## 2 Aircraft Dynamic Model and Vibration Excitation Equation
### Excitation Equation of Aircraft Load Vibration  
Because the tunnel is under an airport runway, the impact of the aircraft load is considerable. Hence, the means for establishing the aircraft dynamic load equation becomes the primary consideration. Based on the provisions of Article 5.0.2.1, \"Specification for Airport Cement Concrete Pavement Design (MHT 5004-2010)\" [29], an equivalent area method is used to simplify the shape of the grounded wheel print of the aircraft wheel. The equivalent area method of this simplification technique is shown in Figure 2.  
In this study, the safety and stability of the underground structure of airport pavement subjected to the loads of different aircraft types are compared, and the weak and damage-prone areas of the tunnel structure under the most unfavorable conditions are observed. For this purpose, A380-800, B-747-400, and B-737-800 aircraft with different landing gear configurations based on the data of MHT 5004-2010 are selected and used in the study. These aircraft are the most massive in terms of load and specifications. The main landing gear configurations are shown in Figure 3; the load application points (center-of-mass points) of the main landing gear of each type are marked. The \"Dload\" subroutine implements aircraft loading in the numerical simulation. Hence, this section is based on Equation (3.0.2) of MHT 5004-2010. The aircraft load excitation equations for the following study are  
\[\text{A380}-800\text{ aircraft single wheel load : }F_{s_{1}}=\frac{\rho G}{n_{c_{2}} \cdot n_{w_{2}}}=\frac{0.97\cdot 5620}{2\cdot 10}=272.6\text{ kN},\]  
Figure 3: The main landing gear configuration. Figure 2: The conversion method of wheel equivalent area. B-747-400 aircraft single wheel load : \(F_{s_{2}}=\dfrac{\rho G}{n_{c_{2}}\cdot n_{u_{2}}}=\dfrac{0.952\cdot 3978}{4.4}=236.7\) kN, B-737-800 aircraft single wheel load : \(F_{s_{3}}=\dfrac{\rho G}{n_{c_{2}}\cdot n_{u_{2}}}=\dfrac{0.95\cdot 792.6}{2 \cdot 2}=188.2\) kN, where is the aircraft's main landing gear load distribution factor; \(G\) is the maximum weight of the aircraft (kN); \(N_{c_{2}}\) is the number of main landing gears; and \(N_{u_{2}}\) is the number of wheels in the main landing gear. The aircraft dynamic load equation can be transformed into a cycle-varying simple harmonic vibration load. The aircraft load equation can be scaled up by considering the effect of aircraft vibration [9; 11]. Accordingly, the aircraft's main landing gear dynamic load equation is  
\[ \begin{array}{c}F_{M}=F_{s}+0.1F_{s}sin(\omega t)\\ f=\frac{v}{\pi\mathcal{D}}\\ \omega=2\pi f\end{array}  \tag{7}\]  
where \(\overline{\omega}\) is the rotational circular frequency of the wheel (known as angular frequency); \(t\) is the time variable (s); and \(f\) is the frequency (Hz). The wheel diameter is 1.5 m, and the aircraft's takeoff speed is considered as 80 m/s. According to the foregoing formula, the circular frequency of wheel rotation is \(f=\frac{80}{1.57\pi}=16.97\) Hz, \(\omega=2\pi f=106.66\) rad/s. Then, the main landing gear dynamic load equations for the three types of aircraft are as follows:  
\[ \begin{array}{c}\text{A380}-800,\ F_{M_{1}}=272.6+27.26 sin(106.66t)\ (\text{kN});\\ \text{B-747-400},\ F_{M_{2}}=236.7+23.67 sin(106.66t)\ (\text{kN});\\ \text{and B-737-800},\ F_{M_{3}}=188.2+18.82 sin(106.66t)\ (\text{kN}).\end{array} \]  
Because the Dload subroutine is loaded in the form of model-applied pressure, the main landing gear dynamic loading equation is transformed into a pressure-time equation using \(P_{M}=\frac{P_{M}}{S}\), where \(S\) is the simplified wheel impression area of the machine wheel; the specific dimensions of the wheel are shown in Figure 3.  
The main landing gear pressure-timescale curves for A380-800, B-747-400, and B-737-800 (unit: kPa) are  
\[ \begin{array}{c}P_{M_{1}}=1456.19+145.62 sin(106.66t)\ (\text{kPa});\\ P_{M_{2}}=1420.77+142.08 sin(106.66t)\ (\text{kPa});\\ P_{M_{3}}=1458.91+145.89 sin(106.66t)\ (\text{kPa}).\end{array}