As a typical nonlinear mechanical system, the combined angular contact ball bearing (C-ACBB) is widely used in various fields, but the theoretical studies on its nonlinear stiffness characteristics are rarely involved. This paper proposed a general mathematical model for the C-ACBB under different configuration forms based on the load-deformation coordination analysis under the multi-coordinate systems, and then a multi-layer nested algorithm based on the homotopy method and Newton-Raphson method for the efficient solution of the nonlinear equilibrium equations of the proposed model is presented. Besides, the complete expressions of the stiffness matrices for the C-ACBB under different configurations are analytically derived, and the relationships between the integral C-ACBB and multiple SR-ACBBs is presented. On the basis, and effects of both the rotating speed and axial preload on the C-ACBB under 5 different configurations and 3 different load conditions are analyzed. The results show that, the C-ACBB under different configurations present various and complex piecewise nonlinear stiffness characteristics, which is affect by both the rotating speed and initial preload. Besides, the axial stiffness variation of the C-ACBB under asymmetric configuration shows obvious directionality, the radial stiffness variation of the C-ACBB is symmetrical and related to the load action point.

Background: Previous researchers have found that cortex is involved in the regulation of treadmill walking. However, cortico-muscular interaction analysis in a ‘fine’ gait phase (such as seven phases of the gait cycle) remains lacking in the time-frequency domain. Methods: In this investigation, we used beta band electroencephalogram (EEG) data to find that eight muscle-related cortices are inconsistent at the end of the swing and stance phases. The eight muscle-related cortices differ at each phase according to gamma band EEG data. Firstly, slope sign change (SSC) and mean power frequency (MPF) features of EEG and surface electromyography (sEMG) were used to recognize the seven gait phases, which are loading response (LR), mid-stance (MST), terminal stance (TST), pre-swing (PSW), initial swing (ISW), mid-swing (MSW) and terminal swing (TSW). Following this, the time-frequency cross mutual information (TFCMI) method, a novel time-frequency analysis method, was applied to examine the eight muscle-related cortices in seven gait phases using beta and gamma band EEG data. Results: We firstly found that the feature set comprising SSC of EEG as well as SSC and MPF of sEMG was available for seven gait phases recognition, and secondly that TFCMI values between each sEMG channel and EEG differed significantly in the seven gait phases. Conclusions: This suggests that analysis of the seven gait phases is beneficial. These insights enrich previous findings from authors carrying out cortico-muscular interaction analysis as well as providing critical information for rehabilitation physicians.