# Research on a Simplified Estimation Method for Wheel Rolling Resistance on Unpaved Runways
## 2 Theory and Methods
### Sobol'-Based Sensitivity Analysis of Rolling Resistance  
#### 2.2.1 Parameter Sensitivity Analysis Methodology  
Equation (11) identifies two distinct classes of parameters influencing wheel resistance:  
* Wheel characteristics: Load \(W\), diameter \(D\), and contact width \(b\);
* Soil properties: Sinkage exponent \(n\), cohesive modulus \(k_{c}\), and frictional modulus \(k_{\varphi}\).  
The multivariate nature of these parameters introduces computational complexity, amplifying both numerical errors and computational workloads. To systematically quantify parameter influences and streamline the resistance calculation framework, this study employs [PERSON]' global sensitivity analysis [26]. This approach enables the identification of dominant parameters and their interaction networks, thereby optimizing computational efficiency without sacrificing model fidelity.  
The [PERSON]' method, a variance decomposition-based global sensitivity analysis technique, is particularly suited for complex models exhibiting nonlinearity, high parameter dimensionality, and interaction effects. Its fundamental principle involves decomposing the variance of model outputs into contributions from individual input parameters and their combinations, thereby enabling quantitative evaluation through first-order sensitivity indices (representing isolated parameter effects) and total sensitivity indices (capturing cumulative effects, including interactions) [27; 28; 29].  
Applied to the theoretical modeling of tire rolling resistance on unpaved surfaces, first-order indices quantify how individual parameter variations independently influence resistance predictions. Total indices serve dual purposes: (1) measuring a parameter's overall contribution to output uncertainty and (2) revealing its interdependencies with other variables. This dual capability proves critical for resistance modeling, where parameters like vertical load and soil deformation indices exhibit strong nonlinear couplings.  
Its core mechanism decomposes the total output variance \(\mathbb{V}(Y)\) of the model \(Y=f(\mathbf{X})\) (where \(\mathbf{X}=\{X_{1},\,\ldots,\,X_{k}\}\)) into orthogonal contributions from individual parameters and their combinations:  
\[Y=f(\mathbf{X})=f_{0}+\sum_{i=1}^{k}f_{i}(X_{i})+\sum_{1\leq i<j\leq k}f_{\bar{y}} \big{(}X_{i},X_{j}\big{)}\ +\ldots\ +f_{1,\ldots,k}(X_{1},\,\ldots,\,X_{k}) \tag{12}\]  
The variance decomposition follows  
\[\mathbb{V}(Y)=\sum_{i=1}^{k}\mathbb{V}_{i}+\sum_{1\leq i<j\leq k}\mathbb{V}_{ \bar{y}}+\ldots\ +\ \mathbb{V}_{1,\ldots,k} \tag{13}\]  
Two key indices quantify parameter influence:  
* First-order sensitivity index: \[S_{i}=\frac{\mathbb{V}_{i}}{\mathbb{V}(Y)}\] (14) quantifies the independent effect of parameter \(X_{i}\) on resistance variance.  
* Total sensitivity index: \[S_{i}=\frac{\mathbb{V}_{i}+\sum_{j\
eq i}\mathbb{V}_{\text{ij}}+\ldots}{\mathbb{ V}(Y)}\] (15) captures the cumulative influence of \(X_{i}\), including all higher-order interactions. The numerical workflow comprises four stages: 3. Quasi-random sampling:
* Generate \(N\times(2k)\) parameter combinations via Sobol' sequences within defined bounds;
* Partition into base matrices \(A\) and \(B\) (\(N\times k\) each). 4. Hybrid matrix construction: For each parameter \(X_{i}\): \[\mathbf{C}_{i}=\left[\mathbf{B}_{(:1:i-1)},\mathbf{A}_{(:i)},\mathbf{B}_{(:i+1:k)}\right]\] (16)
5. Model evaluation: Compute output vectors: * \(f(\mathbf{A})\): Baseline resistance values; * \(f(\mathbf{B})\): Perturbed parameter effects; * \(f(\mathbf{C}_{i})\): Isolated \(X_{i}\) influences. 6. Index calculation: * First-order sensitivity: \[S_{i}=\frac{\frac{1}{N}\sum_{m=1}^{N}f(\mathbf{A})_{m}f(\mathbf{C}_{i})_{m}-f_{0}^{2}} {\mathbb{V}(Y)}\] (17)
* Total sensitivity: \[S_{T}=1-\frac{\frac{1}{N}\sum_{m=1}^{N}f(\mathbf{B})_{m}f(\mathbf{C}_{i})_{m}-f_{0}^{2} }{\mathbb{V}(Y)}\] (18) where \[f_{0}=\frac{1}{N}\sum_{m=1}^{N}f(\mathbf{A})_{m},\ \ \ \mathbb{V}(Y)=\frac{1}{N}\sum_{m=1}^{N}f^{2}(\mathbf{A})_{m}-f_{0}^{2}\]