A geometric and analytic technique for studying multi-DOF planar mechanisms’ dynamics

Most approaches of analytical mechanics (e.g., D'Alembert principle) build dynamic models of mechanisms that are difficult to relate with clear geometric interpretations. This feature is a drawback that confines their use to the implementation of algorithms in multi-body-dynamics software, whose results must be interpreted by using the designer intuition when the mechanism must be modified to match particular design requirements. Here, starting from D'Alembert principle, the dynamic model of any multi-degrees-of-freedom (multi-DOF) planar mechanism (PM) is obtained by suitably combining the dynamic models of all the single-DOF PMs generated by locking all the actuated joints, but one. The dynamic models of the so-generated single-DOF PMs are written by using an approach, presented in a previous paper, which admits a geometric interpretation through diagrams, named “active-load diagrams”, and fully discloses the role of instant centers (ICs) in the dynamic behavior of single-DOF PMs. Accordingly, the resulting dynamic model of the generating multi-DOF PM admits a geometric interpretation that reveals the role of ICs in multi-DOF PMs’ dynamic behavior. Such geometric interpretation is so effective that, formally, the model could be written starting from it without any analytic consideration the same way as the equilibrium equations can be written from free-body diagrams in the Newton-Euler formulation. The presented model is general and, as far as this author is aware, is novel. The proposed model and the associated algorithms for solving dynamic problems are also illustrated through a case study. The obtained results are of interest in mechanism analysis and design.