## 2 Methodology
### Theoretical Analysis of Concrete Pavement Vibration  
Assume an external excitation \(q(x,y,t)\) is applied on a concrete pavement slab, the equations of motion are formulated as:  
\[D\
abla^{2}\
abla^{2}w+\rho h\frac{\partial^{2}w}{\partial t^{2}}=q(x,y,t) \tag{1}\]  
where \(w\), \(D\), \(\rho\), and \(h\) denote the deflection, bending stiffness, density, and thickness of the concrete pavement slab respectively. The solution to Equation (1) is given by:  
\[w(x,y,t)=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}W_{mn}(x,y)\ T_{mn}(t) \tag{2}\]  
where \(W_{mn}(x,y)\) denotes the mode shape of \(mn_{th}\) order, \(T_{mn}(t)\) denotes the weight of \(mn_{th}\) order along with time \(t\). Equation (3) gives the following mode shape equation:  
\[D\
abla^{2}\
abla^{2}W_{mn}=\rho h\omega_{mn}^{2}W_{mn} \tag{3}\]  
where \(\omega_{mn}\) denotes the natural frequency of \(mn_{th}\) order. From Equations (2) and (3), Equation (1) becomes  
\[\frac{d^{2}T_{mn}(t)}{dt^{2}}+\omega_{mn}^{2}T_{mn}(t)=\frac{P_{mn}(t)}{M_{mn}} \tag{4}\]  
where \(M_{mn}\), and \(P_{mn}(t)\) denote the generalized mass, and force of \(mn_{th}\) order respectively and is given by  
\[M_{mn}=\iint\rho hW_{mn}^{2}(x,y)ds \tag{5}\]  
\[P_{mn}(t)=\iint q(x,y,t)W_{mn}(x,y)ds \tag{6}\]  
When the excitation is an impulse load, \(q(x,y,t)\) can be written as  
\[q(x,y,t)=q_{0}(x,y)f_{I}(t) \tag{7}\]  
\[f_{I}(t)=\left\{\begin{array}{ll}P_{I},&0\leq t\leq t_{1}\\ 0,&t>t_{1}\end{array}\right. \tag{8}\]  
where \(q_{0}(x,y)\) denotes the distribution of loading position, \(t_{1}\) denotes the loading time and \(P_{I}\) denotes the magnitude of the impulse load. From Equations (5)-(8), the solution of Equation (4) is given by  
\[T_{mn}(t)=\begin{array}{ll}\frac{\iint q_{0}(x,y)W_{mn}(x,y)ds}{\omega_{mn} M_{mn}}\int_{0}^{t_{1}}f_{I}(\tau)\sin\omega_{mn}(t-\tau)d\tau\\ +a_{mn}\cos\omega_{mn}t+b_{mn}\sin\omega_{mn}t\end{array} \tag{9}\]  
The loading time of an impulse load is below 0.03 s. It could be considered as close to zero. As \(t_{1}{\rightarrow}0\) and the initial conditions of Equation (1) are zero, Equation (9) becomes  
\[T_{mn}(t)=\frac{\iint q_{0}(x,y)W_{mn}(x,y)ds}{\omega_{mn}M_{mn}}\cdot P_{I}t _{1}\cdot\sin\omega_{mn}t \tag{10}\]  
When an impulse load (with magnitude \(P_{A}\)) is applied at \((-x_{0},y_{0})\) on Slab A, as shown in Figure 2, the amplitude of the vibration response of \(mn_{th}\) order (\(A_{mn}\)) can be expressed based on Equation (10):  
\[A_{mn}=\frac{\iint q_{0}(x,y)W_{mn}(x,y)ds}{\omega_{mn}M_{mn}}\cdot P_{I}t_{1 }=\frac{W_{mn}^{A}(-x_{0},y_{0})}{\omega_{mn}^{A}M_{mn}^{A}}P_{A}t_{1}^{A} \tag{11}\]Similarly, when an impulse load (with magnitude \(P_{B}\)) is applied at (\(x_{0}\), \(y_{0}\)) on Slab B, the amplitude (\(B_{mn}\)) is written as  
\[B_{mn}=\frac{W_{mn}^{B}(x_{0},y_{0})}{\omega_{mn}^{B}M_{mn}^{B}}P_{B}t_{1}^{B} \tag{12}\]  
Considering that the joint conditions affect Slab A and Slab B equally, as well as the structural and material properties of these two slabs, are nearly the same, the modal characteristics of the joint area could be approximately regarded as equal  
\[W_{mn}^{A}(-x,y) \approx W_{mn}^{B}(x,y) \tag{13}\] \[\omega_{mn}^{A} \approx\omega_{mn}^{B}\] (14) \[M_{mn}^{A} \approx M_{mn}^{B} \tag{15}\]  
Since the impulse load has equal loading time on Slab A and Slab B (\(t_{1}{}^{A}=t_{1}{}^{B}\)), Equation (12) divided by Equation (11) can be written as  
\[\frac{B_{mn}}{A_{mn}}=\frac{P_{B}}{P_{A}} \tag{16}\]  
From Equation (16), the impulse load (magnitude = \(P_{A}\)) applied at (\(-x_{0}\), \(y_{0}\)) has an equal excitation to Slab B, compared with the impulse load (magnitude = \(P_{A}A_{mn}\)/\(P_{B}\)) applied at (\(x_{0}\), \(y_{0}\)). Therefore, the load transfer efficiency (\(P_{B}\)/\(P_{A}\)) can be reflected by the ratio of vibration response amplitude.