Coupling relationship of the non-ideal parallel mechanism using modified Craig-Bampton method

Abstract

The coupling relationship in the dynamics model of 4-SPS/PS (P-prismatic joint, S-spherical joint) parallel mechanism with a flexible moving platform and clearance spherical joint is investigated. Two primary coupling relationships are studied: (i) the coupling between reference motion and elastic deformation; (ii) the coupling between the elastic vibration and collision. Firstly, the dynamic model of the flexible moving platform is formulated using the floating frame of reference (FFR) formulation. According to the topology of this parallel mechanism, the fixed-fixed reference conditions are selected and imposed at the finite element (FE) analysis stage for the thin-plate element. Also, considering that the conventional Craig-Bampton (CB) method does not impose any reference condition at FE analysis stage, therefore, a modified CB method is proposed to obtain the fixed-fixed modes, which is validated by normal mode approach. Secondly, the clearance is introduced to one of the spherical joints in the parallel mechanism. The normal and tangential contact forces are calculated based on the Lankarani-Nikravesh contact force model and a modified Coulomb friction model, respectively. In addition, the effect of two coupling relationships on the dynamic response of this parallel mechanism with various thickness of the moving platform is discussed. Finally, the simulation results reveal that the coupling between reference motion and elastic deformation can be ignored in certain situations. However, the coupling between the clearance spherical joint and flexible moving platform deserves more attention, wherein, the flexibility of the moving platform leads to more dramatic collision when its deformation is relatively large, oppositely, it behaves like a dashpot damper to make the system more stable when the deformation is small. This phenomenon also depends on different modal representations.