Study of the Geometry of Threads Machined by Plane Rolling
Abstract Generated abstract
This paper examines geometric deviations in helical threads produced by machining with a plane, including flat-die rolling, face milling, and grinding with the flat side of a wheel. Using flat-die thread rolling as the model case, it formulates the interaction between an assumed Archimedean helical surface and the calibrating plane of the tool, deriving expressions for axial deviation, contact conditions, and maximum groove widening. The analysis shows how profile distortion arises from the variation of lead angle over thread height and provides transformed equations using grouped dimensionless coefficients to simplify practical calculation. The resulting method reduces the tabulation needed for standard and nonstandard thread parameters and supports estimation of one-sided widening in precision thread-rolling rollers and related helical surfaces.
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STUDY OF THE GEOMETRY OF THREADS MACHINED BY PLANE ROLLING
P. A. EKATERINYUK
(Presented by the scientific seminar of the Department of Applied Mechanics)
The subgroup of threads produced by rolling with a plane includes three technological methods of machining helical surfaces: rolling the threads of fastening screws with flat dies, milling threads with the end face of a cutter, and grinding during the final machining of helical surfaces of lead screws, worms, and thread-rolling rollers with the flat side of a grinding wheel. If the deviations obtained in the process of rolling fastening threads are of no essential significance, since they fall within the tolerance zone, then the deviations of ground helical surfaces, commensurate with the small manufacturing tolerances, will be rather considerable.
It becomes necessary to determine the amount of widening of the groove for thread-rolling rollers ground with a flat wheel. In order to obtain good finish and high accuracy, the helical surface of the sizing section of the thread-rolling roller is ground. The lead-in cone is ground during periodic regrinding.
If the geometry of the helical surface of the lead-in cone has no effect on the accuracy of the thread of the part, but can only to some extent influence the magnitude and point of application of the forces of resistance of the metal to deformation, then the geometry of the sizing part will have a substantial influence on the accuracy of the rolled part.
The deviations of the sizing thread turns of the roller may be added to or subtracted from the deviations of the thread of the part. Preliminary investigations show that, in practical cases, the deviations will be subtracted:
\[ \delta = \delta_1 - \delta_2, \]
where \(\delta\) is the deviation of the thread of the part rolled by a roller with a ground thread of the sizing section;
\(\delta_1\) is the deviation of the part when rolled by a roller with an Archimedean helical surface;
\(\delta_2\) is the deviation obtained when grinding the sizing section of the thread of the thread-rolling roller.
Let us consider the geometry of a thread obtained by rolling with a plane, using as an example the technological method of machining by rolling with flat dies. The conclusions obtained may also be used for other technological methods of thread machining—grinding with the plane of a wheel and milling with the end face of a cutter.
In rolling a thread with flat dies, the distortion of the thread profile will be less than when rolling the same thread with knurled rollers. In a flat die, the lead angle of the crest does not change with height and will be constant. Distortion of the thread profile will occur owing to the change, over the height of the thread, in the lead angle of the helical line of the thread of the screw being rolled (Fig. 1).
Let us consider the geometry of the thread in a rectangular fixed coordinate system \(x, y, z\). Let us position the workpiece being rolled so that its axis coincides with the \(z\)-axis (Fig. 2), and so that the axial lines of the crests of the knurling flat die are located in the coordinate plane \(yoz\) and are parallel to the coordinate plane \(yox\), forming the angle \(\lambda\).
Fig. 1. Graph of the change in the tangent of the lead angle of the helical line as a function of the change in radius of the cylindrical section: \(ab\)—curve of the change in the tangent of the lead angle of the helical line; \(mn\)—for the plane.
We shall provisionally take, as the “base” surface of the screw, an Archimedean helical surface. On the basis of the construction (Fig. 2), the equation of an Archimedean helical surface of right-hand direction may be expressed as:
\[ \begin{aligned} x &= u \cos \alpha \cos \varphi;\\ y &= u \cos \alpha \sin \varphi;\\ z &= -u \sin \alpha + p\varphi \end{aligned} \tag{1} \]
The calibrating surface of the crest of the flat die is a plane. If the calibrating plane of the crest is extended, it will cut off segments on the coordinate axes \(x, y, z\), respectively \(a, b\), and \(c\). The equation of the plane in intercepts cut off on the coordinate axes is
\[ \frac{x}{a}+\frac{y}{b}+\frac{z_{1}}{c}=1. \tag{2} \]
From the construction (Fig. 2) it follows that
\[ a=\frac{c}{\operatorname{tg}\alpha};\qquad b=\frac{c}{\operatorname{tg}\lambda}; \tag{3} \]
where \(\alpha\) is the thread profile angle;
\(\lambda\) is the lead angle of the mean helical line, and in the flat die—the lead angle of the crest.
Let us express the segments \(a\) and \(b\) through the segment \(c\) and the tangents of the angles \(\alpha\) and \(\lambda\). Substituting the values of \(a\) and \(b\) from equation (3) into the equation of the plane (2), and, as is evident from the drawing (Fig. 2), \(b\) will have a negative value:
\[ x\operatorname{tg}\alpha-y\operatorname{tg}\lambda+z_{1}=c. \tag{4} \]
The numerical value of the segment \(c\) is determined from the third equation of system (1), taking the parameter \(u\) equal to zero; then
\[ z=p\varphi_0. \]
If it is assumed that the generatrix of the helical surface will coincide with the line of intersection of the coordinate plane \(zox\) and the specified plane
Fig. 2. Diagram of the interaction of the specified Archimedean helical surface with the sizing plane of a flat die (right-hand thread)
of the die comb, then \(z\) in this case will be the segment \(c\) cut off by the plane on this axis. Thus,
\[ c=p\varphi_0. \tag{5} \]
Substituting the value of \(c\) into equation (4), we obtain
\[ z_1=p\varphi_0-x\,\operatorname{tg}\alpha+y\,\operatorname{tg}\lambda. \tag{6} \]
The numerical value of the angle \(\varphi_0\) may be taken as
\[ \varphi_0=2\pi n, \tag{7} \]
where \(n\) may be an integer or a fractional number, but for convenience of calculation it may be assigned as an integer \(n=1,2,3\ldots\). Substituting the value of \(\varphi_0\) into equation (6), then
\[ z_1=2\pi n p-x\,\operatorname{tg}\alpha+y\,\operatorname{tg}\lambda. \tag{8} \]
Equation (5) is the condition for intersection along a line located in the coordinate plane \(zox\), of the sizing surface of the flat...
of the flat die, expressed by equation (8), and the helical surface of the workpiece, expressed by the system of equations (1).
In the present case it is more convenient to represent the angle of rotation of the generatrix as consisting of two parts: the constant \(2\pi n\) and the variable \(\varphi\). Then equation (1) may be written as follows:
\[ \begin{aligned} x &= u \cos \alpha \cos (2\pi n \pm \varphi);\\ y &= u \cos \alpha \sin (2\pi n \pm \varphi);\\ z &= -u \sin \alpha + p(2\pi n \pm \varphi). \end{aligned} \tag{9} \]
The value of the angle \(2\pi n\) for the helical surface expressed by the system of equations (9), and for the calibrating plane of the flat die expressed by equation (8), is taken to be one and the same, while the angle \(\varphi\) may be negative or positive. For a positive value of the angle \(\varphi\),
\[ \begin{aligned} x &= u \cos \alpha \cos \varphi;\\ y &= u \cos \alpha \sin \varphi;\\ z &= -u \sin \alpha + p(2\pi n + \varphi). \end{aligned} \tag{10} \]
The segments \(\Delta z\), measured in the direction of the \(z\)-axis and bounded by two intersecting surfaces—the specified “basic” Archimedean helical surface and the calibrating plane of the crest of the die—may be determined as the difference between the values \(z\) and \(z_1\) of equations (10) and (8):
\[ \Delta z = z - z_1 = -u \sin \alpha + p(2\pi n + \varphi) - 2\pi n p + x \operatorname{tg}\alpha - y \operatorname{tg}\lambda . \tag{11} \]
If in equation (11) we substitute the value of \(x\) from the first equation, and \(y\) from the second equation of system (10), we obtain
\[ \Delta z = -u \sin \alpha + u \sin \alpha \cos \varphi - u \cos \alpha \operatorname{tg}\lambda \sin \varphi - p\varphi . \tag{12} \]
The numerical value of \(\Delta z\) may be positive or negative. A positive value of \(\Delta z\) will correspond to penetration of the plane of the crest into the body of the specified “basic” helical surface; a negative value of \(\Delta z\) will denote a clearance between the specified helical surface and the plane of the crest.
Let us take the partial derivative of the deviations \(\Delta z\) with respect to the angle \(\varphi\), assigning \(u\) a constant value, and set it equal to zero.
\[ \frac{\partial \Delta z}{\partial \varphi} = -u \cos \alpha \operatorname{tg}\lambda \cos \varphi - u \sin \alpha \sin \varphi + p = 0. \tag{13} \]
By setting the partial derivative \(\dfrac{\partial \Delta z}{\partial \varphi}\) equal to zero, we have obtained the equation of the maximum deviations \(\Delta z_{\max}\). Equation (13) is also the equation of the projection onto the coordinate plane \(xoy\) of the line of contact of the conjugate helical surface with the plane of the crest of the flat die.
In equation (13) there are two variable quantities, \(u\) and \(\varphi\); by taking a numerical value of one of the variables, we can determine the other. Substituting the found values of \(u\) and \(\varphi\) into equation (12), we find a series of maximum deviations \(\Delta z_{\max}\). Let us replace \(\cos \varphi\) in equation (13) by \(\sin \varphi\) and reduce the equation to canonical form:
\[ u^2 \cos^2 \alpha \left(\operatorname{tg}^2 \lambda + \operatorname{tg}^2 \alpha\right)\sin^2 \varphi - 2pu \sin \alpha \sin \varphi - u^2 \cos^2 \alpha \operatorname{tg}^2 \lambda + p^2 = 0. \tag{14} \]
Solving the quadratic equation (18), for a given \(u\) one can determine the cosine and the angle \(\varphi\). Equations (12), (13), (14) can be somewhat simplified if the variable \(u\) is replaced by \(r\), where \(r\) is the radius vector.
From the drawing (Fig. 2),
\[ r = u \cos \alpha; \qquad \operatorname{tg}\alpha = u \sin \alpha . \tag{15} \]
Substituting the values from equations (15) into equation (14), after substitution and transformation we obtain
\[ r^2(\tg^2\lambda+\tg^2\alpha)\sin^2\varphi-2pr\tg\alpha\sin\varphi-r^2\tg^2\lambda+p^2=0. \tag{16} \]
Solving the quadratic equation, we obtain
\[ \sin\varphi= \frac{p\tg\alpha-\tg\lambda\sqrt{r^2(\tg^2\alpha+\tg^2\lambda)-p^2}} {r(\tg^2\alpha+\tg^2\lambda)}. \tag{17} \]
Having chosen the value of \(r\) and substituting it into formula (17), we determine the value of the angle \(\varphi\). The study can also be carried out in another way. We transform equations (12) and (13), replacing \(u\) by \(r\) from equations (15). After substituting the value of \(u\) into equations (12) and (13),
\[ \Delta z=-r\tg\alpha(1-\cos\varphi)-(r\tg\lambda\sin\varphi-p\varphi); \tag{18} \]
\[ r=\frac{p}{\tg\lambda\cos\varphi+\tg\alpha\sin\varphi}. \tag{19} \]
Equations (18) and (19) may be used in determining the location and size of the contact patches arising on the lead-in part of the die during rolling and depending on the magnitude of the feed, the lift angle \(\lambda\), and the parameter \(p\).
The simultaneous solution of equations (18) and (19) as a system, and the elimination of the variable \(\varphi\) and its functions, gives the equation of the generatrix of the conjugate helical surface in deviations from the rectilinear generatrix \(\Delta z\).
For rapid performance of calculations, one may use tables prepared in advance. In order to cover all standard sizes of threads provided for by GOST standards, as well as nonstandard threads of the lead-in cones of rolling dies, it is necessary, with constant \(\alpha\) and \(p\), to assign various values and regard them as variables.
Equation (17) contains four variable quantities: \(p\), \(\alpha\), \(\lambda\), and \(r\) as arguments, and a fifth, \(\sin\varphi\), as the function. The second equation (18) contains six variable quantities: \(\Delta z\), \(\alpha\), \(\lambda\), \(\varphi\), \(p\), and \(r\).
With such a number of variables it is difficult to compile tables that would cover all parameters of threads used in practice. Such a table would be too bulky and would contain about a million values, many of which would either be repeated or be very close to one another.
Therefore it became necessary to group the variables and replace them by “group coefficients.”
Let us divide all terms of the numerator and denominator of the right-hand side of equation (17) by the common divisor \(r\tg^2\alpha\) and set
\[ \tg\lambda=\frac{p}{r_0}. \tag{20} \]
After substitution we obtain
\[ \sin\varphi= \frac{ \dfrac{p}{r\tg\alpha} -\dfrac{p}{r_0\tg\alpha} \sqrt{ \dfrac{p^2}{r_0^2\tg^2\alpha} +1 -\dfrac{p^2}{r^2\tg^2\alpha} } }{ 1+\dfrac{p^2}{r_0^2\tg^2\alpha} }. \tag{21} \]
Let us set
\[ q_0=\frac{p}{r_0\tg\alpha}, \tag{22} \]
\[ q=\frac{p}{r\tg\alpha}, \tag{23} \]
where \(r_0\) is the radius of the cylinder, the lead angle of whose helical line is equal to the lead angle of the plane of the die; \(r\) is the current radius vector; \(p\) is the magnitude of the displacement in the axial direction when the screw is rotated through an angle equal to one radian; \(\alpha\) is the profile angle; \(q\) and \(q_0\) are group coefficients.
Introduce into equation (21) the group coefficients \(q\) and \(q_0\) from relations (22) and (23)
\[ \sin \varphi = \frac{q - q_0 \sqrt{q_0^2 - q^2 + 1}}{1 + q_0^2}. \tag{24} \]
Set
\[ K = \frac{q_0}{q}. \tag{25} \]
After substituting the values of \(q\) and \(q_0\) from relations (22) and (23) into relation (25) and reducing,
\[ K = \frac{r}{r_0}. \tag{26} \]
Find the value from relation (25) and substitute it into equation (24).
\[ \sin \varphi = \frac{q_0 \left[1 - \sqrt{q_0^2 (K^2 - 1) + K^2}\right]} {K(1 + q_0^2)}. \tag{27} \]
In order to reduce the number of variables, transform equation (18). Divide both sides of the equation by \(r \tan \alpha\) and introduce, in place of \(\tan \lambda\), its value from relation (20)
\[ \frac{\Delta z}{r \tan \alpha} = - (1 - \cos \varphi) - \frac{p}{r_0 \tan \alpha} \sin \varphi + \frac{p}{r \tan \alpha}\,\varphi . \]
Replace the coefficients standing before \(\sin \varphi\) and the value of the angle \(\varphi\) by \(q_0\) and \(q\) from relations (22) and (23), and set
\[ \Delta z_1 = \frac{\Delta z}{r \tan \alpha}. \tag{28} \]
Then
\[ \Delta z_1 = - (1 - \cos \varphi) - q_0 \sin \varphi + q\varphi . \tag{29} \]
Substitute into equation (29), in place of \(q\), its value from relation (25)
\[ \Delta z_1 = - (1 - \cos \varphi) + q_0 \left(\frac{\varphi}{K} - \sin \varphi\right). \tag{30} \]
Solving equations (27) and (30) as a system and assigning values to \(q_0\) and \(K\), we find the value of \(\varphi\). After substituting the value of \(\varphi\) into equation (30), we find the specific amount of widening during rolling of the helical surface by a plane, \(\Delta z\). To find the full amount of one-sided widening, \(\Delta z\), it is necessary to use relation (28), according to which
\[ \Delta z = \Delta z_1 r \tan \alpha . \tag{31} \]
The calculated data can be collected in a table, since the number of values does not exceed fifteen hundred.
REFERENCES
- N. I. Kolchin, F. L. Litvin. Methods of calculation in the manufacture and inspection of toothed products. Mashgiz, 1952.
- F. L. Litvin. New types of cylindrical worm gears. Mashgiz, 1962.