ON THE CALCULATION OF THE SERVICE LIFE OF GEAR WHEELS IN MINING ELECTRIC DRILLS BASED ON WEAR

Yu. S. SEMENOV, V. G. GORBENKO, S. I. SHUBOVICH

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

Mining electric drills are hand-held or portable machines; therefore, an effort is made to design them with the smallest possible dimensions and weight. As a result, the gear wheels of the reduction gears of mining electric drills, as a rule, are made of alloy steels with high hardness of the working tooth surfaces (pinions of steels 18KhGT, 20Kh—cemented and hardened; wheels of steel 40Kh, hardened). Nevertheless, as our calculations have shown, because of their small dimensions the gear pairs operate with a two- to threefold overload in contact strength. This leads not only to pitting, but also to wear of the teeth.

As is known, owing to the complexity of the wear process and the variety of factors affecting this process, there are still no generalized equations that would make it possible to determine the service life of a part from the condition of permissible wear. Such calculations are carried out, with a high degree of approximation, only for a limited number of pairings.

The magnitude of the linear, volumetric, or weight wear of a part over time (the wear rate) is usually expressed as a function of the normal force or contact stress and velocity. This dependence is, as a rule, a power-law one [1], [2], and it may be represented, for example, in the following form:

\[ \gamma = K \cdot P_n^s \cdot V^n \tag{1} \]

or

\[ \gamma = C \cdot \sigma_k^m \cdot V^n, \tag{2} \]

where \(\gamma\) is the wear rate \((\mathrm{mm}/\mathrm{hour},\ \mathrm{mm}^3/\mathrm{hour},\ \mathrm{mg}/\mathrm{hour})\);
\(P_n\) is the normal force \((\mathrm{kg})\);
\(\sigma_k\) is the contact stress \((\mathrm{kg}/\mathrm{cm}^2)\);
\(V\) is the relative sliding velocity \((\mathrm{m}/\mathrm{s})\);
\(s, m\), and \(n\) are exponents determined from experiments on the change of wear rate as a function of load and velocity;
\(K\) and \(C\) are proportionality coefficients depending on the characteristics of the material (determined experimentally).

In some works [3], when deriving the equation for the wear rate of gear-wheel teeth, a directly proportional dependence of it on the magnitude of the load is assumed.

The authors carried out wear tests on gears of reducers of mining electric drills ER-5, ER-16, ER-14D, and ER-18D. All parameters of the gears, including the hardness of their working surfaces, conformed to the technical specifications for electric-drill parts [4].

To reduce weight, the electric motors of all types of electric drills have the limiting synchronous speed (at a current frequency of 50 Hz) of 3000 rpm. The gear ratio of the gear pairs of different types of drills, since the drill-bit speeds are determined by the drilling process and are the same for all hand-held electric drills, varies within small limits (most often about 3). The overall dimensions of the drills also differ little from one another; therefore the circumferential speeds of the gears (and, consequently, the relative sliding speeds of the teeth) in different electric drills differ insignificantly. This makes it possible not to take the effect of speed into account when calculating the wear intensity of the gears of mining electric drills. This is all the more so because, owing to the small dimensions of these gears, their speeds are low, and according to literature data—for example [1]—at low speeds the exponent \(n\) in expressions (1) and (2) is equal to unity. Taking the above into account, in the present work the authors studied the change in the wear intensity of the gears of mining electric drills only as a function of the load magnitude.

Figures 1 and 2 present the graphs obtained as a result of our experiments for the dependence of gear-wear intensity on the magnitude of the normal force in the mesh, the power lost to friction in the mesh, and the contact stress. Since the pinions of the gear pairs of electric-drill reducers are case-hardened, after running-in two wear periods are usually observed in them (normal and accelerated), whereas in the wheels there is one period—the period of normal wear. We explain the presence of the accelerated-wear period in pinions by the wearing away of the case-hardened layer.

Fig. 1. Graphs of the dependence of gear-wear intensity \(\gamma\) on the normal force \(P_n\) (solid lines) and the power expended on friction in the mesh \(N_3\) (dashed lines). 1—period of normal wear of pinions, 2—period of accelerated wear of pinions, 3—period of normal wear of the wheel.

Experiments showed that the dependence of the wear intensity of the electric-drill gears on the magnitude of the normal force in the mesh (and consequently also on the contact stress), as well as on the power expended on friction in the mesh, is nonlinear in character (Fig. 1); when shifting to increased loads, wear rises sharply. The ratio \(\gamma/N_3\) is, in essence, an energy criterion of wear—

is not a constant, since the latter is the ratio of the volume of separated material to the work of friction forces [5]. Consequently, in our case as well the proposition given in [5] remains valid: the energy criterion of wear for a given material or a given pair is not a constant.

As noted, the gear wheels of mining electric drills operate with an overload in contact strength, that is, precisely under increased loads. Consequently, to adopt for calculation a directly proportional dependence of tooth wear on the magnitude of the load, as is done, for example, in [3] with respect to lightly loaded transmissions, in which the contact stresses are considerably below the contact-strength limit, is not possible for electric-drill reducers, since this will lead to gross errors. Graphs of the dependence of the wear intensity of gear wheels on the normal force in the mesh and on the contact stress for gear pairs subjected to testing (Fig. 2) show that a power-law equation of the following type may be used:

\[ \gamma = K \cdot P_n^{s} \tag{3} \]

or

\[ \gamma = C \cdot \sigma_k^{m}. \tag{4} \]

According to equations (3) and (4), for graphs 1, 2, and 3 (Fig. 2) we have, respectively:
\(s_1 = 4.00,\quad s_2 = 3.52,\quad s_3 = 2.97;\quad K_1 = 4.68 \cdot 10^{-8},\quad K_2 = 1.12 \cdot 10^{-6},\quad K_3 = 1.53 \cdot 10^{-5};\quad m_1 = 8.00,\quad m_2 = 7.04,\quad m_3 = 5.94;\quad C_1 = 6.76 \cdot 10^{-32},\quad C_2 = 1.19 \cdot 10^{-27},\quad C_3 = 3.05 \cdot 10^{-23}.\)

Fig. 2. Graphs of the dependence of the wear intensity of gear wheels \(\gamma\) on the normal force in the mesh \(P_n\) and on the contact stress \(\sigma_k\). The values of \(\sigma_k\) were calculated by the Hertz formula without allowance for dynamic load and the nonuniformity of load distribution along the tooth. The numbers 1, 2, and 3 designate the periods of wear of the wheel and pinion indicated in Fig. 1.

The data of the graphs in Figs. 1 and 2 were obtained from experiments in the absence of abrasive in the lubricant. Mining electric drills operate under such conditions that coal dust may penetrate into the reducers and act as an abrasive; therefore, an additional investigation was carried out into the contamination of the lubricant of mining electric-drill reducers by coal dust and into the effect of the latter on gear-wheel wear.

As a result of an inspection of a large number of electric drills in the mines of the Kuzbassugol combine, it was established that in electric drills of older types (ER-5, ER-16, ER-15) the contamination of the lubricant by coal dust averages 6–8%, while in electric drills of newer types (ER-14D, ER-18D), owing to improvement in the design of the seals, it is 1.5–3%.

The results of experiments to determine the effect of the content of coal dust in the lubricant of gear wheels on their wear are presented in the graphs of Fig. 3. In these experiments, for the pinions (as well as for the wheels), after running-in

only a period of uniform wear was observed (i.e., the periods of normal and accelerated wear merged into one continuous period). It should be noted that, even in the absence of abrasive in the lubricant, under increased loads the period of normal gear wear is very short, and the main operating period is the period of accelerated uniform wear [6].

From a comparison of the data in Fig. 3 and graphs 2 and 3 of Figs. 1 and 2, it is seen that, when the content of coal dust in the lubricant is up to 4%, the wear intensity increases only slightly. With an increase in the abrasive content in the lubricant, the wear intensity increases sharply (Fig. 3); in this case the proportionality coefficients \(K\) and \(C\) and the exponents \(s\) and \(m\) in expressions (3) and (4) will differ significantly from those obtained from the graphs in Figs. 1 and 2. For example, solving jointly two equations:

\[ \gamma_1 = C \cdot \sigma_{k1}^{m} \quad \text{and} \quad \gamma_2 = C \cdot \sigma_{k2}^{m}, \]

for \(\sigma_{k1}=6950\ \text{kg}/\text{cm}^2\), \(\sigma_{k2}=9000\ \text{kg}/\text{cm}^2\) and the corresponding values \(\gamma_1\) and \(\gamma_2\) in Fig. 3, at a coal-dust content in the lubricant of 10%, we obtain, for the pinion, \(C=2.76 \cdot 10^{-7}\), \(m=1.7\); for the wheel, \(C=4.76 \cdot 10^{-10}\), \(m=2.3\) (per tooth).

Fig. 3. Change in wear intensity \(v\) of gear wheels of mining electric drills as a function of the coal-dust content in the lubricant. ○%, ○, ◇ — pinion; △, □ — wheel. ○, △ — at contact stress \(\sigma_k=9000\ \text{kg}/\text{cm}^2\); ◇, □ — at \(\sigma_k=6950\ \text{kg}/\text{cm}^2\). The values of \(\sigma_k\) were calculated by the Hertz formula without taking into account the dynamic load and the nonuniformity of load distribution along the tooth.

Thus, it may be concluded that, just as in any calculation method—for example [7]—the power equation \(\sigma_k^x N_\mu=\text{const}\) is used to determine the permissible contact stress \([\sigma_k]\) when calculating tooth durability in terms of contact strength, although the exponent \(x\) in it varies, for different cases, within very wide limits (according to [7], [8], from \(x=2.8\) to \(x=25\)); dependence (4) can be used to determine \([\sigma_k]\) when calculating durability in terms of wear. In this case the latter is unified with the calculation for contact strength in the same way as is done in the TsNIITMASH method [7] for the calculation for scoring. Having adopted the value of the limiting wear of the teeth on the basis of their bending strength (for example, 20% of the tooth thickness measured along the arc of the pitch circle [9]), and having a specified service life for the transmission, one can determine the permissible wear rate \(v\). Then, using dependence (4), the permissible contact stress \([\sigma_k]\) is determined. After that, one can use the usual formulas applied for calculating contact strength, for example those given in [7].

Since, with the modern design of seals and observance of the rules of technical operation of electric drills, the amount of coal dust in the lubricant does not exceed 3%, for gear wheels of electric-drill reducers the values \(C\) and \(m\) in expressions (4) can be assigned on the basis of graphs 2 and 3 of Fig. 2. Rounded off, for the pinion \(C=1.2 \cdot 10^{-27}\), \(m=7\); for the wheel \(C=3 \cdot 10^{-23}\), \(m=6\). With an appropriate selection of the hardness ratio—

of the working surfaces of the gear pair, when equal wear of the teeth of the pinion and the wheel is ensured [6], when calculating \(\gamma\) per one tooth on average one may take \(C = 7 \cdot 10^{-25}\), \(m = 6\).

REFERENCES

1. A. S. Pronikov. Wear and Durability of Machine Tools. Mashgiz, Moscow, 1957.

2. K. P. Chudakov. On the calculation of service lives of gear wheels subjected to abrasive wear. Problems of the Quality and Strength of Gear Transmissions. Coll. 1, TsBTI, Moscow, 1961.

3. I. S. Kuzmin. Load-Carrying Capacity of Open Gear Transmissions. Abstract of Cand. Diss., Leningrad, 1967.

4. Hand-Held Electric Drills ER-4, ER-5, ERP-5. Glavuglemash, Tomsk Electromechanical Plant named after V. V. Vakhrushev, Ugletekhizdat, Moscow, 1956. Hand-Held Electric Drills ER-15, ER-16, ERP-20. TEMZ, 1959. Hand-Held Electric Drills with Remote Control ER-14D, ER-15D, ER-18D, ERP-18D. TEMZ, 1962.

5. I. V. Kragelsky. Friction and Wear. Mashgiz, Moscow, 1962.

6. V. T. Gorbenko, Yu. S. Semenov, S. I. Shubovich. Studies of the influence of certain technological factors and load conditions on the wear of gear wheels of reducers of mining electric drills. Izv. TPI, vol. 147, published by TSU, Tomsk, 1966.

7. Ya. G. Kistyan. Method for calculating gear transmissions for strength. TsNIITMASH, book 107, Mashgiz, Moscow, 1963.

8. B. S. Kovalsky. Principal issues in the strength operation of gear transmissions. Problems of the Quality and Strength of Gear Transmissions, coll. II, TsBTI, Moscow, 1961.

9. I. F. Deryuga, Yu. S. Semenov. A simplified method for checking the bending calculation of worn teeth. Izv. TPI, vol. 183, published by TSU, Tomsk, 1968.