Comparison of non-linear vibration of outer race defective ball bearing using two defect functions

The aim of this article is to study the effects of evolution in defect size on vibration of a ball bearing by simulation of a ball bearing by developing a 2-DOF mathematical model and to compare the vibration responses of defective bearings obtained for two widely used defect functions, viz., rectangular function and half-sine wave function. MATLAB codes are developed to prepare a mathematical model of a ball bearing and to solve the differential equations of the model using the Runge-Kutta method. In the model, the mass supported by the bearing is considered as a lumped mass, and the contact between the races and the balls is considered as a series of springs, whose spring stiffness is obtained by using Hertz’s contact deformation theory. This model considers the contact deformation between the balls and the races and the additional displacement between the balls and the inner race due to radial clearance and defect geometry. The maximum possible radial displacement of the ball into the defect is obtained analytically and graphically from the race-ball-defect geometry. First, the impulses generated due to an outer race defect in the ball bearing are modeled using two different defect functions separately and their vibration responses are compared. Secondly, the effects of increase in defect length on vibration of the bearing are simulated separately for two defect functions, and then their responses are compared and analyzed. The results show that when the defect is modeled with a rectangular defect function, the vibration responses obtained are greater than when the defect is modeled with a half-sine wave defect function. And, vibration responses increase rapidly up to a certain level of defect length and then decrease with a further increase in defect length. The vibration analysis performed for different defect lengths can provide good support to vibration analysts and researchers.