# Dynamic Response Analysis of Asphalt Pavement under Pavement-Unevenness Excitation
## 3 Solution of Multilayer Pavement under Pavement-Unevenness Excitation
### Analytic Solutions Based on Dual Vectors  
Because the basic governing equation of the pavement problem is a partial differential equation, it is necessary to convert the response component into an ordinary differential equation expressed by an expansion coefficient by virtue of the basic property of the vector function system. In the cylindrical coordinate system, the definition and properties of the orthogonal vector function system are as follows:  
\[ \begin{array}{l}\mathbf{L}(r,\theta;\lambda,m)=e_{z}S(r,\theta;\lambda,m)\\ \mathbf{M}(r,\theta;\lambda,m)=\Big{(}e_{r}\frac{\partial}{\partial r}+e_{\theta} \frac{\partial}{r\theta\theta}\Big{)}S(r,\theta;\lambda,m)\\ \mathbf{N}(r,\theta;\lambda,m)=\Big{(}e_{r}\frac{\partial}{r\theta\theta}-e_{\theta }\frac{\partial}{\partial r}\Big{)}S(r,\theta;\lambda,m)\end{array}  \tag{31}\]  
In the formula, \(e_{r}\), \(e_{\theta}\), and \(e_{z}\) are, respectively, the unit vectors of the \(r\), \(\theta\), and \(z\) axes in the cylindrical coordinate system, and \(S\) is the scalar function, which is defined as follows:  
\[S(r,\theta;\lambda,m)=\frac{1}{\sqrt{2\pi}}J_{m}(\lambda r)e^{\mathrm{i}m \theta};m=0,\pm 1,\pm 2,\ldots \tag{32}\]where \(J_{m}(\lambda r)\) represents the Bessel function of order \(m\), \(m=0\) corresponds to the axisymmetric deformation, the variables \(\lambda\) and \(m\) are the transformation variables of \(r\) and \(\theta\) under the vector function system, and \(i\) is the imaginary unit. Therefore, by importing the displacement, stress, and other functions in the basic governing equation into the orthogonal vector function system, we can get  
\[ \begin{array}{l}\mathbf{u}(r,\theta,z)\equiv u_{r}e_{r}+u_{\theta}e_{\theta}+u_{z }e_{z}\\ =\frac{1}{m}\int_{0}^{+\infty}[U_{L}(z)\mathbf{L}(r,\theta)+U_{M}(z)\mathbf{M}(r, \theta)+U_{N}(z)\mathbf{N}(r,\theta)]d\lambda\\ \mathbf{t}(r,\theta,z)\equiv\sigma_{r}e_{r}+\sigma_{\theta}e_{\theta}+\sigma_{z}e_ {z}\\ =\sum\limits_{m}\int_{0}^{+\infty}[T_{L}(z)\mathbf{L}(r,\theta)+T_{M}(z)\mathbf{M}(r, \theta)+T_{N}(z)\mathbf{N}(r,\theta)]\lambda d\lambda\end{array}  \tag{33}\]  
In the formula, \(U_{L}\), \(U_{M}\), \(U_{N}\), \(T_{L}\), \(T_{M}\), and \(T_{N}\) represent the vector-function coefficients corresponding to the displacement and stress, respectively. According to the expansion coefficient in the vector-function system, we can also get the expressions of the stress, displacement, and stress components in the physical domain. Expression for displacement in the conversion domain:  
\[ \begin{array}{l}u_{r}=\sum\limits_{m}\int_{0}^{+\infty}\left(U_{M}\frac{ \partial S}{\partial r}+U_{N}\frac{\partial S}{r\partial\theta}\right)\lambda \mathrm{d}\lambda\\ u_{\theta}=\sum\limits_{m}\int_{0}^{+\infty}\left(U_{M}\frac{\partial S}{r \partial\theta}-U_{N}\frac{\partial S}{\partial r}\right)\lambda\mathrm{d} \lambda\\ u_{z}=\sum\limits_{m}\int_{0}^{+\infty}U_{L}S\lambda\mathrm{d}\lambda\end{array}  \tag{34}\]  
The expression of stress in the transformation domain:  
\[ \begin{array}{l}\sigma_{rz}=\sum\limits_{m}\int_{0}^{+\infty}\left(T_{M} \frac{\partial S}{\partial r}+T_{N}\frac{\partial S}{r\partial\theta}\right) \lambda\mathrm{d}\lambda\\ \sigma_{\theta z}=\sum\limits_{m}\int_{0}^{+\infty}\left(T_{M}\frac{\partial S }{r\partial\theta}-T_{N}\frac{\partial S}{\partial r}\right)\lambda\mathrm{d} \lambda\\ \sigma_{zz}=\sum\limits_{m}\int_{0}^{+\infty}T_{L}S\lambda\mathrm{d}\lambda \end{array}  \tag{35}\]  
By substituting the displacement expression in the transformation domain into the relationship between strain and stress, the following ordinary differential equations for the expansion coefficients of each layer can be obtained.For the N-type, there are  
\[\frac{d}{dz} \begin{bmatrix}U_{N}\\ T_{N}\end{bmatrix} = \begin{bmatrix}0&\frac{1}{c_{11}}\\ \lambda^{2}c_{66}-\rho\omega_{l}^{2}&0\end{bmatrix}\begin{bmatrix}U_{N}\\ T_{N}\end{bmatrix}  \tag{36}\]  
For the LM-type, there are  
\[