# Application of the Random Decrement Technique in Damage Detection under Moving Load
## 4 Results and Discussion
### Normalizing the Arias Intensity along the Structure  
As previously pointed out and shown in Figure 1, most of the damage detection methods are based on estimating modal parameters through RDSs. These parameters are then used as damage indicators to identify the damage locations. However, recent research carried out stipulates how to find damage without detecting the modal parameters [8,13]. Taking this approach, Arias Intensity (AI) is employed as a new damage indicator in this paper to locate the damage without calculating any modal parameter. AI is one of the most famous energy criteria defined by [PERSON] in 1970 [32]. The Arias intensity calculates the energy of the acceleration signal expressed as below:  
\[I_{A}(\frac{m}{s})=\frac{\pi}{2g}\int_{0}^{T}a^{2}(t)dt, \tag{9}\]  
in which \(I_{A}\), \(T\), and \(a(t)\) are AI, duration of acceleration, and acceleration data, respectively. Using this definition, AI is calculated as a scalar and invariant value. Additionally, it contains the full range of frequencies and time duration of the signal. To begin with the proposed method, Equation (9) was applied to all the recorded signals, and their corresponding AI values were determined. For instance, Figure 12 shows the results for each sensor from both auto and cross-RDSs of intact structure individually. Figure 11: (a) Response acceleration data recorded at node 1, (b) auto-RDS of node 1, and (c) cross-RDS of node 5. It is supposed that the damage location can be estimated from the absolute difference value of the AIs calculated from a structure in damage and non-damage conditions. However, as shown in Figure 12, the AI differs from node to node. Thus, it is impossible to find any damage without normalization. To normalize the AI spectrums, the following equations are used:  
\[I_{A-normalized}=\frac{\pi}{2g}\int_{0}^{T}(\beta a(t))^{2}dt, \tag{10}\]  
\[\beta=\sqrt{\frac{1000}{I_{A}}}, \tag{11}\]  
Equation (10) causes the AIs to normalize to 1000. The parameter \(\beta\) is normalization factor and should be calculated only for structures without damage. Figure 13 and Table 4 show the normalized AIs and corresponding \(\beta\)s.  
Figure 12: The AIs calculated from the RDSs for each sensor of the intact structure individually, considering the speed as v = 0.3 m/s. (**a**) Al calculated for intact bridge and (**b**) AI calculated for the intact beam. Figure 13: The AIs after normalization for the structure under moving load considering the speed as v = 0.3 m/s. (**a**) Normalized Al calculated for intact bridge and (**b**) normalized AI calculated for the intact beam. Although Table 4 shows the value of \(\beta\) considered to normalize the AI of RDSs, it is limited to the speed of 0.3 m/s. Considering different weights and velocities leads to a different \(\beta\). Twenty-nine laboratory tests were assumed for the non-damage models, in which twenty-one tests were implemented on the bridge with 3 different weights (90, 110, and 130 kg) and velocities (0.3, 0.4, and 0.5 m/s). The remaining 8 tests were implemented on the beam considering a 27.5 kg weight and three different velocities (0.3, 0.4, and 0.5 m/s). In each test, the parameter \(\beta\) was calculated for all the nodes. Figure 14 shows the \(\beta\) for node 1 of both bridge and beam under moving load at different weights and velocities. Two points can be deduced from Figure 14. First, in our models, \(\beta\) follows a linear pattern and has an inverse relationship with velocity. Secondly, \(\beta\) has no relationship with changes in the weights. For instance, there is no relationship between different weights at speed \(v\) = 0.4 m/s. Using the linear relationship between \(\beta\) and velocity may lead to a new \(\beta\) for other velocities. This linear relationship can be defined as follows:  
\[\beta=x\vartheta+y, \tag{12}\]  
in which \(\vartheta\) is the speed of moving load. The two coefficients \(x\) and \(y\) can be estimated for each node. Figure 14 shows how to use a trend line to estimate \(x\) and \(y\) for node 1.