ACCOUNTING FOR DYNAMIC LOADS IN HIGH-SPEED HELICAL GEARS

A. E. BELYAEV

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

The operation of gear transmissions is substantially affected by oscillatory processes that arise because of manufacturing errors, variations in stiffness, and the like. In individual cases these oscillatory processes can cause additional loads on the teeth comparable in magnitude with the static loads (and, under resonance phenomena or phenomena close to them, even exceeding them). Therefore, accounting for dynamic loads in the calculation of transmissions is mandatory.

Dynamic loads depend on the masses of the rotating gear wheels (and of the parts connected with them), on the magnitudes of manufacturing errors, the stiffnesses of the connections, and the speed, increasing as the listed factors grow.

A number of works have been devoted to the investigation of dynamic loads, most of which concern spur gears (E. Buckingham, A. I. Petrusevich, M. S. Polotskii, and others).

In modern high-speed drives, helical gears have become widespread. Gear transmissions are exciters of two types of vibrations: parametric and kinematic. The parametric exciter of vibrations is the total tooth stiffness, which changes periodically with the angle of rotation of the gear wheel, while kinematic vibrations arise because of manufacturing errors of the gear wheels. These errors, as analysis shows, have a regular periodic dependence on the angle of rotation of the gear wheels. At present, gear wheels are cut mainly by the generating method, carried out with the aid of a worm cutter.

Cylindrical helical wheels cut by the indicated method (in contrast to spur wheels) are much weaker exciters of vibrations because the two above-mentioned types of vibrations are not manifested in them at all (or are manifested to a lesser degree).

In helical gears, the insignificant change in total stiffness with the angle of rotation (in contrast to spur gears, where at the points of mesh alternation the stiffness changes abruptly), as well as the insignificant influence of errors in the setting and manufacture of the tool, is confirmed by a whole series of investigations.

According to the method of A. I. Petrusevich, which has become most widespread in the USSR [4], dynamic loads in helical gears are caused by errors in the circular pitch, i.e., in the position

teeth along the pitch circle of the gear wheel. This occurs as a result of inaccuracy in the kinematic chain of the machine tool. The dynamic load can be determined from the formula

\[ U= \frac{26\cdot \Delta t} {\dfrac{150\cdot d}{d_{\mathrm{ш}}\cdot v^{2}} +\sqrt{\dfrac{150\cdot d}{d_{\mathrm{ш}}\cdot v^{2}}+1} +\sqrt{\dfrac{225\cdot 10^{4}}{z_{\mathrm{ш}}^{2}\cdot v^{2}}+1}} . \tag{1} \]

In this formula:

\(\Delta t\)—the error in the circular pitch, in \(\mu\)m (at speeds greater than 15 m/s, in formula (1), instead of \(\Delta t\), one substitutes \(\Delta t-5\) in the case when \(\Delta t>\mu\)m, and \(\dfrac{\Delta t}{2}\) when \(\Delta t<10\ \mu\)m [4]);

\(d_{\mathrm{ш}}\)—diameter of the pinion, cm;
\(v\)—circumferential speed, m/s;
\(z_{\mathrm{ш}}\)—number of teeth of the pinion;
\(d\)—diameter of that gear wheel (pinion or gear) on which the error causes the greater dynamic load.

According to [4], formula (1) for high-speed transmissions can be somewhat simplified.

If the condition

\[ \frac{\Delta t}{m_s}\cdot z \leq 440 \]

is satisfied, where \(m_s\) is the end-face module, and the gears are cut on machines with accurate indexing transmissions having well-fitted meshing (errors in the circular pitch lead to a smooth distortion of the tooth profile), the denominator of formula (1) is replaced by twice the third term.

To analyze this situation, we write formula (1) as follows:

\[ U=\Omega\frac{\Delta t}{A+B+C}, \tag{2} \]

where

\[ A=\frac{150\cdot d}{d_{\mathrm{ш}}\cdot v^{2}} \]

is the 1st term of the denominator;

\[ B=\sqrt{A^{2}+1} \]

is the 2nd term of the denominator;

\[ C=\sqrt{\frac{225\cdot 10^{4}}{z_{\mathrm{ш}}^{2}\cdot v^{2}}+1} \]

is the 3rd term of the denominator;

\(\Omega\)—coefficient of proportionality.

Most often, the pinion and gear wheel of high-speed transmissions are cut on the same machines; therefore the value of \(d\) may be taken equal to \(d_{\mathrm{ш}}\).

The calculation performed (Table 1) showed that, at circumferential speeds and numbers of pinion teeth characteristic of high-speed transmissions (\(v=35\div100\) m/s, \(z_{\mathrm{ш}}=25\div77\)), the value of the dynamic load calculated by a formula of type 2 differs by no more than 5% from the values calculated by a formula of type 1*)

\[ U=\Omega\frac{\Delta t^{*}}{2\cdot C}. \tag{3} \]

Thus, the additional dynamic load can be determined with sufficient accuracy according to the method of [4] by the formula:

\[ U= \frac{13\cdot \Delta t} {\sqrt{\dfrac{225\cdot 10^{4}}{z_{\mathrm{ш}}^{2}\cdot v^{2}}+1}}, \ \text{kg/cm}. \tag{4} \]

*) Analysis shows that when

\[ A=\frac{150\cdot d_k}{d_{\mathrm{ш}}\cdot v^{2}}=\frac{150\cdot i}{v^{2}}, \]

the final result differs little from that given in Table 1, owing to the smallness of the quantity \(A\) itself.

Table 1

On the basis of formula (4) we constructed a graph (Fig. 1) for determining the quantity $U$ as a function of the parameter $z_{\mathrm{sh}}\cdot v^2$ for various values of $\Delta t$. The ranges of variation of $\Delta t$ and $z_{\mathrm{sh}}\cdot v$ are characteristic of high-speed transmissions. The value of $\Delta t$ may be specified on the basis of the passport data for the cut gear pair.

An analytical study of oscillatory processes in transmissions is difficult, since many parameters are little known, and gearbox installations (for example, GTZA) are complex systems with a wide range of variation in masses and stiffnesses. The development of engineering calculation methods that take into account all the principal factors on which the determination of dynamic loads depends presents very great difficulties. Experimental work in this field is scarce and was begun relatively recently [2, 5, and 7]. Therefore, it should be recommended that additional dynamic loads in high-speed transmissions be determined by the method of [4], which has been successfully applied in the design of many critical gearboxes.

In calculating the contact strength of helical gears (with hardness of the working tooth surfaces $HB \leq 350$), the method of [4] proposes taking into account only one half of the dynamic load caused by errors of the circular pitch. This recommendation is based on the following assumptions:

1. Medium-hard and especially soft (improved) surfaces of gear wheels do not reduce the endurance limit under limited pitting.

2. Pitch errors are partially compensated because of changes in tooth shape resulting from running-in of their working surfaces (Fig. 2).

3. The dynamic load, varying sinusoidally, is sometimes added to the useful load and sometimes subtracted from it (Fig. 3).

However, taking into account only one half of the dynamic-load value raises certain doubts.

1. The first assumption is valid; however, when carrying out experiments and when calculating critical drives, it is desirable to know the actual stresses.

2. The fact of compensating running-in of the transmission, confirmed by a number of researchers [3, 6, and 8], is already taken into account in determining the dy-

Fig. 1. Graph for determining additional dynamic loads

Vertical axis: \(C\), kg/cm

Curve labels: \(\Delta t = 20\), \(\Delta t = 17\), \(\Delta t = 16\), \(\Delta t = 12\), \(\Delta t = 10\), \(\Delta t = 8\), \(\Delta t = 5\)

Right-side label: \(Z_{\omega}^{2} \cdot v^{2}\)

Horizontal scale markings: \(10^{5}\), \(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\), \(9\), \(10^{6}\), \(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\), \(9\), \(10^{7}\), \(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\), \(9\), \(10^{8}\)

dynamic load (in formulas (1) and (4), the value \(\Delta t\) for \(\Delta t > 10\) is taken equal to \(\Delta t - 5\), and for \(\Delta t < 10\), to \(\dfrac{\Delta t}{2}\)).

1. When the transmission operates for a number of cycles exceeding the basic number, the fact that the dynamic load changes is of no significance.

2. In the method of [4], instead of the quantity in the formula one should substitute the value \(U_1 + U\), where \(U_1\) is the additional dynamic load caused by the accumulated pitch error, although this quantity (according to [2]) is not large.

To estimate the magnitude of the dynamic load that must be taken into account in the calculation of gear transmissions, we carried out special experiments. Series of gear wheels were tested with \(\Delta t\) values equal to 13 and 8 microns*).

All experiments were carried out on a high-speed test stand with a closed two-flow circuit in the speed range \(20 \div 60\) m/sec. The load-carrying capacity of the gear wheels was compared at the limiting load, which we selected as the endurance criterion in tests of gear transmissions for contact strength.

Fig. 2. Change in pitch error \((Z = 21)\):
1 — before running-in; 2 — after running-in

Fig. 3. On accounting for dynamic loads: \(P_{\text{pol}}\) — static loads; \(P_{\text{dyn}}\) — dynamic loads; \(\varphi_k\) — angle of rotation of the gear wheels

1. As a result of the experiment, values of the nominal loads \(P_{\text{nom}}\) corresponding to the fatigue limit were obtained. Since the gear wheels were made of the same material with the same hardness of the working surface, the limiting stresses according to Hertz for

* The geometric and design parameters of the compared gear pairs were chosen to be identical: \(m_n = 3\) mm; \(\beta_d = 19^\circ 11'25''\); \(B = 30\) mm; \(i \approx 2\); \(B/t_a = 1.06\).

of both transmissions ($\sigma'$ and $\sigma''$*), caused by the combined action of static and dynamic loads, must be equal.

Since the value of $\sigma$ can be determined as

\[ \sigma = Q \sqrt{P_{\mathrm{st}}} \tag{5} \]

and $\sigma'$ is equal to $\sigma''$, it follows, therefore, that $P_n' = P_n''$, or

\[ P_{\mathrm{st}}' + A_{\mathrm{d}} \cdot P_{\mathrm{dyn}}' = P_{\mathrm{st}}'' + A_{\mathrm{d}} \cdot P_{\mathrm{dyn}}'', \tag{6} \]

where $P_{\mathrm{st}}'$ and $P_{\mathrm{st}}''$ are the nominal loads for the transmissions being compared; $P_{\mathrm{dyn}}'$ and $P_{\mathrm{dyn}}''$ are the additional dynamic loads found for gears with different pitch errors; $A_{\mathrm{d}}$ is a coefficient evaluating the influence of the dynamic load on contact strength.

According to the method of [4], the value of the dynamic load for our gears, cut respectively with accuracies $\Delta t' = 13$ μm and $\Delta t'' = 8$ μm, is equal to

\[ P_{\mathrm{dyn}}' = \frac{13(\Delta t' - 5)} {\sqrt{\dfrac{225 \cdot 10^{4}}{z_{\mathrm{sh}}^{2} \cdot v^{2}} + 1}}, \tag{7} \]

and

\[ P_{\mathrm{dyn}}'' = \frac{13 \cdot \dfrac{\Delta t''}{2}} {\sqrt{\dfrac{225 \cdot 10^{4}}{z_{\mathrm{sh}}^{2} \cdot v^{2}} + 1}}. \tag{7′} \]

Substituting into expression (6) the values $P_{\mathrm{dyn}}'$ and $P_{\mathrm{dyn}}''$ from formulas (7) and (7′), as well as the quantities $P_n'$ and $P_n''$ obtained experimentally for various speeds, we find the values of the coefficient $A_{\mathrm{d}}$ (Table 2).

Table 2

The fact that the coefficient $A_{\mathrm{d}}$ decreases with increasing speed can be explained by the damping properties of the oil wedge increasing as its thickness increases.

Thus, the obtained values of the coefficient $A_{\mathrm{d}}$, which evaluates the influence of the dynamic load on the contact strength of gear transmissions, confirm the need to take into account the full value of the dynamic load. Therefore the dynamic factor $K_{\mathrm{d}}$, used in calculations of gear transmissions for the endurance of the working tooth surfaces, should be determined by the formula $K_{\mathrm{d}} = 1 + \dfrac{U}{P_n}$ (8) instead of the expression $K_{\mathrm{d}} = 1 + \dfrac{0.5\,U}{P_n}$ recommended in [4]. In this case the value $U$ is determined according to the method of [4].

Conclusions

1. Calculations carried out for a large number of high-speed reducers and multipliers show that, in order to determine the additional dynamic load with a sufficient degree of accuracy, one may use the simplified formula (4).

* The stresses in the contact band, obtained by formula (5) for gears with pitch errors of 13 and 8 μm, are denoted by $\sigma'$ and $\sigma''$, respectively.

1. The constructed graph makes it possible, with a minimum expenditure of time, to determine the value of the dynamic load by the method of A. I. Petrusevich.

2. Experiments carried out on gears of various accuracies showed that, in calculating gear transmissions for contact strength, it is necessary to take into account not one-half, but the full value of the dynamic load. In this case the dynamic factor introduced into the calculation is determined by formula (8).

REFERENCES

1. A. E. Belyaev. The influence of speed on the contact strength of gear transmissions. Izv. TPI, vol. 147. Tomsk. 1966.

2. M. D. Genkin, V. K. Grinkevich. Dynamic loads in transmissions with helical gears. Publishing House of the Academy of Sciences of the USSR. Moscow. 1961.

3. V. N. Kudryavtsev. Gear transmissions. Mashgiz. 1957.

4. A. I. Petrusevich. Gear transmissions, Machine Parts, ed. by N. S. Acherkan, book 1. Mashgiz. 1954.

5. A. I. Petrusevich et al. Dynamic loads in gear transmissions with spur gears. Publishing House of the Academy of Sciences of the USSR. Moscow. 1956.

6. A. I. Petrusevich, M. Z. Saburov. Machining of gear wheels and reducers. Mashgiz. 1946.

7. N. V. Skrabelinsky. Dynamic forces in the meshing of high-speed cylindrical transmissions caused by cyclic manufacturing errors of gear transmissions. Department of Technical Services and Information. Moscow. 1959.

8. L. N. Solomin. Increasing the accuracy of high-speed gear transmissions. Department of Scientific and Technical Information No. 140, TsNIITMASH, Moscow. 1959.

9. I. N. Frenkel. Experimental determination of the stiffness of teeth with a helix angle of 30° in cylindrical gear wheels. “Calculation, Design, and Investigation of Transmissions,” III. Publication of the Odessa Polytechnic Institute. 1959.