Vibration transmission through rolling element bearings, part I: Bearing stiffness formulation

Current bearing models, based on either ideal boundary condition or purely translational stiffness element description, cannot explain how the vibratory motion may be transmitted from the rotating shaft to the casing and other connecting structures in rotating mechanical equipment. For example, a vibration model of a rotating system based upon the existing bearing models can predict only the purely in-plane type motion on the flexible casing plate given only the bending motion on the shaft. However, experimental results have shown that the casing plate motion is primarily flexural or out-of-plane type. In this paper this issue is claridied quantitatively and qualitatively by developing a new mathematical model for the precision rolling element bearings from basic principles. A comprehensive bearing stiffness matrix [K]_bm of dimension six is proposed which clearly demonstrates a coupling between the shaft bending motion and the flexural motion on the casing plate. A numerical scheme which involves a solution of non-linear algebraic equations is proposed for the estimation of the stiffness coefficients given the mean bearing load vector. A second method which requires the direct evaluation of these stiffness coefficients given the mean bearing displacement vector is also discussed. Some of the translational stiffness coefficients of the proposed bearing matrix have been verified by using available analytical and experimental data. Further validation of [K]_bm is not possible as coupling coefficients are never measured. Also, parametric studies on the effect of unloaded contact angle, preload or bearing type are included. These results lead to a complete characterization of the bearing stiffness matrix. The theory is used to analyze vibration transmission properties in the companion paper, Part II.