4 S. Cinquemani et al. where c is the radial clearance of the bearing in operating conditions. It is worth observing that, because of the geometrical nature of this expression, it is an actual contact deformation only if ıj is positive. The next step consists in determining the load-deformation relationship. According to Harris and Kotzalas [4], on the basis of laboratory testing of crowned rollers loaded against raceways, Palmgren [6] experimentally determined the following equation for contact deformation ı [mm]: ı D3:84 10 5 Q0:9 = l 0:8 where l is roller effective length [mm], and Qis the normal total force between a rolling element and a raceway [N]. This expression has been adopted since it allows deriving a direct and explicit relationship between the applied load and the consequent displacements, even if it does not allow taking into account some aspects (i.e. curvature variations and ratios). The numerical constant value for specific roller crowning and/or materials can anyhow be determined by means of experimental tests. By solving this equation with respect to Q and taking into account that each roller is in contact with both races, it is possible to derive an “equivalent stiffness”: Keq Dkb 2 n where nD10/9 for roller bearings and kb D7.7652 10 4 l8/9 is the “single stiffness”. As a result, the load acting on a single roller and its deformation are related by the following equation: Qj D j Keq ıj n where j D1 if •j >0, and is zero otherwise. The final step is the summation of components of forces Qj, in order to determine the force exerted by the outer ring on its seat as a consequence of inner race displacement . This approach have been further extended to the so called “lamina model” [9], in order to take also into account roller profile modifications [9]. At the end, for each bearing, it is possible to find out a combination of displacements along the three axes (X, Y, Z) and forces transmitted along the same three directions (FX, FY, FZ). This look-up table can be calculated offline for all the bearings in the database and represents a theoretical stiffness of each bearing, as it relates displacement to forces. Once the look-up table has been calculated, the displacement of the inner ring of the bearing can be obtained iteratively finding the best combination of forces (FX, FY, FZ) that matches with the forces applied on the bearing (FX, FY, Fz). It is important to note that the accuracy of the solution is strictly related to the resolution of the look up table. If the combinations are few, the result of the algorithm could be far from the real solution. 1.3 Models of Defects on Bearings Pits presence on a raceway has been introduced as a reduction of roller contact deformation, i.e. as an increase of bearing radial clearance. In practice, this means that the expression of ıj modifies as follows ıj D x cos j C y sin j– c ˇj.t/ Cd where Cd is pit depth and ˇj(t) is a Boolean time dependant function that is equal to 1 only if the theoretical contact point between roller and inner or outer race at time t is within pit circumferential extension. It is worth observing that this function depends also on which race the pit is located, since the inner race rotates, and hence the absolute position of a pit on it is not constant. To reduce the number of variable, a simple relationship between the depth (Cd) and the width ( d) of the defect has been imposed as shown in Fig. 1.5 To evaluate the effects of damages on bearings, it is possible to set different values of Cd, and then of d. It’s important to note that models of defects are based on the idea that a roller, when it is over a defect, loses some of the load that can transmit. The limit condition occurs when the roller completely loses the contact with the ring and, therefore, it is no more able to transmit any force. In this condition (easy to be reached for bearing under small a load), even increasing the depth of the defect an increases in force associated with the defect would not be observed. Coming to cage wear, it was introduced by assuming that its effect on bearing global model can be considered by changing the circumferential positions (with respect to their evenly spaced theoretical positions) of the points where each roller exerts
RkJQdWJsaXNoZXIy MTMzNzEzMQ==