# Dynamic Deflection Measurement on Stiff Bridges with High Piers by Preloaded Spring Method
## 4 Experimental Validation of Indoor and Field  
### Indoor Experiment  
The measurement principle of the static deflection of the PSM is clear, and the error is related to the parameters of the stretched wire and the compression spring; therefore, no verification is required. The mechanism of dynamic deflection measurement of the PSM is more complex since the vibration equations incorporate both damping and inertia forces. In this paper, the dynamic properties of the PSM were verified by laboratory experiments. In addition, comparative experiments between the PSM and SHM were also conducted. The effect of the SWSS on the bridge is disregarded due to the overall stiffness of the SWSS being smaller than that of the bridge. The dynamic deflection of the bridge was simplified as a harmonic excitation, which could be simulated by a reciprocator. Figure 14a shows the measurement process of the PSM, while Figure 14b presents the data acquisition system. As shown in Figure 14a, the reciprocator was moved in a simple harmonic motion at 3 Hz and the SFM was used to identify the signals from the reciprocator, while the PSM was used to measure the signals transmitted by the stretched wire. In Figure 14b, the acquisition system consists of a signal amplifier, a signal processor, and a minicomputer. The initial preload force of the PSM was 10 N and the mass of the pull bar was 0.1 kg. The stiffness of the compression spring is 50 N/m. In the comparative experiment, the mass of the suspension hammer was 0.8 kg, and the simple harmonic vibration was applied at a frequency of 5 Hz. In the above experiments, the stretched-wire diameters were 0.6 mm and the stretched-wire lengths were 2.5 m. Meanwhile, the FE model of the SWSS was established in Ansys software. Link180, Mass21, and Combine14 elements were used to simulate the stretched wire, material point, and compression spring, respectively. More details of the model can be found in Section 3.1. The measured displacement time-history curve and numerical simulation results are shown in Figure 15.  
The displacement estimation performance is compared in terms of the absolute peak value error (APVE) and normalized root mean square error (NRMSE), defined as follows [57]:  
\[\text{APVE}=\left|\frac{\max(\left|u_{e}\right|)-\max(\left|u_{r}\right|)}{ \max(\left|u_{r}\right|)}\right| \tag{11}\]  
\[\text{NRMSE}=\sqrt{\frac{\sum(u_{e}-u_{r})^{2}}{\sum(u_{r})^{2}}} \tag{12}\]  
where \(u_{r}\) and \(u_{e}\) denote the signal sources and estimated displacements, respectively. The comparison results of the APVE and NRMSE are summarized in Tables 2 and 3. For Excitation 1, the NRMSEs of the measured and simulated results of the PSM are both less than 9% and the APVE is less than 5% compared to the excitation source. In Figure 15a, under the harmonic excitation condition of 3 Hz, the displacement measured by the PSM is 2.7% smaller than the excitation source, which is 1.4% different from the numerical simulation. The numerical simulation and experimental test results are highly consistent. Figure 14: Laboratory experiment: (**a**) measurement process of the SWSS and (**b**) signal acquisition system. Figure 15: Dynamic displacements: (**a**) dynamic displacements of the PSM simulated at 3 Hz and (**b**) dynamic displacements of the PSM and the SHM at 5 Hz. Notes: PSM is the preloaded spring method; SHM represents the suspension hammer method. For Excitation 2, the PSM reduces the NRMSE and APVE by 18.21% and 25.39%, respectively, compared to the SHM technique. When the frequency increased to 5 Hz, as shown in Figure 15b, the maximum displacement measured by the PSM was 2.1% smaller than the displacement of the excitation source, which is 21.0% different from the result of the SHM. However, the hammer generates inconsistent vibration due to the obvious left-right swing, and the results measured by the SHM cannot reflect the real dynamic response of the structure with higher vibration frequencies.