*3.4. Effects on the Quality of the Working Depth*

The vibration of the supports of the cultivator's working organs involves relative movements horizontally, vertically, and laterally. The vibrations of the cultivator are complex; they are not reduced to the elastic vibrations of the working supports; there are also random rigid vibrations of the supporting structure and the working organs. All these types of vibrations have consequences for the main parameters of the work quality: working depth, initial working width, energy consumption, productivity, and comfort of the tractor driver.

In order to give only an ideal picture of the variations in the working depth due to random vibrations, we performed an elementary and ideal calculation on the deformations of the supports of the working bodies in the work process by applying some forces included in the experimental range. Using the finite element method and the model to calculate the first five natural frequencies, we estimated the relative displacements (deformations) in the three directions for the extremity of the support to which the working body is attached. The results are given in Table 2.

**Table 2.** The values of the relative displacements of the tip of the support of the working body are calculated using a linear-elastic structural model built by the finite element method.


The results in Table 2 are slightly exaggerated because the support material is assumed to be perfectly linearly elastic, which is not true. However, they suggest a good value as an order of magnitude for the relative displacements. The dominant overshoots are manifested in the forward and vertical directions on the ground. These values are approximately linearly related. At the average value of the force from the recordings related to the analysed experiment (500 N), the relative vertical displacement reaches 7 cm, which seriously affects the programmed working depth (10 cm). These situations are not frequent, but they are facilitated by exceeding a working speed limit, deformations of the working surface (unevenness), and errors in the working depth regulation system.

In such conditions, the cultivator cannot be a tool to process the soil with high precision (a maximum error of the order of 10% of the theoretical depth) at the working depth. A previously levelled and sufficiently crushed work surface, combined with an adapted work speed, are mandatory conditions for precise work in terms of working depth. A general picture of the quality of the work performed in terms of working depth is shown in Figure 21.

**Figure 21.** Errors in the working depth are calculated according to the measurements made on each channel.

The calculation from this subsection is based on the graphic representation in Figure 21, which uses the hypothesis of the operation of the supports of the working bodies of the MCLS in a linear-elastic regime without damping, which leads to an overestimation of the relative displacements. Direct measurements regarding the working depth and its

monitoring are insufficiently developed in the available experimental techniques; therefore, the estimation method described above was used.

An option to Improve the accuracy of the working depth is to ballast the load-bearing structure by adding some ballast materials. Increasing the precision of the working depth in this way will lead to higher energy consumption and, possibly, a decrease in productivity.

Additionally, a problem that must be solved for this machine is finding the maximum speed at which it can work without manifesting intense vertical oscillations throughout the entire structure, which completely compromises the quality of soil processing. The speed limit, which is determined through tests, is dependent on the characteristics of the soil, so an exact speed limit cannot be indicated a priori.

#### **4. Comments**

According to [51], most random vibration analyses are intended to realistically characterise the behaviour of structural systems excited by random inputs. The responses of real systems are known only when measured during their actual physical loading (and even then, only approximately). The effects caused by the use of simplifying assumptions in the process of numerical simulation of physical systems are rarely evaluated. Because of the depth of our knowledge in the analysis of linear systems subject to stationary media, we often idealise real systems as linear and the inputs as stationary. Practically, all real systems are nonlinear and random to a small or large extent. Therefore, the characterisation of the response can only be approximate. Such reasoning also directed our attention to the experimental or theoretical-empirical research of the working processes of agricultural machines.

Starting from these realities, we tried to formulate the problem within the strict framework of random vibrations. We tried to use as few simplifying assumptions or assumptions from the theoretical field of linear vibrations as possible. For this, we constructed the characteristic functions of the empirical sequences exactly as they result from the definition (Section 2). Probability densities and probabilities are obtained directly by using the numerical sequences and interpolating them with spline functions. We did not use probability density modelling with idealised functions (normal distributions, Student, Fischer, etc.), and consequently, the resulting probabilities are also modelled on real data.

Representations of the spectral frequency distributions of the twelve supports of the working bodies (Section 2.1.4, Figures 15 and 16) can be found in [24] for the chassis of a harvester. The orders of magnitude and the numerical values are also comparable, being characteristic of the working processes of agricultural machines. As values, the forces resulting from the experimental study described in this article are comparable to those found for the working bodies and also for a cultivator by the authors [68]. The obtained frequency spectra are similar to those from [69]. Unfortunately, we could not get accelerometer signals from the experimenter, which was very important. Acceleration recording remains a problem to be solved for MCLS. Some of the forces' values and the frequency spectra's frequencies were also obtained in [41]. In the same register of concerns and spectral values, [26] estimates the effects of frequencies and vertical accelerations on several types of tractors using frequency-acceleration diagrams. Such diagrams were not accessible for this paper, so frequency-load force diagrams on the working body were used instead. However, the principle of selecting deficient or dangerous work regimes is the same. The authors [29] use the same principle of spectral selection for the vibration level on the front axle of the Valtra 800 L tractor. As an estimator of the vibration level, the authors [29] use RMS (root mean square) amplitude [70] for the experimentally recorded acceleration sequences in the version indicated by the authors [29]. The RMS maximums are recorded in the frequency range where the maximum oscillation amplitudes of our force sequences were located. The optimisation of the frame of a precision agricultural machine for sowing vegetables by avoiding resonances, based on the same principles as in this article, is described in [61]. The authors proceed similarly [36] for cultivators with working bodies very similar to those with which the MCLS cultivator was equipped in our

experiments. Similarly, the authors [62] proceed with the study of the vibration of rapeseed seeds, additionally using high-speed photography and image recognition. All the frequency spectra (in this case normalised) are used in [67] to carry out experimental investigations with vertical damping for block-modular aggregates. As in this article, the authors [71] use spectral analysis and finite element analysis to predict fatigue accumulation in the arm of an excavator. The use of statistical analysis to estimate fatigue life dates back a long time [72], even for an agricultural cultivator; this is one of the main applications of the study of random vibrations. A similar approach for the testing of agricultural machines in an accelerated regime is proposed by the authors [73], in the same manner that we also proposed in Section 2.1.2. It is emphasised that not only the value of the load matters but also its frequency. The authors proceed similarly in [74,75]. A solution to reduce the force of resistance to advance for a cultivator by inducing forced vibrations is presented in [34]. Additionally, the return to some classical solutions, which had been partially abandoned, appeared after studies of random vibrations [76].
