Today, parts with helical (screw) surfaces are mainly machined by means of disk tools. If several helical surfaces are present on the part, it is more efficient to use worm mills. However, the design of worm mills for that purpose is a complex task, which has yet to be fully understood.

Tooth-cutting worm mills are often produced on the basis of the properties of a common normal [1]. This method consists of two stages. The first is to determine the profile of the conjugate helical rack. This method is inapplicable if the position of the normal cannot be determined—for example, if the initial profile is specified by the coordinates of individual points or a spatial curve. In addition, it cannot be used for the machining of more complex surfaces, such as tapered helical surfaces.

These problems may be circumvented by employing numerical methods in which the profile of the worm mill is established in a single stage.

The proposed system includes four basic modules; (1) formulation of a model of the surface to be machined; (2) analysis of this model; (3) formulation of a model of the tool surface; (4) analysis of that model.

To solve specific production problems, we need only use one or two modules. For predesign analysis of a new tool for machining a complex part, it is best to use all four modules.

The first module formalizes the numerical representation of the points in the initial surface of the part, on the basis of a coordinate system tied to the tool’s generating surface [2]. The initial data are as follows:

(1) the coordinates xi , yi and the number i of points on the surface of the part;

(2) the number f of coordinate transformations and the number (order) n of each transformation;

(3) the angular displacements xy, yz, zx and linear displacements Ax, Ay, Az characterizing the transformation of the coordinate systems.

The basis of the module is the initial matrix

$$MO = \left( {\begin{array}{*{20}{c}} {\cos (xy)\cos (zx)}&{\sin ( - xy)}&{\sin (zx)}&{Ax} \\ {\sin (xy)}&{\cos (yz)\cos (xy)}&{\sin ( - yz)}&{Ay} \\ {\sin ( - zx)}&{\sin (yz)}&{\cos (zx)\cos (yz)}&{Az} \\ 0&0&0&1 \end{array}} \right).$$
(1)

The program implementing the first module permits automatic formulation of the matrices М1, М2, М3, М4 from the initial matrix МO in Eq. (1). The matrices М1, М2, М3, М4 correspond to successive transformation of the coordinate systems from the profile of the part to the desired tool surface in accordance with the specified values of f and n and the numerical value characterizing each transformation.

In Fig. 1, we show the shaping of the worm mill for machining a part with a helical channel of arbitrary profile. This process includes four coordinate transformations (f= 4) in the following sequence.

Fig. 1.
figure 1

Formulating a three-dimensional numerical model of the worm mills.

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(1) Displacement along the Х axis (n = 1) of coordinate system O0Х0Y0Z0 by a distance Ax = −ricφ, where ric is the radius of the initial cylindrical part to be machined, and consistent rotation (rolling) around the Z axis by an angle xy = φ, which is a variable ensuring the rolling motion of the initial cylinder with respect to the plane N tangential (in terms of the line L–L) to the initial cylinder of a worm mill of radius Rim.

On the basis of the initial data, the program implementing the module formulates transformation matrix М1 from matrix МO in Eq. (1) in accordance with the established algorithm

$$M1 = \left( {\begin{array}{*{20}{c}} {\cos (\varphi )}&{ - \sin (\varphi )}&0&{ - {{r}_{{{\text{ic}}}}}\varphi } \\ {\sin (\varphi )}&{\cos (\varphi )}&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right).$$
(2)

(2) Rotation around axis Z (n = 3) of coordinate system O1Х1Y1Z1 by an angle xy = −ν, which is a variable ensuring the helical motion that creates the machined surface of the part and consistent displacement along this axis by a distance Ax = −pcν. Here pc = rc/tan ω; rc is the external radius of the part; and ω is the inclination of the helical channel produced.

According to the algorithm, the corresponding transformation matrix М2 takes the form

$$M2 = \left( {\begin{array}{*{20}{c}} {\cos (\nu )}&{\sin (\nu )}&0&0 \\ { - \sin (\nu )}&{\cos (\nu )}&0&0 \\ 0&0&1&{ - {{p}_{{\text{c}}}}\nu } \\ 0&0&0&1 \end{array}} \right).$$
(3)

(3) Displacement along the Y axis (n = 2) of the coordinate system O2Х2Y2Z2 by a distance Ay = −A = −(ric + Rim) and rotation by an angle zx = −ε corresponding to the skewing of the axes of the worm mill and the part. On the basis of the initial data, the first module forms transformation matrix М3 from the initial matrix МO in Eq. (1)

$$M3 = \left( {\begin{array}{*{20}{c}} {\cos (\varepsilon )}&0&{\sin (\varepsilon )}&0 \\ 0&1&0&{ - ({{r}_{{{\text{ic}}}}} + {{R}_{{{\text{im}}}}})} \\ { - \sin (\varepsilon )}&0&{\cos (\varepsilon )}&0 \\ 0&0&0&1 \end{array}} \right).$$
(4)

(4) Rotation around axis Z2 (n = 3) of coordinate system O3Х3Y3Z3 by an angle xy = α, which is a variable related to the coordinates x2i, y2i of the point considered, and consistent displacement along this axis by distance Az = pmα, where pm is the helical parameter of the mill. As an example, note that, for point \(S{\kern 1pt} '({{x}_{{2s{\kern 1pt} '}}},{{y}_{{2s{\kern 1pt} '}}})\), tan α = \({{{{x}_{{2s{\kern 1pt} '}}}} \mathord{\left/ {\vphantom {{{{x}_{{2s{\kern 1pt} '}}}} {{{y}_{{2s{\kern 1pt} '}}}}}} \right. \kern-0em} {{{y}_{{2s{\kern 1pt} '}}}}}.\)

According to the algorithm, the corresponding transformation matrix М4 takes the form

$$M4 = \left( {\begin{array}{*{20}{c}} {\cos (\alpha )}&{ - \sin (\alpha )}&0&0 \\ {\sin (\alpha )}&{\cos (\alpha )}&0&0 \\ 0&0&1&{{{p}_{{\text{m}}}}\alpha } \\ 0&0&0&1 \end{array}} \right).$$
(5)

The resulting numerical model MR is a column matrix of the form

$$MR = \left( {\begin{array}{*{20}{c}} {{{x}_{{3i}}}} \\ {{{y}_{{3i}}}} \\ {{{z}_{{3i}}}} \\ 1 \end{array}} \right) = M\Sigma \left( {\begin{array}{*{20}{c}} {{{x}_{i}}} \\ {{{y}_{i}}} \\ 0 \\ 1 \end{array}} \right),$$
(6)

where \(M\Sigma = M4 \cdot M3 \cdot M2 \cdot M1\).

Software has been written for the operations based on the transformation matrices in Eqs. (1)–(5). As a result of those operations, we obtain the numerical model in Eq. (6), where the coordinate y3i is the distance from the axis Z3 to the given point i after the transformations just itemized. The coordinate z3i determines the distance to the coordinate plane O3Х3Y3.

To determine the tool profile, we may use the following shaping algorithm.

(1) Specification of the rolling motion by angle φj in the range from +φmax to −φmax, in increments established on the basis of the required precision (mean value 0.01π).

(2) Specification of the number i of the point with coordinates xi, yi for each value of φj. In Fig. 1, i = 1, …, 5.

(3) Specification of the change in the angle ν in the range from +νmax to −νmax, for each angle φj and value of i. The value of ν depends on the profile of the part, the inclination of the helical channel, and the radius of the initial worm circumference, which is (0.1–0.2)π. In this case, points i = 1, …, S, …, 7 move to the positions i = 1', …, S ', …, 7'. The increment in ν is established on the basis of the required precision (mean value 0.01π).

(4) As a result of displacement of the point S to position S ', say, plane P0 passing through that point is moved to position P1. Correspondingly, the plane is turned by angle α and moved consistently from center О2 over a distance pmα to point О3.

As a result of these actions, when φj = φmax, we note the appearance of curves representing the intersection of the lines 1–1', 2–2', …, S–S ', …, 7–7' (Fig. 2) with plane Р1, which moves helically around axis Z3 with helical parameter pm.

Fig. 2.
figure 2

Graphical formulation of the numerical model of the initial surface with rolling parameter φj= φmax.

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In Fig. 3, we show all the curves for the complete range of angular variation φmin ≤ φj ≤ φmax. Software constructing the envelope of these curves permits determination of the desired worm-mill profile.

Fig. 3.
figure 3

Graphical formulation of the numerical model of the shaped worm mill.

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The proposed approach may be used for the shaping and analysis of worm mills with a protuberance [3]. It sets no constraints on the shape of the generatrix and directrix of the helical surface produced nor on the method used to specify the generatrix and directrix.

The proposed system may be used not only to plot curves and determine the desired profile but also to formulate mathematical expressions for that purpose on the basis of the initial matrix МO in Eq. (1) and the transformation matrices. By that means the profile of the worm mill may be determined in a single stage without determining the profile of the conjugate helical rack.