# Application of the Random Decrement Technique in Damage Detection under Moving Load
## 2 Basic Theories and Method  
The basic concept of RD technique is very simple, especially if the structural response under random load at each time instant, \(t_{i}\), is composed of these two parts:  
1. Deterministic response due to initial displacement and velocity and,
2. Random response due to random excitation. In the mathematical solution provided for RD technique, the input excitation should be stationary, random white noise, Gaussian, and zero-mean [20]. Considering a linear system with multi-degree of freedom, the corresponding equation of motion can be expressed as below:  
\[M\ddot{X}(t)+C\dot{X}(t)+KX(t)=f(t), \tag{1}\]  
in which \(M\), \(C\), and \(K\) are mass, damping, and stiffness matrices, respectively. Parameter _f(t)_ is a random white noise, zero-mean, stationary, and Gaussian input excitation. \(X(t)\) is the displacement recorded in the vertical direction. Assuming that Equation (1) is satisfied at instant \(t_{i}\), a simple shift in time (\(\tau\)) leads to:  
\[M\ddot{X}(t_{i}+\tau)+C\dot{X}(t_{i}+\tau)+KX(t_{i}+\tau)=f(t_{i}+\tau), \tag{2}\]  
It is supposed that the input excitation is zero-mean, which can be easily averaged out using the mean of \(N\)\((N\rightarrow\infty)\) different starting instants. Computing the mean of \(N\) different starting instants can be expressed as below:  
\[\frac{M}{N}\sum_{i=1}^{N}\ddot{X}(t_{i}+\tau)+\frac{C}{N}\sum_{i=1}^{N}\dot{X} (t_{i}+\tau)+\frac{K}{N}\sum_{i=1}^{N}X(t_{i}+\tau)=\frac{1}{N}\sum_{i=1}^{N} f(t_{i}+\tau)=0, \tag{3}\]  
Defining \(Y(\tau)\) as below, Equation (3) turns into a free vibration form as shown in Equation (5). \[Y(\tau)=\frac{1}{N}\sum_{i=1}^{N}X(t_{i}+\tau), \tag{4}\]  
\[M\ddot{Y}(\tau)+C\dot{Y}(\tau)+KY(\tau)=0, \tag{5}\]  
In the literature, \(Y(\tau)\) is known as Random Decrement Signature (RDS) [1; 11; 23; 24; 29]. RD technique is defined as a special averaging method in which \(N\) different starting instants, \(t_{i}\), are chosen using a so-called triggering level, as shown in Figure 2. Some default values for triggering levels recommended by different researchers are listed in Table 1.  
The level-crossing triggering condition was utilized in different studies due to its simplicity. To use the level-crossing triggering condition, the optimum value is assumed as \(a=\sqrt{2}\times\) standard deviation [24]. Then, RDS is simply defined as follows:  
\[Y(\tau)=\frac{1}{N}\sum_{i=1}^{N}X(t_{i}+\tau)|\ X(t_{i})=a, \tag{6}\]  
In the Equation (6), two parameters are questionable, namely \(N\) and \(\tau\). These two parameters can control the consuming time and cost, especially in long-term SHM. The minimum acceptable value for \(N\) is 100, while 1000 seems to give a stable result [30]. For parameter \(\tau\), the RDS should have enough length to represent a full frequency content of the original response. The time and cost in RD technique are mainly affected by three parameters, namely, triggering level (\(a\)), length of RD signature (\(\tau\)), and number of signals. According to Equation (6), triggering level has the greatest effect on the calculation's time and increases the number of segments, especially when the acceleration data are utilized in the analyses. A small change in the parameter \(a\) could increase \(N\) by hundreds. All the above illustrations are only for single time-history data. In order to control the safety and health of infrastructures, such as bridges, a set of sensors distributed along the structure are necessary. Calculating the RDS in multi-channel signals was well-illustrated by [PERSON] [1]. He supposed that if the RDS for a signal is called auto-RDS, the cross-RDS could then be computed using triggering information on the auto-RDS.