A General Framework Of Stiffness And Dynamic Modeling And Analysis For Lower Mobility Parallel Mechanisms

Parallel mechanisms(PM) are core compenents of high speed machining(HSM) for large aluminum-alloy structures, whose rigidty and dynamic behaviors will effect machining accuracy and efficiency directly. However, a stiffness model and a dynamic model of a PM are always established separately. A general framework for stiffness and dynamics has not been proposed yet. In view of this, a general framework for a kind of PMs with lower mobility is proposed in this paper, which can be applied in static stiffness and dynamics evaluation simultaneously. Conclusions are drawn as follows.Firstly, the substructure synthesis technique is applied due to PMs’ structure features. The whole system is divided into a moving platform, a fixed base and three limb substructures. In order to take the compliance and complex structures of a limb, each limb assemblage is modeled as a spatial beam with corresponding cross sections, whose differential equation of motion is derived through finite element formulation. Considering compliance of joints, each joint is simplified into a six degree-of-freedom(6-DOF) virtual lumped springs with equivalent stiffness at their geometric centers. The moving platform is treated as a rigid body due to its high stiffness, and the Newton’s 2nd law is adopted to establish the differential equation of motion. The governing differential equation of motion is synthesized by introducing the deformation compatibility conditions between the moving platform and limbs as well as the fixed base and limbs.In order to verify the accuracy of the proposed model, analytical stiffness and dynamic behaviors of A3 Head are compared with experimental results which were conducted by Tianjin University. Comparisons demonstrate that the proposed general framework is accurate enough, thus can be applied to evaluate the stiffness and dynamic performance of one kind of PMs with lower mobility.Then, taking the Z3 Head, A3 Head, Exechon PM and Exe-Variant PM as examples, stiffness distributions of these four PM modules are evaluated and a parametric analysis is conducted. Results show that stiffness distributions of the four PMs are strongly position-dependent and symmetric due to their structure features. Comparisions of the stiffness distributions depict that stiffness of the Z3 Head is higher than that of A3 Head with the roughly same size and workspace, and stiffness of the Exe-Variant PM is comparative to that of Exechon PM with the roughly same size and workspace. Meanwhile, the parametric analysis shows that dimensional parameters and structural parameters are of great importance on the stiffness performance, which should be considered during the design stage.Finally, inherent characteristics of the four PMs are analysed. Results demonstrate that lower natural frequencies of the four PMs are position-dependent and symmetric. A parametric analysis shows that the dimensional parameters as well as structural parameters have important effects on the dynamic behaviors of PMs. Comparisions of the inherent characteristics depict that the dynamic behaviors of the Z3 Head are superior to those of A3 Head with the roughly same size and workspace, meanwhile the dynamic behaviors of the Exe-Variant PM are better than those of Exechon PM with the roughly same size and workspace.The proposed general frame combines the advantages of FEM and anlytical methods, which has satisfying computation accuracy and efficency. The propsed mode can be extended to other types of PMs with minior revisions. The research conduted in the paper is of great use in geometric optimization, vibration reduction for a kind of PMs with lower mobility.