# Dynamic Response Analysis of Asphalt Pavement under Pavement-Unevenness Excitation
## 2 Solution of Vehicle Load under Pavement-Unevenness Excitation  
### Establishment of Pavement-Unevenness Model  
The establishment of a random pavement-unevenness excitation model is the basis for the study of random road loads exerted by vehicles. In this paper, the filtered white-noise method is used to build a time-domain model of four-wheel pavement unevenness to provide pavement excitation for the solution of maneuvering loads under the seven-degree-of-freedom vehicle model. In this paper, the power-spectral-density function is used to characterize the classification and statistical characteristics of pavement unevenness. The fitting expression of pavement-unevenness spectral density \(G_{q}\) (\(n\)) within the spatial frequency \(n_{1}<n<n_{2}\) is as follows:  
\[G_{q}(n)=G_{q}(n_{0})(\frac{n}{n_{0}})^{-w} \tag{1}\]  
where \(n\)--spatial frequency; \(n_{0}\)--reference spatial frequency, usually \(n_{0}=0.1\) m\({}^{-1}\); \(G_{q}(n_{0})\)--pavement-unevenness coefficient; \(w\)--frequency index, where in general \(w=2\). According to the power spectral density of the road surface, the uneven condition of the road surface is divided into different grades, and the quality of the road surface gradually decreases with \(G_{q}(n_{0})\) increasing [33]. \[\dot{q}_{1}(t)=-2\pi vn_{q}q_{1}(t)+2\pi n_{0}\sqrt{G_{q}(n_{0})v}\cdot\omega(t) \tag{2}\]  
In this paper, the filtered white-noise method is used to generate random pavement-unevenness excitation. Assuming that the left-front wheel, right-front wheel, left-rear wheel, and right-rear wheel are stimulated by road \(q_{1}(t)\), \(q_{2}(t)\), \(q_{3}(t)\), and \(q_{4}(t)\) in turn, Formula (2) represents the time-domain model of the uneven pavement of the left-front wheel. In Formula (2), \(q_{n}(t)\) is the pavement-unevenness excitation of the wheel, \(n_{q}\) is the cutoff space frequency under the pavement space, taking \(n_{q}=0.011\) m\({}^{-1}\); \(\omega(t)\)--white-noise signal; \(v\)--vehicle speed in m/s. Assume that the front- and rear-wheel road excitation are exactly the same but with a lag of some time. The lag time \(t_{d}\) is the ratio of the wheelbase \(L\) of the front and rear wheels to the speed \(v\); that is  
\[t_{d}=\frac{L}{v} \tag{3}\]  
The pavement-unevenness excitation of the front and rear wheels has the following relationship:  
\[q_{3}(t)=q_{1}(t)(t-t_{d}) \tag{4}\]  
Due to the time-lag correlation between the front and rear wheels, the Pade approximation algorithm can be adopted. The transfer function \(G_{31}\) between the input of pavement unevenness of the front and rear wheels is approximately as follows:  
\[G_{31}=\frac{q_{3}(s)}{q_{1}(s)}=\frac{1-\frac{t_{d}}{2}s+\frac{t_{d}^{2}}{12 }s^{2}}{1+\frac{t_{d}}{2}s+\frac{t_{d}^{2}}{12}s^{2}} \tag{5}\]  
The Laplace inverse transformation is performed on the above Equation (5), and the time-domain relationship between the front-wheel road input and the rear-wheel road input is obtained as follows:  
\[\dot{q}_{3}(t)=-\frac{2}{t_{d}}q_{3}(t)+\frac{2}{t_{d}}q_{1}(t)-\dot{q}_{1}(t) \tag{6}\]  
The time-domain function \(q_{1}(t)\) of front-wheel pavement unevenness is substituted into the above formula to obtain the following:  
\[\dot{q}_{3}(t)=(\frac{2}{t_{d}}+2\pi n_{q}v)q_{1}(t)-\frac{2}{t_{d}}q_{3}(t)- 2\pi n_{0}\sqrt{G_{q}(n_{0})v}\cdot\omega(t) \tag{7}\]  
Assuming that the self-spectral density of the left and right wheels of the vehicle is the same and the phase difference is 0, the coherence function of the left-front-wheel road input \(q_{1}(t)\) and the right-front-wheel road input \(q_{2}(t)\) is as follows:  
\[coh(\omega)=\frac{s_{12}(\omega)}{s_{1}(\omega)} \tag{8}\]where, \(s_{12}(\omega)\) is the left- and right-wheel mutual-spectral density function; \(s_{1}(\omega)\) is the self-spectral density function.