Theoretical analysis of shaft vibration supported by a ball bearing with small sinusoidal waviness

This paper describes theoretical analysis of radial vibration of a rigid shaft supported by a ball bearing with form errors. Assuming a small sinusoidal waviness of n-th order on the inner race, outer race or rotating balls, the displacement of the shaft center while rotating is numerically calculated. The frequency and amplitude of the shaft vibration are analyzed by numerical FFT. From consideration of the generating mechanism of the shaft vibration, it is found that the outer race waviness of the order n=jZ+1 (j: integer, Z: total ball number) generates a backward exciting force and vibration with frequency of jZf _c (f_c: cage speed), while that of n=jZ-1 generates forward ones of the same frequency. The inner race waviness of n=jZ+1 generates a forward exciting force and vibration with frequency of jZ(f _r-f_c)+f_r (f_r: inner race speed), while that of n=jZ-1 generates backward ones with frequency of jZ(f_r-f_c)-f_r. It is also found that small amplitude vibrations with frequencies jZfc and jZ(f_r-f _c)±f_r can be generated by the waviness of the order number different from n=jZ±1, when the ball number Z is a prime number, e.g. 7. A good qualitative agreement between calculated and experimental Campbell diagrams for shaft vibration is obtained