# Model-Based Design of Aircraft Landing Gear System
## 3 Dynamic Verification Model
### Landing Gear Model
#### 3.2.1. Wheel Assembly  
As introduced earlier, the wheel assembly of the main landing gear is modeled in the \"Main Wheel LL\" subsystem. This element, which can be seen in Figure 6, receives the following as input:  
* the actuation force \(F_{a}\) of the brake calipers, which is previously sized;
* the longitudinal speed of the aircraft \(V_{long}\);
* the total reaction force of the landing gear \(F_{vert}\), which is computed as the sum of the damping force and spring force from the \"Vertical Dynamics\" subsystem. These inputs are used to compute the following outputs:  
* the braking force on a single wheel, which is intended as \(F_{\mathit{trake}}/N_{\mathit{wheel}}\);
* the longitudinal slip coefficient of the tire, which is defined as \(\delta_{long}\);
* the wheel rotational speed \(\omega_{\mathit{wheel}}\) and the acceleration \(\omega_{\mathit{wheel}}\);
* the brake discs temperature, which is defined as \(T_{\mathit{brake}}\). The first subsystem of the \"Main Wheel LL\", again reported in Figure 6, is dedicated to the interface model between the tire and road. This block implements the rotational equilibrium of the wheel in order to obtain its behavior during landing. Such a model makes it possible, therefore, to add control logic to the braking system, which makes it possible to avoid certain unwanted phenomena, such as the lock of the wheel due to the  
Figure 6. Simulink subsystem for the landing gear assembly. excessive braking torque applied by the brakes. Such an equilibrium is found using the following equation:  
\[\frac{\partial\omega_{wheel}}{\partial t}=\frac{-\frac{F_{wet}}{N_{wheel}}\mu_{ long}R_{main}-\tau_{brake}}{I_{wheel}} \tag{38}\]  
where \(N_{wheel}\) is the number of wheels of the main landing gear, \(R_{main}\) is the effective tire radius, \(\tau_{brake}\) is the braking torque on the single wheel, \(I_{wheel}\) is the moment of inertia of the wheel, and \(\mu_{long}\) is the longitudinal friction coefficient. This last term represents the ratio between the longitudinal force exerted on the wheel and the vertical force applied on it. Therefore, the first term in the numerator of Equation (38) is the torque exerted on the wheel by its interaction with the road. The longitudinal force coefficients can be found through many methods. In the current application, the model used is the Pacejka \"magic formula\" [PERSON] (2015), which makes it possible to find the longitudinal friction coefficient of the wheel based on the slip coefficient. It is defined as follows:  
\[\mu_{long}(\sigma)=D_{dry}\sin(C_{dry}\tan^{-1}(B_{dry}\sigma-E_{dry}(B_{dry} \sigma-\tan^{-1}(B_{dry}\sigma)))) \tag{39}\]  
where \(B_{dry}\), \(C_{dry}\), \(D_{dry}\), and \(E_{dry}\) are coefficients experimentally obtained relative to the road conditions, and \(\sigma\) is the longitudinal slip coefficient, which is defined as follows:  
\[\sigma=\frac{R_{main}\omega_{wheel}-V_{long}}{V_{long}}\qquad\text{for braking conditions} \tag{40}\]  
The road coefficients have been set for \"dry tarmac\" conditions. Due to the unavailability of experimental coefficients specific to the tires found with the preliminary sizing script, the same coefficients have been used for every simulation. All of these equations have been implemented in the subsystem (with Equations (39) and (40) in the \"Slip + Pacejka\" subsystem), thereby iterating the computed longitudinal friction coefficient at every integration step for the computation of the wheel dynamics. The other subsystem composing the landing gear is the \"Brake\", which is in charge of modeling the behavior of the brake pack of a main gear wheel. More specifically, it simulates two main characteristics of the brakes: the braking torque generated (and, consequently the braking force) and the temperature reached by the brake discs during braking.