# Tracking of Stiffness Variation in Structural Members Using Input Error Function Observers
## 5 Numerical Simulation for Stiffness Tracking: Two-Dimensional Three-Bay Truss Structure  
An analytical model of the two-dimensional three-bay truss structure representing a bridge structure is considered here to validate the proposed methodology of optimization of the input error function for stiffness tracking in structural members. Details of the planar truss structure are shown in Figure 7. Excitation is provided to the planar truss with two inputs at the nodes 1 (Y-direction) and 2 (X-direction), as shown in Figure 7.  
Figure 8 plots the stiffness degradation in a truss member # E10 assumed during numerical simulations. As shown in Figure 8, truss member # E10 is assumed healthy till 500 s and has a stiffness degradation of 20% between 500 and 1000 s. Structural degradation continues and truss member # E10 loses another 10% of stiffness between 1000 and 1500 s and a further 10% resulting in total degradation of 40% between 1500 and 2000 s.  
Optimization of the input error function is implemented to track the stiffness variation in truss member # E10. To simulate practical applications, a sensitivity study for the % rms noise is carried out. Noise of 2% rms, 5% rms and 10% rms is added to the output measurements. Figure 9 plots the stiffness variation as tracked by optimization of an input error function corresponding to truss member # E10. For comparison purposes, stiffness ratio estimations for an ideal system without any noise are also included in Figure 9. Significant spikes with reference to the idea scenario is evident for data corresponding to 5% rms and 10% rms noise in output measurements. However, the mean values reported in Table 3 clearly show the robustness of the proposed method. Stiffness estimation is \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**\#Data Points** & **Statistics** & **Spring K1** & **Spring K2** & **Spring K3** \\ \hline \multirow{3}{*}{200} & Mean (N/m) & 79.98 & 60.11 & 75.22 \\  & Error (\%) & 0.03 & 0.18 & 0.29 \\  & Standard Deviation & 1.42 & 1.33 & 1.51 \\ \hline \multirow{3}{*}{400} & Mean (N/m) & 80.13 & 59.91 & 75.04 \\  & Error (\%) & 0.16 & 0.14 & 0.06 \\  & Standard Deviation & 0.64 & 0.61 & 0.76 \\ \hline \multirow{3}{*}{600} & Mean (N/m) & 80.03 & 59.99 & 75.07 \\  & Error (\%) & 0.03 & 0.02 & 0.10 \\  & Standard Deviation & 0.47 & 0.51 & 0.51 \\ \hline \multirow{3}{*}{800} & Mean (N/m) & 80.04 & 59.98 & 75.01 \\  & Error (\%) & 0.06 & 0.03 & 0.02 \\ \cline{1-1}  & Standard Deviation & 0.36 & 0.34 & 0.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Estimation of stiffness with different data points and 5% rms noise in output measurements. Figure 7: Schematics of a planar truss structure. within 2% even for 10% rms noise, though bigger spikes give less confidence in the stiffness estimations. A sensitivity study for the number of data points used for estimation of stiffness in truss member # E10 is also carried out, and stiffness tracking is plotted in Figure 10. Please note that only 5% rms noise is considered here. Relatively bigger spikes can be observed for stiffness ratio estimation values corresponding to 200 data points, whereas stiffness ratio estimations corresponding to 800 data points closely fol  
Figure 8: Stiffness variation in truss member # E10. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Time Interval** & **0\%rms** & **2\% rms** & **5\% rms** & **10\% rms** \\ \hline
0–500 & 1.0000 & 0.9946 & 0.9878 & 0.9830 \\ \hline
500–1000 & 0.8000 & 0.8005 & 0.8048 & 0.8049 \\ \hline
1000–1500 & 0.7000 & 0.6998 & 0.7002 & 0.7156 \\ \hline
1500–2000 & 0.6000 & 0.6004 & 0.6036 & 0.6077 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Estimation of stiffness with different data points and 5% rms noise in output measurements. Figure 9: Stiffness variation in truss member # E10 for different % rms noise. ratio. However, stiffness ratio estimations listed in Table 4 show an accurate stiffness ratio estimation for any number of data points used.