# Study of the Dynamic Response of a Rigid Runway with Different Void States during Aircraft Taxing
## 2 Research Methodology
### Aircraft Taxing Dynamic Loads  
In order to solve the aircraft taxiing dynamic load intuitively and conveniently, the aircraft-runway coupled vibration model was simulated by SIMULINK, according to the state space theory [33], and the aircraft vibration balance equation was converted into the form of state equations. The vertical displacements of the sprung masses, the vertical displacements of the unsprung masses of the left and right main landing gears, the lateral tilt rotation displacements, and the first-order derivatives of these parameters were selected as the state vector components, x, which can be expressed by Equation (15). \[x=(x_{1}\;x_{2}\;x_{3}\;x_{4}\;x_{5}\;x_{6}\;x_{7}\;x_{8})^{T}=(d_{1}\;\dot{d} _{1}\;d_{3}\;\dot{d}_{3}\;\theta\;\dot{\theta}\;d_{0}\;\dot{d}_{0})^{T}. \tag{15}\]  
The output vector, \(y\), is expressed by Equation (16). \[y=(y_{1}\;y_{2}\;y_{3}\;y_{4}\;y_{5}\;y_{6}\;y_{7}\;y_{8}\;y_{9}\;y_{10})^{T}= (d_{1}\;\dot{d}_{1}\;\ddot{d}_{1}\;d_{3}\;\dot{d}_{3}\;\ddot{d}_{3}\;\theta\; \ddot{\theta}\;d_{0}\;\ddot{d}_{0})^{T}. \tag{16}\]  
The runway pavement roughness and its first derivative are represented by the input vector, \(u\), which is expressed by Equation (17). \[u=(q_{1}\;\dot{q}_{1}\;q_{2}\;\dot{q}_{2})^{T}. \tag{17}\]  
The differential equation of the vibrating system can be transformed into a linear state space equation, as shown in Equation (18). \[ \begin{cases}\dot{x}(t)=Ax(t)+Bu(t)\\ y(t)=Cx(t)+Du(t)^{T},\end{cases}  \tag{18}\]  
where \(A,B,C\), and \(D\) matrices; the four matrices can be jointly obtained by Equations (8)-(11) and (15)-(18), as shown in the specific Equations (19)-(22). \[A=\begin{bmatrix}0&1&0&0&0&0&0\\ -\frac{(K_{L1}+K_{R2})}{m_{1}}&-\frac{(C_{L1}+C_{R2})}{m_{1}}&0&0&\frac{K_{L2 }l}{m_{1}}&  
\[B= \begin{bmatrix}0&0&0&0\\ \frac{K_{1,1}}{m_{1}}&\frac{C_{1,1}}{m_{1}}&0&0\\ 0&0&0&0\\ 0&0&\frac{K_{2,1}}{m_{2}}&\frac{C_{2,1}}{m_{2}}\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix} . \tag{20}\]  
\[C= \begin{bmatrix}1&0&0&0&0&0\\ 0&1&0&0&0&0&0\\ -\frac{(K_{1,1}+K_{1,2})}{m_{1}}&-\frac{(C_{1,1}+C_{1,2})}{m_{1}}&0&0&\frac{K_ {2,1}}{m_{1}}&\frac{C_{1,2}}{m_{1}}&\frac{C_{1,2}}{m_{1}}\\ 0&0&1&0&0&0&0\\ 0&0&0&1&0&0&0\\ 0&0&-\frac{(K_{2,1}+K_{2,2})}{m_{2}}&-\frac{(C_{2,1}+C_{2,2})}{m_{2}}&-\frac{K_ {2,2}}{m_{2}}&\frac{K_{2,2}}{m_{2}}&\frac{C_{2,2}}{m_{2}}\\ 0&0&0&0&0&0\\ \frac{K_{2,2}}{m_{2}}&\frac{C_{2,2}}{m_{2}}&-\frac{(C_{2,1}+C_{2,2})}{m_{2}}&- \frac{K_{2,2}}{m_{2}}&-\frac{C_{2,2}}{m_{2}}&\frac{K_{2,2}}{m_{2}}&\frac{C_{2,2 }}{m_{2}}\\ 0&0&0&0&0&0&0\\ \frac{K_{2,2}}{m_{2}}&\frac{C_{2,2}}{m_{2}}&\frac{C_{2,2}}{m_{2}}&0&0&-\frac{(K _{2,2}+K_{2,2})}{m_{2}}&-\frac{(C_{2,2}+C_{2,2})}{m_{2}}\end{bmatrix}  \tag{21}\]  
\[D= \begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ \frac{K_{1,1}}{m_{1}}&\frac{C_{1,1}}{m_{1}}&0&0\\ 0&0&0&0\\ 0&0&\frac{K_{2,1}}{m_{2}}&\frac{C_{2,1}}{m_{2}}\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix} . \tag{22}\]  
Figure 4 shows the simulation model built in SIMULINK. Given the random irrelevance of the pavement position of the main landing gear tires on both sides in the actual situation, the roughness signal was generated by two independent modules using the method described in Section 2.1, and the seeds of two band-limited white-noise modules were set to different numbers to match the actual pavement conditions. The maximum takeoff weight of the A320 is 770 kg; however, considering the safety of taxiing, the weight of taxiing is usually less than its maximum takeoff weight. Table 3 shows the calculation parameters of the aircraft vibration model when the Airbus A320 aircraft is in normal operation [34; 35]. Simulations in SIMULINK were performed to obtain the left and right main landing gear dynamic loads during the taxiing of this model. Figure 5 shows the main landing gear dynamic load generated when the aircraft taxied on the runway with an IRI of 3 at a speed of 20 m/s. The maximum dynamic load on the left main landing gear of the aircraft acting on the pavement was 310 kN, and the dynamic load coefficient (DLC) was obtained as 1.072.