Tool stability analysis for deep hole drilling

A large number of parts have deep holes, therefore, rotary cutting tools, which represent relatively long and thin columns are used for holemaking. In this article we analyze the behavior of such tools under the influence of an axial compression load, in our case, the axial cutting force Fр , which differs fundamentally from the compression of short tools. Moreover, experience shows that when the cutting force FР reaches a certain critical value equal to Fкр , a long straight column becomes unstable. Among the procedures of processing metals by cutting [1-5] deep drilling takes a special place. The unique feature of this operation is that the tool does not have a pre-prepared support and a fixed direction. It is also typical of deep drilling not to be able to directly monitor the progress of the procedure and to experience difficulties with performing this procedure with the help of universal equipment without its thorough preparation [1-5]. A large number of parts have deep holes. Therefore, rotary cutting tools are used for holemaking. These tools represent relatively long and thin columns, in which one or two sizes of the cross-section are small in comparison with the length of the column. The behavior of such tools under the action of an axial compressive load, in our case the axial cutting force F_p, is fundamentally different from the compression of short tools. Experience shows that when the cutting force F_p reaches a certain critical value equal to F_кр, the rectilinear equilibrium form of the long column turns out to be unstable. When FР > Fкр, the tool acquires a new curved shape. This phenomenon is called the loss of stability, which leads to the drift of the drill axis and the decrease of processing accuracy. (Fig. 1). * Corresponding author: lvsedykh@mail.ru © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). MATEC Web of Conferences 224, 01035 (2018) https://doi.org/10.1051/matecconf/201822401035 ICMTMTE 2018

Among the procedures of processing metals by cutting [1][2][3][4][5] deep drilling takes a special place.The unique feature of this operation is that the tool does not have a pre-prepared support and a fixed direction.It is also typical of deep drilling not to be able to directly monitor the progress of the procedure and to experience difficulties with performing this procedure with the help of universal equipment without its thorough preparation [1][2][3][4][5].
A large number of parts have deep holes.Therefore, rotary cutting tools are used for holemaking.These tools represent relatively long and thin columns, in which one or two sizes of the cross-section are small in comparison with the length of the column.The behavior of such tools under the action of an axial compressive load, in our case the axial cutting force F_p, is fundamentally different from the compression of short tools.Experience shows that when the cutting force F_p reaches a certain critical value equal to F_кр, the rectilinear equilibrium form of the long column turns out to be unstable.
When F Р > F кр , the tool acquires a new curved shape.This phenomenon is called the loss of stability, which leads to the drift of the drill axis and the decrease of processing accuracy.(Fig. 1).

Fig. 1. Structural design for conventional deep hole drilling technology
Let us calculate the cutting force for a deep-hole drilling operation: where D -the drill diameter, mm.The values of coefficients, as well as exponent quantities are given in Table 1 [6].The coefficient k p in this case depends only on the material of the raw part being treated.
Let us express the cutting force (1) through the drill diameter D,  [7]:

Calculation of compression stress
Now we determine the compression stress in the drill: where A is the cross-sectional area of the drill, m 2 .
Let us determine the diameter of the drill D, at which the compressive stresses will exceed the allowable stresses [σ] for compression, (let us assume [σ] = 300 MPa): where It is obvious that for real drill sizes, when D < 0,3 mm, the compression stresses in the column are less than the allowable ones.

Calculation of stability
Critical force �F кр � is a load, the exceeding of which causes loss of stability of the original shape of the part.Using the Euler formula, we determine the critical force: where E is the Young's modulus for tool steel, 2 • 10 5 MPa = 2 • 10 11 Н m 2 ; μ -column length reduction coefficient, depending on the fixing condition (Fig. 1), let us assume that μ = 1; l -drill length, mm.

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-axial moment of inertia of the drill section, m 4 .Condition of column stability: We insert the values of the forces according to formulas (4) and (2) into this equation: from where, after simple transformations, we get: where с = l D .By setting the value с, it is possible to determine the critical value of the pair of parameters D − l, when the loss of stability occurs.
For example, setting the value с, we get:  If there is a need to use a drill with parameters outside the shaded zone, i.e. in the case when the nominal dimensions of the tool (diameter D н and length l н ) result in instability of drilling process, measures that eliminate the loss of stability should be applied.
Option 1 The drilling is performed in two manufacturing operations.
In the course of the first manufacturing operation the hole treatment is performed with a drill of diameter D 1 < D н to the depth l 1 smaller than l н and ensuring the drill stability (Fig. 3). Here: where δ is allowance, which is left for hole treatment during the second manufacturing operation During the second manufacturing operation, the drilling is performed with a drill D Н > D 1 (D Н = D 1 + δ) in diameter , and the cutting force F Р for channel drilling is less than the value of the critical force F кр , when the shank bit loses stability (Fig. 4).

Fig. 3 .
Fig. 3. Choice of drill parameters for the first manufacturing operation.

Fig. 4 . 5 MATECFig. 5 .
Fig. 4. Structural design for the second manufacturing operation: а) drilling diagram; b) structural designLater on, when the drill enters the part (Fig.5, а), the boundary conditions of the structural design change (Fig.5, b) as well as the coefficient μ in the Euler formula.At the same time, the critical force value in accordance with the last structural design doubles.

Fig. 6 .
Fig. 6.Structural design for deep-hole drilling technology using a drill steel guide.

Table 1 .
The values of coefficients and exponent quantities in the formulas of torque and axial force when treating with a rotary cutting tool