River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Dynamics of Civil Structures, Volume 2 Kirk Grimmelsman Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics 2021 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 Dynamics of Civil Structures, Volume 2 Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics 2021 Kirk Grimmelsman Editor
Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-4380-012-5 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2022 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 Dynamics of Civil Structures represents one of the nine volumes of technical papers presented at the 39th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics and held on February 8–11, 2021. The full proceedings also include volumes on Nonlinear Structures and Systems; Model Validation and Uncertainty Quantification; Dynamic Substructures; Special Topics in Structural Dynamics and Experimental Techniques; Rotating Machinery, Optical Methods and Scanning LDV Methods; Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting and Dynamic Environments Testing; Topics in Modal Analysis and Parameter Identification; and Data Science in Engineering. Each collection presents early findings from analytical, experimental, and computational investigations on an important area within structural dynamics. Dynamics of civil structures is one of these areas which cover topics of interest of several disciplines in engineering and science. The Dynamics of Civil Structures Technical Division serves as a primary focal point within the SEM umbrella for technical activities devoted to civil structures analysis, testing, monitoring, and assessment. This volume covers a variety of topics, including structural vibrations, damage identification, human–structure interaction, vibration control, model updating, modal analysis of in-service structures, innovative measurement techniques and mobile sensing, and bridge dynamics among many other topics. Papers cover testing and analysis of different kinds of civil engineering structures, such as buildings, bridges, stadiums, dams, and others. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Philadelphia, PA, USA Kirk Grimmelsman v
Contents Particle Filters and Auto-Encoders Combination for Damage Diagnosis on Hysteretic Non-Linear Structures Subject to Changing Environmental Conditions ................................................... 1 Luca Lomazzi, Francesco Cadini, and Marco Giglio A New Benchmark Problem for Structural Damage Detection: Bolt Loosening Tests on a Large-Scale Laboratory Structure .......................................................................... 15 Onur Avci, Osama Abdeljaber, Serkan Kiranyaz, Mohammed Hussein, Moncef Gabbouj, and Daniel Inman Implementation of an Organic Database Structure for Population-Based Structural Health Monitoring 23 Daniel S. Brennan, Chandula T. Wickramarachchi, Elizabeth J. Cross, and Keith Worden Estimation of Blade Forces in Wind Turbines Using Strain Measurements Collected on Blades.......... 43 Bridget Moynihan, Babak Moaveni, Sauro Liberatore, and Eric Hines On Health-State Transition Models for Risk-Based Structural Health Monitoring......................... 49 A. J. Hughes, R. J. Barthorpe, and K. Worden Cointegration for Structural Damage Detection Under Environmental Variabilities: An Experimental Study................................................................................................. 61 J. C. Burgos, B. A. Qadri, and M. D. Ulriksen Footbridge Vibrations and Modelling of Pedestrian Loads .................................................... 69 Lars Pedersen and Christian Frier Multi-LSTM-Based Framework for Ambient Intelligence..................................................... 75 Nur Sila Gulgec, Martin Takácˇ, and Shamim N. Pakzad Operational Modal Analysis and Finite Element Model Updating of a 53-Story Building................. 83 Onur Avci, Khalid Alkhamis, Osama Abdeljaber, and Mohammed Hussein An Overview of Deep Learning Methods Used in Vibration-Based Damage Detection in Civil Engineering .......................................................................................................... 93 Onur Avci, Osama Abdeljaber, and Serkan Kiranyaz Transfer Learning from Audio Domains a Valuable Tool for Structural Health Monitoring .............. 99 Eleonora M. Tronci, Homayoon Beigi, Maria Q. Feng, and Raimondo Betti Experimental Evaluation of Drive-by Health Monitoring on a Short-Span Bridge Using OMA Techniques ........................................................................................................... 109 William Locke, Laura Redmond, and Matthias Schmid Investigation of Low-Cost Accelerometer Performance for Vibration Analysis of Bridges ................ 129 Kirk Grimmelsman Real-Time Human Cognition of Nearby Vibrations Using Augmented Reality.............................. 139 Elijah Wyckoff, Marlan Ball, and Fernando Moreu vii
viii Contents Understanding Errors from Multi-Input-Multi-Output (MIMO) Testing of a Cantilever Beam.......... 147 Fernando Moreu, James Woodall, and Arup Maji Load-Displacement Behavior Clustering of RC Shear Walls Using Functional Data Analysis ............ 153 Hamed Momeni and Arvin Ebrahimkhanlou
Particle Filters and Auto-Encoders Combination for Damage Diagnosis on Hysteretic Non-Linear Structures Subject to Changing Environmental Conditions Luca Lomazzi, Francesco Cadini, and Marco Giglio Abstract Damages may naturally arise in structures within their life span due to the insurgence of phenomena related to normal operation. Their occurrence might also be favored by external boundary conditions the systems experience during their lifetime, such as time-varying environmental and operating conditions. Standard maintenance activities, such as scheduled non-destructive testing (NDT) and corrective maintenance, are typically carried out to improve the health and longevity of such systems, typically entailing long downtimes with significant economic impacts. In recent decades, condition-based maintenance strategies (CBM) or even predictive ones (PM) have increasingly gained popularity since, in principle, they allow to optimally intervene on the structure only when really required by its current conditions. These maintenance schemes require that a deep knowledge of the system current state of health and, possibly, of the main degradation mechanisms be available, which may rely on advanced structural health monitoring (SHM) systems being installed on the structures for performing real-time diagnosis and prognosis. Many approaches to SHM have been formulated, with several applications to mechanical, aeronautical, space, and civil structures. Particle Filters (PFs) have been proposed as a model-based, time-domain tool for estimating hidden, not observable system states, including those normally affected by damage, in particular, when the structure behavior is non-linear and affected by non-Gaussian disturbances and noises. Yet, in case of varying operating and environmental conditions, the SHM task often still turns out to be quite challenging, since the diagnostic features associated with damage can be significantly distorted. To overcome this issue, auto-encoders have successfully been employed to extract damage-related features in presence of such varying external conditions. Thus, this work aims at combining these two methods for developing an original approach to damage detection and localization in structures, robust with respect to changing environmental and operating conditions, capable of leveraging the specific benefits provided by the two aforementioned methodologies. The proposed algorithm is demonstrated with reference to the problem of damage diagnosis on a vibrating n-degrees of freedom system, featuring a non-linear stiffness component characterized by a Bouc-Wen hysteretic behavior and subject to varying temperature conditions. Keywords Structural health monitoring · Damage diagnosis · Changing environmental conditions · Non-linear · Bouc-Wen 1 Introduction Damage may arise within the lifetime of structural systems due to several favorable conditions such as occasional loads, timevarying environmental and operating conditions, and long service times. That is the case, for example, of civil structures, the damages of which are usually determined by aging [1] and by the constantly increasing loads they are exposed to, e.g., the frequency and the mass of vehicles crossing bridges. Those effects are also experienced by mechanical systems, which are as important to society as civil structures. For instance, that is common to wind turbines [2], whose 20 years designed service life in most of the cases has already been reached or will be in the nearest future in Europe. In the general framework of structures, several damages may occur, such as fatigue, creep, erosion, or corrosion. Those may arise and accumulate within the service life not only because of extreme events, such as generic overloads from earthquakes and winds, but also due to aging, which affects structures in normal operating conditions, eventually determining failure conditions. L. Lomazzi · F. Cadini ( ) · M. Giglio Dipartimento di Meccanica, Politecnico di Milano, Milano, Italy e-mail: luca.lomazzi@polimi.it; francesco.cadini@polimi.it; marco.giglio@polimi.it © The Society for Experimental Mechanics, Inc. 2022 K. Grimmelsman (ed.), Dynamics of Civil Structures, Volume 2, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-77143-0_1 1
2 L. Lomazzi et al. In this outlined context, among the many effective ways to increase the return on investment of the existing assets [3], lengthening the lifetime of such systems turns out to be particularly appealing. To this purpose, novel maintenance policies, such as condition-based and predictive maintenance techniques, are constantly gaining importance over time. Those rely on the information automatically gathered and processed by a net of sensors embedded in structures, performing the so-called Structural Health Monitoring (SHM). A quite ample literature is available on time and frequency domain-based methods that may be exploited to identify possible damage-induced deviations of features extracted from measured data, with the purpose of performing damage detection, localization, quantification, and, eventually, prognosis. Envelope spectrum [4], cepstrum [5], high-order spectrum, and coherence function are among the frequency domain-based methods [6]. Moreover, the exploitation of transmissibility functions (TFs) for performing damage detection is worth a particular mention [7, 8]. In fact, those both do not require any prior knowledge of the structural model and have been proved to always be damage sensitive [8]. Additionally, TFs may be exploited to perform a further step of the SHM procedure, i.e., damage localization [9], even though criticalities may arise in this practice [10]. On the other hand, time-waveform indices, orbits, probability density functions, and probability density moments are typical time domain-based methods employed in the SHM framework. By applying such methods a huge amount of data is accumulated over the lifetime of the monitored structure, with both the advantage of leading to detailed damage analyses, and the disadvantage of increasing the required resources for performing damage identification at any complexity level [6]. However, all the classical feature extraction methods are unable to distinguish between damage-driven deviations from the healthy structural behavior and those determined by a normal change in environmental and operating conditions, such as temperature and operative loads variations, thus sometimes leading to false damage detection alarms. This topic is widely discussed in [11, 12] and [13]. With the purpose of solving that issue, many methods known as data normalization [11] have been proposed in the available literature, which aim at suppressing the effects of the aforementioned confounding, yet inevitable, factors. Among those proposed approaches, singular-value decomposition [14], principal component analysis [15], factor analysis [16], cointegration [17, 18], and auto-associative neural networks [19] leverage on some identifiable feature shifts induced by damage, while other methodologies exist which exploit regression analysis to automatically detect the relationship between varying boundary conditions and measured features [20–23]. However, although possibly able to subtract the confounding factors effects from damage-related features, those proposed methods may only detect the presence of a damage, without further analyzing the data to localize and quantify it. An exception may be identified in the work by Limongelli [24], which proposes an interpolation-based damage detection method, leading, however, to frequent false alarms. Additionally, these methods are all completely tailored to a particular setup of interest, thus not allowing the generalization of the proposed approach to the damage assessment in generic structures, with few exceptions, e.g., [25]. More advanced time-domain methods employing the Bayesian framework have been recently exploited to perform structural parameters identification [26] and fault diagnosis [27]. Among those ones, the use of particle filters (PFs) seems to be particularly promising, since those can also cope with non-linear dynamics of the investigated system. For instance, PFs have been exploited in system parameters identification applications, such as the stiffness value identification in case of multiple-degree of freedom (MDOF) systems experiencing seismic-like accelerations [28]. However, to the best of the authors’ knowledge, no exploitation of PFs for assessing structures in time-varying external conditions have been investigated yet for performing damage diagnosis. The PF potentiality of predicting the system dynamic behavior in presence of nonlinear effects is particularly interesting in the civil engineering field. In fact, some civil structures, such as Reinforced Concrete (RC) frames, may show a non-linear behavior, in form of hysteretic behavior, when subject to dynamic loads, such as earthquakes. Many models have been proposed in the literature for accounting for the observed hysteretic behavior of such structures, (e.g., [29–31]). Among those many theories, the most reasonable and popular one is the Bouc-Wen model [32], which is widely adopted in parameters identification-aimed works [33]. Recently, damage diagnosis has been performed by means of neural networks [34, 35], thanks to the great capability of those machine learning tools of not requiring any physics-based model of the event under investigation, thus leading to damage-related features extraction even in case of complex dynamics. In particular, it seems particularly promising the exploitation of auto-encoders for performing damage detection [34, 36, 37]. For instance, those have been implemented in a frequency domain-based framework for automatically processing the information extracted from transmissibility functions built from vibration measurements [38], allowing to perform novelty detection in MDOF-modeled systems. However, as already mentioned, TFs may provide inaccurate results when employed for performing damage localization [39]. In this work, PFs and auto-encoders are combined into an original methodological approach, working in the time-domain, aimed at performing damage detection and localization in MDOF-like structures subject to changing environmental and external conditions, eventually characterized by non-linear dynamics. In particular, processing the information of available vibration measurements, i.e., accelerations, the PF provides an estimate of some not directly observable structural parameters. Auto-encoders are then employed for combining PF posterior estimation and temperature measurements, with the purpose of subtracting the effects of confounding influences from the damage-related features extraction process. Additionally, an
Particle Filters and Auto-Encoders Combination for Damage Diagnosis on. . . 3 automatic statistical threshold is implemented in the proposed algorithm as human-free decision-making tool for identifying the presence of a damage with a certain confidence level. The method is demonstrated with reference to a MDOF system whose parameters are taken from a literature benchmark case study [28]. The case study proposed in this work focuses on performing damage detection and localization considering a structure with a component affected by hysteretic behavior, with the aim of showing the proposed framework performances in presence of eventual non-linearities, which may severely hamper the damage diagnosis task. The paper is organized as follows. In Sect. 2 the generic methodological approach is presented, focusing on the specialization of the selected tools to the MDOF system model. Section 3 presents a case study consisting of a 3-DOF system affected by temperature variations, experiencing an earthquake-like acceleration. In Sect. 4 a critical discussion of the main results achieved in this work is presented, along with the conclusions and some possible future work. 2 Methodology The dynamic behavior of any kind of periodic structure, e.g., multi-span bridges and rotary machines, is commonly modeled exploiting the one-dimensional MDOF system formulation. Periodic structures identify those systems composed of the repetition of the same structural basic elements, i.e., spring, mass, and damper, eventually with different parameters, joint together. Moreover, particular elements may be introduced with the purpose of describing non-linear dynamics. In particular, the one-dimensional MDOF formulation is suitable for describing structures mainly experiencing motion in one single direction. Within this framework, damage is often simulated by reducing the stiffness value of one or more springs in the MDOF system [25, 36, 38]. In this work, it is assumed that acceleration measurements be available from sensors eventually installed on the real structure, thus allowing to estimate some hidden internal states of the structure, i.e., springs stiffnesses, by means of PF posterior estimation. Note, however, that the states estimation procedure would in principle work processing other type of measurements as well, e.g., masses displacements and velocities. The differential equations of motion governing the dynamic behavior of the MDOF system degrees of freedom are implemented in the proposed procedure. Those are exploited for generating the state-space database describing the system observed behavior, thus allowing the PF-based state estimation process. It is assumed that the acceleration value of each degree of freedom of the system be measured at each time step in which the differential equations of motions are discretized. Then, the PF algorithm samples a large number of particles, i.e., plausible system dynamic evolutions, and further assigns to each possible trajectory an importance weight by a Bayesian combination with the observations on the structure under assessment. Finally, properly combining importance weights and particles, a posterior estimation of some hidden states, i.e., the MDOF model structural stiffnesses, is provided. In case no varying operating and environmental conditions affects the scenario, the PF framework by itself may be able to perform damage detection, localization, and quantification [27]. On the other hand, when confounding external agents are present, even the only and simple damage detection procedure by means of the PF estimation procedure would produce unsatisfactory results. Hence, it is proposed in this work an original, fully integrated PF—auto-encoder framework, which leveraging the capability of the two combined methods allows to perform damage detection and localization even in presence of varying external conditions and non-linear system dynamics. A training phase of the auto-encoder with a healthy, temperaturedependent baseline allows the neural network to learn the unknown relationship between temperature and stiffness. Later on, in the so-called operational phase, damage diagnosis is achieved comparing the input data of the auto-encoder to the output data. Moreover, an automatic statistical threshold is introduced, with the purpose of establishing an automatic alarm system independent of any human interpretation of quantitative results. PF algorithms and neural network auto-encoders are briefly introduced herein, focusing on their mutual interaction within the general framework established in this work. For further mathematical details on the individual methods, the interested reader is referred to the vast available literature. 2.1 State-Space Model, Particle Filter an Auto-Encoders The proposed methodology refers to the model shown in Fig. 1, which represents a generic MDOF model of a mass-springdamper system. Here the assumption is made that all the damping values be proportional to a stiffness reference value through a coefficient β, which is common to all the dampers in the system [40]. This assumption of proportional damping is
4 L. Lomazzi et al. Fig. 1 MDOF mass-spring-damper system commonly adopted in modeling structural systems [41]. Moreover, it is assumed that the first degree of freedom is affected by a degrading hysteretic behavior described by the Bouc–Wen’s formula [28]: ˙r(t) = ˙x −βBW|˙x||r| n−1 r −γ ˙x|r|n (1) In order to setup the environment for the PF posterior estimation framework, an augmented state-space representation of the MDOF system considered is introduced. The position and the speed of each mass, the stiffness values, the unique proportionality damping constant, the Bouc-Wen parameters βBW, γ and n, and the hysteretic displacement r are set as state variables. Hence, considering the generic case of ndegrees of freedom: xτ = xτ 1 . . . x τ n y τ 1 . . . y τ n k τ 1 . . . k τ n β βBW γ n r (2) where xτ i is the position of mass i at time step τ, yτ i is the speed of mass i at the generic, discrete time step τ, kτ i is the i-th spring stiffness at time step τ, β is the proportionality constant of damping, according to the relationship cτ i = βkτ i , βBW, γ and n are the Bouc-Wen parameters, and r the hysteretic displacement. The system evolution in time, evaluated at each discrete time step τ, is described by a state-space model which has to be a hidden Markov process of the generic form: xτ =g xτ−1 , wτ−1 (3) where g(·) represents a generic, possibly non-linear function and wτ−1 is the process noise vector value at time step τ −1. The noise considered in this work is a Gaussian noise determined according to a multivariate distribution N(0, w), w = diag(σx1 . . .σβ). In particular, considering the MDOF system affected by degrading hysteretic behavior adopted as reference system in the proposed framework, five state evolutions are identified, according to the state variables listed in (2). The position of each degree of freedom, i.e., mass, of the system evolves in time according to: xτ+1 i =yτ i dt +x τ i +wτ position; ∀i =1, . . . ,n (4) That equation directly follows from the discretization in time of the system of differential equations of motion of the MDOF system, to which a position process noise wτ position is added. As a result of a similar procedure, the state-space equation governing the dynamic behavior of the velocity of each degree of freedom is: yτ+1 i = Fτ i mi dt +yτ i (1− dt mi (cτ i +c τ i+1)) +y τ i−1 dt mi cτ i +y τ i+1 dt mi cτ i+1 −x τ i dt mi (kτ i +k τ i+1)+ +xτ i−1 dt mi kτ i +x τ i+1 dt mi kτ i+1 +r τ i k τ i δi1 +wτ speed; ∀i =1, . . . ,n (5) where Fτ i represents a generic force acting on mass mi, ci is the damping coefficient value of the i-th damper, ri is the hysteretic displacement pertaining to the i-th degree of freedom, andδi1 is the Kronecker delta, which is introduced with the purpose of accounting for the fact that only the first degree of freedom is affected by degrading hysteretic behavior. Finally, wτ speed is the speed process noise. Equation (5) slightly changes for the first and the last degree of freedom of the system, since it must be updated canceling out the contribution of the missing components, i.e., the terms with xτ i−1 and yτ i−1 for i = 1, and those with xτ i+1 , yτ i+1 , kτ i+1 and cτ i+1 for i = n. The discretized dynamic equation governing the Bouc-Wen hysteretic displacement reads: rτ+1 =rτ +dt(y1 −βBW|y1||r τ|n−1 rτ −γ(y1)|r τ|n) +wτ r (6)
Particle Filters and Auto-Encoders Combination for Damage Diagnosis on. . . 5 where wτ r represents the hysteretic displacement process noise. Note that this equation is specialized to MDOF systems with degrading hysteretic behavior on the first degree of freedom only. All the variables which, in a healthy structure, are assumed to be invariant over time with respect to variations of external conditions other than temperature are modeled as an augmented state with constant value in time, to which some process noise is added to represent the effects of unmodeled dynamics. For instance, the equation governing the evolution of the spring stiffness value over time is: kτ+1 i =kτ i +wτ spring; ∀i =1, . . . ,n+1 (7) where wτ spring is the process noise related to stiffness. Similar equations determine the evolution of the other invariant parameters included in the state-space, i.e., β, βBW, γ, andn, with the only difference of replacing the process noise related to stiffness to the corresponding one, i.e., wτ damping, wτ βBW , wτ γ, and wτ n. Moreover, stiffness is the only parameter which is assumed to vary with temperature. Hence, rescaling the relationship reported in [25] in order to comply with the stiffness values considered in this work, the discretized equation governing the stiffness evolution in time due to eventual change in temperature is represented by: ki(T) =(0.1(T) 2 −18T +10000) · ki(0◦C) T ; ∀i =1, . . . ,n (8) where T identifies the external temperature measured in◦C. The equations shown above represent the evolution of the hidden Markov states of the MDOF system considered. Those hidden states are estimated within the PF framework processing the information from some measurements z, which are related to the system states through a model which is assumed to be known, which is determined by the following general function: zτ =h xτ, vτ (9) whereh(·) is a generic, possibly non-linear function andvτ is the Gaussian measurement noise vector at time stepτ,which is assumed to be distributed according to a multivariate distribution N(0, v). Assuming that only acceleration measurements are available, the explicit form of Eq. (9) for the i-th degree of freedom at time step τ is: zτ i = Fτ i mi − yτ i mi (cτ i +c τ i+1) + yτ i−1 mi cτ i + yτ i+1 mi cτ i+1 − xτ i mi (kτ i +k τ i+1) + xτ i−1 mi kτ i + xτ i+1 mi kτ i+1+ − rτ i mi kτ i δi1 +v τ i ; ∀i =1, . . . ,n (10) wherevτ i represents the acceleration measurement noise at time stepτ, so that v =diag(σz1 . . .σzn). When the acceleration equation for the extreme degrees of freedom, i.e., i =1,n, is considered, Eq. (10) has to be modified deleting those terms including xτ i−1 and yτ i−1 for i =1, those withxτ i+1 , yτ i+1 , kτ i+1 and cτ i+1 for i =n. The augmented state-space representation of the MDOF system presented above is considered within the PF posterior estimation framework for performing state estimation. Particle filtering represents a widely used model-based parameters identification and state estimation Bayesian method which can deal with complex scenarios, including non-Gaussian noise and non-linear dynamics. In particular, the selected PF algorithm is the sample importance resampling (SIR) algorithm, the main operative steps of which have already been described by the authors [42]. For brevity’s sake, in this work no detailed mathematical treatment of the topic is given. The interested reader is referred to the works in [28, 43, 44] for further details about the functioning of the PF SIR algorithm. However, as already mentioned above, exploiting PF posterior estimation only may not reveal successful in performing damage detection in case varying external and operational conditions arise, leading to inaccurate results. Thus, it is employed the original, unique framework already presented in the work in [42], to which the reader is referred to for the detailed description of the procedure adopted in this work. That framework consists of the integration of the PF posterior estimation with neural network auto-encoders, which allow to offset the effects of confounding factors from the damage detection and localization procedure. Auto-encoders are deep neural networks characterized by a symmetric structure of layers, consisting of one dimensionality reduction side, which reduces the input dimensionality, followed by a reconstructing side, which aims at producing in the output layer the same values fed in input. In particular, one auto-encoder per spring in the MDOF system is considered in this work, all sharing the same structure: one input layer, three hidden layers, i.e., mapping, bottleneck, and demapping layers, and one output layer (Fig. 2). The relationship between layers is set up during the so-called training phase,
6 L. Lomazzi et al. Acceleration measurements Stiffness estimates PF Temperature measurements Input layer Output layer Mapping layer Demapping layer Bottleneck layer Fig. 2 Proposed auto-encoder structure during which the implicit relationship between the inputs of the auto-encoder, i.e., temperature and stiffness, is learned by the network by analyzing supervised examples from a healthy baseline. During the operational phase of the damage assessment procedure, the auto-encoder receives as input the temperature value measured at the location of the respective spring, along with the PF-based estimate of the stiffness of that spring. The input signals are reconstructed in the output layer of the autoencoder according to the model learned during the training phase. In case no damage occurs, the healthy relationship between temperature measurement and spring stiffness entirely drives the stiffness value change, allowing to perfectly reconstruct the input values in the output network layer. Differently, in case damage occurs, which is modeled as a temperature-uncorrelated stiffness value change, the auto-encoder may be likely to giving rise to some discrepancies between the input and the output data. The reconstruction error between output and input data is processed to detect eventual anomalies in the MDOF system that affects the healthy baseline correlation between temperature and stiffness values. Hence, as already mentioned above, for performing damage localization of the anomalies, in the general case of an n-degrees of freedom MDOF model, nautoencoders are associated with thensprings in the system. Indeed, localization is allowed since only the reconstruction error of the auto-encoder associated with a damaged spring would be affected by the presence of the anomaly. According to previous literature works [36, 45, 46], the squared error between the inputs and the reconstructed outputs of the auto-encoders is defined as: SEτ i =( ˆT τ i − ˆˆT τ i ) 2 +(ˆkτ i − ˆˆk τ i ) 2 (11) where SEτ i is the error between the input variables (i.e., the temperature measurement ˆTτ i and the PF stiffness estimate ˆkτ i ) and the output variables ( ˆˆTτ i and ˆˆkτ i ) recorded at time step τ on the i-th spring of the system. Neural networks need the data fed in input to be normalized in order to perform properly, in particular, this is vital when more inputs characterized by different units of measurement are analyzed. In this work, the normalization procedure of the input data is achieved by means of the following linear transformation: ˆTτ i −(−50◦C) 120◦C−(−50◦C) (12) ˆkτ i 10N m (13) Hence, the squared error defined above can vary in the range SEτ i ∈ [0, 2]. In what follows, the values ˆTτ i and ˆkτ i directly refer to the already normalized input data.
Particle Filters and Auto-Encoders Combination for Damage Diagnosis on. . . 7 2.2 Definition of the Diagnostic Indicators The squared error value related to each spring in the system has to be further processed in order to eventually trigger a diagnostic alarm. Those alarms may be given comparing the SEτ i value with a properly defined threshold, as suggested in [38]. However, the threshold value definition is arbitrary and may lead to inaccurate results. Moreover, in case of a not properly defined threshold, false alarms may be given since noise affects the inputs signals to the auto-encoder, due to a combination of acceleration measurement noise, process noises simulating unmodeled dynamics and Monte Carlo errors. Hence, in order to try to solve these issues, the authors proposed an original method consisting of combining an automatic threshold definition module with the construction of both a deterministic and a probabilistic indicator [42]. In particular, each spring i =1, . . . ,n+1 is given one threshold value according to the following relationship: ATSE 99 i =p99(SE 1:τ0 i ) (14) wherep99(SE 1:τ0 i ) is the 99th percentile of the probability density function of the squared error defined in Eq. (11), computed over a fixed time window of widthτ0 time steps taken at the beginning of the operational life of the structure, when the system is assumed to be in healthy conditions. An on/off deterministic fault indicator is introduced for each spring i, which is built up, at each time step i, comparing the ATSEi 99 threshold value with the moving average over a sliding window of widthτ0 time steps of the squared errors for each spring i, i.e., μ SE τ−τ0+1,τ i . Note that the time window within which the automatic thresholds are defined coincides with the one over which the moving average is computed. Thanks to the properties of the moving average operation, the wider the window width, the more robust the algorithm to false alarms from error fluctuations, with the drawback of increasing the anomaly detection time. The deterministic fault indicator is defined according to the following relationship: Id τ−τ0+1,τ SE,i = ⎧ ⎨ ⎩ 1 if μ SE τ−τ0+1,τ i >ATSE 99 i 0 otherwise (15) Hence, the fault indicator is activated, i.e., is set to 1, whenever the signal μ SE τ−τ0+1,τ i exits the healthy region defined by the respective threshold. Therefore, only significant variations of the signal μ SE τ−τ0+1,τ i may trigger alarms, filtering out the confounding effects coming from noise and disturbances. Moreover, an additional, probabilistic fault indicator is introduced in order to further damp out possible false alarms triggered by its deterministic counterpart: Ip τ−τ0+1,τ SE,i = τ j=τ−τ0+1 Id j SE,i τ0 (16) where Idτ SE,i represents the instantaneous counterpart at time step τ of Id τ−τ0+1,τ SE,i , which is defined as: Idτ SE,i = 1 if SEτ i >ATSE 99 i 0 otherwise (17) 3 Case Study The case study presented in this work is developed considering a three degrees of freedom MDOF system on the basis of the one shown in Fig. 1. The system parameters are taken from the work in [28], according to which mi =1kg ∀i =1, . . . , 3, kj =9 N m andcj =0.25 N· s m (i.e., β =27.8·10−3) ∀j =1, . . . , 3). The first degree of freedom is assumed to be affected by non-linear hysteretic degradation with parameters βBW =2, γ =1 and n =2. The structural state variables are affected by normally distributed process noise characterized by zero mean and variances σ 2 w,position =10−16 m2, σ 2 w,speed =10−16m2 s2 , σ 2 w,spring =1.62·10−5N2 m2 , andσ 2 w,damping =8·10−13 for position, velocity, stiffness, and damping proportionality constant,
8 L. Lomazzi et al. respectively. The process noise related to the Bouc-Wen model adimensional parameters is characterized by variances σ 2 w,βBW = 9.6 · 10−7, σ 2 w,γ = 2.4 · 10−7 and σ 2 w,n = 9.6 · 10−7, while the hysteretic displacement noise is assigned the variance σ 2 w,r =10−16 m2. Similarly, the observation noise related to the only measured variable considered in this work, i.e., acceleration, is taken as σ 2 v = 1 · 10−5m2 s4 . The widely adopted practice of simulating the response to earthquakes using white noise accelerations to generate the forces acting on the system degrees of freedom is applied in this case study [33]. Northridge earthquake-like forces are considered as forcing terms, distributed as white noise with maximum excitation frequency of 30 Hz [28], absolute value acceleration amplitude in the range [0g;1.8g] [47] and time duration equal to the analysis time. The analysis time horizon is Thor =16.7 s, discretized in Nt =3· 10 4 time intervals of width dt =0.556ms each. The assumption is made that the structural elements are in equilibrium with the external temperature, which linearly decreases from the initial value 40◦C to the final one 10◦C within the analysis time horizon Thor. Damage is introduced in spring number 3 as a progressive linear degradation of the corresponding stiffness value, occurring at time t =8.3, up to a final stiffness value at the analysis time horizon equal to the 70% of the corresponding healthy one at the same temperature, i.e., at 10◦C. The sampling frequency is fs =1/dt =1800 Hz, which turns out to satisfy the Shannon sampling theorem, being sufficiently larger than the forcing function frequency fforce =30Hz. APFwith 3· 104 particles is used for estimating the whole state vector xτ at each time step τ. The particles’ values are initialized as follows: • the position and velocity states estimates are initialized according to a normal distribution with zero mean and variance σ 2 w,position =10−16 m2 and σ 2 w,speed =10−16 m 2 s2 , respectively; • the stiffnesses, the damping coefficient, the Bouc-Wen model parameters, and the hysteretic displacement are initialized to a random number between the 95% and the 105% of the actual state value to be estimated. The auto-encoders employed in the simulation need to be trained with a healthy baseline acquired on the system under assessment. That baseline is built up simulating the non-damaged system and estimating the model state-space with the PF. A time horizon Tbaseline hor =20 s is considered for the baseline acquisition, simulated by means of Nbaseline t =3· 10 4 time steps of width dtbaseline =0.667 ms each. The structural elements are assumed to be in instantaneous equilibrium with the external temperature value, common to all the elements in the system, linearly increasing from−50 to 120◦C. This wide temperature range is selected for making sure that in the operational phase of the SHM framework the auto-encoder is fed with temperature values inside the range it has been trained with. Indeed, this is only possible when a real temperaturecontrolled environment is considered or in artificially generated case studies. In real applications with no temperature control available, a long baseline needs to be acquired to ensure the possible temperature range is sufficiently represented. Since all the springs are identical and experience the same temperature value at the same time instant, i.e., no temperature spatial gradient is considered, it turns out it is enough to train one auto-encoder considering any spring in the MDOF system. Note that, in the operational phase, one copy of the trained auto-encoder per spring is considered, in order to allow damage localization. The temperature observations are generated at time step τ using ˆTτ = Tτ +wτ T, where wτ T ∼ N(0, 0.1)◦C is the corresponding observation noise considered herein. Thus, a total of Nbaseline t examples are generated and used for training the auto-encoder exploited in this work, the layout of which is described in Sect. 2.1. The MATLAB Deep Learning Toolbox™ is employed for building and training the auto-encoder. After the training phase it is possible to put on-line one auto-encoder per spring and to start with the operational phase. The state-space estimates of the spring stiffnesses are shown in Fig. 3, where the estimates are identified with red lines and the corresponding reference values with black ones. As already mentioned above, the estimated value of each stiffness is not initialized to the actual one. It can be seen that the PF estimate goes to the reference value with quite fast settling time, with satisfactory parameters identification performance. The same consideration also holds for the stiffness degradation due to damage, which is clearly identified. Note that in the present case study it is possible to identify the damaged spring just by looking at the PF estimate value (Fig. 3c). The framework further processes the PF-estimated stiffness values and the temperature measurements, corrupted with noise according to the healthy baseline construction procedure outlined above, which are then fed in real time to the three auto-encoders. The squared error SEτ i between the input and output values of the auto-encoder (solid lines), evaluated by means of Eq. (11), its average over the time window of width τ0 = 5000 time steps, corresponding to a time horizon of 2.8s, i.e., μ SE τ−τ0+1,τ i (black, dashed lines), and the corresponding automatic thresholdATSE 99 i (red dotted line), for the stiffnesses k1, k2, andk3 are shown in Fig. 4. The thresholds are only shown after the first τ0 time steps, which are required to automatically identify their values, under the assumption that the system is healthy. Figure 5 presents the deterministic fault indicator time history for each spring in the system, i.e., Id τ−τ0+1,τ SE,i , while the probabilistic counterparts Ip τ−τ0+1,τ SE,i are shown in Fig. 6. Spring i =1 is well characterized by the proposed framework. According to the fact that no damage is introduced on that
Particle Filters and Auto-Encoders Combination for Damage Diagnosis on. . . 9 Fig. 3 Temporal evolution of the particle filter stiffness estimates (red) and true stiffness values (black) of spring i =1 (a), i =2 (b), i =3 (c) spring, the mean squared error SEτ 1 stays within the healthy region defined by the thresholdATSE 99 1 (Fig. 4a). Accordingly, the deterministic index Id τ−τ0+1,τ SE,1 is never triggered (Fig. 5a). The probability of damage occurrence Ip τ−τ0+1,τ SE,1 is always equal to zero, except from a slight increase from time t =10.1s to t =12.9 s, determined by the fact that there is a peak instantaneous value of the squared error related to spring i = 1 that temporarily exits the healthy region at t = 10.1s. A similar behavior is identified in the other healthy spring, i.e., spring i = 2. However, in the last second of analysis the mean squared error SEτ 2 exits the corresponding healthy region (Fig. 4b), determining a false alarm in the deterministic fault indicator Id τ−τ0+1,τ SE,2 . Due to two wide oscillations of the instantaneous squared error value associated with spring i = 2, which occur at time instants t = 12.3s and t = 14.7 s, the probabilistic damage indicator Ip τ−τ0+1,τ SE,2 starts identifying a possible failure at t = 12.3 s. However, no sensible failure probability is reached, being the maximum registered value equal to 39%. Similarly, also spring i =3 is correctly identified by the proposed methodology. The progressive degradation damage in introduced in the model at time t =8.3 s and identified at t =11.5 s. In fact, at that time instant the mean squared error value SEτ 3 overcomes the safety threshold ATSE 99 3 triggering the deterministic fault indicator Id τ−τ0+1,τ SE,3 , which is kept on the alert level until the end of the analysis. Accordingly, the probabilistic damage indicator Ip τ−τ0+1,τ SE,2 reaches the peak value, i.e., 100% failure probability, at time t =13.1 s, without decreasing as time goes by.
10 L. Lomazzi et al. Fig. 4 Temporal evolution of the squared error SEτ i between input and output layers of the auto-encoder (solid lines) for k1, k2 andk3, the colored dashed lines indicate the corresponding automatic threshold. (a) Spring 1. (b) Spring 2. (c) Spring 3 4 Conclusions An original framework combining PF and auto-encoders is proposed in this study, with the aim of providing a tool suitable for performing damage diagnosis in structures subject to changing environmental conditions. The framework is specialized to the general case of mono-dimensional MDOF systems characterized by non-linear hysteretic behavior, and it is applied in Sect. 3 to a 3DOF system, proving that it satisfactorily provides both real-time estimation of each spring stiffness and information about the presence of eventual damages, along with their position. Eventual changes in the stiffness value of a spring in the system are provided by the particle filter embedded in the framework; however, it is not able to provide any insight in the cause of those variations, which may be caused either by damages or by changing external conditions. Hence, one auto-encoder per spring receives the stiffness estimate from the PF and offsets eventual temperature variation effects from it. In order to do so, the auto-encoder needs to be trained with a healthy baseline properly acquired to describe well the range of operating conditions the system is supposed to work in. In the operational phase, the squared error mean value between the input and the output of each auto-encoder is employed to detect any anomaly which may arise from eventual damages. The squared error mean value computed over a properly set
Particle Filters and Auto-Encoders Combination for Damage Diagnosis on. . . 11 Fig. 5 Temporal evolution of fault indicator Id τ−τ0+1,τ SE,i for springs 1 (a), 2 (b) and3 (c) Fig. 6 Temporal evolution of the probabilistic fault indicator Ip τ−τ0+1,τ SE,i
12 L. Lomazzi et al. moving time window is compared to an automatically set threshold for properly turning on the fault indicator Id τ−τ0+1,τ SE,i . Moreover, in order to damp out the false alarms due to eventual oscillations of the PF estimate, since increasing the particles involved in the simulation would significantly increase the required computational resources, a more robust probabilistic fault indicator Ip τ−τ0+1,τ SE,i is also computed for each spring i in the MDOF system, with overall satisfactory performances. As shown in the case study reported in Sect. 3, the deterministic fault indicator occasionally gives false alarms, while the probabilistic indicators provide more reliable information about eventual damages in the structure assessed. Further work still has to be done in this field. Expanding the proposed framework for considering changing external conditions different than temperature would provide interesting, more general results. 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