PSI - Issue 24

514 5 rotational frequency 0 of the system itself (1X). Therefore, resonances with peaks in the response occur when a critical speed is reached. The resultant of the fluid dynamic pressure field acting on the impeller in the radial direction is not constant; it can be evaluated by CFD simulations: the fundamental frequency of this fluctuating force is 8 0 (8X). At the same time, defects on bearings may introduce vibrations in the system and can be divided in two main categories: distributed and localized. The first one occurs when the rotating parts do not follow a perfect circular path because of imperfections such as waviness and misalignment; this type of defects is related to the class of tolerance of the bearing. The second one is, instead, characterized by a small damage on a component that, during the bearing life, can have different causes such as excessive loads, overheating, fatigue, corrosion, poor lubricating condition and so on. These defects increase the vibration level hugely and reduce the bearing life faster if they involve amplification of the dynamic response of the system. Even if the complete frequency content and the whole spectrum of the excitation can be different, for both localized and distributed defects the fundamental frequency of the produced vibration is the same and depends only on which component is damaged (outer ring, inner ring, rolling elements and cage). The theoretical values of the characteristic frequencies of the defects can be computed considering the ideal kinematics of the bearing and they are well known from the literature (Tab.1). Vincenzo D’Addio et al. / Procedia Structural Integrity 24 (2019) 510–525 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

Table 1. Characteristic frequencies of the bearing components, Ehrich (1998).

Z =number of rolling elements

Outer race (BPFO) BPFO = 0 2 Z (1 − D d cos α) BPFI = 0 2 Z (1 + D d cos α) Rolling Elements (BFS) BSF = 0 1 2 D d (1 − ( D d cos α) 2 ) Cage (FTF) FTF = 0 1 2 (1 − D d cos α) Inner race (BPFI)

BPFO represents the frequency of the ball passing on a point of the outer race, BPFI represents the frequency of the ball passing on a point of the inner race. BFS represents the frequency of the ball passing on a point of the inner and outer race alternatively. FTF represents the rotational frequency of the cage. If the point corresponds to a defect the contact with the ball is a source of periodic impulsive excitation. As one can easily observe, the values of these characteristic frequencies are functions only of geometrical features of the components of that specific bearing. It must be underlined that while the defective point on the outer race is fixed in space the others rotate (the one on the inner race at f 0 , the one on the ball and on the cage at the cage frequency). In the former case the periodic excitation can be seen as the sum of two counter rotating vectors of equal magnitude, rotating at ±BPFO respectively, thus capable of exciting both forward and backward rotor whirling modes at such frequency. In the case of a rotating defect, such a rotation makes the excitation frequency vary with respect to the characteristic frequency. Considering, for example, a defect on the inner race rotating at f 0 , it produces a periodic excitation that can be seen as the sum of two counter rotating vectors rotating at a frequency of ±BPFI+ f 0 respectively, again capable of exciting both forward and backward rotor whirling modes but at frequencies affected by f 0 .