## 2 Background
### Dynamic Characterization and Processing Techniques  
The classic dynamic response is defined by a physic model considering the mass, stiffness, and damping properties of a linear time-invariant structural system to an external force. The following second-order differential Equation (1) represents the harmonic vibration response of viscous damping multi-degree-of-freedom structural systems [18]:  
\[[M]\{\dot{x}\}+[C]\{\dot{x}\}+[K]\{x\}=\left\{ \begin{array}{c}p(t)=0\\ p(t)=p_{0}sin(wt)\end{array} \right\} \tag{1}\]  
where \(M\) is the mass matrix, \(C\) is the damping matrix, \(K\) is the stiffness matrix, \(x\) is the vector of displacement of the system, \(p(t)\) is known as the external dynamic force, \(p_{0}\) is the force amplitude, \(w\) is the exciting frequency, and \(t\) is the time instant. For the estimation of the damping properties, the free-vibration response of the system only considers the response thereof to initial conditions of displacement and velocity as the external dynamic force disappears (\(p(t)\) is 0 at Equation (1)). Thus, the damping ratio \(\zeta\) takes a relevant role in the decaying phenomena of the motion. As outlined in [18], the relation between acceleration \(\ddot{x}\), at \(t\) time, is independent of time history, so this definition enables the damping ratio \(\zeta\) calculation through logarithmic decrement as:  
\[\delta=Ln\left(\frac{\dot{x}_{1}}{\dot{x}_{j+1}}\right)=\frac{2\pi i\zeta}{ \sqrt{1-\zeta^{2}}} \tag{2}\]  
where \(j\) is the number of vibration periods proceeded. The modal analysis consists of a methodology for determining the dynamic characteristics of a system in terms of natural frequencies, mode shapes, and damping ratios in order to formulate a mathematical model that estimates the dynamic response of a structure. Concerning this subject, the modal analysis framework considered in this paper involves operational modal analysis (OMA), adopting the frequency domain decomposition (FDD) technique and performing the vibration mode correlation by modal assurance criterion (MAC). Furthermore, the time-frequency analysis was conducted to study the dynamic performance of the system across time. FDD technique relies on the singular value decomposition (SVD) of the power spectral density matrix \(G_{yy}\). The power spectral density matrix \(G_{yy}\) is decomposed into auto-power spectral density functions corresponding to a single-degree-of-freedom system [19]. Then, it is possible to estimate natural frequencies and modes shapes of vibration modes as [20]:  
\[G_{yy}(w_{i})=U_{i}S_{i}U_{i}^{H} \tag{3}\]  
where, for every frequency line \(w_{i}\), \(U_{i}\) is the matrix of the \(m\) singular vectors \(u_{il}\), \(S_{i}\) is the diagonal matrix of the \(m\) singular values \(s_{il}\) considering \(l\) from 1 to \(m\), and \(H\) denotes the transpose conjugate complex. This approach is accurate when the input excitation is white noise, the structure has light damping properties, and mode shapes from close vibration modes are well separated [21]. To compare the identified mode shapes, MAC is performed \begin{table}
\begin{tabular}{l l l} \hline
**Monitored Structure (Year)** & **Type of SHM System** & **Estimated Cost of SHM System (USD)** & **Estimated Cost per Node (USD)** \\ \hline Ising Ma Bridge (1997) & Traditional & 58,000,000 & 526,667 \\ Bill Emerson Memorial Bridge (2005) & Traditional & 51,500,000 & 517,857 \\ Golden Gate Bridge (2006) & Wireless network & 538,400 & 5600 \\ Aquila Tower (2008) & Wireless network & 51429 & 589 \\ Jindo Bridge (2009) & Wireless v & 535,000 & 5500 \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of costs for SHM systems.