# Study of the Dynamic Response of a Rigid Runway with Different Void States during Aircraft Taxing
## 2 Research Methodology
### Runway Pavement Roughness Time-Domain Model  
Airport runway pavement roughness is the vertical deviation of the runway surface relative to the ideal plane, which corresponds to the longitudinal roughness of the runway surface [25]. It has been shown that pavement roughness can be considered to obey a normal distribution with zero mean. The pavement roughness's vertical speed power spectrum density (PSD) is consistent with the white-noise power spectrum. Therefore, the vertical speed input of pavement roughness can be simulated by white noise. Establishing the time-domain model of pavement roughness is feasible using the filtered white noise method [26]. According to the international standard ISO 8608 recommendation [27], the pavement roughness PSD function can be expressed by Equation (1). \[G_{q}(n)=G_{q}(n_{0})\bigg{(}\frac{n}{n_{0}}\bigg{)}^{-2}, \tag{1}\]  
where \(n\) is the spatial frequency (m\({}^{-1}\)), \(n_{0}\) is the reference frequency (\(n_{0}\) = 0.1 m\({}^{-1}\)), and \(G_{q}\) (\(n_{0}\)) is the pavement PSD at the reference frequency (m\({}^{3}\)). Figure 1: Research methods and steps. The aircraft taxiing speed is \(v\) (m/s), and the time frequency is \(f=vn\); then, the PSD function with time as the reference is expressed by Equation (2). \[G_{q}(f)=G_{q}(n_{0}){n_{0}}^{2}\frac{v}{f^{2}}. \tag{2}\]  
Considering the actual situation, by introducing a lower cutoff frequency \(f_{0}\) to the pavement roughness model, Equation (2) can be rewritten as Equation (3). \[G_{q}(f)=G_{q}(n_{0}){n_{0}}^{2}\frac{v}{\left(f+f_{0}\right)^{2}}. \tag{3}\]  
The filtered white noise method equates the pavement roughness signal \(q(t)\) to the first-order linear system response of the unit white-noise signal \(w(t)\). The frequency response function of the system is obtained from Equation (4). \[H(j\omega)=\frac{2\pi n_{0}\sqrt{G_{q}(n_{0})v}}{j\omega+\omega_{0}}=\frac{2 \pi n_{0}\sqrt{G_{q}(n_{0})v}}{j\omega+2\pi n_{1}v}, \tag{4}\]  
where \(\omega\) is the circular frequency (rad/s), and \(n_{1}\) is the cutoff frequency of the pavement space (m\({}^{-1}\)). The first-order differential equation to establish the pavement roughness using filtered white noise is expressed by Equation (5). \[\dot{q}(t)=-2\pi f_{0}q(t)+2\pi n_{0}\sqrt{G_{q}(n_{0})v}w(t), \tag{5}\]  
where the lower cutoff time frequency is \(f_{0}=vn_{1}\), and the filtered white noise pavement roughness time-domain model can be further defined as Equation (6). \[\dot{q}(t)=-2\pi n_{1}vq(t)+2\pi n_{0}\sqrt{G_{q}(n_{0})v}w(t). \tag{6}\]  
The general pavement roughness wavelength is from 0.1 to 100 m; hence, the lower cutoff spatial frequency \(n_{1}\) in the above equation is 0.01 m\({}^{-1}\). The International Roughness Index (IRI) can evaluate airport runway leveling grades [28; 29], and the IRI grading criteria are shown in Table 1.  
The corresponding PSD \(G_{q}\) (\(n_{0}\)) is obtained by converting the IRI. According to existing studies, \(G_{q}\) (\(n_{0}\)) and IRI can be interconverted using Equation (7) [30], and the IRI grading criteria and their corresponding power spectral densities are shown in Table 2.  
\[\text{IRI}=0.78a_{0}\sqrt{G_{q}(n_{0})}, \tag{7}\]  
where, \(a_{0}=10^{3}\) m\({}^{-1.5}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Evaluation Level** & **Good** & **Medium** & **Poor** \\ \hline IRI average (m/km) & \textless{}2.0 & 2.0–4.0 & \textgreater{}4.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Airport pavement IRI grading standard. The corresponding module was built in the MATLAB/SIMULINK environment based on Equation (6). A band-limited white-noise output with a unilateral power of 1 was required in the simulation. A bilateral power spectrum was assumed in the SIMULINK band-limited white-noise module [31]; therefore, this module's noise power spectral density was set to 0.5.