# Modeling Movement Stability of Machine-Tractor Units Based on Modular Type Tractor
## 4 Results and Discussion
While checking the mathematical model (1) for adequacy, we found that the maximum value of the difference in resistance of the MTU's side cultivators (\(\Delta P_{l}\)) was approximately 1000 N. Since this MTU = 4.2 m, the value of the turning moment \(\Delta P_{l}\cdot S_{o}\) was 4200 N\(\cdot\)m. The influence of the latter with a frequency of 2.5 s\({}^{-1}\) and its amplification level of 1.1 \(\times\) 10\({}^{-5}\) rad/N\(\cdot\)m causes a change in the heading angle of the MTU with a free TM hinge by an angle of approximately 3\({}^{\circ}\). When the hinge is blocked, the impact of the same turning moment with the same frequency but with a gain level of 0.39 \(\times\) 10\({}^{-5}\) rad/N\(\cdot\)m, leads to a change in the parameter \(\varphi\) by only 1\({}^{\circ}\), i.e., 3 times less. Figure 6: AFC of the dynamic system with unlocked (—) and locked (- -) states of the TM vertical hinge. In addition to AFC, phase frequency response (PFC) is an important characteristic of a dynamic system. They show the delay in the system's response to input influences. When working out the disturbing influence, the ideal PFC should tend to infinity [23; 29]. In other words, the later a dynamic system reacts to a disturbance, the better. In the case we are considering, with the vertical hinge TM in a free state, the phase shift in the unit's processing of the turning moment \(\Delta P_{t}\)-\(S_{o}\) has the shape of a logistic curve (Figure 7). In this case, the phase shift in the MTU response to disturbance does not exceed \(-202^{\circ}\). Blocking the vertical hinge of the TM contributes to an increase in the phase shift in the response of the dynamic system by \(-240^{\circ}\) or more. At the same time, the growth rate of this process is significantly lower than that of the version with a free state of the TM hinge. As a result, we observed the following pattern. When the oscillation frequency of the external disturbance increases by more than 1.5 s\({}^{-1}\), the difference between the obtained PFCs (\(\Delta\)PFC) increases. However, such an assessment of the resulting PFCs is not entirely convenient. It is more practical and visually convenient to represent the difference \(\Delta\)PFC in terms of time (s). It is defined as follows:  
\[\Delta PFC=PFC\cdot\pi/(180\cdot\omega) \tag{10}\]  
where \(\omega\)--oscillations frequency of the disturbing influence--moment \(\Delta P_{t}\cdot S_{o}\) (s\({}^{-1}\)). Analysis of the process \(\Delta PFC=f(\omega)\) showed the following: At \(\omega\) = 0.5 s\({}^{-1}\), blocking the vertical hinge of the TM increases the delay in the MTU response to fluctuations in the disturbing moment \(\Delta P_{t}\cdot S_{o}\) by 1.9 s, which is a significant result (Figure 8). The subsequent increase in the oscillation frequency of the external disturbance to a value of 1.5 s\({}^{-1}\) is represented by a decrease in the difference \(\Delta\)PFC to 0.6 s. A slight drop in the value of the \(\Delta\)PFC parameter characterizes a further increase in \(\omega\). According to our research, at least 95% of the variance in the different oscillations in traction resistance of side machines, aggregated with the modular tractor, is concentrated in the frequency range 0-0.6 Hz or 0-3.7 s\({}^{-1}\). As we can see, this frequency range fits nicely into the one considered in this analysis (0-5.5 s\({}^{-1}\)). Considering this fact and the data from Figures 6 and 8, we can conclude that blocking the TM's vertical hinge is advisable to increase the movement stability of wide-width MTUs based on a modular tractor. Figure 7: PFC of the dynamic system with unlocked (—) and locked (- -) states of the TM vertical hinge. It should be noted that, unlike a three-axle (or multi-axle) vehicle, the result obtained for a modular tractor is not obvious and therefore requires study. For specific values of the \(S_{o}\) parameter (Figure 3), the sum of the values \(l+b\) (Figure 1), the mass of the machines used, etc., the blocking of the vertical hinge TM may be insufficient.