Ball bearings - Calculations. Hertz pressure and friction torque

Bearing catalog forms generally allow static load capacity and service life to be calculated using simplified formulas and a few abacuses. The basic concepts for a quick calculation are presented in the article [BM 5 371] . Advanced concepts leading to the exact calculation of the static load carrying capacity of ball bearings are developed in this article.

Knowing the axial and radial loads acting on a bearing, the load distribution on the balls is expressed using one-parameter integrals, the parameter of which is the load distribution factor. These integrals have no analytical solution, but their numerical values are tabulated. The force exerted on each ball is then deduced. Hamrock and Anderson's simplified solution can then be used to calculate the maximum Hertz pressure generated at the center of the ball/track contact.

The case of pairing two O- or X-mounted bearings with a rigid preload deserves special attention, as precision bearings are used, and exact calculation of forces and service life becomes necessary. First, the radial force exerted on each bearing is calculated from the force torsor exerted on the pairing. However, this radial force generates an induced axial force that is difficult to calculate, as it depends on the number of balls under load resulting from the static equilibrium of the pairing, and must be calculated for each of the two bearings. For this purpose, the author of this article proposes an original method of resolution, based on the use of two auxiliary functions (named φ(ε) and ψ(ε) by the author). The result is a system of two non-linear equations whose unknowns are the load distribution factors. Once this system has been solved, the induced axial forces are easily deduced.

Another difficulty in calculating preloaded bearings is the variation of the contact angle with the load, which is why calculations are usually carried out on a computer. This article proposes an analytical solution, obtained from a limited development, which enables the new angle to be calculated by solving a second-degree trinomial.

The last section gives formulas for estimating the friction torque of various types of ball bearings.

A table of notations and acronyms is provided at the end of the article.