Numerical simulations performed on linear and nonlinear systems demonstrate the applicability of the algorithms for real-time damage detection. 203 6.7.2 Case study for the 5 DOF B-W system excited using El Centro ground motion 205 6.7.3 Performance check of the proposed method against RPCA for a strong non-.
Aims and Objectives
RPCA, RSSA and RPCA-TVAR, RCCA algorithms come under the umbrella of first-order eigenperturbation (FOEP) techniques. Specifically, the performance of the hybrid algorithm in identifying spatio-temporal damage patterns will be studied in the context of real-time damage detection.
Organization of the thesis
- Model based damage detection methods
- Drawbacks of the model based methods
- Response based damage detection methods
- Drawbacks of the response based methods
2.1, the location and size of the damage inflicted on the system can be determined. The EVD of the matrix RX¯(p) can be expressed in the spectral decomposition format satisfying the relation, VX¯RX¯(p)VXT¯ =λX¯.
Recent trends in damage detection
PCA and structural damage detection
Each individual element of the covariance matrix formed by the POCs can be obtained from the following steps. The details of the RPCA-based approach can be found in Chapter 3 of this thesis.
Singular spectrum analysis (SSA)
This problem can be addressed by developing a real-time damage detection strategy that uses the input from only a single sensor and provides accurate estimation results. The new detection strategy RSSA, which is based on the basic principles of EVD (as in PCA), uses only the input of a single sensor to identify damage; the details of which are discussed in detail in Chapter 4.
Basic SSA
Let xij be the elements of the Hankel matrix, where i denotes the row number and j the column number. By subsequently applying SVD to the covariance matrix, the PCs of the time series are elicited.
First order eigen perturbation technique
This data provides a covariance matrix via the signal's trajectory matrix in an offline framework. The above expressions provide a central idea of the data-driven nature of the FOEP approach.
Summary
This chapter presents the theoretical development of a new method called recursive principal component analysis (RPCA). Finally, a comparison between the traditional PCA and its recent variant, RPCA, is provided, highlighting the superiority of the RPCA in the exact identification of immediate damage.
Motivation
Problem formulation
This algorithm is based on a rank-one update of the eigenspace of the covariance matrix applied to the data vectorxk. The covariance matrix can be interpreted as consisting of eigenvalues and eigenvectors, and when new sampled data become available, the eigenstructure is updated as a whole instead of directly updating the covariance matrix, thus ensuring an immediate update of the eigenvalue and eigenvector matrix. vectors in a recursive manner.
Recursive covariance estimation and FOEP
RPCA: Theoretical development using POMs
From equation 3.12, it is clear that Ωk−1 can be understood as a sum of QTQ and first-order error terms [88]. This can be solved by sorting the basis vectors in descending order of the corresponding eigenvalues in Ωk.
Damage detection using real time condition indicators
- Recursive residual error (RRE)
- Recursive eigen vector change
- Outlier detection using correlation coefficient (ρ)
- Local damage detection
Therefore, tracking the change in the eigenvector updates is expected to show deviations that may correspond to the moment of damage. It is essential to note that there is no change in the sign of the correlation coefficient due to the presence of outliers.
Proposed Algorithm
3.1, it is clear that the RPCA algorithm can process data as and when it flows in (online) and provides a recursive update of the eigenstructure at each data point, which is further used by the recursive CIs to track the moment of damage. It is also clear that there are no parameters that control the operation of the algorithm, i.e.
Numerical Example
- Structural model and simulation parameters
- Results for White Noise
- Results for Underdetermined case-White Noise Excitation
- Comparative study with batch PCA
- Results for El Centro ground excitation
The constant κ controls the non-linearity introduced into the equation of motion of the system (through the non-linear force term). However, the proposed RPCA-RRE approach indicates the exact moment of damage by a change in the mean level of the plot.
Practical implementation studies
Experimental study
A realistic damage case used in this study involves a change in the nonlinear state of the system, caused by the sudden rupture of the rubber band at a certain instant of time. Rupture of the rubber band causes an overall reduction in the stiffness of the model and is interpreted as damage to the system.
Case study of the UCLA Factor building
The table shows that the percentage change in RRE decreases (from 57.39% to 33.33%) with the decrease in the number of sensor data available as input to the algorithm. The figure shows that εRR −Yi deviates from the environmental regime around t=380s and significantly near t=410s, indicating the occurrence of damage.
Summary
The RPCA-TVAR method is a direct extension of the RPCA-RRE approach discussed in detail in the previous chapter. First, the fundamental concepts of the RPCA-TVAR approach, which are key to the development of the proposed methodology, are reviewed.
Motivation
Background
Therefore, a relatively low model order of the time series model [51, 53] is sufficient to capture the dynamics of the structure in the transformed domain. The online damage detection framework, excluding basic data, is used to identify the damage event in the monitored system through the time-varying coefficients of the TVAR models.
RPCA and structural dynamics: A POC based formulation
From the above expression, RQ = N1QQT can be identified as the covariance matrix of the modal responses. Since the term (k−1)Υk−1+ ˜ψkψ˜kT is diagonally dominant, the eigenvalues can be assumed to be the diagonal entries of the matrix.
TVAR modeling
This can be addressed by ordering the basis vectors in descending order of the corresponding eigenvalues in Υk. The following equation is the discrete representation of the coefficients ai(t) and w(k) is the process noise with variance σ2w and covariance, Pw=Ip×pσw2.
Damage sensitive features
Time varying auto-regressive coefficients
In the context of the present study, the TVAR coefficients are treated as the main DSFs. The proposed method tracks the TVAR coefficients online, which are used to ascertain the damage caused to the structure by a change in the top of the plot, at the exact moment of the damage.
Recursive statistics on TVAR coefficients
Although a1 and a2 are expected to vary slightly at each time point, the moment of damage is characterized by sudden changes in the overall behavior of the VATR coefficients after damage.
Proposed Algorithm
First, batch PCA is used on some initial data points to estimate the initial eigenvector and eigenvalue matrices, that is, the initial eigenspace. It should be noted that the aforementioned proposed algorithm has the following few features: (i) the data is processed at every moment, as and when it becomes available, i.e. the algorithm works online. ii) to detect the moment of damage, a reference value (baseline) is not necessary, i.e. iii) there are no parameters governing the operation of the algorithm, hence its parameter free.
Numerical example
Temporal damage detection results
- Spatial Damage Detection Results
- Results for El Centro ground excitation
- Results for time-diluted damage
- Results for underdetermined case-White Noise Excitation
4.4(a) and 4.4(b) it is clear that there is a significant change in the estimate of the recursive mean. Once the moment of damage is detected, the spatial RRE in the vicinity of the damage is examined (say 29s to 33s).
Experimental study
Detection results for the experimental case
Along with this figure is the plot of the other TVAR coefficient (a2(t)) which clearly outlines the accurate damage instantaneously at 33s, indicating the occurrence of damage. To validate these findings, the HOM (ζai(t)) plot shows a clear change in the mean level, thereby corroborating the accurate moment of the rubber snap, an event that closely corresponds to a real life injury.
Case study for the UCLA factor building
Consequently, it can be concluded that the proposed method provides quality detection results even when the nature of the excitation is predominantly non-stationary. The temporal RRE diagrams illustrated in the previous chapter (Fig..) could not depict significant deviations at the onset of the earthquake at 380s, but could clearly capture the prominent damage moment around the 410s mark.
Summary
To the best of the author's knowledge, concepts using RSSA in the context of real-time structural damage detection have not been investigated in the literature so far. Since most of the established damage detection algorithms work offline in a batch process, the development of online algorithms becomes necessary in the context of real-time structural damage identification [75, 96].
Problem formulation
RSSA: Theoretical development
This can be mitigated by using the FOEP approach [134], which provides recursive updates of the eigensubspace from the previous eigenspace of the data at a given instant in time. Depending on the relative contribution of the eigenvectors, the number of PCs required for reconstruction can be automated.
Recursive damage indices
Damage detection using recursive eigen ratio difference
The principle behind the use of this marker is that structural damage or instability induces a low-frequency component in the signal, resulting in a separation between the two most important computers. At the moment of injury, the recursive ERD shows a change in its mean level, indicating a separation between the two most important singular values.
Multichannel singular spectrum analysis (MSSA)
Recursive multichannel singular spectrum analysis (RMSSA)
The main steps in the formulation include the update of the covariance matrix formed at each timestamp and the provision of eigendecomposition to tailor the basic MSSA algorithm to its recursive version. As the data evolves from the zero mean process, along similar lines of evolution, the eigendecomposition of the covariance matrix at the kth time can be written as Ck = WkΩkWTk.
Proposed RSSA based framework
The RSSA algorithm then works online ingesting streaming data in real time. Using the recursive gain depth parameter, the covariance estimate of the Hankel matrix at the current instant is derived using the covariance estimate at the previous time instant.
Hybrid FOEP based RPCA-RSSA framework
The RPCA algorithm then operates online on the real-time input of the stream data. Using the FOEP approach, the recursive updates of the eigenvector and the eigenvalue matrices are updated at each time instant and a new set of transformed responses is obtained, evident from the Eq.
Numerical studies
Numerical case studies using RSSA
Detection results using RSSA
Temporal damage detection results for the 5 DOF B-W system using RSSA 126
The change in the mean level of the AR plot occurs at 40s from the start. The response from the first floor, when supplied as input to the algorithm, only shows changes at 40 seconds from the start of the event.
Performance of the RSSA algorithm against recently established damage de-
As evidenced by previously elucidated case studies, the damage detection potential of the RPCA algorithm is TH. This confirms the superiority of the proposed RSSA algorithm over the aforementioned damage detection schemes.
Results for the SDOF Duffing oscillator model
5.11 (a) and (b) show a noticeable change in the mean level of the AR plots at the time of damage. Based on these interpretations, it becomes very clear to explain the sudden change in the average level of the ERD graph shown in Figure 1.
Detection results using RMSSA
To further validate the damage moment given by the TVAR coefficients, the ERD is applied on the processed data and tracked in real time, as discussed in the previous sections. 5.12 (c)) further proves the damage moment to be the same as that observed from the AR plots.
Numerical case studies using hybrid RPCA-RSSA algorithm
Description of the 2-storey modeled with Duffing oscillator on both floors
Each individual element of the matrix Ω˜ indicates Duffing parameter value to contribute to each floor. By solving the equation of motion numerically, the statistics of the response vectors x1(t) and x2(t) can be obtained.
Description of the 2-storey modeled with a base Duffing oscillator
The system information contained in the mathematical model of the dynamic system expressed using the aforementioned discretization scheme leads to the development of the real-time damage detection strategy determined by the hybrid algorithm. Being an inherently weak nonlinear system, the B-W system retains its temporal change in the nonlinear parameter progressing at a relatively slower rate due to the higher number of DOF.
Temporal damage detection results using the hybrid algorithm
Detection results for the 2-storey Duffing oscillator model using hybrid algorithm140
5.16, it is clear that the algorithm detects the exact moment of damage at 35 s by changing the mean level of the DSF graphs. While approaching the moment of injury, the graphs show a sudden change in the mean TH level.
Temporal damage detection results for the 5 DOF B-W system
Therefore, it can be concluded that the hybrid method is efficient in determining global damage from 10% for nonlinear systems in a recursive framework. The performance of the FOEP-based example in detecting spatial damage in a concurrent recursive framework is described in the next section.
Spatial damage detection results for the B-W system
Since the moment of damage is now ascertained from the AR plot, the spatial RRE is examined near the vicinity of the damage (e.g. 25s-35s). This confirms the fact that damage has indeed occurred in the third floor of the structure.
Results for El Centro ground excitation
This is a clear illustration of damage localization, manifested through a deviation in spatial RREs for the structure. In the present scenario, spatio-temporal damage detection incorporated in a single recursive framework for non-stationary excitation for a 25% damage case is successfully reported.
Comparison of all the developed FOEP based techniques
Performance check using B-W system
The basis of the comparison of the algorithms is to detect a change in the spatial orientation of the system in real time. In accordance with the above findings, a tabular comparison of the algorithms with respect to the percentage change in linear storey stiffness is given in Table 5.4.
Performance check using a 5 DOF structure with the third storey modeled as
Although the RSSA-TVAR algorithm shows a certain amount of distortion at the average plot level in TH. A clear change in the average level of the graph of the TVAR coefficient clearly distinguishes the efficiency of the proposed method compared to its contemporaries.
Experimental verifications
Experimental verifications using RSSA
These slowly varying processes may hinder the applicability of the proposed algorithm toward real-time implementation and may lead to masking of environmental variations. The possible implementations of the proposed algorithm range from real-time damage detection of civil structures to data-driven mechanical and aerospace applications where the acquisition of time series data is possible.
Experimental verifications using the hybrid approach
Upon removal of the impactor after a period of 20 seconds, the linear vibration of the beam due to external excitation continued for 30 seconds. 5.31, the TVAR plots show significant activity in the region from 0-20s, indicating a non-linear state of the system.
Practical case studies
Case study for the UCLAFB using RSSA
The vibration response obtained from the database server indicates that a significant occurrence of nonstationary activity in the vicinity of t=360s indicates the onset of the Parkfield earthquake. From the 48 responses obtained from the dataset, the translation responses are used for online processing using the proposed algorithm.
Case study for the UCLAFB using the hybrid RPCA-RSSA approach
It is normal to expect that these two moments differ from each other because it takes a limited time for the structure to undergo significant changes, reflected by a change in stiffness.
Summary
However, it is worth noting that most of the available damage detection schemes are offline, requiring windowing of the data in order to compare the newest response set with baseline values. Additionally, a numerically simulated 7 DOF B-W system is adopted to evaluate the performance of the proposed scheme against increasing DOF.
Background
Before going into the details of the proposed methodology, it is necessary to review the concepts of PCA through a structural dynamics perspective, discussed in depth in Chapter 2. Readers are advised to familiarize themselves with the FOEP formulations described earlier, before entering on the theoretical attributes of the present recursive approach.
RCCA: Detailed derivation
Recursive damage sensitive features
Proposed algorithm
Detection results using proposed algorithm
Temporal damage detection studies for the B-W systems excited using white
Important acronyms
Important acronyms
Damage index for varying levels of nonlinearity
Global RREs for numerical modeling (using white noise)
Global RREs for experimental case
Spatial RREs for the UCLA factor building
Important acronyms
Important acronyms
Comparison of existing damage detection methods with the proposed algorithm
Summary of real time damage detection results
Comparison of the existing damage detection methods with the proposed algorithm 156
Percentage changes in statistical mean of TVAR coefficients
Oakland bridge after San Francisco earthquake
Summary of FOEP methods
Flowchart for the proposed method
Force-displacement curves for various levels of nonlinearity
Acceleration plots for white noise excitation for different cases of non linearity
Damage detection using condition indicators for 50% non linearity change
Damage detection using residual errors for varying cases of non linearity change
Damage detection using RE and scatter plots for 50% non linearity
Damage detection using RRE1 for various non linearity change at different time in-
Comparison between spatial and temporal damage for 50% change
Comparison between spatial and temporal damage for 35% change
Cumulative contribution of principal components
CIs for Underdetermined case- Global Damage, 25% change in nonlinearity
Comparison between batch PCA and RPCA
Residual error and scatter plots for El Centro excitation
Details of the experimental setup, courtesy [157]
