Justification for Selecting a Machining Scheme for Chip Flutes of Disk Relieved Milling Cutters
Abstract Generated abstract
This paper compares two machining schemes for milling chip flutes in disk relieved cutters, focusing on accumulated circular pitch error that can later increase cutting-edge runout during relieving. It analyzes the effects of system rigidity and skew between the arbor axis and the dividing-head spindle axis, deriving relationships between spindle rotation and workpiece rotation and a formula for the greatest accumulated pitch error over one revolution. The argument indicates that mounting the arbor in the taper of the dividing head provides higher rigidity and avoids additional skew-related errors found in center-mounted setups. It also recommends clamp designs that minimize the offset parameter, since reducing this value correspondingly reduces the maximum accumulated circular pitch error.
Full Text
JUSTIFICATION FOR SELECTING A MACHINING SCHEME FOR CHIP FLUTES OF DISK RELIEVED CUTTERS
A. A. Asmus
(Presented by the scientific seminar of the Departments of Machine Tools and Metal Cutting and of Mechanical-Engineering Technology)
In milling the chip flutes of disk relieved cutters, two different machining schemes are used:
- Machining on arbors mounted in the centers of the dividing head and tailstock (Fig. 1).
Fig. 1.
1 — center of the dividing head, 2 — driving dog, 3 — clamp, 4 — center of the tailstock
Fig. 2.
1 — special tailstock center, 2 — split bushing
- Machining on arbors that are fastened in the taper of the dividing head (Fig. 2).
It is very important that the greatest accumulated error of the circular pitches after milling be minimal, so as not to have increased runout of the cutting edges in the subsequent operation—relieving the profile.
The error of the circular pitch depends on a number of factors, the principal ones being the following: the accuracy of the dividing mechanism, eccentricity of the workpiece relative to the spindle axis of rotation, rigidity of the SPID system, and skewing of the axis of rotation of the workpiece relative to the spindle axis of the dividing mechanism.
Analyzing the machining schemes indicated above, it should be noted that, when the scheme is changed, only the rigidity of the SPID system and the errors caused by skewing of the axes change. The remaining factors do not change.
Let us examine the influence of each factor.
Rigidity of the system will be greater when the scheme in Fig. 2 is used, since in this case there is no movable joint between the center of the fixture and the center hole of the arbor.
In addition, in order to increase rigidity, it is advisable to use the tailstock design according to variant II in Fig. 2.
Errors of the circular pitch caused by skewing of the axes
Figure 3 shows the case in which there is skewing of the axes of the arbor and of the spindle of the dividing head.
Fig. 3. Diagram of transmission of rotation from the spindle of the dividing head to the arbor.
\(A\)—point of contact of the clamp with the driver
and of the spindle of the dividing head. It is clear from the diagram that, when the spindle of the dividing head is turned through an angle \(\varphi\), the arbor with the workpiece will turn through an angle \(\psi\), different from the angle \(\varphi\).
The circular-pitch error \(\Delta \varphi_{\mathrm{p}}\) caused by skewing of the axes is determined from the relation
\[ \Delta \varphi_{\mathrm{p}} = \psi - \varphi . \]
To establish the dependence between the angles \(\psi\) and \(\varphi\), let us analyze the diagram in Fig. 3.
\[ \psi=\beta_1+\beta_2 \tag{2} \]
\[ \tg\varphi=\frac{E'B'}{E'C'};\qquad \tg\beta_1=\frac{E''B''}{E''C''}; \]
\[ E''B''=E'B';\qquad E''C''=\frac{E'C'}{\cos\gamma}; \]
then
\[ \tg\beta_1=\frac{E'B'}{E'C'}\cos\gamma,\quad \text{i.e.}\quad \tg\beta_1=\tg\varphi\cdot\cos\gamma. \tag{3} \]
\[ \sin\beta_2=\frac{O''F''}{O''B''};\qquad O''F''=C''O''\cdot\sin\beta_1;\qquad C''O''=CO=a\cdot\tg\gamma. \]
\[ O''F''=a\cdot\tg\gamma\cdot\sin\beta_1, \]
\[ O''B''=O''A''=OA=k, \]
i.e.
\[ \sin\beta_2=\frac{a}{k}\cdot\tg\gamma\sin\beta_1; \]
for small values of the angles \(\beta_2\) one may take: \(\beta_2\simeq\sin\beta_2\), hence,
\[ \beta_2=\frac{a}{k}\cdot\tg\gamma\cdot\sin\beta_1 \]
in radians, or in seconds
\[ \beta_2=206\cdot10^3\cdot\frac{a}{k}\cdot\tg\gamma\cdot\sin\beta_1 \tag{4} \]
Finally we have
\[ \Delta\varphi_{\text{p}}=\psi-\varphi=\beta_1+\beta_2-\varphi. \tag{5} \]
\(\beta_1\) is determined from formula (3),
\(\beta_2\) is determined from formula (4).
Special case 1. \(\gamma=0\).
Then \(\beta_1=\varphi\); follows from formula (3),
\(\beta_2=0\); follows from formula (4).
\[ \Delta\varphi_{\text{p}}=\beta_1+\beta_2-\varphi=\varphi-\varphi=0, \]
i.e., for \(\gamma=0\) there will be no error in the transmission of rotation.
Special case 2. \(a=0\).
Then \(\beta_2=0\); follows from formula (4),
\[ \Delta\varphi_{\text{p}}=\beta_1-\varphi. \]
For small angles \(\gamma\), the error \(\Delta\varphi_{\text{p}}\) may be considered a quantity of second-order smallness, which follows from formula (3).
The value \(a\) is selected from design considerations; consequently, when designing clamps one should strive for the value \(a\to0\). The error \(\Delta\varphi_{\text{p}}\) will then be minimal.
Let us analyze for what values of the angles \(\varphi\) the error \(\Delta\varphi_{\text{p}}\) will be maximal.
It follows from formula (5) that \(\Delta\varphi_{\text{p}}\) will assume its maximum value at \((\beta_1-\varphi)_{\max}\) and \(\beta_{2\max}\).
For small angles \(\gamma\), the difference \(\beta_1-\varphi\) is a quantity of second-order smallness, which follows from formula (3).
The angle \(\beta_2\) assumes its maximum value at \(\beta_1 = 90^\circ\) (see formula 4).
From formula (3), at \(\beta_1 = 90^\circ\) it follows that \(\beta_1 = \varphi\); then the greatest error \(\Delta\varphi_{\text{p max}} = \beta_1 + \beta_2 - \varphi = \beta_2\); bearing in mind that at \(\beta_1 = 90^\circ\)
\[ \beta_2 = 206 \cdot 10^3 \cdot \frac{a}{k}\,\operatorname{tg}\gamma \ \text{sec}, \]
we finally have:
\[ \Delta\varphi_{\text{p max}} = 206 \cdot 10^3 \cdot \frac{a}{k}\cdot \operatorname{tg}\gamma \ \text{sec}. \tag{6} \]
It should be borne in mind that formula (6) determines the greatest positive error.
At \(\varphi < 180^\circ\) the error \(\Delta\varphi_{\text{p}}\) changes sign.
The greatest negative error will occur at \(\beta_1 = 270^\circ\);
\[ \Delta\varphi_{\text{p max}} = -206 \cdot 10^3 \cdot \frac{a}{k}\cdot \operatorname{tg}\gamma \ \text{sec}. \]
Consequently, the greatest accumulated error within one revolution, i.e., when the angle \(\varphi\) changes from 0 to \(360^\circ\), will be determined by the formula
\[ \Delta\varphi_{\text{p max}_{\Sigma}} = 412 \cdot 10^3 \cdot \frac{a}{k}\cdot \operatorname{tg}\gamma \ \text{sec}. \tag{7} \]
Analogous investigations were carried out by the scientific and technical staff of the public enterprise Carl Zeiss, Jena [1].
According to [1], the relationship between the angles \(\varphi\) and \(\psi\) is expressed by the formula
\[ \operatorname{tg}\psi = \frac{\sin\varphi}{\cos\gamma \cdot \cos\varphi + \frac{a}{k}\sin\gamma}. \tag{8} \]
When determining the greatest error within one revolution by formula (8), errors are obtained that are half as large as those obtained in calculation by formula (7). This is explained by the fact that relation (8) does not take into account the change in sign of the error at \(\varphi > 180^\circ\).
Conclusions
- The two schemes currently used (Fig. 1 and Fig. 2) for machining chip flutes of disk relieved cutters do not provide the same accuracy with respect to the accumulated error of the circular pitch.
Larger errors arise when the scheme of Fig. 1 is used, since, in comparison with the scheme of Fig. 2, scheme 1 has lower rigidity and additional errors arise, caused by the skewing of the axes of rotation of the spindle and the arbor.
-
To determine the greatest accumulated error under skewing of the axes, formula (7) has been derived; it gives more accurate results in comparison with formula (8) [1].
-
The accumulated error of the circular pitch in scheme 2 can be reduced by means of a rational design of the clamp. The size \(a\) of the clamp should be chosen as small as possible, since as \(a \to 0\), \(\Delta\varphi_{\text{p max}} \to 0\).
LITERATURE
- Designs of precision instruments. Translated from German by N. K. Derevenko. Mashgiz, Moscow, 1960.