Experimental Determination of Spalling Areas on the Working Surfaces of Helical Gear Teeth

A. E. Belyaev

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

Determining the load-carrying capacity of gear transmissions is associated with adopting a definite criterion that evaluates the strength of the working surfaces of the teeth. Among the large number of criteria proposed by researchers (noise, change in efficiency and surface roughness, etc.), the most definite is the adoption of some degree of damage to the working surfaces of the teeth by spalling.

At present, the majority of various researchers consider it possible to determine the load capacity of transmissions on the basis of allowing spalling (it should be noted that recognizing the possibility of normal operation of helical transmissions with a certain degree of spalling is very important). The point is that denial of this fact would sharply reduce the load-carrying capacity of helical gears. The following propositions of Prof. V. N. Kudryavtsev may serve as the basis for this.

In the near-pole zone of a spur-gear transmission, the actual length of the contact line is equal to the width of the gear rim (i.e., one pair of teeth is in operation at this time). In a helical transmission, the contact lines are located on the tooth surfaces at an inclination to the generatrix of the base cylinder. Therefore (at \( \frac{B}{t_a} = 1 \) and more), theoretically no fewer than two pairs of teeth are in mesh. The total load transmitted by the mesh is distributed between the addendum and dedendum in accordance with the length of the contact lines passing over the addendum and dedendum of the tooth.

Owing to the inevitable manufacturing errors inherent even in fairly precise transmissions, during the initial period of time contact may be absent in some regions of the working surfaces of the teeth and, consequently, the specific loads may prove greater than those for which the transmission was designed. These increased loads cause spalling in the gear transmission, which quickly ceases as a result of the running-in of the pair and the redistribution of pressure along the contact lines.

Such spalling of the dedenda, as the weaker elements, may occur under loads that are quite permissible for a spur-gear transmission. If, as was indicated above, the criterion of inoperability is taken to be the appearance of spalling, then such a transmission may be regarded as having failed, and in this sense, with respect to load-carrying capacity, it would be equivalent to, or even worse than, a spur-gear transmission. However, fur-

a further increase in the load will not lead to an increase in spalling. Moreover, nonprogressive spalling of the tooth root, which will occur at an even higher load, likewise does not put this transmission out of service.

Long experience in the operation of gear transmissions (spur, helical, and herringbone), as well as observations of gearboxes at UZTM and of hoisting machines in the mines of the Kuzbass (the latter carried out by us in the late 1950s), confirm the validity of the opinion expressed above: gears subjected to fatigue spalling continue to operate for a very long time.

However, the opinions of researchers differ when it comes to specifying the criterion associated with damage to the working surfaces of the teeth (the degree of this damage). Some associate the failure of a gear with a certain area of damage to the working surface, others with the number of pits appearing on it, and still others with the number of teeth affected by spalling.

A. I. Belyanin [3], in the course of conducting an experiment, compared the condition of the working surfaces of the teeth of the test gear with the condition of a reference gear, the tooth roots of which had been affected by pitting by 30%. The test ended when at least 70% of all teeth were affected by the same degree of spalling. The criterion mentioned was the same for both spur and helical gears.

V. Kalkert [9] measured the load-bearing area of the flank surface \((V_G)\) damaged by pitting:

\[ V_G=\frac{\text{area covered by cavities}}{\text{total (active) contact area}}\,100\%. \]

The area measurements were performed using a measuring magnifier with 25× magnification and a reticle scale. As the magnitude of permissible spalling, the value \(V_G=0.6\%\) was adopted (at various specific pressures).

G. Niemann and V. Richter [10, 11], for all gear transmissions tested, take as the permissible magnitude of spalling the value

\[ G=G_1+G_2=2\%, \]

where

\[ G_1=\frac{F_{G_1}}{F_{A_1}}\cdot 100\% \quad \text{and} \quad G_2=\frac{F_{G_2}}{F_{A_2}}\cdot 100\%, \]

In the last expressions:

\(F_{G_1}\) and \(F_{G_2}\) — the area of cavities on the working surfaces of the teeth of the pinion and the gear, respectively;

\(F_{A_1}\) and \(F_{A_2}\) — the total area of the working surfaces of the pinion and the gear, respectively.

The areas \(F_G\) and \(F_A\), when there was a significant number of teeth, were determined on no fewer than 30 teeth, and when there were few teeth, with all teeth taken into account. The indicated fatigue criterion was adopted by the authors when testing both spur and helical gears.

Ya. G. Kistyan believes that in helical transmissions made of improved steels, when there is a small difference in the hardness of the working surfaces of the teeth of the pinion and the gear, only nonprogressive fatigue spalling of the tooth roots in one of the two gears may be permitted.

As the fatigue criterion in work [5], damage by 5–6 spalling pits on the surface of a tooth was adopted, provided that the spalling affected no fewer than three quarters of the teeth of the gear being tested.

Davidson and Ku [8] determined and recorded the appearance of pitting on 1 tooth, on 3 teeth, and on 50% of the total number of teeth \((Z = 28)\). The criterion adopted was the appearance of spalling on any 3 teeth (provided that they were not adjacent).

The author of [7] considered the criterion for reaching the fatigue-spalling limit to be the presence of not fewer than 10 large or 20 small pits, uniformly distributed over the entire length of the tooth. As the main indicator of whether the transmission (spur or helical) was operating below or above the fatigue limit, the absence or presence of progressive spalling on the working surfaces of the gear teeth was taken. In this case it was assumed that destruction of the tooth surface along the entire length of the near-pitch zone indicated that the given transmission was operating under a load exceeding the fatigue-spalling limit.

In [6], when testing helical gear transmissions, a fatigue criterion was established that recorded not the fact of fatigue spalling on one of the gears, but the appearance of spalling pits both on the tooth root and on the tooth tip, or on the roots of both mating gears.

Fig. 1. Schematic diagram of the instrument for determining spalling areas

Analysis of the endurance criteria considered shows that all of them are, to a certain degree, conventional for two reasons: the subjectivity of the selected criterion and the possible accuracy of recording it. This leads to the fact that the magnitude of the permissible pitting damage lies in the interval from 0.6 [9] to 10–25% [4, 6] of the total area of the working surfaces, which makes it difficult to compare the load-carrying capacity of gear transmissions.

In the course of our experiments in the Gear Transmission Laboratory (LZP) [1], the condition of the working tooth surfaces was objectively recorded, beginning with the first pits and up to surface damage at limiting loads [2].

Thus, the entire development of the spalling process was under constant control. At the end of the test, a conclusion was drawn as to the presence or absence of progressive spalling of the tooth surfaces (the development of the spalling process on teeth that had a load concentration along the face width was not taken into account).

As indicated above, the evaluation of the adopted spalling criteria (spalling areas) by various researchers was carried out either with the aid of multiple-magnification magnifiers with a gridded scale [9], or by eye—for example, by comparison with a specified reference gear [3].

These methods for determining the absolute areas of spalling are, to a considerable degree, subjective and labor-intensive, and also—and this is the main point—produce a multiple error, failing to meet modern requirements for measurement accuracy.

Using the method of taking impressions by means of adhesive tape applied to the sooted surface (proposed by G. Niemann), we evaluated both the absolute and the relative degree of spalling of the working surfaces of the teeth with the aid of an instrument developed at the LZP (Fig. 1)*.

The design of the instrument is based on measuring the magnitude of the electrical signal from a photocell with electrical measuring instruments. The electric current in the photocell (1) arises under the action of a light beam from a stabilized light source (2), passing through a negative (3) or reflected from a photograph (3) obtained from the tooth impression. The tooth impression is obtained by applying transparent or white adhesive tape to the sooted surface of the tooth, as a result of which the soot located in the depressions of the spalled areas does not adhere to the tape. To focus the image in front of the photocell, an objective lens [4] is installed. The sharpness of the image is checked on a ground glass (5), which, by means of a special rotating device (6), is placed

\(N_{\mathrm{c.k}} = 1.04 \cdot 10^{7};\quad N_{\mathrm{total}} = 1.78 \cdot 10^{7};\quad K_{\mathrm{n}} = 17\ \mathrm{kg/cm^{2}}.\)

\(N_{\mathrm{c.k}} = 1.02 \cdot 10^{7};\quad N_{\mathrm{total}} = 2.92 \cdot 10^{7};\quad K_{\mathrm{n}} = 24.3\ \mathrm{kg/cm^{2}}.\)

\(N_{\mathrm{c.k}} = 1.05 \cdot 10^{7};\quad N_{\mathrm{total}} = 5.27 \cdot 10^{7};\quad K_{\mathrm{n}} = 26.3\ \mathrm{kg/cm^{2}}.\)

\(N_{\mathrm{c.k}} = 2.18 \cdot 10^{7};\quad N_{\mathrm{total}} = 4.5 \cdot 10^{7};\quad K_{\mathrm{n}} = 28.8\ \mathrm{kg/cm^{2}}.\)

Fig. 2. Impressions taken from teeth at different stages of destruction of their working surfaces. \(N_{\mathrm{c.k}}\) is the number of stress-change cycles of the wheel; \(N_{\mathrm{total}}\) is the total number of stress-change cycles; \(K_{\mathrm{n}}\) is the load-carrying-capacity coefficient of gear transmissions

in place of the photocell. On the same rotating device there is a reference photocell (7), which serves to monitor the stability of the readings of the working photocell and the constancy of the brightness of the light source (2).

Calibration of the electrical measuring instrument (microammeter) and construction of the calibration curve were performed using photographs or negatives with deliberately known spalling areas,

* The instrument was created by I. M. Klimov, V. L. Levanov, and G. V. Efimov.

from which a stabilized beam of light is reflected or through which it passes. For this purpose, with the aid of a microammeter, a reading is taken from photocell (1) corresponding to the given absolute or relative known area of coloring. The scale of the electrical measuring instrument is calibrated both in absolute and in relative areas.

Using the instrument described above, processing was carried out on the impressions obtained (at speeds of 20 and 40 m/s on gear transmissions with a module of 2.5 mm).

Figure 2 shows photographic prints of impressions of the surface of a new tooth and impressions of the same tooth with a gradually increasing degree of damage to the working surface caused by an ever-increasing load. (The results of processing the areas are given in Table 1.)

Table 1

| | \multicolumn{4}{c}{Circumferential speed, m/s} | \multicolumn{4}{c}{Circumferential speed, m/s} |
|---|---:|---:|---:|---:|---:|---:|---:|---:|
| | \multicolumn{4}{c}{20} | \multicolumn{4}{c}{40} |
| | \multicolumn{8}{c}{Impression indices} |
| Absolute area of coloring \(S_{\text{col}}\), % | 1-1-1 | 2-1-1 | 1-2-2 | 2-2-2 | 2-2-1 | 1-2-1 | 2-1-2 | 2-2-1 |
| Absolute area of coloring \(S_{\text{col}}\), % | 7.86 | 6.72 | 7.85 | 7.05 | 6.92 | 7.44 | 13.2 | 10.5 |

As follows from the table, the value of the areas covered by coloring, under the criterion adopted by us and at the limiting load, fluctuates within the range of 6–15%, and the absolute value in % at 20 m/s is on average 1.3 times smaller than at a speed of 40 m/s.

Although these results are far from exhaustive, it may nevertheless be noted that the criterion of fatigue coloring adopted in works [9, 10, 11], which is associated with damage to the working surfaces of teeth, substantially underestimates the load-carrying capacity of gear transmissions.

LITERATURE

1. A. E. Belyaev. On the contact endurance of teeth at different speeds. NIIMinformtyazhmash. Collection of reports at the conference in Saratov, Moscow, 1967.

2. A. E. Belyaev. On the choice of an endurance criterion for helical gears operating at high speeds. Izv. TPI, No. 188, 1974.

3. A. I. Belyanin. Investigation of factors affecting the load-carrying capacity of gear transmissions. TsKVIA im. Mozhaiskogo, L., 1957.

4. V. N. Kudryavtsev, I. M. Kilimov. On the question of evaluating the load-carrying capacity of gear transmissions limited by resistance to fatigue coloring. Collection of works of LMI, No. 34, 1963.

5. V. P. Murashko, K. I. Zablonskii. Contact endurance of heavily loaded gear wheels under a stepped loading regime. Publ. OPI, Sh., 1959.

6. Z. N. Pavlov. Influence of the hardness of the teeth of mating wheels on the load-carrying capacity of a transmission. Publ. OPI, 1, 1959.

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

8. T. F. Davidson, P. M. Ku. Lubrication Science and Technology, vol. 7, No. 1, 1961.

9. W. T. Kalkert. Industrie Anzeiger, No. 19, III, 1961.

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

11. W. Richter. Konstruktion, Heft 5, Maj, 1960.