УДК 625.76.08 : 517.938 ANALYSIS OF MOTOR-GRADER LOADING ON THE BASIS OF FRACTAL DIMENSION

The possibility of fractal dimension application for analysis of grader loading modes is considered in the article. The fractal dimensions of experimental dependences of load on the coupling pin of the grader at its different working conditions are calculated. It is determined that the magnitude of the fractal dimension allows to estimate the highest and lowest load of the grader.


Introduction
The main working body of any motor-grader (grader) is a fully steerable blade with knives mounted at an angle to its longitudinal axis. Depending on the soil structure the grader blade is under the influence of alternating dynamic loads. In its turn the impact of loads can lead to the machine failures due to the appearance of cracks in metal structures. In addition to the soil structure, the load blade grader is also influenced by the parameters of its work, such as grader blade deflection, turning the angle of the blade, rotation frequency of the motor shaft, etc. The influence of these parameters can be investigated by using the vibration diagnostic methods. Various parameters of the machine lead to different modes of dynamic loads. The level and nature of the vibrations acting on the machine construction due to alternating loads can be estimated by analyzing the time-stress dependences on the grader coupling pin.

Analysis of publications
Investigation of earth-moving machines parameters is carried out to predict the possible malfunctions of their work. In this case various vibration diagnostics methods are used and some of them are presented in [1]. The research results presented in [2] show that it is possible to estimate the dynamic forces that act on the metal structure by using the results of dynamics of earth-moving machines analysis with a sharp increase in the resistance of movement. To carry out such an analysis it is necessary to obtain the original data about loads on the elements of machines received in different modes of their operation. In [3] the possibility of the phase portraits application for classification of grader load modes is considered. On the basis of experimental data describing the loads on the grader coupling pin the phase portraits for different modes of its work were constructed.

Purpose and problem statement
It is expediently the determination of grader load modes to carry out on the basis of experimentally obtained signal implementations which describe the load on the grader coupling pin. In this case, different load modes lead to various forms of recorded signals. The value of the fractal dimension (FD) [4] can be the characteristic of signal shape and the contrast of these shapes due to the loading on the grader coupling pin, in turn, leads to different values of FD. Therefore, the analysis of the possibility of FD using for estimation of load on grader coupling pin is of practical importance.
The aim of the article is estimation of fractal dimension usability for determination of grader load mode changes.

Obtaining the experimental data
An essential element of grader is a ball coupling pin through which the tractive forces from the drive wheels to the grader blade during performing of work operations are transmited. Therefore, it is advisable to estimate the loads acting on this element of grader. For this purpose, the field measuring experiments were carried out on the test area of our university. As a research facility the grader DZK-251 produced at Kriukov's Railway Car Building Plant was used. The procedure of experimental research conducting and measuring system that was used at the same time are described in [5]. Measurements of stress on the grader pin were performed with using of strain gauge transducers (sensors).
Changing parameters of grader during the experiments are shown in Fig. 1. During the experiments the influence of various positions of the grader blade at the stress arising on the grader pin was evaluated. The grader parameters were changed as follows: grader blade deflection (R) was equal to 0 m, 0,7 m and 1,4 m; blade angle of rotation (α) -40°, 60° and 80°; number of the motor shaft revolutions (f) -900 rev/min, 1100 rev/min and 1300 rev/min. Indications of the sensors in the form of digital data from the measurement system were recorded in the permanent memory of the computer. In [3], all the time dependences of the stress (σ, MPa) on the grader pin from variable parameters outlined above were presented and detailed analysis of these temporary implementations was carried out. In this paper, for example, we'll adduce only temporary dependences of stresses ( shows that at the first 2-3 seconds grader was working without interaction with the ground and the load on the pin was minimal. The stress on the pin was increasing abruptly during the interaction of the blade and the ground. Later the load signal on the pin was changed irregularly and had an indented character. It should be noted that the indented nature of the stress changing is practically independent of the grader blade deflection.
Thus, from the analysis of the stresses on the pin it can be concluded if there was a load on the motor grader or it was working without load. However, over the time realizations of stresses it is practically impossible to determine under what parameters of grader the load is maximal since the maximum value of the signal amplitude is almost the same and the nature of its changes is indented at any parameters of grader.
As well as in the study of stresses at the grader blade deflection during fulfillment of working operations at different turning angle of the blade the stress amplitude on the pin is increased and has an indented character when the blade interacts with a soil.
The character of time realizations of the stress on the grader pin at different frequency of the motor shaft which is given in [3] doesn't differ from the character of time realizations that has been shown in the Fig. 2. Initially, the work operations are done with no load and then the stresses on the pin are increasing abruptly and have an irregular character.
Thus it can be easily defined the time of motorgrader loading occurrence with the help of time realizations of loadings (by an abrupt increase in the stress amplitude), however, it is hardly to determine the relationship between motor-grader loading conditions and its parameters as a part of the stress time realizations of working operations. For analysis of signal forms that describe the load on the grader pin we'll use the value of the fractal dimension.

Calculation of fractal dimension
In practice, the dimension of Hausdorf-Besicovitch D [6] is often used for estimation of fractal characteristics of various structures where N(ε)number of covering elements; εthe side length of covering element.
All existing FD calculation methods include the calculation of volume, area or length of the fractal shape and its changes during scaling.
The method of the fractal dimension determining with using of signals covering by squares comprises the following steps [7].
1. Some value of ε is defined, the time domain of the source data existence is divided into squares with a side ε and the number of squares that covered all the known points (Fig. 3) are calculated. As a result, one value N(ε) is obtained. Assume that the calculations of ( ) N  were performed for different lengths of the side ε (at Fig. 3 these values are ε 1 , ε 2 = ε 1 /2, ε 3 = ε 1 /4). As it follows from the definition of FD [4], for small values of ε the number of the covering elements ( ) N  should be equal to ~ε -D and in this case log N(ε) = -D•log ε. Now, with using of the obtained data the dependence log ( ) N  versus 1 log ε       (Fig. 4) is plotted. It should be noted that the choice of the most linear area in this algorithm is a difficult thing to formalize. Approximation of a linear part of the plot with using of LSM does not always produce reliable results. The straight line plotted on the basis of linear approximation for 10 points (LSM straight line) is shown in Fig. 4. In addition, another straight line is depicted at the same figure according to the selected 7 points when choosing the linear range of the plot (straight line area). It can be seen that the slopes of the lines do not differ significantly, however, FD is calculated more precisely when choosing linear range. Thus it needs to calculate the slope of approximating line using linear range of plot of log N(ε) as a function of log 1/ε.

Analysis of grade loads using the fractal dimension
Let's consider the possibility of FD using for the analysis of grader load. For this purpose, we calculate the values of the FD for the time realizations of stresses on the pin at different grader parameters (grader blade deflection, grader blade rotation angle, frequency of the motor shaft). The fractal dimension was calculated by using of the method described above.
The   However, the FD behavior differs from the cases examined above when grader blade deflection value is R = 0,7 m (Fig. 7).  Fig. 8. Fig. 8 shows that with the in-creasing of grader blade deflection the FD value also increases due to greater unevenness of measured signals. The maximum FD values and consequently the big loads were fixed at f = 900 rev/min. Meanwhile, when the grader blade rotation angle is α = 60°, the FD minimum value occurs at grader blade deflection value R = 0,7 m for any frequency of the motor shaft value (Fig. 9) and FD minimum value -at f = 1100 rev/min. Thus, the minimum loads of grader will be received when its blade deflection R = 0,7 m and f = 1100 rev/min.  The significant differences in FD values behavior occur at grader blade rotation angle value α = 80° (Fig. 13). In this case the spread of FD values is insignificant, i. e. it depends less on frequency of the motor shaft. In addition, it's difficult to estimate the maximum and minimum FD values at various frequency of the motor shaft and grader blade deflection values, because practically they don't differ.

Conclusions
The calculations of FD can be used for numerical estimation of irregularities of signals received from the sensors mounted on motorgrader pin.
Analysis of FD of experimental stress signals measured on motor-grader pin showed that its value depends on grader parameters when work operations are in progress.
Analysis of dependences of FD values on blade rotation angle, grader blade deflection and frequency of the motor shaft showed that when work operations are in progress the minimum loadings occur at α = 60°, R = 0,7 m и f = 1100 rev/min and the maximum loadings -at α = 40°, α = 60°, R = 1,4 m и f = 900 rev/min.
During further research it is advisable to consider the possibility of the fractal dimension using for the analysis of phase portraits of stress signals on motor-grader pin.
In the further work it is necessary to assess the possibility of using of the work results to improve the methods for determining grader oper-