# Multi-Body Model of Agricultural Tractor for Vibration Transmission Investigation
## 2 Materials and Methods
### Type Model  
PAC2002 and FTire are [PERSON]'s two most mature tyre models [38]. PAC2002 was designed to describe the behaviour of tyres travelling over relatively smooth road surfaces. It is also called the Magic Formula tyre model because it uses a mathematical formula capable of expressing essential tyre characteristics surprisingly well despite no physical basis for this structure of equations. This Magic formula has two primary forms, sine and cosine [40]:  
\[\mathbf{y}(\mathbf{x}) =\mathbf{D}\,\mathbf{sin}[\mathbf{C}\,\mathbf{arctan}\{\mathbf{B}\mathbf{x}-\mathbf{E}(\mathbf{B} \mathbf{x}-\mathbf{arctan}\,\mathbf{B}\mathbf{x})\}], \tag{2}\] \[\mathbf{y}(\mathbf{x}) =\mathbf{D}\,\mathbf{cos}[\mathbf{C}\,\mathbf{arctan}\{\mathbf{B}\mathbf{x}-\mathbf{E}(\mathbf{B} \mathbf{x}-\mathbf{arctan}\,\mathbf{B}\mathbf{x})\}], \tag{3}\]  
where \(\mathbf{y}\) is the dependent variable (force or moment), \(\mathbf{x}\) is the independent variable (slip parameter), and \(\mathbf{B}\), \(\mathbf{C}\), \(\mathbf{D}\), and \(\mathbf{E}\) are fitting constants, which represent stiffness, shape, peak, and curvature factor. The sine form describes lateral force as a function of the lateral slip or longitudinal force as a function of the longitudinal slip. The cosine form describes a self-aligning moment. PAC2002 is a semi-empirical model based on non-linear mathematical approximations of tyre forces and moments, but it also contains physically modelled components. A physical approach models transient tyre behaviour. This model uses one point-follower contact method by default (for 3D spline roads), which was insufficient for modelling tyres over short wavelength irregularities [41]. However, when the user sets  
Figure 5: Stiffness characteristic of the driver seat. the 3D enveloping contact method in the tyre property file, the performance of the tyre model on short road obstacle wavelength is highly improved. It is recommended to use PAC2002 for handling simulations, such as steady-state cornering, single or double lane change, braking, and turning manoeuvres [38]. On the other hand, a physics-based model, FTire, developed by [PERSON], is suitable for vehicle comfort investigations and other vehicle dynamics on even or uneven roads. It is the best model for testing durability [42]. A tyre belt is described as an extensible and flexible ring with bending stiffnesses, elastically founded on the rim by distributed stiffnesses in radial, tangential, and lateral directions as it is shown in Figure 6[42]. A finite number of point masses numerically approaches the ring; these belt elements are coupled with their direct neighbours by stiff springs and bending stiffnesses both in-plane and out-of-plane. The radial stiffness between a single belt element and the rim is refined by the parallel connection of a spring with a spring-damper series connection to allow dynamic stiffening of the tyre radial stiffness at high speeds. To every belt element, a set of mass-less tread blocks is associated, and the blocks carry non-linear stiffness and damping properties in radial, tangential, and lateral directions. To take full advantage of the precision of the model, there should be a minimum of 1000 steps per second of simulation time. The main advantages of using FTire are that the model is fully non-linear, and its accuracy is very high when passing single obstacles, such as cleats or potholes. FTire is sufficient for modelling tyres over short wavelength irregularities. The disadvantage is that many parameters are required to parametrise the tyre model. However, once appropriately parametrised, the FTire model can predict tyre's behaviour over smooth road surfaces and rough terrain [43]. Extensive research has been performed to model the tractor tyres precisely. In [44], a regression model of tractor tyres' vertical stiffness and damping was developed, showing non-linearity in these characteristics. The need for a non-linear description of tyres' vertical characteristics for very heavy vehicles is mentioned in [21].