ON THE INFLUENCE OF SPEED ON THE CONTACT STRENGTH OF GEAR TRANSMISSIONS

A. E. Belyaev

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

In most existing methods for calculating gear transmissions for contact strength, the permissible load on a tooth decreases with increasing circumferential speed. This phenomenon is explained mainly by the growth of dynamic loads with increasing circumferential speed, caused by manufacturing errors of gear transmissions (pitch errors, profile errors, accumulated and cyclic errors). The indicated reduction in permissible loads is usually taken into account by so-called speed coefficients.

Fig. 1. Speed coefficient \(C_v\): curve 1—according to the Norwegian Veritas; curve 2—according to the French Veritas

As an example, Fig. 1 gives the speed coefficients currently used in the calculation of marine gear transmissions, according to the Norwegian and French Veritas [3, 6]. Both speed coefficients are constructed according to a dependence of the type

\[ C_v=\left(\frac{a}{V_{\text{circ}}}\right)^n, \]

where
\(a\) is the coefficient of proportionality,
\(V_{\text{circ}}\) is the circumferential speed, m/sec,
\(n\) is the exponent.

The character of the indicated curves is analogous to the speed coefficients previously recommended in the USA.

Table 1 gives the values of speed coefficients (according to the British standard) by which the permissible shear stress recommended for 1 rpm must be multiplied in order to obtain the permissible stress at a given number of revolutions.

Analysis of the data of Table 1 shows that the stresses in a transmission operating at \(n = 20000\) rpm amount to only one third of the permissible stresses at \(n = 1\) rpm.

Meanwhile, recent investigations of turbine-type gear transmissions abroad have shown that generally accepted ideas about the influ-

...of circumferential speed on contact strength are incorrect. Thus, pitting damage on the second stages of marine reduction gears amounts to approximately 75% of the total damage from spalling.

In a single experiment described in [8], a sixfold increase in speed led to an increase in the load-carrying capacity of the transmission by more than a factor of 2. A. Sykes [9], after examining a large number of high-speed transmissions, recommends increasing the load as the sum of the numbers of revolutions \((n_1+n_2)\) increases.

Table 1

In the USSR, isolated experiments carried out by B. S. Shuleiko on gear transmissions of the 7th–8th accuracy grades, and detailed experiments by R. R. Galper on rollers, showed that with increasing speed the resistance to spalling also increases.

The small number of experimental data obtained both in the USSR and abroad determined the need to carry out corresponding investigations.

Gear transmissions of the 4th–5th accuracy grades with modules \(m = 2 \div 4\) mm were selected as test objects; moreover, all parameters of the experimental pairs were chosen with allowance for their greatest applicability in high-speed gear reducers. Before the start of the tests, the gear wheels underwent stepwise running-in under load at low speeds directly on the test stand. All experiments were carried out in the speed range \(20 \div 60\) m/s. As the base number of cycles, \(N_{\text{ts}} = 4 \div 5 \cdot 10^7\) cycles of load variation on the pinion was adopted.

In the course of the experiment, the load-carrying capacity of gear transmissions was recorded in the indicated speed interval according to the endurance criterion for the working surfaces of the teeth chosen by us. It is known that attempts to record the moment at which dangerous forms of fatigue failure of the working surfaces of teeth occur, on the basis of noise characteristics or changes in the efficiency of the mesh even at low speeds (see, for example, [5]), were not successful: over a wide range of changes in the condition of the working surfaces of the teeth, from microscopic pitting to damage of the entire surface, the mentioned characteristics remained unchanged.

On the basis of a number of works [1], carried out under the direction of Prof. V. N. Kudryavtsev, we adopted the following endurance criterion. The limiting load was taken to be such a load under which the transmission can operate for a very large number of stress cycles (far exceeding the conventional base number of cycles for the contact endurance limit) without noticeable development of failures of the working surfaces of the teeth (as an example, Fig. 2 shows the working surface of a gear wheel that had operated for two base periods under the specified load). When it is exceeded by 25–35%, over a time corresponding approximately to the base number of cycles, the degree of failure of the working surface is such that further operability of the transmission becomes impossible.

The processing of the results obtained was carried out with allowance for the dynamic loads in the mesh, determined by the method of [4], and with allowance for the concentration of load at the pitch point.

In most methods for calculating gear transmissions, the values of the contact-fatigue limit are taken to be proportional to the hardness number of the working surface of the teeth,

\[ \sigma_k = K \cdot HB, \]

where \(\sigma_k\) are the contact stresses,
\(HB\) is the hardness of the working surfaces of the teeth,
\(K\) is the coefficient of proportionality.

The validity of the above relationship is confirmed by a number of investigators. Thus, instead of determining the dependence of the contact stresses on the total rolling speed \(\sum V\) (as it increases), one may investigate a dependence of the form

Fig. 2. Condition of the working surface of the teeth under the action of the limiting contact load, \(N_{\mathrm{ts}} = 9.5 \cdot 10^7\)

\[ K = \frac{\sigma_k}{HB} = f(V_\Sigma). \]

The choice of such a method for evaluating the experimental data also makes it possible to use data from experiments by other investigators.

Figure 3 shows the variation of the coefficient \(K\), equal to the ratio of the limiting contact stresses to the hardness of the working surfaces of the teeth, as a function of speed. In the above-mentioned range of speeds, the monotonically increasing dependences are described by the equation

\[ K = 15.5 \div 16 \cdot V_\Sigma^{0.2 \div 0.21} \tag{1} \]

independently of the module of engagement of the tested gear wheels. Processing these same experiments by the method of mathematical statistics [2, 7] made it possible to obtain the correlation equation of dependence 1:

\[ \lg K = 0.2025 \lg V_{\Sigma} + 1.1994. \tag{2} \]

Fig. 3. Dependence of the quantity \(K\) on speed

Conclusions

1. With increasing speed, the contact strength of gear transmissions increases.

2. The indicated influence of speed can be taken into account by means of dependences 1 or 2.

LITERATURE

1. V. N. Kudryavtsev, I. M. Kilimov. On the question of evaluating the load-carrying capacity of gear transmissions limited by resistance to fatigue pitting. Collection of Papers of LMI, No. 34, 1963.

2. A. K. Mitropolsky. Technique of Statistical Computations, Fizmatgiz, 1961.

3. Norwegian Veritas. Rules for the Classification and Construction of Steel Ships. L., 1960.

4. K. Trubin. Contact fatigue of the teeth of spur gears. TsNIITMASH, book 37, Mashgiz, 1950.

5. A. I. Petrusevich. Gear transmissions. Collection “Machine Parts.” Edited by N. S. Acherkan, Mashgiz, 1954.

6. French Veritas. Rules for the Construction and Classification of Ships. L., 1961.

7. M. Ya. Shashin. Method for determining mean probable values of cyclic durability. Factory Laboratory, No. 6, 1952.

8. A. Fisher. Use of speed factors in calculation load carrying of helical gears. Machinery, vol. 98, 8, 1961.

9. A. Sykes. Allowable load on helical gears, Engineer, November 13, 1959.