EFFECT OF WEAR ON THE BENDING STRENGTH OF GEAR TEETH

V. P. STESHENKO

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

In a number of cases it becomes necessary to calculate the bending strength of worn teeth of gear wheels.

The existing methods for calculating worn teeth [2], [5] are based on the following assumptions:
1) the normal pressure acting on the tooth is considered constant in magnitude and direction; 2) the profiles of worn teeth are geometrically similar to the profiles of unworn teeth; 3) the critical section does not change its position along the height of the tooth and remains at the fillet at the tooth root. On the basis of these assumptions, when calculating worn teeth, correction factors [1], [7] are introduced into the calculation formulas for new teeth, by means of which the reduction in tooth strength due to wear is taken into account.

Thus, the reduction in the bending strength of worn teeth has until now been explained only by a decrease in the cross section of the teeth.

In reality, however, when teeth wear, the working profile loses its original involute form [1], [3], [4], [8], as a result of which the magnitude and direction of the normal-pressure force, as well as the position of the critical section of the tooth, change as wear increases. Therefore the initial premises in the above-mentioned methods for taking account of the influence of tooth wear on its bending strength do not have sufficient justification.

The purpose of the present article is to show the influence of wear on the bending strength of a tooth, taking into account the change in the tooth profile.

Studies carried out at the Tomsk Polytechnic Institute [6] made it possible to reveal certain basic regularities in the change of the profile form of worn teeth.

On a closed test rig, involute gear wheels of reducers for the ER-5 mining electric drill were worn. The wheels, made of 40Kh steel, with hardness of the working surfaces from \(R_c = 30\) to \(R_c = 40\), had 28 teeth. The pinions, made of 20Kh steel, with hardness of the working surfaces from \(R_c = 55\) to \(R_c = 60\), had 17 teeth. The module of engagement was \(m = 2\).

It was established that, under different loads and different combinations of hardnesses and initial cleanliness of the tooth surfaces, the basic regularities in the change of the geometric form of worn teeth

remain approximately the same. The profiles of the worn teeth of the experimental gears are not involute, and tooth wear occurs nonuniformly at different points of the profile along the height of the tooth.

The parameters characterizing involute meshing (the pressure angle, line of action, and contact ratio) become variable and change as wear increases. The line of action changes from a straight line for involute meshing into a curve for the worn pair; the pressure angle for the meshing of worn teeth is a variable quantity and, in individual portions of the mesh, may be greater or smaller than the pressure angle for involute teeth. The arc of action decreases, and the contact ratio rapidly approaches unity. As the meshing parameters of worn teeth change, the magnitude and position of the normal pressure force acting on the mating tooth also change. On the basis of these experiments, we found the law governing the change in the pressure angle as a function of tooth wear, where tooth wear was expressed as a percentage of the thickness of the unworn tooth along the pitch-circle arc, Fig. 1, a.

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

where \(\Delta S_i\) is the amount of tooth wear along the pitch-circle arc, in %;

\(S\) is the thickness of the unworn tooth along the pitch-circle arc, in mm.

\(S_i\) is the thickness of the worn tooth along the pitch-circle arc, in mm.

From Fig. 1, b it is seen that this dependence is almost rectilinear in the range of wear up to 35%. For practical calculations, the most important case is the calculation of teeth having wear up to 30–35%, since already at 30% wear of the teeth of mating gears the contact ratio becomes close to unity. According to Fig. 1, b, the dependence of the change in the pressure angle on tooth wear, for wear of the latter up to 35%, has the following form:

\[ \delta_i = \delta_e + 0.155 \Delta S_i; \]

where \(\delta_i\) is the pressure angle of the worn tooth, in degrees (the angle between the direction of the normal pressure force applied to the tooth tip and the normal to the axis of symmetry of the tooth);

\(\delta_e\) is the pressure angle for a new involute tooth, in degrees;

\(\Delta S_i\) is tooth wear along the pitch-circle arc, in percent.

As the wear of the profiles of mating teeth increases, the pressure angle increases. Since the circumferential component of the normal pressure force, directed perpendicular to the axis of symmetry of the tooth, remains constant when a constant torque is transmitted by a worn pair of gears, it follows that, with an increase in the pressure angle and in the normal force, the radial component compressing the worn tooth increases. In the sections of the worn tooth, a redistribution of the total stresses from bending and compression occurs. In the tensile zone, the total stresses decrease because of the increase in the compressive force.

Calculations carried out using the formulas of strength of materials for plane sections showed that, as tooth wear proceeds, the dangerous section shifts from the tooth root to its tip, and the tooth remains as strong as the unworn one in the tensile zone for wear up to 20% of the initial thickness measured along the pitch-circle arc. In Fig. 1, a are shown diagrams of the distribution of maximum stresses along the height of a tooth worn by 30% (for the tensile zone).

As can be seen from Fig. 1, a, the maximum resultant stresses in the critical section differ only slightly from the maximum resultant stresses in the critical section of an unworn tooth, and they remain close to one another both below and above the near-pitch zone.

Fig. 1. Distribution of stresses over the height of a worn tooth and graph of the change in the pressure angle under wear.

With wear of more than 35%, the calculated critical section shifts to the very top of the tooth addendum, since the transverse section of the tooth addendum decreases sharply.

To confirm the conclusions obtained regarding the calculated stresses in the critical sections of worn teeth, we used the photoelastic method to investigate the resultant stress on

transparent models made of isotropic material. The tooth models were made of optically sensitive ED-6 material with a thickness of 5 and 7 mm. After annealing and final machining, the epoxy-resin tooth models were loaded by means of a special device with a constant normal force. The device had a special mechanism by means of which the pressure angle was varied in accordance with the graph shown in Fig. 1, b. The normal pressure force was determined by calculation and, during the tests, was measured with a clock-type dynamometer. The models of the loaded teeth were illuminated in a polarization apparatus, and the patterns of the loaded states of the models were photographed. To obtain a black-and-white image of the stress-state patterns, a special liquid light filter was used.

For comparison of the patterns of stress distribution, we used both models of worn teeth and models of unworn involute teeth.

Fig. 2. Loaded new wheel tooth made of optically active material in polarized light.

Fig. 2 shows a loaded tooth made of optically active material and having an involute profile. The dark bands—isochromatics—are the loci of points at which the difference of the principal stresses is constant in magnitude. From the number of bands that have passed through points of the external contour of the tooth during its gradual loading, it is possible to determine the total stresses at any point of the contour. As can be seen from Fig. 2, at the base of the tooth, in the region of the transition curve, the greatest number of isochromatics passed through the profile points, and they are most densely spaced. This indicates the location of the critical section at the beginning of the transition curve of the tooth and the presence of stress concentration in the region of the tooth transition curve. The maximum total stresses in the compressed zone are 40% greater than in the tensile...

tensile zone. The location of the critical section in the “pictures” of the stressed state of the tooth coincides exactly with the calculated position of the critical section.

Figure 3 shows loaded teeth with different degrees of wear. The pictures of the stressed state of worn teeth differ from the picture for an unworn tooth the more, the greater the wear of the teeth.

Fig. 3. Loaded worn gear teeth made of optically active material in polarized light.

From Fig. 3 it is evident that, with wear up to 10%, the stressed state of the worn tooth differs hardly at all from the stressed state of an unworn tooth. The maximum resultant stresses, both in the tensile zone and in the compressed zone, remained unchanged, and the stress concentration occurs at the beginning of the tooth fillet curve.

The critical section of the tooth did not change its position and corresponds exactly to the position determined by calculation.

With tooth wear of 20%, the maximum resultant stresses in the tensile zone increase by only 6–8% and already occur at two locations of the tensile contour. The stresses along the contour are distributed more uniformly, while the stress concentration at the location of the fillet curve has a greater magnitude than on the worn portions

profile. The maximum total stresses in the compressed zone increased by 15%. The positions of the critical sections agree quite accurately with the calculated data.

For teeth having wear from 20 to 30%, the maximum total stresses increased by no more than a factor of 1.5. The patterns of the stressed states of teeth having wear of 20% and more differ greatly from the calculated data obtained. According to the calculation, as wear increases the critical section of the tooth gradually shifts toward the tooth tip. From the patterns of the stressed states of worn teeth it is seen that, when the tooth is worn by more than 20%, two clearly expressed zones of stress concentration appear on the tensile contour. The maximum total stresses on the tensile contour occur at the fillet of the worn part of the tooth leg.

When tooth wear exceeds 30%, and up to the wear at which sharpening of the tooth head occurs, the place of greatest stress concentration remains the fillet on the worn part of the tooth leg. However, the maximum total stresses in the sections of the entire tooth head differ little from the maximum total stresses in the section located at the fillet of the worn part of the tooth. Hence it follows that, with considerable tooth wear, teeth may break not only at the fillet of the worn part of the tooth leg, but also considerably above the fillet. Such tooth breakages are encountered in the operation of gear wheels.

It is interesting to note that, in worn gear wheels that have intensive pitting, tooth breakages occur not only at the fillet of the worn part of the tooth leg, but also at the places of intensive pitting.

Fig. 4. Character of fracture of models of worn teeth.

Apparently, pitting not only weakens the transverse section of the tooth, but also creates additional centers of stress concentration, which are the cause of an increase in the maximum total stresses in these sections of the worn tooth.

To determine the location of fractures of worn teeth, we tested a number of tooth models under static loading. The tooth models

with a module equal to 40 mm were cast from aluminum and then machined from the end surfaces in order to obtain teeth of constant thickness. Aluminum was chosen as the material for the following reasons: 1) the manufacture and machining of tooth models is considerably simplified; 2) both ductile and brittle teeth can be obtained by adding different amounts of copper to the alloy.

The aluminum tooth models were mounted on a special fixture and loaded on a hydraulic press to failure by a static force applied to the tooth tip at the pressure angle. Figure 4 shows characteristic fractures of teeth having different wear. Unworn teeth break at the beginning of the tooth fillet curve. The fracture section is not flat, but has a curvilinear form.

Teeth worn by more than 20% break along an inclined flat section from the point of rounding of the worn part of the dedendum to the point of the fillet curve on the compressed zone of the tooth.

The character of fracture obtained for worn teeth confirms the patterns of the stressed states of the teeth and shows that fracture of a worn tooth occurs along an inclined flat section passing through the points of maximum total stresses of the tensile and compressive zones.

Conclusions

1. As the teeth wear, their profiles change, which leads to a change in the total stresses in the critical section of the worn teeth.

2. With wear greater than 20%, the position of the critical section of the tooth changes. The critical section gradually shifts from the beginning of the fillet curve toward the tooth head.

3. Fracture of worn teeth ($\Delta S_i > 20\%$) that do not have zones of intensive pitting occurs along an inclined flat section passing through the points of maximum total stresses in the tensile and compressive zones. In the case of intensive pitting, fracture of a worn tooth along a section located in the pitting zone is possible. Thus, pitting may be an indirect cause of tooth fracture.

4. The bending strength of gear teeth does not decrease with wear of approximately up to 20%.

References

1. Dobrovol’skii V. A. et al. Machine Parts. Mashgiz, 1960.

2. Zaslavskii V. I. Calculation of gear meshes for strength and wear by a new method. Vestnik inzhenerov i tekhnikov, No. 3, 1936.

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

4. Kozhevnikov S. N. Theory of Mechanisms and Machines. Mashgiz, 1954.

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

6. Semenov Yu. S. Results of an experimental study of wear of gears of mining electric-drill reducers. Proceedings of the Tomsk Polytechnic Institute, vol. 108.

7. Chasovnikov L. D. Mesh Transmissions. Mashgiz, 1961.

8. Maschmeyer A. H. Wear Life of Aluminum Gears. Product Engineering, No. 9, 1956.