# Model-Based Design of Aircraft Landing Gear System
## 3 Dynamic Verification Model
### Aircraft Dynamics  
The aircraft is considered to be a point-mass system. Due to the lack of aircraft rolling motion in the model, each wheel can be considered in the same exact conditions, and, therefore, the simulation of just one wheel, when properly scaled in terms of forces, is sufficient to capture the entire system. Even without the inclusion of the rolling motion, the need for an analysis of the dynamics of the aircraft is clear, at least in terms of longitudinal and vertical behavior. Due to the phase we are considering, the dynamics can be compared to car dynamics, since we are dealing with only the longitudinal and vertical motion [24]. To this end, two subsystems in Simulink(r) have been included, which are dedicated to the simulation of such characteristics of the aircraft. These elements are described in Figure 4.  
As can be noticed, a few variables are computed in these blocks. For what concerns the longitudinal dynamics, three are the forces considered: the aerodynamic resistance, which is defined as  
\[F_{drag}=\frac{1}{2}\rho_{air}S_{wing}C_{drag}V_{long}^{2}; \tag{32}\]  
the rolling resistance, which is defined as  
\[F_{roll}=M_{land}\cdot g\cdot f_{roll}; \tag{33}\]  
and the braking force \(F_{brake}\), which will be described later. These forces provide the longitudinal equilibrium of the aircraft during landing, which is defined as follows:  
\[\frac{\partial\dot{x}_{land}}{\partial t}=\frac{-F_{brake}-F_{drag}-F_{roll}}{M _{land}} \tag{34}\]  
which computes the longitudinal acceleration \(a_{long}\) and, thanks to a first and second integration, makes it possible to obtain the longitudinal velocity \(V_{long}\) and the landing distance \(x_{land}\) of the aircraft. For the sake of simplicity, the thrust of the propellers have not been taken into account, considering that the engines when landing should be at idle--if not providing a reverse thrust--or, in case of engine failure, the aircraft should be capable of stopping even without reverse thrust.  
In a similar manner, the vertical dynamics block computes the forces involved in the equilibrium of the aircraft, from which its motion can be described. The forces involved in this case are the aerodynamic lift, which is defined as  
\[F_{lift}=\frac{1}{2}\rho_{air}S_{wing}C_{lift}V_{long}^{2}; \tag{35}\]  
Figure 4: Simulink® subsystems dedicated to the simulation of the longitudinal (**top**) and vertical (**bottom**) aircraft dynamics.  
the weight of the aircraft, which is equal to  
\[F_{weight}=M_{land}\cdot g; \tag{36}\]  
and the reaction forces due to the presence of the landing gear, which act as a spring-damper system that is characterized by a damping coefficient \(c_{shock}\) and a spring stiffness \(k_{shock}\), which have been computed by the sizing script. These forces define the following equilibrium:  
\[\frac{\partial\dot{x}_{vert}}{\partial t}=\frac{F_{weight}-F_{lift}-c_{shock} \cdot V_{vert}-k_{shock}\cdot x_{vert}}{M_{land}} \tag{37}\]  
where the vertical acceleration \(a_{vert}\), the vertical velocity \(V_{vert}\), and the vertical displacement \(x_{vert}\) can be found and where \(c_{shock}\cdot V_{vert}=F_{damp}\) is the damping force and \(k_{shock}\cdot x_{vert}=F_{spring}\) is the spring force. It can be noticed that the system considered in the vertical direction has a single degree of freedom, which is equivalent to the one defined in Figure 5. As such, the tires of the landing gear have been considered as rigid bodies for simplicity.