Theoretical model design for processing internal surfaces of machine parts with free abrasives that ensures the reliability of technological processes in digital manufacturing

. The paper presents the research results of vibroabrasive processing of variously located part surfaces. An analysis of technological possibilities of processing with free abrasives in digital manufacturing was made. The dependability issue of technological processing is considered. Refined dependences were obtained to determine the mass of metal removed during processing and the steady-state roughness of the internal surface. The processing features of internal surfaces are revealed, taking into account the different ratio of length to the size of the cross section. In the course of experi-mental studies, the adequacy of the proposed dependencies was confirmed. A dependability as-sessment of the designed technological process at the stage of technological preparation was made, taking into account the possible spread of technological parameters. The results obtained make it possible to design highly efficient technological processes for processing parts with free abrasives and carry out technological preparation of digital production in an automated mode.


Introduction
Over the past 10 years, digital technologies have developed to such a level that they have been applied everywhere in all fields of human activity.Back in 2011, the concept of industry 4.0 [1] was developed in Germany.Digital technologies started actively integrating into manufacturing processes.The deepening of this integration makes it possible to significantly increase the efficiency of machine-building industries.The problem of integration such technologies is justified not only by high costs but also by the need to revise the entire structure of production [2].Both Russian corporations (Roskosmos, Rostec, Rosatom) and foreign ones (Digital Europe Program (Europe), Made in China 2025 (China), Advanced manufacturing initiative (USA)) are engaged in the study of digital transformation.
The positive aspects of digital transformation should be noted:  increasing the automation of production;  reducing time for technological preparation;  the possibility of monitoring and managing the life cycle of the product.However, most of the works are conceptual, since there is still a need to design mathematical models for the technological production preparation and the possibility to carry it out in an automated mode.From this contradiction, a scientific problem arises: how to ensure the required quality of the product at the minimum cost.
To solve this problem, based on many years of scientific experience in the field of "mechanical engineering" there are various finishing methods.Significant part of these methods are abrasive processing.It can be divided into two groups: processing with fixed abrasive and free abrasive.
Processing methods with a fixed abrasive are grinding (round, flat, centerless), polishing, lapping, honing.Processing with a fixed abrasive has both a number of advantages (high performance, accuracy, low part surface roughness) and disadvantages (restriction in the processed surface shape, difficulties in processing non-stiffness parts, high temperature and complexity of automation).
The processing methods with free abrasives are vibroabrasive, centrifugal-rotary, turboabrasive, jet-abrasive, magnetic-abrasive.The listed methods have the same advantages as processing with a fixed abrasive but are devoid of some of their shortcomings.
One of the widely used methods is vibroabrasive machining (VaM).This processing method has been introduced at many manufactures, as it has high technological capabilities [3].The main advantages include simplicity of equipment design, the ability to process parts from different materials, various shapes, sizes, including low stiffness.It is especially important in digital manufacturing as it produces various parts with individual properties.
VaM is a well-studied processing method.Many researchers have received mathematical models describing the process [4][5][6][7][8][9].Dependencies were obtained to determine the main processing parameters, such as metal removal and surface roughness.Despite numerous studies, the issue of processing internal surfaces remains inadequately studied.It should be noted that scientists have argued about the differences in the processing of external and internal surfaces, but so far no mathematical models have been obtained.In this regard, the existing dependencies do not fully reveal the abrasive treatment mechanism of internal part surface, which does not allow accurate engineering calculations and application in digital manufacturing.
Taking into account the feature of digital manufacturing, the dependability issue of the technological processing with free abrasives should be considered.Most of the existing works are devoted to the technological equipment dependability, very few works are devoted to the technological process dependability and even fewer works are devoted to the processing dependability with free abrasives [9].This raises the issue of ensuring the dependability of technological processes in digital manufacturing.
The research aim is to develop a theoretical model for processing the internal surfaces of parts with free abrasives to improve the technological preparation efficiency of digital manufacturing.
Research objectives: 1. design a refined model of abrasive granule single interaction with the internal part surface; 2. research the processing features of the internal surfaces; 3. evaluate the technological process dependability with free abrasives; 4. develop technological recommendations for the design highly efficient technological processes with free abrasives in digital manufacturing.

Materials and methods
VaM is the removal of the smallest metal particles from the workpiece surface by repeated impact interaction with an abrasive granule [3].The main advantage of VaM is a high degree of automation, since the workpieces are loaded in bulk.This advantage allows using this method in digital manufacturing.Magnetic drums or vibratory unloaders are used to separate parts from abrasive granules.Processing occurs with a continuous supply of process fluid to flush out wear products and prevent corrosion.Figure 1 shows the processing scheme.An analysis of preliminary studies showed that the processing of internal surfaces differs from the processing of external surfaces [3,4,8].The hypotheses were put forward for possible reasons for the differences in processing: 1. increased effective impact speed of the abrasive granule with the internal surface, expressed by the greater impact energy of the abrasive granule due to interaction with neighboring granules (the effect of the added mass).Babichev A.P. and Tamarkin M.A. obtained dependence (1) to determine the effective speed.The coefficient of effective speed ef k is introduced, which takes into account the effect of the added mass; A -amplitude, ω -oscillation frequency; 2. additional impulse transmitted from the internal surface of the part to the abrasive media.During processing external surfaces, the impulse is transmitted from the wall of the working chamber to the first granule.Then, the impulse is transmitted along the chain between the granules to the granule in contact with the processing surface.During processing internal surfaces, the part itself transmits an additional impulse to the granules located in the internal cavity along a shorter chain; 3. a larger interaction angle of the abrasive granule with the processing surface in  .Babichev A.P. determined the average probabilistic interaction angle of an abrasive granule with the external surface to be 28° [3].Suppose that the interaction angle with the internal surface will be close to 90°, but such a value will lead to a low tangential component of the cut force and there will be no microcutting.Therefore, for our calculations we will take an angle in the range of 76-82°.Using Nepomniachtchi E.F. methodology describing the interaction of a spherical particle with a deformable half-space [10], make the following assumption: an abrasive granule of radius R collide with the surface of the part at an angle  with a speed 0 V , which makes it possible to provide a force sufficient to form an imprint.
The main provisions of the sliding theory of a rigid sphere over a plastically deformable half-space [11] allow calculate the local metal volume deformed along the sliding path dx: Vd -deformed volume, when the granule interacts with the surface, l -average contact patch diameter: Substituting ( 3), ( 4) into (2): Integrating the granules over the sliding path: To determine the limits of integration introduce dimensionless coordinates: ε -dimensionless penetration, ξ -dimensionless sliding, hmax -maximum penetration depth of the granule.
With an increase in the plasticity of the material, the amount of deformable metal increases along the edges of the scratch and at the same time the amount of material removed in the form of chips decreases.This phenomenon is described by the chip formation coefficient Vc -volume of metal removed as microchips, Vsc -theoretical scratch volume.
Considering the above: The interaction of a granule with a deformable half-space is described by the equation: The well-known relation from the sliding theory of a spherical granule over a deformable half-space [11], it is possible to solve the first equation of system (12): The surface of an abrasive granule has significant differences compared to a smooth sphere.This is expressed not only by the geometric shape, but also by the presence of abrasive grains on the surface, which form additional irregularities.The author [3] introduced the granularity coefficient R k , which takes into account the actual contact area of the abrasive granule with the treated surface.
After substituting (13) into (12), taking into account the coefficient R k , obtain: Taking into account , tangential force is due only to friction write:  0 d  mark "+"increase penetration;  0 d  mark "-"decrease penetration.Taking into account (20) it is possible to integrate (11).Define the limits of integration: when the slide stops.
In the first case: Second case: , included in expression (11) will depend on the value *  and will be determined by the numerical value f tg  , several options should be considered.Taking into account that the value of the sliding friction coefficient of abrasive granules on the most common materials of workpieces is 0.2 -0.3 [11].Conferences 402, 10025 (2023) https://doi.org/10.1051/e3sconf/202340210025TransSiberia 2023 3) The paper [3] considers the case of the angle of contact with the surface of the workpiece, which does not exceed 45° (during VaM the average probabilistic value for the external surfaces is 28°) Based on the available experimental studies, it is known that the circulation of the abrasive media inside the part is different, which in turn leads to different processing intensities [4].Preliminary results of experimental studies showed that the removal from the internal surfaces is higher than from the external ones.Taking into account the assumptions put forward above, the angle of interaction of the abrasive granule with the internal surfaces in  is in the range of 76-82° in accordance with this write-up This integral can be transformed through the Gamma-function: Substituting ( 17), ( 27) into (11) we obtain the dependence for calculating the volume of metal removed in one cycle of interaction of the granule from the internal surface of the part: 8,9 sin 0,87 0, 47 3 Accordingly, the removal of material during a single interaction of the abrasive granule with the internal surface of the part will be determined by: The total metal removal is determined by the metal removal during a single interaction of the granule multiplied by the number of such interactions per unit time.It must be taken into account that with an increase in the length of the internal surface of the parts, the processing intensity will decrease, since there will be "stagnation" zones where the processing process will be disrupted, respectively, the removal will decrease.Analyzing the above, in order to correct the formula, it is necessary to introduce a coefficient that takes into account the ratio of the length to the size of the cross section of the machined surface of the part sr k .Let us consider the case of packing of granules on the treated surface in a square.Taking into account the above, the amount of metal removed will be determined The value of the introduced coefficient will be determined in the course of experimental studies.
It is known that when processing in granular media, the change in roughness is exponential [3].When processing with free abrasives, the initial roughness is removed and a new microprofile is formed.After a certain time, such a profile begins to be reproduced, and its height and pitch parameters do not change, but depend on the processing mode.Such a microprofile is called "steady-state roughness".
The author [3] obtained a dependence for determining the steady-state roughness during processing with free abrasives.With an increase in the angle of interaction of the granule with the treated surface and an increase in the effective speed, the maximum depth of penetration of the abrasive granule into the treated surface increases For experimental studies, samples of the "sleeve" type were used from the material steel 30KhGSA with a different combination of external diameter, internal diameter and length:  internal diameter Ø25, external diameter Ø30, length 20, 40, 60;  internal diameter Ø45, external diameter Ø50, length 20, 40, 60;  internal diameter Ø65, external diameter Ø70, length 20, 40; To determine the removal from the internal surfaces of the part, electrical tape was applied to the external surface.
Processing was carried out on a vibration machine model UVG 4×10.As an abrasive media, the battle of abrasive wheels was used, granulation R ~ 0.0057 m, grain size 16 and triangular prisms, granulation R = 0.0057, grain size 8, 12.Every 30 minutes, the processing was stopped, the electrical tape was removed, the samples were washed and dried.Weighing was carried out on an AD 200 analytical balance, roughness was assessed using a portable profilometer SURFTEST SJ-210.The treatment was carried out with a continuous supply of a 0.2% soda ash solution to flush and prevent corrosion.

Results
The coefficient of effective impact velocity with internal surfaces .эф в k was determined by comparing the results of theoretical calculations and experimental studies.Samples with a length of 20 mm were used to exclude the appearance of stagnant zones.The following values of technological parameters were used in the calculations: A = 2.5 mm, ω = 30 Hz, ρgr = 3900 kg/m 3 , R = 0.0057 m, βin = 79°, c = 3, f = 0.25 kc = 0.9.The calculations were carried out using the averaged values of 5 experiments.Table 1 presents the results of the calculation.The results of the experiments are presented in Figures 2-3.The theoretical calculations are marked with a solid line, and the experimental results are plotted as points with confidence intervals.The confidence level is 95%.Experimental studies of the internal surfaces processing features were carried out using a different ratio of length to cross-sectional size.Figure 4 shows a diagram of removal from a unit area of the part surface.The determination of the size ratio coefficient sr k was carried out by comparing the experimental results and theoretical calculations of the specific metal removal Table 2 shows the bank of coefficients, and in Figure 5 is a graph of the influence of the ratio of the length to the diameter of the internal surface on the value of the coefficient of the ratio of dimensions ksr.As mentioned above, ensuring the reliability of technological operations is of great importance for digital production.We will assess the reliability according to the methodology proposed in GOST 27.202-83 "Reliability in engineering.Technological systems.Methods for assessing reliability in terms of quality parameters of manufactured products" [12].Usually, to determine reliability, it is necessary to have a large sample.Based on the obtained theoretical dependencies for theoretical calculations of reliability, we will make the following assumption: during processing, the amplitude A, the oscillation frequency ω and granulation of working media R can vary within a range 5, 10 and 15% and we will calculate the roughness of the machined surface.Figure 6 shows the influence of the spread of technological parameters (amplitude, frequency, granulation) and the tolerance of the controlled parameter (roughness Ra) on the value of the reliability indicator (accuracy margin factor з K ).The technological operation is considered reliable when the condition з K > 0 is met.
As the results of the calculations show, the reliability of the technological operation of vibroabrasive processing can be ensured only in a certain range of modes, and the tolerance on the controlled parameter has a significant effect.

Conclusion
After analyzing the results obtained, the following conclusions can be drawn: 1. the development and adequacy of the refined theoretical model for processing the internal surfaces of machine parts with free abrasives, which improves the efficiency of technological preparation of digital production, have been developed and proved; 2. a refined model of a single interaction of an abrasive granule with the internal surface of a part was obtained, which taking into account the peculiarities of the processing conditions; 3. the volume of metal removed per single interaction of the granule and the maximum depth of penetration of the granule when processing internal surfaces is higher than when processing external surfaces; 4. with an increase in the ratio of the length to the size of the cross section, a decrease in the specific removal of metal from the internal surfaces, caused by the appearance of stagnation zones, is observed;


-abrasive grain density, m k -coefficient taking into account the influence of neighboring granules during processing, t -processing time, N P -normal component of the cutting force, P  -tangential component of cutting force.

1 P 2 P
-the probability that any point of the packing square is covered by a contact spot in one cycle of exposure to a mass of abrasive media, -probability of microcutting at a single interaction, p S -processing surface area.

L
max h .Accordingly, for internal surfaces, it is necessary to adjust the maximum penetration depth max.in h by taking into account the effective velocity coefficient for internal surfaces .ef in k .Taking into account the above, the dependence for determining the steady-state roughness for internal surfaces will have the following form: -unit length, 0 z -nominal number of grains per unit surface of an abrasive granule.It should be noted that the value of the effective speed coefficient in the processing of the internal surfaces of the part .ef in k will be determined in the course of experimental studies.

Fig. 5 .
Fig. 5.The value dependence of size ratio coefficient sr k on the ratio of length to cross-sectional

Table 1 .
Calculation of the effective speed coefficient for internal surfaces Dependence of removal from a unit area on the ratio of length to cross-sectional size

Table 2 .
Size ratio coefficient