STUDY OF THE GEOMETRY OF THREADS ROLLED BY ROLLERS WITH AN ARCHIMEDEAN HELICAL SURFACE

P. A. EKATERINYUK

(Presented by the scientific seminar of the Department of Applied Mechanics)

In the manufacture of threaded products by rolling with thread-rolling rollers, the geometry of the helical surface of the product differs from the geometry of the helical surface of the rolling roller even in the case when the screw and the roller have not only the same pitch and number of starts, but also the same initial thread diameters. The magnitude of the deviation at the given points of the mating helical surfaces from the preliminarily specified ones will be affected by the difference between the helix angles of the helical lines of the product and of the rolling roller, \(\Delta\lambda\), the thread profile angle \(\alpha\), and other factors.

\[ \Delta\lambda = \lambda_b - \lambda_p, \]

where \(\lambda_b\) and \(\lambda_p\) are, respectively, the helix angles of the helical line of the product and of the tool, whose tangent is determined by the formula

\[ \operatorname{tg}\lambda = \frac{t}{\pi d}, \tag{1} \]

\[ \lambda = \operatorname{arctg}\frac{t}{\pi d}, \tag{2} \]

where \(\lambda\) is the helix angle of the helical line;
\(t\) is the pitch in mm;
\(d\) is the diameter in mm.

From formulas (1) and (2) it is evident that the tangent of the helix angle of the helical line, \(\operatorname{tg}\lambda\), will vary inversely proportionally to the diameter \(d\). The helical line at the root of the screw thread has the maximum helix angle \(\lambda_{\max}\), while the helical line at the crest of the screw thread has the minimum helix angle \(\lambda_{\min}\).

When rolling with rollers, the elements of the helical surface located at the crest of the roller thread turn, with the minimum helix angles of the helical line, mate with the elements of the helical surface located at the root of the screw being rolled, which have maximum helix angles; and, conversely, the elements of the helical surface located at the root of the roller thread turn, which have maximum helix angles of the helical lines, during rolling mate with the elements located at the crest of the screw thread, with minimum helix angles of the helical lines. Only the helix angles of the initial lines of the roller and the screw coincide in magnitude.

A discrepancy in the magnitudes of the lead angles of the helical lines (Fig. 1) is the reason why the geometry of the helical surface of the product will not correspond to the initial geometry of the helical surface of the rolling roller; the thread root will be somewhat spread out; the contact areas of the deforming metal, located on both sides of the thread being formed, may be somewhat displaced along the axis of the thread and will not coincide with one another, which produces bending stresses dangerous to the strength of the roller thread.

Fig. 1. Diagram of the change in the lead angles of the helical lines in the screw and the roller as a function of the ratio of their radii

The discrepancy between the lead angles of the helical lines of the roller and of the screw being rolled may be the reason why unrounded edges at the crest of the thread, at small angles of the thread profile, can produce shaving, while the thin chip cut off is partially pressed into the surface of the thread being formed and worsens its finish.

In order to ensure axial feed of the screw being rolled, the roller diameter is taken larger or smaller than the diameter of the screw being rolled, while maintaining the same pitch value for the product and the roller. Therefore the lead angles of the product and the roller differ even more from one another (Fig. 1), and the deviation of the geometry of the conjugate helical surface from the specified one will be more significant.

The magnitude of the deviations, the shape of the generatrix of the helical surface, and also the size, shape, and location of the contact areas can be determined if the equations are derived using the method we have developed.

In a rectangular coordinate system associated with fixed space, let us specify the Archimedean helical surfaces of the rolling roller and of the product being rolled. The specified Archimedean helical surfaces of the rolling roller and the product will not be two conjugate surfaces; they will intersect as a result of the variation of the lead angles over the height of the threads of the rolling roller and of the product threads.

In studying the geometry of helical surfaces obtained by rolling, two problems may be posed.

First problem. Given the geometry of the helical surface of the rolling roller, find the geometry of the product conjugate to it, obtained as a result of rolling during thread rolling.

Second inverse problem. Given the geometry of the helical surface of the product, find the conjugate geometry of the helical surface of the rolling roller.

The system of equations of the Archimedean helical surface of the right-hand rolled screw, expressed in a fixed system of rectangular coordinates, may be represented in the following form:

\[ \begin{aligned} x &= u \cos \alpha \cos \varphi;\\ y &= u \cos \alpha \sin \varphi;\\ z &= A \tg \alpha - u \sin \alpha + p\varphi. \end{aligned} \tag{3} \]

The equations of the Archimedean helical surface of the rolling roller with a left-hand thread in the same fixed coordinate system are:

\[ \begin{aligned} x &= A - u_1 \cos \alpha \sin \varphi_1;\\ y &= u_1 \cos \alpha \sin \varphi_1;\\ z_1 &= u_1 \sin \alpha + p\varphi_1; \end{aligned} \tag{4} \]

where \(A\) is the center distance;
\(u\) and \(u_1\) are the lengths of the generatrices of the helical surfaces of the screw and the roller;
\(\alpha\) is the thread profile angle;
\(\varphi\) and \(\varphi_1\) are the angles of rotation of the generatrices of the helical surface of the product and of the helical surface of the rolling roller;
\(p\) is the axial displacement of the screw when rotated through an angle equal to 1 radian.

With this expression of the system of equations of the helical surfaces, in the axial section the generatrices of both helical surfaces will coincide along their entire length. To the right and to the left of the axial section, the helical surfaces will either intersect, with one helical surface cutting into the other, or will move away from each other, forming gaps between them. There will be no intersections along the mean helical lines, whose lead angles are equal.

The deviation from the Archimedean helical surface may be represented by the equation

\[ \Delta z = z - z_1 = A \tg \alpha - u \sin \alpha + p\varphi - u_1 \sin \alpha + p\varphi_1. \tag{5} \]

Equation (5) is obtained as the difference of the third dependences of equations (3) and (4).

The right-hand side of the equation contains four variables: \(\varphi, u, \varphi_1\), and \(u_1\). Let us eliminate the superfluous unknowns.

Take an arbitrary point \(M\) on the helical surface and project it onto a plane perpendicular to the axial lines of the product and the roller. On the projection plane, the projections of the generatrices of the helical surfaces of the product \(u\) and of the roller \(u_1\), and the center-distance line \(OO_1\), form an oblique triangle \(OMO_1\).

From the oblique triangle (Fig. 2),

\[ u \cos \alpha \cos \varphi + u_1 \cos \alpha \cos \varphi_1 = A. \tag{6} \]

Equating the right-hand sides of the second dependence of systems (3) and (4), we find

\[ u_1 = \frac{u \sin \varphi}{\sin \varphi_1}. \tag{7} \]

Substituting into equation (6) the value of \(u_1\) from equation (7), we obtain

\[ \tg \varphi_1=\frac{u\cos\alpha\sin\varphi}{A-u\cos\alpha\cos\varphi}; \tag{8} \]

\[ \varphi_1=\arctg \frac{u\cos\alpha\sin\varphi}{A-u\cos\alpha\cos\varphi}. \tag{9} \]

Fig. 2. Diagram of the skewing of the generatrix of an Archimedean helical surface and the generatrix of the helical surface of the rolling roller

From the oblique triangle (Fig. 2), by the law of cosines,

\[ u_1=\frac{\sqrt{A^2+u^2\cos^2\alpha-2Au\cos\alpha\cos\varphi}}{\cos\alpha}. \tag{10} \]

Substitute the values of relations (9) and (10) into equation (5):

\[ \Delta z=A\tg\alpha-u\sin\alpha+p\varphi-\sqrt{A^2+u^2\cos^2\alpha-2Au\cos\alpha\cos\varphi}\tg\alpha- \]

\[ -p\arctg \frac{u\cos\alpha\sin\varphi}{A-u\cos\alpha\cos\varphi}. \tag{11} \]

We have obtained the equation of deviations along the \(z\)-axis from the specified Archimedean helical surface.

Using equation (11), one can find the deviations \(\Delta z\), but in order to study the geometry of the helical surface it is necessary to take only the maximum deviations, which will determine the generatrix of the helical surface. To determine the maximum value of the deviation \(\Delta z_{\max}\), we take the partial derivative \(\dfrac{\partial \Delta z}{\partial \varphi}\) and set it equal to zero:

\[ \frac{\partial \Delta z}{\partial\varphi} = p- \frac{Au\sin\alpha\sin\varphi} {\sqrt{A^2+u^2\cos^2\alpha-2Au\cos\alpha\cos\varphi}} - \]

\[ - \frac{pAu\cos\alpha\cos\varphi-pu^2\cos^2\alpha} {A^2+u^2\cos^2\alpha-2Au\cos\alpha\cos\varphi} =0. \tag{12} \]

Let us carry out a further transformation:

\[ p\left(A^{2}-3Au\cos\alpha\cos\varphi+2u^{2}\cos^{2}\alpha\right)= Au\sin\alpha\sin\varphi\sqrt{A^{2}-2Au\cos\alpha\cos\varphi+u^{2}\cos^{2}\alpha}. \tag{13} \]

Squaring both sides of the equation, replacing \(\sin\varphi\) by \(\cos\varphi\), and reducing to canonical form, we obtain:

\[ \begin{aligned} &2A^{3}u^{3}\sin^{2}\alpha\cos\alpha\cos^{3}\varphi -A^{2}u^{2}\left(A^{2}\sin^{2}\alpha+u^{2}\sin^{2}\alpha\cos^{2}\alpha+\right.\\ &\left.\quad +9p^{2}\cos^{2}\alpha\right)\cos^{2}\varphi +2Au\left(6p^{2}u^{2}\cos^{3}\alpha+3p^{2}A^{2}\cos\alpha-\right.\\ &\left.\quad -A^{2}u^{2}\sin^{2}\alpha\cos\alpha\right)\cos\varphi +A^{2}u^{2}\sin^{2}\alpha\left(A^{2}+u^{2}\cos^{2}\alpha\right)-\\ &\quad -p^{2}\left(A^{2}+2u^{2}\cos^{2}\alpha\right)^{2}=0. \end{aligned} \tag{14} \]

For more convenient use in equations (11) and (14), let us make a substitution, using relation (15),

\[ \begin{aligned} u\cos\alpha&=r;\\ u\sin\alpha&=r\tg\alpha; \end{aligned} \tag{15} \]

where \(r\) is the variable magnitude of the radius vector of the helical surface of the workpiece.

After substituting expressions (15) into equations (11) and (14), we obtain

\[ \Delta z= \left(A-r-\sqrt{A^{2}+r^{2}-2Ar\cos\varphi}\right)\tg\alpha +p\varphi -p\arctg\frac{r\sin\varphi}{A-r\cos\varphi}. \tag{16} \]

\[ \begin{aligned} &2A^{3}r^{3}\tg^{2}\alpha\cos^{3}\varphi -A^{2}r^{2}\left(A^{2}\tg^{2}\alpha+r^{2}\tg^{2}\alpha+9p^{2}\right)\cos^{2}\varphi+\\ &\quad +2Ar\left(6p^{2}r^{2}+3p^{2}A^{2}-A^{2}r^{2}\tg^{2}\alpha\right)\cos\varphi -p^{2}\left(A^{2}+2r^{2}\right)^{2}+\\ &\quad +A^{2}r^{2}\tg^{2}\alpha\left(A^{2}+r^{2}\right)=0. \end{aligned} \tag{17} \]

The simultaneous solution of equations (16) and (17) makes it possible to determine the deviations from the specified Archimedean helical surface \(\Delta z_{\max}\). Relation (17) is the equation of the contact line of two conjugate helical surfaces, projected onto the coordinate plane \(xoy\). The spatial curve of the contact line is expressed here as the plane curve of its projection.

Equation (17) contains two arbitrary variables—\(r\) and \(\varphi\). By assigning one arbitrary variable and solving the equation, we can determine the other. Substituting the found values of \(r\) and \(\varphi\) into equation (16), we determine the deviation from the specified helical surface \(\Delta z\). Thus, the helical surface specified in advance serves, as it were, as a kind of curvilinear coordinate surface, from which the segments \(\Delta z\) are laid off.

Each of the obtained maximum deviations belongs to its own generator. In order to determine the form of the generator of the conjugate helical surface, it is necessary to lay off all the obtained maximum deviations \(\Delta z_{\max}\) from one generator at the corresponding distance from the axis \(r=u\Cos\alpha\). Here we are, as it were, combining the fan of generators into one by a rotational motion with its center on the screw axis.

If the deviation \(\Delta z\) is set equal to zero, then equation (16) will be the equation of the line of intersection of the helical surfaces of the tool and the specified “base” surface of the workpiece:

\[ \left(A-r-\sqrt{A^{2}+r^{2}-2Ar\cos\varphi}\right)\tg\alpha +p\varphi -p\arctg\frac{r\sin\varphi}{A-r\cos\varphi}=0. \tag{18} \]

The large number of variables makes it difficult to create calculation tables by which ready-made solutions can be found quickly. Such tab-

the faces will have more than a million values, which will be repeated.

To eliminate these shortcomings, we introduce group coefficients that make it possible to reduce the number of variables. Let us divide all terms of equation (16) by the common divisor \(A \tg \alpha\), and the numerator and denominator of the arctangent by \(A\); then

\[ \frac{\Delta z}{A \tg \alpha} = -\sqrt{1-2\frac{r}{A}\cos\varphi+\frac{r^{2}}{A^{2}}} +1-\frac{r}{A} - \frac{p}{A \tg \alpha}\arctg \frac{\frac{r}{A}\sin\varphi}{1-\frac{r}{A}\cos\varphi} + \frac{p}{A \tg \alpha}\cdot \varphi . \tag{19} \]

Let us denote by \(K\) and \(q\) the “group” coefficients:

\[ K=\frac{r}{A}, \tag{20} \]

\[ q=\frac{p}{A \tg \alpha}, \tag{21} \]

\[ \Delta z_{1}=\frac{\Delta z}{A \tg \alpha}. \tag{22} \]

After substituting the values (20), (21), and (22) into equation (19),

\[ \Delta z_{1} = -\sqrt{1-2K\cos\varphi+K^{2}} +1-K -q\left( \arctg \frac{K\sin\varphi}{1-K\cos\varphi} -\varphi \right). \tag{23} \]

Let us divide all terms of equation (17) by \(A^{6}\tg^{2}\alpha\):

\[ 2\frac{r^{3}}{A^{3}}\cos^{3}\varphi - \frac{r^{2}}{A^{2}} \left( 1+\frac{r^{2}}{A^{2}}+9\frac{p^{2}}{A^{2}\tg^{2}\alpha} \right) \cos^{2}\varphi + 2\frac{r}{A} \left( 6\frac{p^{2}}{A^{2}\tg^{2}\alpha}\cdot \frac{r^{2}}{A^{2}} + 3\frac{p^{2}}{A^{2}\tg^{2}\alpha} - \frac{r^{2}}{A^{2}} \right) \cos\varphi - \frac{p^{2}}{A^{2}\tg^{2}\alpha} \left( 1+2\frac{r^{2}}{A^{2}} \right)^{2} + \frac{r^{2}}{A^{2}} \left( 1+\frac{r^{2}}{A^{2}} \right) =0 . \tag{24} \]

Substitute from relations (20) and (21) the values of \(K\) and \(q\) into equation (24):

\[ 2K^{3}\cos^{3}\varphi - K^{2}(1+K^{2}+9q^{2})\cos^{2}\varphi + 2K(6q^{2}K^{2}+3q^{2}-K^{2})\cos\varphi - q^{2}(1+2K^{2})^{2} + K^{2}(1+K^{2}) =0 . \tag{25} \]

Using equations (23) and (25), the results of the calculation can easily be reduced to tables. The tables will have a small number of values, since they are compiled for two variables—the coefficients \(q\) and \(K\). We shall call coefficient \(K\) the size coefficient. In fact, \(K\) is the radius of the cylindrical section of the screw–roller system in which the distance between the axes of the screw and roller is equal to unity. \(\Delta z_{1}\) is the deviation in a system with unit center distance. We shall call coefficient \(q\) the screw-shape coefficient. \(q\) takes account of changes in the geometry of the system simultaneously in terms of 3 parameters. From relation (21) it is evident that coefficient \(q\) depends on \(p\), which characterizes the magnitude of the helix angle of the mean helical line; on the tangent of the profile angle \(\alpha\); and on the center distance \(A\).

To determine the deviation \(\Delta z\) from the tabular data, it is necessary to multiply the tabular value \(\Delta z_{1}\) by \(A \tg \alpha\), since from relation (22)

\[ \Delta z=\Delta z_{1}A \tg \alpha . \tag{26} \]