Modeling of cross-spring pivots subjected to generalized planar loads

Cross-spring pivots, formed by crossing two identical flexural beams at their midpoint, have been broadly used in precision engineering and aerospace fields. Many researches have been conducted on modeling and analysis of cross-spring pivots. However the influence of application position and magnitude of the external loads on the load-rotation and parasitic motion characteristics has not yet been discussed. In order to reveal the effect of the external loads, this paper develops the accurate load-rotation and center shift models of cross-spring pivots, with generalized planar loads applied including bending moment, horizontal and vertical forces. Firstly, by using the energy method, the load-displacement models of the pivot are derived with the assumption of small rotational angles. Based on the models, the influence of generalized planar loads on the load-rotation relationship is discussed, which shows that both application position and magnitude of the vertical and horizontal forces influence the load-rotation behaviors. Then the accurate center shift expressions of the pivot with generalized planar loads are developed, which shows that the rotational angle is the dominant term for both components of the center shift while the vertical and horizontal forces are small. Finally, the accuracy of the proposed model is validated by finite element analysis(FEA). Comparing the model data with the results obtained from FEA, the relative error of the load-rotation is less than 6% even if the rotational angle reaches 20°; the relative errors of the two components of center shift are less than 5% and 10% respectively when the rotational angle reaches 10°. The proposed model and analytical conclusions can be used to analyze and preliminarily design the compliant mechanisms containing cross-spring pivots.