# Multi-Body Model of Agricultural Tractor for Vibration Transmission Investigation
## 2 Materials and Methods  
### Building the Model  
When building the model, it is essential to consider the problem the model should solve. The ideal model is one with minimum complexity but capable of solving the problem [38]. More complex models demand more experimental data in the model validation stages, which takes time and money and the risk of errors. In [21], a system schematic is a recommended first step, helpful for understanding the mechanism and estimating degrees of freedom in the model.  
Adams multi-body modelling software was chosen to create the model. The scheme of the proposed multi-body model is shown in Figure 3. The tractor body, cabin, the suspended part of the driver seat, and the driver are modelled as rigid bodies. The mass and inertia were prescribed based on CAD data and experimental data provided by the manufacturer (see Table 1). The stiffness and damping properties of suspension elements in the model are in Table 2. The connection between the body and the cabin in the front part is provided by a flexible connection--bushing. The bashing is defined by stiffness and damping for translational relative movement of connected parts.  
Figure 3: The scheme of the proposed multi-body model.  
The rear part of the cabin is suspended passively by a spring-damper element. In Adams, the spring-damper connection is defined in one axis. Translational stiffness and damping are defined as non-linear functions (see Figure 4). The data were acquired by measuring a physical spring-damper on a specialised test rig. Small bushings used for spring-damper mounts are modelled using small dummy parts.  
A Panhard rod is also modelled as a rigid body, and it is connected to the tractor body and the tractor cabin by bushings. A driver seat suspension is modelled only in the vertical axis via a spring-damper element combined with the translational joint. The seat cushion effects are also modelled by this combination (see Figure 5). The mass and inertia properties of the driver were taken from [39].  
Another advantage of multi-body simulation models is the possibility of implementing and comparing various design alternatives. The independent front axle suspension model is displayed in Figure 3. It consists of an axle body modelled as a rigid body connected  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & & & \multicolumn{4}{c}{**Front axle**} & \multicolumn{4}{c}{**Dummy**} & \multicolumn{4}{c}{**Realial Stiffness**} & \multicolumn{4}{c}{**Stiffness**} & \multicolumn{4}{c}{**Radial**} & \multicolumn{4}{c}{**Radial**} \\  & & & \(\mathbf{[N\cdot m^{-1}]}\) & \(\mathbf{[N\cdot m^{-1}]}\) & & & \(\mathbf{[N\cdot m^{-1}]}\) & & \(\mathbf{[N\cdot m^{-1}]}\) & & \(\mathbf{[N\cdot m^{-1}]}\) & & \(\mathbf{[N\cdot m^{-1}]}\) \\ \hline \multirow{4}{*}{
\begin{tabular}{c} **Stiffness** \\ **S  
by a revolute joint to the tractor body. Wishbones are connected to the tractor body and wheel hub by revolute joints. Spring-dampers connect lower wishbones with the axle body. With this implementation, it is easy to achieve the model of the unsuspended front axle by setting the stiffness of these spring dampers to a very large value.  
It should be noted that the presented configuration of the joints mimics the front axle design and enables easy extension of the model by the replacement of rigid bodies with flexible bodies. However, from the perspective of the MBS system, it contains six redundant constrained equations (caused by multiple revolute joints at the front axle). Utilising 2 rotational joints, 1 cylindrical joint, and 1 spherical joint is correct according to MBS theory and would not produce redundant constraints. The configuration of 4 revolute joints results in equivalent dynamic behaviour of the tractor, and since the forces in revolute joints are not investigated, it was possible to use these joints.