# Wireless Accelerometer Architecture for Bridge SHM: From Sensor Design to System Deployment
## 4 Laboratory Tests Results
### Metrological Verification
#### 4.2.3 Synchronization Tests  
Synchronization Error Analysis Between Two Sensors  
To quantify the synchronization error between two sensors, let us define the following variables:  
* \(t_{11}\): Time when the acquisition is triggered for Sensor 1. * \(t_{12}\): Time when the data are ready for Sensor 1. * \(t_{21}\): Time when the acquisition is triggered for Sensor 2. * \(t_{22}\): Time when the data are ready for Sensor 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Axis** & **Offset [LSB]** & **Gain [g/LSB]** \\ \hline \multirow{3}{*}{Sensor 1} & X & 5412 & \(3.84\times 10^{-6}\) \\  & Y & 3412 & \(3.89\times 10^{-6}\) \\  & Z & \(-2345\) & \(3.87\times 10^{-6}\) \\ \hline \multirow{3}{*}{Sensor 2} & X & \(-2981\) & \(3.82\times 10^{-6}\) \\  & Y & 1134 & \(3.87\times 10^{-6}\) \\ \cline{1-1}  & Z & \(-4325\) & \(3.82\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Offsets and gains determined for two different sensors. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Frequency (Hz)**} & \multirow{2}{*}{**Acceleration (m/s\({}^{2}\))**} & \multicolumn{2}{c}{**Error (\%)**} & \multirow{2}{*}{\(\mathbf{U_{s}}\) (\%)**} \\ \cline{3-4}  & & **Z Axis** & & **Y Axis** \\ \hline
1 & 0.5 & 3.4 & 1.6 & 0.5 \\ \hline
2 & 1 & \(-0.1\) & 1.0 & 0.4 \\ \hline
5 & 3 & \(-0.8\) & 1.0 & 0.4 \\ \hline
10 & 3 & 0.0 & 2.0 & 0.4 \\ \hline
20 & 2 & 0.0 & 2.3 & 0.4 \\ \hline
40 & 1 & \(-3.0\) & 5.2 & 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Measurements obtained during the INRIM certification of the sensor. The effective time of data acquisition for Sensor 1, \(t_{1}\), lies within the interval \([t_{11},t_{12}]\), while the effective time for Sensor 2, \(t_{2}\), lies within \([t_{21},t_{22}]\). The time difference between data acquisition for the two sensors is given by  
\[\Delta t=|t_{1}-t_{2}|.\]  
To compute the maximum synchronization error, we consider two main sources of uncertainty:  
1. The synchronization uncertainty between the trigger signals from the two sensors. 2. The delay between the trigger signal and the data-ready signal for each sensor. Dataset Analysis  
Two datasets are analyzed to compute the synchronization error:  
1. The first dataset contains the time differences between trigger signals from different sensors. 2. The second dataset contains measurements of the time between a trigger signal and the corresponding \"data-ready\" signal for one sensor. Figure 4 illustrates examples from these datasets:  
Synchronization Uncertainty and Variance Computation  
From the first dataset, the synchronization uncertainty between trigger signals is computed as  
\[\sigma_{1}^{2}=\frac{\text{Var}(\Delta t_{\text{trigger}})}{2},\]  
where the factor \(\frac{1}{2}\) accounts for the dataset measuring differences between two independent realizations of the trigger signal timing. From the second dataset, the mean (\(\mu_{2}\)) and variance (\(\sigma_{2}^{2}\)) of the time between the trigger signal and the data-ready signal are extracted. Maximum Synchronization Error  
Combining the synchronization uncertainty and the data-ready delays, the worst-case synchronization error is given by  
\[T=\mu_{2}+3\cdot\sqrt{\frac{\sigma_{1}^{2}}{2}+\sigma_{2}^{2}}.\]  
Here, we having the following:  
* \(\mu_{2}\): Mean value of the time to data readiness from the second dataset. * \(\sigma_{1}^{2}\): Variance of the time differences between trigger signals from the first dataset. * \(\sigma_{2}^{2}\): Variance of the time to data readiness from the second dataset. Figure 4: On the left, an example of two trigger signals acquired from different sensors. On the right, an example showing a trigger signal (yellow line) together with the subsequent data-ready signal (blue line). Results  
Using the datasets, we obtain the following:  
* From the first dataset, the synchronization uncertainty (\(\sigma_{1}\)) was calculated. * From the second dataset, the mean (\(\mu_{2}\)) and variance (\(\sigma_{2}^{2}\)) of the data-ready delay were determined.