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

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 $\lambda \notin \mathrm{Spec}{\mathcal{K}}_{l,\alpha ,k}$, 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 ${\mathcal{K}}_{l,\alpha ,k}$.

Changes: 1. ${\mathbf{Y}}_{\lambda }(x)$ are defined for $\lambda$ and $x$ such that $\det {\mathbf{X}}_{\lambda }(x)\ne 0$. 2. The boundary condition for infinitely long beam in Introduction. 3. Some sign corrections in (B.9). 4. Inclusion of the new item [4] in References