From the Rolls-Royce experimental archive: a quarter of a million communications from Rolls-Royce, 1906 to 1960's. Documents from the Sir Henry Royce Memorial Foundation (SHRMF).
Technical document detailing the formulas and principles for ball bearing design.
| Identifier | ExFiles\Box 20\7\ Scan007 | |
| Date | 23th August 1911 | |
| THE MOTOR TRADER August 23, 1911 presses of varying intensity, since each ball as it passes around the semi-circle which represents the loaded half is subjected to a variation of direct pressure which (theoretically) varies from 0 to W and back to 0. (3) A tension stress which results from the tendency of the outer ring or race to deform from a true circle under the unequal application and distribution of the load. Formula for Ball Bearing Design.—Stresses Nos. 1 and 2 are greatest at the point of greatest intensity of direct load, which is generally at or near the bottom of the bearing, while the third, or tensile, stress is probably greatest on the opposite side of the ring. Referring to Fig. 9:— Let d = diameter of ball in inches. φ = radial angle between centres of adjacent balls. P = safe load capacity of one ball. E = elastic limit of steel used. S = safe working stress for the steel under the conditions of load and speed to be provided for. m = minimum thickness of metal in the outer race. m¹ = minimum thickness of metal in the inner race. c = effective width of race. D = outside diameter of the bearing. D¹ = diameter ball path in outer race (or the enclosing circle). D² = diameter of circle through ball centres. D¹¹¹ = diameter ball path in inner race. R, R¹, R¹¹, and R¹¹¹ = radii corresponding to D, D¹, D¹¹, and D¹¹¹. W = total load capacity of bearing. 9.—Illustrating Application of Formula for Design of a Ball-bearing. Application of Formula.—For high-grade alloy steels, such as are generally used for ball bearings, the elastic limit E is generally about 100,000 pounds per square inch; therefore, with this class of material the safe working stress S may be taken as follows :— S = 100,000 / 5 = 20,000 for bearings under quiescent load, or for very slow speed and steady load. S = 100,000 / 10 = 10,000 for variable live loads and speeds. S = 100,000 / 27 = 5,000 for bearings subject to shocks. Minimum Metal Thickness.—The minimum thickness of metal (M and m) to be provided in the races of a ball bearing which is designed for stresses at the maximum safe load capacity of the balls will be: M = √(6 P R¹² 2 sin ½φ / 10 S c) m² = √(6 P D¹¹¹ sin ½φ / 10 S c) m = √(6 P D¹¹¹ sin ½φ / 10 S c) In terms of d, for the maximum safe load capacity, the minimum thicknesses of M and m may be determined by the following: M = d √(12,000 D¹ sin ½φ / 10 S c) for 2-point type. M = d √(8,000 D¹ sin ½φ / 10 S c) for 4-point type. Thus in bearings of equal dimensions and with the same number of balls, the four-point type of bearing requires less thickness of metal in the races, because the safe working load capacity for balls for that form of contact is only about two-thirds as large as for two-point grooved races, and the total load capacity of the four-point type of bearing must be further reduced by multiplying the apparent load capacity by the cosine of the angle of inclination of the race surface. (Cos 0 deg. = 1, cos 90 deg. = 0, cos 45 deg. = .7.) Maximum Metal Thickness.—To determine the minimum thickness of metal (M and m) for any desired working load W, the following formulæ may be used: M = √(W D¹ sin ½φ / 5 S c K) Or, M = sin ½φ √(3 W D¹ / 5 S c) m = √(3 W D¹¹¹ sin ½φ / 5 S c K) Effect of Varying Number and Diameter of Balls. Increasing the number of balls by using balls of smaller diameter permits the use of heavier races within the limiting outer diameter of the bearing and decreases the maximum load permitted. In the formula W/K, which determines the maximum load which comes upon a single ball, the denominator K increases as the number of balls is increased (being approximately one-third of n); hence inversely as the diameter of the balls is decreased; the numerator W, or total safe load capacity of the bearing (= P x K), increases as a variable function of the square of the diameter of balls used; therefore, the largest size of balls permitted by the limiting bore and diameter of housing will give the largest rated load capacity for the bearing. To increase the size of balls to be used in a ball bearing, however, also involves an increase in the thickness of metal required for the races; therefore, the determination of the proper elements of a ball bearing for the maximum results, under any certain limiting conditions, involves a series of calculations with variable factors and the process of elimination by 'trial and error'. Through over-zeal on the part of designers, some of the large load capacities designated for standard sizes of ball bearings in various catalogues, are based upon a selection of maximum ball diameters, with corresponding maximum rated load capacities, which are attained at the expense of races which are too light for the service for which the bearings are rated and ostensibly recommended. Effect of Speed upon Load Capacity. To determine the true relation of speed to the load capacity of ball bearings would require an extended series of experiments with different sizes and types of bearings, in which several bearings of each size and type, made of the same materials, were run side by side, under the same load, but at different speeds, and similar sets of bearings side by side, with different loads, with the duration of the tests limited only by the ultimate failure of the bearings from 'fatigue'. Rule Experimentally Suggested by Tests.—The writer’s limited range of experiments indicate that the load capacity of a radial four-point contact type of ball bearing decreases in a somewhat more rapid ratio than the cube-roots of the speeds and at a less rapid rate than the square-roots of the speeds; therefore, in the absence of data admitting of a greater degree of refinement, the following empirical rule is suggested: | ||