5.13 Comparison of all the developed FOEP based tech-
method with the established real time damage detection schemes is that the individual essence of real time updating of the matrices are done sequentially, viz., updating the sample covariance esti- mate at each time stamp followed by the updating the Hankel covariance matrix upon obtaining the transformed response from the RPCA algorithm. The key results using the newly proposed method
15 20 25 30 35 40 45
7
RRE
8RPCA
15 20 25 30 35 40 45
a (t)
2 5RPCA+TVAR
15 20 25 30 35 40 45
2 2.4
a
2(t)
RSSA+TVAR
10 15 20 25 30 35 40 45 50
Time (s)
0.480.5 0.52
a
2(t)
RPCA+RSSA+TVAR Moderate change at damage
instant
Negligible change at damage instant
Significant change at the instant of damage
No change at damage instant
Figure 5.25: Comparison of the recently established damage detection methods with the proposed algorithm
are summarized and compared against the recently established damage detection strategies. In the present context, the comparative study is conducted on the 5 DOF B-W system under a zero mean Gaussian white noise excitation of 50s duration. A change in the third storey linear stiffness by 10% at 31s induces a local damage to the structure. The basis of comparison of the algorithms is to detect a change in the spatial orientation of the system in real time. As evident form the literature, the damage detection potential of the RPCA algorithm is confined to the order of 25% for spatial damage [41]. In addition, recently published works on RSSA towards damage detection indicate a lower limit of 15% for detecting local damage scenarios. As observed from the Fig. 5.25, the efficacy of RPCA-RRE, RPCA-TVAR and RSSA-TVAR algorithms is slightly questionable for 10% dam- age. While the RRE and AR plots corresponding to both RPCA and RSSA based algorithms fail TH-1989_156104031
to show any significant deviation in the mean level at the instant of damage, the proposed method captures the instant of damage accurately and show a significant change in the mean level of the DSF plot. This confirms the superiority of the proposed algorithm over the established damage detection schemes. In line with the above findings, a tabular comparison of the algorithms towards percentage change in linear storey stiffness is provided in Table 5.4. From the results, it can be concluded that the hybrid algorithm is advantageous over the individual RPCA and RSSA based damage detection strategies in terms of an improved detectability of spatial damage in real time.
Table 5.4: Comparison of the existing damage detection methods with the proposed algorithm Local damage (%) RPCA-RRE RPCA-TVAR RSSA-TVAR Proposed algorithm
35.00 Yes Yes Yes Yes
25.00 Yes Yes Yes Yes
20.00 No Yes Yes Yes
15.00 No No Yes Yes
10.00 No No No Yes
5.13.2 Performance check using a 5 DOF structure with the third storey modeled as Duffing Oscillator
In this section, the FOEP improvised examples are applied a 5 DOF system modeled with a Duffing oscillator at the third floor. The purpose of modeling the third DOF with a Duffing spring is to induce nonlinearity to the structure through a storey stiffness and to ensure the robustness of the algorithm towards a more practical scenario as well. In the previous assumptions, the nonlinearity was confined at the base of the model that allowed the change in the nonlinear term to induce a global damage to the structure. As the damage induced affected the entire structure as a whole, the induced damage was reported to be global in nature whose affects did not wear out over the progression of time. Following similar thoughts, the present numerical model is also excited with a zero mean Gaussian white noise. The governing equation of motion is modeled as Ito’s stochastic differential equation and subsequently discretized using Taylor’s 1.5 strong scheme [155] considering a step size ∆= 0.01s. The governing equation of motion can be written as:
MX¨ +CX˙ +KX+ΨZ =F(t) (5.26)
TH-1989_156104031
15 20 25 30 35 40 45 1
a (t) 2 2
RSSA+TVAR
15 20 25 30 35 40 45
-0.5 0 0.5
RR-1
RPCA
-1
a (t) 2 -0.8
RPCA+TVAR
10 15 20 25 30 35 40 45 50
Time (s)
0.8 0.9 1
a 2 (t)
RPCA+RSSA+TVAR Moderate change at damage
instant
Significant change at damage instant
No change at damage instant No change at damage instant
Figure 5.26: Comparison between the methods for a strongly nonlinear system
where M, C and K follow the similar notations as before. Ψ represents the location of DOF modeled with the Duffing oscillator and its neighboring DOF involved towards the formulation of the equation of motion. The vectorZ represents a curtailed version of the operation (X2−X3)3 that arises due to the coupling of the DOF while obtaining the equation of motion. Interested readers are kindly referred to APPENDIX E for a brief derivation of the response using Taylor’s scheme and the dynamical considerations of the same.
From Table 5.4, it is ensured that RPCA-RRE algorithm fails to detect spatial damage finer than 25%. Considering an even comparison among the established online damage detection methods for a 20% spatial damage, it is safe to conclude that RPCA-RRE algorithm can be ignored from the comparison scheme. As the system is strongly nonlinear, finer levels of damage detection is difficult, especially when applied in an online framework. The performance of the participating algorithms towards a localized event corresponding to a 20% damage is illustrated in Fig. 5.26. As observed from the figure, RPCA-TVAR algorithm does not indicate a significant change at the instant of damage.
Although RSSA-TVAR algorithm shows a certain amount of distortion in the mean level of the plot at TH-1989_156104031
the damage instant, the associated caveat with the methodology is the optimum placement of sensors that necessitates sensor instrumentation at the damage-induced DOF for capturing local damage.
However, the proposed hybrid algorithm surpasses the performance of the established methods both in terms of detection potential in real time as well as placement of sensors, since a complete set of sensor response is adequate to detect local damage in the structure. A clear change in the mean level of the plot of the TVAR coefficient clearly distinguishes the efficacy of the proposed method as compared to its contemporaries. Thus, it can be concluded that the proposed methodology is advantageous over the established real time detection schemes even for systems having nonlinearity at individual storey level, in terms of an improved detectability of spatial damage in real time.