Variation of Gear Meshing Parameters under Wear

Yu. S. Semenov

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

Changes in the geometry of tooth profiles during wear and the associated characteristics of gear meshing have a substantial influence on the intensity of wear and the contact strength of gear transmissions.

Below is an analysis of changes in meshing parameters under wear of the gear wheels of the reduction gears of ER-5 mining electric drills that were subjected to wear tests. The conditions under which the tests were carried out, the method for assessing wear, and the characteristics of the experimental gear wheels are described in [1].

Fig. 1. Meshing of the experimental gear wheels before wear and after wear.

The tests [1] showed that, when gear wheels wear, the tooth profiles, as a rule, deviate from the involute, and the general cha-

the character of the regularities of these deviations is preserved for all types of loads and for various combinations of the hardnesses of the wheels and pinions; only a difference in the absolute magnitude of wear is observed. When the shape of the tooth profiles changes, the positions of the normals at individual points of the profile curve change. Therefore the pressure angle \(\alpha\) and the normal force depending on it at the contact points of the interacting teeth vary in magnitude within the limits of the path of contact; the line of action is transformed from the straight line \(ab\) into the curve \(KK_1K_2PK_3K_4K_5\) (Fig. 1)\(^{1}\), and, as a result, the distribution of specific sliding and of the specific work of friction along the profiles of these teeth changes.

As is known, the specific sliding can be expressed by the following relations [2]:

\[ \lambda_{12}=1-\frac{V^t_{k_2}}{V^t_{k_1}};\qquad \lambda_{21}=1-\frac{V^t_{k_1}}{V^t_{k_2}}, \]

where \(\lambda_{12},\ \lambda_{21}\) are the specific slidings, respectively, on the tooth profile of the first and second wheels.

\(V^t_{k_1},\ V^t_{k_2}\) are the tangential components of the velocities of the point of tangency of the tooth profile of the first and second wheels.

After some transformations one can obtain:

\[ \lambda_{12}=1-\frac{r_2\sin\alpha \pm x}{r_1\sin\alpha \pm x}\cdot\frac{1}{i_{12}}; \tag{1} \]

\[ \lambda_{21}=1-\frac{r_1\sin\alpha \pm x}{r_2\sin\alpha \pm x}\cdot i_{12}, \tag{2} \]

where \(r_1,\ r_2\) are the radii of the pitch circles of the wheels,

\(x\) is the distance from the point of contact \((K,\ K_1,\ldots K_5)\) to the pitch point,

\(\alpha\) is the pressure angle when the teeth are positioned at the corresponding point of contact on the line of action,

\(i_{12}\) is the gear ratio.

In equations (1) and (2), the plus sign corresponds to the case when the point of contact lies on the profile curve of the tooth addendum; the minus sign, on the tooth dedendum.

In Fig. 1 it may be noted that, when points located at the same height along the profile of a worn and an unworn tooth come into contact, the distances \(x\) are different. When the pressure angle \(\alpha\) of the worn profiles is greater than the pressure angle of the initial (involute) profiles \(\alpha_e\), the value of \(x\) for the worn teeth is smaller, and when \(\alpha<\alpha_e\) it is greater than the corresponding distances in involute meshing. Taking into account this dependence of \(x\) on \(\alpha\) and expressions (1) and (2), it is easy to conclude that when \(\alpha>\alpha_e\), the specific sliding and the specific work of friction associated with it decrease; when \(\alpha<\alpha_e\), they increase.

The distribution of the values of specific sliding and specific work of friction for the meshings (Fig. 1) is shown in Fig. 2, from which it is evident that, with wear, there occurs a redistribution (over the tooth profile) of the values of specific sliding and specific work of friction\(^{2}\), and their maxima decrease and shift toward the pitch point—

\(^{1}\) The curve \(K \ldots K_5\) in Fig. 1 was constructed graphically for the tooth profiles of worn wheels. The profiles were copied with the aid of a projector [1].

\(^{2}\) The curves of specific work of friction in Fig. 2 were constructed for identical values of the coefficient of friction and without taking into account the simultaneous meshing of two pairs of teeth.

...of meshing. This can explain the fact that, as tooth wear increases, the wear intensity of the tooth tips and roots decreases; in the near-pitch zone, however, it increases.

Here it is appropriate to note that the above-mentioned displacement of the maxima of the specific sliding and of the specific work of friction may be so great that the place of most intense wear of the tooth profile will be located in the immediate vicinity of the pitch point of meshing. An illustration of such a case may be

Fig. 2. Diagrams of the distribution of the values of specific sliding and specific work of friction along the tooth height.
1 and 2—specific sliding and specific work of friction of involute (unworn) meshing, respectively, for the first and second wheels; 3 and 4—specific sliding for worn meshing of the same wheels; 5 and 6—specific work of friction for the same worn profiles.

Fig. 3, which shows the meshing of heavily worn wheels. It is not difficult to see (Fig. 3) that at point \(K_2\) of the line of action the pressure angle will be minimal and, consequently, the specific sliding and the specific work of friction for the corresponding points on the tooth profiles located in the near-pitch zone of the profile will be maximal.

Thus, in contrast to involute (unworn) meshing, the specific sliding and the specific work of friction in the near-pitch zone of worn profiles are not close to zero, as noted in [3], but may have maximum values.

Fig. 3. Meshing of heavily worn experimental gear wheels.

As the teeth wear, the velocities of relative sliding of the profiles change. As is known [2], the magnitude of the relative sliding velocity is directly proportional to the magnitude of the distance \(x\) from the point of contact of the profiles to the pitch point of meshing. Using the values of the segments \(x\) (Fig. 1) for unworn and worn profiles, one can construct diagrams of these velocities. On

Fig. 4 shows such diagrams of sliding velocities as functions of the position of the profiles on the arcs of the base circles. It follows from these diagrams that, as the teeth wear, the maximum value of the sliding velocities decreases, which in turn leads to a reduction in the power lost to friction in the mesh, \(N_T\).

In Fig. 5 are presented diagrams of \(N_T\), constructed by the known method [4] for the mesh under study, with an unchanged value of the coefficient of friction and with allowance for the actual contact ratio. As is evident from Fig. 5, the mean value of the friction losses in the mesh decreases with wear and, consequently, the efficiency of the mesh increases. The latter is also confirmed by our experiments [1].

Fig. 4. Diagrams of the sliding velocities of the profiles of the teeth of the meshes shown in Fig. 1.

Along with the curvature of the line of action, the length of the arcs of action and the contact ratio dependent on it change. In Fig. 1 it can be seen that the length of the arcs of action of the involute profile before wear is \(ee_3\) and \(cc_3\); after wear, however, the new length of the arcs of action, \(qq_3\) and \(uu_3\), is already a smaller quantity. This leads to a reduction of the contact ratio. Thus, for example, the mesh (Fig. 1) before wear had a contact ratio \(\varepsilon_3 = 1.37\), while after wear \(\varepsilon = 1.07\).

The reduction of the contact ratio leads, as is known, to a decrease in the smoothness of operation of the transmission, contributes to an increase in dynamic load and to an intensification of noise. In addition, the reduction of the contact ratio leads to an increase in the zone of single-pair meshing. Thus, if in involute meshing (Fig. 1), before wear, the zone of single-pair meshing on the base circles was expressed by the arcs \(e_1 e_2\) and \(c_1 c_2\), then after wear it is expressed by the arcs \(q_1 q_2\) and \(u_1 u_2\). Accordingly, on the tooth profiles the zone of single-pair meshing before wear was bounded by the radii \(R_{a11}\) and \(R_{b11}\) for the first wheel and

Fig. 5. Diagrams of the magnitude of the power lost to friction in the mesh: \(N_{T3}\)—for the meshing of involute tooth profiles; \(N_T\)—for the meshing of worn tooth profiles; \(N_{T3\,\mathrm{av}}\) and \(N_{T\,\mathrm{av}}\)—mean values of the power lost to friction in the involute and worn meshes.

by the radii \(R_{a12}\) and \(R_{b12}\) for the second gear; after wear, by the radii \(R_{k21}\) and \(R_{k31}\) for the first gear and by the radii \(R_{k22}\) and \(R_{k32}\) for the second gear.

A decrease in the contact ratio, an expansion of the zone of single-pair meshing, and also an increase in the normal pressure of the interacting teeth (and consequently of the friction force) leads to an expansion of the zone of minimum contact strength on the tooth surface. This, for example, can explain the appearance of pitting in sections of the tooth profile remote from the pitch point [1].

From the analysis carried out it follows that, as tooth wear proceeds, the maxima of the specific sliding and of the specific work of friction decrease and are favorably redistributed; the efficiency of the meshing increases, while the contact ratio decreases, which leads to an expansion of the zone of single-pair meshing and to the spread of the region of minimum contact strength of the tooth surface to sections remote from the pitch point of engagement (to the tooth tip and root).

Since a reduction in the contact ratio entails a reduction in the contact strength of the teeth and a deterioration of the operating conditions of the transmission as a whole, in a number of cases the maximum permissible amount of tooth wear must be limited by the minimum permissible value of the contact ratio, and not only by the minimum permissible tooth thickness under the condition of bending strength.

REFERENCES

1. Yu. S. Semenov. Results of an experimental study of wear of gear wheels of mining electric-drill reducers. Izvestiya GPI, vol. 108, study of drilling machines, Metallurgizdat, Sverdlovsk, 1959.

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

3. A. D. Nevsky. On the question of tooth wear. Scientific Reports of Higher Education Institutions, Machine Building and Instrument Making, No. 2, 1959.

4. V. N. Kudryavtsev. Gear Transmissions. Mashgiz, Moscow—Leningrad, 1957.