Spectral analysis for the class of integral operators arising from well-posed boundary value problems of finite beam deflection on elastic foundation: characteristic equation

TL;DR

This paper develops spectral analysis techniques for integral operators associated with boundary value problems of finite beam deflection on elastic foundations, linking boundary conditions to spectral properties and revealing their influence on operator spectra.

Contribution

It introduces a novel eigencondition framework connecting boundary conditions to the spectra of integral operators in beam deflection problems.

Findings

Eigenconditions isolate boundary condition effects on spectra.

Spectra of operators can be arbitrarily complex, not necessarily positive or contractive.

Existence of boundary conditions for arbitrary spectral values outside known spectra.

Abstract

We consider the boundary value problem for the deflection of a finite beam on an elastic foundation subject to vertical loading. We construct a one-to-one correspondence $\Gamma$ from the set of equivalent well-posed two-point boundary conditions to $\mathrm{gl}(4,\mathbb{C})$. Using $\Gamma$, we derive eigenconditions for the integral operator $\mathcal{K}_\mathbf{M}$ for each well-posed two-point boundary condition represented by $\mathbf{M} \in \mathrm{gl}(4,8,\mathbb{C})$. Special features of our eigenconditions include; (1) they isolate the effect of the boundary condition $\mathbf{M}$ on $\mathrm{Spec}\,\mathcal{K}_\mathbf{M}$, (2) they connect $\mathrm{Spec}\,\mathcal{K}_\mathbf{M}$ to $\mathrm{Spec}\,\mathcal{K}_{l,\alpha,k}$ whose structure has been well understood. Using our eigenconditions, we show that, for each nonzero real $\lambda \not \in \mathrm{Spec}\,\mathcal{K}_{l,\alpha,k}$, there exists a real well-posed boundary condition $\mathbf{M}$ such that $\lambda \in \mathrm{Spec}\,\mathcal{K}_\mathbf{M}$. This in particular shows that the integral operators $\mathcal{K}_\mathbf{M}$ arising from well-posed boundary conditions, may not be positive nor contractive in general, as opposed to $\mathcal{K}_{l,\alpha,k}$.