Selection of the Minimum Number of Teeth in Gears with an Intermediate Wheel
Abstract Generated abstract
The paper analyzes how to select the minimum tooth numbers in a three gear transmission with an intermediate wheel while avoiding tooth undercutting during simultaneous meshing. It derives limiting geometric conditions for three possible undercutting cases and formulates expressions for the minimum number of teeth on the first gear as functions of transmission ratio, pressure angle, and addendum height coefficients. For the special case of equal addendum heights and a 20 degree pressure angle, the author tabulates and plots the resulting tooth numbers, showing that the controlling undercutting case changes near a ratio of about 4. The study concludes that the smallest overall transmission dimensions occur near this ratio, while ratios close to unity require sharply increasing tooth numbers and should generally be avoided for normal meshing with an intermediate wheel.
Full Text
N. P. Shubin
Selection of the Minimum Number of Teeth in Transmissions with an Intermediate Wheel
With simultaneous meshing of three wheels (Fig. 1), undercutting of the teeth of the wheels is possible in three cases:
- Undercutting of the teeth of wheels II and III, when point \(E\)—the intersection of the addendum circle of wheel II with the generating straight line—is located outside the segment \(HL\).
- Undercutting of the teeth of wheels I and III, when point \(B\)—the intersection of the addendum circle of wheel III with the generating straight line—is located outside the segment \(MN\).
- Undercutting of the teeth of the same wheels when point \(A\), the intersection of the addendum circle of wheel I with the generating straight line, is located outside the segment \(MN\).
Fig. 1.
For the first case, the conditions for correct transmission are determined by the position of point \(E\) (Fig. 2).
Let us draw from the centers of the wheels \(O_1\) and \(O_2\) the perpendiculars \(O_1K\) and \(O_2L\) to the generating straight line.
In the absence of undercutting, point \(E\) must lie outside the segment \(KL\)
\[ LE > O \]
Let us consider the limiting case, which determines the beginning of undercutting,
\[ LE = O . \]
From Fig. 2
\[ LE = KE - LK \tag{a} \]
From triangle \(KEO_1\)
\[ KE = \sqrt{O_1E^2 - OK^2} = \sqrt{R_{a2}^{2} - R_2^{2}\cos^{2}\alpha} \tag{b} \]
Drawing 2.
where:
\(R_{a_2}\) — radius of the addendum circle of wheel II,
\(R_2\) — radius of the pitch circle of wheel II,
\(\alpha\) — angle of inclination of the generating straight line.
\[ LK = O_2G = (R_2 - R_{a_2})\sin\alpha. \]
Replacing the radius of wheel III by the radii of wheels I and II, we obtain
\[ LK = \left(R_2 - \frac{R_2 - R_1}{2}\right)\sin\alpha = \frac{R_1 + R_2}{2}\sin\alpha \tag{c} \]
From equations (a), (b), and (c) it follows, for \(LE = O\),
\[ \frac{R_1 + R_2}{2}\sin\alpha = \sqrt{R_{a_2}^{\,2} - R_2^{\,2}\cos^2\alpha}. \]
or
\[ \frac{(R_1 + R_2)^2}{4}\,\sin^2\alpha = R_{a_2}^{\,2} - R_2^{\,2}\cos^2\alpha. \]
Substituting
\[ R_1 = \frac{Z_1 m}{2};\qquad R_2 = i\,\frac{Z_1 m}{2};\qquad R_{a_2} = R_2 + h_2' = i\,\frac{Z_1 m}{2} + x_2 m, \]
where:
\(i\) — transmission ratio, \(i = \dfrac{Z_2}{Z_1}\),
\(x_2\) — ratio of the addendum height of wheel II to the module,
we obtain:
\[ \left(\frac{Z_1 m}{2}+i\frac{Z_1 m}{2}\right)^2\sin^2\alpha = 4\left(i\frac{Z_1 m}{2}-x_2 m\right)^2 - 4\left(i\frac{Z_1 m}{2}\right)^2\cos^2\alpha . \]
After reduction by \(\left(\dfrac{Z_1 m}{2}\right)^2\), we determine \(Z_1\)—the minimum number of teeth on wheel I required in order to avoid undercutting of the teeth of wheels II and III, which is possible when point \(E\) is incorrectly positioned.
\[ Z_{1(E)} = \frac{2x_2} {-\sqrt{\frac{1}{4}(1+i)^2\sin^2\alpha+i^2\cos^2\alpha}+i} \tag{1} \]
In the absence of undercutting in the second case, the condition must be satisfied that point \(B\) lies within the segment \(CN\).
For the limiting case, in which undercutting begins,
\[ BN=0. \tag{d} \]
From Fig. 2,
\[ BN=MN-MB. \]
From the triangles \(O_1CN\), \(O_2CM\), \(MO_2B\), and equation (d), it follows:
\[ (R_1+R_3)\sin\alpha-\sqrt{R_a^2-R_3^2\cos^2\alpha}=0, \]
where:
\(R_3\)—radius of the pitch circle of wheel III,
\(R_a\)—radius of the addendum circle of wheel III,
\(R_1\)—radius of the pitch circle of wheel I.
Let us replace the radius of the intermediate wheel by the radii of wheels I and III.
\[ R_3=\frac{R_2-R_1}{2}; \qquad R_a=R_3+h_3=\frac{R_2-R_1}{2}+x_3m, \]
where \(x_3\) is the ratio of the addendum height of the intermediate wheel to the module.
\[ \left(R_1+\frac{R_2-R_1}{2}\right)\sin\alpha = \sqrt{ \left(\frac{R_2-R_1}{2}+x_3m\right)^2 - \left(\frac{R_2-R_1}{2}\right)^2\cos^2\alpha }. \]
or
\[ \frac{(R_1+R_2)^2}{4}\sin^2\alpha = \left(\frac{R_2-R_1}{2}+x_3m\right)^2 - \left(\frac{R_2-R_1}{2}\right)^2\cos^2\alpha . \]
After substituting
\[ R_1=\frac{Z_1m}{2}; \qquad R_2=i\frac{Z_1m}{2} \]
and reducing by
\[ \frac{1}{4}\left(\frac{Z_1m}{2}\right)^2 \]
we obtain:
\[ (1+i)^2\sin^2\alpha = \left[(i-1)+\frac{4x_3}{Z_1}\right]^2 - (i-1)^2\cos^2\alpha . \]
From this equation we determine \(Z_1\)—the minimum number of teeth on wheel I required in order to avoid undercutting of the teeth of wheels I and II, which is possible when point \(B\) is incorrectly positioned.
\[ Z_{1(B)} = \frac{4x_3} {\sqrt{(1+i)^2\sin^2\alpha+(i-1)^2\cos^2\alpha}-(i-1)} \tag{2} \]
Finally, for the third case the conditions of correct transmission are determined by the position of point \(A\) between the base of the perpendicular dropped from center \(O_3\) onto the generating line (point \(M\)) and the pole of engagement \(C\).
For the limiting case
\[ AM = 0. \tag{e} \]
From Fig. 2
\[ AM = MN - AN \]
From the triangles \(O_1CN\), \(O_2CM\), \(O_1AN\) and equation (e) it follows:
\[ (R_1 + R_3)\cdot \sin \alpha - \sqrt{R_{a_1}^{\,2} - R_1^{\,2}\cdot \cos^2 \alpha} = 0, \]
where \(R_{a_1}\) is the radius of the addendum circle of wheel I.
Replacing, analogously to the preceding case, the radius of the intermediate wheel by the radii of wheels I and II,
\[ R_3 = \frac{R_2 - R_1}{2}; \qquad R_{a_1} = R_1 + x_1 m, \]
where \(x_1\) is the ratio of the addendum height of wheel I to the module,
\[ \frac{R_1 + R_2}{2}\cdot \sin \alpha = \sqrt{(R_1 + x_1 m)^2 - R_1^2\cos^2 \alpha} \]
Substituting the values \(R_1\) and \(R_2\), we obtain the value of the least number of teeth on wheel I, in the absence of undercutting of the teeth of wheels I and II, possible with an improper position of point \(A\),
\[ Z_{1(A)} = \frac{2x_1}{ \sqrt{\frac{1}{4}(1+i)^2\cdot \sin^2\alpha + \cos^2\alpha} - 1 } \tag{3} \]
The equations obtained (1, 2, 3) make it possible to take a critical approach to assigning the number of teeth on the first wheel and to ensure the minimum dimensions of the entire transmission, under the condition that undercutting of the teeth of all three wheels is absent.
The number of teeth on the first wheel, for a given transmission ratio and accepted values \(x_1, x_2, x_3\), must be chosen so that it is not less than the greatest value of \(Z_1\) obtained from the three equations given above.
The value \(Z_1\) is determined most simply for the case when the addendum heights of the teeth of all three wheels are equal. For this special case a table has been compiled, in which the values \(Z_1, Z_2, Z_3\) have been calculated for:
\[ x_1 = x_2 = x_3 = 1 \quad \text{and} \quad \alpha = 20^\circ. \]
In the second, third, and fourth columns are given the values of the numbers of teeth of the first wheel for various transmission ratios, calculated from equations (1), (2), and (3).
For each transmission ratio three values of \(Z_1\) are given:
\(Z_{1(E)}\)—the minimum number of teeth of the first wheel guaranteeing the absence of undercutting of the addendum of a tooth of wheel II and the dedendum of a tooth of wheel III.
\(Z_{1(B)}\)—the same—for the addendum of a tooth of wheel III and the dedendum of a tooth of wheel I.
\(Z_{1(A)}\)—the same—for the addendum of a tooth of wheel I and the dedendum of a tooth of wheel III.
To avoid undercutting in all three cases, the number of teeth on the first wheel must be chosen so that it is not less than the greatest value among \(Z_{1(E)}\), \(Z_{1(B)}\), and \(Z_{1(A)}\).
Minimum number of teeth of wheels in transmissions with an intermediate wheel, under the condition of absence of undercutting
| Transmission ratio \(i\) | Minimum number of teeth on the first wheel, by eq. (1) \(Z_{1(E)}\) | Minimum number of teeth on the first wheel, by eq. (2) \(Z_{1(B)}\) | Minimum number of teeth on the first wheel, by eq. (3) \(Z_{1(A)}\) | Minimum number of teeth on the second wheel \(Z_2\) | Minimum number of teeth on the third wheel \(Z_3\) |
|---|---|---|---|---|---|
| 1.0 | \(\infty\) | 5.85 | \(\infty\) | \(\infty\) | — |
| 1.1 | 343.11 | 6.41 | 335.96 | 377.42 | 17.15 |
| 1.2 | 177.54 | 6.96 | 163.86 | 213.05 | 17.75 |
| 1.3 | 120.19 | 7.47 | 114.19 | 156.25 | 18.03 |
| 1.4 | 91.34 | 7.95 | 78.70 | 127.88 | 18.27 |
| 1.5 | 73.92 | 8.48 | 61.78 | 110.88 | 18.48 |
| 1.6 | 62.26 | 8.83 | 50.47 | 99.62 | 18.68 |
| 1.7 | 53.86 | 9.21 | 42.55 | 91.56 | 18.85 |
| 1.8 | 47.53 | 9.58 | 36.59 | 85.55 | 19.01 |
| 1.9 | 42.56 | 9.91 | 31.98 | 80.87 | 19.15 |
| 2.0 | 38.57 | 10.22 | 28.32 | 77.14 | 19.29 |
| 2.1 | 35.29 | 10.50 | 25.34 | 74.11 | 19.41 |
| 2.2 | 35.53 | 10.77 | 22.88 | 71.57 | 19.52 |
| 2.3 | 30.18 | 11.01 | 20.80 | 69.42 | 19.62 |
| 2.4 | 28.17 | 11.24 | 19.04 | 67.61 | 19.72 |
| 2.5 | 26.41 | 11.52 | 17.52 | 66.03 | 19.81 |
| 2.6 | 24.87 | 11.65 | 16.21 | 64.66 | 19.89 |
| 2.7 | 23.50 | 11.83 | 15.05 | 63.44 | 19.97 |
| 2.8 | 22.27 | 12.01 | 14.03 | 62.36 | 20.04 |
| 2.9 | 21.17 | 12.17 | 13.13 | 61.40 | 20.11 |
| 3.0 | 20.18 | 12.32 | 12.32 | 60.53 | 20.18 |
| 3.5 | 16.36 | 13.04 | 9.32 | 57.26 | 20.50 |
| 4.0 | 13.77 | 13.46 | 7.39 | 55.09 | 20.66 |
| 4.5 | 11.90 | 13.85 | 6.07 | 62.31 | 24.23 |
| 5.0 | 10.48 | 14.16 | 5.11 | 70.80 | 28.32 |
| 6.0 | 8.47 | 14.64 | 3.84 | 87.84 | 36.60 |
| 7.0 | 7.11 | 14.98 | 3.03 | 104.86 | 44.94 |
| 8.0 | 6.12 | 15.24 | 2.48 | 121.92 | 53.34 |
| 9.0 | 5.38 | 15.44 | 2.10 | 138.99 | 61.78 |
| 10.0 | 4.80 | 15.60 | 1.81 | 156.07 | 70.23 |
| 15.0 | 3.12 | 16.10 | 1.06 | 241.50 | 112.70 |
| 20.0 | 2.31 | 16.35 | 0.74 | 326.96 | 155.30 |
| 30.0 | 1.52 | 16.60 | 0.46 | 497.92 | 240.66 |
| 40.0 | 1.13 | 16.72 | 0.33 | 668.78 | 326.03 |
In the table, the values of the smallest number of teeth on the first wheel for various gear ratios are outlined.
In the fifth and sixth columns of the table are indicated the numbers of teeth of wheels II and III, calculated at the smallest value of \(Z_1\)
\[ Z_2=iZ_1;\quad Z_3=\frac{Z_2-Z_1}{2}. \]
All calculations in the table have been made without rounding the numbers of teeth to an integer.
The data of the table, characterizing the change in the smallest number of teeth on the first wheel (columns 2, 3, 4) as a function of the gear ratio, are represented graphically in Fig. 3. Along the abscissa axis are plotted the gear ratios (on a logarithmic scale), and along the ordinate axis—the numbers of teeth of the first wheel.
Curve \(E\) gives the dependence of the smallest number of teeth on the first wheel on the gear ratio in the absence of undercutting of the tooth head of wheel II and of the tooth root of wheel III.
Curve \(B\)—the same, in the absence of undercutting of the tooth head of wheel III and of the tooth root of wheel I.
Curve \(A\)—the same, in the absence of undercutting of the tooth head of wheel I and of the tooth root of wheel III.
From the diagram it is evident that, for gear ratios from 1 to 4, the most probable location of undercutting is the tooth head of wheel II and the tooth root of wheel III. For these gear ratios the smallest number of teeth on the first wheel must be chosen according to curve \(E\) (segment \(DC\)).
For gear ratios greater than 4, the smallest number of teeth of the first wheel is determined by undercutting of the tooth head of wheel III and of the tooth root of wheel I, and must be taken according to curve \(B\) (segment \(CF\)).
Thus, the number of teeth of the first wheel, under the condition that there is no undercutting of the teeth of all three wheels at various gear ratios, is determined by the region of the diagram located above curve \(DCF\).
It should also be noted that curve \(A\), which governs undercutting of the tooth head of wheel I and the root of wheel III, has no influence on the choice of the smallest number of teeth of the first wheel, since it is located inside curve \(DCF\).
A characteristic point in the diagram is point \(C\), the intersection of curves \(B\) and \(E\). At the gear ratio corresponding to this point (approximately 4), the number of teeth on the first wheel will be the smallest in comparison with the number of teeth at other gear ratios.
As the gear ratio increases, the number of teeth of the first wheel grows comparatively slightly and is determined by curve \(B\) (segment \(CF\)).
Conversely, as the gear ratio decreases, the number of teeth on the first wheel increases sharply along curve \(E\) (segment \(DC\)), approaching infinity in the limit when the gear ratio is equal to unity.
The influence of the gear ratio on the dimensions of the entire transmission is shown in Fig. 4, where the gear ratios are plotted along the abscissa axis, and the numbers of teeth along the ordinate axis.
Curve \(DCF\) has been transferred from Fig. 3 and gives the smallest number of teeth on the first wheel \(Z_1\).
Curve \(KLM\) has been constructed from the data of the fifth column of the table and represents the boundary of the smallest number of teeth of the second wheel \(Z_2\).
The ordinates between these curves give the doubled smallest numbers of teeth on the third wheel \(2Z_3\) and correspond to the doubled numbers of teeth of the sixth column of the table.
Fig. 3.
From the character of the curve \(KLM\) it follows that the transmission will have the smallest dimensions when the gear ratio is equal to four (point \(L\)). In this case the minimum numbers of teeth on the wheels must be taken as not less than:
\[ Z_1 = 14 \quad \text{(according to table 13.77)} \]
\[ Z_2 = 56 \quad (55.09) \]
\[ Z_3 = 22 \quad (20.66) \]
For all other gear ratios, the dimensions of the transmission, provided the modules are the same, will be larger.
Fig. 4.
With an increase in the transmission ratio, the dimensions of the transmission increase and are determined by the smallest number of teeth on the 2nd gear according to the curve \(LM\).
For example, at \(i = 10\),
\[ Z_1 = 16 \quad (15{,}60), \]
\[ Z_2 = 160 \quad (156{,}07), \]
\[ Z_3 = 72 \quad (70{,}23). \]
When the transmission ratio is decreased, the dimensions of the transmission also increase along the curve \(LK\), at first comparatively slowly, and then, beginning approximately
at \(i = 2.0\text{--}2.5\), they begin to increase rapidly, approaching infinity.
| Smallest actual numbers of teeth | Smallest actual numbers of teeth | Smallest actual numbers of teeth | Smallest actual numbers of teeth | Smallest numbers of teeth according to the table | Smallest numbers of teeth according to the table | Smallest numbers of teeth according to the table | Smallest numbers of teeth according to the table | |
|---|---|---|---|---|---|---|---|---|
| \(i =\) | 2.5 | 2.0 | 1.5 | 1.2 | 2.5 | 2.0 | 1.5 | 1.2 |
| \(Z_1 =\) | 28 | 40 | 76 | 180 | 26.41 | 38.57 | 73.92 | 177.54 |
| \(Z_2 =\) | 70 | 80 | 114 | 216 | 66.09 | 77.14 | 110.88 | 213.05 |
| \(Z_3 =\) | 21 | 20 | 19 | 18 | 19.81 | 19.29 | 18.48 | 17.75 |
The curve \(KLM\) also shows that transmissions with a small gear ratio will, in their dimensions, be equal to transmissions with a large gear ratio when the modules are the same.
For example, a transmission at \(i = 2.0\) will have approximately the same dimensions as a transmission at \(i = 5.4\). A transmission at \(i = 1.5\) will correspond in dimensions to a transmission at \(i = 7.0\).
This peculiar increase in the dimensions of transmissions of this kind when the gear ratio is reduced must be taken into account when selecting transmissions with an intermediate wheel; and, for gear ratios close to unity, the use of such transmissions with normal meshing should be avoided.