# Model-Based Design of Aircraft Landing Gear System
## 2 Preliminary Sizing Model
### Brakes Sizing  
While the tires and wheels of the landing gear provide support to the aircraft when in contact with the ground, it is fundamental for the aircraft to be able to come to a complete stop in due time. This is true not only when the aircraft is landing, but also during taxi maneuvers and in case of a rejected takeoff. The main system in charge of this operation is the landing gear braking system, an example of which is reported in Figure 2.  
The most common type of braking system employs disc rotors, which are integral to the rotating wheels, which, when the braking action is required, are compressed between stator discs, fixed to the landing gear, by applying hydraulic pressure through a pressure plate. The friction generated by the discs converts the kinetic energy of the aircraft into thermal energy, thus slowing it down. The energy to be converted during landing can be easily computed as follows:  
\[E_{land}=\frac{1}{2}M_{land}V_{long}^{2} \tag{14}\]Since mechanical brakes convert kinetic energy into thermal energy, the main issue that must be taken into account is the brake temperature. Based on the material, the thermal characteristics of the brake pack can largely vary. For instance, steel brakes (very common and cheaper solution) cannot withstand temperatures higher than 1000 \({}^{\circ}\)C; meanwhile, carbon discs are a more expensive solution but can manage temperatures of 1800 \({}^{\circ}\)C [18]. Therefore, the energy requirement during landing with respect to the minimum thermal mass of the brakes can be found depending on the material employed:  
\[m_{brake,req}=\frac{E_{land}}{C_{b}(T_{b,des}-T_{amb})} \tag{15}\]  
where \(C_{b}\) is the specific heat capacity of the brake material, and \(T_{b,des}\) is the design brake temperature (i.e., the desired brake temperature at the end of the braking phase). Although the required brake mass provides an indication as to the characteristics of the brakes to be designed, it does not allow for the actual sizing of the system. To this end, [PERSON] [13] has provided formulas that make it possible to estimate the characteristic diameters of the brake discs, based on the dimensions of the housing wheel, which are previously sized:  
\[Rotor\;outer\;diameter=D_{r,out}=0.788D_{rim}+2.322\;[\mathrm{in}] \tag{16}\]  
\[Rotor\;inner\;diameter=D_{r,in}=0.6645D_{rim}-2.361\;[\mathrm{in}] \tag{17}\]  
\[Stator\;outer\;diameter=D_{s,out}=0.7091D_{rim}+2.286\;[\mathrm{in}] \tag{18}\]  
\[Stator\;inner\;diameter=D_{s,in}=0.417D_{rim}+0.391\;[\mathrm{in}] \tag{19}\]  
Based on these equations, the lateral areas of the brake rotors and stators are calculated follows:  
\[A_{b,r}=\pi\frac{D_{r,out}^{2}-D_{r,in}^{2}}{4}\qquad A_{b,s}=\pi\frac{D_{s, out}^{2}-D_{s,in}^{2}}{4} \tag{20}\]  
Then, the thickness of the discs can be computed. Since the disc must fit inside the wheel rim, a good approximation would be to consider the thickness of the disc pack \(t_{b,pack}=0.75W_{flarge}\). The thickness of each disc is then found though the expression  
Figure 2: Example of a multidisc brake pack of an aircraft. \(t_{b,disc}=t_{b,pack}/(N_{b,r}+N_{b,s})\), where \(N_{b,s}\) is the number of stator discs, which is assumed to be \(N_{b,r}+1\). These dimensions allow us to find the actual mass of the brakes:  
\[m_{brake,act}=(\rho_{b,r}A_{b,r}N_{b,r}+\rho_{b,s}A_{b,s}N_{b,s})t_{b,disc}N_{ wheel} \tag{21}\]  
which should be higher than the requested mass computed in Equation (15). In order to evaluate the performance of the designed brake pack, the lining loading may be computed as follows:  
\[E_{land}=\frac{1}{2}M_{land}V_{long}^{2} \tag{14}\]0  
where \(A_{b}\) is the contact surface between a rotor and stator discs pair, and \(N_{b,i}=2N_{b,r}\) is the number of braking interfaces. It represents the amount of mechanical energy converted into thermal energy per unit of braking surface, thus indicating the stress applied on the brakes [19]. The last sizing parameter for the braking system is the actuation force required to stop the aircraft.