Research of the Milling Process of a Cylindrical Surface by an Oriented Instrument

The object of research is the milling process with the crossed axes of the cylindrical surface and the tool. During the research, general modular three-dimensional models of the tool surface, the processes of removing the allowance and the shaping of the cylindrical surface are used on the basis of three unified modules: tool, shaping and orientation. Computer simulation is also used to build a three-dimensional model of the milling process of a cylindrical surface with an oriented tool. A graphic scheme of milling a cylindrical surface with an oriented tool has been created. The developed cylindrical module for shaping the tool surface, which is described by the product of the displacement matrices along the corresponding axes and the surface of the machined part, is represented by the product of the radius of the tool vector and its orientation module in the shaft coordinate system. The resulting graph of the distribution of the specific productivity of the milling process along the tooth profile of the tool during processing with crossed axes of the cutter and part. An analysis of this graph shows that the milling method with an oriented tool makes it possible to increase the accuracy of the shaping process due to uniform wear of the tool. The intersection angle of the cylindrical surface and the tool is also determined, the value of which is taken from the condition of ensuring the maximum removal of the material layer with uniform loading of the end part of the cutter. For this, a three-dimensional model of the process of milling a cylindrical surface with crossed axes of the tool and the part is developed, in which rough milling is carried out by the end part of the tool, and the finish – by the peripheral. In the course of the research, it is found that when finishing milling, the value of the rotation angle of the cutter is taken from the condition that the peripheral part of the cutter is fully loaded. Improving the processing efficiency is achieved by crossing the axes of the tool and the part, which allows to program the intersection point, and uniform wear of the cutter, which improves the quality of the machined surface. It is also possible to use high-speed milling to provide increased processing productivity.


Introduction
One of the most common parts that are produced by engineering enterprises includes parts that have cylindrical surfaces of revolution. In order for the parts to fulfill all the technical conditions, their critical surfaces must satisfy the necessary requirements for quality and accuracy. Milling is a productive method for processing cylindrical parts. The study of the milling process of the cylindrical surface makes it possible to increase the processing productivity while maintaining the necessary quality and accuracy of the surfaces of the parts. So, in [1], the results of a study of milling parts with cylindrical surfaces using a set of disk milling cutters are presented. And in [2], a method for milling cylindrical parts that have a large diameter is considered. In this method, two end mills are used, the axes of which are crossed with the axis of the part and a longitudinal feed of the tool is carried out along the axis of the part. A study of the milling process with the crossed axes of cylindrical parts and tools was carried out in [3]. In addition, the work [4,5] was devoted to the study of the milling process of cylindrical surfaces. In [6,7], threedimensional models of various methods of milling materials were studied. Thus, the study of the milling of a cylindrical surface with an oriented tool is an urgent task. Thus, the object of research is the milling process with the crossed axes of the cylindrical surface and the tool. The aim of research is to develop a three dimensional model of the process of milling a cylindrical surface with the end and the periphery of the cutter with crossed axes of the tool and part.

Methods of research
During the research, the results of [8,9] are applied. Fig. 1 shows a diagram of a milling process with crossed axes of a mill 1 and a shaft 2.
To obtain a threedimensional tool surface, it is neces sary to define a cylindrical shaping module that takes into account changes in the position of the coordinate along the profile of the tool tooth and the rotation angle j of the tool: TECHNOLOGY AUDIT AND PRODUCTION RESERVES -№ 3/1(53), 2020 where R t -radius vector of the tool points; M Z R C t t t t ⋅ ⋅ α -cylin drical tool shaping module; e 4 -radius vector of the initial coordinate [8].
The cylindrical tool surface shaping module is described as the product of the displacement matrices M2, M3 along the Y t та Z t axes of the M6 rotation matrix around the O t Z t axis: where Z H t = 0... -linear coordinate along the periphery of the tool tooth, varies from 0 to the value of the cutter width H; R t -outer tool radius; α t =°0 360 ... -angular coordinate along the tool profile. The product of the radius vector of the tool and its orientation module in the shaft coordinate system, as well as the part shaping module, have a machined part surface: where -cylindrical mo dule that describes the movement of the tool relative to the part; Z p -part feed along the axis O p Z p ; β p -rotation angle of the part; h -center distance of the shaft and tool; 4 5 -spherical module of tool orientation in the coordinate system of the part; j -intersection angle of the axes of the cutter and the part; ψ -rotation angle of the tool in the coordinate system of the shaft [8,9].
Given equations (1) and (2), a cylindrical surface when processing with an oriented tool will look like: The contact condition of the cutter profiles and parts at different points in time [3] is used to determine the profile of the machined shaft.
A modular threedimensional model of the tool R t (equa tion (1)) is used to determine the specific processing pro ductivity Q: where Vn -projection of the relative speed vector of the mill on the direction of the normal to its surface; α α t t 1 2 , angular coordinates along the profile of the tool.
A graph of the distribution of the specific processing productivity Q along the profile of the cutter tooth when processing with an oriented tool is shown in Fig. 2.
In the graph (Fig. 2), the section from i rend to i rst corresponds to the cutter end, and i rst starts from the beginning the peripheral part of the tool. So, the removal of the rough allowance falls on the end face of the tooth of the instrument, and the finishing allowance -on the peripheral section. While the specific processing produc tivity Q in the direction from i rend to i rst decreases, and after that i rst takes the smallest values. So, the milling process of a cylindrical surface with an oriented tool pro vides high accuracy of shaping the part surface due to the small wear of the final peripheral parts of the tool tooth.
Therefore, rough milling of a cylindrical surface with an oriented tool is possible to carry out with the end part of the cutter, and the finish is peripheral. This processing method will provide the necessary quality and accuracy of the machined surfaces of the parts.
To ensure the removal of the maximum allowance while maintaining the condition of uniform loading of the end face of the mill during rough processing of the cylindrical surface, the tool is rotated through an angle j relative to the part: where m -processing allowance [9]; s p -longitudinal feed of the part; r z -radius of the workpiece. When finishing processing, the cutter 1 must be oriented relative to the workpiece 2 at an angle at which the end point і rend of the curvature radius r of the cutter tooth will be placed on the radius of the workpiece r z (Fig. 3). This placement of the tool will provide the condi tion for a full load of the peripheral section of the cutter during finishing of parts. This will contribute to an even distribution of forces and wear of the cutter teeth. TECHNOLOGY AUDIT AND PRODUCTION RESERVES -№ 3/1(53), 2020 ISSN 2664-9969

Research results and discussion
Using the results of studies in [10], a threedimensional model of the process of milling a cylindrical surface 2 ( Fig. 4) with an oriented tooth of a mill 1 in the form of a tetra hedral plate is created. The universal software ABAQUS is used to create this model. The finite element mesh is applied using the Mesh module. The shape of the elements is chosen hexahedral.
The resulting stress distribution model makes it possible to analyze the state of the surface layer of the part in the process of milling with an oriented tool. So, according to Fig. 4, the highest stress values (4390 MPa) fall on the inser tion zone of the cutting insert into the part, and when ap proaching the periphery of the tool they decrease (295 MPa). 2 1 Fig. 4. Three-dimensional model of the process of milling a cylindrical surface with an oriented tool: 1 -oriented mill tooth; 2 -processed shaft When milling a cylindrical surface with crossed tool axes by the end face of the cutter tooth, the rough stock is removed, and the periphery is responsible for removing the final stock and the final shaping of the part surface.
This distribution of the allowance provides an increase in processing productivity and accuracy of shaping due to the uniform minimum wear of the final section of the tool (Fig. 2). The performance of the milling process with an oriented tool is affected by the width of the tool, the number of cutting inserts and their material.
In this case, roughing is carried out with the orien tation angle of the cutter (equation (6)), which ensures uniform loading of the tool face and removal of the maxi mum allo wance. A fine milling -with the rotation angle of the tool, which provides the condition for a full load of the pe riphery of the tool (Fig. 3).

Conclusions
A threedimensional milling model has been developed that makes it possible to study the processes of removing allow ance and shaping of the part surface. The proposed method improves the processing productivity due to the intersection of the axes of the tool and the part, as well as the shaping accuracy due to the possibility of programming the point of intersection of the axes of the mill and the cylindrical part. Roughing with the end face of the tool, and finishing with its periphery ensure uniform wear of the tool. At the same time, for effective roughing, the rotation angle of the cutter is taken from the condition of uniform loading and maximum removal of the allowance with the end face of the tool. And during finishing -with a full load of the peripheral part of the cutter. The resulting stress distribution model makes it possible to analyze the state of the surface layer of the part in the process of milling with an oriented tool.