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

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 $. W e p r o v e t h a t a l l t h e a s s o c i a t e d e i g e n v a l u e s$ $\la_{n}$ {n\geq1} $a r e a l g e b r a i c a l l y s i m p l e, f u r t h e r m o r e t h e c o r r e s p o n d i n g e i g e n f u n c t i o n s$ $ϕ_{n}$ {n\geq1} $s a t i s f y$ \phi_n’{T}{\phi}{n}(-1)>0 $a n d$ \phi_n’{T}{\phi}{n}(1)<0 $f o r a l l$ n\geq1$. These results give a key to the solutions of various control and stability problems related to this system.

q(x)y_x $. W e p r o v e t h a t a l l t h e a s s o c i a t e d e i g e n v a l u e s$ $\la_{n}$ {n\geq1} $a r e a l g e b r a i c a l l y s i m p l e, f u r t h e r m o r e t h e c o r r e s p o n d i n g e i g e n f u n c t i o n s$ $ϕ_{n}$ {n\geq1} $s a t i s f y$ \phi_n’{T}{\phi}{n}(-1)>0 $a n d$ \phi_n’{T}{\phi}{n}(1)<0 $f o r a l l$ n\geq1$. These results give a key to the solutions of various control and stability problems related to this system.

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

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

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

文献摘要

讨论

|讲解博客

|AI智能校验

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

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|AI智能校验

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