English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We analyze the eigenstructure of the integral operator \(\mathcal{K}_{l, \alpha, k}\) which arise naturally from the beam deflection equation on linear elastic foundation with finite beam. We show that \(\mathcal{K}_{l, \alpha, k}\) has countably infinite number of positive eigenvalues approaching 0 as the limit, and give explicit upper and lower bounds on each of them. Consequently, we obtain explicit upper and lower bounds on the \(L^{2}\)-norm of the operator \(\mathcal{K}_{l, \alpha, k}\). We also present precise approximations of the eigenvalues as they approach the limit 0, which describes the almost regular structure of the spectrum of \(\mathcal{K}_{l, \alpha, k}\). Additionally, we analyze the dependence of the eigenvalues, including the \(L^{2}\)-norm of \(\mathcal{K}_{l, \alpha, k}\), on the intrinsic length \(L = 2 l \alpha\) of the beam, and show that each eigenvalue is continuous and strictly increasing with respect to L. In particular, we show that the respective limits of each eigenvalue as L goes to 0 and infinity are 0 and \(1/k\), where k is the linear spring constant of the given elastic foundation. Using Newton’s method, we also compute explicitly numerical values of the eigenvalues, including the \(L^{2}\)-norm of \(\mathcal{K}_{l, \alpha, k}\), corresponding to various values of L.

Spectral analysis of the integral operator arising from the beam deflection problem on elastic foundation II: eigenvalues