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

Boundary value problems

Sung Woo Choi¹ 

[1] Department of Mathematics, Duksung Women’s University, Seoul, Republic of Korea

关键词: beam;    deflection;    elastic foundation;    integral operator;    eigenvalue;    \(L^{2}\)-norm;    34L15;    47G10;    74K10;   

DOI  :  10.1186/s13661-014-0268-2

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

We analyze the eigenstructure of the integral operatorKl,α,k$\mathcal{K}_{l, \alpha, k}$which arise naturally from the beam deflection equation on linear elastic foundation with finite beam. We show thatKl,α,k$\mathcal{K}_{l, \alpha, k}$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 theL2$L^{2}$-norm of the operatorKl,α,k$\mathcal{K}_{l, \alpha, k}$. We also present precise approximations of the eigenvalues as they approach the limit 0, which describes the almost regular structure of the spectrum ofKl,α,k$\mathcal{K}_{l, \alpha, k}$. Additionally, we analyze the dependence of the eigenvalues, including theL2$L^{2}$-norm ofKl,α,k$\mathcal{K}_{l, \alpha, k}$, on the intrinsic lengthL=2lα$L = 2 l \alpha$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 and1/k$1/k$, 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 theL2$L^{2}$-norm ofKl,α,k$\mathcal{K}_{l, \alpha, k}$, corresponding to various values of L.

【 授权许可】

CC BY