Structural damage identification by a cross modal energy sensitivity based mode subset selection strategy
Introduction
Because of increased safety requirements and financial implications, structural health monitoring (SHM) for offshore engineering structures, such as offshore jacket platforms, offshore wind turbines, and booster stations et al. has been of great significance in the last few years. [1] Vibration-based damage identification (VBDI), as a hot topic of SHM, has been used in some situations [2] and emerged as one of the most promising technologies. [3], [4], [5], [6], [7] This term generally refers to the process of implementing a damage identification strategy for the structure of interest by observing the dynamic response measurements at periodic intervals, extracting damage sensitive features from these measurements, and statistically analyzing these features to determine the current state of structural health. [8] The VBDI is a kind of global damage identification method that does not require that the vicinity of damage is known a priori and that the portion of the structure being inspected is readily accessible. In addition, it is well-suited to automation. [9] However, there are some issues remaining open in the practical application of the vibration-based methods to offshore engineering structures, such as the low sensitivity of dynamic features to structural damage, insufficient measurement data of the structure and the uncertainties that may exist in damage identification, including measurement noises, environmental and operational variations, [10], [11] modeling errors, [12] etc. This study only focuses on the first issue with the aim of enhancing the sensitivity of dynamic features to damage. Other issues will be discussed in future studies.
Among VBDI technologies, the MSE-based methods have been paid more attention because of the excellent sensitivity to damage. [13] The basic idea of MSE-based methods is that, when damage occurs, the distribution of strain energy originally stored in the structure will change in a more pronounced manner in the damaged areas. Therefore, changes in the strain energy distributions of the undamaged and damaged structures can be used to identify the existence, location and extent of damage. [8] Dessi and Camerlengo [14] overviewed and compared several MSE-based methods by evaluating their performances on a damaged Euler–Bernoulli beam. Guidelines for selecting the most appropriate technique for damage identification in practical cases were provided. Wang and Xu [13] compared four best-known MSE-based methods for damage localization by numerical simulation and experiment validation and argued that the modal strain energy decomposition (MSED) index is most effective for locating damage under the effects of noises and spatial incomplete measurements. Hu et al. [15] proposed a MSE-based method, i.e., the cross modal strain energy (CMSE) method to identify simultaneously the location and the extent of damage within the structural system. Unlike the generalized sensitivity analysis methods, this method is strictly derived, thus overcoming the truncation errors due to Taylor series expansion. [15] Then, by considering both the CMSE and the cross-modal kinetic energy (CMKE) of the structure, this method is extended as a model updating technique called CMCM method. [16] The adjective “cross” here indicates that ME-like terms are product terms extending over various modes, thus enlarging the number of available modes and alleviating the indeterminate inverse identification arising from insufficient mode order. Another advantage of the CMCM method is that the stiffness and mass of structural elements can be updated simultaneously. Damage can be defined as the changes of material and/or geometric properties that influence the dynamic behaviors of the structure. Thus both the changes in stiffness and mass can regarded as a kind of damage. A common assumption of most of the VBDI methods reported in the technical literature to date is that the mass of the structure does not change appreciably due to damage. Therefore, the damage can be simulated as a reduction of stiffness of in a local region of the structure. However, this assumption is not valid for offshore engineering structures because the mass distribution of these structures will change due to structural reinforcement, marine biofouling and other environmental and operational condition variations. In this study, damage detection is implemented using the CMCM method under stiffness and mass change simultaneously.
Like many other damage identification methods,[17], [18], [19] the CMCM method involves identifying damage by solving an overdetermined system of linear equations. The coefficient matrix of the CMCM system collects the CME of the suspiciously damaged elements. In order to identify damage more effectively, it is necessary to select the best subset of the analytical and measured modes. In general, many modes do not undergo significant modifications due to the changes in stiffness and mass. In return, they only contribute to computational burden instead of the damage identification analysis. In addition, these redundant modes often lead to a complex CMCM system with the inherently ill-conditioned problem. Because of these two limitations, it is important to have a systematic criterion to evaluate how the employed modes are indicative of stiffness and mass changes.
Up to now, the MSSS for identifying the changes in structural stiffness and mass has been rarely studied, although it is extremely important to improve the damage detection accuracy. Kashangaki [20] proposed a modal sensitivity parameter (MSP), which combined the sensitivity of eigenvalues and eigenvectors to changes of structural physical parameters. The MSP can serve as a pre-test tool to indicate which eigenvalue–eigenvector pairs are most sensitive to damage. However, it is very expensive to calculate, because it needs iterative process. Doebling et al. [21] examined three MSSSs and found that the maximum MSE-based strategy outperforms that based on minimum frequency. Nevertheless, choosing the maximum MSE cannot guarantee the maximum sensitivity of modes with respect to structural damage.
This paper presents a MSSS specialized for ME-based methods. In order to apply the proposed scheme to a more general situation, the CME sensitivity instead of the usual ME sensitivity is derived and designated as the sensitivity index. According to this index, the MCs that are highly sensitive to structural stiffness and mass changes are selected. The CMCM method is taken as an example to evaluate the effectiveness of the MSSS. The performance of the CMCM method in conjunction with the MSSS is tested by analyzing two beams and an offshore platform structure.
Section snippets
Formulas of CME sensitivity
In this section, the CMSE and CMKE sensitivities with respect to both of the changes in stiffness and mass are derived. The derivations will be used to define a sensitivity index to select suitable MCs for damage identification. This starts by considering a structure that is discretized into finite elements. The CMSEs associated with the mth MC of the lth structural element and the whole structural system are respectively defined as where the superscript ‘*’ denotes
Numerical examples
In this section, numerical investigations are conducted by contemplating a Clamped–clamped beam and a real-world offshore platform structure to demonstrate the effectiveness of the proposed MSSS. To achieve this purpose, analyses by using the CMCM method combined with and without the proposed MSSS are compared.
Experimental setup
A cantilever beam was experimentally tested to evaluate the effectiveness of the proposed MSSS for a pure damage detection problem. The cantilever beam, as displayed in Fig. 28, had length of 200 cm, width of 5.0 cm and thickness of 2.8 cm. It was simulated by twenty Euler–Bernoulli beam elements.
Twenty accelerometers were vertically instilled at every 10 cm on the beam to collect its acceleration time-histories. The accelerometer used for tests was Model 2220-005 of SILICON DESIGNS with an
Conclusions
In this paper, the CME sensitivities with respect to both stiffness and mass changes were derived and designated as a pre-test tool to select damage-sensitive modes. The CMCM method was taken as an example to illustrate its practical values. Numerical and experimental studies were carried out by considering beam and offshore platform structures to prove the effectiveness of the proposed scheme. Two main aspects are emphasized by the results of the analyses. On one hand, the performance of the
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 authors disclose receipt of the following financial supports for the research, authorship, and/or publication of this article: This work is supported by the National Science Fund for Distinguished Young Scholars, China (51625902), the National Key Research and Development Program of China (2019YFC0312404), the Major Scientific and Technological Innovation Project of Shandong Province, China (2019JZZY010820), and the Taishan Scholars Program of Shandong Province, China (TS201511016).
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