On the Simplicity of Eigenvalues of Two Nonhomogeneous Euler-Bernoulli Beams Connected by a Point Mass

Jamel Ben Amara; Hedi Bouzidi

On the Simplicity of Eigenvalues of Two Nonhomogeneous Euler-Bernoulli Beams Connected by a Point Mass

Spectral Theory 2018-04-18 v1 Optimization and Control

Authors: Jamel Ben Amara , Hedi Bouzidi

Abstract

In this paper we consider a linear system modeling the vibrations of two nonhomogeneous Euler-Bernoulli beams connected by a point mass. This system is generated by the following equations\bea &&\rho(x)y_{tt}(t,x)+(\sigma(x)y_{xx}(t,x))_{xx}-(q(x)y_{x}(t,x))_{x}=0,~t>0,~x \in (-1,0)\cup(0,1), &&M y_{tt}(t,0)= ${T}y(t,x)$ _{\mid_{x=0^-}}- ${T}y(t,x)$ _{\mid_{x=0^+}},~~t>0,\eea with hinged boundary conditions at both ends, where ${T}y = (\se(x)y_{xx})_{x} - q(x)y_x$ . We prove that all the associated eigenvalues $$\la_n$_{n\geq1}$ are algebraically simple, furthermore the corresponding eigenfunctions $$\phi_n$_{n\geq1}$ satisfy $\phi_n'{T}{\phi}_{n}(-1)>0$ and $\phi_n'{T}{\phi}_{n}(1)<0$ for all $n\geq1$ . These results give a key to the solutions of various control and stability problems related to this system.

Keywords

euler-bernoulli beams point mass eigenvalue simplicity hinged boundary conditions control problems stability problems

Cite

@article{arxiv.1804.06172,
  title  = {On the Simplicity of Eigenvalues of Two Nonhomogeneous Euler-Bernoulli Beams Connected by a Point Mass},
  author = {Jamel Ben Amara and Hedi Bouzidi},
  journal= {arXiv preprint arXiv:1804.06172},
  year   = {2018}
}

Comments

Euler-Bernoulli beams, point mass, algebraic simplicity, subwronskians

On the Simplicity of Eigenvalues of Two Nonhomogeneous Euler-Bernoulli Beams Connected by a Point Mass

Abstract

Keywords

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