Dynamic modeling and model order reduction of compliant mechanisms

In this work, a novel approach towards statical and dynamical modeling of compliant mechanisms is presented which serves as an origin for model order reduction procedures to provide small, efficient and accurate approximations. As 3D finite element modeling of compliant mechanisms results in very large-scale systems, both model reduction and real-time controlling of the mechanism are not possible. This and the fact that in compliant mechanisms mostly the flexure hinges contribute to the overall performance motivates a procedure which is based on partitioning the structure into elastically deformable hinges and rigid linkages. Unlike common modeling techniques, the reproduction of the non-negligible nonlinear behavior is assured and contributes to precise approximation as well as the fact that not only the flexure itself deforms but also adjacent structures. Furthermore, the proposed methodology is applicable to all kinds of flexure hinges with concentrated compliances and compliant mechanisms of complex geometric shape as well as spatial loading cases. First, an analysis of the inserted flexure hinges yields the significantly deformed region of which corresponding master models with connecting nodes are created. With their geometric properties the mechanism is further divided into remaining stiff sections. Their spatial centroid location and moments of inertia about mass centroid are calculated and according point masses are generated. A finite element model of the mechanism is then developed by rigidly linking the master models with the point masses. Therewith, an accurate 3D model with appreciable less degrees of freedom arises, named significant region model. The system matrices K, M in combination with the input matrix B and the output matrix C yield a closed-form description of the mechanism. In a second step, a considerable decrease of system size is performed by applying modern model order reduction techniques, namely Krylov subspace reduction. A comparison of the new two step approach with full 3D finite element modeling reveals only marginal deviations of less than 6% regarding the static displacements and less than 0.0001% relating to the frequency response of an exemplary mechanism with a substantial lower number of degrees of freedom.