Kinematics of Zigzagged Articulated Parallelograms with Articulated Braces (ZAPAB) mechanisms

We study the kinematics of Zigzagged Articulated Parallelograms with Articulated Braces (ZAPAB) mechanisms, a modular-based bars linkage characterized by a single Lagrange parameter, aimed to supply the microstructure for a third-gradient planar one-dimensional continuum after homogenization. We choose a specific geometry for the constituent planar modules of the ZAPAB mechanism, where each module is built upon rigid bars and hinges (nodes). The hinges are arranged in three layers: upper, middle, and lower. Thus, when perfect constraints are imposed, the placement and the length of each module depend uniquely on the selected Lagrangian parameter: the distance between adjacent, nonconnected nodes associated with the middle layer hinges. In this way, we introduce a mechanism in which a designated set of material points belong to a family of circumferences parameterized by a unique degree of freedom, when neglecting the global rigid motions. We prove, following a symmetry argument, that the allowed configurations for the considered ZAPAB mechanism are circumferences of different radii. Furthermore, given a reference length of the mechanism, we prove that, in the limit of a large number of modules, the length of the mechanism does not vary as the Lagrange parameter changes: the mechanism is inextensible. We give the analytical expression for the curve traced by the terminal point of ZAPAB mechanism. Therefore, this configuration of ZAPAB structure is a candidate to represent a synthesis for a particular class of third-gradient one-dimensional continua: those that are inextensible and whose deformation energy depends on the derivative of the curvature, with respect to its curvilinear abscissa.