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

Relating geometric interpretations to analytic procedures for solving mechanical problems improves their comprehension and gives a deeper insight on how to eliminate unwanted behaviors of the studied machine. This aspect is well stated in Newton-Euler formulation of mechanism dynamics. Unfortunately, many approaches of analytical mechanics build dynamic models of mechanisms that are difficult to put in relation with clear geometric constructions. D'Alembert principle, which is based on the virtual work principle, is one of these approaches. Here, a systematic way for modeling single-degree-of-freedom (single-DOF) planar mechanisms through the D'Alembert principle and the instant centers’ (ICs’) positions is presented which is related to geometric interpretations of the terms appearing in it. The proposed geometric interpretation is so effective that, formally, the model could be written starting from it without any analytic consideration as same as the equilibrium equations can be written from free-body diagrams in the Newton-Euler formulation. The resulting dynamic model is novel and general. The proposed model and the associated solution algorithms of the dynamic problems are also illustrated through a case study. The obtained results are of interest in mechanism analysis and design and, due to their simple theoretical bases, can be presented in courses for undergraduate or graduate students.