# Wireless Accelerometer Architecture for Bridge SHM: From Sensor Design to System Deployment
## 5 Results from on Field Test
### Operational Modal Analysis  
The covariance-driven stochastic subspace identification (SSI-COV) algorithm is a robust time-domain methodology employed for identifying modal parameters of engineering structures, particularly in output-only configurations. This algorithm is well suited for applications where excitation forces are unknown, such as OMA of civil structures under ambient or operational vibrations [26].  
The SSI-COV algorithm is rooted in the stochastic state-space model, which represents the dynamics of the system as  
\[x_{k+1} =Ax_{k}+w_{k} \tag{3}\] \[y_{k} =Cx_{k}+v_{k} \tag{4}\]  
where \(x_{k}\) denotes the state vector at time k, \(y_{k}\) represents the measured output vector (e.g., accelerations or velocities), and \(A\) and \(C\) are the discrete state and output matrices, respectively. The terms \(w_{k}\) and \(v_{k}\) account for process noise and measurement noise, encapsulating uncertainties and disturbances inherent to the system.  
Modal parameters are derived by decomposing the discrete state matrix \(A\), whose eigenvalues yield natural frequencies and damping ratios, while the eigenvectors of the output matrix \(C\) provide the corresponding mode shapes. The effectiveness of SSI-COV depends on two key input parameters [27]:  
* Model Order (\(n\)): A crucial parameter linked to the degrees of freedom in the system. This is typically set conservatively, often exceeding twice the expected number of modes in the analysis frequency range, to ensure all significant modes are captured.
* Correlation Points (\(i\)): Defined by the number of block rows in the Toeplitz covariance matrix, this parameter influences the stabilization diagram's resolution. Its value is determined based on the sampling frequency \(f_{s}\) and the lowest frequency of interest (\(f_{0}\) that corresponds to the first natural frequency of the bridge), using the following rules: \[i_{min}\geq\frac{f_{s}}{2f_{0}}\] (5)  
The SSI-COV algorithm was successfully applied to the monitoring of road and railway bridges, utilizing minimal sensor configurations. The algorithm's output was further refined through stabilization diagrams, where stable poles representing true physical modes were distinguished from spurious poles based on frequency, damping ratio, and mode shape consistency by using the procedure described in [28].  
For the bridge under consideration, 100 Hz acquisition of 30 min for the six accelerometers during the first acquisition day is considered for the SSI-COV analysis. Specifically, all sensors are treated as biaxial, disregarding the direction parallel to the longitudinal axis of the bridge. In this case, the parameter \(n\) range from 20 to 150, increasing by a step of 2, while \(i\) values ranges from 20 to 60, with step of 10. The resulting stability diagram, after removing non-physical poles and clustering the remaining ones, is shown in Figure 6.  
Additionally, in Figure 6, the first two principal components of the frequency domain decomposition (FDD) [29] are represented to check the correctness of the SSI-COV estimation. In Figure 7, the resulting mode shapes are represented.  
Figure 6: Stability diagram.  
Figure 7: Natural frequencies, damping and mode shape of the five vibration modes determined by the SSICOV algorithm.