2D shape optimization of leaf-type crossed flexure pivot springs for minimum stress

Leaf-spring crossed flexure pivots are used in an increasingly large number of applications, many of which require the fatigue life and/or the rotational capability of the pivots to be maximized. Since these two qualities are determined by the endurance limit or the yield stress of the leaf-springs’ material, maximizing them will necessarily require reducing the stress levels arising in the aforementioned springs during operation. Partly for the sake of simplicity, partly due to manufacturing constraints, constant thickness leaf-springs have traditionally been used in crossed flexure pivots. However, minimizing stress simply by increasing the length or by decreasing the thickness of the leaf-springs is often not viable, either due to spatial constraints or to the attendant degradation in spring performance. This paper investigates the scope for stress reduction through shape optimization of the leaf-springs. To this end, a procedure combining a linear strain energy formulation, a parametric thickness profile definition and a series of optimization algorithms is employed. The resulting optimized thickness profiles are proven to be not only independent of the angular rotation at which the pivot operates, but also linearly scalable to leaf-springs of any length, minimum thickness and width. Validated using non-linear finite element analysis, the results show very significant reductions in relative maximum stress, of up to as much as 24% in the case of some pivot configurations. The optimized profiles and their corresponding constant thickness counterparts are also compared in terms of stiffness, strain energy and parasitic motion characteristics. It is concluded that shape optimization offers great potential for extending the fatigue life and/or rotational range of crossed flexure pivots.