# Grousers Effect in Tracked Vehicle Multibody Dynamics with Deformable Terrain Contact Model
## 3 Methods: Multibody Modelling
### Grousers Modelling  
In order to evaluate the drawbar pull developed by a tracked machine the contribution in terms of traction force offered by grousers must be considered. The traction force equation presented in the previous section, and implemented for each track link, considered only the shear stress between the track flat surface and the soil. Focusing on rubber tracks with lugs, the cutting action of grousers sunk into the soil must be considered. In fact, grousers generate a greater traction effort contribution. This contribution varies with soil characteristics and grousers number and dimensions. In the presented model grousers as large as the track and 25 mm height were considered, one for each link. For the sake of simplicity, no new rigid bodies were introduced into the MTB model and the link shape was not modified to include the grousers. The virtual grouser position is shown in Figure 13. Traction and Bekker force equations referred to a single grouser were properly related to its size, even if grouser body was not geometrically modelled. Fg in Figure 14 indicates the traction force developed by a grouser. The sinkage behaviour of the whole track changes with the presence of grousers because they are the first parts coming into contact with the soil. No new force elements were introduced but Bekker equation referred to the grouser base was introduced in the force formulation previously written for each link. If the base of the grouser sinks into the ground but the latter does not touch the link base, the track portion receives an upward force equal to:  
Figure 12: Forces acting on the single link. Figure 13: Virtual grouser position. \[F_{B}=A_{g}\left(\frac{k_{c}}{b}+k_{\phi}\right)\cdot y_{g}^{n}+A_{g}c\dot{y}_{g} =A_{g}k_{eq}y_{g}^{n}+A_{g}c\dot{y}_{g}, \tag{12}\]  
in which \(A_{g}\) is the grouser base area (Figure 13). If also the link base sinks into the soil the track portion receives an upward force equal to:  
\[F_{B}=A_{link}k_{eq}y^{n}+A_{link}c\dot{y}+A_{g}k_{eq}y_{g}^{n}+A_{g}cy_{g}^{n}, \tag{13}\]  
in which \(A_{link}\) is the link base area (Figure 13); \(y_{g}\) differs from \(y\) by the grouser height. Also, for the traction force developed by grousers no new force elements were introduced into the MTB model. The expression of the new force contribution was integrated into the existent traction force routine implemented for each track link. The resultant equation was a parametric equation in which grousers number and dimensions could be changed. The traction force offered by a grouser is related to the passive soil pressure and expressed by Equation (6), where \(q\) is different from zero only if the link base touches the soil. In fact, it represents the surcharge acting on the terrain surface portion under the grouser. For this reason, it can be set equal to the Bekker pressure acting on the link base. The grouser height sunk into the soil \(h_{b}\) depends on soil characteristics and Bekker equation. To sum up, the whole traction force equation implemented on each link force element depended on two contribution: the shear stress between link and/or grouser base and soil, and the grousers cutting action. If the whole grouser sank into the terrain and the latter touches the link base, the traction force expression for the link is reported in Equation (14). \[F_{t}=A\left(c+p(y)\tan\phi\right)\cdot\left[1+\frac{K}{ib}\left(e^{-ib_{1}/K} -e^{-ib_{2}/K}\right)\right]+F_{g}\left(1-e^{-i\pi/K}\right)_{q\
eq 0}, \tag{14}\]  
where \(A\) is the total link base area (which includes the grouser base area) and \(y_{g}\) is reasonably approximated with \(y\) values. If only part of the grouser sank into the soil, the traction force expression for the link which the grouser was attached to was expressed as in Equation (15).