GRAPHICAL METHOD FOR THE STUDY OF HELICAL SURFACES

P. A. EKATERINYUK

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

The method we have developed for the graphical study is intended for determining the magnitude of the opening of the helical groove of a rolled product, determining the form of the generatrix of a helical surface, and the magnitude and location of the contact areas. By graphical means it is possible to establish how the magnitude of the contact areas and their location change as a function of changes in the feed value and in the profile angle of the entry cone.

By applying the graphical method of calculation, one can prescribe in advance for the designed rolling mechanism such operating conditions as will ensure good accuracy of the product, sufficient strength of the tool, and high labor productivity.

In the analytical study we derived the dependence (1) of the variation of the segments cut off by the working surface of the machining tool and by the prescribed Archimedean helical surface of the product.

\[ \Delta z=(A-r-\sqrt{A^{2}-2Ar\cos\varphi+r^{2}})\operatorname{tg}\alpha+p\varphi-p\operatorname{arctg}\frac{r\sin\varphi}{A-r\cos\varphi}, \tag{1} \]

where \(A\) is the center distance of the screw–rolling-tool system;
\(r\) is the radius;
\(\alpha\) is the thread profile angle;
\(p\) is the axial displacement of the screw when rotated through an angle equal to one radian;
\(\varphi\) is the angle of rotation of the generatrix.

Equation (1) decomposes into a number of independent dependences. If the parameter of the helical surface is taken equal to zero, then dependence (1) becomes the equation of the conical surface of the tool, expressed in cylindrical coordinates,

\[ z_{1}=(A-r-\sqrt{A^{2}-2Ar\cos\varphi+r^{2}})\operatorname{tg}\alpha. \tag{2} \]

The axis of the cylindrical coordinate system is made to coincide with the axis of the product.

Suppose that the profile angle of the helical surface is equal to zero; then equation (1) becomes the equation of the segments cut off by two rectangular helical surfaces: the prescribed rectangular one,

helical surface of the workpiece and the rectangular helical surface of the tool,

\[ \Delta z = p\varphi - p \operatorname{arctg} \frac{r \sin \varphi}{A - r \cos \varphi}. \tag{3} \]

Using relation (3), one can study a thread with ribbon-like turns. Relation (3), in turn, decomposes into the equation of the helical line of the workpiece

\[ z_2 = p\varphi \tag{4} \]

and the equation of the helical line of the thread of the tool in cylindrical coordinates with the \(z\)-axis concentric with the axis of the workpiece,

\[ z_3 = p \operatorname{arctg} \frac{r \sin \varphi}{A - r \cos \varphi}. \tag{5} \]

Thus, analysis of equation (1) shows that it decomposes into three independent relations (2), (4), and (5), the values of which can be computed on electronic machines and entered into tables.

Relations (2) and (5) together give the equation of the Archimedean helical surface of the rolling tool

\[ z = \left(A - r - \sqrt{A^2 - 2Ar \cos \varphi + r^2}\right)\tg \chi - p \operatorname{arctg} \frac{r \sin \varphi}{A - r \cos \varphi}. \tag{6} \]

The graphical construction is carried out in a rectangular coordinate system. Along the vertical axis, segments are laid off in the selected scale \(M_z\). Along the horizontal axis of the coordinate system, the angle \(\varphi\) is laid off in radians or degrees in the specified scale \(M_\varphi\).

In a graphical study, a number of sections are made by cylindrical surfaces concentric with the axis of the screw being machined. The section includes the helical surface of the workpiece and the machining tool conjugate with it. In a graphical study, the cylindrical sections are developed onto a plane, and the construction is performed in the selected coordinate system \(z\)—\(\varphi\). In the construction, numerical data previously computed on an electronic machine and entered into tables may be used.

By the method we have developed, it is possible to investigate and carry out a graphical calculation of a helical surface conjugate with a machining tool, of any geometry used in practice and for any arrangement of the axes.

The construction in the graphical study of the helical surface is performed as follows. The development of the curve onto the plane of the cylindrical section of the tool is constructed. In Fig. 1 this curve is denoted \(C_2PD_2\). The helical line of the section of the machined helical surface by a concentrically located cylinder is developed onto the plane in the form of a straight line. In the drawing (Fig. 1) these lines are denoted \(A_0B_0\), \(AB\), \(A_1B_1\), and \(A_2B_2\).

The curve of the tool section \(C_2PD_2\) is constructed point by point. The segments \(z'_1, z''_1, \ldots, z^{(n)}_1\), calculated by one of the formulas, are laid off on the graph from the horizontal axis at the points \(\varphi'\), \(\varphi''\), \(\ldots \varphi^{(n)}\). The upper points of the segments are connected by a smooth curve.

The position of the development of the helical line of the section of the workpiece thread \(AB\) (Fig. 1) is determined by two points. The first point is the origin of coordinates (point \(p\)); the second point is the top of the segment (point \(D\)).

\[ z = \frac{p\varphi}{\omega_z}, \tag{7} \]

where \(p\) is the parameter of the helical surface,
\(\varphi\) is an arbitrarily assigned angle in radians,
\(M_z\) is the adopted scale.

The segment \(z\) is laid off perpendicular to the horizontal axis from the selected point \(\varphi\).

Parallel to the straight line \(AB\), the tangent \(A_0B_0\) is drawn to the curve of the tool section \(C_2PD_2\). The straight line \(AB\) is the development onto the plane

Fig. 1. Construction used in the graphical calculation of the geometry of the threads and the contact areas of the threads of a rolling tool with the threads of the screw being rolled

of the helical line of the section of a previously specified Archimedean helical surface. The straight line \(A_0B_0\) is the development of the helical line of the helical surface conjugate with the tool. The segment \(PA_0^1\), cut off by the straight lines \(AB\) and \(A_0B_0\) on the \(z\)-axis, is the value of the one-sided groove clearance \(\Delta z\). The point of tangency of the straight line \(A_0B_0\) with the curve \(C_2PD_0 - C_0\) is the point of contact of the two conjugate surfaces; the coordinates of the point are determined by the section radius \(r\) and the angle \(\varphi\).

By graphical means one can quickly determine the axial feed—\(S_0\). Since a helical line develops onto a plane as a straight line, in order to plot on the graph (Fig. 1) the trace of the intersection by a cylinder of the helical surface of the thread of the \(i\)-th pass, it is necessary, on the drawing, to draw the line \(A_1B_1\) parallel to the line \(A_0B_0\). The distance between the lines \(A_0B_0\) and \(A_1B_1\), measured in the direction of the \(z\)-axis, will be the feed—\(S_0\). The line \(A_0B_0\) is the development of the helical line of pass \(i+1\).

The line \(A_0C_0D_1B_1\) is the development onto a plane of the section by a cylinder of the transition portion of the helical surface. The line \(A_0C_0D_1B_1\) consists of three portions. The portion \(A_0C_0\) is the development of the helical line after pass \(i+1\). The curvilinear portion \(C_0D_1\) is common to the tool and the workpiece: in this portion plastic deformation of the workpiece metal takes place. The portion \(D_1B_1\) is the development of the helical line after the \(i\)-th pass. \(C_0D_1D_1^1\) is the zone of metal deformation; \(D_1D_1^1\) is the feed.

\[ S_0 = D_1D_1^1 \]

Along the horizontal axis, the segment bounded by the points \(\varphi_0\varphi_1\) will be the angle of contact; the arc of contact \(l_k'\) is found if the angle of contact is multiplied by the radius \(r\):

\[ l_k' = r(\varphi_1-\varphi_0)+l. \tag{8} \]

The deformation zone on the second side of the groove is shown at the left (Fig. 1) and is denoted \(C_0C_1C_1^1\). The angle of contact is bounded by the points \(\varphi_0\varphi_1^1\); the arc of contact is

\[ l_k'' = r(\varphi_1^1+\varphi_0)+l, \tag{9} \]

where \(l\) is the increase in the width of the contact area due to the formation of a bulge and the elastic deformation of the metal. When the feed is increased, it is necessary on the graph to draw a new straight line \(A_2B_2\) at a distance from \(A_0B_0\) equal to \(S_0^1\). Then the deformation zones will respectively be \(C_0D_2D_2^1\) and \(C_0C_2C_2^1\); the arcs of contact will also increase and will be

\[ l_{2k}' = r(\varphi_2' + \varphi_0')+l, \tag{10} \]

\[ l_{2k}'' = r(\varphi_2''+\varphi_0'')+l. \tag{11} \]

In order to investigate the geometry of the helical surface by the graphical method, it is necessary to take 3–4 sections by cylinders of different radii. In Fig. 2 two sections are shown.

Let us find the dimensions of the contact area. On the drawing (Fig. 3) we draw, from the center \(O_1\), circles with radii \(r_1\) and \(r_2\). From this same center \(O_1\) we draw rays making, with the line of centers, the angles \(\varphi_0'\) and \(\varphi_0''\), to their intersection with the circles of radii \(z_1\) and \(z_2\). We denote the points of intersection respectively by \(a_1\) and \(a_2\). Next we draw a ray at the angle \(\varphi_1^1\) to its intersection with the circle \(r_1\)—we denote the point of intersection by \(b_1\); we draw a ray at the angle \(\varphi_1''\) to its intersection with the circle of radius \(r_2\)—we denote the point of intersection by \(b_2\). Through the points obtained \(a_1, a_2 \ldots a_n, b_1, b_2 \ldots b_n\) we draw smooth curves, which will bound the contact area on two sides. On the other two sides the contact area is bounded by the circles of the crests of the roller and of the screw being worked.

The dimensions and shape of the contact area located on the second side of the tool thread (Fig. 4) are determined in the same way as in the first case. The only difference is that the angles \(\varphi_0'\) and \(\varphi_0''\) are laid off on the left side of the axial line \(O_1O_2\), and the angles \(\varphi_2'\) and \(\varphi_2''\) on the right side of the axial line. The line of centers passes through the contact area.

Determining the dimensions and positions of the contact areas makes it possible to determine the magnitudes of the forces acting on the threads of the knurling roller.

Fig. 2. Construction for determining the contact areas on the left and right sides of a turn

Screw

Roller

Fig. 3. Graphical determination of the dimensions and shape of the contact patch

Fig. 4. Graphical determination of the dimensions and shape of the contact patch on the concave side of the thread

Conclusions

1. The graphical method for studying threads can be used in the rolling processing of large lead screws and power screws, blanks for worm cutters, and worms.

2. By graphical means it is possible to establish how the sizes of the contact patches and their position relative to the section by the axial plane of the workpiece–tool system change as a function of changes in the feed rate and in the profile angle of the thread of the lead-in cone.

3. The graphical method for studying helical surfaces makes it possible to determine the amount of widening of the helical groove of the workpiece being machined and the shape of the generator of the helical surface.

4. Using the graphical method of calculation, one can assign in advance to the designed rolling mechanism such parameters and operating conditions as will ensure good accuracy of the workpiece, sufficient tool strength, and high labor productivity.