Design of High Speed Gears

Abstract

The surface stresses on teeth are calculated in terms of radius of curvature of the tooth flank. Upon this is based a method to determine the maximum energy that can be safely transmitted from one to another during one revolution. This quantity of energy, divided by the square of the diameter of the pinion and a constant is called power-function of the gear and is a dimensionless quantity. I t can be easily determined by a tooth form layout to any enlarged scale. This power function is calculated with various pressure angles, addenda, and ratios. D a successful reduction to handle power outputs of modern aircraft engines in excess of 1000 hp. has proven to be a very difficult problem. Fortunately, considerable work has been done to guide the design; that of Buckingham* is outstanding. A formula by which the loading of teeth due to the impact of the teeth can be calculated, allowing both for speed and tooth errors, is given in this reference. The Lewis formula gives a measure of the strength of the tooth. In the average high speed application, however, the conventional tooth strength is not the controlling factor. The final step is in the so-called Wear Factor. This is a function of the surface stresses on the tooth flank. The purpose of the present paper is to supplement this last and most important phase of the problem. It has been observed that high speed gears have a tendency to show wear at or near the base circle of the teeth. As a result, the use of short addendum and dedendum teeth has been introduced with considerable success, such as 10/12 pitch; this rneans the pitch itself is 10 P., but the addendum and dedendum correspond to 12 P. As shown later in this paper, this is a move in the right direction, but better results can be expected in any individual case by the method suggested below. The stress on the tooth surface depends upon the radii of curvature as worked out by Hertz. With an involute these radii are determined by the instantaneous position of the point contact. I t will be shown that, for a given stress, the load-carrying capacity is a maximum at the pitch point with 2 equal gears. This load carrying capacity decreases along the line of contact, and is equal to zero at both base circles. Later it is proven that one definite length of tooth contact allows a maximum of energy to be transmitted per tooth contact. This maximum times the number of teeth is the energy transmitted per revolution of the gear. By reducing the latter quantity to a dimensionless function a so-called power function is introduced. In this paper no limitations regarding cutters, etc., are taken as criteria; only involute curves are used. First, two equal gears are considered and the power function calculated with various pressure angles and; tooth numbers. Due consideration is given to the number of teeth in contact. In a graph it is shown that a 30° pressure angle is much better than 20°; also large tooth numbers give better results; in other words with a certain pitch diameter, a finer pitch is; better. Next, the other extreme, a pinion with a rack, again varying the pressure angle and number of teeth, is investigated. Here it is shown that a small pressure angle is superior. Large tooth numbers are only slightly better. Other ratios, including internal gearing, are dealt with graphically and the results are shown for one particular number of teeth in the pinion and 3 pressure angles. The capacity of internal gearing is very large. When two cylinders are in contact as shown in Fig. 1 the well known equation of Hertz applies: P = k YXYII(YX + r2) (1> Received January 26, 1939. * Buckingham, Earl, Dynamic Loads on Gear Teeth, A.S.M.E. Research Publication. 1932. where P k F I G . 1. Two cylinders pressed against each other are stressed according to Hertz equation. load carrying capacity per 1 in. width; radius of cylinder 1, 1 in. wide; radius of cylinder 2, 1 in. wide; constant dependent upon the material only; D E S I G N O F H I G H S P E E D G E A R S = compressive strength in lbs. per square inch. In Fig. 2 a set of involute gears are shown in contact. In this figure, Ro = base radius of both gears; a = the pressure angle; N = number of teeth of both gears; C = center distance. The point of contact travels along the line of action GF from B to D (see Fig. 2). At point B the radius