## 2 Mathematical Model of the Wheel Loader
### Revolute Joint Loads  
By performing a virtual cut of the kinematic chain at the joint \(O_{i}\) (\(i=3,4\)) and applying the equilibrium conditions to the separated segment, the resultant loads at the center of the joint \(O_{i}\) are determined using the vector of the total reduced force and moment (Figure 2):  
\[\mathbf{F}_{i}=\mathbf{F}_{ri}+\mathbf{F}_{pi}\quad\forall\;i=3,4 \tag{3}\]  
\[\mathbf{M}_{i}=\mathbf{M}_{ri}+\mathbf{M}_{pi}\quad\forall\;i=3,4 \tag{4}\]  
where \(\mathbf{F}_{ri}\) and \(\mathbf{M}_{pi}\) are the vectors of the total force and total moment of all forces acting on the separated part of the kinematic chain of the manipulator, while \(\mathbf{F}_{pi}\) and \(\mathbf{M}_{pi}\) are the vectors of the force and moment of the drive mechanism of the kinematic pair at the joint \(O_{i}\). The total force and total moment vectors of all forces acting on the separated part of the manipulator's kinematic chain are determined by the following equations:  
\[\mathbf{F}_{ri}=\mathbf{F}_{gi}+\mathbf{F}_{i}+\mathbf{W}\quad\forall\;i=3,4 \tag{5}\]  
\[\mathbf{M}_{ri}=\mathbf{M}_{gi}+\mathbf{M}_{i}+\mathbf{M}_{W}\quad\forall\;i=3,4 \tag{6}\]  
here, \(\mathbf{F}_{gi}\) is the resultant of gravitational forces acting on the members of the separated part of the chain, \(\mathbf{F}_{i}\) is the resultant of inertial forces caused by the motion of the members of the separated part of the chain, \(\mathbf{W}\) is the resistance force of the material loaded by the bucket, \(\mathbf{M}_{gi}\) represents the resultant moment of gravitational forces, \(\mathbf{M}_{i}\) is the resultant moment of inertia due to the motion of the members of the separated part of the chain, and \(\mathbf{M}_{W}\) is the resistance moment of the material loaded by the bucket. The components of the total reduced force at the joint \(O_{i}\) in the local coordinate system \(O_{i}x_{i}y_{i}z_{i}\) of the driven member \(L_{i}\) of the kinematic pair are determined by the following equation:  
\[F_{ix}=\mathbf{F}_{i}\cdot\mathbf{i}_{i},\;\mathbf{F}_{iy}=\mathbf{F}_{i}\cdot\mathbf{j}_{i},\; \mathbf{F}_{iz}=\mathbf{F}_{i}\cdot\mathbf{k}_{i} \tag{7}\]These components load the joint elements with the components \(F_{ix}\) and \(F_{iy}\), which are normal to the joint axis \(O_{i}z_{i}\), generating friction between the joint elements. The components of the total reduced moment at the joint \(O_{i}\) in the local coordinate system \(O_{i}x_{i}y_{i}z_{i}\) of the member \(L_{i}\) of the kinematic pair are determined by the following equations:  
\[M_{ix}=\mathbf{M}_{i}\cdot\mathbf{i}_{i},\ \mathbf{M}_{iy}=\mathbf{M}_{i}\cdot\mathbf{j}_{i},\ \mathbf{M}_{iz}=\mathbf{M}_{i}\cdot\mathbf{k}_{i}=0 \tag{8}\]  
These components load the joint elements, except for the component \(\mathbf{M}_{iz}\), since the load moment component \(\mathbf{M}_{rz}\) is counteracted by the required moment of the drive mechanism, such that \(\mathbf{M}_{pi}=-\mathbf{M}_{rz}\). In the further analysis of the influence of mechanism parameters on the tribological characteristics of the joints, the resultant force \(F_{in}\), composed of the components \(F_{ix}\) and \(F_{iy}\) acting normal to the joint axis, is considered as resulting from all applied loads:  
\[F_{in}=\left(F_{ix}^{2}+F_{iy}^{2}\right)^{\frac{1}{2}} \tag{9}\]  
The basic structural parameters of the revolute joint elements are the pin diameter \(d_{is}\) (Figure 2) and the sleeve length \(l_{is}\). Due to the action of the resultant normal force \(F_{in}\), the primary stresses in the elements of the kinematic pairs of the manipulator mechanism are shear, surface pressure, and bending. Shear stresses arise primarily from the transmission of tangential forces through the pin, while surface pressure is generated at the contact interfaces between the pin and the sleeve, influencing wear and fatigue life. Bending stresses occur due to eccentric loading or misalignment of the joint elements, which can induce additional moments on the pin.