Mathematical model of machining errors on CNC lathes

. An analysis of the conditions of operation of CNC lathes. A mathematical model of machining errors has been constructed, which allows one to get an idea of the nature of changing the dimensions of parts in the order of their sequential processing on the machine, which is an important aspect in solving the problem of controlling machining accuracy.


Introduction
In machine-building production, due to the increased requirements for the accuracy of manufacturing parts, CNC machines embedded in GPS are widely used.
CNC lathes are designed for processing all kinds of parts such as bodies of rotation with grooves, chamfers, threaded sections and surfaces with a curved forming in one or more passes with one or more cutters in a closed automatic cycle.
The workpieces are processed with fixing in cartridges, cam, collet, in some cases the workpieces are supported by the rear center and mandrels are used. The largest diameter of the blanks installed above the caliper on the selected machines does not exceed 200 mm, the length of the blanks is also not more than 200 mm.
The maximum recommended processing diameter is 125 mm. In cartridges, it is recommended to process blanks with a length of no more than 100 mm. The machines of the selected type according to the passport data provide diametrical processing accuracy in the range of 0.005-0.010 mm; surface roughness Ra in the range of 0.32-1.25 microns; shape error in cross-section no more than 0.0025 mm, in longitudinal -no more than 0.0050 mm.

Methods
It is advisable to start analyzing the operating conditions of machine tools by studying the nomenclature of parts, which is characterized by constancy over a sufficiently long period of time. Based on the study of the nomenclature, productivity and output can be determined, as well as the degree of flexibility and readjustability of the production system. At the same time, it is necessary to consider the time spent on manufacturing and their impact on productivity and output, as well as measures to reduce the main and auxiliary time, to concentrate operations, to use cutting blades, tools and technological transitions in parallel.
Based on the generalization in the systematization of the results of the analysis, the following can be concluded: 1. When processing on lathes of the selected type, more often there are parts with diametrical dimensions of the order of 100 mm, long dimensions of the order of 40 mm, while parts with a small length-to-diameter value are larger; 2. Processed parts are stepped, parts with narrow steps are more common; 3. The parts have internal cylindrical surfaces, and more parts with internal diameters of about 60 mm; 4. The masses of the processed workpieces are small and in most cases are 2-4 kg; 5. The workpieces are delivered to the machine in small batches, more often 100 pieces. The release programs are small, programs designed for the release of 100 parts per year are more likely; 6. Piece time during turning is about 5 min.
High precision requirements are imposed on the processed parts. To ensure the specified accuracy, depending on the size of the parts being processed, the option of basing the workpiece and the corresponding equipment necessary for solving the technological problem is selected.
Analysis of the operating conditions of lathes shows that a wide variety of technological situations can arise in production conditions. In some cases, ensuring the required accuracy becomes difficult and leads to a significant increase in the piece processing time and a decrease in productivity. Therefore, there is a need to find methods to ensure the specified processing accuracy without reducing productivity.
A mathematical model of the processing process makes it possible to obtain characteristics of processing errors and the main factors that generate these errors. However, the model does not give an idea of the nature of the change in the dimensions of parts in the order of their sequential processing on the machine, knowledge of these patterns acquires a special role in solving precision control problems. The mathematical model of processing errors allows us to identify these patterns.
To build a mathematical model of processing errors, a batch of 31 40X steel blanks was turned [1,2].
In the order of vanishing from the machine, the diameters of the treated surfaces were measured using a digital raster measuring system of the model I9000, TУ2-034-206-83 .
The results of the experiment are summarized in Table 1 and presented in Fig. 1.  From the graph, it is possible, albeit roughly, to judge the behavior of the processing process over time. The displacements of the scattering centers (adjustment level) determine the systematic component of the total processing error, and the size dispersion relative to the displacement line of the scattering centers is a random component.
To mathematically describe the behavior of the processing process, the deviation of the dimensions of sequentially processed parts should be considered as some random function Y(t) of the serial number of the part tn=n. In cases where the parameter t can take only discrete values t=tn=nh, h=1, the random function Y(t) is a random sequence, which we denote {yn}, n=1,2,…., assuming Y(t)={yn}.
With this approach, the dot diagram should be considered as the realization of a random sequence of deviations in the dimensions of the machined parts {yn}, n=1,2,..., N, where N is the number of machined parts. In this case, the size deviations of each part act, in turn, as the realization of some random variable Yn corresponding to the number of the processing cycle n; the size deviations of different parts yn and yn+1, manufactured in different cycles, are the values of different random variables. Such an approach fully meets the essence of the matter: indeed, every nth processing cycle occurs under a unique set of conditions [3][4][5][6][7][8][9][10].
Therefore, strictly speaking, each processing cycle should have its own distribution with a certain mathematical expectation of size and its own variance. Then it is appropriate to consider the correlation between the random variables YnYn+τ-deviations in the dimensions of parts processed in the nth and n+τ cycles. y n , mm Taking into account the correlation between size deviations in different processing cycles significantly increases information about the processing process and allows us to establish a natural criterion for dividing processing errors into systematic and random.
Thus, there is a convenient criterion for the quality of the separation of total processing errors -the degree of correlation of deviations from the systematic component.
This criterion is satisfied by the known approximate methods of separation of processing errors [1].
The most common way to isolate the systematic component is to smooth out the initial implementation of the size deviations {yn} by a polynomial using the least squares method.
In this case, the systematic component of un can be represented as un = A{n}; yn=zn+un (1) where A{} is the transformation operator, yn is the initial sequence, n is the sequence number of the processed part.
The problem is simplified under the assumption that the systematic component changes according to a linear law. Then the expression for the systematic component can be written as a linear regression equation: un= ao+a1n (2) The coefficients a0, a1 can be estimated using the least squares method, while the mean square deviation from the systematic component can be estimated using the formula In this case, when the condition of uncorrelated deviations from the straight line is met, this method gives the only solution to the error separation problem. In this case, the value of έz coincides with the variance of the instantaneous error distribution.
As a result of processing experimental data, the following mathematical model of machining errors on CNC lathes was obtained: { = + 0.0034879 + 0.0000988 ; = 0.0018.

Conclusion
Thus, the constructed mathematical model of processing errors makes it possible to estimate the values of the systematic and random components of the total error, to determine the ratio between them, which contributes to increasing the reliability of the choice of the control principle when solving the problem of increasing the accuracy of machining.