Selection of the Gear Ratio in the Design of Modular Gearboxes for Boring Housings
Abstract Generated abstract
This paper addresses the choice of gear ratios in modular multi-spindle boring gearboxes, where using equal gear ratios simplifies manufacture but can produce large differences in cutter life among spindles. Using a gearbox boring example, it compares equal shaft speeds, equalized cutting speeds, and gear ratios calculated from the condition of equal tool life while accounting for hole diameter and depth of cut. The analysis shows that the conventional unity-ratio scheme can yield a 25-fold difference in tool life, whereas the proposed calculation reduces the discrepancy to within practical accuracy limits. A formula is given for selecting gear ratios based on diameter and cutting-depth ratios, with the suggestion that it may be implemented graphically for design use.
Full Text
SELECTION OF THE GEAR RATIO IN THE DESIGN OF MODULAR GEARBOXES FOR BORING HOUSINGS
V. D. VARLAKOV
(Presented by the scientific seminar of the Department of Mechanical-Engineering Technology, Machine Tools, and Metal Cutting)
When considering the question of designing boring modular heads, it is necessary to dwell on one important question, in our opinion. Rational operation of multi-spindle boring heads takes place when the tool life of the tools on all spindles is approximately the same.
The Sibtiazhmash plant has adopted a distribution-gearbox scheme in which the gear ratios between the shafts are taken equal to unity. This ratio is convenient in that all gears of the gearbox can be made identical—their manufacture is simplified. There is equality of minute feeds and feeds per spindle revolution.
However, this scheme also has very substantial shortcomings:
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The gear diameter is limited by the smallest center distance; the mounting hole of the gear is determined by the diameter of the shaft boring the hole of the smallest diameter. This leads to the fact that the shafts transmitting torques to larger holes operate under increased stresses.
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The most substantial shortcoming of this scheme is that it results in an enormously differing life of the cutting tools.
Let us consider this using the example of boring holes in gearbox R-03:
\[ d_1 = 200 \ \text{mm} \quad t_1 = 5 \ \text{mm}, \qquad s_1 = 1.2 \]
\[ v_1 = 44 \ \text{m/min}, \qquad n_1 = 70 \ \text{rev/min}; \]
\[ d_2 = 120 \ \text{mm} \quad t_2 = 3, \qquad s_2 = 1.2, \]
\[ v_2 = 26.3 \ \text{m/min}, \qquad n_2 = 70 \ \text{rev/min}. \]
The tool life will be determined from the equation
\[ T_1 = \sqrt[m]{\frac{C_v}{v s^y t^x}} = \sqrt[0.2]{\frac{147}{44 \cdot 1.2^{0.4} \cdot 5^{0.2}}} = 52 \ \text{min}, \]
\[ T_2 = \sqrt[0.2]{\frac{147}{26.3 \cdot 1.2^{0.4} \cdot 3^{0.2}}} = 1310 \ \text{min}. \]
The difference is 25-fold.
To reduce the difference in tool life, one can proceed by equalizing the cutting speeds. In this case, different feeds per spindle revolution are obtained.
For our example, the ratio of the numbers of revolutions must be equal to the ratio of the diameters. Then the machining conditions will be:
\[ d_1 = 200\ \text{mm}\quad t_1 = 5\ \text{mm},\qquad s_1 = 1.2\ \text{mm/rev},\qquad v_1 = 44\ \text{m/min}, \]
\[ n_1 = 70\ \text{rev/min}, \]
\[ d_2 = 120\ \text{mm}\quad t_2 = 3\ \text{mm},\qquad s_2 = 0.72\ \text{mm/rev},\qquad v_2 = 44.4\ \text{m/min}, \]
\[ n_2 = 118\ \text{rev/min}, \]
\[ T_2 = \sqrt[0.2]{\frac{147}{44.4 \cdot 0.72^{0.4} \cdot 3^{0.2}}} = 251\ \text{min}. \qquad i = \frac{d_1}{d_2} = \frac{200}{120} = 1.67. \]
The difference from the most heavily loaded cutter is by a factor of 4.82.
The best results will be obtained if the calculation of gear ratios is carried out from the equality of tool life.
The gear ratios, proceeding from equality of tool lives with allowance for the changing diameters and depths of cut, can be determined by the formula
\[ i = \frac{n_2}{n_1} = \frac{s_1}{s_2} = \sqrt[1-y]{\frac{d_1}{d_2}\cdot \frac{t_1^x}{t_2^x}}; \tag{1} \]
\[ i = \sqrt[1-0.4]{\frac{200 \cdot 5^{0.2}}{120 \cdot 3^{0.2}}} = 2.82. \]
The operating conditions in this case will be:
\[ d_1 = 200\ \text{mm}\qquad t_1 = 5\ \text{mm},\qquad s_1 = 1.2\ \text{mm/rev}, \]
\[ v_1 = 44\ \text{m/min},\qquad n_1 = 70\ \text{rev/min}; \]
\[ d_2 = 120\ \text{mm}\qquad t_2 = 3\ \text{mm},\qquad s_2 = 0.42\ \text{mm/rev}, \]
\[ v_2 = 74\ \text{m/min},\qquad n_2 = 197\ \text{rev/min}, \]
\[ T_2 = \sqrt[0.2]{\frac{147}{74 \cdot 0.42^{0.4} \cdot 3^{0.2}}} = 65.8\ \text{min} \]
a difference by a factor of 1.26.
This is within the accuracy limits of calculations of cutting conditions. The tool life may be regarded as equal in both cases, 60 minutes.
Thus, the calculation of gear ratios must be carried out with allowance for the ratios of diameters and depths of cut according to the formula (1) given above; for solving it, a graph may be constructed.