# Multi-Body Model of Agricultural Tractor for Vibration Transmission Investigation
## 1 Introduction
### Vehicle Suspension Models  
A quarter-vehicle model is often used initially to study a vehicle suspension design [21; 22; 23; 24]. This model consists of a few parameters: tyres' stiffness and damping, unsuspended mass, suspension elements stiffness and damping, and suspended mass. The equation of motion is derived by applying [PERSON]'s second law of motion [25]. This model with two degrees of freedom is known for a good description of the vertical dynamics of a car [26].  
Because of the body arrangement of a standard agricultural tractor, a quarter model is not applicable. The next step, a simple half model of the tractor, is plotted in Figure 2. The tractor body, cabin, front axle, and moving part of the driver seat with the driver are represented as rigid bodies. Suspension properties of tyres, front axle, cabin suspension, and seat suspension are modelled as translational stiffness \(\mathrm{k_{x}}\) and damping \(\mathrm{c_{x}}\). The equations of motion for this model based on the second Newton Law can be written as:  
\[\mathbf{M}\ddot{\mathbf{x}}+\mathbf{C}\dot{\mathbf{x}}+\mathbf{K}\mathbf{x}=\mathbf{F_{y}}(\mathbf{t}), \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are matrices of the mass, damping, and stiffness, \(\mathbf{F_{y}}(\mathbf{t})\) is the force excitation vector, and \(\mathrm{x}\) is the displacement vector, which contains variables for degrees of freedom (DOF) of the model. The displayed model has six DOF (xFA, xB, xC, xDS, \(\theta\)B, \(\theta\)C), but it can be expanded or simplified depending on the model objectives. When different linear stiffness and damping properties are used, and an assumption about small angular displacements is made (\(\theta=\sin(\theta)\)), the model is suitable for numerical optimisation of suspension parameters based on particular criteria for displacement, velocities, or accelerations.  
Figure 1: Front unsuspended tractor axle (**right**) and front axle with independent suspension (**left**).  
A half model of the agricultural tractor was presented in [6]. This model was reduced to a two degree of freedom model of a passive cabin suspension. Parameters of the passive cabin suspension were then optimised with an analytical approach. In [8], a half model of the tractor was reduced to four DOF by neglecting the front axle and driver with the driver seat. The model was used to design a semi-active suspension control algorithm to improve ride comfort in the tractor cabin. Several half-tractor models were utilised in [9] to determine the effects of the suspension system configuration.  
The possibility of using optimisation techniques is the main advantage of these simple linearised models. They can predict trends, and their robustness enables them to be used in active suspension control algorithms [20]. They are insufficient for a complex study of whole system behaviour under various loading conditions. In [27], a scissor seat suspension was optimized using an improved NSGA-II algorithm to find the best spring stiffness and damper damping combination for reducing vibrations.