# Dynamic Deflection Measurement on Stiff Bridges with High Piers by Preloaded Spring Method
## 2 Preloaded Spring Method and the Related System
### Principle of Dynamic Deflection Measurement  
When vehicles move across a bridge, the stretched wire transfers the vibration signals to the pull bar. The dial gauge then indirectly obtains the dynamic deflection of the bridge by measuring the displacement of the pull bar in real time. Figure 5 shows the mechanical model of the vehicle-bridge-SWSS coupled system. It is assumed that the material of the stretched wire is uniform and continuous and that the density, cross-sectional area, and elastic modulus are kept constant during vibration. The stretched wire and compression spring are simply equivalent to damped springs, and the pull bar is simplified to a material  
Figure 4: Static model of the stretched-wire-spring system. Site A represents the upper end of the stretched wire, site B represents the lower end of the stretched wire, Site B\({}^{\prime}\) represents the lower end of the compression spring, C represents the upper end of the compression spring, and Site C\({}^{\prime}\) represents the ground fixed platform. point. For a five-axle space vehicle, ignoring the local vibrations inside it, the vehicle body is assumed to be rigidly coupled and supported on a suspension-spring system and wheels. Both the wheels and the suspension system are considered as elastomers whose damping is proportional to the vehicle's speed. Ultimately, the vehicle is simplified to a spring-mass model with 15 independent DOFs. Taking the Euler-Bernoulli beam as an example, according to conventional vehicle-bridge coupled vibration theory, the vibration equation of the vehicle can be expressed as follows:  
\[M_{V}\ddot{\mathbf{Z}}+\mathbf{C}_{V}\dot{\mathbf{Z}}+\mathbf{K}_{V}\mathbf{Z}=\mathbf{F}_{vb} \tag{3}\]  
where \(M_{V}\), \(\mathbf{C}_{V}\), and \(\mathbf{K}_{V}\) represent the mass, damping, and stiffness matrices of the vehicle, respectively; \(\mathbf{Z}\), \(\dot{\mathbf{Z}}\), and \(\ddot{\mathbf{Z}}\) are the displacement, velocity, and acceleration of the vehicle in the \(y\)-direction, respectively; \(\mathbf{F}_{vb}\) refers to the wheel-to-road contact force vector of the vehicle and bridge. The coupled vibration equations of the bridge and SWSS can be written as follows:  
\[\left\{ \begin{array}{l}EI\frac{\dot{a}y}{\dot{a}x}+c\dot{y}+m\ddot{y}=-m_{p}g+ c_{w}(\dot{y}_{p}-\dot{y}_{b})+k_{w}(y_{p}-y_{b})\\ m_{p}y_{p}+c_{w}(\dot{y}_{p}-\dot{y}_{b})+k_{w}(y_{p}-y_{b})-c_{s}\dot{y}_{p}-k _{s}y_{p}=0\end{array} \right. \tag{4}\]  
where \(y\) is the abbreviated form of \(y(x,t)\), and represents the dynamic deflection of a bridge subjected to a vertical load in the \(y\)-direction; \(m_{p}\) is the mass of the simplified material point, which is related to the pull bar; \(y_{p}\), \(\dot{y}_{p}\), and \(\ddot{y}_{p}\) are the displacement, velocity, and acceleration of the simplified material point in the \(y\)-direction, respectively; \(y_{b}\) and \(\dot{y}_{b}\) are the displacement and velocity of the measurement point of the bridge in the \(y\)-direction, respectively; \(k_{w}\) and \(c_{w}\) are the stiffness and damping of the stretched wire, respectively; \(f_{p}\) is the initial preload force of the compression spring; \(k_{s}\) and \(c_{s}\) are the stiffness and damping coefficient of the compression spring. The initial static balance state of the vehicle and the SWSS is used as a starting state for each dynamic balance equation. The vibration equations of the vehicle-bridge-SWSS coupled system can be obtained by introducing the vehicle-bridge coupled force while  
Figure 5: Dynamic model of the vehicle-bridge-SWSS coupled system. considering a road roughness profile. After combining Equations (3) and (4), the final coupled equations can be described as follows:  
\[ \begin{bmatrix}M_{\mathbf{v}}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&M_{\mathbf{b}}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&m_{p}\end{bmatrix}\begin{bmatrix}\tilde{Z}\\ \tilde{\mathbf{x}}\\ \tilde{y}_{p}\end{bmatrix} +