2.1.4. Spectral Analysis

The frequency spectrum of the signals collected from the twelve working bodies provides interesting information, especially from identifying possible deterministic components that appear in all the examined sequences, which indicate possible deficiencies of the working regime, abnormal clearance, or defects. Deterministic components can also appear in agricultural soil processing machines that produce programmable vibrations, which is not the case for the structure examined in this paper.

Another application of the frequency spectrum of the random sequence is the estimation of the risk of resonant vibrations produced by random components on the elements of the cultivator structure. Components of the excitation that produce resonances are unlikely because the coincidence of a constant excitation with one of the first natural frequencies of the structure (here we refer only to the support of the working organ; in the general case, the calculation is very voluminous) is difficult to manifest. Even if it happens, the damping from the metal material and the ground on which it moves severely diminishes the resonant effect. Therefore, resonance effects are probably due to the improper functioning of some internal organs or a work speed that, combined with the soil profile, can produce them. However, a comparison between the spectrum of the supports of the working body and the spectrum of the excitations can be made to investigate the resonance problem at the supports of the working body or some more complex sub-assemblies. Figure 13 shows an example of a study of the distribution of the natural frequencies of the studied structure (here, the supports of the working body) in relation to the excitation spectrum (signals from the supports), obtained using the Fourier transform. In Figure 13, with solid grey bars, the spectra of the twelve supports of the working bodies are drawn in the 14–16 Hz range, in which the fundamental frequency of the support falls. The red bar with a circle at the upper end marks the position of the fundamental natural frequency of the supports in the chosen frequency range. The fundamental Eigen frequency and the other Eigen frequencies are calculated and shown in Figure 14 (the first five natural frequencies), but they were not experimentally determined as they should have been when calibrating the structure.

**Figure 13.** Representation of the frequency spectrum of the twelve signals from the supports of the working bodies, compared to the fundamental frequency of the support, calculated by the finite element method (represented by the thin red bar with a circle at the upper end).

**Figure 14.** The first five natural frequencies of the supports of the working bodies of the MCLS in three variants of fixation on the load-bearing structure and the deformed forms of the structure in the vibration mode correspond to the fundamental frequency.

Since we had no data about the operation in the non-linear elastic or plastic domains or about the damping capacity of the steel from which the supports are built, we worked in the linear elastic domain, which makes the displacement values exaggerated when calculating the natural frequencies. For this reason, we have not given numerical data relative to the deformed shapes corresponding to the five fundamental natural frequencies.

The value of the fundamental natural frequency represented by the red bar with a circle at the upper end in each of the twelve graphs in Figure 13 corresponds to the support or semi-hard fixation of the support (15.151 Hz). Overlaps of the fundamental frequency over local maxima of the excitation spectrum occur for channels ch4, ch23, ch24, ch3, ch20, ch2, and ch1. However, there are no reasons for concern because the peaks in the spectra of the supports have small values and the damping capacity of the material of the supports, and especially of the soil in which the working body moves, is high.

In addition, when calculating the spectra of all the twelve signals coming from the twelve measurement locations, it is found that they are all highly correlated, with the minimum value of the correlation of two such numerical sequences being 0.979. The most intensely correlated signals correspond to channel pairs: ch2 and ch17, ch2 and ch1, and ch1 and ch21. At the lowest intensities, we find correlations between channels ch4 and ch1, and between ch4 and ch17.

A better understanding of the distribution of the frequency spectrum of all twelve signals coming from the measurement location of the supports of the working body of the cultivator in relation to the natural (calculated) frequencies of the supports is possible using the graphic from Figures 15 and 16.

**Figure 15.** The frequency spectrum for the twelve signals (in the 0–50 Hz window).

**Figure 16.** The frequency spectra of all twelve signals in narrow windows around the first three natural frequencies of the support of the working organ (see Figure 14).

In Figure 15, the spectral distribution is plotted for all signals in the range (window) 0–50 Hz, which includes the first three fundamental frequencies. Details in narrow intervals around the first three fundamental frequencies (see Figure 15) are plotted in Figure 16. It is found that, around the first three natural frequencies of the supports of the active organs, there are few frequencies from the signal spectrum, and, in addition, their magnitude is very small (when comparing the magnitudes of the frequencies in Figure 17 with those of the spectrum components in Figure 15).

**Figure 17.** The von Mises stress map on the section with the maximum load on the support of a working body of the MCLS. The load values were 518 N (**left**) and 1086 N (**right**).

#### 2.1.5. RMS and the Average Value

In many studies of random vibrations, RMS (root mean square) is calculated as a measure of amplitude extremes. The frequent use of RMS in the study of vibrations is also due to its direct connection with the energetic content of vibrations and implicitly with their destructive capacity [51]. RMS is related to the vibration amplitude and its average value.

Table 1 gives the RMS values and the average value for each sequence recorded in the examined experiment. It can be seen that the two measures of the force amplitude are close. Moreover, the correlation between the two matrices (that of the RMS and the average values) is very high: 0.999.

**Table 1.** RMS and average values for the numerical sequences corresponding to the twelve measurement points (see Figure 6), organised in the table according to the order of the physical structure.


### **3. Results**

The main results that will be presented are applications for the design, execution, and operation of agricultural machines such as the MCLS complex cultivator, which is presented in Section 2. The results refer to the calculation of the probability of the occurrence of dangerous peaks, the selection and counting of the peaks of force that produce fatigue accumulation (in the supports of the active organs), the identification of design defects or deficiencies or of the work regime, and effects on the quality of the work.

### *3.1. The Probability of the Occurrence of Dangerous Loads*

To identify and count the dangerous peaks of force in support of the working organ, the characteristic resistance limits of the material from which it is constructed will be used: the bending fatigue limit stress, *σ*−1; the yield stress (plasticisation), *σY*; and the breaking or yielding limit stress, *σr*. To be able to solve the proposed problem, the finite element method was used to determine the state of equivalent tension in support of a working body as a result of the application of forces with experimentally determined values. The material used has the following limit characteristics: *σ<sup>Y</sup>* = 620.422 MPa, *σ<sup>r</sup>* = 723.826 MPa, modulus of elasticity *E* = 210,000 MPa, Poisson's ratio *ν* = 0.28, mass density *ρ* = 7700 kg/m3, and the fatigue resistance limit by bending, *σ*−<sup>1</sup> = 380 MPa. The structural model used is described in [16,17]. The occurrence of damage phenomena (irreversible deformations, cracks, or breaks in the material) is determined with the help of the structural model by comparing the values of the equivalent stress (von Mises) in the structure with the critical limit stresses

of the material. In Figure 18, the colour maps of the state of equivalent tension resulting from the application of the average (518 N) and maximum (1086 N) forces to the support of the working body are shown.

**Figure 18.** The variation in the maximum von Mises stress in the material supporting the active organs depends on the loading of the organ compared to the critical limits of the material.

It can be observed that the maximum equivalent stress for the load of 518 N has a value of 428.5 MPa (with a maximum resultant relative displacement of 77 mm), and for the load of 1086 N, it has a value of 942.3 MPa. Therefore, the average load of 518 N does not pose damage problems, and the support works in the field of linear elasticity with a safety coefficient of approximately 1.45. In the case of the maximum stress (1086 N), the equivalent stress in the structure reaches maximum values higher than the breaking limit stress of the material. However, the organ will not yield yet because the core of the bar from which the support is built presents an appreciable area that works in the linear elastic domain. For the maximum value of 1086 N, the yield stress is exceeded in an appreciable area (Figure 17), and even the breaking stress is exceeded in narrow areas in strips near the longer sides of the cross-section. However, the section works in linear elastic mode in the central area, so a failure is not directly recorded. However, in this working regime, high peaks lead to the accumulation of fatigue and possibly premature failure.

Using the results provided by the structural analysis of the support of the working body and the probabilities interpolated with cubic spline functions, it is possible to calculate the probability of exceeding the limit forces that cause the critical limits of the material to be exceeded: fatigue, plasticisation, and breaking. First, based on the hypothesis of the linear elastic behaviour of the support material of the working body, the graphic representation is shown in Figure 18.

Note: To calculate the equation of the oblique line in Figure 18, we accepted the hypothesis of linear elastic behaviour of the support material upon bending up to the plasticisation limit. Thus, the fatigue limit is included in the elastic bearing. The equation of the oblique line in Figure 18 is:

$$
\sigma = \mathbf{Kx} \tag{7}
$$

where the elastic constant *K* is calculated using the result of the structural analysis for the average value (518 N) of the force applied on the twelve sequences of analysed force values: *K* = 0.827 N/Mpa.

The probability that the loading force corresponding to each of the twelve supports of the active organs will exceed the critical values of flow, rupture, and fatigue (characteristics of the material of the supports) is shown in Figure 19.

**Figure 19.** The probability that the loading force will exceed the limits of flow, rupture, and fatigue of the material of the support of the working body.

#### *3.2. Counting the Peaks That Produce Fatigue*

One of the immediate applications of this research is the possibility of counting the "cycles" that produce fatigue in the support material of the working body. Using the data and the graph in Figure 18, one can find that the "part" of the random sequence of force values that requires the support of the working body can produce the accumulation of fatigue, cracks, and breaks in the material.

A stress sequence filter is the simplest selection method for the part of the stress that produces fatigue accumulation, plastic yielding, or breaks. The expression of such a filter is given by Formula (8):

$$\mathfrak{g}\_{i} = \begin{cases} \mathfrak{s}\_{i\prime} & \mathfrak{s}\_{i} > \sigma\_{-1} \\ \sigma\_{-1\prime} & \mathfrak{s}\_{i} \le \sigma\_{-1} \end{cases}, i = \mathbf{1}, \ldots, N \tag{8}$$

where *s* is the sequence of experimental data in force values converted into values of the equivalent stress (von Mises) in the material, according to Formula (7).

In Figure 20, the load peaks of the support of the active body from the extreme left of the first line of working bodies (ch4) after the back of the tractor are highlighted. Prominent spikes produce fatigue buildup in the support material.

**Figure 20.** Filtering the sequence of forces that produce the accumulation of fatigue in the signal corresponding to channel ch4.

The wear of the support in 40 s will be calculated by structural engineers using the above data. During this time, the vehicle covers approximately 30 square metres of land. The result allows the estimation of the life span in time (neglecting the wear caused by the action of water or other environmental factors) in hectares of ploughed land or seasons or years of exploitation. In order to determine the lifetime of the supports of the working bodies, any structural analysis programme that has this facility can be used (for example, [52,53]).
