STUDY OF THE STRESS STATE OF WORN TEETH OF SPUR GEARS UNDER BENDING

I. F. Deryuga

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

In a number of cases, when it is necessary to decide whether worn gear wheels are suitable for further operation, it becomes necessary to determine the actual bending strength of the teeth of these wheels. However, in solving this problem, owing to the present lack of sufficiently reliable methods for calculating worn teeth, difficulties arise that are associated with determining the true stresses in the teeth. The use, in calculations of worn teeth, of formulas obtained for new teeth by introducing correction factors is only a partial and far from complete solution of this problem, since it does not take into account the great variety of factors that affect the bending strength of worn teeth, nor does it take into account the change of these factors over time.

Existing methods for calculating worn teeth for bending [2, 4] are based on the assumption that, under wear, the involute profile of the working side of the tooth is preserved, the magnitude and line of action of the force are unchanged, and the critical section of the tooth does not change its position and remains at the fillet at the tooth root. In other words, the reduction in the bending strength of teeth under wear is associated only with a reduction in tooth thickness.

In works [1, 3] it is noted that, as teeth wear, their working profile loses its initial involute form. As a result, the magnitude and line of action of the force applied to the tooth, as well as the position of the critical section of the tooth, change as wear increases. For this reason, the above-mentioned methods for calculating worn teeth for bending lack sufficient justification.

The influence of wear on the bending strength of teeth, taking into account the change in their profile, is described in work [6]; however, the friction forces occurring in the meshing of the teeth were not taken into account there.

The aim of the present work was to determine to what extent friction forces affect the bending strength of worn teeth and, in addition, to determine numerical values of the theoretical stress-concentration factor for two zones on the tensile side of the tooth contour.

As the object of investigation, the gear wheels of the reducer of an ER-5 mining electric drill [6] were adopted. The principal regularities governing the change in the profile shape of the worn teeth of this reducer are described in work [5].

Tooth wear was calculated along the arc of the pitch circle and expressed as a percentage.

\[ \Delta S_i = 100\left(1-\frac{S_i}{S}\right)\%, \]

where \(\Delta S_i\) is the amount of tooth wear along the arc of the pitch circle, in %;
\(S\) is the thickness of a new tooth along the arc of the pitch circle, in mm;
\(S_i\) is the thickness of the worn tooth along the arc of the pitch circle, in mm.

The pressure angle was determined by the formula

\[ \gamma = \delta \pm \rho, \]

where \(\gamma\) is the pressure angle taking into account the friction force, in degrees;
\(\delta\) is the pressure angle without taking into account the friction force, in degrees;
\(\rho\) is the friction angle, in degrees.

Fig. 1. Graph of the change in pressure angle with wear

Figure 1 shows a graph of the change in the pressure angle \(\delta\) as the teeth wear. For tooth wear of approximately up to 30%, the relation between the pressure angle \(\delta\) and the magnitude of relative wear \(\Delta S\) may be represented as follows:

\[ \delta_i = \delta_e + 0.15\Delta S_i, \]

where \(\delta_i\) is the pressure angle of the worn tooth, in degrees;
\(\delta_e\) is the pressure angle of the new tooth, in degrees.

The values of the coefficient of friction for the experimental gear wheels were taken from work [7].

The analytical calculations associated with determining the magnitude of the nominal stresses on the tooth contour were carried out using the well-known formulas of strength of materials for plane sections. The stress-distribution diagrams over the tooth height constructed on the basis of these calculations made it possible to establish that, as tooth wear progresses, the dangerous section shifts from the tooth root to its head, and the bending strength of the tooth does not undergo substantial changes for wear up to 20%. The latter is explained by the fact that, with an increase in the pressure angle and in the force acting

on the tooth, the tangential component of the force remains constant (when transmitting a constant torque), while the radial component increases, causing a corresponding increase in compressive stresses. In the sections of the worn tooth, on the tensile side, the stresses from bending and compression are redistributed in such a way that, with tooth wear of up to 20%, the total stress in the critical section at the fillet in the worn part of the tooth practically does not differ from the maximum total stress in the critical section of a new tooth.

To clarify the pattern of stress distribution in the teeth, an experimental study was carried out by applying the photoelastic method. For this purpose, transparent models of teeth were made at a scale of 15:1 from the isotropic, optically sensitive material ED-6. To load the tooth models, a special loading device was designed and manufactured, making it possible to create the required magnitude of force and at the same time to provide the necessary pressure angle. The experiment was conducted on a PPU-7 polarization-projection apparatus, with geometric and force similarity observed. To obtain the fringe pattern, polarized monochromatic light with wavelength \(\lambda \simeq 546\,m\mu\) was used; the tests were carried out under conditions of circular polarization. To decode the fringe pattern and calculate the numerical value of the stresses, calibration was performed.

Fig. 2. Interference fringe pattern of a model of a new tooth: left—loaded only by the normal force; right—loaded additionally by the friction force.

Figure 2 (left) shows a model of a new wheel tooth loaded only by the normal force. Beside it, for comparison, the same model is shown loaded additionally by the friction force. Comparing the patterns of the stressed states, it was found that the magnitude of the total stress on the tensile side in this case decreased by approximately 10%, while on the compressive side it correspondingly increased by the same amount. For pinion teeth, since the friction force in this case is directed away from the pitch point, the opposite phenomenon will evidently be observed, i.e., a certain increase in the total stress on the tensile side and a corresponding decrease on the compressive side. It was also established that the patterns of the stressed sta-

states of worn teeth differ from the pattern for an unworn tooth, the more so the greater the wear. With wear of approximately up to 20%, the dangerous section of the tooth practically does not change its position and corresponds to the position determined by calculation; stress concentration occurs at the beginning of the transition curve of the tooth root. With tooth wear of more than 20% and up to sharpening of the tooth, two clearly expressed zones of stress concentration appear on the tensile side of the contour; in this case the maximum total stress occurs in the fillet on the worn part of the tooth root.

In [6] it is noted that, when models of worn teeth made of a brittle aluminum alloy were loaded by a static force, the latter fractured along an inclined flat section from the location of the fillet on the worn part of the root to the location of the transition curve on the compressed side of the tooth. Thus, the obtained nature of fracture of worn teeth confirms the patterns of the stressed states of the teeth. The nature of failure of worn teeth can also be explained on the basis of the values we obtained for the theoretical stress-concentration factor, calculated for two characteristic concentration zones on the tensile side.

Fig. 3. Interference pattern of the field of a tooth model with wear \(\Delta S_i = 43\%\), loaded taking into account the friction force

Conclusions

1. As tooth wear progresses, beginning at approximately 20% wear, the dangerous section of the tooth changes its position, moving from the tooth root toward its head.

2. As wear increases, the magnitude of the maximum total stress on the tensile side gradually increases; in this case, the bending strength of the tooth does not decrease substantially for wear up to 20%.

3. When calculating the magnitude of the maximum total stress without taking into account the friction axis in the mesh, an error occurs, amounting to approximately \(\pm 10\%\).

4. With tooth wear of more than 20%, two clearly expressed zones of stress concentration occur on the tensile side of the contour, the more dangerous of which is the fillet zone on the worn part of the tooth.

References

1. V. A. Dobrovolsky, K. I. Zablonsky, S. L. Mak, A. S. Radchik, L. B. Erlikh. Machine Parts. Mashgiz, 1960.

2. V. I. Zaslavsky. Calculation of gear meshes for strength and wear by a new method. Bulletin of Engineers and Technicians, No. 3, 1936.

3. V. N. Kudryavtsev. Gear Transmissions. Mashgiz, 1957.

4. A. K. Sulimov. On the problem of calculating the maximum permissible wear and service life of gear transmissions. Scientific Notes of the Voronezh Forestry Engineering Institute, vol. 16, 1959.

5. Yu. S. Semenov. Results of an experimental study of wear of gear wheels of reducers of mining electric drills. Proceedings of TPI, vol. 108, 1959.

6. V. P. Steshenko. The influence of wear on the bending strength of teeth of gear wheels. Proceedings of TPI, vol. 114, 1964.

7. Yu. S. Semenov, V. T. Gorbenko. Wear and cleanliness of the working surfaces of teeth of gear wheels. Proceedings of TPI, vol. 114, 1964.