Fuzzy-driven strategy for fully automated modal analysis: Application to the SMART2013 shaking-table test campaign

https://doi.org/10.1016/j.ymssp.2020.107388Get rights and content

Highlights

  • A robust fuzzy-driven approach is derived for fully IO/OO automated modal analysis.

  • The methodology is applicable using any growing model-order identification method.

  • A new indicator called nMTN is derived for emphasizing weakly excited modes.

  • The efficiency of the strategy is demonstrated on actual shaking-table tests results.

Abstract

A crucial step when identifying the modal signature of systems using growing order parametric methods consists in discriminating spurious modes from physical modes. In this paper, a three-stages clustering strategy is presented in a fuzzy framework for automating this selection process in the context of Input/Output and Output-Only identification. The novelty and strong point of the approach lies in the first stage where, after computation of single mode validation indicators, a modified fuzzy c-means clustering procedure is developed for performing a first partition. It is shown how the membership function obtained for the cluster of physical modes can be interpreted as a new synthetic modal indicator and helps with pole-splitting detection, outlier rejection and generally improves the final modal parameters estimation. The developed methodology does not involve any user-specified threshold and can be used for discriminating modes produced by any methodology consisting in fitting a growing order model to experimental data of any type. In this paper, accelerations measured during the SMART2013 shaking-table test campaign are processed using data-driven state-space identification algorithms. The automated selection process is used for tracking the modal signature of a trapezoidal shaped reinforced-concrete specimen using in turn stochastic and combined deterministic-stochastic algorithms, defining for the latter the movement of the shaking table as input. Variations in the modal signature are then correlated to the damage actually observed on the specimen and a comparison between Output-Only and Input/Output results is made in order to estimate the interaction between the specimen and the whole shaking table device.

Introduction

Analyzing the response of real structures is an essential preliminary step to modeling and control of mechanical systems. For linear structures, the modal signature composed of frequencies, damping ratios and modeshapes summarizes this response. In order to characterize this key feature, measurements performed on real structures in operational context or on specimen under experimental dynamic loading are processed using identification methods described either in a time or frequency framework (see e.g. [1], [2] to only mention one of the most popular references of each case and [3] for a more complete overview). Those methods lie in fitting a linear model to time or frequency data; the modal signature is then extracted after eigenvalue decomposition of the matrix describing the dynamic behavior of the system. The order n of the model to be fitted being unknown, a current practice consists in performing identification for increasing values of n. One then needs to distinguish the sought physical modes from mathematical artifacts that arise owing to model-order over-specification.

This crucial discrimination step is generally done with the help of stabilization diagrams where physical modes can be detected as columns in a frequency versus model-order plot. However, a careful inspection of stabilization diagrams still requires time and the trained eye of an experimented user. This paper then focuses on automating the physical modes selection process among poles identified for large model order range based on a single data set. This automated modal analysis must be distinguished from modal tracking that relies on existing knowledge on modal parameters (based on numerical model or reference data) to detect changes in the modal signature that can further be related to global or local damage [4], correlated with environment conditions [5], [6] or be used for instability or fault detection [7], [8]. At the end of this paper, the evolution the modal signature of a reinforced concrete specimen is sought on the basis of repeated analysis on different data sets; no online monitoring is performed.

The attempts for automating or at least assisting the interpretation of stabilization diagrams are not new (see e.g. [9], [10], [11] for a more extensive bibliographic review). Early references relied on the computation of modal indicators: plotting modes with high indicator values leads to clearer stabilization diagrams. Among the most effective indicators, one can mention the Modal Phase Collinearity (MPC) [12], the Modal Transfer Norm (MTN) [13], the Modal Coherence Indicator (MCI) [14] and the Mean Phase Deviation (MPD) [9]. Other authors [15], [16], [8], [17] rely on uncertainty computation for discarding spurious modes; data sets are divided in (statistically independent) blocks and uncertainty bounds on modal parameters are obtained from perturbation theory. The poles whose modal properties have the largest standard deviations are declared spurious. Different clustering techniques have also been used for automating the selecting process usually referred to as hierarchical clustering, partitioning methods and histogram analysis (see Ref. [9] for details). Among the most effective and followed attempts, one can cite [18] that defined a methodology based on hierarchical clustering for grouping similar modes clusters and successfully performed the modal identification of a bridge.

The main drawbacks of the aforementioned references is the systematic presence of user-specified thresholds on modal distances for achieving proper clustering or on modal indicators for discarding spurious modes (see also [19]). Those thresholds depend on the application, on the amount and quality of available data, on the level of solicitation and must sometimes be relaxed depending on (expected) modeshapes complexity. Such methodologies have proven effective in the case of long-term monitoring where single structures are instrumented with permanent sensing devices; in that case the time spent for correctly adapting the thresholds to the problematic is of less importance compared to monitoring time. However, the seismic tests conducted in the TAMARIS/CEA facility impose a different context: the zoology of experimental systems is vast (reinforced concrete building [20], steel-frame structures [21], [22], timber-frame construction [23], tanks [24], etc.), the ground motion tests numerous, involving possibly different configurations and a wide variety of sensors. Manual user-dependent threshold calibration is a huge obstacle to automation and might be intractable in the case of specimens exhibiting important modal signature variations throughout test sequence. To overcome this obstacle, Reynders et al. in [9] proposed an efficient strategy for automating the mode selection process in three stages:

  • (i)

    Several modal indicators are first computed for each mode and a k-means (k=2) algorithm allows to remove the modes interpreted as certainly spurious modes,

  • (ii)

    A hierarchical clustering stage is then carried out on the subsisting family containing possibly physical modes for detecting groups of similar modes identified for different values of n,

  • (iii)

    A last partition stage is conducted for retaining only the most populated groups generated at stage (ii). This last group contains the physical modes because they are expected to be identified for every n.

The maximum within-clusters distance between representations of the same physical mode for different system orders is the decisive criterion for stopping the hierarchical clustering procedure in stage (ii). This criterion, that had to be specified by the user in [18], is derived from the statistics of the group selected after stage (i) in [9].

This strategy offers a high degree of automation, however some limitations have been experienced. The k-means algorithm used in stage (i) is not necessarily the best option for building the first partition, as it was also noticed in [25]. For most indicators computed in stage (i), the cluster of possibly physical modes resembles an exponential distribution whereas the cluster of spurious modes tends to be normally distributed. The k-means algorithm is always dominated by the variables with highest variance and, when applied to normally distributed data, the algorithm tends to split datasets in approximately equally sized clusters [26]. For an optimal partitioning, one would thus have to choose the maximum order n from the number of expected observable modes (that is not known in advance) or at least specify a sufficiently large maximum n in order to guarantee that no possible physical modes will be discarded after stage (i). Moreover, when plotting the respective cardinals of the groups identified after hierarchical clustering at stage (ii), in most applications, no clear jump is noticeable, i.e. no clear partition between ’heavily populated groups’ and ’poorly populated groups’ can be made by the k-means algorithm that is used at stage (iii) for the final selection. Once again, k-means will tend to build clusters of equal size.

The present contribution proposes an enhancement of the three-stages selection strategy described in [9] with the idea of overcoming the above-mentioned limitations. The key enhancement consists in replacing the k-means algorithm of stage (i) by a more adaptive fuzzy c-means algorithm that is modified to bring a better insight in the data to be split. The membership function, computed at stage (i), can then be interpreted as a synthetic modal indicator and will help the partitioning at stage (iii). At the end of the proposed algorithm, the means and standard deviations will be given for physical clusters in a fuzzy sense (weighted by the membership function), what will improve the overall quality of the modal estimates, empowering the most reliable candidates of each group and will help dealing with outliers and pole splitting phenomenon in the same time.

The robustness of the improved algorithm is illustrated using the measurements of the SMART2013 test campaign [20] during which a reduced scale Reinforced Concrete (RC) building was submitted to a sequence of shaking table tests. The sequence is composed of seismic tests of increasing level, that gradually activate non-linear mechanisms on the specimen, and intermediate low-level broad-band tests. The accelerations measured during intermediate tests are processed using subspace-based identification algorithms and the new pole selection strategy is carried out for an automated tracking of the modal signature along the test campaign. The evolution of the modal signature will be correlated to the damage actually observed on the specimen and a comparison between Output-Only (OO) and Input/Output (IO) results will be made in order to estimate the interaction between the specimen and the shaking table device.

The paper is organized as follows. Section 2 gives a brief recap of the SMART2013 test campaign, describes the processed dataset and details how subspace-based algorithms are tuned. The single mode validation criteria used in this paper are also defined, paying particular attention to a newly derived indicator, called normalized Modal Transfer Norm (nMTN) difference, that is developed for promoting the detection of weakly excited modes. Section 3 describes the fuzzy-enhanced three stages procedure that is central to this contribution. A specific focus is made on the modified fuzzy c-means procedure that has been derived at stage (i) for making a first rejection of artifacts. After quantifying its performance on clusters of known distribution, the procedure is carried out for modal analysis purpose. The way the resulting membership function is reused to enhance the procedure is illustrated using OO measurements performed at the beginning of the SMART2013 test campaign where the RC specimen is considered healthy. An thorough discussion on the comparative use of k-means vs. fuzzy-c-means algorithms at stage (i) is also present. The effect of this first partition on yielding hierarchical clustering threshold used at stage (ii) is particularly studied and this for different modal indicators-sets whose influence is also investigated. Section 4 presents the modal analysis results and the evolution of the modal signature throughout the test campaign, obtained from repeated stochastic OO and combined deterministic-stochastic IO identification. Section 5 finally gives conclusions.

Section snippets

The SMART2013 shaking table test campaign

End 2013, within the framework of the EDF-CEA SMART project, a three-story RC specimen was tested on the six degrees of freedom AZALEE shaking table of the TAMARIS/CEA facility. The singular trapezoidal design of the RC specimen (see Fig. 1) was chosen such as to emphasize torsional effects to which constructions are subjected during seismic loading. The 6×6m2 AZALEE shaking table is equipped with eight 1000 kN maximum capacity hydraulic MTS actuators and can reproduce complex seismic loading

Three-stages automated selection strategy

The strategy, initially proposed by [9] and used and improved in this paper, aims at separating the physical modes from the spurious modes produced by a growing model-order identification algorithm. Somehow, this strategy tries to mimic the classification that would be done manually by an expert user and is composed of three stages:

  • stage (i)

    : Based on the preliminiary calculation of modal indicators detailed in Section 2.3 this stage consists in partitioning the modes of the diagram into two families

Output-only framework

Table 4 represents the modeshapes associated to the six modes identified by the FCM-X[1:8] procedure and collected in Table 3. Those six modes dominate the dynamic behavior of the system Z on the low frequency range (see also Fig. 14). On each line of the table, spatial representations of the modeshapes are given for the sensors on the RC specimen and the participation of the accelerometers on the hydraulic actuators (not represented) are plotted as histograms with full scale harmonized with

Conclusion

This paper describes a fuzzy-driven multistage clustering strategy for automating the time consuming modal selection process when performing growing model-order system identification. Just like earlier references, in the continuity of which this work is part, the presented automated selection strategy does not contain any case-dependent manually-chosen threshold nor parameter and the fuzzy framework has proven to bring more robustness to the methodology in general and particularly with respect

CRediT authorship contribution statement

P.-É. Charbonnel: Conceptualization, Methodology, Software, Validation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The tandem EDF-CEA is deepfully acknowledged for funding this research activity and giving access the SMART2013 test campaign database. The work reported in this paper has been also supported by the SEISM Institute. F. Voldoire (EDF) is more personally thanked for the fruitful discussions while following up this research activity. C. Feau (CEA) and A. Le Maoult (CEA) are also thanked for being such valuable beta-testers: their permanent feedback has been an asset for the development of many of

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