A matrix method to determine infinitesimally mobile linkages with only first-order infinitesimal mobility. (June 2020) - Wu, Liheng; Müller, Andreas; Dai, Jian S.
Highlights: A matrix method in screw coordinates is provided to determine linkages with only first-order infinitesimal mobility, which relies on the definitiveness test of a constructed quadratic form. Second-order kinematic constraints of multi-loop linkages are written in explicit form in terms of screw coordinates with the help of kinematic graph representation. Some physical interpretations of the constructed matrix expressions are given which are related to prestress stability. Results for a special 3-UU mechanism are presented which is a first-order infinitesimal linkage but not prestress-stable. Abstract: Immobile linkages admitting only (possibly higher-order) infinitesimal mobility are shaky structures. In the past, determination of the order of mobility or shakiness was usually approached in a purely kinematic way namely by the higher order kinematic constraint analysis, involving solutions of higher-order kinematic constraints. In this paper, in terms of screw theory and an appropriate representation of kinematic topology, a matrix method is provided to test whether a multi-loop linkage is immobile and only possesses first-order mobility, without the need to solve the second-order constraint equations. The corresponding linkages are called first-order infinitesimal linkages. To this end, the first- and second-order kinematic constraints of multi-loop linkages are firstly formulated explicitly in matrix form, in terms of a Jacobian matrix and Hessian matrix, respectively, and are combined to a quadratic form. The definitiveness of this quadratic form then provides a sufficient condition for being a first-order infinitesimal linkage. This is related to the concept of prestress-stability. The method is applied to several immobile closed-loop linkages with only infinitesimal mobility. A special example is the 3-UU mechanism, which is a first-order infinitesimal linkage but not prestress-stable. Since higher-order derivatives of screws can be obtained explicitly with Lie brackets, a matrix method may be established, in which higher-order kinematic constraints may be analyzed in a more qualitative way. This paper is a first step towards a matrix method for determination of higher-order infinitesimal linkages.