# Grousers Effect in Tracked Vehicle Multibody Dynamics with Deformable Terrain Contact Model
## 3 Methods: Multibody Modelling
### Deformable Ground Contact Model  
The mathematical model of deformable soil was integrated into the MTB environment to correctly simulate the contact and the behaviour of the track on agricultural terrain. An efficient track-terrain contact model was developed implementing soil mechanics equations within routines defining different force elements applied to each rigid link of the track. Resulting forces acting on a link depend on ground characteristics, although they do not affect soil body deformation. The following routines description refers to smooth tracks (without grousers). Bekker equation Equation (1) was implemented in the routine related to sinkage behaviour of track links. The soil was modelled as a rigid body which may present irregularities such as peaks or valleys. Equation (1) multiplied for the link contact area was implemented in a non-linear Single-component-force (S-force) applied to the center of mass of each link and moving together with it (Figure 8). Thus each track link correctly sink into the ground body. The routine was defined to solve the Bekker force equation only when a contact between the soil body and the link was detected. The distance between the free surface of the soil and the contact base of a track link (\(y\) in Equation (1)) must be known at each time step of the solution. The terrain was modelled in the MTB environment as a rigid body with the desired shape to replicate the real undeformed one. To solve the contact equations, the information of the undeformed shape of the terrain was taken into account using a particular modelling strategy. A set of small and with low mass spheres forced to move with each track links but free to follow normally the terrain profile was used. The spheres CM position was evaluated at each time step, thus the information of the terrain geometry was available to the developed routine. Figure 7: Tensioning system. The sinkage value \(y\) was constantly given as the difference between the base of the track link and the lowest contact point of each sphere (function of the sphere CM position and its radius). In other words, the marker centred in the sphere CM was one side of the non-linear spring defined by Equation (1) and applied to the track link surface. As a result, the tracks sinkage and the machine position depended on the terrain characteristics and shape (Figure 9). To be sure to not interfere with the track link movement, but to correctly sense the terrain shape, a frictionless contact model was implemented between each sphere and the ground body. Each sphere had to be vertically aligned with the corresponding link. For this reason, two motion laws were imposed. The first (between each sphere and the corresponding link), had a constant null displacement in \(z\) direction, and no constraints in the other directions. In the second motion law (between each sphere and the terrain) the sphere speed value in horizontal direction was set to be equal to the corresponding track link speed in the same direction. Soil damping was considered into account using typical values of terrain damping coefficients [44] and modifying Bekker equation as follows. \[F=A\left(\frac{k_{c}}{b}+k_{\phi}\right)y^{n}+Ac\dot{y}=Ak_{eq}y^{n}+Ac\dot{y}. \tag{7}\]  
The complete algorithm can be summarized as in Figure 10 in which \(F_{B}\) and \(F_{t}\) are the Bekker force and the traction force respectively. Figure 8: Bekker force acting on a single-track link. Figure 9: Auxiliary spheres following the undeformed shape of the terrain profile. The contribution of each track portion in terms of traction force was obtained employing Janosi-Hanamoto equation in a specific routine inserted in another Single-component-force (S-force) defined for each track link. The internal shear stress - shear displacement law is reported below. \[\tau=\tau_{max}\cdot\left(1-e^{-j/K}\right)=\left(c+p(x)\tan\phi\right)\cdot \left(1-e^{-j/K}\right) \tag{8}\]  
As explained in Section 2, Equation (8) was implemented both for loose soil and compact agricultural terrain.