A unified element stiffness matrix model for variable cross-section flexure hinges in compliant mechanisms for micro/nano positioning

  • Yanling Tian
  • Mingxuan Yang
  • Fujun WangEmail author
  • Chongkai Zhou
  • Xingyu Zhao
  • Dawei Zhang
Technical Paper


This paper presents a unified analytical model of element stiffness matrix for variable cross-section flexure hinges in plane deformation which is most frequently employed in planar-motion compliant mechanisms for micro/nano positioning. The unified analytical model is derived for flexure hinges in plane deformation based on the principle of virtual work. The derivation is generalized such that it can be applied to various geometries. To optimize and simplify element stiffness matrix model, four coefficients with curve integral are introduced. The results of the analysis were validated in part by modeling several previously investigated configurations, namely flexible beam hinge and right circular hinge. To further validate the proposed analytical model, finite element analysis and experimental testing were used. And the experimental testing shows the proposed unified model is more precise than pseudo-rigid-body method. The proposed model provides a concise and generalized solution to derive the element stiffness matrices of flexible hinges in plane deformation, which will have excellent applications in design and analysis of variable cross-section flexible hinges in compliant mechanisms.



This research is supported by National Natural Science Foundation of China (Grant nos. 51675376, 51675371, and 51675367), National Key R&D Program of China (nos. 2017YFB1104700, 2017YFE0112100, and 2016YFE0112100), China-EU H2020 FabSurfWAR (no. 644971) and MNR4SCell (no. 734174).


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, School of Mechanical EngineeringTianjin UniversityTianjinChina

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