A matrix method to determine infinitesimally mobile linkages with only first-order infinitesimal mobility

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.