BI-BCM: A closed-form solution for fixed-guided beams in compliant mechanisms

A fixed-guided beam is one of most commonly used flexible segments in compliant mechanisms such as bistable mechanisms, compliant parallelogram mechanisms, compound compliant parallelogram mechanisms and thermomechanical in-plane microactuators. In this paper, we split a fixed-guided beam into two elements, formulate each element using the Beam Constraint Model (BCM) equations, and then assemble the two elements' equations to obtain the final solution for the load-deflection relations. Interestingly, the resulting load-deflection solution (referred to as Bi-BCM) is closed-form, in which the tip loads are expressed as functions of the tip deflections. The maximum allowable axial force of Bi-BCM is the quadruple of that of the BCM. Bi-BCM also extends the capability of the BCM for predicting the second buckling mode of fixed-guided beams. Besides, the boundary line between the first and the second buckling modes of fixed-guided beams can be easily obtained using a closed-form equation. Bi-BCM can be immediately used for quick design calculations of compliant mechanisms utilizing fixed-guided beams as their flexible segments (generally no iteration is required). Different examples are analyzed to illustrate the usage of Bi-BCM, and the results show the effectiveness of the closed-form solution.