Journal
MECHANISM AND MACHINE THEORY
Volume 148, Issue -, Pages -Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.mechmachtheory.2019.103776
Keywords
Infinitesimal mechanisms; Shakiness; Prestress-stability; Local mobility; Second-order kinematic constraints; Topological graph representation
Categories
Funding
- Austrian COMET-K2 program of the Linz Center of Mechatronics (LCM)
- Engineering and Physical Sciences Research Council UK [EP/P026087/1, EP/S019790/1]
- Natural Science Foundation of China [51535008, 51721003]
- International Centre for Advanced Mechanisms and Robotics at Tianjin University
- [B16034]
- EPSRC [EP/P026087/1, EP/S019790/1] Funding Source: UKRI
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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. (C) 2020 Elsevier Ltd. All rights reserved.
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