMAC5 depends on the values
8 L. Balsamo et al. of the difference between healthy and damaged flexibilities at the fifth floor as well as at the four preceding floors, which can then mask the presence of damage. Secondly, the threshold employed to take into account the structural characteristic variations caused by external factors can cover the presence of damage, when this is too low. This raises an important issue, not often addressed in the literature, i.e. the question of how to tackle with the problem of model uncertainties caused by diverse factors including environmental and loading conditions, as well as the intrinsic uncertainties introduced by the modeling process itself. 1.5 Conclusions A statistical damage detection algorithm was proposed. The proposed technique attempts to combine the model and non-model based approaches, which are often incorrectly considered as mutually exclusive. The recent advances in the field of system identification permit to extract the modal characteristics from the structural vibration response delivering accurate model of the system at hand, as the case being for ECCA, the stochastic subspace identification algorithm employed in this work. Nonetheless, the trend of using the modal parameters to generate a deterministic model of the system is questionable, as uncertainties associated to the model itself and to the effects of external factors, such as environmental and loading conditions, require the problem to be treated in a probabilistic framework. This is, indeed, the approach that has been proposed in this work, where modal analysis is used to evaluate the damage sensitive feature eFPCOMAC, while the damage detection algorithm is developed according to the training and testing phases typical of a statistical pattern recognition framework. The algorithm is validated by constructing the training model through a database obtained by simulating the response of a system in different operational conditions characterized by a variation in its structural properties during the different simulation sessions. In addition, a methodology apt to normalize the mode shape matrices identified through output only algorithms to be proportional to the mass normalized ones enables the proposed damage detection routine to be used when output-only data are available. The method proved to be effective in detecting and locating the damage for a shear-type system. The results obtained by performing numerous tests employing simulated response of a 7 DOFs system suggest that, even for relatively low damage magnitudes, the algorithm is able to detect the damage location, although some false acceptance error are recorded when low damage occurs at high DOFs. This outcome is due to the way the damage sensitive feature is defined: the eFPCOMAC evaluated at the higher DOF is influenced by the values of all of the preceding flexibilities, i.e. the deviation from the threshold may not be significant for damage scenarios where damage is located in a region far from the ground. On the other hand, the threshold defined during the training phase sometimes could not encompass properly the healthy system behavior, leading to the manifestation of Type II kind of error for relatively small damage. The latter consideration highlights the necessity of making the issue of how to deal with uncertainties correlated to the training model the primary focal point of future research. References 1. Sohn H, Farrar CR, Hunter NF, Worden K (2001) Structural health monitoring using statistical pattern recognition techniques. J Dyn Syst Meas Control 123:706–711 2. Gul M, Catbas FN (2011) Structural health monitoring and damage assessment using a novel time series analysis methodology with sensor clustering. J Sound Vib 330:1196–1210 3. Mukhopadhyay S, Lus H, Hong AL, Betti R (2012) Propagation of mode shape errors in structural identification. J Sound Vib 331(17):3961–3975 4. Hong A (2010) Model order determination in stochastic system identification for civil infrastructure systems. Columbia University, New York 5. Mukhopadhyay S, Betti R, Lus H (2013) Output only structural identification with minimal instrumentation. In: Proceedings of the 31st international modal analysis conference, Garden, California, USA
Chapter2 Automated Selection of Damage Detection Features by Genetic Programming Dustin Harvey and Michael Todd Abstract Robust damage detection algorithms are the first requirement for development of practical structural health monitoring systems. Typically, a damage decision is made based on time series measurements of structural responses. Data analysis involves a two-stage process, namely feature extraction and classification. While classification methods are well understood, no general framework exists for extracting optimal, or even good, features from time series measurements. Currently, successful feature design requires application experts and domain-specific knowledge. Genetic programming, a method of evolutionary computing closely related to genetic algorithms, has previously shown promise as an automatic feature selector in speech recognition and image analysis applications. Genetic programming evolves a population of candidate solutions represented as computer programs to perform a well-defined task such as classification of time series measurements. Importantly, genetic programming conducts an efficient search without specification of the size of the desired solution. This preliminary study explores the use of genetic programming as an automated feature extractor for two-class supervised learning problems related to structural health monitoring applications. Keywords Structural health monitoring • Genetic programming • Feature extraction • Damage detection • Supervised learning 2.1 Introduction 2.1.1 Structural Health Monitoring Structural health monitoring (SHM) aims to replace ad-hoc inspection programs and preventative maintenance of civil and aerospace structures with on-line systems providing real-time structural performance and damage state information. One of the biggest roadblocks to implementing these systems is the development of specialized signal processing for each individual application. Robust damage detection algorithms are the first step toward practical structural health monitoring systems. Typically, a damage decision is made based on time series measurements of structural responses. Data analysis involves a two-stage process, namely feature extraction and classification. While classification methods are well understood, no general framework exists for extracting optimal, or even good, features from time series measurements. Currently, successful feature design requires application experts and domain-specific knowledge. D. Harvey ( ) Graduate Student, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA 92093, USA e-mail: dyharvey@ucsd.edu M. Todd Professor University of California, San Diego, 9500 Gilman Dr., La Jolla, CA 92093, USA e-mail: mdtodd@ucsd.edu R. Allemang et al. (eds.), Topics in Modal Analysis, Volume 7: Proceedings of the 31st IMAC, A Conference on Structural Dynamics, 2013, Conference Proceedings of the Society for Experimental Mechanics Series 45, DOI 10.1007/978-1-4614-6585-0 2, © The Society for Experimental Mechanics 2014 9
10 D. Harvey and M. Todd 2.1.2 Genetic Programming Genetic Programming (GP), formally introduced by Koza [1], employs search methods based on the biological processes of natural selection and evolution to solve optimization problems. GP is a unique variant of evolutionary computation methods as candidate solutions are represented as computer programs, hence the name. Importantly, Genetic Programming conducts an efficient search with minimal specification of the size and structure of the desired solution. GP has previously produced human-competitive and patentable solutions in the areas of circuit design, image and signal processing, industrial process control, bioinformatics, financial trading, and others [2]. Genetic Programming is well suited for problems with the following properties [2]: 1. Poorly understood relationships between variables. 2. Solution size and shape unknown. 3. Large datasets. 4. Good solutions are easy to judge but hard to find. 5. No analytic solutions available. 6. Approximate solutions are acceptable. 7. Small performance improvements are beneficial. 2.1.3 Paper Overview This preliminary study explores the use of genetic programming as an automated feature extractor for two-class supervised learning problems related to structural health monitoring applications. This paper is organized into six sections including the current introduction. Section 2.2 briefly covers details of the Genetic Programming system developed for this work. The reader is directed to [1, 2] for deeper understanding of GP concepts and terminology. In Sect. 2.3, GP performance is evaluated on three signal detection problems with known optimal results. Section 2.4 summarizes the findings of this study and proposes future work. 2.2 Genetic Programming System 2.2.1 Solution Structure The current problem involves binary classification of vector time series measurements. Classification is a natural task for GP; however, input data to GP typically contains multivariate scalar measurements not structured data such as signals and images. A number of approaches to handling structured data have been proposed such as strongly-typed GP, automatically-defined iterations, and the scanning approach adopted here. As demonstrated in [3], by recursively feeding in samples of the input vector, a type of non-linear digital filter is created that scans along a time series. For input Xi withi D0;1; : : : ;N 1, each output, Yi , is a function of the current input, past output, and sample index with the final output, YN 1, saved as the output feature for classification. Yi Df.Xi ;Yi 1; i/ (2.1) Genetic programming is used to search for the function, f./, that optimizes a fitness measure. The function is represented as a GP tree. A tree is a hierarchical structure where each node has one and only one parent node but an unspecified number of children. In a GP tree, function nodes have as many children as the arity of the function. The end of each branch is a terminal node representing input data or a constant. The tree is evaluated starting at the bottom with outputs from each function passed as inputs to the next level of the tree until the root is reached. For example, the function Yi Dsin X2 i CXi CYi 1 (2.2) is produced by the tree in Fig. 2.1. The corresponding program code is y[i]=sin((x[i]*x[i]))+x[i]+y[i-1].
2 Automated Selection of Damage Detection Features by Genetic Programming 11 Fig. 2.1 Example GP tree for function in Eq. 2.2. White nodes are functions, and black nodes are terminals 2.2.2 Function and Terminal Sets The function and terminal sets define the programming language available to build the solution trees. The function set used here consists of the standard addition, subtraction, multiplication, and division operations plus four additional functions. Absolute value, sine, and cosine functions are included as common signal processing operators. Lastly, a three-input sigmoid function defined as sigmoid(a,b,c)=tanh(c*(a-b))provides a degree of switching functionality. The terminal set is composed of the current input, past output, sample index, and ten constants selected to cover four orders of magnitude. 2.2.3 Fitness Fitness of the candidate solutions is measured by area under the receiver operating characteristic’s curve (AUC) with the feature YN 1 used to detect to which of two classes a signal belongs. The receiver operating characteristic’s (ROC) curve defines probability of detection (PD) as a function of probability of false alarm (PFA) and fully characterizes the performance of a detector. The AUC provides a scalar indicator of detector performance bounded between 0 and 1 and is therefore a suitable fitness measure. To ease computation requirements, the detector output for each class was assumed to be normally distributed allowing AUC to be calculated analytically from the mean and standard deviation of the two classes. For performance evaluation of the final solution, the normality assumption was removed. Fitness case subsampling was employed to avoid overfitting and further decrease computation requirements with less than 10% of the fitness cases randomly selected for fitness computation during each generation. 2.2.4 Breeding The GP search process operates by preferentially breeding the better solutions from one generation to populate the next generation with offspring through various genetic operators. The crossover operator replaces a subtree in one parent with a randomly selected subtree from another parent. The mutate operator, here implemented as a headless-chicken crossover, replaces a subtree in the parent with a random subtree. Finally, reproduction allows an individual to pass unchanged to the next generation. The three operations of crossover, mutation, and reproduction occur with probabilities of 80, 10, and 10%, respectively. Parents are selected through deterministic four individual tournaments. 2.2.5 Genetic Programming Summary Table 2.1 summarizes the GP system and chosen parameters for this study. For most problems, fifty runs were performed in parallel on six processing cores of a single workstation. A flexible Python implementation of Genetic Programming was developed for this project and future studies. Python was chosen due to its open-source licensing, speed, and parallel processing capabilities.
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