# Grousers Effect in Tracked Vehicle Multibody Dynamics with Deformable Terrain Contact Model
## 3 Methods: Multibody Modelling
### Deformable Ground Contact Model
Shear displacement \(j\) can be defined as the product \(ix\), being  
\[i=\frac{v_{t}-v}{v_{t}}=\frac{v_{j}}{v_{t}}. \tag{9}\]  
\(i\) is also known as slip coefficient, \(v_{t}\) is the theoretical speed of the vehicle, \(v\) is the actual one and \(v_{j}\) is the relative speed between track portion and soil. In other words, \(v_{j}\) is the absolute link speed in \(x\) direction: when no slip occurs the track link in contact with the terrain is still. In this approach, the \(x\) coordinate is correctly identified as in Figure 11 and the theoretical speed is defined by Equation (10)  
Figure 10: Complete multibody (MTB) code scheme  
\[v_{t}=\omega_{sprocket}\cdot\left(r_{sprocket}+s_{track}\right). \tag{10}\]  
\(\omega_{sprocket}\) and \(r_{sprocket}\) are the angular speed and the radius of the driven wheel, respectively, and \(s_{track}\) is the track thickness. The syntax of the traction force was written into a specific routine which solved it only if contact between soil body and links was detected. The traction force expression was obtained integrating Equation (8) along the length of each track link, for the portion of the track in contact with the soil. \[F=l\int_{b_{1}}^{b_{2}}\left(c+p(y)\tan\phi\right)\cdot\left(1-e^{-ix/K}\right) dx=A\left(c+p(y)\tan\phi\right)\left[1+\frac{K}{ib}\left(e^{-ib_{1}/K}-e^{-ib_{2}/K} \right)\right], \tag{11}\]  
where \(A\) was the link base area, \(b\) was the link length and \(b_{1}\) and \(b_{2}\) were the limits of integration for a single link (Figure 12). They were moving limits of integration because the \(x\) coordinate was referred to the moving reference frame centred on the front idle wheel. Each link varied its position with respect to the idle wheel during vehicle motion as shown in Figure 11. The traction force routine was implemented into each link S-force element (Figure 12). Traction force sign was always opposite to the sign of the corresponding link speed (slip speed) coherently to the machine movement. The product between the normal pressure and the link contact area (term \(Ap(y)\) in Equation (11)) coincided with the Bekker force previously implemented. Within the traction force formulation, the term \(Ap(y)\) was replaced by the Bekker force definition. In this way the traction force evaluation depended on the current normal pressure under the track link (influenced by the terrain shape) and the approximation of the normal pressure with the overall average one was avoided. To sum up, soil mechanics equations were implemented only within force elements applied to the track links. No other equations related to soil stress and deformation were computed during simulations. However, tracks sinkage and developed drawbar pull were always related to deformable terrain characteristics and shape by mean of moving spheres constrained to each link as shown in Figure 12, where the sphere A is the sphere corresponding to the highlighted link. \[F=A\left(\frac{k_{c}}{b}+k_{\phi}\right)y^{n}+Ac\dot{y}=Ak_{eq}y^{n}+Ac\dot{y}. \tag{7}\]0 Coordinate system for traction force equation integration.