## 2 Analytical Model of the Drive and Transmission Mechanism
### Analytical Calculation of the Vehicle's Dynamic Parameters  
The basis for the design and verification of the drive and transmission mechanism of a vehicle is in its dynamics. Many various forces and moments act on the vehicle during driving. The vehicle therefore represents a complex mechanical system of objects with several degrees of freedom, and the detailed solution of the behavior of such a system is a very complex task. For the solution of the external dynamic conditions, it is possible to simplify the calculation model and to consider the vehicle as a single object. For this reason, it was not necessary to analyze in detail the internal forces and moments that act among the particular internal components of the vehicle. The external moments, forces and reactions used for the calculations are shown in Figure 1 and involve \(F_{air}\)--air resistance, \(F_{ff}\)--rolling resistance of the front wheels, \(F_{rr}\)--rolling resistance of the rear wheels, \(F_{i}\)--the vehicle's inertia, \(F_{s}\)--terrain resistance, \(G\)--force of gravity, \(F_{rt}\)--trailer drag resistance, \(F_{tf}\)--driving force of the front wheels, \(F_{tr}\)--driving force of the rear wheels, \(F_{Nf}\)--normal reaction of the front wheels, \(F_{Nr}\)--normal reaction of the rear wheels, \(\gamma\)--road/terrain gradient, \(L\)--wheelbase, \(I_{1}\)--distance of the front wheels from the center of gravity, \(I_{2}\)--distance of the rear wheels from the center of gravity, \(I_{g}\)--height of the center of gravity and \(r_{d}\)--dynamic wheel radius. During the analysis of the vehicle's behavior in straight-line driving, the dependence between the resistances and the driving effect of the engine and the entire drive and \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Gear** & **1** & **2** & **3** & **4** & **5** & **6** & **R** \\ \hline _sgn1_ & \(-\)0.3425 & \(-\)0.3425 & \(-\)0.3425 & \(-\)0.0933 & 0.1282 & N/A & \(-\)0.3425 \\ _sgn2_ & 1.0000 & 0.3090 & N/A & \(-\)0.1421 & 0.1328 & 0.3090 & 1.0000 \\ _sgn3_ & 1.0000 & 0.5509 & N/A & \(-\)0.1424 & N/A & N/A & N/A \\ \(i_{k}\) & 4.1717 & 2.3400 & 1.5210 & 1.1427 & 0.8672 & 0.6910 & \(-\)3.4013 \\ \(i_{F}\) & 3.8569 & 2.2478 & 1.5038 & 1.1289 & 0.8607 & 0.6838 & \(-\)3.2519 \\ \(\eta_{A}\) & 0.9245 & 0.9606 & 0.9887 & 0.9880 & 0.9925 & 0.9896 & 0.9561 \\ \hline \end{tabular}
\end{table} 
Table 1: Analytically determined values of the gearbox mechanical efficiency. Figure 1: Forces and moments acting on the vehicle while driving. transmission mechanism of the vehicle were monitored. This dependence determines whether and how the vehicle moves, and thanks to this analysis, it is possible to also determine the values of the acceleration or deceleration of the vehicle and related forces and moments. In general, this dependence can be analyzed by applying relation (5) for the balance of power:  
\[P_{e}\eta_{m}=v(F_{r}+F_{s}+F_{air}+F_{i}+F_{T}) \tag{5}\]  
or the balance of forces (6):  
\[F_{T}=F_{r}+F_{s}+F_{air}+F_{i}+F_{T} \tag{6}\]  
where \(F_{T}\) is traction force, \(F_{r}\) is rolling resistance, \(F_{s}\) is terrain resistance, \(F_{air}\) is air resistance, \(F_{i}\) is the vehicle's inertia, \(F_{T}\) is trailer resistance, \(P_{e}\) is engine power, \(\eta_{m}\) is mechanical efficiency, and \(v\) the velocity of the vehicle. To verify the correctness of the numerical model of the drive and transmission mechanism created in Matlab R2024b software and its modules, it was necessary to analyze the characteristics of the several quantities characterizing the movement and dynamic properties of the vehicle. With the increasing complexity of the model and the increasing simplifications applied during the calculations using the analytical method, it is necessary to increase the tolerance between the characteristics and values obtained using analytical and numerical methods.