ON THE CAUSES OF INCREASED CONTACT STRENGTH OF GEAR TRANSMISSIONS WITH INCREASING CIRCUMFERENTIAL SPEED

A. E. BELYAEV

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

Experimental data [1] obtained on gear transmissions (with parameters characteristic of the drives of power installations) convincingly show that contact strength increases with increasing speed.

There are several hypotheses explaining the influence of speed on contact strength.

The most widespread hypothesis is that, as the rolling speed increases, the thickness of the oil film between the teeth of the transmission increases. In this case the contact area increases substantially, which leads to a more uniform distribution of pressure. Consequently, the maximum pressure, as well as its distribution over the contact strip, becomes significantly smaller than in metallic contact. This follows from the theoretical work of Academician P. L. Kapitsa [6]. The author of that work was the first to apply the classical hydrodynamic theory of lubrication to the calculation of surfaces with double curvature, taking into account the dependence of oil viscosity on pressure. He also showed that, as applied to rolling bearings, the stresses in the contact zone determined on the basis of the theory of elasticity (without taking into account the influence of lubrication) have a very large magnitude, lying beyond the limits of elastic deformation (concentration of loads on a small contact area). The work shows that, with increasing speed, the favorable action of the oil film between the contacting surfaces increases. Considering this question as applied to bearings, the author of [6] considers it possible to extend its conclusions also to gear transmissions.

The author of [8] believes that, since the oil film separating the working surfaces acquires the properties of a solid body* because of the high pressures, this film significantly increases the real contact area, owing to the transition from contact of asperities to full contact, and thereby reduces the acting specific loads.

Studies by H. Niemann and Olendorf [15] have shown that, as a result of contact between the peaks of asperities, only part of the normal force \(P_n\) is transmitted hydrodynamically (\(P_{\text{hydr}}\)), while the remaining

* According to the data of [12], this can occur only if the time of load application is less than the relaxation time of the fluid \(\tau\) (for oils \(\tau \simeq 10^{-6} \div 10^{-5}\) sec).

part by direct contact of the surfaces \((P_{\text{Hertz}})\). Proceeding from this, the total power of friction losses in meshing will consist of friction losses under direct contact of the working surfaces of the teeth and friction losses in the oil layer. In the work under consideration, experimental curves were obtained for the variation of the relative power of losses in meshing as a function of circumferential speed, transmitted force, and other factors.

The good agreement with experiment of the curves of the coefficient of relative power of losses in meshing as a function of circumferential speed, obtained theoretically, made it possible to draw interesting conclusions. The main conclusion is that the part of the load transmitted hydrodynamically, i.e., through the lubricant layer, increases continuously with increasing circumferential speed of the gears. In this, in the opinion of the authors of [15], lies the basis of the mechanism of the favorable influence of increasing circumferential speed on contact strength. Calculations show that, when the circumferential speed changes from 2 to 20 m/s, the fraction of the normal force transmitted through the oil layer increases from \(0.38P\) to \(0.8P\), i.e., by more than a factor of two. At a fixed circumferential speed, the fraction of the normal force transmitted hydrodynamically is the greater, the lower the roughness of the tooth surfaces and the greater the viscosity of the oil and the radii of curvature of the profiles.

The creative works of K. Weber and G. Niemann show that the reason for the favorable influence of increasing circumferential speed on the load-carrying capacity of transmissions (from the standpoint of contact strength) is the different nature of the distribution of contact stresses when the load is transmitted hydrodynamically and when it is transmitted directly by contact of the surfaces. This proposition is illustrated by Fig. 1. It is evident (as a first approximation) that only that part of the load which is transmitted by the teeth through metallic contact may be associated, as the initial quantity for calculation, with fatigue pitting.

Fig. 1. Nature of the distribution of contact stresses under hydrodynamic transmission of load \((P_{\text{hydr.}})\) and under transmission of load directly by metallic contact \((P_{\text{Hertz}})\)

The opinion that, with increasing speed, the fraction of the load transmitted directly by metallic contact decreases is also expressed by Borzov [11]. He believes that at high speeds the time of load application is insufficient for complete squeezing-out of the lubricant; moreover, the higher the speed, the larger the amount of lubricant remaining in the contact.

The hypothesis presented above, explaining the connection between contact strength and speed, is also supported by Sykes [17] and others.

A rather logical explanation of this connection may also meet with certain objections. Thus, from the works of A. N. Grubin [5] and A. I. Petrusevich [9] it follows that the presence of lubrication does not change the basic stressed state (if the local pressure peak upon leaving contact* and friction forces are not taken into account), and in calculations one may use the Hertz–Belyaev formula.

In this case, if this hypothesis were correct, a change in rolling speed would affect deep strength. If, with increasing speed, the stresses on the contact area decrease, then the stresses beneath the layer should also change, since they are a function of the surface stresses. Consequently, between speed and the deep strength of a material with a hard coating there should exist the same dependence as that established between speed and the contact strength of the working surfaces of teeth: with increasing speed, the endurance of the working surfaces of teeth increases. However, the experiments of R. R. Galper [3] showed that the limit of deep strength does not depend on speed. From this it may be concluded that changes in the basic surface stresses do not occur.

The essence of the second hypothesis is that, with an increase in the frequency of load application, the material is strengthened, the mechanical characteristics increase, and, consequently, the contact strength rises. An increase in the mechanical properties of materials should naturally also increase the bending strength of gears.

However, the results of a large number of experiments aimed at identifying an increase in mechanical characteristics with an increase in the frequency of load application are highly contradictory, and this applies equally to results obtained on specimens and to results obtained directly on gears.

According to the data of V. Weibull [2], the fatigue limit of metal usually does not depend on frequency up to 10,000 cycles/min, but increases with an increase in the test frequency above this value.

The author of work [2] indicates that raising the frequency from 20 to 2000 cycles/min leads to a 1.5–5.5-fold increase in the number of cycles to failure.

G. Lomas, I. Reith, I. Ward, and E. Kolbek [2], as a result of their experiments in 1956 with 8 different grades of steel, showed a steady increase in the fatigue limit with increasing frequency up to a maximum value between 1200 and 1800 Hz, and then a progressive decrease in it as the frequency increased to 25,000 Hz.

Table 1 presents the results of processing a number of experiments, the analysis of which shows that the bending fatigue limit increases only slightly with increasing frequency.

Table 1

Change in the bending fatigue limit with increasing frequency of load application

In addition, a twentyfold change in the frequency of load application (from 5 to 105 Hz) in the experiments of various authors did not affect the limit of deep strength (Table 2).

The author of work [14] asserts that, for metals operating under condi-

* The author of work [7] did not observe the indicated phenomenon upon leaving contact.

under conditions of pure rolling, the frequency of stress variation does not have a substantial effect on surface fatigue: when the machine speed was varied from 1040 to 3200 rpm, the results of the experiment on rollers proved similar. In [16], however, the opinion is expressed that contact strength increases as the frequency of tooth contact increases.

Table 2

Frequency of load application in experiments according to data from various authors

Thus, this hypothesis too has not yet been sufficiently tested, and its existence will depend on the further accumulation of experimental data.

And, finally, the third hypothesis explaining the phenomenon under discussion is the connection between rolling speed and the coefficient of friction of the material. A. I. Petrusevich [9] shows that the strength of a material will be the higher, the lower the coefficient of friction. The decrease in the values of the coefficient of friction with increasing speed has been confirmed theoretically and experimentally by a large number of researchers (e.g., [5] and others).

Fig. 2. Effect of the viscosity and type of lubricant on the endurance pitting limit of teeth

An increase in the endurance pitting limit was also recorded in the experiments of G. K. Trubin [10] (Fig. 2), and the author reduces the influence of viscosity on the contact endurance of transmissions to the influence of friction in connection with the change in the magnitude of contact stresses: the favorable effect of increasing lubricant viscosity is explained by a decrease in the magnitude of contact stresses due to a reduction of friction forces on the teeth.

This conclusion, which applies equally to gear wheels made of soft and surface-hardened mate-

...materials is valid, in the author’s opinion, only for transmissions operating under conditions of semi-fluid friction, in which an increase in viscosity causes a decrease in the coefficient of friction. For high-speed transmissions, which operate under conditions of purely fluid friction*, an increase in the viscosity of the lubricant in the contact should, on the contrary, increase friction.

Thus, A. I. Petrusevich’s hypothesis qualitatively explains the increase in the strength of the material with increasing speed. However, calculations carried out by R. R. Galper showed that the analytical dependences presented in [9] do not yield quantitative agreement with experiment. Thus, if in these dependences the coefficient of friction is varied from 0.08 (as the maximum possible for gear transmissions according to [4]) to 0, then the contact-fatigue limit increases by approximately 20%. According to the data of [3], in fact the fatigue limit, when the total rolling speed changes from 1 to 60 m/sec, increases by more than a factor of two, and according to our data [1] by slightly less than a factor of 2. Thus, in addition to friction forces, which do not affect the strength of the contacting material, a number of other factors also act which, changing together with the rolling speed, also change the fatigue limit.

Moreover, as noted in R. R. Galper’s experiments, the influence of these factors is limited to the thin surface layers of the metal, since the rolling speed did not affect the limit of subsurface strength.

The most probable assumption, in our opinion, is that, along with friction forces, the thickness of the oil film also plays an enormous role. In this connection it is expedient to note that the decrease in the coefficient of rolling friction in the contact with increasing speed and the simultaneous increase in the thickness of the oil film are phenomena that are evidently closely related to each other; therefore, an explanation of the phenomenon under consideration—the influence of speed on contact strength—will probably be more plausible if the first and third hypotheses are combined.

LITERATURE

1. A. E. Belyaev. The influence of speed on the contact strength of gear transmissions. Izv. TPI, No. 157. Tomsk, 1968.
2. V. Weibull. Fatigue Testing and Analysis of Its Results. Moscow, 1964.
3. R. R. Galper. Contact Strength of High-Speed Gear Transmissions. Dissertation, Leningrad, 1964.
4. M. D. Genkin. Questions of Scuffing of Gear Wheels. Academy of Sciences of the USSR, Moscow, 1959.
5. A. N. Trubin. Fundamentals of the Hydrodynamic Theory of Lubrication of Heavily Loaded Cylindrical Surfaces. (TsNIITMASH, book 30), Mashgiz, 1949.
6. P. L. Kapitsa. Hydrodynamic theory of lubrication in rolling. Tekhnicheskaya fizika, section “Mechanics,” vol. 25, issue 4, 1955.
7. D. S. Kondir. Contact-Hydrodynamic Theory of Lubrication. Kuibyshev, 1963.
8. A. V. Orlov. The Role of Dynamic Loads in the Contact Strength of Gear Transmissions. Izd. “Nauka,” Moscow, 1964.
9. A. I. Petrusevich. Principal conclusions from the contact-hydrodynamic theory of lubrication. Izvestiya AN SSSR, OTN, No. 2, 1951.
10. G. K. Trubin. Contact Fatigue of Materials for Gear Wheels. Mashgiz, 1962.
11. W. N. Borsoff. Transaction of ASME, Ser. D, III, 1959.
12. H. Brugger. VDI—Zeitschrift, Bd. 103, No. 6, 1961.
13. H. Christensen. Teknisk ukeblad, 109, No. 13, 1962.
14. I. D. Cram. Handbook of Mechanical Wear, 1961.
15. G. Niemann. Maschinenelemente, II, 93, 1960.
16. L. Rozeanu; O. Prepţescu. Studii si cercetări de mecanică aplicată extras, vol. 8, No. 1, I–III, 1957.
17. A. Sykes. The Engineer, Nov., No. 5416, 1959.

* Machine-building practice, in a number of convincing examples [13], has shown the presence of a continuous lubricating layer between the tooth profiles of gear wheels. Cases have repeatedly been noted of many years’ operation of transmissions working over a wide range of speeds, with machining marks from their manufacture preserved on the working surfaces of the teeth.