arXiv subjects
Publications and source records attributed to Tamás Baranyai.
We give a necessary condition of generic 3 -rigidity of graphs relying on partitioning the edges into 3 subsets; such that each subset-pair gives a generically 2-rigid graph, either by themselves or after an appropriate edge-deletion. Notably, as pointed out by Dewar and Gallet, the condition is still not sufficient.
We present the foundations of a projective geometric theory of elasticity, as well as outline a few possible application possibilities. We give the description of the Cauchy stress and infinitesimal strain tensors compatible with coordinate description of projective geometry and derive their transformation rules under projective transformations. We identify these tensors with generalized conics and quadrics; which shows that the material law cannot be directly given as a correlation, and that the absence of normal stresses and normal strains are projective invariants. The transformation of the graph of the Airy stress function is given in a point-wise projective three dimensional way, which preserves Pucher's equation corresponding to a unidirectionally loaded membrane shell. The paper closes with an example considering tapered beams, showing how to find a statically possible stress distribution in trapezioidal plates relying on the Euler-Bernoulli beam theory for rectangular beams.
In his work on stress functions Maxwell noted that given a planar truss the internal force distribution may be described by a piecewise linear, $C^0$ continuous version of the Airy stress function. Later Williams and McRobie proposed that one can consider planar moment-bearing frames, where the stress function need not be even $C^0$ continuous. The two authors also proposed a discontinuous stress function for the analysis of space-frames, which however suffers from incompleteness. This paper provides a discontinuous stress function for $n$-dimensional space frames that is complete and minimal, along with its derivation from an $n$-dimensional continuous stress function.
One of the most influential early works in graphic statics is one of Maxwell, where he introduced the idea of a discontinuous stress function and the use of a 3D projective polarity for a planar problem. A recent work gave an analytical description with the goal to provide explaining power, treating planar forces as linear functionals (moment functionals) forming a 3D vector space, so force and form diagrams of planar problems can be interpreted in a 3-dimensional way. The linear combination of these moment functions can naturally be used as a discontinuous stress function since the Airy stress function is known to correspond to moments of planar forces. The main contribution of this work is to present a dimension independent way of treating forces as functions, that returns the known stress-functions of planar and spatial graphic statics. This is done by relying on the multi-linear function definition of Grassmann-algebra.
Elasticity ellipses or central ellipses have been long used in graphic statics to capture the elastic behaviour of structural elements. The paper gives a generalisation the concept both in dimensions and in the possibility of degenerate conics / quadrics. The effect of projective transformations of these quadrics is also given, such that the entire mechanical system can be transformed preserving equilibrium and compatibility between its elements.
In engineering practice one often encounters planar problems, where the corresponding vector space of forces, velocities or (infinitesimal) displacements is three dimensional. This paper shows how these spaces can be factorized, such that the arising equivalence classes correspond to points and lines of action of the forces / velocities / displacements in the (projective) plane. It is shown how the study of projective transformations and dualities of planar mechanical systems is closely related to the study of linear maps of these spaces. A few past results are analysed and sometimes extended to show the power of this description.
A pair of ovservations on the role of projective geometrical dualities in graphic statics.
Graphic statics is undergoing a renaissance, with computerized visual representation becoming both easier and more spectacular as time passes. While methods of the past are revived and tweaked, little emphasis has been placed on studying the details of these methods. Due to the considerable advances of our mathematical understanding since the birth of graphic statics, we can learn many interesting and beautiful things by examining these old methods from a more modern viewpoint. As such, this work shows the mathematical fabric joining different aspects of graphic statics, like dualities, reciprocal diagrams and discontinous stress functions.
Dualities have been known to map space trusses and plate structures to each other since the 1980-s. Yet the computational similarity of the two has not been used to solve the unfamiliar plate structure with the methods of the well known truss. This paper gives a method to find the forces and displacements of a plate structure with rigid plates and elastic edges, using a dual truss. It is applicable for both statically determinate and indeterminate structures, subjected to both statical and kinematical loads.
The motion of a rigid, spinning disk on a flat surface ends with a dissipation-induced finite-time singularity. The problem of finding the dominant energy absorption mechanism during the last phase of the motion generated a lively debate during the past two decades. Various candidates including air drag and various types of friction have been considered, nevertheless impacts have not been examined until now. We investigate the effect of impacts caused by geometric imperfections of the disk and of the underlying flat surface, through analysing the dynamics of polygonal disks with unilateral point contacts. Similarly to earlier works, we determine the rate of energy absorption under the assumption of a regular pattern of motion analogous to precession-free motion of a rolling disk. In addition, we demonstrate that the asymptotic stability of this motion depends on parameters of the impact model. In the case of instability, the emerging irregular motion is investigated numerically. We conclude that there exists a range of model parameters (small radii of gyration or small restitution coefficients) in which absorption by impacts dominates all preiously investigated mechanisms during the last phase of motion. Nevertheless the parameter values associated with a homogenous disk on a hard surface are typically not in this range, hence the effect of impacts is in that case not dominant.
Ideally rigid objects establish sustained contact with one another via complete chatter (a.k.a. Zeno behavior), i.e. an infinite sequence of collisions accumulating in finite time. Alternatively, such systems may also exhibit a finite sequence of collisions followed by separation (sometimes called incomplete chatter). Earlier works concerning the chattering of slender rods in two dimensions determined the exact range of model parameters, where complete chatter is possible. We revisit and slightly extend these results. Then the bulk of the paper examines the chattering of three-dimensional objects with multiple points hitting an immobile plane almost simultaneously. In contrast to rods, the motion of these systems is complex, nonlinear, and sensitive to initial conditions and model parameters due to the possibility of various impact sequences. These difficulties explain why we model this phenomenon as a nondeterministic discrete dynamical system. We simplify the analysis by assuming linearized kinematics, frictionless interaction, by neglecting the effect of external forces, and by investigating objects with rotational symmetry. Application and extension of the theory of common invariant cones of multiple linear operators enable us to find sufficient conditions of the existence of initial conditions, which give rise to complete chatter. Additional analytical and numerical investigations predict that our sufficient conditions are indeed exact, moreover solving a simple eigenvalue problem appears to be enough to judge the possibility of complete chatter. \keywords{contact dynamics \and chattering \and Zeno behavior \and common invariant cone
This paper presents a partial reconstruction of the rotational dynamics of the Philae spacecraft upon landing on comet 67P/Churyumov-Gerasimenko as part of ESA's Rosetta mission. We analyze the motion and the events triggered by the failure to fix the spacecraft to the comet surface at the time of the first touchdown. Dynamic trajectories obtained by numerical simulation of a 7 degree-of-freedom mechanical model of the spacecraft are fitted to directions of incoming solar radiation inferred from in-situ measurements of the electric power provided by the solar panels. The results include a lower bound of the angular velocity of the lander immediately after its first touchdown. Our study also gives insight into the effect of the programmed turn-off of the stabilizing gyroscope after touchdown; the important dynamical consequences of a small collision during Philae's journey; and the probability that a similar landing scenario harms the operability of this type of spacecraft.