ON THE BENDING FRACTURE STRENGTH OF WORN TEETH

I. F. DERYUGA, S. I. SHUBOVICH

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

In the domestic and foreign literature, as a rule, the results are given of studies of the bending fatigue strength of the teeth of new (unworn) gear wheels, and no attention is paid to the strength of worn teeth, whereas in practice, in a number of cases, in order to increase the service life of a transmission, the operation of gear wheels with relatively great wear is permitted. In the literature, for example, the influence of the amount of wear and of pitting, which in some cases accompanies wear, on the fatigue strength of a tooth is not covered. Nor is there clear information on the permissible degree of tooth wear from the condition of ensuring the bending strength of the latter.

The purpose of the present article is to partially fill this gap.

As the object of investigation, the authors took hardened steel spur cylindrical wheels of the gearbox of the ER-5 mining electric drill. The basis for this choice was that the gear transmissions of mining electric drills, because of overload in contact stresses caused by restrictions on the dimensions and weight of the drill and the penetration of coal dust into the gearbox housing, wear heavily under operating conditions. The functional purpose of these transmissions is such that wear and limited pitting of the working surfaces of the teeth are entirely permissible, and, consequently, the service life of the wheels must be determined from the condition of maintaining sufficient bending strength of the teeth under wear.

The wearing-in of the investigated wheels was carried out on a closed test stand under loads and operating conditions close to those in service. The experimental wheels were made of 40Kh steel with hardness \(HRC = 28—45\) and had 28 teeth of normal mesh; the pinions—of 20Kh steel with hardness of the working surfaces \(HRC = 55—60\)—had 17 teeth. The module of the gear mesh was \(m = 2\) mm.

An analysis of the mesh geometry of a large number of worn gear pairs showed that, as wear proceeds, the original involute profiles are regularly distorted and remain conjugate until the contact ratio decreases to a value close to unity. The line of action is curved in such a way that the pressure angle at the beginning and at the end of engagement of one pair of teeth increases sharply in comparison with its initial value [1]. This phenomenon leads to the fact that the tangential component of the normal pressure force applied at the tooth tip decreases, while the radial component increases—

For this reason, in the sections of a worn tooth the ratio of bending and compressive stresses changes. The total stress on the tensile side of the tooth decreases because of an increase in compressive stresses and a decrease in bending stresses.

Calculations carried out using formulas based on the hypothesis of plane sections showed that, as the tooth wears, the dangerous section shifts from the tooth leg toward its tip, while the magnitude of the stresses in these sections remains practically constant during the first period of wear and then increases sharply. It was established that, when the tooth is worn to 20% of its initial thickness, measured along the pitch-circle arc, the total stress remains practically unchanged and, consequently, for wear up to this degree the bending strength of the tooth should not decrease.

Thus, the amount of wear constituting 20% of the initial tooth thickness may be regarded as an allowable amount of wear from the condition of ensuring the bending strength of the tooth.

It is appropriate to note that, for an amount of wear close to 20%, two dangerous sections are observed: at the tooth root at the fillet and somewhat higher, at the step formed as a result of wear (Fig. 1).

Fig. 1. Profile of a tooth worn by 20%, and stress diagrams

To verify these calculations and to clarify the pattern of stress distribution, a study of the stress state of a worn tooth was carried out by the photoelastic method [2]. The results of these studies confirmed the conclusions made above and made it possible to determine the theoretical stress-concentration factors at the dangerous points. Thus, for example, it was established that the step formed on the tooth profile as a result of wear causes approximately the same stress concentration as the fillet at the tooth root. Therefore it may be concluded that, at 20% tooth wear, tooth breakage during operation of the transmission is equally probable both at the tooth root and in the section at the step formed by wear.

For the purpose of determining the fatigue strength of a worn tooth and clarifying the negative influence of pitting marks, which were observed on worn wheels heavily overloaded by contact stresses, full-scale tests of unworn

Fig. 2. Fatigue curves of wheel teeth with hardness HRC = 45

Fig. 3. Fatigue curves of wheel teeth with hardness HRC = 28

and worn wheels (by 20% of the initial tooth thickness) for tooth fracture under cyclic alternating load. These tests were carried out on a specially designed pulsator, which provided loading according to the “gear wheel—rack” scheme with a cycle asymmetry coefficient close to zero.

The results of the fatigue tests, processed using methods of mathematical statistics, are presented in Figs. 2 and 3.

Figure 2 presents the fatigue curves in logarithmic coordinates for an unworn wheel with hardness \(HRC = 45\) (line 1) and a wheel of the same hardness with teeth worn by 20% in thickness (line III), which showed no signs of contact failure. It can be seen that, in this case, the presence of wear did not cause a decrease in the unlimited endurance limit in tooth bending.

Figure 3 presents the results for wheels of low hardness (\(HRC = 28\)). During wear of wheels of this hardness, limited pitting occurred; therefore, the working surfaces of the worn teeth in the pitch-zone area were affected by pits. Comparing the unlimited endurance limits (Fig. 3) of the unworn wheel (line II) and the worn wheel (line IV), it can be seen that, in this case, the fatigue strength of the tooth decreases under wear by approximately 20%, i.e., by an amount approximately equal to the magnitude of the relative tooth wear. The reason for the decrease in strength in this case is the stress concentration caused by pitting cavities. This explanation is supported by the fact that the fatigue crack developed from the zone affected by pitting.

Thus, on the basis of the conducted investigation, it may be concluded that, in the absence of pitting on the working surfaces of the teeth, wear of up to 20% of their initial thickness is safe from the standpoint of bending strength. A decrease in fracture strength may occur only when pitting is present on the working surfaces of the teeth.

REFERENCES

1. Yu. S. Semenov. Change in the parameters of gear meshing under wear. Izv. TPI, vol. 96, issue 2, 1961.

2. I. F. Deryuga. Investigation of the stress state of worn teeth of spur gears in bending. Izv. TPI, vol. 173 (in press).