On Vertex Conditions In Elastic Beam Frames: Analysis on Compact Graphs

TL;DR

This paper extends the analysis of vertex conditions in elastic beam frames modeled as compact graphs, relaxing assumptions on vertex rigidity and mass concentration, and explores spectral properties and operator decompositions in various geometric configurations.

Contribution

It introduces generalized vertex matching conditions for elastic beam frames, including mass effects, and analyzes the spectral and decomposition properties of the associated differential operators.

Findings

Operators are shown to be self-adjoint.

Decomposition into scalar operators occurs under specific symmetry conditions.

Limit conditions for parameter space lead to operator decompositions.

Abstract

We consider three-dimensional elastic frames constructed out of Euler-Bernoulli beams and describe extension of matching conditions by relaxing the vertex-rigidity assumption and the case in which concentrated mass may exists. This generalization is based on coupling an (elastic) energy functional in terms of field's discontinuities at a vertex along with purely geometric terms derived out of first principles. The corresponding differential operator is shown to be self-adjoint. Although for planar frames with a class of rigid-joints the operator decomposes into a direct sum of two operators, this property only holds for a special class of the proposed model. Application of theoretical results is then discussed in details for compact frames embedded in Euclidean spaces with different dimensions. This includes extension of the established results for rigid-joint case on exploiting the symmetry present in a frame and decomposing the operator by restricting it onto reducing subspaces corresponding to irreducible representations of the symmetry group. Derivation of characteristic equation based on the idea of geometric-free local spectral basis and enforcing geometry of the graph into play by an appropriate choice of the coefficient set will be discussed. Finally, we prove the limit conditions in parameter space which results in decomposing of vector-valued beam Hamiltonian to a direct sum of scalar-valued ones.