# Asphalt Pavement Transverse Cracking Detection Based on Vehicle Dynamic Response
## 2 Theoretical Modeling  
Suppose the vertical displacement between the wheel and the body is \(Z_{1}\) and \(Z_{2}\), respectively. The coordinate origin is the equilibrium position between the wheel and the body, shown in Figure 1. \(m_{1}\) and \(m_{2}\) denote the quality of the wheel and body, respectively. \(k_{1}\) and \(k_{2}\) denote the stiffness of the tire and spring, and \(c_{2}\) denotes the damping coefficient of the shock absorber. In this study, we see the tire as an undamped linear spring system. For a quarter vehicle-normal pavement system, the equation of motion is constructed based on the d'Alembert principle [22], which is shown in Equations (1) and (2):  
\[m_{2}\dot{Z_{2}}+c_{2}\;(\dot{Z_{2}}-\dot{Z_{1}})+k_{2}\;(Z_{2}-Z_{1})=0 \tag{1}\]  
\[m_{1}\dot{Z_{1}}-c_{2}\;(\dot{Z_{2}}-\dot{Z_{1}})-k_{2}\;(Z_{2}-Z_{1})+k_{1}Z_{ 1}=k_{1}q \tag{2}\]  
where the first-order derivation of the vertical displacement denotes the vertical speed and the second-order derivation of the vertical displacement indicates the vertical acceleration. The equation of motion for a quarter vehicle-cracked pavement system is constructed following Equations (3) and (4). \[m_{2}\dot{Z_{2}}+c_{2}\;(\dot{Z_{2}}-\dot{Z_{1}})+k_{2}\;(Z_{2}-Z_{1})=0 \tag{3}\]  
\[m_{1}\dot{Z_{1}}-c_{2}\;(\dot{Z_{2}}-\dot{Z_{1}})-k_{2}\;(Z_{2}-Z_{1})+k_{1}Z_{ 1}=k_{1}q^{\prime} \tag{4}\]  
In Equations (2) and (4), the system's input is expressed as \(q\) (normal pavement) and \(q^{\prime}\) (cracked pavement), which denotes the pavement's surface profile. They can be represented in the complex exponential form:  
\[q=q_{0}^{i\omega t} \tag{5}\]  
\[q^{\prime}=q_{0}^{i\omega t}+\delta(\Delta t) \tag{6}\]  
where \(\omega\) denotes angular frequency, and \(q_{0}\) is the initial profile. \(\delta\) denotes the transient impulse caused by transverse cracking, which is affected by the crack's cracking degree. \(\Delta t\) is the response time that the wheel passes by the transverse crack. Figure 1: A quarter vehicle–pavement system. The tire-cracked pavement contact process is shown in Figure 2a. During actual driving, the tire and the pavement are always in contact with a certain geometry, defined as ground contact (GC). In addition, the contact shape between transverse cracking and the tire is not a point but a certain geometric area, defined as the Areas of Crack Action (ACA). The cracks are located at the junction of the ground surface (GC1) and the action surface when first touching the transverse cracked pavement (t = t1). The action area is virtual, defined as the Potential Action of Crack Action (PACA), since the tire has not yet begun to adequately feel the excitation of the transverse crack. As the wheel rolls forward (t = t2), the transverse crack and the tire act continuously, giving a sufficient transient shock. GC2 continues to iterate forward and overlaps each other, which makes up the action areas. The action areas are defined as the Actual Action of Crack Action (AACA). By the end of the excitation of the crack (t = t3), the crack is at the AACA's end and the end of the next ground contact (GC3). It can be seen that the displacement between the crack and the tire is the length of the AACA along the direction of the vehicle (L), which is shown in Figure 2b. The geometric size of the GC is certain when the tire pressure and spring load mass are certain. As a result, the grounding area remains unchanged. While passing by different pavement conditions (transverse cracks, potholes, bumps), the AACA differs due to the significant differences in interaction characteristics caused by the changed space shape. Therefore, the tire-cracked pavement action time can be calculated by the length (\(L\)) and the speed (\(v\)) of the grounded rectangle, which is shown in Equation (7). \[\Delta t=\frac{L}{p} \tag{7}\]  
Figure 2: The tire-cracked contact model. (**a**) Tire-cracked pavement contact process; (**b**) AACA and its action length.