## 2 Theoretical Research on Indirect Measurement Method Based on Vehicle-Bridge Coupling
### Theoretical Analysis of Detection Method Based on Vehicle Response  
Indirect measurement of bridge damage using vehicle response begins with the bridge frequency indirect measurement method proposed by [PERSON] and [PERSON] et al., and the first-order frequency of the bridge can be identified by performing an FFT transformation on the acceleration response of the vehicle. [PERSON] et al. derived the vehicle acceleration response [20]:  
\[\ddot{q}_{v}(t)=\sum_{i=1}^{n}\bigg{\{} \begin{array}{c}P_{i}cos(\omega_{v}t )+Q_{i}cos(2\omega_{ci}t)+\\ X_{i}cos[(\omega_{ci}-\omega_{i})t]+Y_{i}cos[(\omega_{ci}+\omega_{i})t]\end{array}  \bigg{\}} \tag{12}\]  
where \(P_{i}\), \(Q_{i}\), \(X_{i}\), and \(Y_{i}\) is a time-independent coefficient, \(\omega_{i}\) is the \(i-\)th natural frequency of the bridge, \(\omega_{v}\) is the natural vibration frequency of the vehicle, and \(\omega_{ci}=i\pi v/L\) is the vehicle loading frequency. The load vector \(\{P_{i+1}\}\) at the right end of the vehicle vibration differential equation can be expressed by the bridge displacement and speed, and the load vector \(\{P_{i+1}\}\) of \(1/4\) vehicle model is:  
\[\{P_{i+1}\}=\left( \begin{array}{c}0\\ c_{v1}f_{z}^{\prime}(x)+k_{v1}f_{z}(x)\end{array} \right) \tag{13}\]  
where \(c_{v1}\) is wheel damping, \(k_{v1}\) is the wheel stiffness, \(f_{z}(x)\) is the bridge displacement, and \(f_{z}^{\prime}(x)\) is the bridge speed. With the tractor test truck model, both the tractor and the inspection vehicle can be reduced to a concentrated force compared to the bridge span, as shown in Figure 2.  
The deflection at the position of the detection vehicle can be calculated using the superposition principle:  
\[f_{z}(x)=\frac{F_{c}x^{2}(x-l)^{2}}{3 EIl}+\frac{F_{q}}{EI}\cdot\left[-\frac{(l- (x+\Delta a))x^{3}}{6l}+\left(\frac{(x+\Delta a)l}{6l}+\frac{(x+\Delta a)l}{3} -\frac{(x+\Delta a)^{2}}{2}\right)x\right] \tag{14}\]  
where \(\Delta a\) is the distance between the tractor and the center of mass of the test vehicle. Replace Equations (13) and (14) into Formula (11) to detect the displacement vector of the vehicle at the driving position \(\{u_{i+1}\}\):  
\[\{u_{i+1}\}=\frac{1}{b_{2}b_{1}-b_{3}^{2}}\left[ \begin{array}{c}b_{1}(b_{4}+ b_{6})+b_{3}(c_{v1}f_{z}^{\prime}(x)+k_{v1}f_{z}(x)+b_{5}+b_{7})\\ b_{3}(b_{4}+b_{6})+b_{2}(c_{v1}f_{z}^{\prime}(x)+k_{v1}f_{z}(x)+b_{5}+b_{7}) \end{array} \right] \tag{15}\]  
where:  
\[ \begin{array}{c}b_{1}=a_{0}m_{v1}+a_{1}(c_{v1}+c_{v1})+(k_{u1}+k_{v1})\\ b_{2}=a_{0}m_{v}+a_{1}c_{v1}+k_{u1}\\ b_{3}=a_{1}c_{u1}+k_{u1}\\ b_{4}=m_{v}(a_{0}u_{i1}+a_{2}i_{i1}+a_{3}i_{i1})\\ b_{5}=m_{v1}(a_{0}u_{i2}+a_{2}i_{i2}+a_{3}i_{i2})\\ b_{6}=c_{u1}(a_{1}u_{i1}+a_{4}i_{i1}+a_{5}i_{i1})-c_{u1}(a_{1}u_{i2}+a_{4}i_{ i2}+a_{5}i_{i2})\\ b_{7}=(c_{u1}+c_{v1})(a_{1}u_{i2}+a_{4}i_{i2}+a_{5}i_{i2})-c_{u1}(a_{1}u_{i1}+a _{4}i_{i1}+a_{5}i_{i1})\end{array}  \tag{16}\]  
Replace Equation (15) with Equation (8) to detect the acceleration vector of the car at the driving position \(\{\ddot{u}_{i+1}\}\)  
\[\{\ddot{u}_{i+1}\}=\left[ \begin{array}{c}\frac{a_{0}}{b_{2}b_{1}-b_{3}^{2}} \bigg{(}b_{1}(b_{4}+b_{6})+b_{3}\bigg{(}\begin{array}{c}c_{v1}f_{z}^{\prime} (x)\\ +k_{v1}f_{z}(x)+b_{5}+b_{7}\end{array}\bigg{)}\bigg{)}-a_{0}u_{i1}-a_{2}\dot{u} _{i1}-a_{3}\ddot{u}_{i1}\\ \frac{a_{0}}{b_{2}b_{1}-b_{3}^{2}}\bigg{(}b_{3}(b_{4}+b_{6})+b_{2}\bigg{(} \begin{array}{c}c_{v1}f_{z}^{\prime}(x)\\ +k_{v1}f_{z}(x)+b_{5}+b_{7}\end{array}\bigg{)}\bigg{)}-a_{0}u_{i2}-a_{2}\dot{u} _{i2}-a_{3}\ddot{u}_{i2}\end{array} \right] \tag{17}\]  
The acceleration response of the car body is:  
\[a_{v1}=\frac{a_{0}}{b_{2}b_{1}-b_{3}^{2}}\bigg{(} \begin{array}{c}b_{1}(b_{4} +b_{6})+b_{3}(c_{v1}f_{z}^{\prime}(x)\\ +k_{v1}f_{z}(x)+b_{5}+b_{7})\end{array} \bigg{)}-a_{0}u_{i1}-a_{2}\dot{u}_{i1}- a_{3}\ddot{u}_{i1} \tag{18}\]  
Figure 2: Schematic diagram of simplified calculation for tractor inspection vehicle.