## 2 The Vehicle-Pavement-Foundation System: Modeling and Solution
### Represent the Pavement with the Timoshenko Beam Model  
The differential equation governing the uniform motion of an infinitely long Timoshenko beam resting on a viscoelastic Winkler foundation and subjected to tire forces is described by the following equation:  
\[ \begin{split}&\rho A\frac{\partial^{2}w}{\partial t^{2}}+\lambda AG \left(\frac{\partial\varphi}{\partial x}-\frac{\partial^{2}w}{\partial x^{2} }\right)+k_{1}w+k_{3}w^{3}+c\frac{\partial w}{\partial t}-G_{p}\frac{ \partial^{2}w}{\partial x^{2}}=F(t)\delta(x-vt),\\ &\rho I\frac{\partial^{2}\varphi}{\partial t^{2}}-EI\frac{ \partial^{2}\varphi}{\partial x^{2}}+\lambda AG\left(\varphi-\frac{\partial w }{\partial x}\right)+k_{f}\varphi+c_{f}\frac{\partial\varphi}{\partial t}=0, \end{split}  \tag{20}\]  
where the coefficients \(E\), \(I\), \(\rho\), \(A\), \(\lambda\) and \(G\) are related to the properties of the beam: \(E\) for the beam's modulus of elasticity, \(I\) for the beam's cross-sectional moment of inertia, \(\rho\) for the beam's density, \(A\) for the beam's cross-sectional area, \(\lambda\) for the shear coefficient of the effective area, and \(G\) for the beam's shear modulus. The subgrade properties are associated with the coefficients \(G_{p}\), \(k_{1}\), \(k_{3}\), \(c\), \(c_{f}\) and \(k_{f}\), which, respectively, denote the shear deformation coefficient, linear stiffness, nonlinear stiffness, viscous damping coefficient, torsional damping coefficient, and torsional stiffness of the subgrade. \(F(t)\) represents the tire contact force, \(v\) denotes the vehicle's velocity, and \(\delta(\cdot)\) refers to the Dirac delta function. The vertical displacement of the beam is given by \(w\), while \(\varphi\) describes the beam's rotational angle. The system's initial and boundary conditions are defined as follows:\[w(x,t)\Big{|}_{t=0}=\left.\frac{\partial w(x,t)}{\partial t}\right|_{t=0}= \left.\varphi(x,t)\right|_{t=0}=\left.\frac{\partial\varphi(x,t)}{\partial t} \right|_{t=0}=0, \tag{21}\] \[\lim_{x\to\pm\infty}\frac{\partial^{n}w(x,t)}{\partial x_{n}}= \lim_{x\to\pm\infty}\frac{\partial^{n}\varphi(x,t)}{\partial x_{n}}=0\quad(n=0,1,2).\]  
Consistent with the previous approach, using the Adomian decomposition method, it is assumed that the solution to Equation (20) can be written in the form of a series  
\[w =w_{0}+w_{1}+w_{2}+\cdots \tag{22}\] \[\varphi =\varphi_{0}+\varphi_{1}+\varphi_{2}+\cdots\]  
The nonlinear term in Equation (20) is also expanded into an infinite series form  
\[k_{3}w^{3}=k_{3}\sum_{n=0}^{\infty}B_{n}, \tag{23}\]  
where the sequence \(B_{n}\) is determined according to Equation (15). By substituting Equations (22) and (23) into Equation (20), the solution for the nonlinear Equation (20) is derived by summing the solutions of the ensuing sequence of linear equations  
\[ \begin{split}&\rho A\frac{\partial^{2}w_{0}}{\partial t^{2}}+\lambda A G \bigg{(}\frac{\partial\varphi_{0}}{\partial x}-\frac{\partial^{2}w_{0}}{ \partial x^{2}}\bigg{)}+k_{1}w_{0}+c\frac{\partial w_{0}}{\partial t}-G_{p} \frac{\partial^{2}w_{0}}{\partial x^{2}}=F(t)\delta(x-vt),\\ &\rho I\frac{\partial^{2}\varphi_{0}}{\partial t^{2}}-EI\frac{ \partial^{2}\varphi_{0}}{\partial x^{2}}+\lambda AG\bigg{(}\varphi_{0}-\frac{ \partial w_{0}}{\partial x}\bigg{)}+k_{f}\varphi_{0}+c_{f}\frac{\partial \varphi_{0}}{\partial t}=0\\ &\rho A\frac{\partial^{2}w_{n}}{\partial t^{2}}+\lambda AG\bigg{(} \frac{\partial\varphi_{n}}{\partial x}-\frac{\partial^{2}w_{n}}{\partial x^{ 2}}\bigg{)}+k_{1}w_{0}+c\frac{\partial w_{n}}{\partial t}-G_{p}\frac{\partial ^{2}w_{n}}{\partial x^{2}}=-k_{3}B_{n-1},\\ &\rho I\frac{\partial^{2}\varphi_{n}}{\partial t^{2}}-EI\frac{ \partial^{2}\varphi_{n}}{\partial x^{2}}+\lambda AG\bigg{(}\varphi_{n}-\frac{ \partial w_{n}}{\partial x}\bigg{)}+k_{f}\varphi_{n}+c_{f}\frac{\partial \varphi_{n}}{\partial t}=0,\end{split}  \tag{24}\]  
where \(n=1,2,\cdots\). By computing the initial several terms of \(w_{n}\), one can approximate the vertical deflection of the Timoshenko beam  
\[