SPECTRUM ANALYSIS OF THE UNDERLYING OPERATOR FOR A 2D PIEZOELECTRIC BEAM WITH MAGNETIC EFFECT ON AN ANNULAR DOMAIN, SECOND VERSION
Abstract
The aim is to determine the eigenvalues of locally coupled equations modelling a 2D piezoelectric beam with magnetic effect with only one local viscous damping on an annular domain without geometric conditions. The eigenvalues are proved to lie in a vertical band defined explicitly in terms of the physical parameters of the problem. They are also the roots of a polynomial of degree 4 whose coefficients are explicit functions of the same parameters and also of a sequence of complex numbers depending on the roots of an explicit trigonometric function. The asymptotic behaviour of the large (complex) eigenvalues is obtained. The exponential stability of the system is proved using Routh-Hurwitz criterion. Except from the sequence of complex numbers which can be computed numerically via a root-finding algorithm, all the computations are explicitly done. Numerical examples are given which lead to conjecture that the decay rate is small which is physically coherent since different time scales are involved.
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