On Non-rigid Linkages

Abstract

Linkage mechanisms and pin-jointed truss structures have been central to engineering design for centuries, providing the fundamental models through which motion transmission and structural rigidity have been understood. Conventional kinematic theory assumes perfectly rigid links joined by ideal revolute joints, while structural mechanics treats bar-and-joint frameworks as load-bearing assemblies with negligible internal deformation. Between these extremes exists a broad class of systems — deployable structures, multistable trusses, origami-inspired architectures, and compliant mechanisms — that exhibit purposeful motion, elastic deformation, and geometric constraint simultaneously. Existing analytical tools do not fully describe such systems, particularly in regimes involving large shape change, changing constraint sets, or transitions between rigid and non-rigid modes of motion. This thesis challenges the assumption of rigidity in classical mechanism analysis and small deformation in structural mechanics. By doing so, we propose a general theoretical framework for analysing non-rigid linkage mechanisms: mechanical systems that combine discrete linkage topologies with large internal deformations, and which therefore lie outside the classical separation between rigid mechanisms and pin-jointed structural frameworks. Beginning with the four-bar linkage, we replace the rigid-link assumption with internal elasticity and show that compatibility conditions can be re-expressed entirely in terms of inter-nodal distances. This yields potential non-rigid deformation paths without introducing additional degrees of freedom. Extending the analysis to Baranov trusses and Assur groups, we adapt bilateration methods from distance geometry to map how structural topology governs the emergence of multiple equilibria and formulate our own novel multistability framework. This reveals a direct correspondence between topological motifs and multistable energy landscapes, supported by physical prototypes. The framework is then generalised to multi-loop linkages, where we identify topological limit-point configurations that act as gateways between distinct geometric configurations and motions, enabling systematic classification of reconfigurable behaviours. Dynamic effects are incorporated through a Lagrangian formulation capable of resolving time-dependent actuation, different loading conditions, and transitions between distinct motions. This exposes how inertia, damping, and elastic energy storage shape the long-term response of non-rigid mechanisms. Overall, the thesis presents the first cohesive treatment of non-rigid structural mechanisms across static, quasi-static, topological, and dynamic regimes. By unifying geometric constraint analysis with mechanical modelling, it offers new insight into how mobility, stability, and elasticity interact — thereby providing foundational principles for the design of deployable structures, adaptive mechanisms, and architected materials.