# Research on a Simplified Estimation Method for Wheel Rolling Resistance on Unpaved Runways
## 2 Theory and Methods  
### Rolling Resistance Calculation Model Based on Wheel-Soil Interaction Mechanism  
#### 2.1.1 Wheel-Soil Interaction Model  
The operational dynamics of aircraft on unpaved runways are fundamentally governed by soil deformation under wheel loading, a distinctive feature that differentiates earthen surfaces from rigid pavements such as concrete. This phenomenon introduces complex rolling resistance characteristics that necessitate a rigorous mechanical model of wheel-soil interaction. Among existing modeling approaches, semi-empirical methods dominate due to their validated practicality.  
The foundational work by [PERSON] [5] established critical assumptions for soil-wheel interaction, proposing the seminal pressure-sinkage relationship expressed as follows:  
\[p=kz^{\mu} \tag{1}\]  
where \(p\) denotes vertical pressure per unit contact area (kPa), \(z\) represents sinkage depth (m), \(k\) signifies the soil deformation resistance coefficient (kN/m\({}^{2}\)), and \(\mu\) is a dimensionless soil property exponent. Both \(k\) and \(\mu\) are derived through empirical curve fitting for specific soil types.  
Building upon this framework, [PERSON] [6] advanced a more comprehensive pressure-sinkage formulation applicable to both cohesive and granular soils:  
\[p=\bigg{(}\frac{k_{c}}{b}+k_{\varphi}\bigg{)}z^{n} \tag{2}\]  
Here, \(b\) indicates wheel contact width (m), \(n\) is the sinkage exponent, while \(k_{c}\) (kN/m\({}^{n+1}\)) and \(k_{\varphi}\) (kN/m\({}^{n+2}\)) represent [PERSON]'s cohesive and frictional soil deformation moduli, respectively. These parameters (\(n\), \(k_{c}\), \(k_{\varphi}\)) collectively characterize the nonlinear pressure-sinkage behavior.  
Subsequent refinements include [PERSON]'s model [16], which incorporates soil unit weight (\(\gamma\), kN/m\({}^{3}\)) and cohesion (\(c\), kPa):  
\[p=\bigg{(}ck_{c}^{\prime}+\gamma bk_{\varphi}^{\prime}\bigg{)}\bigg{(}\frac{z }{b}\bigg{)}^{n} \tag{3}\]  
where \(k_{c}^{\prime}\) and \(k_{\varphi}^{\prime}\) are dimensionless soil constants. [PERSON] [17] further extended the theoretical framework by introducing velocity-dependent sinkage effects:  
\[p=p_{0}\bigg{(}\frac{u}{u_{0}}\bigg{)}^{m} \tag{4}\]  
Here, \(p_{0}\) denotes reference pressure (kPa), \(u\) and \(u_{0}\) represent instantaneous and reference sinkage velocities (m/s), respectively, with \(m\) as the velocity sensitivity exponent.  
Comparative analyses of these models [18; 19; 20; 21; 22] reveal that Equations (2) and (3) demonstrate superior correlation with experimental data for both sandy and clayey soils. Notably, [PERSON]'s formulation exhibits advantages for large-diameter wheels (\(D>0.5\) m) under limited sinkage conditions, achieving prediction accuracies within 12-15% of measured values [20; 21; 22]. The model's parametric efficiency (\(n\), \(k_{c}\), \(k_{\varphi}\)) compared to [PERSON]'s additional variables (\(c\), \(\gamma\)) and [PERSON]'s velocity-dependent parameters (\(u_{0}\), \(m\)) enhances its practicality for runway applications.  
While [PERSON]'s original formulation accounts for bulldozing resistance in the rolling direction [6], this component becomes negligible for large-diameter wheels operating on well-compacted unpaved runways [23]. Therefore, bulldozing resistance is excluded from the current analytical framework to focus on dominant resistance mechanisms.