# Model-Based Design of Aircraft Landing Gear System
## 2 Preliminary Sizing Model
### Wheels and Tires Sizing
Based on this approach, [PERSON] [4] has provided a formula to estimate the diameter of the wheel rim:  
\[D_{rim}=1.4\bigg{(}F_{main,ret}\frac{2.20468}{g}\bigg{)}^{1/4}\text{ [in]} \tag{2}\]  
The other fundamental values can be found utilizing the Goodyear tire catalog [16] as a database to create second-order polynomial regression models with respect to the fundamental parameter, i.e., the tire rated load, for each dimension:  
\[Tire\text{ }outer\text{ }diameter=D_{tire,out}=-0.0264F_{load}^{2}+2.0033F_{load}+15.8 532\text{ [in]}\quad R\text{-}square=89.15\% \tag{3}\]  
\[Tire\text{ }width=W_{tire}=-0.0056F_{load}^{2}+0.6993F_{load}+5.0774\text{ [in]}\quad R\text{-}square=89.53\% \tag{4}\]  
\[Tire\text{ }ply\text{ }index=N_{ply}=-0.0236F_{load}^{2}+1.5917F_{load}+7.3648 \quad R\text{-}square=90.95\% \tag{5}\]  
where \(F_{load}=F_{main,ret}\cdot 10^{-5}\). With the tire outer diameter, it is also possible to obtain the distance between the wheel rim flanges through another regression model:  
\[W_{flanges}=0.0025D_{tire,out}^{2}+0.1010D_{tire,out}+1.9183\text{ [in]}\quad R\text{-}square=99.08\% \tag{6}\]  
Upon obtaining these dimensions, it is immediately possible to estimate the mass of the tires and the wheels. Again, [PERSON] [4] has provided two useful formulas:  
\[m_{tire}=\frac{D_{tire,out}N_{ply}W_{tire}}{107}\text{ [bs]}\qquad m_{wheel}=0.1 \pi\bigg{(}D_{rim}W_{tire}+\frac{D_{rim}^{2}}{4}\bigg{)}\text{ [bs]} \tag{7}\]  
These masses then make it possible to compute the inertia of the wheel assembly through the following [17]:  
\[I_{wheel}=I_{tire}+I_{rim}=m_{tire}\bigg{(}\frac{D_{tire,out}}{2}\bigg{)}^{2}+ \frac{3m_{wheel}}{4}\bigg{(}\frac{D_{rim}}{2}\bigg{)}^{2} \tag{8}\]  
Figure 1: Example of a wheel tire assembly section of an aircraft. It is very important noticing that, once the dimensions of the wheel assembly have been found, it is possible to compute the tire rated load \(RL_{tire}\), which makes it possible to verify the load capacity of the new tire compared to the aircraft mass. Regarding this purpose, [PERSON] [3] has provided an equation to find such a value:  
\[RL_{tire}=A_{cp}(P_{tire}+P_{tire,c})\ [\text{lbs}] \tag{9}\]  
where \(A_{cp}\) is the contact patch area, \(P_{tire}\) is the tire pressure index, and \(P_{tire,c}\) is the equivalent pressure of the tire carcass. First, the contact patch can be found as follows:  
\[D_{rim}=1.4\bigg{(}F_{main,ret}\frac{2.20468}{g}\bigg{)}^{1/4}\text{ [in]} \tag{2}\]0  
where \(d=b(D_{tire,m}-D_{rim})/2\) indicates the static deflection of the tire, \(D_{tire,m}=D_{tire,out}-D_{rim}\) is the mean tire diameter, and \(b\) is the fractional tire deflection, which is usually assumed to equal 0.32 [3]. Then, the pressure index can be computed:  
\[D_{rim}=1.4\bigg{(}F_{main,ret}\frac{2.20468}{g}\bigg{)}^{1/4}\text{ [in]} \tag{2}\]1  
where \(R_{e}=1.475-0.331L_{r}\) is a parameter that is dependent on the tire lift ratio, \(L_{r}=D_{tire,m}/D_{rim}\), which is related to the tire construction, \(T_{o}\), which is equal to 4.4 and is a tire type-related parameter; \(N_{e}=N_{plies}-0.4\), \(S=\left(\frac{D_{tire,m}-D_{rim}}{4}\right)\left(2.5+\frac{D_{rim}}{2D_{tire,m }}\right)\) and \(F_{o}=-1.623104\times 10^{-7}D_{rim}^{5}+1.463062\times 10^{-5}D_{rim}^{4}-5.60 7522\times 10^{-4}D_{rim}^{3}+0.01288401D_{rim}^{2}-0.197904D_{rim}+2.567982\) express the tire operating factor. Finally, the tire carcass equivalent pressure can be found as follows:  
\[D_{rim}=1.4\bigg{(}F_{main,ret}\frac{2.20468}{g}\bigg{)}^{1/4}\text{ [in]} \tag{2}\]2  
If the computed rated load of the tire \(RL_{tire}\) is higher than the load applied on each tire, the tire/wheel combination found may be considered satisfactory. Finally, it is possible to estimate the elastic stiffness of the sized tire from the following equation:  
\[D_{rim}=1.4\bigg{(}F_{main,ret}\frac{2.20468}{g}\bigg{)}^{1/4}\text{ [in]} \tag{2}\]3  
where \(S_{tire,rest}=\frac{1}{3}\left(\frac{D_{tire}}{2}-\frac{D_{rim}}{2}\right)\) is the assumed tire deflection at rest [4].