Technological Capabilities of a Special Strategy for Processing Cycloidal Helical Surfaces with a Non-Profiled Tool

. The working bodies of screw volumetric machines are complex-profile cycloidal screw surfaces. This article deals with the problem of technological preparation for the production of such surfaces, associated with the uncertainty of the technological capabilities of a special processing strategy. To solve this problem, an analytical expression of the criterion was proposed, which makes it possible to assess the applicability of the processing strategy for the given geometric parameters. The adequacy of the proposed criterion is confirmed by machine and natural experiments.


Introduction
Сomplex-profile сycloidal screw surfaces found application in various fields of industry. The most typical representatives are the working bodies of single-screw volumetric machines for pumping high-viscosity oil products ( Fig. 1) [1,2]. Such cycloidal screw surfaces are one of the most labor-intensive surfaces to manufacture. The processing of these surfaces using modern computer numerical control (CNC) equipment can significantly reduce the labor intensity of technological preparation of production in comparison with methods based on the use of profiled tools, in which the shaping of the surface occurs by copying [3,4,5]. The trend of the last few years is the use of single-screw volumetric machines for the needs of high-precision dosing of high-viscosity substances. To solve this technological problem, a new class of equipment was proposed: high-precision single-screw dispensers [6]. The working body of these dispensers is a precision miniature gerotor pair, consisting of a polymer cage and a steel rotor (Fig.  2). A feature of this pair is the increased requirements for shape accuracy, as well as the miniature dimensions of these surfaces: the cross-sectional diameter of the screw rotor often ranges from 3 to 8 mm. Earlier it was shown that the shaping of these miniature surfaces by the methods usual for larger-sized products is practically impossible due to the low rigidity of the workpiece being processed [7,8]. However, in [9,10], a special strategy was proposed for processing cycloidal helical surfaces with different types of cross-section, which allows ensuring the required shape accuracy for given cross-section diameters (Fig. 3).

Formulation of the problem
The previously proposed processing strategy allows obtaining a tool path according to a formatted algorithm that uses data on the geometry of the tool and the machined surface in a parametric form. On the basis of this algorithm, an automated programming system was created that allows obtaining the G-code of the control program for a helical surface with arbitrary parameters. To obtain the control program code in this way, a three-dimensional model of the processed surface is not used, which makes it possible to reduce the labor intensity of technological preparation of production. However, such an analytical calculation algorithm does not allow checking the trajectory for the absence of penetration into an already processed surface. The experience of using this algorithm has shown that at certain ratios of the section diameter, pitch, eccentricity of the helical surface and the cutter diameter, the specified surface profile may not be provided during processing. Fig. 4a shows the result of processing a double-helical surface, corresponding to the specified characteristics. In Fig. 4b and 4c, you can see the cross-section of a double-helical surface with obvious deviations from the given cross-sectional profile. The results obtained showed that the proposed mathematical model makes it possible to obtain an adequate trajectory not for any input parameters. There are geometric restrictions on the extreme relationship between the section diameter, pitch, eccentricity of the helical surface and the cutter diameter, at which the degeneration of a given profile of the cycloidal helical surface will not occur. Determination of these constraints will allow timely identification of situations when the selected design and technological parameters go beyond the technological capabilities of the strategy.

Identification of technological capabilities
For the analytical determination of the limiting conditions under which the algorithm works correctly, it is necessary to refer to the mathematical processing model proposed earlier [10]. Fig. 5 and Fig. 6 schematically shows the relative position of the cutter and the cross-section of the treated surface. Analyzing this scheme, the following necessary condition for the adequacy of the mathematical model can be distinguished:  where Rcutter radius, rsection radius, eeccentricity, hhelical pitch.
Substituting the corresponding parameters of the helical surface and the radius of the cutter into expression (2), it is possible to determine, without carrying out experimental processing, whether the resulting sectional shape will correspond to the specified parameters. Applying the same expression with known geometric parameters of the cycloidal surface, it can be determined the maximum possible cutter diameter for machining, thereby ensuring the maximum rigidity of the technological system.

Experimental confirmation of the obtained limitations
To confirm the efficiency of the proposed criterion, we will conduct a series of machine experiments using the G-code verifier. For arbitrary geometric parameters of the helical surface and various values of the cutter diameter the resulting surface profiles are modeled, their geometry is analyzed, and the criterion is calculated. The simulation results are presented in Table 1 and Table 2, where the calculated criterion is set in accordance with the obtained profile of the screw surface. Analysis of the experimental data presented in the tables shows that the geometrically correct profiles of the helical surface (highlighted by a frame) correspond to the positive values of the criterion . Thus, we can conclude that the obtained analytical expression of the criterion is valid.
Also, a natural experiment was carried out, during which samples from the V95T aluminum alloy were processed with the following geometric characteristics: cross-sectional diameter 9.8 mm, eccentricity 3.4 mm, pitch 36 mm. Processing was carried out on a 4-axis horizontal CNC milling machine equipped with a rotary table. A series of 4 samples was processed (Fig. 7) using cutters of different diameters -3, 4, 5 and 6 mm. The diameters of the cutters were selected using the proposed criterion. As a result, the geometry of the obtained parts in all 4 cases corresponded to the specified parameters. Fig. 7. Samples of a helical surface, machined with a tool of various diameters

Conclusions
As a result of the study, a formalized criterion for assessing the compliance of the specified geometric parameters of the cycloidal helical surface with the technological capabilities of a special processing strategy was proposed. The expression obtained analytically for this criterion was confirmed using a machine experiment, during which the treatment of surfaces with various parameters was simulated, and the results were compared with the calculated values of the criterion. The efficiency of the criterion was also shown during experimental processing of a series of samples on a CNC machine. The formalized criterion can be integrated into an automated system for calculating the tool path for processing cycloidal helical surfaces, which will significantly reduce the labor intensity of technological preparation of production.

Ethical Approval
The authors declare that this article has not been submitted to other journals for simultaneous consideration. The submitted work is original and not has been published elsewhere in any form or language (partially or in full). This study has not be split up into several parts to increase the quantity of submissions and submitted to various journals or to one journal over time. Results have been presented clearly, honestly, and without fabrication, falsification or inappropriate data manipulation. The presented data, text and theories are original, not copied from the works of other authors and were developed by the authors of this article. The authors declare that they have permission for the use of software used in the article. The article cites appropriate and relevant literature, inappropriate self-citation and coordinated efforts among several authors to collectively self-cite were not used. Text of the article does not contain untrue statements about an entity or descriptions of their behavior. Submitted research cannot be misapplied to pose a threat to public health or national security. Author group, the Corresponding Author, and the order of authors are all correct at submission.