Stiffness matrix formulation for double row angular contact ball bearings: Analytical development and validation

• New theory developed for double row angular contact ball bearings.
• Diagonal elements of the proposed stiffness matrix are verified with a commercial code.
• Changes in stiffness coefficients are investigated.
• Calculated natural frequencies of shaft-bearing experiment are compared with measurements.

Though double row angular contact ball bearings are widely used in industrial, automotive, and aircraft applications, the scientific literature on double row bearings is sparse. It is also shown that the stiffness matrices of two single row bearings may not be simply superposed to obtain the stiffness matrix of a double row bearing. To overcome the deficiency in the literature, a new, comprehensive, analytical approach is proposed based on the Hertzian theory for back-to-back, face-to-face, and tandem arrangements. The elements of the five-dimensional stiffness matrix for double row angular contact ball bearings are computed given either the mean bearing displacement or the mean load vector. The diagonal elements of the proposed stiffness matrix are verified with a commercial code for all arrangements under three loading scenarios. Some changes in stiffness coefficients are investigated by varying critical kinematic and geometric parameters to provide more insight. Finally, the calculated natural frequencies of a shaft-bearing experiment are successfully compared with measurements, thus validating the proposed stiffness formulation. For double row angular contact ball bearings, the moment stiffness and cross-coupling stiffness terms are significant, and the contact angle changes under loads. The proposed formulation is also valid for paired (duplex) bearings which behave as an integrated double row unit when the surrounding structural elements are sufficiently rigid.