5.11 Temporal damage detection results using the hybrid algorithm
5.11.1 Detection results for the 2-storey Duffing oscillator model using hybrid algorithm140
The results for damage detection for the 2 storey modeled with duffing oscillator on both floors using the hybrid approach is presented in this section. In this method, the damage is induced through a change in the nonlinear parameter α1 as shown in Eqn. 5.23. As previously described, the change in the nonlinear parameter at a particular instant of time induces a global damage to the structure that does not wear off as time progresses. For the current study, the damage is induced to the model TH-1989_156104031
through change of nonlinearity by 10%, 15% and 25% at a particular time instant of 35s from the start.
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a
1(t)
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1 2 3
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2(t)
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Time (s) 0.2
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RR-1
Damage instant identified
Figure 5.13: DSFs for 25% change in nonlinearity for 2 DOF Duffing oscillator
The detection results for 25% damage is shown in Fig. 5.13. It is evident from the figure that the plots of AR-1 and AR-2 clearly indicate an accurate damage instant at 35s, through a discernible change in the mean level at the instant of damage. The figure also shows the performance of χRR−1
towards indicating the damage instant. It is imperative to note that the temporal RRE plots serve as a visual confirmation for the exact damage instant for the system. From the AR-2 plot, it can be observed that there occurs a period of activity even after 35s, that could be solely attributed to the nonlinear nature of the response and not a possible event indicating damage, which is verified from the use of the RRE plots that show distortion only at the damage instant. The detection results for 15% are clearly observable from fig. 5.14. The change in the mean level of the AR plots at 35s indicate the possible damage event for the system. From the AR plots, the presence of spurious peaks indicate of period of activity prior to the instant of damage. As previously explained, this arises due to the instabilities present in the system due to the nonlinearities involved. The distortion TH-1989_156104031
in the temporal RRE plot at 35s confirms the exact instant of damage to the system corresponding to a 15% nonlinear change. Thus, it can be safely concluded that albeit these instabilities, the algorithm is equipped to address detection scenarios for nonlinear systems as well. The absolute percentage change in the mean level before and after damage is provided in Table 5.3. It can be observed that the absolute percentage change in the mean level decreases as the damage becomes finer, thereby indicating the efficacy of the proposed algorithm towards detecting lower percentage of damage in real time.
Time (s)
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a 1(t)
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0.8 a)
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χ RR-1 0 0.5 1 1.5
c)
Damage instant
Figure 5.14: DSFs for 15% change in nonlinearity for 2 DOF Duffing oscillator
Previous damage identification studies have reported real time damage detection of the order of 15% for a single sensor data input using a FOEP based algorithm, RSSA. The FOEP based method, RPCA, is not expected to perform well for such a strongly nonlinear system and hence the detection results using RPCA have not been reported here for brevity. In addition, RPCA-TVAR FOEP based algorithm provides successful identification results for weakly nonlinear systems up to 15% global damage cases. Therefore, it can be well understood that the proposed hybrid approach fares better than the recently established online damage detection methods, extensively dealt with in chapters 2 and 3. The present study successfully identifies real time damage of the order as low as 10%, which is difficult, especially in an online framework. To the best of the knowledge of the author, this level of damage detection has not been reported in real time damage detection literature. The TH-1989_156104031
results of damage detection for a 10% damage are displayed in Fig. 5.15. Although there appears to be a certain amount of activity before the instant of damage, the DSFs detect the exact instant of damage through a change in the mean level of the plots at 35s. As evident from the figure, the distortion in the RRE plot further confirms the efficacy of the DSFs as a potential tool for detecting real time damage as low as 10%. Although it can be understood from the figure that the efficacy of the RREs for lower percentage of damage is slightly questionable, it does not pose any impediment towards the current framework.
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1(t)
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1 1.5 2
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2(t)
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RR-1
Damage instant identified
Figure 5.15: DSFs for 10% change in nonnlinearity for 2 DOF Duffing oscillator
5.11.2 Temporal damage detection results for the 2-storey modeled with Duffing oscillator at the base
Following the same lines of development, damage detection for changes in 20%, 15% and 10%
nonlinear force term are sequentially presented. An important note to be considered here is the fact that the changes to the system are global in nature that do not wear off over time which indicates that the system has certainly not repaired from the effects of the damage as time progresses. The TH-1989_156104031
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a 1(t)
a)
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a 2(t)
b)
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Time (s) 0
0.5 1
RR-1
c)
Figure 5.16: Damage detection using AR coefficients for 20% nonlinearity change
FOEP based example is applied to the raw vibration data in order to identify the exact instant of damage. Consider the first case where the damage is simulated for a 20% change at 35s. After the streaming data is processed by the algorithm, the DSFs are tracked online, recursively, in order to identify the exact instant of damage. Upon examination of the identification results shown in fig.
5.16, it is clear that the algorithm detects the exact instant of damage at 35s through the change in the mean level of the DSF plots. To verify these findings, the RRE plot clearly shows distortions at exactly 35s, thereby rendering the instabilities observed from the AR plots before damage as indicators of nonlinearities associated with the system. Thus, it can be safely concluded that the hybrid method detects damage exactly for the aforesaid percentage change in nonlinearity. While a change in the mean level of the TVAR plot verifies the damage instant to be at 35s, a distortion in the RRE plot indicating the exact instant of damage further reinforces the efficacy and robustness of the FOEP based hybrid approach. Through these discussions, it can be well understood that the proposed algorithm gives provides better detectability for an increased percentage of damage. The results, therefore, have not been reported here for brevity.
Proceeding to the next case, detection results for a 15% change in nonlinearity is provided in Fig.
5.17. As observed from the figure, the TVAR plots serve as a robust DSF for identifying damage.
While approaching the damage instant, the plots show a sudden change in the mean level of the TH-1989_156104031
graph that indicates a possible damage instant at 35s. The RRE plot shows a distortion at 35s, thereby validating the exact instant of damage for the system. To illustrate the potential application of the hybrid approach for a finer level of detection, the algorithm is applied for a 10% change in the nonlinear force term of the system. Through the TVAR plots, it can be seen that the exact instant of damage is identified at 35s for even a 10% change in the nonlinearity, which now establishes a finer level of detection as compared to the existing literature. Although the presence of certain system instabilities account for a gradual change in the mean level of the plot before the instant of damage is reached, the exact damage instant can be clearly observed at 35s from the plots. The RRE plot further verifies the damage instant through a sharp peak indicating the distortion in the subspace that occurs due to the presence of damage.
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a 1 (t)
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a 2 (t)
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RR-1
Exact instant of damage
Figure 5.17: Recursive DSF plot for 15% damage
From the previously explored online damage detection studies, it is clear that RPCA-RRE based algorithm does not provide successful identification for such a low percentage of real time damage. It can also be perceived from prior case studies that the RPCA-TVAR algorithm fails to detect damage lower than 15% nonlinearity change. As the system under consideration is strongly nonlinear, the TH-1989_156104031
RPCA-TVAR algorithm is not expected to perform under the specified changes in nonlinearity. From the present results, it can be concluded that the hybrid method provides better detectability than reported in the currently available literature [51]. It is well understood that the instant of damage can be accurately detected by the algorithm for the aforementioned changes in the nonlinear force term.
However, it should be noted that a fine level of detectablility (such as 10%) for a nonlinear system is a difficult feat considering the essentially online nature of the implementation. The consistent evolution of the TVAR coefficients and the distinct change in the mean level of the plots at the instant of damage validates the efficacy of the said DSF. It can be observed from Table 5.3 that the absolute percentage change in the mean values lower gradually corresponding to the finer levels of damage. This provides ample evidence regarding the efficiency of the proposed method in detecting finer levels of real time damage. The time taken for a complete run of the algorithm is found to be 2.16s, that approximately emulates a real time process. The recursive implementation of the RPCA-RSSA module takes 10ms which is 0.46% of the total time consumed for a single iteration.
However, the time consumed depends on the computational power of the system used and is more efficient for systems with superior processing power of the console used for computation.
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a 1 (t)
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a 2 (t)
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Exact instant of damage
Figure 5.18: Recursive DSF plot for 10% damage TH-1989_156104031