Bifurcation of a special line and plane symmetric Bricard linkage

Chen, Yan; Chai, Woon Huei

doi:10.1016/j.mechmachtheory.2010.11.015

Cited by 60 publications

(10 citation statements)

References 23 publications

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“…However, such nonlinear features can also create problems 10 , 13 – 15 . For example, self-folding origami, despite the name, has an exponential number of misfolding pathways that meet at a “branch point” at the flat state 13 , 16 – 19 , making it nearly impossible to fold into the desired folding mode 14 , 15 , 20 , 21 . Similar “branch points” in mechanical linkages pose challenges in robotics and other applications 22 – 24 .…”

Section: Introductionmentioning

confidence: 99%

Shaping the topology of folding pathways in mechanical systems

2018

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Disordered mechanical systems, when strongly deformed, have complex configuration spaces with multiple stable states and pathways connecting them. The topology of such pathways determines which states are smoothly accessible from any part of configuration space. Controlling this topology would allow us to limit access to undesired states and select desired behaviors in metamaterials. Here, we show that the topology of such pathways, as captured by bifurcation diagrams, can be tuned using imperfections such as stiff hinges in elastic networks and creased thin sheets. We derive Linear Programming-like equations for designing desirable pathway topologies. These ideas are applied to eliminate the exponentially many ways of misfolding self-folding sheets by making some creases stiffer than others. Our approach allows robust folding by entire classes of external folding forces. Finally, we find that the bifurcation diagram makes pathways accessible only at specific folding speeds, enabling speed-dependent selection of different folded states.

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Section: Introductionmentioning

confidence: 99%

Shaping the topology of folding pathways in mechanical systems

2018

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“…The displacement equations of Bennett and Goldberg mechanisms are usually expressed by exterior angles of adjacent links [29][30][31][32][33][34][35][36]. To reflect more intuitively and clearly the geometric relationship between the links in the analysis of multiple loops, interior angles of adjacent links are adopted in this paper.…”

Section: Traditional and New Methods For Expressing Displacement Equamentioning

confidence: 99%

Structure Parameters and Displacement Equations of Goldberg Mechanisms With the Rank More Than sixStructure Parameters and Displacement Equations of Goldberg Mechanisms With the Rank More Than Six

Zhang¹,

Ling²,

Liu³

et al. 2020

Preprint

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An infinite number of N-bar Goldberg mechanisms with the rank more than six are proposed. A single closed-loop eight-bar mechanism is first developed by removing the common side links of adjacent loops from the five-loop Bennett mechanism in which all loop frames are collinear. An analysis of trajectory circles of the links shows that the mechanism developed maintains the original constraint conditions and kinematic characteristics, and that its degree of freedom (DOF) is also one. This novel single-loop mechanism is named the eight-bar Goldberg mechanism because of its similar constraint characteristics with the five-bar and six-bar Goldberg mechanisms. The eight displacement parameters of this mechanism are analyzed by using the interior angles of adjacent links, and the five cosine equations for the tangent product of the stagger angles between adjacent links are established in together with one angular displacement equation of odd links and one angular displacement equation of even links. This novel mechanism has one DOF, eight displacement parameters and seven independent displacement equations. The loop rank of the eight-bar Goldberg mechanism is seven, which revises the existing mechanism theory of single closed-loop mechanisms with the rank no more than six. Similarly, a nine-bar Goldberg mechanism is constructed, and its nine displacement parameters are analyzed. This mechanism has eight independent displacement equations and the loop rank is eight, proving that the mechanisms with the rank more than six are not rare. Then, according to the characteristics of the independent displacement equations of the eight-bar and nine-bar Goldberg mechanisms, the N (N>7) displacement parameters of N-bar Goldberg mechanisms are analyzed, and the (N–1) independent displacement equations are deduced with odd and even Ns. It is proved that the loop rank of N-bar Goldberg mechanisms is (N–1), so there exist an infinite number of mechanisms with the rank more than six

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“…This case leads to a linkage that is both line-and plane-symmetric. This example was analyzed before in [28] and the linesymmetric case was investigated by the authors of this paper in another work yet to be published. The other solutions that do not degenerate the toroids imply y = 0.…”

Section: Tangent Intersections Of Concentric Singular Toroidsmentioning

confidence: 99%

“…In regard to Bricard linkages that lead to different motion branches, the following contributions have already been presented: the analysis of reconfiguration of a plane-and line-symmetric Bricard linkage by means of geometric constraints and screw-system variations [18], spatial triangle formed joint and variable axis joint are used to obtain a linesymmetric linkage that can behave as a Bennett linkage [27], the analysis of a plane-and line-symmetric Bricard linkage with different motion branches in order to avoid singularities [28] and a line-symmetric Bricard linkage evolved from a metamorphic 8R linkage [29]. In these studies, a theory of the reconfigurability of Bricard loops that can help the design of these linkages has not been studied thoroughly.…”

Section: Introductionmentioning

confidence: 99%

Branch Reconfiguration of Bricard Linkages Based on Toroids Intersections: Plane-Symmetric Case

López-Custodio

Dai

Rico

2018

Journal of Mechanisms and Robotics

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This paper for the first time reveals a set of special plane-symmetric Bricard linkages with various branches of reconfiguration by means of intersection of two generating toroids, and presents a complete theory of the branch reconfiguration of the Bricard plane-symmetric linkages. An analysis of the intersection of these two toroids reveals the presence of coincident conical singularities, which lead to design of the plane-symmetric linkages that evolve to spherical 4R linkages. By examining the tangents to the curves of intersection at the conical singularities, it is found that the linkage can be reconfigured between the two possible branches of spherical 4R motion without disassembling it and without requiring the usual special configuration connecting the branches. The study of tangent intersections between concentric singular toroids also reveals the presence of isolated points in the intersection, which suggests that some linkages satisfying the Bricard plane-symmetry conditions are actually structures with zero finite degrees-of-freedom (DOF) but with higher instantaneous mobility. This paper is the second part of a paper published in parallel by the authors in which the method is applied to the line-symmetric case.

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Bifurcation of a special line and plane symmetric Bricard linkage

Cited by 60 publications

References 23 publications

Shaping the topology of folding pathways in mechanical systems

Shaping the topology of folding pathways in mechanical systems

Structure Parameters and Displacement Equations of Goldberg Mechanisms With the Rank More Than sixStructure Parameters and Displacement Equations of Goldberg Mechanisms With the Rank More Than Six

Branch Reconfiguration of Bricard Linkages Based on Toroids Intersections: Plane-Symmetric Case

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