A semi-analytic strategy for the system-level modelling of flexibly supported ball bearings

This work presents a semi-analytic approach that allows to efficiently solve the large sliding or rolling contact problems characteristically associated with the dynamic simulation of flexibly supported ball bearings. The approach separates the bulk deformation, represented by a reduced order model, and the analytically described nonlinear local Hertzian deflections at the contact zone. The interacting raceways of the bearing are represented by B-spline surfaces in order to alleviate the issues of non-smoothness. A solution is introduced that allows an efficient redefinition of these deformed interacting surfaces based on the reduced order model, without necessitating iterative procedures or back-projections to the nodal coordinates. Instead, the deformed B-splines are reconstructed by linear combinations of the control points, utilizing the participation factors of the affected mode shapes in the reduced order model. These B-spline surfaces are exploited by a novel ball bearing specific, and spline-based, contact formulation. The numerical results demonstrate the performance and accuracy of the novel technique for various reduction spaces representing the bulk deformation. Moreover, the developed bearing framework offers both accurate stiffness and displacement patterns with respect to the interfaces to the other bodies in the flexible mechanical system. Through a proper selection and adjustments of the mode shapes, even statically and dynamically complete results can be obtained.