Analysis of compliant mechanisms with series and parallel substructures through the ellipse of elasticity theory

Compliant mechanisms with complex hybrid configurations have been designed to meet the requirements of specific applications demanding high performance. Kinetostatic analysis, fundamental at the early stage of design, can become difficult for compliant systems characterized by series and parallel substructures. In the present paper, the ellipse of elasticity method is implemented for the analysis of a compliant mechanism with hybrid topology. Firstly, the ellipses associated to the different flexure hinges, characterized by uniform or non-uniform cross-sections, and by constant or variable initial curvature, are determined. Then, the unique ellipse representing the compliant mechanism is obtained by means of series and parallel compositions. By exploiting the antiprojective polarity properties of the ellipse, the kinetostatic analysis of the compliant system is reduced to a geometric problem with a straightforward solution. Linear and nonlinear finite element analyses and experimental tests are performed to verify the theoretical results.