Waviness on the bearings and its' effect on the bearing vibrations

Bearings in widespread use in so many areas, are indispensable machine elements in industry at the present time. Since the bearings directly affect the performance of the machines, under today's competitive environment, the bearing customers demand better quality and higher levels of performance from bearings. This situation has led to positive changes in bearing manufacturing techniques by the help of developments in the manufacturing technologies. It is well known that, a slightest manufacturing defect in a bearing directly and adversely affects the operating performance of machines using faulty bearings. Furthermore, manufacturing defects in bearings may lead to malfunctioning or complete failure of the machine due to bearing-induced vibrations in the system. The manufacturing processes of raceways of the inner and outer rings play the most decisive role in the quality of the ball bearings. The final grinding process of the raceways is probably the most important processes as it forms the final form or shape of the raceways. During this process, the relative vibrations between the grinding wheel and the inner or outer rings occur and this situation leads to a characteristic surface defects called waviness in the raceways of inner and outer rings of bearing. This defect which is a direct result of the manufacturing process causes vibrations in the bearings and the machines where those bearings are used, and adversely affects the performance of the machine. The performance of the bearings is very important for bearing manufacturers under today's competitive conditions. Therefore, they are aiming to understand various parameters that are affecting the quality of the bearings. One of such parameters is the waviness of the raceways which initiated many studies in the literature aiming to understand its effects on vibrations of the bearing and/or machines. This is also the primary objective of this thesis. More specifically, this thesis aims to develop first a mathematical model of a test system that is used to measure the vibrations of bearings and then, using this model, to make predictions about the effects of waviness on the vibrations of ball bearings. In line with this goal outlined above, first of all, the existing standards about how to define and how to quantify waviness by using measurement systems are researched. It is found that waviness is obtained by processing data acquired via surface finish measurements by using appropriate band-pass filters. Then, the published work in the literature dealing with modelling the waviness and the effects on the bearing vibrations are reviewed. The systems studied in the published work are individually described and the results obtained from these studies are outlined. In these published works, the effects of waviness of the inner and outer raceways and/or balls on bearing vibrations are studied. Some published works point out a relation between the number of balls in a ball bearing and the orders of waviness, which cause severe vibrations. According to xxvii these published works, when the number of balls in a ball bearing and the orders of waviness are equal to each other, there would be severe vibrations due to the fact that, in this case, there is a symmetry of loading and all balls tend to vibrate in phase. It is also found that some published works indicate that the high radial vibrations are expected at certain orders of waviness which are related to the number of balls in a ball bearing. Then, the mathematical models developed/used by the published articles are studied and scrutinised. During this critical review, different mathematical models ranging from two degrees of freedom to higher degrees of freedom have been identified. Consequntly, the existing mathematical models have been classified according to the number of degrees of freedom. Similarities and differences of these models are determined and explained. The significant differences between them are found to be the variety of external forces included in the models and how the deformation of the balls in the bearing are modelled by using various number of degrees of freedom. Furthermore, all the mathematical models in the literature, are found to created by assuming that the outer ring is fixed and the inner ring is moveable in the radial direction. A new mathematical model for the investigation of the effect of waviness on bearing vibrations is developed in this thesis. The new mathematical model presented in this thesis is developed in accordance with the vibration measurement test apparatus specified in BS ISO 15242-2 standard. Some assumptions are made during the development of the mathematical model, the important assumptions worth mentioning here are that there is no slipping of balls as they roll on the surface of races, the races are flexurally rigid and undergo only local deformation due to the stresses in contacts and balls are positioned equi-pitched around the inner race and there is no interaction between them. In the model, small level of damping is assumed to present damping due to friction. The model is created by assuming that the inner ring is rotating, but it is fixed in the radial direction and the outer ring is capable of vibrating along horizontal and vertical directions. Also, it is assumed that the ball, inner and the outer races and the cage have motions in the plane of the bearing only. Any motion in the axial direction is eliminated due to this assumption. Furthermore, this model is different from existing models in the literature in the sense that it allows the predictions of the outer ring vibrations due to the waviness of the inner and the outer raceways. In the bearing model proposed in this thesis, the effect of the waviness on the bearing vibration is modeled by considering the Hertzian contact theory. It is considered that the inner and outer rings may have waviness. However, the waviness related to bearing balls is neglected. The numerical solution of the mathematical model with two degrees of freedom is obtained by utilising Runge-Kutta iterative method. For numerical solution of the mathematical model, the forces due to the deflection of a ball are determined and the total force acting on a bearing is calculated; these forces are utilised in obtaining the solutions of the equations of motion for the vibrations of the ball bearing. Then, the results obtained from the numerical solution of the mathematical model are recorded as time-domain records and their Fourier transforms are computed to obtain the spcetrums of vibrations in the frequency domain. Investigations of the effect of waviness on bearing vibrations are carried out in MATLAB enviroment using a code developed for this purpose. The analyses are carried out for a bearing (6205) with 8 and 9 balls while it is rotating at a speed of 1800 rpm. The waviness amplitude was set to 0.2 μm for inner and outer races and the xxviii number of waves round the inner and outer races circumference is varied for each analysis. Predictions made using the mathematical model developed in this thesis are validated by using the available findings in the literature. Then, after detailed investigations carried out within the scope of this thesis, a new relationship between the waviness and the bearing vibrations is established and presented.