# Grousers Effect in Tracked Vehicle Multibody Dynamics with Deformable Terrain Contact Model
## 1 Introduction  
Numerical simulations are very useful tools to evaluate complex mechanical systems behaviour during their early design stage as well as during product development. Multibody models are widely used to predict vehicle kinematics and dynamics [1], giving the ability to identify/predict safety critical working conditions to be avoided [2; 3]. Focusing on wheeled or tracked vehicles, machine-soil contact modelling is relevant to assess its performance. Several mathematical models have been proposed in the literature to characterize wheel or track-terrain interaction [4; 5]. A comprehensive contact model is a particularly complex task to be accomplished. The identification of the terrain mechanical characteristics is strongly affected by its composition and humidity content. A terrain can be described as an elasto-plastic medium which exhibits a strong non-linearity in the force-deformation characteristic. The behaviour of different types of soil has been characterized by means of several empirical parameters for terrain cohesion, internal friction angle and stiffness [6; 7; 8]. Several numerical models have been proposed in the literature to study and simulate wheel-soil interaction [9; 10; 11] as well as the experimental test rigs to characterize them [12]. Similarly, empirical methods, discrete elements and finite elements methods as well as reduced order models have been proposed to model the terrain-type and terrain track interaction response when in contact with machines wheels or tracks modelled as rigid or flexible bodies [13; 14; 15]. Two mathematical modelsare usually considered to represent the track(or wheel)-soil interaction--the Bekker's model and the Janosi-Hanamoto's law. [PERSON]'s equation relates the normal pressure exerted on the ground with the vertical sinkage of a circular or rectangular plate used for the soil characterization procedure through a non-linear elastic relationship. When modelling the sinkage behaviour of the tracked machine also the soil damping contribution should be considered.Janosi-Hanamoto equation relates shear stress to the corresponding shear displacement of the soil. The difference between the actual shear stress and the maximum one admissible by the terrain strongly affects the availability of tractive force to be developed by the tracked vehicle. Both the models depend on empirical parameters to be derived through specific terrain characterization procedures [16]. A novel soil-tyre model has been recently proposed in Reference [17] integrating soil mechanics behaviour with a soft soil-elastic tyre interaction model. The tyre and soil deformations are computed iteratively solving the equilibrium problem as implemented by [PERSON] et al. in Reference [18] to model an off-road vehicle. On the other hand, different procedures have been adopted to model a deformable terrain in multibody (MTB) codes. [PERSON] and [PERSON] [19] modelled the terrain shape with rigid bodies whereas in other MTB codes discrete elements method (DEM), finite element or mesh free methods [20] have been proposed. A springs-damper-oil sand slots connected by spherical joints terrain modelling approach was proposed by [PERSON] and [PERSON] [21; 22]. Soil modelling through triangular meshes or discrete elements was also considered in the Adams Tracked Vehicle Toolkit (ATV) [20; 23; 24]. In other numerical environments, soil behaviour under load has also been investigated thanks to Finite Element Models (FEMs) [25] or Semi-empirical Contact Models (SCMs) [26]. There are two different types of tracks classified as rigid or flexible. Rubber tracks, reinforced with a steel core, are usually classified as flexible whereas tracks made of steel links are classified as rigid. The latter are composed of rigid links [27] usually connected in multibody models with rotational joints [28] or bushing elements [29]. Bushing elements can also damp the system to limit numerical errors [30]. In fact, simple rotational joints without damping can lead to high vibration phenomena of the track links [31].