Proceedings of the Tomsk Polytechnic Institute
of the Order of the October Revolution and the Order
of the Red Banner of Labor
named after S. M. Kirov

Vol. 188　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1974

INVESTIGATION OF THE LOAD-CARRYING CAPACITY OF GEAR TRANSMISSIONS WITH A HIGH HARDNESS DIFFERENTIAL AT VARIOUS SPEEDS

A. E. BELYAEV

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

One way of increasing the load-carrying capacity of gear transmissions is to use a high hardness differential between the pinion and the wheel. The present investigations were carried out in order to determine the effectiveness of using a high hardness differential at speeds of 20–60 m/s. Analysis of the operation of such gear transmissions at speeds not exceeding 20 m/s (for example, [5]) showed that the use of a pinion with a hard surface, working in a pair with a wheel made of improved steel, increases the load-carrying capacity of the gear transmission.

This is explained by the different resistance of the tooth roots and tips to pitting, which is associated with the action of friction forces that increase the strength of some regions and reduce the strength of others [7, 8, 11].

At the same time, as follows from theoretical and experimental works [4 and 8], the effect of a high hardness differential of the working surfaces of the teeth of the pinion and wheel should decrease as the peripheral speed increases. Thus, the author of work [4], in his experiments on rollers, found that the increase in strength of the roots and tips occurs up to the value \(V_{\Sigma} \approx 45\) m/s (\(V_{\mathrm{okr}} \approx 60\) m/s). Having estimated the load-carrying capacity of transmissions as proportional to the square of the contact-fatigue limit, the author [4] gives a dependence for determining the ratio between the strength of the tooth tips and roots at different speeds

\[ \beta = \frac{K_{\mathrm{g}}}{K_{\mathrm{n}}} = 4.7 / V_{\mathrm{okr}}^{0.26}. \]

In this case, the load-carrying capacity of the roots and tips converges with increasing speed. This dependence confirms the conclusion of A. I. Petrusevich [10], made on the basis of theoretical studies, and the difference in the values of \(\beta\) in the cited works is quite insignificant.

Processing the results of our experiments with gear transmissions made of improved steels also confirms this proposition. The loads recorded at various stages of pitting of the working surfaces of the teeth make it possible to judge the difference in load-carrying capacity of different regions of the tooth. The ratio of the torques corresponding to the onset of pitting of the tooth tip (\(M_{\mathrm{g}}\)) and root (\(M_{\mathrm{n}}\)) tends to decrease with increasing peripheral speed (Fig. 1), and the value

\[ \frac{M_{\mathrm{g}}}{M_{\mathrm{n}}} \]

at \(v_{\mathrm{okr}} = 60\) m/s proves to be close to 1.5. Processing of the resul-

Fig. 1. Load-carrying capacity of tooth tips and roots at various speeds: ● — according to experimental data; × — according to the results of tests with rollers [4]

of this series of experiments by the method of mathematical statistics [12], in order to determine the regularity of the change in the ratio $\dfrac{M_\mathrm{r}}{M_\mathrm{n}}$ at various speeds for transmissions with different geometric parameters, is shown in Fig. 1.

The dependence obtained experimentally,

\[ \frac{M_\mathrm{r}}{M_\mathrm{n}}=\frac{3.8}{v_{\mathrm{okr}}^{0.233}}, \tag{1} \]

differs from that given by the author of work [4] (Fig. 1, upper curve).

Denoting by $\mu_\mathrm{k}$ the ratio of the permissible contact stresses for the surfaces of the tooth tips and roots, respectively, $[\sigma_\mathrm{k}]_\mathrm{r}$ and $[\sigma_\mathrm{k}]_\mathrm{n}$, we have

\[ \mu_\mathrm{k}^{2}=\frac{[C_\mathrm{k}]_\mathrm{r}}{[C_\mathrm{k}]_\mathrm{n}}>1, \]

where

$[C_\mathrm{k}]_\mathrm{r}$ and $[C_\mathrm{k}]_\mathrm{n}$ are the permissible values of the contact-stress coefficients for the surfaces of the tooth tips and roots.

The coefficient introduced here is $[C_\mathrm{k}]=0.918[\sigma_\mathrm{k}]^2/E_{\mathrm{pr}}$. Values of the coefficient $\mu_\mathrm{k}^{2}$ for speeds $v_{\mathrm{okr}}<20$ m/sec according to data from various authors are given in Table 1.

As was already indicated above, according to the work [4] and the results of the first series of experiments with improved transmissions [2] (column 4, Table 2), the value of \(\mu_k^2\) decreases with increasing speed in the speed range \(20\text{–}60\ \mathrm{m/sec}\).

To confirm the results obtained and to investigate the effect of the hardness differential at various speeds, comparative experiments were carried out on a high-speed test rig with a closed power loop, using gear transmissions having improved and nitrided pinions (the geometric parameters of all the transmissions compared were the same: \(m_n = 3\ \mathrm{mm}\), \(z_1 = 28\), \(i = 2.035\), \(\alpha = 20^\circ\), \(\beta_d = 19^\circ 11'17''\); see also Table 2)*.

The tests carried out showed that the load-carrying capacity of transmissions with a hardness differential is somewhat higher than that of identical transmissions made of improved steel (the values of the limiting torques \(M_1\) and \(M_{1n}\) for the transmissions compared are given in Table 3). At the same time it should be noted that, in this series of experiments, only the surfaces of the soft tooth were damaged by pitting, and the character of the development of this process did not differ in any way from the process of failure in gear transmissions made of improved steels.

Table 1

Recently, a proposition has been developed concerning an increase in the load-carrying capacity of a transmission through the use of the tip effect [5]. The point is that, in a normal uncorrected mesh, the load carried by the contact lines passing along the tip (\(L_g\)) and the root (\(L_n\)) of the tooth is different (Fig. 2), owing to the higher strength of the tooth tips as the leading surfaces [11]. The above-mentioned tip effect can be realized by creating an off-pitch mesh.

Table 2

Some parameters of experimental gear wheels

* Column 2 gives the parameters of gear wheels with a high hardness differential; column 4 gives improved gear wheels with identical hardness of the working tooth surfaces.

** Hardness according to HV; in parentheses, core hardness according to HB.

Fig. 2. Load-carrying capacity of the heads and roots of teeth in uncorrected (a) and corrected (b) gear wheels: \(\varepsilon_g\) and \(\varepsilon_n\) are contact ratios corresponding to the heads and roots of the teeth; \(t_{0s}\) is the base pitch of engagement.

If, in the case of a hardness differential, negative correction is introduced on the pinion with a surface-hardened tooth and positive correction on the wheel, then contact will occur between the head of the wheel tooth, made of a material of low hardness, and the hardened root of the pinion tooth. According to a number of researchers [5, 6, 10, 11], this makes it possible to increase the power transmitted by the gear set (without increasing its dimensions) by 20–60%. However, according to [4], such an increase in the load-carrying capacity of offset gear transmissions at high speeds is hardly possible because of a decrease in the coefficient \(\beta\).

To verify these propositions, comparative experiments were carried out on offset gear transmissions with a high hardness differential. The parameters of the transmissions compared are given in Table 2 (parameters of offset transmissions—column 3; parameters of uncorrected transmissions—column 2).

Experiments carried out at speeds of 40 and 60 m/sec showed that the load-carrying capacity of offset transmissions is substantially higher than that of transmissions made with a hardness differential without correction.

Table 3

Principal results of the investigation

* At these loads, fatigue fractures of the pinion teeth occurred.

(table 3). It should be noted that at both speeds (under the loads given in Table 3), fatigue fractures of the pinion teeth occurred.

The working surfaces of the pinion teeth, however, remained completely undamaged, while the surfaces of the teeth of the soft wheel were covered with pitting cavities slightly above the pitch line, or the part of the tooth located at the root was completely pitted out (Fig. 3).

Fig. 3. Condition of the working surface of the gear wheel

When comparing off-center gears and gears of the usual design, it is necessary to note that the coefficient of load concentration at the pole, adopted on the basis of our investigations [1], should not be taken into account in the case of an off-center design of the gear pair. The validity of this proposition is also confirmed by the authors of works [3, 10]. Thus

Thus, the ratio $\dfrac{M_1}{M_{1v}}$ (Table 3) at speeds of 40 and 60 m/s proves to be, respectively, equal to 1.58 and 1.47. The values obtained agree very well with the previously considered values of the ratio $\dfrac{M_r}{M_n}$, and the increase in the load-carrying capacity of external gear transmissions amounts to approximately 50%.

Conclusions

1. With increasing speed (in the range 20–60 m/s), the load-carrying capacity of gear transmissions with a high hardness differential between the pinion and the wheel decreases.

2. The ratio of the load-carrying capacity of the tooth tips and roots decreases with increasing speed; therefore, use of the hardness-differential effect cannot provide the same gain in load that is possible in low-speed transmissions.

3. The load-carrying capacity of transmissions made with a high hardness differential (unlike the load-carrying capacity of corrected external transmissions) differs little from that of equal-hardness improved transmissions.

References

1. A. E. Belyaev. Distribution of load along the contact line of helical wheels of ship reduction gears. Tekhnologiya sudostroeniya, No. 6, 1965.

2. A. E. Belyaev. Influence of speed on the contact strength of gear transmissions. Publ. TPI, No. 147, Tomsk, 1966.

3. R. R. Galper. Contact strength of high-speed gear transmissions with surface hardening. LD NTP, 1964.

4. L. S. Borovich. Investigation of the effectiveness of correction of cylindrical gear transmissions. Trudy TsNIITMASH, 28, Moscow, 1962.

5. I. M. Klimov. Increasing the load-carrying capacity of gear transmissions by using the hardness-differential effect. Vestnik mashinostroeniya, No. 3, 1959.

6. V. N. Kudryavtsev, A. I. Belyanin. Vestnik mashinostroeniya, No. 10, 1957.

7. N. F. Kuzmin. On the coefficient of friction in a heavily loaded contact. Vestnik mashinostroeniya, No. 5, 1954.

8. A. I. Petrusevich. Principal conclusions from contact-hydrodynamic lubrication. Izv. AN SSSR, OTN, No. 2, 1951.

9. A. I. Petrusevich. The role of the hydrodynamic oil film in the stability and durability of contact surfaces of machine parts. Vestnik mashinostroeniya, No. 1, 1963.

10. M. S. Polotskii, Z. P. Pavlov. Load-carrying capacity of a gear external mesh of the second type. In: Increasing the Load-Carrying Capacity of Gear Transmissions and Reducing Weight. Mashgiz, 1956.

11. G. K. Trubin. Contact fatigue of materials for gear wheels. Mashgiz, 1962.

12. M. N. Shashin. Method for determining mean probable values of cyclic durability. Zavod. labor., XVII, No. 6, 1952.