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Mathematical models for the elastic beam with structural damping. (English) Zbl 0793.34046

A mathematical model for the elastic beam is studied in order to prove its structural damping property. An oscillatory system is said to exhibit a structural damping if, asymptotically at least, the damping rates of the natural modes of vibration are linearly proportional to the frequency of the modes. Under suitable conditions on the coefficients the resulting equation is given by: \[ a(x)u_{tt} (t,x)-b(x)u_{txx} (t,x)+\bigl( c(x)u_{xx} (t,x) \bigr)_{xx}=0, \quad t \geq 0,\;O \leq x \leq L. \] This problem is studied through its abstract form \(u''(t)+Mu(t)+Nu(t)=0\) with \(M\) generator of an analytic semigroup in a Banach space.
Reviewer: G.Di Blasio (Roma)

MSC:

34G10 Linear differential equations in abstract spaces
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35Q72 Other PDE from mechanics (MSC2000)
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References:

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