Diagnostics of the Excessive Wear of the Machine on a Model of Developmental, Progressive Interactions Between Diagnostic Signals and Rotation Speed Signals


 In the machine operating process, there are certain interactions between its operational use and wear. The current wear is increased by the current intensity of operational use, and usable potential is reduced by the current wear rate. In the diagnostic inference process, static characteristics and trajectories of state from the experiment are compared with different matrices determined for various assumed configurations of changes. As a result, the approximated diagnosis is created. This method is not universal. It applies only to the continuous progressive state, more or less increased wear rate of the machine.


Introduction
In the machine operating process (assemblies, elements), there are two closely related phenomena: operational use (related to the quality and intensity of machine operation, requirements of its operation, intensity and quality of usage, maintenance, etc.) that generates a set of rotation speed signals and wear (connected with changes in the technical condition of the machine, damages, degradation, deterioration, wear, etc.) that generates a set of diagnostic signals . These processes (operational use and wear) have a different physical nature. Thus, signals and are any physical quantities: functional, defectoscope, thermovision, vibroacoustic, i.e., determined and probabilistic. Due to this, studying the relationship between these signals is complex and is often treated in a simplified way.
In operational practice, it is often assumed that these processes are autonomous. With this assumption, the processes (phenomena) of usage (operation) and wear (diagnostic) can be described by separate (unrelated) equations of state [1,2,5]: where: ,̇ -complex rotation speed signal (operation, functioning) and its derivative -parameter of the machine's usable potential ,̇ -complex diagnostic signal (of wear and its derivative) -parameter of the wear rate (degradation) of the machine.
We cannot adopt the autonomy of wear and operational processes [3,5,6]. It should be mentioned that the method and intensity of machine operation has a significant impact on the speed of change of its technical condition (wear), and reversely, machine wear has an impact on its operation (possibilities of using it). The relations between the rotation speed signals and the diagnostic signals under mutually progressive (developmental, dangerous) interactions can be described by typical linear equations of state applied in automation control engineering [2,4,5]: Where additionally: bU -intensity parameter of the impact of the wear state D on operational state U bD -intensity parameter of the impact of the wear state U on operational state D.
By analyzing equations (3) and (4), it can be concluded that identification of the system described by related formulas (3) and (4) can come down to defining the determined relations between signals { } and { } in the form of tangential characteristics, trajectory of state as well as the values and proportions of parameters aU, bU, aD, bD. In this way, a necessary information to conduct the diagnosing process of a machine (assessment of degradation degree and quality of operation) will be determined.

Theoretical basics for analyzing the equations of interaction
A diagnosed machine can have different operating states. It can have a small usable potential (low value| |); low sensitivity to usable potential (low value b U ). The object can have significant wear (high value ) and high sensitivity to operational and control conditions (high value ). There may be plenty of such combinations. Thus, the analysis of equations (3) and (4) must be extensive, complex, and deep and include the entire static and dynamic identification and actively use the elements of algebraic topology and variational calculus (combinatorics) [1,5,7].

Statical identification of the object described by the equation of progressive interaction
The relations between signals D and are determined for steady states, which are characterized by the fact that derivatives ̇ and ̇ are zero. Relations D = f(U) for steady states are presented by the static characteristics. To determine the static characteristics, equations (3) and (4) should be added by equations. This operation of variational calculus is often used to solve these problems [1,5]. After adding sides of the equation, the following is obtained: after the arrangement, we receive the following: After assuming ̇= 0 and ̇= 0 , we have: and further: and finally: And further: After assuming = = and , we have: Static characteristics based on formula (10) taking into account (11), (12), and (13) were exhibited in fig. 1.
The process of diagnostic inference can depend on testing the location of static characteristics determined from the current diagnostic test in relation to pattern characteristics (reference) ( fig. 1). Thus, when | | increases, then static characteristics becomes steeper, and when increases, then it becomes flatter (11). Further, when decreases, then the characteristics will become very steep, and when b U decreases, then static characteristics will be flatter (9). It should be mentioned that in the examined operational state distinguished by permanent, progressive process of machine destruction, and always increase (wear rate and usage degree of usage potential are on the increase) and also increase before they reach the following state:

Dynamic identification of the object described by the equation of progressive interaction
Relation between signals and can be described as a trajectory, which is obtained from equations (3) and (4) after removing time by a direct integration method [5]. Thus, Equation (5), which is a sum of Equations (3) and (4) after making certain conversions, takes the following form: The right side of the equation is converted to the form that enables us to apply a direct integration method: After integration, we obtain: Equation (16) can be converted to the following form: and further: and finally, after performing substitutions, − = ; − = we receive: According to topology rules, a metric can be defined for the selected characteristic model states consisting of trajectories described by (19).
Equation (19) can be rewritten in different forms: Many exemplary trajectories are determined by different assumptions. It is adopted that = = . Then, from (19), we obtain the following: After conversions (20), the following is received: Additionally, when it is assumed that = = = 2, then: It can be adopted that > 2, and next < 2. After that, a set of straight lines parallel to the straight line described by (23) is obtained from (22). The question arises what is the relation between D = f(U) for U = 0.
From equation (19a), we get: From equation (24), we obtain: For U > 0 equation (19c) takes the following form: Furthermore, by using the known formulas for solving the quadratic equation: From (25) for every adopted U > 0, 2 radicals are determined: Since the experiment shows that signals and are positive, then only radical 1 is analysed (radical 2 is always negative). After that, additional characteristic trajectory points can be determined, taking into account the appropriate assumptions, e.g. that is a few times bigger than , and then that , is a few times bigger than , and it enables to get information needed to interpret formulas (22) and (23), where the assumption = was applied.
Therefore, it can be adopted that: for = 0; < ; = 0,3, then, from equation (26) we get: for = 0,1; < ; = 0,3, then, from equation (26) we get: (26) we get: Moreover, by assuming values for = 1, 2, 4 appropriately bigger from the value of parameter = 0,3 we get points of the selected, sought model reference trajectories. The points of these trajectories are presented in tab. 1. After that, the change 1 = ( ) is analyzed for different values = 1, 2, 4 , which are appropriately bigger from = 0,3, so for the case when the condition > is met by fulfilling the condition that a D > 0; a U > 0.
Therefore, it can be written that; for = 0; < ; = 0,3, then from equation (26) we obtain: Furthermore, by assuming values for = 1, 2, 4, which are appropriately bigger than parameter = 0,3, the points of the selected and sought model reference trajectories are received.
The points of these trajectories are presented in tab. 2. Based on data from tables 1 and 2, characteristic trajectories for the solutions of equations (3) and (4)   They can also be bundles of straight lines, which have different origin, and for U = 0 we have the value 2 for = 0,3. Thus, it is visible that for U = 0 we can estimate and further also from from trajectory path, so it is possible to identify the wear of the object.

Diagnostic matrix
The diagnostic matrix is created from the virtual static characteristics and trajectories determined for the clearly identified wear mode described for it by an adjusted interaction model. Thus, the static characteristics of fig. 1 and the trajectories of fig. 2 belong to the diagnostic matrix that enables interpreting the waveforms of signals and assuming that they were defined for an excessively worn machine. The matrix can be successfully used applied in the process of diagnostic inference.
Signals (effective value) and (rev/min) were determined experimentally. They are shown in figs. 3a, b, c, d [8].  A scale for measuring and is extended to a signal scale, which was used in the matrix. Thus, when in figs. 3 we see that changes from 0 ÷ 6000, and changes from 6 ÷ 1,6 * 10 4 , then after taking into account the matrix scale (tables 1 and 2), we obtain the following: changes from 0 ÷ 30, and from 0 ÷ 8. Then, the waveforms in figs. 3 and after time removal, figs. 4 takes the same form as figs. 5. They are static trajectories and characteristics, which clearly describe the steady and unsteady states of machine operation. The results of the experiment in the form of the trajectory and statistical characteristics (figs. 5) are overlaid with a "matrix" constructed from the model calculated static characteristics ( fig. 1) and trajectory (fig. 2). Then, a picture of the relationship between the D and U signals for the experiment and for the pattern (matrix) is obtained, which allows to interpret the current technical condition of the machine, which is illustrated in fig. 6.  fig. 6, we see that the time waveforms, e.g., from fig. 3a, are synthetically expressed by means of the static characteristics and trajectory. Their analysis can be performed successfully by the appropriate use of a static pattern characteristic D = U and a trajectory for AD=2 and AU=0,3. It is shown here the excessive unstable machine wear, which can also be assessed based on the waveform of static characteristics AU/AD >1 and the path of phase trajectory near the pattern when AU is big, and AD is small.

Summary
In the machine operation process, relations between diagnostic signals and rotation speed signals are analyzed. To assess its technical condition, it is essential to determine the character of wear. It is also examined whether we have excessive wear or not. To conduct such analyses, static characteristics and machine trajectory are used. They are determined directly from the recorded waveforms of signals and . Their interpretation can be efficiently made after defining the theoretical static characteristic and trajectory from coupled equations of state:

̇= − −
for the assumed modern machine states. Assuming that ( ) and ( ) are constant, positive and characteristics as well as the trajectory are in the first quarter of the Cartesian coordinate system and there is no indication of singularities, then they become a sufficient premise for assessing the degree of excessive wear of the machine. Further research with the use of subsequent matrices will be helpful in the precise assessment of machine technical condition [1, 3, 5].