Assessment of the Load Capacity of a Helical Gear Compared to a Spur Gear
Abstract Generated abstract
This paper examines how the contact-strength load capacity of helical gearing should be assessed relative to spur gearing, challenging calculation methods that treat the two as equivalent. Using a coefficient relating transmitted moments, the author discusses the roles of contact ratio, contact-line variation, helix angle, and load concentration at the pitch point, drawing on theoretical considerations and prior experimental data. The analysis suggests that helical gears have a higher load-carrying capacity than spur gears, with a typical coefficient of about 1.3 to 1.35 for commonly used helix angles, and that at high speeds smaller helix angles may be advantageous. The paper also notes practical design implications, including possible replacement of herringbone gears by helical gears with thrust washers in some high-speed drives.
Full Text
ESTIMATION OF THE LOAD-CARRYING CAPACITY OF A HELICAL GEARING IN COMPARISON WITH A SPUR GEARING
A. E. BELYAEV
(Presented by the scientific seminar of the Department of Applied Mechanics)
In a number of methods for calculating gear transmissions (for example, [3]) that are widely used, helical gearings, in terms of load-carrying capacity limited by contact strength, are equated to spur gearings. Estimating the load-carrying capacity of uncorrected spur \((M_1)\) and helical \((M_{1k})\) gearings by the coefficient \(\vartheta_k = \dfrac{M_{1k}}{M_1}\), according to the method of [4], it can be shown that
\[ \vartheta_k=\frac{\lambda_{\min}\cdot \varepsilon_s}{K_p\cdot \cos^2\beta_0}, \tag{1} \]
where \(\lambda_{\min}\) is the coefficient characterizing the fluctuation of the lengths of the contact lines in a helical gearing;
\(\varepsilon_s\) is the contact ratio;
\(K_p\) is the coefficient taking into account the concentration of load at the pitch point;
\(\beta_0\) is the helix angle of the tooth on the base cylinder.
The equivalence of the load-carrying capacities of the transmissions being compared is explained by the concentration of load at the pitch point, which is caused by the greater total stiffness of the teeth at the pitch point, as well as by the reduction of wear of the working surfaces as they approach the pitch point, in accordance with the decrease in sliding velocity. In this connection reference is made to the fact that the factors mentioned lead to pitting at the pitch point under loads considered permissible for spur gearings. However, V. N. Kudryavtsev showed [5] that the serviceability of spur and helical gearings, proceeding from the condition of their working surfaces, cannot be assessed by identical criteria. Pitting arising at the pitch point will lead to a redistribution of contact loads toward an increase in the regions distant from the pitch point, and subsequently the helical gearing may carry a much greater load. This served as the basis for theoretically and experimentally substantiating the value \(K_p=1.2\) (instead of \(K_p=1.5 \div 1.6\), previously adopted), which realistically reflects reality. The same value was also theoretically established by us for crowned helical gearings [2], although some authors believed that in the latter case the values of \(K_p\) should be considerably higher.
The experiments conducted with helical gears [1] in the speed range of 20–60 m/sec confirmed the higher load-carrying capacity of helical gears in comparison with spur gears. In Fig. 1 the values of \(\vartheta_k\) (according to data from various authors) vary in the range \(\sim 1.1 \div 1.4\), and there is no single view on the dependence of \(\vartheta_k\) on the angle \(\beta_{\text{д}}\). In this connection, the data recalculated in terms of rolling speed (for the possibility of comparing them) are plotted as curve 7. The analysis shows that the value of \(\vartheta_k\) may be taken as \(1.3 \div 1.35\) for the range of angles with which helical gears are most often designed. In this case, the character of the change in \(\vartheta\) remains practically the same for all speeds.
Fig. 1. Value of the coefficient \(\vartheta_k\), according to data from various authors:
1 — according to formula (1),
2, 3, 4, 5, 6 — dependences constructed according to the data of work [4];
7 — according to our experimental data (average for three speeds);
8 — according to data of A. I. Khoteev [6]
The experiments conducted also make it possible to suppose that at high speeds the gears with small angles \(\beta_{\text{д}}\) have the greater load-carrying capacity (in contrast to the opinions of the authors of Soviet and foreign works), based on the condition of contact strength.
In this connection, in some cases designers of high-speed drives may be recommended to replace herringbone gears with helical gears, using special (thrust) washers to take axial loads. By abandoning the traditional design, one can reduce axial dimensions and simplify gear cutting.
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A. I. Khoteev. Experimental determination of the influence of the tooth helix angle on the load-carrying capacity of cylindrical gear transmissions. In: Ways of Reducing the Dimensions and Weight of Gear Transmissions, Leningrad, 1959.