# Study of the Dynamic Response of a Rigid Runway with Different Void States during Aircraft Taxing
## 2 Research Methodology
### Establishment of Coupled Vibration Model of Aircraft-Runway  
The main landing gear of most civil airliners has an airframe weight distribution coefficient of about 0.95 [32]; thus, only the interaction between the main landing gear of the aircraft and the road surface when bearing 95% of the sprung mass needs to be considered. Thus, the vibration model of the aircraft could also be simplified by dividing the upper sprung mass and the lower unsprung mass. Both the cushioning device of the main landing gears and the tires were treated as consisting of a spring with the corresponding stiffness coefficient and a damping device with the corresponding damping coefficient. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Road-Surface Evaluation Grade** & **IRI** & \(G_{q}\) (\(n_{0}\)) (\(10^{-6}\) m\({}^{3}\)) \\ \hline Good & 1 & 1.6436 \\ Medium & 2 & 6.5746 \\ Medium & 3 & 14.7929 \\ Medium & 4 & 26.2985 \\ Poor & 5 & 41.0914 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The PSD of different IRI grades. Figure 2: The roughness curve of pavement and the power spectrum density (PSD) of pavement were verified. Based on the vertical motion and lateral tilt rotation of the fuselage, the vertical motion of the left landing gear, and the vertical movement of the right landing gear while the aircraft taxies on the runway, a four-degrees-of-freedom aircraft vibration model was established, as shown in Figure 3. For the fuselage, \(m_{0}\) is the sprung mass of the fuselage, the vertical displacement is \(d_{0}\), and the lateral tilt rotation inertia and the rotational displacement are \(I_{z}\) and \(\theta\) (side-turning rotation around the \(z\)-axis). For the left and right main landing gears, the unsprung masses are \(m_{1}\) and \(m_{2}\), the unsprung mass stiffness values are \(K_{L1}\) and \(K_{R1}\), the unsprung mass damping values are \(C_{L1}\) and \(C_{R1}\), the sprung mass stiffness values are \(K_{L2}\) and \(K_{R2}\), the damping values of sprung mass are \(C_{L2}\) and \(C_{R2}\), the vertical displacements at the corresponding fuselage are \(d_{2}\) and \(d_{4}\), the vertical displacements of unsprung mass are \(d_{1}\) and \(d_{3}\), and the excitations of unsprung mass by runway pavement roughness are \(q_{1}\) and \(q_{2}\). The distances from the left and right main landing gears to the \(z\)-axis are \(l_{1}\) and \(l_{2}\), where \(l_{1}=l_{2}=l\), while, from the established model, it is known that \(d_{2}=d_{0}+l\cdot\theta\), \(d_{4}=d_{0}-l\cdot\theta\). The four-degrees-of-freedom vibration balance equation of the aircraft model can be expressed by Equations (8)-(11). \[m_{0}\ddot{\ddot{d}_{0}}+K_{L2}(d_{2}-d_{1})+C_{L2}(\dot{d}_{2}- \dot{d}_{1})+K_{R2}(d_{4}-d_{3})+C_{R2}(\dot{d}_{4}-\dot{d}_{3})=0. \tag{8}\] \[m_{1}\ddot{d}_{1}+K_{L2}(d_{1}-d_{2})+C_{L2}(\dot{d}_{1}-\dot{d} _{2})+K_{L1}(d_{1}-q_{1})+C_{L1}(\dot{d}_{1}-\dot{q}_{1})=0.\] (9) \[m_{2}\ddot{d}_{3}+K_{R2}(d_{3}-d_{4})+C_{R2}(\dot{d}_{3}-\dot{d} _{4})+K_{R1}(d_{3}-q_{2})+C_{R1}(\dot{d}_{3}-\dot{q}_{2})=0.\] (10) \[I_{z}\ddot{\theta}+l\left[K_{L2}(d_{2}-d_{1})+C_{L2}(\dot{d}_{2} -\dot{d}_{1})\right]-l\left[K_{R2}(d_{4}-d_{3})+C_{R2}(\dot{d}_{4}-\dot{d}_{3} )\right]=0\;. \tag{11}\]  
The random dynamic load, \(F_{d}\), exerted by the left and right landing gears on the runway surface is represented by Equation (12). \[\left\{ \begin{array}{l}F_{d1}=K_{L1}(d_{1}-q_{1})+C_{L1}(\dot{d}_{1}-\dot{q} _{1})\\ F_{d2}=K_{R1}(d_{3}-q_{2})+C_{R1}(\dot{d}_{3}-\dot{q}_{2})\end{array} \right.. \tag{12}\]  
Figure 3: Simplified 4-DOF aircraft runway vibration model. Equation (13) expresses lift force \(F_{s}\) generated by the wing when the aircraft taxies. \[F_{s}=\frac{1}{2}\rho v^{2}c_{y}s, \tag{13}\]  
where \(\rho\) is the air density (kg/m\({}^{3}\)), \(v\) is the speed when the aircraft taxies (m/s), \(c_{y}\) is the lift coefficient, which is considered equal to the lift coefficient corresponding to the aircraft stopping angle (\(-1^{\circ}\)) before taxing head-up, and \(s\) is the wing area (m\({}^{2}\)).