A Numerical Analysis of the Bearing Vibration Response under Varying Outer Race Defect Positions

A dynamic model of a ball bearing with an outer race spall defect was formulated to investigate the influence of the defect angular position on the vibration response. The model integrates Hertzian contact deformation, elastic energy principles, and a mathematical representation of the spall geometry within the bearing system. The governing equations of motion were derived using the Lagrangian approach and solved numerically through a fourth-order Runge–Kutta method. Defect orientations at 0°, 90°, 180°, and 270° were examined to capture the variations in system behavior. The time-domain analysis revealed periodic impact signals corresponding to the successive rolling elements striking the defect. The most severe vibratory amplitudes occurred at 270°, where the defect coincides with the zone of maximum radial load. In contrast, positions away from this load zone exhibited weaker responses due to the reduced contact forces. The envelope analysis highlighted modulation at the Ball Pass Frequency of the Outer Race (BPFO) and its higher harmonics, confirming the diagnostic significance of defect-induced excitations. The results demonstrated that defect orientation plays a crucial role in shaping vibration signatures, and accurate fault detection depends on accounting for angular position effects in the condition monitoring of rotating machinery.