Rotating machinery dynamics simulation. I. Rigid systems with ball bearing nonlinearities and outer ring ovality under rotating unbalance excitation

Abstract

The radial clearance in rolling bearing systems, required to compensate for dimensional changes associated with thermal expansion of the various parts during operation, may cause dimensional attrition and comprise bearing life, if unloaded operation occurs and balls skid [D. Childs and D. Moyer, ASME J. Eng. Gas Turb. Power 107, 152-159 (1985)]. Also, it can cause jumps in the response to unbalance excitation. These undesirable effects may be eliminated by introducing two or more loops into one of the bearing races so that at least two points of the ring circumference provide a positive zero clearance [D. Childs, Handbook of Rotordynamics, edited by F. Ehrich (McGraw-Hill, NY, 1992)]. The deviation of the outer ring with two loops, known as ovality, is one of the bearing distributed defects. Although this class of imperfections has received much work, none of the available studies has simulated the effect of the outer ring ovality on the dynamic behavior of rotating machinery under rotating unbalance with consideration of ball bearing nonlinearities, shaft elasticity, and speed of rotation. To fill this gap, the equations of motion of a rotor-ball bearing system are formulated using finite-elements (FE) discretization and Lagrange's equations. The analyses are specialized to a rigid-rotor system, by retaining the rigid body modes only in the FE solution. Samples of the results are presented in both time domain and frequency domain for a system with and without outer ring ovality. It is found that with ideal bearings (no ovality), the vibration spectrum is qualitatively and quantitatively the same in both the horizontal and vertical directions. When the ring ovality is introduced, however, the spectrum in both orthogonal planes is no longer similar. And magnitude of the bearing load has increased in the form of repeated random impacts, between balls and rings, in the horizontal direction (direction of maximum clearance) compared to a continuous contact along the vertical direction (direction of positive zero clearance). This underlines the importance of the vibration measuring probe's direction, with respect to the outer ring axes, to capture impact-induced vibrations. Moreover, when the harmonic excitation is increased for a system with ideal bearings, the spectral peaks above forcing frequency have shifted to a higher-frequency region, indicating some sort of a hard spring mechanism inherent in the system. Another observation, is that for the same external excitation, vibration amplitude at forcing frequency in the bearing force spectrum is the same for systems with or without outer ring ovality.

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

“…El-Sayed [28] used his experimental results that show radial stiffness of a DGBB under only a radial dead load to increase nonlinearly as load increases and derived an emperical equation for stiffness calculation. El-Saeidy [30] showed that an increase in the rotating unbalance load of a rigid rotor-ideal ball bearings system will shift its frequencies other than forcing (unbalance) frequency to a higher frequency region; i.e. system's nonlinearity is of the hard spring type.…”

“…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 ).…”

“…However, equivalent viscous damping of constant = 200 N.s/m is used for deep groove ball bearing [36,50]. Further, different authors neglected bearing damping, see for example [1][2][3][4] and [30]. In Ref.…”

Dynamics of a Rigid Rotor Linear/Nonlinear Bearings System Subject to Rotating Unbalance and Base Excitations

“…Components of the vector 1 in the abc frame are (see El-Saeidy (1998, 2000a1 see also the derivation of 1 components in terms of Euler angles by, for example, El-Saeidy (1998)):…”

Some characteristic parameters affecting the natural frequency of a rotating shaft supported by defect-free ball bearings

“…The parameters affecting the natural frequency of the system are found to be not very easily identi ed by different researchers [20,21]. Akturk et al [4] showed that increasing the preload would force the system to oscillate with a higher stiffness bearing, and therefore the natural frequency of the system would be increased.…”