**1. Introduction**

The phenomena and work processes encountered in agriculture involve, for the most part, important areas of the field of biology (environment, soil, plants, animals, etc.), which is "living". According to all the assessments in the literature, the field of life is a field of unpredictability, uncertainty, randomness, and probability. According to [1], "understanding randomness is essential for modern life, as it underpins decisions under uncertainty".

Random phenomena are treated in physics through theories different from deterministic ones, using the concepts of probability theory and statistics. In [2], it is shown that "in mechanical engineering, random vibration is motion that is non-deterministic, meaning that future behaviour cannot be precisely predicted. The randomness is a characteristic

**Citation:** Cardei, P.; Constantin, N.; Muraru, V.; Persu, C.; Sfiru, R.; Vladut, N.-V.; Ungureanu, N.; Matache, M.; Muraru-Ionel, C.; Cristea, O.-D.; et al. The Random Vibrations of the Active Body of the Cultivators. *Agriculture* **2023**, *13*, 1565. https://doi.org/10.3390/ agriculture13081565

Academic Editors: Jin He and Caiyun Lu

Received: 25 April 2023 Revised: 26 July 2023 Accepted: 31 July 2023 Published: 4 August 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

of the excitation or input, not the mode shapes or natural frequencies. Some common examples include an automobile riding on a rough road, wave height on the water, or the load induced on an airplane wing during flight. Structural response to random vibration is usually treated using statistical or probabilistic approaches. Mathematically, random vibration is characterized as an ergodic and stationary process".

In [3], it is explained that an experiment is random when the answer cannot be precisely and unambiguously predicted, as several answers are possible.

According to [2], the main way of approaching random vibrations is through "a measurement of the acceleration spectral density (ASD), which is the usual way to specify random vibration. The root mean square acceleration (Grms) is the square root of the area under the ASD curve in the frequency domain. The Grms value is typically used to express the overall energy of a particular random vibration event and is a statistical value used in mechanical engineering for structural design and analysis purposes." Additionally, in [2], it is stated that "while the term power spectral density (PSD) is commonly used to specify a random vibration event, ASD is more appropriate when acceleration is being measured and used in structural analysis and testing." Such specifications and assessments can be found in [4–9]. In the case of the study whose results are presented in this article, we worked with sequences of specific deformations measured and recorded, which were then converted into loading forces on the supports of the active working bodies.

As the scientific literature shows, work processes in agriculture have random characteristics [10–20]. Much of the specified literature has agro-economic or bio-economic origins, many of which emphasise management. Uncertainties in agricultural work processes are also caused by management system uncertainties [21]. In [22], it is stated that "problems related to agriculture are, in essence, stochastic because of the uncertain nature of their parameters. Many systems in this sector are affected by yield uncertainty caused by factors such as climatic conditions. Uncertainty and imperfect information involved therein challenge decision-making, as decision-makers are led to make decisions before observing the realization of the random factors. Traditional approaches to dealing with agricultural problems do not integrate the risks and uncertainties involved therein, while it is relevant for efficient managerial decision-making to consider uncertainties and respond to opportunities and threats." The authors [23] show that "the second law of thermodynamics states that entropy or randomness in a given system will increase with time". This is shown in science, "where more and more biological processes have been found to be independent" and "randomness is the fundamental and overarching principle that helps to explain how traits are independently passed from parent to offspring". It is the presence of randomness in all biological systems that this paper aims to highlight.

In the exploitation of agricultural machines, the study of vibrations is carried out more and more experimentally. The results are processed using mathematical statistics in the spirit of the theory of random vibrations [24–30]. The classical theory of vibrations cannot effectively study the complex vibrations of agricultural machines due to objective reasons: the difficulty of estimating model constants, nonlinear behaviours, and excitations, most often of a random nature. Apart from these aspects, the influence of games with a functional character, which is impossible to model in a deterministic way, is manifest. Concrete cases are harvesters [24–26], farm tractors in transport or various agricultural works [25,27–33], as well as cultivator subsoilers [34–36].

In the field of agricultural processes that directly involve the soil, the random character of the process is due for the most part to the random properties of the soil. In agricultural machines intended for soil processing, the soil-working body contact excites the analysed structures. The highly random properties of the soil induce the excitation of the process (especially contact forces) to have a deep random character. In general, the dynamic processes produced by the contact between the soil and the working body of agricultural machines intended for soil processing are well included in the category of random vibrations. Sometimes the machines themselves include vibration sources with adjustable, programmable characteristics to achieve a certain type of soil processing. In

the latter case, in the experimentally recorded signals, it is natural for the deterministic vibration elements to appear in the spectrum, even if they have been altered to some extent. The random vibrations of agricultural machines intended for soil processing are another example of random vibrations encountered in the technique, apart from those given in [2]. We recall the definition of the agricultural cultivator as given in [37], that is, an agricultural machine that serves to shred and loosen the soil, to destroy weeds from crops of creeping plants, etc. This definition was added because the study material presented in this article has as a main component a working variant of a complex cultivator with variable working width. According to [38], the cultivator can be understood as an agricultural machine used for surface soil work to loosen and destroy weeds without overturning the furrow.

The original aspect of this study is the presentation of a research method based on the theory of random vibrations using the capabilities of a complex modulated cultivator with a variable working width, MCLS. MCLS performance testing is presented simultaneously as a research method for other similar machines or for machines obtained from MCLS by modulating or fitting different active bodies that the structure can use.

Also, the paper tries to show the researchers the benefits produced by the study of vibrations in an experimental framework with the help of the theory of random functions. Most researchers are trained in the scientific spirit of the deterministic theory of vibrations, which is why their first tendency is to look for answers in terms of the deterministic theory. From there, it comes to mathematical modelling and simulation, which are easy to develop on the computer. After the models are run, however, it is found that it is difficult to give answers to the real problems, especially to motivate and validate the hypotheses and mathematical models. Advanced designers in the field of such phenomena have been using the path of experimentation and the theory of random functions for decades. The deterministic approach tries to force physical reality into the model's pattern, while modelling in the experimental framework seeks to take as much as possible from the real conditions and extract a minimum of useful conditions in conception, design, execution, and exploitation. The most difficult problems with this last option are the generalisation (which is low for the time being) and the generalisation costs (which are very high).

#### **2. Material and Methods**

Research on the MCLS soil tillage machine (a variant with a working width of 1 m) is the subject of this article. The presented results describe the experiments and the statistical processing of the experimental data. The MCLS complex cultivator was initially conceived, designed, and realised to facilitate research in the field of soil processing [39,40]. The machine was designed for three working widths: 4 m and 2 m (with an 80 HP tractor) and 1 m (for a 45 HP tractor). Additionally, the supporting structure can support the installation of 5–10 types of working bodies to make comparative estimates as complete as possible. After conception, design, and execution, the idea of using the machine in farm operations also appeared. Thus, the machine can be used simultaneously for farm applications and research. Images from the experimental activity with the MCLS machine in all working variants are given in Figures 1–4.

The experiments were carried out in Romania, in the Bucharest-Ilfov area, in an experimental plot owned by the INMA Bucharest Institute, which has a transitional temperate continental climate. The morphological characteristics of the soil were: red-yellow clay loam, in a dry state; medium and small angular agglomerate structures; slightly loosened. The soil moisture at the measurement time was between 20 and 35 percent (Delta-T HH2 digital moisture meter), while the average soil compaction level was 45 KPa.

Traction is provided by agricultural tractors New Holland 80 TD and New Holland 50 TD. The working speeds were between 0.78 and 2.15 m/s. The results presented in this article refer to the experiment carried out at a constant speed of 0.789 m/s.

The experiments were carried out on a plot of 500 square metres, marked with benchmarks. For each test, 10 m were set for the acceleration and 10 m for the deceleration of the tractor, and 30 m were set for the evaluation of the cultivator (see Figure 4).

**Figure 1.** The MCLS complex cultivator operates in the version with a working width of 4 m.

**Figure 2.** MCLS in operation in the 2 m working width variant.

**Figure 3.** MCLS in operation in the 1 m working width variant.

The QuantumX MX1615B data acquisition system was used for the strain gauge measurements, with 32 measurement channels and KFG-6-120-C1-11 N15C2 strain gauge sensors (120 Ω). The strain gauge amplifiers, QuantumX MX1615B, are suitable for precise data acquisition in full-bridge, half-bridge, and quarter-bridge configurations, as well as for strain gauge-based transducers, potentiometers, resistance thermometers (PT100), or normalised voltage (+/−10 V).

The measurement of the forces on each support of the working body was performed by the strain gauge methodology, with the procedure including a calibration stage of the indications of the deformation sensors. Although it is not the most precise measurement method, it is convenient and accessible, especially considering that only values measured by the same procedure are compared and are, therefore, hypothetically, affected by comparable errors.

**Figure 4.** The location and parcelling of the land on which the experiments were carried out with the MCLS machine.

The experiments took place in the testing ground of INMA Bucharest, where the soil was classified as reddish-brown from the forest and pre-processed with autumn ploughing (coarse). Figure 5 depicts the location of the experimental polygon in detail with reference to the Bucharest-Baneasa airport runway.

**Figure 5.** Indexing of measurement locations on the structure.
