1. Introduction

2. Material and Methods

2.1. The Subject of Research

Approaching the soil processing phenomenon using an MCLS-type structure involves statistical processing of experimental results (descriptive and inferential statistics, random process theory). Descriptive and partial inferential statistics were discussed in a previous article [42]. In this article, the focus will be on approaching the problem within the theory of random processes. I demonstrated in the article [42] that all the data strings that are generated by the contact between the ground and the working body have a random character and another comparable to the pseudorandom strings generated by special programs or to the strings of prime numbers of the same length.

In order to approach it within the theory of random functions, the phenomenon of ground processing by each organ was considered a phenomenon of random vibrations. The main random component is the excitement of the process. Excitation is generated by the interaction of the working body with the ground. In the range of demands considered, the vibrations can be considered linear-elastic.

In this article, the analysis will be made only for the MCLS variant with a working width of 1 m because in the case of this variant, signals were collected from all the supports of the working bodies throughout the experiment. The analysis carried out on the simple version with a width of 1 m can later be extended to the versions with higher working widths, for which the load on each support cannot yet be measured, and other random parameters also appear (for example, the clearances in the truss with a working width of 4 m).

The indexing of the measurement locations and the signal transmission channels from the deformation sensors to the acquisition board is shown in Figure 5. An image of the MCLS variant in experimental work, on the parcelled track of the polygon, with the complete data acquisition equipment, is shown in Figure 6.

2.1.1. The Formulation of the Problem in Terms of the Theory of Random Vibrations

According to [18], physical processes that cannot be characterised by deterministic functions, i.e., vibrations whose instantaneous values cannot be predicted as functions of time, are called random. The random nature of the vibrations of the investigated structure was demonstrated in [42], where even the degree of randomness was quantified. A relatively simple criterion for experimental recognition of the randomness of a phenomenon that is experimental is formulated, for example, in [18]: if, in several experiments organised under identical controllable conditions, the measured quantities differ only by quantities of the order of errors of measurement, the phenomenon can be considered deterministic; it is reproducible, so its development after the moment of measurement is predictable. Otherwise, the phenomenon is random. Apparently, the criterion seems simple, but in the case of soil processing and, in addition, using non-conventional and insufficiently verified structure load measurement systems (strain gauge methodology), this criterion is difficult to use. Given the findings from [42], we opted for the random vibration variant to describe the working process of the 1 m working width version of the MCLS complex cultivator.

The main characteristic of a random variable is the distribution function [19,20]. This is defined according to (1).

$${\mathit{F}}_{\mathit{X}}\left(\mathit{x}\right)=\mathit{P}\left(\mathit{X}<\mathit{x}\right), \mathit{x}\in \mathbb{R}$$

(1)

where $F_X$ is the distribution function of the random variable$\mathrm{x}$, $x$ is a real number, and $\mathbb{R}$ is the set of real numbers. In definition (1), some authors use non-strict inequality.

2.1.2. Distribution Functions

In Figure 8, in the column of graphs on the left, the temporal variations of the forces that load the supports of the twelve working bodies of the version with a working width of 1 m of the MCLS are shown. Also, in Figure 8, the histogram of the frequencies of the force values is represented in the right column. The polygon of frequencies is built using the histograms,. When the number of classes in the histograms becomes very large (theoretically to infinity), the frequency polygon tends to reflect the real distribution density of each force. The frequency histograms are constructed with 64 classes using the formula of Mosteller and Tukey [74]. We took into account the length of the data strings obtained experimentally in calculating the number of classes of the histograms.

The elementary formula (2) was used to calculate frequency histograms.

$${\mathit{f}}_{\mathit{i}}=\sum _{\mathit{j}=1}^{\mathit{N}}{\mathit{H}}_{\mathit{i}}\left({\mathit{x}}_{\mathit{j}}\right), \mathit{i}=1,\dots , {\mathit{n}}_{\mathit{h}}, \mathit{j}=1,\dots ,\mathit{N}$$

(2)

where:

$${\mathit{H}}_{\mathit{i}}\left({\mathit{x}}_{\mathit{j}}\right)=\left\{\begin{array}{c}1, {\mathit{h}}_{\mathit{i}}\le {\mathit{x}}_{\mathit{j}}<{\mathit{h}}_{\mathit{i}+1}\\ 0,{\mathit{x}}_{\mathit{j}}<{\mathit{h}}_{\mathit{i}} or {\mathit{x}}_{\mathit{j}}\ge {\mathit{h}}_{\mathit{i}+1} \end{array}, \mathit{i}=1,\dots ,{\mathit{n}}_{\mathit{h}}, \mathit{j}=1,\dots ,\mathit{N}\right.,$$

(3)

$${\mathit{H}}_{{\mathit{n}}_{\mathit{h}}}\left({\mathit{x}}_{\mathit{j}}\right)=\left\{\begin{array}{c}1, {\mathit{h}}_{{\mathit{n}}_{\mathit{h}}-1}\le {\mathit{x}}_{\mathit{j}}\le {\mathit{h}}_{{\mathit{n}}_{\mathit{h}}}\\ 0,{\mathit{x}}_{\mathit{j}}<{\mathit{h}}_{{\mathit{n}}_{\mathit{h}}-1} or {\mathit{x}}_{\mathit{j}}>{\mathit{h}}_{{\mathit{n}}_{\mathit{h}}} \end{array}, \mathit{j}=1,\dots ,\mathit{N}\right.$$

Article

(4)

$N$ is the number of observations from the examined random sequence, and $n_h$ is the number of classes in the frequency histogram. Then the probability density histogram was calculated according to formula (5),

$${\mathsf{\rho }}_{\mathrm{i}}=\frac{{\mathrm{f}}_{\mathrm{i}}}{\mathrm{N}},\mathrm{i}=1,\dots ,{\mathrm{n}}_{\mathrm{h}},$$

(5)

where ${\left\{{\rho }_i\right\}}_{i=1,\dots ,n_h}$ is the series of probability densities. The series of probabilities (or cumulative probabilities) is calculated using the formula (6).

$${\mathit{p}}_{\mathit{i}}=\sum _{\mathit{k}=1}^{\mathit{i}}{\mathit{\rho }}_{\mathit{k}}, \mathit{i}=1,\dots ,{\mathit{n}}_{\mathit{h}}$$

(6)

Figure 7. The time dependence of the load in the supports of the active bodies (left column) and the frequency distribution of the load values (right column). The four rows of figures correspond to the four lines of active organs, starting from the top with the ones closest to the tractor.

Figure 7. The time dependence of the load in the supports of the active bodies (left column) and the frequency distribution of the load values (right column). The four rows of figures correspond to the four lines of active organs, starting from the top with the ones closest to the tractor.

In Figure 8, the probability density and the probability that the random variable (the series of numerical values over time recorded at each of the twelve measurement locations) will take different values in the experimental range are shown.

Figure 8. The probability density and the probability of taking certain values from the experimental range for the forces recorded in the twelve measurement locations.

Figure 8. The probability density and the probability of taking certain values from the experimental range for the forces recorded in the twelve measurement locations.

The probability densities calculated as shown above do not resemble those found in statistics books or in articles in which the experimental results are approximated, by hypothesis, with one of the classical statistical distributions (normal, exponential, Student, Fischer, etc.). Probability distributions differ from the statistical densities with which the experimental curves are usually approximated. Precision is lost by approximating with the classical statistical distributions of the probability densities, and we cannot precisely estimate the local error in relation to the experimental data. For this reason, we preferred, considering our goals, to interpolate the empirical probability densities through cubic spline functions, since we wanted maximum precision and were not interested in using the results in immediate theoretic models. Regarding the probability curves, they have shapes very similar to the classical ones, but for the same reasons, we proceeded with the probabilities by interpolating with spline functions. In Figure 8, the experimental curves interpolated by spline functions are given in each graph for the twelve series of 4000 numerical data points.

The probabilities interpolated with spline functions are used to calculate the probability of the occurrence of a force greater than a fixed force (related to the limit resistance characteristics of the material of the supports of the working body).

The distribution functions of the twelve random variables described by the numerical sequences in Figure 7 are calculated in the same way as the probability densities and probabilities whose graphic representations are given in Figure 8. Therefore, the distribution functions are expressed as interpolations by cubic spline functions, for which the graphic representations are given in Figure 9. The graphic representations of the twelve distribution functions of the sequences of numerical values coming from the measurement points located on the supports of the twelve working bodies are grouped three by one on the graphic, corresponding to the three working bodies in each line, with the counting starting from the back of the tractor.

For complete agreement with the theoretical definition, the interpolated distribution functions are extended with the value zero to the left of the minimum value and with value one to the right of the maximum value of the corresponding random variable.

2.1.3. Characteristic Functions of Random Vibrations

From a theoretical point of view, the first tested properties of random vibrations are stationarity and ergodicity. The definitions of these notions can be found in all the literature dedicated to random vibrations, for example, [18,19,21,22,23,24,25,26,27,28]. In general, the response of structures to random vibrations is studied through experiments, and they have strictly elaborated procedural standards [29].

A synthetic characterisation of random signals is given in [43]. The non-deterministic signal has specific characteristics: mean, dispersion, global average, global dispersion, histogram, power spectral density, etc. The signal can have a certain degree of predictability in its evolution over time. Depending on specific characteristics, the non-deterministic signal can be:

Stationary, if the mean and dispersion do not depend on time but are constant;

Ergodic if the mean per portion does not differ from the global mean;

White noise has a constant power spectral density throughout the frequency band.

Testing the stationarity and ergodicity of the experimental signals (sequences of finite length) cannot be done using the definitions of the mean value and the autocorrelation function of the random variables, for example, in [18]. The definitions in [18] assume processes for crossing the limit after the number of samples. We do not have an infinite number of samples, and if we did, we would be constrained by costs to limit the duration of the experiences as much as possible. For this reason, the calculations are made for a finite number of samples, limited by the number of samples in the entire sequence.

In Figure 10, we see the behaviour of the average value of the random variable given by the registration with the ch4 code (coming from the working body in the first line after the tractor, on the extreme left). The tendency to decrease the amplitude is observed with the increase in the number of samples in the sequence considered for calculating the average value. Therefore, one can only suspect asymptotic behaviour, but such behaviour cannot be stated strictly theoretically.

In Figure 11, the terms below the limit of the definition of the autocorrelation function are calculated for the subsequence of the ch4 signal, with the graphic representation being made only for five values of the gap. There is a tendency to decrease the amplitude of the autocorrelation function with the gap value and a weak asymptotic tendency, which does not allow us to make statements about the stationarity and/or ergodicity of the signal.

In Figure 12, the autocorrelation curves are given for each of the twelve signals collected from the supports of the working bodies of the 1 m working width version of the MCLS. The curves are calculated using the lcorr function of the programme [69]. The calculation can also be done directly by programming the three simple formulas given in [70]. These curves can be considered substitutes in the field of real signals of finite length for the assessment of the stationarity or ergodicity of signals. In Figure 12, it is first observed that all twelve signals tend to asymptote in time towards the value 0. First, this means that the signals are weakly autocorrelated (another argument for their randomness). The fact that they are not constant in time shows that the signals are neither stationary nor ergodic.

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 indicates 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 of 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 being very voluminous) is difficult to manifest. Even if it happens, the damping from the metal material and the ground in 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 own 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 window 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.

Since we had no data about the operation in the non-linear elastic domain or in the plastic domain 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 the correlations between channels ch4 and ch1, respectively, 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 Figure 15 and Figure 16.

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 (compare the magnitudes of the frequencies in Figure 17 with those of the spectrum components in Figure 15).

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 [68]. 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.

3. Results

4. Comments

5. Conclusions

References

1. Ingram, J. ; Randomness and probability: exploring student ‘teachers’ conceptions, Mathematical Thinking and learning, Taylor and Francis Group, 2022, pp. 1–19. [CrossRef]
2. Costake, N. ; Cu privire la conceptul de Management societal informatizat, Revista Romana de Statistica, Trim I/2013 – Supliment, pp. 153–301.
3. Kawtar El, K.; Driss, M. ; Stochastic Programming for Agricultural Planning. Journal of Positive School Psychology 2022, 6, 1938–1944. [Google Scholar]
4. Raun, W.R.; Dhillon, J.; Aula, L.; Eichoff, E.; Weymeyer, G.; Figueirdeo, B.; Lynch, O.; Nambi, E.; Oyebiyi, F.; Fornah, A. ; Unpredictible Nature of Environment on Nitrogen Supply and Demand. Agronomy Journal 2019, 111, 2633–3403. [Google Scholar] [CrossRef]
5. Nyamekye, A.B.; Nyadzi, E.; Dewulf, E.; Werners, S.; Van Slobbe, E.; Biesbroek, R.G.; Termeer, C.J.A.M.; Ludwig, F. ; Forecast probability, lead time and farmer decision-making in rice farming systems in Northern Ghana. Climate Risk Management 2021, 31. [Google Scholar] [CrossRef]
6. Hamsa, K.R.; Bellundagi, V. ; Review on Decision-making under Risk and Uncertainty in Agriculture. Economic Affairs 2017, 62, 447–453. [Google Scholar] [CrossRef]
7. Buschena, D.E.; Zilberman, D. ; What Do We Know About Decision Making Under Risk and Where Do We Go from Here? Journal of Agricultural and Resource Economics 1994, 19, 425–445. [Google Scholar]
8. Kumari, P.L.; Reddy, P.M. ; Decision Making under Uncertainty in Agriculture-A Case Study. International Journal of Current Microbiology and Applied Sciences 2021, 10, 188–194. [Google Scholar]
9. Eisele, M.; Troost, Ch. , Berger Th., How Bayesian Are Farmers When Making Climate Adaptation Decisions? A Computer Laboratory Experiment for Parameterising Models of Expectation Formation. Journal of Agricultural Economics 2021, 72, 805–828. [Google Scholar] [CrossRef]
10. Magda, D.; Girard, N.; Angeon, V.; Cholez, C.; Raulet_Croset, N.; Sabbadin, R.; Salliou, N.; Barnaud, C.; Monteil, C.; Peyrard, N.; A Plurality of Viewpoints Regarding the Uncertainties of the Agroecological Transition. In: Bergez, JE.; Audouin, E.; Therond, O. (eds) Agroecological Transitions: From Theory to Practice in Local Participatory Design. Springer, Cham., 2019, pp. 99–120. [CrossRef]
11. Spash, C.L. ; Strong uncertainity, Ignorance and indeterminacy, Greenhouse Economics: Value and Ethics, London: Routledge, 2002, pp. 120–152.
12. Litre, G.; Bursztyn, M. Climatic and socio-economic risks and adaptation strategies among livestock family farmers in the pampa biome. Ambiente&Sociedade 2015, XVIII, 53–79. [Google Scholar]
13. Coletta, A.; Gianpietri, E.; Santeramo, F.G.; Severini, S.; Triestini, S. ; A preliminary test on risk and ambiguity attitudes, and time preferences in decisions under uncertainty: towards a better explanation of participation in crop insurance schemes. Bio-Based and Applied Economics 2018, 7, 265–277. [Google Scholar]
14. Bunyasiri, I.N. ; Impacts of income uncertainty and diversified agriculture on savings of thai agricultural households. The Business and Management Review 2017, 8, 311–322. [Google Scholar]
15. Burgman, M. ; Risks and Decisions for Conservation and Environmental Management, Cambridge University Press, 2005.
16. Cardei, P.; Constantin, N.; Sfiru, R.; Muraru, V.; Condruz, P. ; Modular load-bearing structures for the study of interactions between the soil and working parts of agricultural machines. Journal of Physics Conference Series 2021, 1781, 012063. [Google Scholar] [CrossRef]
17. Cardei, P.; Nicolae, C.; Sfiru, R.; Muraru, V.; Muraru, S.L. ; Structural analysis of a modulated load-bearing structure designed to investigate the interaction between soil and the working parts of agricultural machines. Journal of Physics Conference Series 2021, 1781, 012064. [Google Scholar] [CrossRef]
18. Buzdugan, Gh. , Fetcu, I.; Radeş, M.; Vibratii mecanice, Editura Didactica si Pedagogica, Bucuresti, 1982.
19. Tiron, M. ; Analiza preciziei de estimare a functiilor aleatoare, Editura Tehnica, Bucuresti, 1981.
20. Iosifescu, M.; Moineagu, C.; Trebici, V.; Ursianu, E. ; Mica Enciclopedie de Statistica, Editura Stiintifica si Enciclopedica, Bucuresti, 1985.
21. Kun II, P. ; Fundamentals of Probability and Stochastic Processes with Applications to Communications, Springer, 2018.
22. Available online: https://en.wikipedia.org/wiki/Random_vibration.
23. Crandall, S.H. ; Random Vibration, New York:MIT Press/Wiley, 1958.
24. Crandall, S.H.; Mark, W.D. ; Random Vibration in Mechanical Systems, New Yorlk: Academic Press, 1963.
25. Bolotin, V.V. ; Random Vibrations of Elastic Systems, The Hague, The Nederlands: Martinus Nijhoff Publishers, 1994.
26. Elishakoff, I. ; Probabilistic Methods in the Theory of Structures: Random Strength of Materials, Random Vibration, and Buckling, World Scientific, Singapore, 2017.
27. Elishakoff, I. ; Solution Manual to Accompany Probabilistic Methods in the Theory of Structures: Problems with Complete, Worked Through Solutions, World Scientific, Singapore, 2018.
28. Elishakoff, I. ; Dramatic Effect of Cross-Correlations in Random Vibrations of Discrete Systems, Beams, Plates, and Shells, Springer, Nature, Switzerland, 2020.
29. Tricker, R.; Tricker, S. ; Environmental Requirements for Electrochemical and Electronic Equipment, 8.4.4.5 Other Standards, 1999. Available online: https://www.sciencedirect.com/topics/earth-and-planetary-sciences/random-vibration.
30. Yao, Y.; Li, X.; Yang, Z.; Li, L.; Geng, D.; Huang, P.; Li, Y.; Song, Z. ; Vibration Characteristics of Corn Combine Harvester with the Time-Varying Mass System under Non-Stationary Random Vibration. Agriculture 2022, 12. [Google Scholar] [CrossRef]
31. Garciano, L.; Torisu, R, Takeda, J. ; Sakai,K.; Random, quasi-periodic and chaotic vibrations of farm tractors. Journal of the Japanese Society of Agricultural Machinery 2002, 64, 91–98. [Google Scholar]
32. Vladut, V.; Biris, S.S.; Petre, A.A.; Voicea, I.; Cujbescu, D.; Ungureanu, N.; Matei Gh, Popa, D. ; Boruz, S.; Constantin, A.M.; Vasile, C.; Comparative sudy of vibrations in apparatus combinations tangential Threshing. Analele Universităţii din Craiova, seria Agricultură – Montanologie – Cadastru (Annals of the University of Craiova - Agriculture, Montanology, Cadastre Series) 2021, 51, 615–627. [Google Scholar] [CrossRef]
33. Matache, M.G.; Voicu, Gh. , Epure, M.; Voicea, I.F.; Gageanu, I.; Ghilvacs, M.; Measuring vibrations level during transportation work for electrical tractor. Acta Technica Corviniensis-Bulletin of Engineering 2020, XIII, 27–30. [Google Scholar]
34. dos Santos, V.C.; Monteiro, L. De, A.; Macedo, D.; X.; S.; Costa, E.; Whole body vibration in operators using agricultural soil preparation equipment. Ciência Rural, Santa Maria 2019, 49, 1–9. [Google Scholar]
35. Villbor, G.; P. ; Santos, F.L.; de Queiroz, D.M.; Guedes, D.M.; Vibration levels on rear and front axles of a tractor in agricultural operations. Acta Scientarium. Technology 2014, 36, 7–14. [Google Scholar] [CrossRef]
36. Hosseinpour-Zarnaq, M.; Omid, M.; Biabani-Aghdam, E. ; Fault diagnosis of tractor auxiliary gearbox using vibration analysis and random forest classifier. Information Processing in Agriculture 2022, 9, 60–67. [Google Scholar] [CrossRef]
37. Ghazali, M.; H.; Rahiman, W.; Vibration Analysis for Machine Monitoring and Diagnosis: A Systematic Review. Shock and Vibration, Hindawi 2021, 1-25.
38. Li, J.; Liu, Y.; Xing, Zh. , Zeng, F.; Rotating machinery anomaly detection using data reconstruction generative adversarial networks with vibration energy analysis. AIP Advances 2022. [Google Scholar]
39. Harris, C. ; M.; Crede, C.; E.; Socuri si vibratii, Editura Tehnica, Bucuresti, 1969.
40. Miki, D.; Demachi, K. ; Bearing fault diagnosis using weakly supervised long short-term memory. Journal of Nuclear Science and Technology 2020, 57, 1091–1100. [Google Scholar] [CrossRef]
41. Ebrahimi, E.; Javadikia, P.; Jalili, M.H.; Astan, N.; Haidari, M.; Bavandpour, M. ;, Intelligent Fault Detection of Retainer Clutch Mechanism of Tractor by ANFIS and Vibration Analysis. Modern Mechanical Engineering, 2013, 3, 17–24. [Google Scholar] [CrossRef]
42. Cardei, P.; Constantin, N.; Muraru, V.; Persu, C.; Sfiru, R.; Estimation of the random intensity of the soil tillage draft forces in the supports of the working bodies of a cultivator, preprint RG. 2023. [CrossRef]
43. Available online: http://www.mobilindustrial.ro/current_version/online_docs/COMPENDIU/semnale_deterministe_si_ned eterministe__aleatorii_htm.
44. Marin C., Vibratiile structurilor mecanice, Editura Impuls, Bucuresti, 2003.
45. Xin, J.; Kaikang, Ch. , Jiangtao, J.; Kaixuan Zh., Xinwu, D.; Hao, M.; Intelligent vibration detection and control system of agricultural machinery engine. Measurement 2019, 145, 503–510. [Google Scholar]
46. Choi, W.-S.; Pratama, P.S.; Supeno, D.; Woo, J.-H.; Lee E-S. ; Chung, S.-W.; Park Ch.-S.; Dynamic vibration characteristics of electric agricultural vehicle based on finite element method. Globa Journal of Engineering Science and Researches 2017, 4, 27–31. [Google Scholar]
47. Iacob, C.; Cheorghita, S. ;I.; Soare, M.; Dragos, L.; Dictionar de mecanica, Editura Stiintifica si Enciclopedica, Bucuresti, 1980.
48. Wang, P.; Gao, M.; Sun, Y.; Zhang, H.; Liao, Y.; Xie, Sh. , A vibration-powered self-contained node by profiling mechanism and its application in cleaner agricultural production. Journal of Cleaner Production 2022, 366. [Google Scholar] [CrossRef]
49. Wang, J.; Xu, Ch. , Xu, Y.; Wang, J.; Zhou, W.; Wang, Q.; Tang, H.; Resonance Analysis and Vibration Reduction Optimization of Agricultural Machinery Frame—Taking Vegetable Precision Seeder as an Example. Processes 2021, 9, 1–19. [Google Scholar] [CrossRef]
50. Zhan, G.; Ma, L.; Zong, W.; Liu, W.; Deng, D.; Lian, G. ; Study on the Vibration Characteristics of Rape Plants Based on High-Speed Photography and Image Recognition. Agriculture 2022, 12, 1–14. [Google Scholar] [CrossRef]
51. Sheng, J.; Zeng, Y.; Liu, G.; Liu, R. ; Determination of Weak Knock Characteristics for Two-Stroke Spark Ignition UAV Engines Based on Mallat Decomposition Algorithm. Mathematical Problems in Engineering 2021, 1–11. [Google Scholar] [CrossRef]
52. Wang, S.; Lu, B. ; Detecting the weak damped oscillation signal in the agricultural machinery working environment by vibrational resonance in the duffing system. Journal of Mechanical Science and Technology 2022, 36, 5925–5937. [Google Scholar] [CrossRef]
53. Jun, L.; Xueping, L.; Zhou, Y.; Tiansheng, H.; Kunpeng, X.; Wei, L. ; Transversal vibration analysis of resonance condition and frequency for orchard chain ropeway system. Transactions of the Chinese Society of Agricultural Engineering 2014, 30, 50–57. [Google Scholar]
54. Farrokhi Zanganeh, N.; Shahgholi Gh., Agh, S.; Studying the effect of balancer on engine vibration of MaSSey Ferguson 285 tractor. Acta Technologica Agriculturae 1, Nitra, Slovaca Universitas Agriculturae Nitriae 2021, 14-19. [CrossRef]
55. Bulgakov, V.; Kuvachov, V.; Ivanovs, S.; Melnyk, V. ; Experimental investigations in vertical vibration damping of agricultural aggregate of block-modular type, Engineering for Rural Development. Proceedings of the International Scientific Conference (Latvia), Jelgava, No. 20, 2021, pp. 635–642.
56. Esehaghbeygi, A.; Abedi, M.; Razavi, J.; Hemmat, A. ; Field evaluation of a vibrating bent-share cultivator. Research in Agricultural Engineering 2020, 66, 123–130. [Google Scholar] [CrossRef]
57. Paez, T. L. ; Random Vibrations: Assessment of the State of the Art, Sandia National Laboratories Albuquerque, New Mexica, 1999.
58. Abbaspour-Gilandeh, Y.; Fazeli, M.; Roshanianfard, A.; Herandez-Hernandez, J.L.; Penna, A.F.; Herrera-Miranda, I. ; Effect of Different Working and Tool Parameters on Performance of Several Types of Cultivators. Agriculture 2020, 10, 1–19. [Google Scholar] [CrossRef]
59. Gong, Y.; Ren, L.; Han, X.; Gao, A.; Jing, Sh. , Feng Ch., Song, Y.; Analysis of Operating Condition for Vibration of a Self-Propelled Monorail Branch Chipper. Agriculture 2023, 13, 1–22. [Google Scholar]
60. Available online: https://vru.vibrationresearch.com/glossary/rms-root-mean-square/.
61. Zhao, G.; Xiao, J.; Zhou, Q. ; Fatigue Models Based on Real Load Spectra and Corrected S-N Curve for Estimating the Residual Service Life of the Remanufactured Excavator Beam. Metals 2021, 11. [Google Scholar] [CrossRef]
62. Harral, B.B. ; The application of a statistical fatigue life prediction method to agricultural equipment. International Journal of Fatigue 1987, 9, 115–118. [Google Scholar] [CrossRef]
63. Wen, Ch. , Xie, B.; Li Zh., Yin, Y.; Zhao, X.; Song Zh., Power density-based fatigue load spectrum editing for accelerated durability testing for tractor front axles. Biosystems Engineering 2020, 200, 73–88. [Google Scholar] [CrossRef]
64. Yeter, B.; Garbatov, Y.; Soares, C.G. ; Spectral Fatigue Assessment of an Offshore Wind Turbine Structure under Wave and Wind Loading, Developments in Maritime Transportation and Exploitation of Sea ResourcesPublisher: Francis & Taylor Group London, UK Editors: Guedes Soares, C. & Lopez Pena, F.
65. Paraforos, D. S.; Griepentrog, H.W. ; Surface profiles acquisition for assessing fatigue life of agricultural machinery in test facilities, The 18th International Conference of the ISTVSAt: Seoul, Korea, 2014.
66. Esehaghbeygi, A.; Abedi, M.; Razavi, J.; Hemmat, A. ; Field evaluation of a vibrating dual bent-share cultivator. Research in Aggricultural Engineering 2020, 66, 123–130. [Google Scholar] [CrossRef]
67. Badescu, M.; Croitoru, St. , Petcu, A.; Marin, E.; Boruz, S.; Ivan Gh., Vladut, V.; Matache, M.; Cujbescu, D.; Voicea, I.; Construction evolution of deep soil loosening equipment in romania in the country specific soil conditions, Analele Universităţii din Craiova, seria Agricultură – Montanologie – Cadastru (Annals of the University of Craiova - Agriculture, Montanology, Cadastre Series), Vol. XLIV, 2014, pp. 25-34.
68. Available online: https://blog.endaq.com/vibration-measurements-vibration-analysis-basics.
69. Mathcad 2001. Available online: https://www.mathcad.com/en.
70. Available online: https://real-statistics.com/time-series-analysis/stochastic-processes/autocorrelationfunction/.
71. Available online: https://dexonline.ro/definitie/cultivator.
72. Available online: https://www.xn--dicionar-qxb.com/Cultivator.
73. Polushkin, O.A.; Ignatenko, V.I.; Ignatenko, I.V.; Vyalikov, I.L.; Bogdanovich, V.P. ; Dynamic models of cultivator spring tine performance, MATEC Web Conferences 226, 2018, pp. 1–7.
74. Doğan, N.; Doğan, I. ; Determination of the number of bins/classes used in histograms and frequency tables: a short bibliography. Journal of Statistical Research 2010, 7, 77–86. [Google Scholar]
75. SOLIDWORKS. Available online: https://www.solidworks.com/.
76. SOLIDWORKS. Available online: https://help.solidworks.com/2021/english/SolidWorks/cworks/c_Fatigue_Analysis.htm.
77. Cardei, P.; Constantin, N.; Gradinaru, V.; Marin, E.; Manea, D.; Matache, M.; Muraru, V.; Muraru, C.; Pirna, I.; Siru, R.; Sorica, C.; Stanciu, L.; Vladut, V. ; Structural analysis and materials focused on mechanics, mechatronics, maintenance and operation of machinery for agriculture and food industry, Editura Terra Nostra, Iasi, 2012, pp. 166-171.