## 3 New Method Development
For example, at tg\(\alpha=20^{\circ}\), the radial force is \(F_{r1}=3880.151\) N, which ensures effective gearing of the kinematic gearwheel pair of the pump transmission shaft. With the same mechanical wear, the deviation of the gearing angle is tg\(\alpha=19^{\circ}40^{\prime}\) the value of the radial force \(F_{r1}=3838.04\) N is reduced by 47 N. Wear of the contact surface leads to a change in the structural gaps \(\Delta i\) between the teeth, and under the influence of dynamic moments of forces and angular velocities \(\omega\), a dynamic impact occurs in the contact gearing surface. The maximum values of the impact load can be observed at the beginning of the movement and the stop of the gear. This is a short time interval when the element gains acceleration and deceleration corresponding to the times t\({}_{1}\) and t\({}_{2}\). Local fatigue stress zones are formed by the action of dynamic loads causing microcracks, cracks, and fractures of the teeth. Even the process of partial tooth destruction leads to a distortion of the gearing angle by at least 1\({}^{\circ}\). For example, subject to a partially painted tooth, the gearing angle changes to tg\(\alpha=19^{\circ}\), and the radial force changes to \(F_{r1}=3670.67\) N, which is 210 N less than the effective force \(F_{r1}=3880.151\) N. Analyzing the calculations, it can be observed that reducing the forces in the gearing reduces the torque, which ensures effective gearing and operation of the pump. However, operating at a lower torque of the tooth gearing in practice actually shows a progressive degradation of the contact surface, resulting in chipping and breaking off the teeth. This phenomenon can be explained by the fact that the amplitude of the destructive stresses of the material acting on the contact area of the worn surface increases under the action of small moments in the gearing, but in short time intervals t\({}_{1}\), t\({}_{2}=0.3\) s.  
When calculating the gear shaft, it is necessary to consider the actual distribution of forces and moments over the entire length of the shaft during operation, taking into account all of its structural elements. In particular, it is necessary to take into account the role of bearings in the effective load distribution in different sections and planes of the studied shaft-gear. To do this, when calculating the forces in the engagement, it is necessary to take into account the degree of influence of the reaction of the forces and moments transmitted from the bearings. This will describe the process of redistributing forces and torques throughout the entire \"Bearing-Shaft-Gear-Gear\" system under study. From the calculation, it can be seen that if the reaction forces in the bearings are affected, then there is a significant increase in the transverse forces in the vertical and horizontal planes of the gear engagement. If we consider the actual operating conditions of the pump with structural elements of bearings, the values of the same forces are reduced several times from 8100 N to 100 N, which is more consistent with the real load and confirms the effectiveness of the bearings. This means that their action and role in the redistribution of loads is of great importance for the methodology of calculating the loads acting in the gearing and calculating the durability of the gear. If the reaction forces of the bearings are neglected, overestimated numerical values of the load are formed, which will lead to a significant error in the design calculations. Thus, a scientific problem arises, namely why (with the existing system of distribution of forces and moments in the vertical and horizontal planes) wear is observed in the gear engagement at small values of forces and moments. A theoretical assumption is made that the change in the durability of the gear and the gear wheel is largely influenced not by the magnitude of the acting force and moment, but by its direction vector, which is proposed to be characterized as the coefficient of displacement of the contact spot relative to the designed axis of symmetry.