# Research on a Simplified Estimation Method for Wheel Rolling Resistance on Unpaved Runways
## 2 Theory and Methods
### Parameter Determination Based on Field Tests
Practical limitations arise due to the empirical nature of \(n\), which assumes idealized homogeneous soil conditions rarely achievable in field environments. For the sake of simplifying the input parameters, the unit of \(m\) is unified to tons, and the unit of \(F\) is unified to newtons. To address these uncertainties and rigorously evaluate the exponent's influence, a dual calibration methodology was employed:  
1. Data-driven parameter optimization: The exponent \(N\) was inversely derived by fitting resistance measurements to refine \(n\), ensuring alignment with experimental observations;
2. Parametric sensitivity exploration: The exponent \(N\) was systematically varied across 1.3-1.7 (0.1 increments) to quantify its impact on predictive accuracy. This dual-strategy calibration ensures adaptability to real-world soil variability while preserving the mechanistic foundation of the original formulation. Then six candidate equations with distinct exponents were evaluated (Table 4 and Figure 6), assessing their ability to reconcile theoretical predictions with measured resistance values. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Fitting Equation** & \(R^{2}\) & **80 kN** & **100 kN** & **120 kN** & **200 kN** & **500 kN** & **1000 kN** & **1500 kN** \\ \hline Measured Value & — & 3.27 & 4.42 & 5.73 & — & — & — \\ \(y=88.1x^{1.7}\) & 0.99632 & 3.02 & 4.42 & 6.02 & 14.35 & 68.11 & 221.30 & 440.89 \\ \(y=111.2x^{1.6}\) & 0.99579 & 3.10 & 4.43 & 5.93 & 13.42 & 58.14 & 176.24 & 337.17 \\ \(y=140.2x^{1.5}\) & 0.97742 & 3.17 & 4.43 & 5.83 & 12.54 & 49.57 & 140.20 & 257.56 \\ \(y=176.7x^{1.4}\) & 0.99972 & 3.25 & 4.44 & 5.73 & 11.71 & 42.25 & 111.49 & 196.68 \\ \(y=181.2x^{1.39}\) & 0.99977 & 3.26 & 4.45 & 5.73 & 11.66 & 41.66 & 109.18 & 191.83 \\ \(y=223.1x^{1.3}\) & 0.99513 & 3.33 & 4.45 & 5.64 & 10.96 & 36.07 & 88.82 & 150.46 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Predicted rolling resistance for different fitting equations under various loads (kN). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Load** & **Maximum (kN)** & **Minimum (kN)** & **Mean (kN)** & **Standard Deviation (kN)** \\ \hline \multirow{4}{*}{80 kN} & 7.20 & 0.01 & 3.22 & 2.25 \\  & 4.76 & 0.22 & 2.93 & 1.12 \\  & 6.64 & 0.07 & 3.67 & 1.90 \\  & 5.85 & 0.05 & 3.25 & 1.44 \\ \hline \multirow{4}{*}{100 kN} & 7.41 & 0.83 & 4.59 & 1.86 \\  & 9.04 & 0.04 & 4.50 & 2.31 \\  & 9.82 & 0.11 & 4.23 & 2.63 \\  & 9.47 & 0.20 & 4.36 & 2.42 \\ \hline \multirow{4}{*}{120 kN} & 8.70 & 2.13 & 5.04 & 1.59 \\  & 9.79 & 2.99 & 6.14 & 1.58 \\ \cline{1-1}  & 12.23 & 0.06 & 6.02 & 3.20 \\ \cline{1-1}  & 10.27 & 0.01 & 5.73 & 2.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Rolling resistance values under different vertical loads. As demonstrated in the analytical results, the optimized exponent \(N\) derived from resistance data fitting converges to 1.39. Within the 80-120 kN load range, predictive outputs across candidate equations exhibit marginal discrepancies, with peak accuracy occurring at \(N\) = 1.39 and progressively diminishing as \(N\) deviates bidirectionally. However, predictive divergence amplifies significantly under elevated loads: at 1500 kN, the maximum predicted value almost triples the minimum, with the \(N\) = 1.5 model overestimating resistance by 34% compared to the \(N\) = 1.39-based formulation. Consequently, according to the test, for soil in loess regions, the following refined equation is proposed to estimate wheel rolling resistance:  
\[F=181.2m^{1.39} \tag{22}\]