Proceedings of the Tomsk Polytechnic Institute named after S. M. Kirov

Vol. 188 1974

STUDY OF THE LOAD CAPACITY OF GEAR TRANSMISSIONS WITH AN INCREASED PRESSURE ANGLE AT VARIOUS SPEEDS

A. E. BELYAEV

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

Involute gear transmissions have considerable reserves for increasing load-carrying capacity and reducing overall dimensions and weight through modifications of the initial contour. Works of recent years [5, 6, 8, 11] show that the load-carrying capacity of gear transmissions can be increased by using a cutting tool with a profile angle \(\alpha_0 > 20^\circ\).

\[ \beta=\frac{\rho}{b\,d_0} \]

Fig. 1. Load-carrying capacity of gear wheels according to the data of works [1, 2]

In a number of works, data are given according to which major foreign organizations and firms producing reducers (for example, “Laval,” “Allen,” etc.) are currently using, for cutting gear transmissions, a tool with profile angles of \(22^\circ 30'\) and \(25^\circ\). However, the literature data on the question of studying the load-carrying capacity of spur gear transmissions cut by a tool with a modified profile angle are few and contradictory. We are not aware of experiments conducted at high speeds.

Theoretical studies by V. N. Kudryavtsev [7, 9, 10] and experimental works at LPZ and the Mozhaisky LVVIA, as well as the few experiments of G. Niemann [12, 13], indicate that the use—

The use of nonstandard basic racks for helical gear transmissions makes it possible to increase their load-carrying capacity in comparison with identical transmissions (in accuracy, heat treatment, and materials) with standard meshing. Figure 1 shows a comparison of the load-carrying capacity of gear transmissions cut by standard and nonstandard cutting tools (G. Niemann’s experiments).

In G. Retting’s work [14], the load-carrying capacity of gear transmissions with angle $\alpha_0 = 28^\circ$ over the entire range of tested angles was almost equivalent to transmissions cut with a tool having the standard profile angle.

However, for example, in [2] the opinion is expressed that the use of basic racks with $\alpha_0 > 20^\circ$ for helical transmissions has an adverse effect on load-carrying capacity because of a noticeable decrease in the total length of the contact lines.

At present, both in our country and abroad, studies at speeds of $3 \div 15\ \text{m/sec}$ have been carried out on gear transmissions corresponding to basic racks with increased values of $\alpha_0$. When analyzing nonstandard basic racks, one should evaluate the influence of their parameters on the load-carrying capacity of transmissions in terms of contact and bending strength. This evaluation was made using the Hertz formula, which, according to the method of [9] (as applied to helical transmissions), for a nonstandard and a standard basic rack may be written in the form

\[ [M_1]^{\mathrm{ns}} = \Omega \cdot \frac{[C_k]\cdot d_{1}\cdot B_p\cdot \rho_{\mathrm{pr.r}}^{\mathrm{ns}}\cdot \lambda_{\min}^{\mathrm{ns}}\cdot \varepsilon_s^{\mathrm{ns}}} {k\cdot (k_{\mathrm{np}}^\beta)^{\mathrm{ns}}} \cdot \cos \alpha_{on}^{\mathrm{ns}*}; \tag{1} \]

\[ [M_1] = \Omega \cdot \frac{[C_k]\cdot d_{1}\cdot B_p\cdot \rho_{\mathrm{pr.r}}\cdot \lambda_{\min}\cdot \varepsilon_s} {k\cdot k_{\mathrm{np}}^\beta} \cdot \cos \alpha_{on}. \tag{2} \]

In these formulas

\[ [M_1]^{\mathrm{ns}},\ [M_1] \]
is the permissible torque on the pinion cut, respectively, with a nonstandard and a standard basic rack;

\[ [C_k] \]
is the permissible value of the contact-stress coefficient;

\[ B_p \]
is the working face width of the gear rim;

\[ d_1 \]
is the diameter of the pitch circle of the pinion;

\[ \rho_{\mathrm{pr.r}}^{\mathrm{ns}},\ \rho_{\mathrm{pr.r}} \]
are the reduced radii of curvature at the pitch pole, respectively, for nonstandard and standard transmissions;

\[ \varepsilon_s^{\mathrm{ns}},\ \varepsilon_s \]
are the overlap ratio in the transverse plane, respectively, for pairs cut with a nonstandard and a standard basic rack;

\[ \lambda_{\min}^{\mathrm{ns}},\ \lambda_{\min} \]
is the coefficient of reduction of the length of the contact lines for nonstandard and standard transmissions;

\[ (k_{\mathrm{np}}^\beta)^{\mathrm{ns}},\ k_{\mathrm{np}}^\beta \]
is the coefficient of concentration of the specific load in the zone of minimum contact strength of the teeth;

\[ k \]
is the coefficient taking into account the influence of the nonuniformity of load distribution along the tooth length and additional dynamic loads in the transmission;

\[ \alpha_{on}^{\mathrm{ns}},\ \alpha_{on} \]
are the profile angles of the nonstandard and standard tool in the normal section;

\[ \Omega \]
[[unclear: definition not visible on this page]].

\[ \overline{\phantom{xxxx}} \]

* Here and below, the superscript “ns” will refer to gear transmissions cut with a tool having an increased angle of the basic rack.

\(\Omega\) is the proportionality coefficient.

Let us estimate the influence of geometry on the load-carrying capacity of a transmission by means of the coefficient \(\chi^*\).

\[ \chi=\frac{[M_1]^{\mathrm{ns}}}{[M_1]} = \frac{ \rho_{\mathrm{pr.r}}^{\mathrm{ns}}\cdot \cos \alpha_{on}^{\mathrm{ns}}\cdot \varepsilon_s^{\mathrm{ns}}\cdot \lambda_{\min}^{\mathrm{ns}}\cdot K_{\mathrm{np}}^{\beta} }{ \rho_{\mathrm{pr.r}}\cdot \cos \alpha_{on}\cdot \varepsilon_s\cdot \lambda_{\min}\cdot \left(K_{\mathrm{np}}^{\beta}\right)^{\mathrm{ns}} }. \tag{3} \]

By the last relation one can estimate the effect on load-carrying capacity of:

1) the magnitude of the ratios of the radii of curvature

\[ \frac{\rho_{\mathrm{pr.r}}^{\mathrm{ns}}}{\rho_{\mathrm{pr.r}}} = \frac{\sin \alpha_{on}^{\mathrm{ns}}}{\sin \alpha_{on}} \cdot \frac{\cos^2 \beta_0}{\left(\cos^2 \beta_0\right)^{\mathrm{ns}}}, \tag{4} \]

where

\(\beta_0^{\mathrm{ns}}\) and \(\beta_0\) are the helix angles on the base cylinders of gear wheels manufactured, respectively, with a nonstandard and a standard basic rack profile,

2) the magnitude of the ratio of the total lengths of contact lines

\[ \frac{\varepsilon_s^{\mathrm{ns}}}{\varepsilon_s}\cdot \frac{\lambda_{\min}^{\mathrm{ns}}}{\lambda_{\min}}, \]

3) the magnitude of the ratio of the coefficients of concentration of specific load

\[ \frac{k_{\mathrm{np}}^{\beta}}{\left(k_{\mathrm{np}}^{\beta}\right)^{\mathrm{ns}}}, \]

4) the magnitude of the ratio of the normal forces in the mesh

\[ \frac{\cos \alpha_{on}^{\mathrm{ns}}}{\cos \alpha_{on}}. \]

Because the evaluation of the load-carrying capacities of transmissions is comparative in nature, the influence of the coefficients \(\left(k_{\mathrm{np}}^{\beta}\right)^{\mathrm{ns}}\) and \(k_{\mathrm{np}}^{\beta}\) is not taken into account. Taking into consideration also that \(B_p/t_a=2\), we assume \(\lambda_{\min}^{\mathrm{ns}}=\lambda_{\min}\). Under these conditions formula 3 takes the form

\[ \chi= \frac{\cos 2\alpha_{on}^{\mathrm{ns}}}{\cos 2\alpha_{on}} \cdot \frac{\varepsilon_s^{\mathrm{ns}}}{\varepsilon_s}. \tag{5} \]

Calculation by formula 5 shows that, for our experimental transmissions, the load-carrying capacity (for equally hard improved transmissions), upon changing to the profile angle \(\alpha_0=27^\circ\), may increase by approximately 8–10% (Fig. 2, \(a\)).

The results of G. Niemann’s experiments and the relation proposed by him do not agree with the data obtained at LZP. The doubled value of the quantity \(B=P/(B_p\cdot d_p)\) (Fig. 1), recorded when testing transmissions with \(\alpha_0=28^\circ\) (in comparison with standard twenty-degree gears), raises doubt.

* The comparison is made for improved helical transmissions with identical dimensions and technological parameters (\(z_1=28\); \(m_n=3\); \(i\approx 2\), \(B_p=30\) mm; \(\beta_d=19^\circ 11'17''\); \(\xi=0\), HB \(=250\div262\), accuracy grade—fifth). Only the meshing angle is different: \(\alpha_0=20^\circ\) and \(\alpha_0^{\mathrm{ns}}=27^\circ\).

Fig. 2. \(a\)—results of calculation by formulas (3, 5); \(b\)—values of the coefficients \(\chi'\) and \(\chi\): 1—by formula 6, 2—by the method of [9]

Calculation of the load capacity of transmissions according to the method of [13] is carried out using a coefficient characterizing the ratio of load capacity, which for the experimental transmissions will take the form

\[ \chi' = \frac{ (\sin^3 \alpha_{on})^{\mathrm{nc}} \cdot (\cos^2 \alpha_{on})^{\mathrm{nc}} \cdot \varepsilon_s^{\mathrm{nc}} }{ \sin^3 \alpha_{on} \cdot \cos^2 \alpha_{on} \cdot \varepsilon_s }. \tag{6} \]

A comparison of the results of calculation by formulas 5 and 6 is shown in Fig. 2, \(b\).

It follows from the foregoing that the use of increased profile angles in transmissions leads to an increase in their load capacity under conditions of contact strength.

The use of modified basic racks \((\alpha_0 > 20^\circ)\) also has a positive effect on the bending strength of transmissions. In this case, the increase in bending strength is due to an increase in the tooth thickness in the danger—

meshing. The calculations performed for the tooth-form factors, found on the basis of A. V. Verkhovsky’s hypothesis, as well as experiments on pulsators, confirm the statement given above.

An experimental study of gear transmissions cut with a tool having a nonstandard basic rack shows that their load capacity is 18–22% higher in comparison with identical transmissions made with \(\alpha_0=20^\circ\). It should be noted that a similar increase in the load capacity of gear transmissions was recorded in earlier experiments at speeds up to \(15\ \text{m/sec}\) at LZP.

Fig. 3. Results of an experimental study of gear transmissions with \(\alpha_0=20^\circ,\ f_0=1\) and \(\alpha_0=27^\circ,\ f_0=0.9\)

In Fig. 3 are given the values of the coefficient \(K=\dfrac{\sigma_k}{HB}\), obtained as a result of processing the experimental data (taking into account the quantities \(\rho_{\text{pr.r}}\) and \(L_k\)); moreover, the ratio of the value \(K\) for transmissions with \(\alpha_0=27^\circ\) to the value \(K\) for standard transmissions remains approximately constant over the entire speed range.

The load capacity obtained experimentally is higher than follows from formula 3. This discrepancy can be explained as follows. With an increase in the pressure angle, the total rolling velocity \(v_\Sigma\) also increases, which, as was shown in [1], has a positive effect on the contact strength of gear transmissions. Taking into account the dependence obtained by us in studying the effect of speeds on the strength of the working tooth surfaces, one can compare transmissions cut with standard and nonstandard basic racks. This dependence has the form

\[ K=\theta\cdot v_\Sigma^{0.2}, \]

where

\[ \theta \]

is the proportionality coefficient. Noting that \(v_\Sigma=2v_{\text{okr}}\cdot\sin\alpha_s\), we find that the ratio of the coefficients for the racks being compared will depend only on \(\alpha_s\).

\[ \frac{K_{\alpha=27}}{K_{\alpha=20}} = \frac{\left(\theta\cdot v_\Sigma^{0.2}\right)_{\alpha=27}} {\left(\theta\cdot v_\Sigma^{0.2}\right)_{\alpha=20}} = \frac{(\sin\alpha_s)_{\alpha=27}} {(\sin\alpha_s)_{\alpha=20}} . \tag{10} \]

For the values of \(\alpha_0\) and \(\beta_d\) adopted by us, this ratio proves to be equal to 1.05. Thus, the increase in the load capacity of the transmissions tested by us with the value \(\alpha_0=27^\circ\) due solely to the increase in the rolling velocity of the profiles reaches 5%.

Taking into account that the increase in load capacity due to the geometric parameters of the transmissions (formula 5) is 9%, the overall increase in load-carrying capacity should be approximately equal to \(1.05 \cdot 1.09 = 1.15\). The test results obtained by us confirm the calculations given above.

Conclusions

1. The load capacity of transmissions cut with a tool having \(\alpha_0 > 20^\circ\) increases in comparison with transmissions cut with a standard tool.

2. The sharp increase in the load capacity of gears (\(\sim\) by a factor of 2), advertised by some foreign authors, is not confirmed.

3. The load capacity of gear transmissions cut with a tool having a nonstandard basic rack profile increases with increasing speed.

4. The experimentally recorded load-carrying capacity of gear transmissions cut with a tool having \(\alpha_0 = 27^\circ\) and \(f_0 = 0.9\) in the speed range \(20 \div 60\ \text{m/sec}\) is greater than the load capacity of standard transmissions by 18–22%.

5. The difference between the calculated load capacity of transmissions with \(\alpha_0 = 27^\circ\) and \(f_0 = 0.9\) and that obtained experimentally is 12–13%. This discrepancy disappears if the influence of the speed of rolling on contact strength is taken into account.

References

1. A. E. Belyaev. The influence of speed on the contact strength of gear transmissions. Izv. TPI, No. 147, Tomsk, 1966.

2. E. B. Vulgakov. On certain reserves in the load-carrying capacity of involute gear meshing. “Problems of the Quality and Strength of Gear Transmissions,” TsBTI, Moscow, 1961.

3. E. B. Vulgakov. On the load capacity of gear transmissions cut with a nonstandard profile. Moscow, 1962.

4. V. A. Gavrilenko. Some geometric aspects of gear transmissions that increase the load capacity of teeth. “Problems of the Quality and Strength of Gear Transmissions,” P, TsBTI, Moscow, 1961.

5. I. M. Kilimov. Experimental study of the load capacity of helical transmissions with involute meshing. “Collection of Works of LMI,” No. 15, 1960.

6. A. I. Korablyov. On the use of involute gears with a modified basic rack profile in civil aviation aircraft engines. Collection “Problems of the Geometry and Dynamics of Gear Transmissions,” “Nauka,” N., 1964.

7. V. N. Kudryavtsev. Gear Transmissions. Mashgiz, 1957.

8. V. N. Kudryavtsev. Improvement of gear transmissions. Leningrad Industry, No. 4, 1958.

9. V. N. Kudryavtsev. Simplified calculations of gear transmissions. Mashgiz, 1960.

10. V. N. Kudryavtsev. On strength calculations of gear transmissions. “Collection of Works of LMI,” No. 23, 1962.

11. L. Müller, Przeglad mechaniczny, XX, No. 5, 1961.

12. G. Niemann, W. Richter, Konstruktion, Heft 6, Juni, 1960.

13. G. Nieman, W. Richter, Konstruktion, Heft 10, Okt., 1960.

14. H. Rettig, Maschinenbautechnik, Januar, Heft, 1, 1958.