# Research on a Simplified Estimation Method for Wheel Rolling Resistance on Unpaved Runways
## 2 Theory and Methods
### Sobol'-Based Sensitivity Analysis of Rolling Resistance
#### 2.2.2 Sobol' Sequence and Numerical Simulation Analysis
For aircraft operations on established unpaved runways, where soil characteristics remain constant, vertical load \(W\) emerges as the predominant factor governing rolling resistance. This dominance underscores the necessity of prioritizing \(W\)-resistance relationships in tire performance studies. To streamline resistance estimation, Equation (11) is simplified as follows:  
\[F=Am^{N} \tag{19}\]  
\[A=\frac{\left(\frac{3\varphi}{\sqrt{D}}\right)^{\frac{2n+2}{2n+1}}}{(3-n)^{ \frac{2n+2}{2n+1}}(n+1)b^{\frac{1}{2n+1}}\left(\frac{k_{c}}{b}+k_{\varphi} \right)^{\frac{1}{2n+1}}} \tag{20}\]  
\[N=1+\frac{1}{2n+1} \tag{21}\]  
where \(A\) denotes the composite coefficient encompassing soil-tire interaction characteristics, \(N\) represents the power exponent determined by the soil's sinkage index \(n\), and \(m\) is the aircraft mass (kg). When both aircraft configuration and soil conditions are specified, \(A\) becomes a constant, while \(n\) serves as a soil-specific empirical parameter. This sinkage index \(n\) functions as a terrain-dependent constant across most analytical models and can be predetermined through standardized plate-sinkage tests [20; 21; 22]. The selection of aircraft mass \(m\) instead of vertical load \(W\) in this simplification strategically emphasizes takeoff mass as the primary operational variable. Although vertical load \(W\) fluctuates during taxiing due to aerodynamic lift variations, this study's methodology determines low-speed resistance under static load conditions (\(W\approx m\cdot g\)) for input into empirical takeoff models that inherently account for lift effects during acceleration [30]. The gravitational component derived from \(m\) remains constant, making it a more stable parameter for resistance modeling. The simplified model operates under two foundational assumptions to balance accuracy and practicality. First, it equates vertical load to gravitational force, presuming minimal surface irregularities that could induce dynamic excitation [31]. This idealized loading condition holds best on well-compacted surfaces with low roughness. Second, gravitational acceleration is standardized as \(g=10\) m/s\({}^{2}\) to streamline calculations. Central to the model's predictive capability are two parameters: the soil-dependent sinkage index \(n\) and the composite coefficient \(A\). Determining \(n\) requires controlled penetration tests to characterize soil deformation properties, while \(A\) synthesizes five in  
Figure 3: Sensitivity indices under fixed soil conditions. terdependent variables (\(b\), \(k_{c}\), \(k_{\varphi}\), \(D\), \(n\)) into a single constant. Traditional methods for quantifying \(A\) accumulate uncertainties from multiple sources, including measurement errors in soil cohesion (\(k_{c}\)) and friction (\(k_{\varphi}\)) moduli, empirical approximations during curve fitting, and stress distribution assumptions tied to tire diameter. These compounded errors highlight the advantage of treating \(A\) as a unified parameter rather than deriving it through sequential measurements. The model's strength lies in its balanced compromise between simplicity and technical rigor, serving as an accessible tool for initial aircraft-terrain interaction studies when detailed soil data acquisition proves challenging. After transforming the formula, the rolling resistance of the wheel is represented in a simple and universal manner, which facilitates higher accuracy for this model.