Stability Analysis of a Rotating System Due to the Effect of Ball Bearing Waviness

Abstract

This research presents an analytical model to investigate the stability due to the ball bearing waviness in a rotating system supported by two ball bearings. The stiffness of a ball bearing changes periodically due to the waviness in the rolling elements as the rotor rotates, and it can be calculated by differentiating the nonlinear contact forces. The linearized equations of motion can be represented as a parametrically excited system in the form of Mathieu’s equation, because the stiffness coefficients have time-varying components due to the waviness. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as the simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving Hill’s infinite determinant for these algebraic equations. The validity of this research is proven by comparing the stability chart with the time responses of the vibration model suggested by prior research. This research shows that the waviness in the ball bearing generates the time-varying component of the stiffness coefficient, whose frequency is called the frequency of the parametric excitation. It also shows that the instability takes place from the positions in which the ratio of the natural frequency to the frequency of the parametric excitation corresponds to i/2 i=1,2,3,….

Time-Varying Total Stiffness Matrix of a Rigid Machine Spindle-Angular Contact Ball Bearings Assembly: Theory and Analytical/Experimental Verifications

“…The works [1][2][3][4][5][6][7][8][9]15,[17][18][19][21][22][23][24][25][26][27][28][29][30][32][33][34][35][36][38][39][40][41][42][43][44]48,[50][51][52] use summation over the bearing rolling elements in contact to compute bearing total loads. Houpert [10,11] replaced this summation by integration using Sjovall's [12] axial and radial load distribution integrals (J a , J r ).…”

“…K b is computed using shaft motions at the bearing location (Section 2.4) and then used to derive the total stiffness matrix of the rigid shaft-bearings assembly (K cg ), see Section 2.5. To the author's best knowledge the studies related to K cg are limited to [32][33][34], see Section 1.2.…”

A general method for impact dynamic analysis of a planar multi-body system with a rolling ball bearing joint

“…To simplify the computational model, only the normal contact forces in bearing multi-point contacts were considered by the authors. On the other hand, the rolling friction in ball and raceway contact is usually quite small and has a smaller effect on the dynamic response of such bearing systems: many researchers [42][43][44][45][47][48][49] examining rolling element bearing dynamics neglect frictional effects.…”

“…All of these determine the complexity of the modelling and solution of the mechanical multi-body system with joint rolling element bearings. In recent years, and with regards the dynamic modelling of rolling element bearings, Jang and Jeong [42][43][44] presented a non-linear model to analyse the ball bearing vibration due to waviness in a rigid rotor supported by two or more ball bearings. Bai and Xu [45] presented a 5-DOF dynamic model to study the dynamic performance of ball bearings due to the effect of both internal clearance and waviness at high speed, where the centrifugal force and gyroscopic moment from balls were taken into account.…”

Application of analytical method in the evaluation of induction motors bearings’ reliability

“…In case of the bearings, the lack of sufficient data for defining the repartitions regarding each type of damage causes, for the moment, the ordinary calculation of bearings to distinctly consider, in relation to reliability, the contact fatigue, [1]- [6], or the abrasive wear [5], [7]- [9]. For the rest are established limit conditions, recommendations and proper specifications.…”