Mathematical modeling of milling cutters with variable radius for processing holes with an equiaxed contour

. The aim of the study is to model the cutting edges, front and rear surfaces, the pitch of the teeth and the cutting surface of a milling cutter with a variable radius for processing holes with an equiaxed contour. In the process of modeling, the method of compiling mathematical models of technological processes for obtaining surfaces of parts by cutting was used. As a result, the equations of the cutting edges of the milling cutter, the equations of the producing surface were obtained, parametric equations of the rear and front surfaces of the cutting teeth were constructed, and conclusions were drawn.


Introduction
In the process of designing a milling cutter with a variable radius, an important step is the modeling of cutting elements, which can be performed using the geometric theory of surface formation [1][2][3].
Figure 1 shows the structure of the elements of the design model of a milling cutter with a variable radius in the form of a graph, the elements of which are: the producing surface of the hole ( 0  0  0 ), milling cutters (      ), i-oh cutting edge (      ) ,front (      ), back surface (      ) and transition matrices.

Methodology
To build a model of the process of forming RC-profile holes by milling with a tool with a constructive feed, it is necessary to determine the equation of the producing surface  4 ̄(θ,  3 ) included in (1): where   −1 (θ 1 , ω,  1 , ) -matrix of the forming system of technological equipment; ̄0(θ, ) -equation RK-3 of the profile hole; where  6 (θ 1 ) -the matrix corresponding to the rotation of the spindle assembly with the fixed workpiece  1 () -a matrix that takes into account the offset along the axis Х;  3 ( 1 ) -the matrix of movement of the longitudinal support;  6 (−ω) -spindle rotation matrix with milling cutter.On parameters   , ,  1 ,  the connections included in Equation 1 must be superimposed (Fig. 2), then we take   In order to move from the coordinate system of the workpiece to the coordinate system of the tool, it is necessary to perform the reverse transformation, i.e. solve the inverse matrix and move from the coordinate system (Fig. 2)    to the coordinate system  1  1  1 , where    -the coordinate system of the workpiece; 1  1  1 -the coordinate system of the tool.
Any inverse matrix has the property: where E -the unit matrix The unit matrix also has the property: where М -coordinates of a point in matrix form ].
After superimposing these links ( 2) is converted to the form On the parameter  we will impose an envelope relation, which we will determine by solving with respect to the parameter  equations, where В -matrix of partial derivatives vector ̄4( 1 , , ).by parameters  1 , , .
Thus, the vector ̄4( 1 , , ).let 's bring it to the form ̄4(, ).Based on the equation of the generating surface ̄4(, ) setting the parameter discretely we obtain the equation of the cutting edge i-th tooth of the milling cutter ̄4  (  , ), where   -a parameter that determines the angular position of the cutting edge.
The front surface of the blade is in contact with the removable layer and chips during cutting and is a very important parameter in the design of the cutting tooth.
The construction of a model of the front and back surfaces is associated with the determination of their normal vectors.To determine the equation of the vectors of the normal to the front  ̅  and the back surface  ̅  and-the cutting edge of the cutter: 1. Calculate the matrix   setting the coordinate system of a radial secant plane with the center passing through the origin of the coordinate system       and the point of the cutting edge ̅ 4 (  , )| = 0, by vectors specifying the positive direction of the axis Y where  4 (−  ) -matrix of rotation of the plane of the front surface around the axis X on the corner −  ; The structure of the matrices of generalized variables is presented in Table 1.

X Y Z
Translational along the axis Rotational around the axis 3. We obtain the equation of the vector of the normal and the back surface as where  5 (−  )matrix of rotation of the plane of the rear surface around the axis Y on the corner −  ; the value of the rear angle.
In order to determine the non-zero elements of the matrix  3 it is necessary to solve a system of vector equations with respect to them (Fig. 3.4).Let 's use the properties described in the equations 6-7.
where  ̅  vector defining the position of the axis  1 , represented in the milling cutter coordinate system ̅  vector defining the position of the axis  1 , represented in the milling cutter coordinate system where  The pitch of the teeth of a milling cutter with a variable radius is defined as where  the number of teeth of the cutter.The cutting surface is represented in the following form  ̄(  , , ) =  6 (  ()) ⋅  1 ( − ) ⋅  6 (−  ()) ⋅  4 (, ), where  6 (  ()) ⋅  1 ( − ) ⋅  6 (−  ()) − the matrix of the transition from the coordinate system of the workpiece to the coordinate system of the producing surface of the milling cutter during milling; () the parameter of the rotation motion of the workpiece (see Fig. 2, Fig. 5) where  initial parameter milling cutter rotation;;  the speed of rotation of the milling cutter, rpm.

Fig. 1 .
Fig. 1.Graph of the milling cutter design model with variable radius.

𝑧 3 =
,where   -the average radius of the milling cutter with a constructive feed.The angle of rotation of the cutter is assumed to be equal to  =  1 .
and  they are connected by a relation that determines the dependence of the rotation speeds of the tool and the workpiece.

Fig. 2 .
Fig. 2. Mathematical model of the PK-3 milling process of a profile hole with a milling cutter with a variable radius.

Fig. 5 .
Fig. 5. Diagram of machining holes with an equiaxed contour with a variable radius milling cutter on a lathe.

Table 1 .
Matrices of generalized variables.