Simplified Method for a Check Calculation of Worn Teeth in Bending

I. F. Deryuga, Yu. S. Semenov

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

In engineering practice it is sometimes necessary to perform a check calculation in bending for the teeth of a worn gear transmission in order to assess the suitability of the gear wheels for further operation. However, at present there is no sufficiently reliable and scientifically substantiated method for calculating worn gear meshing.

In works [1, 2, 3], allowance for tooth wear in calculating teeth in bending is based on the assumption that, under wear, the geometric form of the tooth profile is preserved; tooth wear occurs over its entire height; the pressure angle and the normal pressure force remain constant under wear; and the position of the dangerous section of the tooth along the height is unchanged. Proceeding from these assumptions, the reduction in the bending strength of teeth under wear is explained exclusively by the reduction in the dimensions of the dangerous section of the tooth at its base (Fig. 1,a). The weakening of the dangerous section of the tooth as a result of wear,

Fig. 1. Calculation schemes of worn teeth: a—according to the existing method, b—according to the proposed method

and consequently the reduction in the bending strength of the tooth, is taken into account by the wear coefficient \(K_{\mathrm{u}}\), which is determined as follows:

\[ K_{\mathrm{u}}=\frac{\sigma'}{\sigma}=\frac{W}{W'}=\left[\frac{a}{a'}\right]^2, \tag{1} \]

where \(\sigma,\ \sigma',\ W,\ W',\ a,\ a'\) are the maximum stress in the critical section, the section modulus of the critical section, and the tooth size in the critical section, respectively, for new (unworn) and worn teeth.

The values of the wear coefficient \(K_{\mathrm{u}}\), for relative tooth wear from 0% to 30%, vary within the range from 1.0 to 2.0. Thus, for example, for relative tooth wear corresponding to 15%, the coefficient \(K_{\mathrm{u}}\) is equal to 1.38. This means that, in comparison with a new (unworn) tooth, the magnitude of the maximum stress in a worn tooth increases by almost 40%.

On the basis of experiments carried out in the laboratory of the Department of Applied Mechanics of the Tomsk Polytechnic Institute [4, 5, 6], it was established that the profile of a worn tooth differs from the initial involute profile; that the tooth does not wear over its entire height; that, with wear, the pressure angle and the normal pressure force change; and that the critical section of the tooth shifts toward the tip under wear. A typical form of the worn tooth profile is shown in Fig. 1, b.

The gears were worn on test rigs of both closed and open type, under both constant and variable loads, at various circumferential speeds, with the test gearboxes filled with both liquid and viscous lubricant, without abrasive in the lubricant and with abrasive present.

The tested gears of involute profile with zero correction were made of various materials, with different hardness of the tooth working surfaces, and had different modules and different numbers of teeth.

The conditions and regimes under which the gear tests were conducted were highly diverse, which made it possible to draw certain generalized conclusions and to consider, with sufficient justification, the question of estimating the bending strength of worn teeth.

Below a simplified method is set forth for a verification calculation of worn teeth of straight-tooth cylindrical gears for bending strength, as applied to nonreversible transmissions.

The formula on the basis of which the calculation of new (unworn) teeth of cylindrical straight-tooth gears for bending strength is performed, without taking into account load concentration, dynamic factors, and friction forces, has the form

\[ \sigma=\frac{P}{B\cdot m\cdot y'_i}\leq[\sigma], \tag{2} \]

where \(\sigma\) is the maximum stress in the critical section of tooth I—I;
\(P\) is the circumferential (tangential) force on the tooth;
\(B\) is the tooth length;
\(m\) is the module;
\(y'_i\) is the tooth form factor, calculated for the critical section I—I, under a load applied at the tooth tip;
\([\sigma]\) is the allowable bending stress of the tooth material for the corresponding load cycle.

The value of the form factor \(y'_i\) for section I—I is determined by the formula

\[ y'_i=\frac{1}{\dfrac{m}{\cos\alpha}\left[\dfrac{6l\cos\gamma}{a^2}-\dfrac{\sin\gamma}{a}\right]}, \tag{3} \]

where: \(\alpha\) is the pressure angle,
\(l\) is the arm of the bending moment,

\(\gamma\) — pressure angle,
\(a\) — tooth dimension in the dangerous section.

Let us denote

\[ \frac{1}{\dfrac{6l\cos\gamma}{a^2}-\dfrac{\sin\gamma}{a}}=Y'_1 \tag{4} \]

and call the quantity \(Y'_1\) the geometric coefficient. Then from formula (3) we obtain

\[ Y'_1=y'_1\cdot\frac{m}{\cos\alpha}. \]

On the basis of formula (5) it may be concluded that the geometric coefficient \(Y'_1\) is a variable quantity, since it is a function of the variable quantities \(l\), \(a\), \(\gamma\), and \(\alpha\).

Static-bending tests of models of worn teeth, made of a brittle aluminum alloy and of organic glass, as well as tests of worn full-scale gear wheels on a pulsator, made it possible to establish that, when tooth wear exceeds 20%, fracture—in the first case of the models, and in the second of the teeth—occurs along a section passing through the undercut zone on the worn part of the tooth. This means that, with wear, the dangerous section of the tooth has shifted and taken up a certain new position II—II (Fig. 1, b). The conclusion regarding the change in the location of the dangerous section with wear also follows from analysis of the stress-variation diagrams along the tooth height, constructed for teeth with different degrees of wear [6]. It should be noted, however, that beginning with wear approximately equal to 20%, the dangerous section of the tooth, having shifted to a new position, thereafter almost does not change its location up to wear of approximately 30%. For our purposes this change may be considered insignificant and therefore may be neglected.

We shall denote the geometric coefficient of a new (unworn) tooth corresponding to section II—II by \(Y'_{\mathrm{II}}\). Then formula (1) for calculating stresses in sections I—I and II—II takes the form

\[ \sigma_{\mathrm{I,II}}=\frac{P}{B\cdot Y'_{\mathrm{I,II}}\cos\alpha}\leq[\sigma]. \tag{6} \]

As applied to a worn tooth, we shall agree to denote the geometric coefficient determined for sections I—I and II—II by \(Y'_{u\mathrm{I}}\) and \(Y'_{u\mathrm{II}}\).

It was established by calculation that, when the teeth wear from 0% to 20%, the coefficient \(Y'_{u\mathrm{I}}\) changes very little (on the average by 5%); therefore one may assume \(Y'_{u\mathrm{I}}\approx\mathrm{Const}\). (In Fig. 2 the graph of the change in the coefficient \(Y'_{u\mathrm{I}}\) is represented as straight line 1, parallel to the abscissa axis.)

The numerical values obtained for the coefficient \(Y'_{u\mathrm{II}}\) make it possible further to construct the graph of variation of this coefficient (curve 2 in Fig. 2).

To simplify subsequent calculations, we replace the graph obtained by two rectilinear segments \(AB\) and \(BC\). The error from such a replacement will be: on segment \(AB\)—5%, on segment \(BC\)—2%.

On the basis of the graph \(Y'_{u\mathrm{II}}=f(\Delta S_u)\), the following relationships may be obtained:

\[ \frac{\text{For }\Delta S_u=0\%-15\%}{Y'_{u\mathrm{II}}=Y'_{\mathrm{II}}-0.0013\cdot\Delta S_u.} \tag{7} \]

\[ \frac{\text{For }\Delta S_u=16\%-30\%}{Y'_{u\mathrm{II}}=Y'_{\mathrm{II}}-0.00155\cdot\Delta S_u.} \tag{8} \]

For calculating worn teeth without taking friction forces into account, formula (6) takes the form

\[ \sigma_{\mathrm{I},\mathrm{II}}= \frac{P}{B\cdot Y'_{\mathrm{I},\mathrm{II}}\cdot \cos\alpha} \leq [\sigma]. \tag{9} \]

The point of intersection \(D\) of curves 1 and 2 (in Fig. 2) corresponds to the case when \(Y'_{\mathrm{uI}}=Y'_{\mathrm{uII}}\). This means that, with tooth wear of approximately up to 17%, the magnitude of the maximum stress changes almost not at all and the bending strength of the tooth does not undergo substantial changes. This conclusion is also confirmed by experiments on transparent models of worn teeth illuminated with polarized light [6].

Fig. 2. Curves of the change in the geometric coefficient with wear

In some cases (for example, for open transmissions), friction forces may become considerable and may influence the strength of the teeth. It is known that friction forces in meshing have an unequal effect on the strength of the teeth of driving and driven wheels. The friction force on the tooth of the driving wheel acts in such a way that it increases the bending moment and, consequently, worsens the working conditions of the tooth. For the teeth of the driven wheel, the opposite is observed. For this reason, when calculating worn meshing with allowance for friction forces, it is necessary to evaluate the bending strength separately for the teeth of the pinion and for the teeth of the wheel.

Taking friction forces into account, the geometric coefficient can be determined by means of the approximate empirical formula

\[ Y=\frac{Y'}{1\pm 0.02\cdot \rho}, \tag{10} \]

where \(\rho\) is the friction angle in degrees.
The sign \((+)\) applies to driving wheels, and the sign \((-)\) to driven wheels.

Thus, a checking calculation of worn teeth of spur cylindrical gears for bending, taking into account the principal factors affecting tooth strength, can be carried out by the formula

\[ \sigma_{\mathrm{I},\mathrm{II}}= \frac{P\cdot K_{\mathrm{d}}\cdot K_{\mathrm{knc}}} {B\cdot Y'_{\mathrm{I},\mathrm{II}}\cos\alpha} \leq \left[ \frac{\sigma_0} {n\cdot K_{\mathrm{z}\,\mathrm{I},\mathrm{II}}\cdot K_{\mathrm{p}}\cdot K_{\rho}} \right], \tag{11} \]

where

\[ K_{\mathrm{d}} \]
is the dynamic coefficient;

\[ K_{\mathrm{knc}} \]
is the coefficient of load concentration along the tooth length;

$K_{\sigma \mathrm{I},\mathrm{II}}$ — effective stress-concentration factors in zones I and II;

$K_{\mathrm{p}}$ — coefficient taking into account the condition of the working surface of the tooth;

$K_{\rho}$ — coefficient taking into account the damaging effect of the loads under which the tooth was worn down (reserve-loss coefficient);

$\sigma_0$ — endurance limit of the tooth material under a pulsating cycle;

$n$ — safety factor.

At present, as applied to worn gear meshing, there is almost no information on the values of the coefficients $K_{\mathrm{p}}$, $K_{\mathrm{kn}}$, $K_{\sigma \mathrm{I},\mathrm{II}}$, $K_T$, and $K_{\rho}$, which is explained by the insufficient study of this question. A way out of this difficulty can be found only after comprehensive theoretical and experimental investigations devoted to questions of the geometry, kinematics, and dynamics of worn gear meshing have been carried out.

Conclusions

1. The currently existing method for taking into account the influence of wear on the bending strength of teeth is not confirmed by calculations and experiments.

2. The bending strength of teeth does not undergo substantial changes with wear: for pinion teeth—approximately up to 15%, and for gear-wheel teeth—approximately up to 20%.

References

1. V. A. Dobrovolsky. Machine Elements. Mashgiz, 1954.

2. L. D. Chasovnikov. Gear Transmissions. Mashgiz, 1961.

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

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

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

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