ON THE CHOICE OF AN ENDURANCE CRITERION FOR HELICAL GEARS OPERATING AT HIGH SPEEDS

A. E. Belyaev

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

For the experimental determination of the load-carrying capacity of gears it is necessary to adopt a definite criterion by means of which one could evaluate the magnitude of either the permissible or the limiting load, limited by the strength of the working surfaces of the teeth.

The methods existing in the technical literature for determining a criterion of surface damage that would be most acceptable in tests for obtaining comparable values of resistance to surface failure are very diverse.

It is known [6] that attempts to record the moment of onset of dangerous forms of fatigue failure of the working surfaces of teeth by noise characteristics or by changes in the efficiency in the mesh, even at low speeds \((2 \div 15 \text{ m/sec})\), were not successful: over a wide range of changes in the condition of the working surfaces of the teeth—from microscopic pitting to damage of the entire tooth flank—the characteristics mentioned remained unchanged. This circumstance can evidently be explained by two reasons: either these characteristics generally change little under such damage to the working surface of the teeth, or the apparatus with which the measurements were made did not detect small changes in noise.

As applied to tests of high-speed transmissions, the use of the above-mentioned criteria is also unacceptable for the following reason. Recording an increased noise level under severe damage (of course, at loads far exceeding the limiting ones) is unacceptable for critical high-speed gear transmissions, since it is necessary for us, first of all, to record permissible, or in the extreme case limiting, loads.

Thus, adoption of the criteria indicated above, which make it possible to evaluate various stages of fatigue spalling of the working surfaces of teeth, does not appear possible either for low-speed or for high-speed transmissions.

Other criteria of surface damage to parts have also been proposed [10]: the change in the weight of a part occurring as a result of surface damage during an experiment, and the change in the roughness of the surface of the part. In the first case, the permissible percentage decrease in weight (determined by direct weighing before and after the experi-

... (the increment) is, for this method, the criterion of surface damage. However, for gear transmissions of comparatively large dimensions ($d_a = 300$ mm), determining the degree of wear by such a method does not appear possible for three reasons: the change in weight during testing is insignificant in comparison with the weight of the part; accurate weighing is possible only for parts of small weight; and, finally, weighing the gear wheel under test requires removing it from the stand, which is unacceptable both on grounds of the reliability of the results obtained and for operational reasons.

The broad application of the second criterion, despite progress in the technique of measuring roughness, is limited by the fact that, when gear pairs are tested for contact strength, excessively sharp changes in surface roughness sometimes occur because relatively large metal particles are removed. Thus, these criteria too prove unacceptable in determining the contact strength of gear transmissions.

In this connection, the desire of a number of researchers to adopt as the fatigue criterion a certain degree of damage to the working surfaces of the teeth by pitting becomes understandable.

Operating experience with gear transmissions convincingly shows that the opinion, very widespread in the past and still current at present, that any pitting is inadmissible (for transmissions with a hardness of the working tooth surfaces of HB $\leq 350$) is incorrect.

This is confirmed by the following examples. Inspections of the reducers and roll gears of rolling mills, carried out by TsNIITMASH personnel in 1949–1950 at the Zaporizhstal, UZTM, MShZ, and other plants, showed that gear wheels subjected to intensive fatigue pitting continue to operate for years and even decades. In this case the surfaces of the mating teeth, if the endurance limit for them has not been exceeded, as a rule do not react to damage of the teeth meshing with them and remain clean.

The staff of the Tomsk Polytechnic Institute, who inspected hoisting machines at 25 mines of the Kuzbass, where foreign firms had installed reducers in the 1930s that are still operating today [1], arrived at similar conclusions.

Observations of the operation of gear transmissions [1, 5] show that limited pitting does not reduce the service characteristics of a transmission and, in a whole number of cases, plays the same positive role as running-in wear, since it promotes a more favorable distribution of the load among the contact lines and thereby improves the operating conditions of the transmission.

Thus, it may be considered established that at present the majority of researchers allow the possibility of normal operation of gear transmissions with a certain degree of pitting damage to the working surface of the teeth.

However, the question of the degree of damage to the working surface is a very complex one; moreover, different authors adopt, as the fatigue criterion, different degrees of pitting damage to the working surfaces of the teeth. These endurance criteria can be divided into two groups. One of them relates the attainment of contact fatigue to the number of pits on the working surface of a tooth (Trubin, Murashko, Zablonsky, Pavlov, Stepanov, Belyanin, Niemann, Kalkert) or to the number of teeth affected by pitting (Davidson and Ku). Obviously, these criteria are conditional in character, and their reliability will cause...

certain doubts. Thus, while considering G. K. Trubin’s [7] choice of the principal indicator of transmission operation to be correct, one cannot agree with the endurance criterion he proposed.

R. Johnson and S. Crane [11], referring to Standard No. 11002, “Wear and destruction of gear teeth,” published by the American Gear Manufacturers Association, believe that the appearance of a chain of small pits along the pitch circle of helical gears (even immediately after the beginning of their operation) is a normal phenomenon. A similar opinion is held by the authors of works [4, 5, 6, 9, 10].

Fig. 1. On determining the quantity \(S\)

According to the data of work [13], in accordance with American standards, the endurance criterion for the working surfaces of teeth is taken to be damage to a tooth by three pits, each of which has an area not exceeding 0.001 square inch \((0.025\ \text{cm}^2)\). The inadequacy of such a criterion becomes obvious if two gear transmissions are compared: the first with \(m_n = 1\ \text{mm}\) and \(b = 8\ \text{mm}\), and the second with \(m_n = 10\ \text{mm}\) and \(b = 80\ \text{mm}\).

This led the author of work [13] to propose the following criterion for damage to the working surfaces of teeth,

\[ S = \sum_{i=1}^{n} x_i \cdot b, \]

where

\(x_i\) is the linear size of a pitting cavity (Fig. 1);

\(b\) is the working width of the toothed rim.

When summing the dimensions \(X\), all pitting cavities are taken into account. As an indicator of limiting surface damage, the quantity \(S_{\max}\) was adopted. The author considers this quantity to be an increasing function of the number of cycles of load variation, with \(S_{\max} \le 1\). On the basis of a number of investigations, the author proposed \(S_{\max} = 02\). This group of endurance criteria is also unacceptable for the following reasons.

It is known that pitting, even in transmissions manufactured with a high degree of accuracy (grade 6 and higher), is distributed very unevenly among the teeth. Therefore, achieving comparatively uniform damage to the greater part (for example, 70% [3]) of the teeth of the tested wheels within strict limits (a specified number of pits on each tooth) becomes almost impossible. Moreover, counting the number of pits on the teeth is a very painstaking operation, the labor input of which depends on the size of the gear wheels. In view of the circumstances indicated above, the application of these endurance criteria should evidently be regarded as irrational.

As for the permissible area of pitting damage (the second group of criteria [e.g., 12, 14]), as far as we know, no universal relationship connecting it with the contact endurance limit exists at the present time.

In light of the foregoing, the fatigue criterion proposed in [6] and confirmed by numerous experiments in the strength laboratory of Lensovnarkhoz and the A. F. Mozhaisky LKVVIA, as applied to helical transmissions carrying a load constant in magnitude and having hardness \(HB \le 350\), becomes understandable.

All that has been said above undoubtedly shows that assigning a fatigue criterion even for a very narrow range of transmissions (i.e., for helical gears with hardness of the working tooth surfaces \(HB \leq 350\)) is a difficult matter, since the stage of pitting at which the transmission can still operate for the specified time, or is close to failure, is assessed subjectively. In addition, the choice of a fatigue criterion is influenced by such factors as the type of lubrication, the accuracy of manufacture and assembly, the criticality of the gear transmissions, etc.

Fig. 2. Condition of the working tooth surfaces after operation under loads exceeding the limiting loads

It becomes clear that the very fact that transmissions with pitting can operate normally is still far from resolving the question of the fatigue criterion, since the opinions of Soviet and foreign researchers differ sharply when establishing a definite (allowable or limiting) degree of damage to the working tooth surfaces by fatigue pitting. From the works listed it follows that the magnitude of allowable damage by pits ranges from 0.6% [12] to 10–25% [6, 10] of the total area of the working tooth surfaces. All this greatly complicates (or even sometimes makes impossible) the comparison of data obtained by different authors.

The fatigue criteria considered for helical gears with \(HB \leq 350\) were proposed for gear transmissions operating at pitch-line speeds not exceeding 15–20 m/sec. As far as we know from the literature sources, there are no proposals concerning the adoption of possible fatigue criteria at speeds greater than 20 m/sec. In this connection, the question of a fatigue criterion for high-speed transmissions is an even more complex problem. Without claiming to solve it, we shall cite only some considerations that formed the basis for adopting the fatigue criterion when testing high-speed transmissions [2].

Pitting in high-speed transmissions does nevertheless occur, although to a considerably lesser extent than in slow-speed transmissions (or, correspondingly, in the second stages of high-speed reducers) [8]. However, the conditions under which these transmissions operate, and the degree of their criticality, require a very serious approach to the question of assigning a pitting criterion. Obviously, simply adopting the aforementioned

criteria proposed for quiet-running transmissions is hardly possible.

An analysis of the works shows that, until a sufficient quantity of experimental data has been accumulated, the following proposal should be regarded as the most appropriate. As the limiting load, one should take the greatest load at which the transmission can operate for a very large number of stress-change cycles (for example, \(2 \div 3\,N\) bases) without noticeable development of damage to the working surfaces of the teeth, recorded in the course of operation. Experiments have shown that exceeding this load by \(25 \div 30\%\) over a time corresponding to the base number of cycles or less led to rapid progression of the damage and failure of the transmissions (Fig. 2). If, in this case, a small concentration of pitting cavities is observed on several teeth, their development should not be taken into account.

LITERATURE

1. A. E. Belyaev et al. Inspection of the condition of the mechanical part of hoisting machines that had operated in the mines of the Kuzbass for more than 20 years. Abstracts of a report, Tomsk, 1959.

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

3. A. I. Belyanin. Study of factors influencing the load-carrying capacity of gear transmissions. Mozhaisky LKVVIA, 1957.

4. Ya. G. Kistyan. Methodology for calculating gear transmissions for strength. TsNIITMash, book 107, Mashgiz, 1963.

5. V. N. Kudryavtsev. On strength calculations of gear transmissions. Collected works, LMI, No. 23, 1962.

6. V. N. Kudryavtsev, I. M. Klimov. On the question of evaluating the load-carrying capacity of gear transmissions limited by resistance to fatigue pitting. Collected works of LMI, No. 34, 1963.

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

8. S. Archer. Some Teething Troubles in Post-war Reduction Gears, The Institute of Marine Engcers, vol. XVIII, No. 9, 1956.

9. I. D. Grahan. Pitting of Gear Teeth, Handbook of Mechanical Wear, 1961.

10. R. Howard. Criteria for Surface Failure, The Engineer, vol. 185, Febr., No. 4805, 1948.

11. R. Jonson, S. Craine. Iron and Steel, No. 9, 1953.

12. W. T. Kalkert. Lebens dauer-versuchungen an breiten ungehaerteter Zahnrädern, Industrie Anzeigen, No. 19, III, 1961.

13. T. Lewandowski. Przeglad mechaniczny, No. 2, 1961.

14. G. Nieman, W. Richter. Konstruktion, Heft 6, Juni, 1960.