Mobility of ‘ N -loops’: bodies cyclically connected by intersecting revolute hinges

Abstract

The mobilities of many objects from toys and molecular models to large-scale deployable structures can be understood in terms of N -loops: sets of N bodies, cyclically connected by pairs of intersecting revolute hinges. A symmetry-extended version of the Grübler criterion for counting kinematic degrees of freedom is used to explain and rationalize the observed mobilities of N -loops with small N . Compared with simple counting, the symmetry-based approach gives improved detection and visualization of mechanisms and states of self-stress. It can also give a better account of the differing mobilities of conformers occupying different regions of the phase space, such as the rigid chair and flexible boat forms of cyclohexane.

On regular seven-membered loops in $${{\mathbb{R}}^3}$$ R 3 with arbitrary join angle

“…and we comment that the only integer solutions to (2) are N = 7 for d ∈ {3, 6} and N = 6 for d = 4. Due to symmetry, there may exist other one-parameter families of solutions, and it is known, for example, that in the case of N = 6 in d = 3, there exists a solution with one degree of freedom [2].…”

“…Bricard analyses concave octahedra which are flexible but have been shown to be equivalent to a non-regular six-membered loop [2]. While some of the interest has been more mathematical and even recreational in focus [3,4], there has also been connections of these mathematical objects to the conformers of ring molecules, particularly cycloalkanes such as cyclohexane and cycloheptane [5][6][7][8].…”

“…Much of the focus in the literature has been on the six-membered case, with passing comment on the seven-membered case, and such analysis of loops in general have been for where the join angle is greater than or equal to π/2 (see, for example, [2]). This is motivated at least in part, because in cycloheptane, the angle between bonds on the carbon backbone in that molecule is assumed to be the that of the tetrahedral structure θ = cos −1 (−1/3) ≈ 109.5 • .…”

Bi-stiffness property of motion structures transformed into square cells

“…In this respect, bar-and-joint frameworks are valid for representing the concise deformations of these materials by mathematical or numerical analysis: for example, the rigidity or flexibility of the frameworks modelled on repetitive cells [42], zeolites [43,44], auxetic materials [45] and the loop structures connected by revolute joints [46]. The aim of this study is to explore structural materials with bi-stiffness, using the abovementioned modelling approach.…”

Symmetry Perspectives on Some Auxetic Body-Bar Frameworks

“…Examples of this symmetry-based approach include symmetry-adapted versions of the Maxwell Rule [5], the mobility criterion for body and joint assemblies [6], and for bar-body systems [7]. Symmetry analyses have dealt with classes of system such as rotating rings of tetrahedra [8], [N ]-loops [9] toroidal deltahedra [10] and various mechanical toys and models [11][12][13]. Symmetry arguments have been used to derive conditions for the existence of isostatic frameworks [14] and to discuss the flexibility of protein molecules [15].…”