# 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  
A global sensitivity analysis of tire rolling resistance was conducted based on the established framework. Six parameters influencing rolling resistance were assumed to follow uniform distributions within their respective ranges, as detailed in Table 1 [25]. A Sobol' sequence was employed for parameter sampling within these ranges. To balance computational efficiency and sensitivity stability, 4000 samples were generated. Figure 2 presents the calculated first-order and total sensitivity indices for each parameter \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Physical Meaning** & **Lower Bound** & **Upper Bound** \\ \hline \(n\) & Soil deformation index & 0.2 & 1.2 \\ \(b/\text{m}\) & Wheel width & 0.235 & 0.375 \\ \(k_{c}/(\text{kN}/\text{m}^{n+1})\) & Cohesive deformation modulus & 0 & 80 \\ \(k_{\varphi}/(\text{kN}/\text{m}^{n+2})\) & Frictional deformation modulus & 0 & 6000 \\ \(\text{W}/\text{kN}\) & Vertical load & 0 & 2000 \\ \(D/\text{m}\) & Wheel diameter & 0.5 & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter distributions for rolling resistance analysis. As shown in Figure 2, the horizontal axis represents six parameters influencing rolling resistance. The analysis reveals that the soil deformation index \(n\) exhibits the highest combined first-order and total sensitivity indices, indicating its dominant influence on tire resistance. Following \(n\), parameters \(W\), \(k_{\varphi}\), and \(D\) demonstrate moderate sensitivity, collectively contributing significantly to resistance variations. In contrast, the sensitivity indices for \(b\) and \(k_{c}\) remain below 0.05, suggesting negligible impacts on resistance. Notably, the total sensitivity indices for all parameters exceed their corresponding first-order values, highlighting the presence of coupling effects among variables. Particularly for the four influential parameters (\(u\), \(k_{\varphi}\), \(W\), and \(D\)), the substantial gaps between total and first-order sensitivity indices reflect strong interactive effects with other parameters. For operational scenarios at established unpaved runways, where soil properties (\(n\), \(k_{c}\), \(k_{\varphi}\)) are typically fixed, further sensitivity analysis of aircraft-specific parameters (\(b\), \(W\), \(D\)) becomes essential. To address this, three representative soil types--clay soil, lean soil, and sandy loam--were selected based on their mechanical properties (Table 2). Under fixed soil conditions, the total and first-order sensitivity indices for \(b\), \(W\), and \(D\) were calculated and illustrated in Figure 3.  
The sensitivity analysis reveals distinct patterns among aircraft-related parameters across different soil types. As demonstrated in Figure 3, the vertical load \(W\) consistently exhibits the highest sensitivity, followed distantly by tire diameter \(D\), while tire width \(b\) shows negligible influence (sensitivity \(\approx\) 0). Notably, the sensitivity magnitudes of these parameters vary slightly between soil types, with fluctuations on the order of \(10^{-2}\). A critical observation lies in the reduced disparity between total and first-order sensitivity indices for \(b\), \(W\), and \(D\), which diminishes compared to the six-parameter analysis in Figure 2. This attenuation suggests that the interactive effects among the aircraft-specific \begin{table}
\begin{tabular}{c c c c} \hline
**Soil Type** & \(n\) & \(k_{c}\)/(kN/m\({}^{n+1}\)) & \(k_{\varphi}\)/(kN/m\({}^{n+2}\)) \\ \hline Clay soil & 0.5 & 13.19 & 692.15 \\ Lean soil & 0.2 & 16.43 & 1724.69 \\ Sandy loam & 0.9 & 52.53 & 1127.97 \\ \hline \end{tabular}
\end{table} 
Table 2: Soil deformation parameters. Figure 2: Parameter sensitivity indices. parameters (\(b\), \(W\), \(D\)) are relatively limited, while a strong coupling remains between the aircraft parameters and soil properties (\(n\), \(k_{c}\), \(k_{\varphi}\)).