THE EFFECT OF GEAR TOOTH WEAR ON THEIR BENDING STRENGTH

I. F. DERYUGA, A. E. BELYAEV, S. I. SHUBOVICH

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

In a number of cases, during the operation of gear transmissions, considerable tooth wear occurs, as a result of which the tooth thickness decreases and their original geometric shape changes. Observations of the operation of worn gear transmissions show that they sometimes continue to perform their functions and have satisfactory operating characteristics.

The bending strength of worn teeth has been little studied; therefore, the data given in the literature on the values of the maximum permissible depth of tooth wear vary over a wide range \((0.15—0.6)\,m\). The currently existing method for assessing the bending strength of worn teeth is based on the assumptions that the tooth wears over its entire height while preserving its involute form; that the position of the critical section along the tooth height remains unchanged; and that the pressure angle and the normal pressure force are constant quantities [1]. On the basis of these assumptions, the change in tooth bending strength under wear is explained solely by the reduction in the dimensions of the critical section of the tooth at its base and is evaluated by the empirical wear coefficient \(K_i\), which is defined as follows:

\[ K_i=\frac{\sigma'}{\sigma}=\frac{W}{W'}=\left(\frac{a}{a'}\right)^2, \]

where \(\sigma, W, a\) and \(\sigma', W', a'\) are, respectively, the stress in the critical section, the section modulus of the critical section, and the tooth dimension in the critical section for a new (unworn) and a worn tooth. Thus, for example, for a tooth having a relative wear of \(20\%\), the wear coefficient is taken to be \(K_i \approx 1.5\), which corresponds to an increase in the total stress in the critical section, compared with the initial value, by 50%.

Studies of the regularities of wear of gear wheels of the reducers of ER-5 mining electric drills, carried out at the Department of Applied Mechanics of the Tomsk Polytechnic Institute named after S. M. Kirov, showed that the involute profiles of the teeth become distorted during wear, and at the same time such important parameters as the angle of engagement and the normal pressure force change, while the position of the critical section does not remain constant along the tooth height [2].

To study the question of how the bending strength of teeth changes under wear, and to identify the principal factors affecting the strength of worn teeth, the authors carried out theoretical and experimental investigations.

Hardened steel spur cylindrical gears of the reducer of the ER-5 mining electric drill \((m = 2\ \text{mm};\ z = 28)\) were chosen as the object of study. The basis for this choice was that the gear transmissions of mining electric drills, because of overload in contact stresses caused by the limited dimensions and weight of the drill and by the penetration of coal dust into the reducer housing, wear severely during operation. The functional purpose of these transmissions is such that tooth wear is entirely permissible and, consequently, the service life of the gears must be determined from the condition of maintaining sufficient bending strength of the teeth.

Fig. 1. Tooth profile with relative wear \(\Delta S_i = 20\%\), stress patterns, and isochromatic map

Calculations associated with determining the bending stresses on the tensile side of the contour of worn teeth, performed using the hypothesis of plane sections, showed that when the tooth is worn by up to 20% of its original thickness, measured along the arc of the pitch circle, the total stress at the critical point remains practically unchanged and, consequently, for wear up to this degree the bending strength of the tooth should not decrease. At a relative wear value of approximately 20%, the critical section of the tooth changes its position and shifts from the fillet zone at the base into the fillet zone formed as a result of wear (Fig. 1).

To verify these calculations and to clarify the pattern of stress distribution, the stressed state of a worn tooth was investigated by the photoelastic method. The results of these investigations confirmed the conclusions drawn above and made it possible to determine the theoretical stress-concentration factors in the critical zones. It was established that, at 20% wear, the fillet formed on the tooth profile as a result of wear causes approximately the same stress concentration as the fillet at the tooth base. Therefore it may be concluded that, at this degree of wear, tooth fracture is equally probable both at the tooth base and along the section passing through the fillet on the worn part of the tooth.

In order to determine the fatigue strength of the worn tooth and to clarify the adverse effect of pitting cavities that had occurred

Comparative tests of full-scale gear wheels for tooth fracture under cyclic variable loading were carried out on the working surfaces of teeth overloaded with respect to contact stresses [3].

The test program provided for an experimental study of two groups of gear wheels: the first group—a new (unworn) wheel and a wheel with 20% wear, without signs of fatigue pitting; the second group—a new (unworn) wheel and a wheel with 20% wear, with the presence of pitting cavities. The teeth of each group of wheels had the same hardness, obtained by through hardening. The tests were carried out on a specially built mechanical pulsator. The experimental data were processed using methods of mathematical statistics.

The results of fatigue tests and analysis of the fatigue curves showed that teeth with relative wear of 20%, without signs of contact failure, had the same endurance limit as new (unworn) teeth. Comparison of the endurance limit of the teeth of the second group established a reduction in the fatigue strength of the worn teeth. The reason for the reduction in strength in this case was the stress concentration caused by pitting cavities. This explanation was confirmed by the fact that the fatigue crack developed from the zone affected by pitting cavities.

Thus, it may be concluded that, in the absence of pitting on the working surfaces of the teeth, wear of up to 20% is safe from the standpoint of bending strength.

On the basis of the studies carried out, the conclusion was drawn that teeth having relative wear up to 20% should be calculated for bending strength without taking into account the reduction in tooth thickness, i.e., the coefficient \(K_{\mathrm{i}}\) should be taken equal to unity.

For teeth with relative wear from 20% to 30%, a simplified relationship was obtained for determining the tooth-form factor in the dangerous section, of the following form:

\[ Y_{\mathrm{in}} = Y_{\mathrm{II}} - 0.0073 \Delta S_{\mathrm{i}}, \]

where \(Y_{\mathrm{in}}\) is the form factor of the worn tooth corresponding to the dangerous section II–II;

\(Y_{\mathrm{II}}\) is the form factor of a new (unworn) tooth in section II–II;

\(\Delta S_{\mathrm{i}}\) is the relative wear of the tooth, measured along the arc of the pitch circle, in %.

Calculations of the fracture strength of worn teeth, performed according to the generally accepted method using the tooth-form factor calculated from the above relationship, agree well with the experimental data.

References

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

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

3. I. F. Deryuga, S. I. Shubovich. Results of tests of worn gear wheels for fatigue fracture of teeth. Proceedings of the MNIIPTMASH Institute, issue 5, Kemerovo, 1968.