# Modeling Movement Stability of Machine-Tractor Units Based on Modular Type Tractor
## 4 Results and Discussion  
During the experiment, the unit under study moved in the forward and reverse directions at a mean velocity of 2.05 m\(\cdot\)s\({}^{-1}\). Analysis of the data obtained showed the following. MTU heading angle oscillations' theoretical and experimental normalized spectral densities have the same cut-off frequency of 3.2 s\({}^{-1}\) (Figure 5). In both cases, the maximum variance values of the \(\varphi\) parameter oscillations also occur at the same frequency, being equal to 0.25 s\({}^{-1}\). A F-test comparison of the values of the obtained variances showed the following. The actual value of this statistical indicator is = 2.9 \(\times\) 10\({}^{-3}\)/2.5 \(\times\) 10\({}^{-3}\) = 1.16. Its tabulated value (\(F_{t}\)), accepted for a statistical significance level of 0.05, equals 1.39 [28]. Since \(F_{r}<F_{t}\), with a confidence probability of 95%, it can be stated that the null hypothesis about the equality of the compared variances is not rejected. The resulting arithmetic difference between the values of these statistical parameters is random and insignificant. Figure 5: Experimental (—) and theoretical (- -) normalized spectral densities of MTU heading angle (D\({}_{1}\) and D\({}_{2}\) are experimental and theoretical heading angle variances, respectively). As a result, we found that both in terms of frequency composition and variance, the theoretical and experimental normalized spectral densities represent the same general population of data. This indicates the adequacy of the developed mathematical model (1). It can then be used to adequately analyse the modular tractor-based MTU's driving stability. During the modeling process, the moment (N\(\cdot\)m) created by the difference in traction resistance (\(\Delta P_{l}\), N) of the side cultivators at a distance \(S_{o}\) (Figure 3b) was taken as a disturbing influence on the MTU. The output parameter of the dynamic system under consideration is the heading angle of the machine-tractor unit (\(\varphi\)). Analysis of the amplitude-frequency characteristics obtained during modeling showed the following. When the vertical hinge of the TM is in a free state (see Figure 3b), AFC has a clearly manifest maximum, which occurs at a frequency of 2.5 s\({}^{-1}\) (Figure 6). Blocking the vertical hinge leads to an increase in the longitudinal base of the modular tractor. When this joint is free, the turning moment \(\Delta P_{l}\cdot S_{o}\) acting from the side cultivators (see Figure 3b) is mainly opposed by EM. Its wheelbase (\(L\)) is 2.40 m (Table 2). When the TM hinge is locked, the EM and TM resist the action of the specified turning moment. Their longitudinal base (\(L+l+b=2.40+1.30+1.30=5.00\) m) increases by 2.08 times. As a result, the AFC of a modular tractor has significant differences. Firstly, it does not have any resonant peaks at all. Secondly, its value, starting from the disturbance frequency of 0.65 s\({}^{-1}\), is significantly less. The obtained result is desirable based on the following considerations. When working out a disturbing influence (i.e., interference), the best is the AFC, which has smaller values. The ideal AFC in the entire frequency range of the disturbance is 0 [23]. Further, in the zone of the resonance peak, the AFC of the unit with a free hinge of the technological module is equal to 1.1 \(\times\) 10\({}^{-5}\) rad/N\(\cdot\)m (Figure 6). If the hinge is blocked, the AFC value decreases to 0.39 \(\times\) 10\({}^{-5}\) rad/N\(\cdot\)m, that is, 2.82 times. Let us express this result through changes in the heading angle \(\varphi\).