Published in:
SIAM J. Discrete Math., vol. 27, no. 4, pp. 1710-1726, 2013.
Let $(G,p)$ be an infinitesimally rigid $d$-dimensional
bar-and-joint framework
and let $L$ be an equilibrium load on $p$.
The load can be resolved by appropriate stresses $w_{i,j}$, $ij\in E(G)$,
in the bars of the framework.
Our goal is to identify the following
parts (zones) of the framework:
(i) when the location of an unloaded joint $v$ is
slightly perturbed, and the same load is applied, the stress will
change in
some of the bars. We call the set of these
bars the {\it influenced zone of} $v$ (with respect to $L, p$ and the
modified configuration $p'$),
(ii) let $S$ be a designated set of joints and suppose that
each joint
with a non-zero load belongs to $S$.
The {\it active zone} of $S$ (with respect to $p$ and $L$)
is the set of
those bars in which the
stress, which resolves $L$, is non-zero.
We show that if $(G,p)$ is generic and $d=2$ then, for almost all
loads, these zones
depend only on the graph $G$ of the
framework and can be computed
by efficient combinatorial methods.
Bibtex entry:
@techreport{egres-11-12,
AUTHOR | = | {Jord{\'a}n, Tibor and Domokos, G{\'a}bor and T{\'o}th, Krisztina}, |
TITLE | = | {Geometric Sensitivity of Rigid Graphs}, |
NOTE | = | {{\tt www.cs.elte.hu/egres}}, |
INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
YEAR | = | {2011}, |
NUMBER | = | {TR-2011-12} |