3.6 Numerical Example
3.6.2 Results for White Noise
where
K1
K0
is defined as the ratio between the secant modulus (K1) associated with a changed level of nonlinearity (damaged state) to the initial secant modulus(K0) of the pristine (undamaged) state. For cases of no damage, the ratio
K1
K0
becomes unity, thereby making the damage index zero. The values of the DI corresponding to various levels of change in nonlinearity along with the corresponding κ values are shown in Table 3.2. From the results in Table 3.2, it is clear that the change in the nonlinear force parameter has a direct correspondence and follows a linear relationship with the DI.
Table 3.2: Damage index for varying levels of nonlinearity Change in nonlinearity (%) κ Damage index (DI)
25.00 0.75 0.32
30.00 0.70 0.38
50.00 0.50 0.51
75.00 0.25 0.66
0 5 10 15 20 25 30 35 40 45 50 -1
0
1 25% nonlinearity change at 31 s
0 5 10 15 20 25 30 35 40 45 50
-1 0
1 50% nonlinearity change at 31 s
Time (s)
0 5 10 15 20 25 30 35 40 45 50
Acceleration (m/s2 )
-1 0
1 75% nonlinearity change at 31 s
0 5 10 15 20 25 30 35 40 45 50
-1 0
1 30% nonlinearity change at 31s
Figure 3.3: Acceleration plots for white noise excitation for different cases of non linearity plot of the transformed responses of the system at pre and post time instants of 30s and 32s is also provided. It can be observed from Fig. 3.4 that EVC is capable of detecting sudden damages. While the RREs show damage by a significant change at that instant and then hitting a plateau region for the rest of the duration of the excitation, the scatter plot between the transformed response show a definite change in orientation which can be depicted easily from the plot (Fig. 3.4(d)). By looking at Fig. 3.4(b) and 3.4(c), it can be seen that there is a significant rise in the estimate of the residual errors χRR−1 and χRR−2. It can also be observed from Fig. 3.4(a), that EVC could capture the essence of fault detection which shows up as a significant peak at the damage instant. The EVC however, shows a period of activity around the 0-8s mark and also shows a false detection at 37s.
This shows that although EVC is useful in certain cases, its utility diminishes corresponding to lower percentage of damage cases (results not reported for brevity). Fig. 3.5 shows the performance of χRR−1 and χRR−2 for various cases of damage. It is worth noting from the figure that the efficacy of the RREs for less than 25% damage detection is slightly questionable which indicates that the current online framework is less reliable when the extent of damage suffered is low (i.e. less than 25%).
The scatter plot between the recursive transformed responses (Ytri vs Ytrj) can also serve as a TH-1989_156104031
10 20 30 40 50 0
0.5 1 1.5
EVC
(a) Eigenvector Change
10 20 30 40 50
0 1 2 3 4
RR-1
(b) Recursive Residual Error 1
10 20 30 40 50
Time (s) 0.4
0.6 0.8 1 1.2
RR-2
(c) RRE 2
-1 -0.5 0 0.5 1
Y transformed 4 -1
-0.5 0 0.5 1
Y transformed 3
(d) Scatter plot : pre and post damage
30.01 s 32.01 s False Detection
Figure 3.4: Damage detection using condition indicators for 50% non linearity change
robust visual CI for detecting damage. Since the transformed responses update recursively, showing point-wise scatter plots is time and memory consuming. To demonstrate the efficacy of scatter plots, data are considered in growing windows with initial window size of 10s and at increments of 10s before damage and a slightly smaller increment in the vicinity of damage (31s). It can be clearly observed from Fig. 3.6 that there is a significant change in orientation of the scatter, also reflected by the change in the signs of the correlation coefficients, between the successive windows immediately before and after damage. Considering the time of damage as 31s (as obtained from RRE plot), the 4th window clearly indicates the damage instant for the specified level of non linearity.
Fig. 3.7 shows the robustness of the RREs for various instants of damage for 30%, 40%, 50%
and 75% percent damage cases. As seen from Fig. 3.7, the RRE shows distinct change for higher percentage change in non linearity. For lower percentages of damage, the changes is slightly less distinct. However, this is not an impediment so far as current online damage detection framework is considered, since any event of change can always be verified using the scatter plot to finally decide on the occurrence of damage. The relative change in global RREs corresponding to different levels of damage is shown in Table 3.3. It is clear from the results in Table 3.3 that the percentage change TH-1989_156104031
10 20 30 40 50 Time (s)
10 20 30 40 50
Time (s)
RR-1 RR-2
X: 31.02 Y: 10.87
X: 31.02 Y: 6.865
50%
75%
50%
30%
75%
30%
25% 25%
Figure 3.5: Damage detection using residual errors for varying cases of non linearity change in RRE increases with the level of damage.
Table 3.3: Global RREs for numerical modeling (using white noise) Change in nonlinearity (%) Pre-damage RRE Post-damage RRE % change
25.00 0.74 0.98 32.43
30.00 0.71 0.97 36.62
35.00 0.77 1.06 37.17
40.00 0.68 1.02 50.15
50.00 0.69 1.27 84.06
Spatial damage detection results
Damage, in the present context is simulated by a change in linear stiffness of an individual storey at a particular time instant. The streaming data is processed by the algorithm online in order to find out the exact time and location of damage through a change in local RRE. For the simultaneous temporal and spatial damage case (online spatio-temporal damage), 50% and 35% changes in linear stiffness of 3rd storey are considered. The nonlinearity level κ is kept at 1 (fully nonlinear). From TH-1989_156104031
Time (s)
10 20 30 40 50
χ RR-1
0 1 2 3 4
Y-Transf 4
Y-Transf 1
11 s 21 s 30 s 33 s 43 s
ρ14=-0.59 ρ14 = -0.65 ρ14 = -0.61 ρ14 = 0.96 ρ14 = 0.98
Figure 3.6: Damage detection using RE and scatter plots for 50% non linearity
Fig. 3.8, it can be shown that using the CI χRR−2 a clear damage instant is found at 31s. Once the damage instant is detected, attention is turned to the spatial RREεRR−Yi in a small neighborhood in the vicinity of damage (29 to 33s). It can be clearly observed from Fig. 3.8 thatεRR−Y3 shows a significant change at 31s compared to the other responses which indicates that damage has occurred in the third storey. It should be understood at this stage that both the RREs are tracked online recursively and εRR−Yi is shown separately represented in the neighborhood of the damage instant to capture the spatial effect of the damage visually. From Fig. 3.9, it is evident that the spatial RRE for 35% damage indicates a visible change at t=31s. However, as the extent of stiffness degradation reduces, detecting spatial damage detection gets increasingly difficult compared to the detection of the temporal damage alone.