ON THE LOAD-CARRYING CAPACITY OF GEAR TRANSMISSIONS AT DIFFERENT SPEEDS

A. E. Belyaev

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

The principal branches of industry in which high-speed gear transmissions are used are aviation, shipbuilding, and power engineering. A high efficiency and comparatively small overall dimensions make the gear drive irreplaceable at high speeds. This circumstance is the reason for the displacement of electric drives by gear transmissions in certain ship mechanisms.

As is known, one of the main causes of failure of high-speed gear wheels is damage to the working surfaces of the teeth as a result of pitting. To evaluate the contact strength of high-speed gear transmissions, the load-carrying-capacity coefficient \(K_{\mathrm{н}}\), determined by the formula

\[ K_{\mathrm{н}}=\frac{p}{d_{\mathrm{д}1}}\cdot \frac{i+1}{i}, \tag{1} \]

is at present widely used, where \(p\) is the specific load, kg/cm;
\(d_{\mathrm{д}1}\) is the diameter of the pitch circle of the pinion, cm;
\(i\) is the gear ratio.

It is very convenient to examine operating transmissions and to compare transmissions being designed in terms of the value of the coefficient \(K_{\mathrm{н}}\), since the indicated coefficient makes it possible to compare practically any cylindrical gear transmissions.

The value of the coefficient \(K_{\mathrm{н}}\) recorded for gearboxes in service is much smaller than the nominal value \([K_{\mathrm{н}}^{\mathrm{н}}]\), i.e., the value of the coefficient characterizing the load-carrying capacity of the gear rim in the absence of nonuniform load distribution over the width of the gear rim, dynamic loads, elevated temperatures of the working surface, and other factors reducing load-carrying capacity.

The ratio of the coefficients

\[ \frac{K_{\mathrm{н}}^{\mathrm{н}}}{[K_{\mathrm{н}}^{\mathrm{н}}]} \]

has especially low values in high-speed gear transmissions. For the majority of high-speed gearboxes \((v>20\ \text{m/sec})\), the coefficient \(K_{\mathrm{н}}\) (with gear-wheel hardness \(\mathrm{HB}<350\)) does not exceed \(7\text{—}8\ \text{kg/cm}^2\). At the same hardnesses of the working...

surfaces of the teeth, but at speeds of 3–15 m/sec general-purpose reducers have a coefficient \(K_{\mathrm{H}}\) equal to 15–20 kg/cm\(^2\).

The relatively low loads incorporated in the calculations of high-speed gear transmissions are explained, in our opinion, by the following factors:

1. The absence of reliable experimental data making it possible to take into account the influence of speed and geometric parameters on the load-carrying capacity of gear transmissions.

2. Incorrect allowance for the influence of speed on the contact strength of gear transmissions.

3. Irrational values of

\[ q_{\mathrm{c}}=\frac{B}{d_{\mathrm{w}}}, \]

adopted in a number of cases in the design of high-speed drives. Thus, according to the data of P. F. Balyunov, K. I. Zablonskii, B. N. Akulenko, and L. M. Garkavi, the value of the coefficient accounting for the nonuniformity of load distribution over the width of the gear rim, for proportions widely used in high-speed gear transmissions, may reach two and more.

1. The high degree of responsibility of the drives.

2. The greater thermal stressing of these transmissions in comparison with low-speed ones.

Owing to the great complexity of the working conditions in real gear wheels and to the considerable number of factors (see above) that influence the load-carrying capacity of high-speed transmissions, it is expedient to determine the coefficient \(K_{\mathrm{H}}\) experimentally.

A study of the load-carrying capacity of improved gear wheels (HB \(\leqslant 350\)) with various geometric parameters (\(\alpha_0=20^\circ\), \(m_n=2 \div 4\) mm, \(\beta_{\mathrm{d}}=12 \div 35^\circ\), \(z_1=22 \div 44\), \(i\approx2\)) was carried out in the speed range 20–60 m/sec.

The gear wheels were lubricated with “Turbine L” oil at an inlet temperature of 35–40°.

As the limiting load, such a load was adopted at which the transmission could operate for a very long time (approximately twice the conditional number of cycles \(N_{\mathrm{c.bas}}\)) without noticeable development of surface failures and deterioration of its operating characteristics. At the same time, increasing the indicated load by 25–30% led to a sharp progression of failures and rapid failure of the transmission.

In testing the gear wheels, a method of stepwise loading was adopted, which makes it possible, comparatively quickly and with a small saving of test specimens, to obtain the endurance limit. This type of loading has become especially widespread in the testing of large transmissions both in the USSR and abroad.

According to the views currently existing, the load-carrying capacity of gear wheels decreases with increasing speed. This is explained by the growth of dynamic loads (caused by manufacturing errors) with increasing speed. As an example, Fig. 1 (curve \(K_{\mathrm{H}}^{1}\)) shows the dependence \(K_{\mathrm{H}}=f(v_{\mathrm{circ}})\), obtained according to the currently most widespread method [3]*). Meanwhile, experimental verification of gear transmissions at different speeds has shown that the generally accepted opinion about the decrease in the load-carrying capacity of transmissions with increasing speed is incorrect. Figure 1 shows a comparison of the load-carrying capacity (curve \(K_{\mathrm{H}}^{3}\)) of one of the experimental gear

*) According to other methods for calculating gear transmissions for endurance of the working surfaces of the teeth (for example [2]), this decrease in load-carrying capacity with increasing speed proves to be still stronger.

pairs with the calculated load-carrying capacity in the speed range of 20–60 m/s, while the allowable value \(K'_n\) (the load-carrying capacity coefficient) was determined according to the procedure of [3]. The value of the proportionality coefficient \(\Omega\), which is the ratio of the load-carrying capacity coefficients obtained on the basis of the experiments performed to the coefficient \(K'_n\) [3], lies in the interval 1.1–2; here the smaller discrepancy between the experimental and calculated values (coefficient value 1.1) corresponds to the lower speed limit (20 m/s).

Fig. 1

Fig. 2

To obtain a general relationship reflecting the ratio of the aforementioned values \(K^e_n\) and \(K'_n\) at different speeds and characteristic of the gear pairs tested by us with different geometric parameters, all experimental results were processed according to the method of [4]. The results of the calculations are given in Tables 1 and 2 and in Fig. 2. Analysis of the tables mentioned shows that the increase in the coefficient \(\Omega\) (and, consequently, in the load-carrying capacity of the gear wheels) with increasing speed can be described by the correlation relationship

\[ \lg \frac{K^e_n}{K'_n} = 0.4410 \lg v_{\mathrm{okr}} - 0.4785 . \]

Table 1

Statistical processing of the results of tests of high-speed gears

In this case the values of the ratio $\dfrac{K_{\mathrm{n}}^{\mathrm{e}}}{K'_{\mathrm{n}}}$, found as the arithmetic mean and the mean probable value, are sufficiently close (Table 2).

Table 2

Basic data and results of tests of high-speed transmissions

Fig. 3

Labels in the figure: $K_{\mathrm{n}}$; $v$, m/sec. “According to experimental data, HBW = 250–290; HBK = 242–255.” “For IzA multipliers (1957–1962), HBW = 290–340; HBK = 220–270.”

The recorded increase in the load-carrying capacity of gear wheels is explained by the positive influence of speed on the contact strength of transmissions ([1]), while the experimental data on limiting values—

... the values of the coefficients \(K_{\mathrm{н}}\) differ sharply from the same coefficients for high-speed machines in service. Figure 3 shows that the values of \(K_{\mathrm{н}}\) obtained for experimental pairs*) (even with hardnesses of the working tooth surfaces lower than those stipulated by the technical specifications of enterprises producing high-speed drives, for example the Nevsky Machine-Building Plant—NZL) are approximately 3–4 times higher than the values of \(K_{\mathrm{н}}\) currently assigned by designers (the lower region of \(K_{\mathrm{н}}\) values). Moreover, the comparison was made taking into account a safety factor with respect to the limiting load of \(n = 1.5\).

Conclusions

1. The load-carrying capacity of modern high-speed drives (both those in service and those being designed) is significantly below the limiting values.

2. As speed increases, the load-carrying capacity of gear transmissions increases, which is explained by the positive influence of speed on contact strength.

References

1. A. E. Belyaev. The Influence of Speed on the Contact Strength of Gear Transmissions. TPI Proceedings, No. 147. Tomsk, 1966.

2. Ya. G. Kistyan. A Method for Calculating Gear Transmissions for Strength. TsNIITMASh, Book 107. Mashgiz, 1963.

3. V. N. Kudryavtsev. Simplified Calculations of Gear Transmissions. Mashgiz, 1960.

4. A. K. Mitropolsky. The Technique of Statistical Computations. Fizmatgiz. Moscow, 1961.

*) The upper curve was constructed from the results of processing 42 experimental points obtained in tests with more than 250 gear wheels.