The use of modal derivatives in determining stroke-dependent frequencies of large stroke flexure hinges

Nowadays, a lot of use is made of large stroke flexure hinges in precision engineering. However, these large stroke flexure hinges typically lose stiffness in supporting direction during deflection. The lowest natural frequency is a commonly used measure for this property. Therefore, in shape and topology optimization, the decrease of this first parasitic frequency is often minimized. These optimizations are typically very time consuming, due to the large number of design evaluations. In this paper, a method is presented for determining stroke-dependent frequencies of large stroke flexure hinges. This method makes use of derivatives of mode shapes with respect to modal coordinates. Therefore, geometrical nonlinearities can be taken into account. Using these modal derivatives, frequency derivatives can be determined, making it possible to determine natural frequencies for any given deflection without having to linearize for every load step. For demonstration, the method is used to determine the first parasitic frequency of a single leaf spring as a function of the deflection. The results show that the decrease of this parasitic frequency has the shape of a bell-shaped curve, as commonly described in literature and found in experiments.