A new analytical stiffness model for the double row angular contact ball bearings is proposed since the current methods do not provide stiffness matrix formulations for double row bearings except for self-aligning (spherical) bearings in which angular deflections and tilting moments are negligible. The moment stiffness terms and the cross-coupling stiffness elements in double row angular contact ball bearings are significant; and the stiffness coefficients are highly dependent on the configuration of the rolling elements. Also, unlike roller-type bearings, the contact angle of ball-type bearings depends on the static load(s). The five-dimensional bearing stiffness matrix is first developed for three configurations (face-to-face, back-to-back, and tandem) from basic principles. The diagonal and off-diagonal (cross-coupling) elements of the matrix are calculated from the explicit expressions given the mean bearing load or displacement vector. Modeling approaches between a double row bearing vs. two single row bearings are also analyzed from statics and stiffness perspective. The proposed stiffness matrix is valid for duplex (or paired) bearings assuming all structural elements (such as the shaft and bearing rings) are sufficiently rigid.
Next, a new modal experiment consisting of a vehicle wheel bearing assembly with a double row angular contact ball bearing in a back-to-back arrangement is designed. The bearing is subjected to axial or radial preloads in a controlled manner. Modal experiments with two preloading mechanisms (under non-roatating conditions) show that the nature and extent of bearing preloads considerably affect the natural frequencies and resonant amplitudes, thus influencing the vibration behavior of the shaft-bearing assembly. A five-degree-of freedom vibration model of the shaft-bearing assembly (including the proposed bearing stiffness matrix) is developed to describe the modal experiment. Two alternate (preload-dependent and preload-independent) viscous damping models are then proposed to describe the effect of bearing preload on the resonant amplitudes, similar to those observed experimentally. The proposed bearing stiffness model is then validated by comparing predicted natural frequencies and accelerance spectra with modal measurements.
Finally, four calculation methods are comparatively evaluated by critically examining bearing loads, deflections and stiffness elements; predicted modal properties of the shaft-bearing assembly using each method are also compared with measurements. In particular, the diagonal elements of the proposed stiffness matrix are compared with a commercial code; and, the effects of critical geometric and kinematic parameters on the stiffness coefficients are explored. A finite element based contact mechanics tool is employed to verify certain assumptions of the new matrix formulation. Preliminary modal experiments with a faulty bearing are included to motivate further research.