Beams with A Varying Cross Section in the Generalized Strain Formulation for Flexure Modelling

Flexure joints are rapidly gaining ground in precision engineering because of their predictable behavior. However their range of motion is limited due to a stress limitation and a loss of support stiffness in deformed configurations. The support stiffness can be significantly increased by using leafsprings of which the width and thickness vary over the length of the leafspring. This paper presents formulations for two beam elements with a varying cross section that can be used for the efficient modelling of these types of leafsprings. One of these beam-formulations includes the modelling of the warping due to torsion, which is shown to be essential for accurate modelling. The 90% accuracy in stiffness results and 80% accuracy in stress results, in comparison with results of finite element analyses, are sufficient for the evaluation of concept-designs. Optimizations shows that the support stiffness of two typical flexure joints can be increased by a factor of up to 4.0 keeping the same range of motion, by allowing the cross section to vary over the length of the leafspring. In these two flexure joints, 98% of this improvement can already be obtained by only varying the thickness, keeping a constant width.