## 6 Evaluation of Compaction Resistance Considering Slip Sinkage  
Slip sinkage affects compaction resistance, a type of off-road tracked vehicle motion resistance that occurs across the entire soil-track interface. The slip sinkage equation proposed in this study was used to assess off-road tracked vehicle compaction resistance, taking into account the effect of slip sinkage. As bulldozing resistance is caused by ground deposited at the front of the track system where slip sinkage does not occur, its dependency on the slip sinkage is not dealt with in this section. As mentioned earlier (refer to Figure 1), slip sinkage is a function of the longitudinal distance (\(0\leq x\leq 1\)) traveled by the track system, which is expressed in Equation (3) as a double integral. \[R_{cl}=w\int_{0}^{l}\int_{0}^{z_{t}}(W/wl)dz\;dx \tag{8}\]  
According to [PERSON] [3], (\(W/wl\)) is a function of \(k\) (sinkage modulus) and \(n\) (sinkage exponent), which are empirical pressure-sinkage parameters determined using the bevameter test. \[W/wl=kz^{n} \tag{9}\]  
Substituting Equations (7) and (9) in Equation (8) results in the following formula:  
\[R_{c}=\frac{wkz_{0}^{n+1}}{l(n+1)}\int_{0}^{l}\Bigl{(}3.78-2.78\mathrm{e}^{(- 0.009(ix)^{1.77})}\Bigr{)}^{n+1}dx \tag{10}\]  
Moreover, assuming that \(R_{c0}\) represents compaction resistance when only static sinkage occurs, the integration range of Equation (3) can be defined as static sinkage, as follows. \[R_{c0}l=wl\int_{0}^{z_{0}}(W/wl)dz \tag{11}\]  
Substituting Equation (9) into Equation (11) provides the following formula:  
\[R_{c0}=\frac{wkz_{0}^{n+1}}{n+1} \tag{12}\]  
By substituting Equation (12) into Equation (10), the compaction resistance of the off-road tracked vehicle considering slip sinkage can be expressed as a function of \(R_{c0}\), sinkage exponent \(n\), soil-track contact length \(l\), and slip ratio \(i\). The following equation can be numerically integrated with \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Model (a)** & **Model (b)** \\ \hline Shape Ratio, \(l_{l}\)/\(h\) & 2 & 5 \\ \hline Width, \(w\) (mm) & 154.8 & 154.8 \\ \hline Length, \(l_{t}\) (mm) & 89.2 & 223.0 \\ \hline Height, \(h\) (mm) & 44.6 & 44.6 \\ \hline Thickness, \(t\) (mm) & 9.0 & 9.0 \\ \hline Vertical Stress, \(\alpha_{q}\) (kPa) & 7.0 (9.7 kg) & 7.0 (24.2 kg) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Material properties of model track systems used for the verification experiments. respect to given driving conditions to quantitatively assess the compaction resistance of the off-road tracked vehicle. \[R_{c}=\frac{R_{c0}}{l}\int_{0}^{l}\left(3.78-2.78e^{(-0.009(ix)^{1.77})}\right)^{ n+1}dx \tag{13}\]  
Based on the proposed evaluation method for compaction resistance, we parametrically investigated the effects of \(n\), \(l\), and \(i\) on compaction resistance considering the slip sinkage. In this study, \(R_{c}\) was normalized with respect to \(R_{c0}\) by dividing both sides of Equation (13) by \(R_{c0}\). A parametric study was then conducted. Considering that \(n\) of silty sand ranges from approximately 0.5 (loose state) to 2.0 (dense state) [2; 32; 33; 34], we set \(n\) as 0.5, 1.0, 1.5, and 2.0. \(l\) of a commercial crawler tractor approximately ranges from 2000-5000 mm. Thus, \(l\) was set to 2000, 3000, 4000, and 5000 mm for the parametric study. According to [PERSON] [2], it is preferable to maintain an off-road tracked vehicle slip ratio of below 0.2. Moreover, a slip ratio of 0.2 is usually used as a common basis to compare the performances of off-road tracked vehicles. Accordingly, we considered \(i\) from 0-0.2. The parameter ranges considered in this study are summarized in Table 5.  
Figure 14 presents plots of the normalized compaction resistance considering the slip sinkage \(R_{c}\)/\(R_{c0}\) with respect to \(i\) when \(l\) = 2000, 3000, 4000, and 5000 mm, with \(n\) = 0.5 (Figure 14a), 1.0 (Figure 14b), 1.5 (Figure 14c), and 2.0 (Figure 14d). Under all conditions, \(R_{c}\)/\(R_{c0}\) increased gradually from 1 as \(i\) increased.